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Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

This document focuses on the mathematical concepts of rotations, particularly in the context of rigid bodies in mechanics. It covers the computation of rotation matrices, the relationship between quaternions and Euler angles, and the application of theorems related to rigid body motion, such as Euler's and Chasles' theorems. Additionally, it includes formulas and discussions on angular velocity, differential equations of rotation, and the properties of rotation matrices.

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1 NOTES ON ROTATIONS SOLO HERMELIN INITIAL INTERMEDIATE FINAL Updated: 3.03.07 Run This http://www.solohermelin.com
2 ROTATIONS TABLE OF CONTENT SOLO Rotation of a Rigid Body Mathematical Computation of a Rotation Rotation Matrix Computation of the Rotation Matrix Consecutive Rotations Decomposition of a Vector in Two Different Frames of Coordinates Differential Equation of the Rotation Matrices Computation of the Angular Velocity Vector from .AB←ω  ( ) ( )nRtC x B A ˆ,33 θ−= Computation of and as functions of .AB←ω  td dθ θ = td nd n ˆ ˆ = • Quaternions Computation of the Rotation Matrix Definition of the Quaternions Product of Quaternions Rotation Description Using the Quaternions
3 ROTATIONS TABLE OF CONTENT (continue – 1) SOLO Rotation as a Multiplication of Two Matrices Relations Between Quaternions and Euler Angles Description of Successive Rotations Using Quaternions Differential Equation of the Quaternions Computation of as a Function of the Quaternion and its Derivatives ( ) ( )t B AB←ω  Computation of as a Function of , and their Derivatives( ) ( )t B AB←ω  θ nˆ Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I Euler Angles The Piogram Successive Euler Rotations 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→
4 ROTATIONS TABLE OF CONTENT (continue – 2) SOLO Cayley-Klein (or Euler) Parameters and Related Quantities Rotation Matrix in Three Dimensional Space Euler Parameters Elementary Features of the 2x2 Rotation Matrix Gibbs Vector Differential Equation of Gibbs Vector References
5 ROTATIONS Rotation of a Rigid Body SOLO 23r 31r 12r1 3 2 P P 1 2 331r 23r 12r A rigid body in mechanics is defined as a system of mass points subject to the constraint that the distance between all pair of points remains constant through the motion. To define a point P in a rigid body it is enough to specify the distance of this point to three non-collinear points. This means that a rigid body is completely defined by three of its non-collinear points. Since each point, in a three dimensional space is defined by three coordinates, those three points are defined by 9 coordinates. But the three points are constrained by the three distances between them: 313123231212 && constrconstrconstr === Therefore a rigid body is completely defined by 9 – 3 = 6 degrees of freedom.
6 ROTATIONS Rotation of a Rigid Body (continue – 1) SOLO We have the following theorems about a rigid body: Euler’s Theorem (1775) The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Chasles’ Theorem (1839) The most general displacement of a rigid body is a translation plus a rotation. Leonhard Euler 1707-1783 Michel Chasles 1793-1880
7 ROTATIONS Rotation of a Rigid Body (continue – 2) SOLO Proof of Euler’s Theorem O – Fixed point in the rigid body A,B – Two point in the rigid body at equal distance r from O.       == rOBOA __________ A’,B’ – The new position of A,B respectively. Since the body is rigid rOBOA == __________ '' Therefore A,B, A’,B’ are one a sphere with center O. α – plane passing through O such that A and A’ are at the same distance from it. β – plane passing through O such that B and B’ are at the same distance from it. PP’ – Intersection of the planes andα β The two spherical triangles APB and A’PB’ are equal. The arcs AA’ and BB’ are equal. That means that rotation around PP’ that moves A to A’ will move B to B’. q.e.d.
8 ROTATIONS Mathematical Computation of a Rotation SOLO A B C O θ φφ nˆ v  1v  We saw that every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters.nˆ • The angle of rotation , defines the third parameter.θ Let rotate the vector around by a large angle , to obtain the new vector → = OAv  nˆ θ→ =OBv1  From the drawing we have: →→→→ ++== CBACOAOBv1  vOA  = → ( ) ( )θcos1ˆˆ −××= → vnnAC  Since direction of is: ( ) ( ) φν sinˆˆ&ˆˆ =×××× vnnvnn  and it’s length is: AC → ( )θφ cos1sin −v ( ) θsinˆ vnCB  ×= → Since has the direction and the absolute value CB → vn  ׈ θφsinsinv ( ) ( ) ( ) θθ sinˆcos1ˆˆ1 vnvnnvv  ×+−××+=
9 ROTATIONS Computation of the Rotation Matrix SOLO We have two frames of coordinates A and B defined by the orthogonal unit vectors and{ }AAA zyx ˆ,ˆ,ˆ { }BBB zyx ˆ,ˆ,ˆ The frame B can be reached by rotating the A frame around some direction by an angle .nˆ θ We want to find the Rotation Matrix that describes this rotation from A to B. ( )θ,ˆ33 nRC x B A = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθ θθ θθ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx ×+−××+= ×+−××+= ×+−××+= Let write those equations in matrix form. ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ θθ AAAA B nnnx [ ]( )           − − − =× 0 0 0 ˆ xy xz yz A nn nn nn n [ ] 0ˆ =×ntrace Rotation Matrix
10 ROTATIONS Computation of the Rotation Matrix (continue – 1) SOLO ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ θθ AAAA B nnnx ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 1 0 sinˆ 0 1 0 cos1ˆˆ 0 1 0 ˆ θθ AAAA B nnny ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 1 0 0 sinˆ 1 0 0 cos1ˆˆ 1 0 0 ˆ θθ AAAA B nnnz ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ Rotation Matrix (continue – 1)
11 ROTATIONS Computation of the Rotation Matrix (continue – 2) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )θθθ ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B =×+−××+==↑ The matrix has the following properties:[ ]( )A n׈ [ ]( ) { } [ ]( )ATA nn ×−=× ˆˆ [ ]( ) [ ]( ) =             −− −− −− =           − − −           − − − =×× 22 22 22 0 0 0 0 0 0 ˆˆ yxzyzx zyzxyx zxyxyz xy xz yz xy xz yz AA nnnnnn nnnnnn nnnnnn nn nn nn nn nn nn nn T x zzyzx zyyyx zxyxx nnI nnnnn nnnnn nnnnn ˆˆ 000 010 001 33 2 2 2 +−=             +           −= [ ]( ) [ ]( ) ( ) 213ˆˆ −=+−=×× AA nntrace [ ]( ) [ ] [ ] nn nn nn nn nnnnn xy xz yz zyx AT ˆˆ000 0 0 0 ˆˆ ×==           − − − =× [ ]( ) [ ]( ) [ ]( ) ( )[ ]( ) [ ]( ) [ ]( ) [ ]( )AATAAT x AAA nnnnnnnnInnn ×−=×+×−=×+−=××× ˆˆˆˆˆˆˆˆˆˆˆ 22 [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) ( )T x AAAAAA nnInnnnnn ˆˆˆˆˆˆˆˆ 33 +−−=××−=×××× skew-symmetric Rotation Matrix (continue – 2)
12 ROTATIONS Computation of the Rotation Matrix (continue – 3) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( ) ( ) B Axx AAA x TATATA x TA B CnRnR nnnI nnnIC =−=−= ×−−××+= ×+−××+= θθ θθ θθ ,ˆ,ˆ sinˆcos1ˆˆ sinˆcos1ˆˆ 3333 33 33 Note The last term can be writen in matrix form as Therefore In the same way End Note In fact is the matrix representation of the vector product:[ ][ ]vnn  ×× ˆˆ ( ) ( )vInnvvnn x T  33ˆˆˆˆ −→−⋅ ( ) ( ) ( ) ( ) vvnnnnvvnnvnn  −⋅=⋅−⋅=×× ˆˆˆˆˆˆˆˆ [ ][ ] T x nnInn ˆˆˆˆ 33 +−=×× ( )[ ] ( )[ ] [ ][ ][ ] [ ]×−=×××→×−=−⋅×=××× nnnnvnvvnnnvnnn ˆˆˆˆˆˆˆˆˆˆˆ  ( )[ ]{ } ( ) [ ][ ][ ][ ] [ ][ ]××−=××××→××−=×××× nnnnnnvnnvnnnn ˆˆˆˆˆˆˆˆˆˆˆˆ  Rotation Matrix (continue – 3)
13 ROTATIONS Computation of the Rotation Matrix (continue – 4) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( ) [ ]( ) ( )( ) [ ]( ) ( ) θθ θθ θθ sin0cos123 sinˆcos1ˆˆ sinˆcos1ˆˆ 33 33 −−−= =×−−××+= =×+−××+= AAA x TATATA x B A ntracenntraceItrace nnnItracetraceC Therefore θcos21+= B ACtrace Let compute the trace (sum of the diagonal components of a matrix) of B AC Also we have [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( )( ) [ ]( ) { } [ ] ( ) [ ]( ) { }=×−−+= =×−−+−+= =×+−××+= θθθ θθ θθ sinˆcos1ˆˆcos sinˆcos1ˆˆ sinˆcos1ˆˆ 33 3333 33 AT x AT xx TATATA x B A nnnI nnnII nnnIC ( ) θθθ sin 0 0 0 cos1cos 000 010 001 2 2 2           − − − −−             +           = xy xz yz zzyzx zyyyx zxyxx nn nn nn nnnnn nnnnn nnnnn Rotation Matrix (continue – 4)
14 ROTATIONS Computation of the Rotation Matrix (continue – 5) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ Therefore we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −+−−+− +−−+−− −−+−−+ = θθθθθθ θθθθθθ θθθθθθ cos1cossincos1sincos1 sincos1cos1cossincos1 sincos1sincos1cos1cos 2 2 2 zxzyyzx xzyyzyx yzxzyxx B A nnnnnnn nnnnnnn nnnnnnn C We get ( )1 2 1 cos −= B AtraceCθ two solutions for θ If ; i.e. we obtain0sin ≠θ πθ ,0≠ ( ) ( )[ ] ( )θsin2/2,33,2 B A B Ax CCn −= ( ) ( )[ ] ( )θsin2/3,11,3 B A B Ay CCn −= ( ) ( )[ ] ( )θsin2/1,22,1 B A B Az CCn −= Rotation Matrix (continue – 5)
15 ROTATIONS Consecutive Rotations SOLO - Perform first a rotation of the vector , according to the Rotation Matrix to the vector . v  ( )1133 ,ˆ θnR x 1v  - Perform a second a rotation of the vector , according to the Rotation Matrix to the vector . 1v  ( )2233 ,ˆ θnR x 2v  ( )vnRv x  11331 ,ˆ θ= ( ) ( ) ( ) ( )vnRvnRnRvnRv xxxx  θθθθ ,ˆ,ˆ,ˆ,ˆ 3311332233122332 === The result of those two consecutive rotation is a rotation defined as: ( ) ( ) ( )1133223333 ,ˆ,ˆ,ˆ θθθ nRnRnR xxx = Let interchange the order of rotations, first according to the Rotation Matrix and after that according to the Rotation Matrix . ( )2233 ,ˆ θnR x ( )1133 ,ˆ θnR x The result of those two consecutive rotation is a rotation defined as: ( ) ( )22331133 ,ˆ,ˆ θθ nRnR xx Since in general, the matrix product is not commutative ( ) ( ) ( ) ( )2233113311332233 ,ˆ,ˆ,ˆ,ˆ θθθθ nRnRnRnR xxxx ≠ Therefore, in general, the consecutive rotations are not commutative. Rotation Matrix (continue – 6)
16 ROTATIONSSOLO INITIAL INTERMEDIATE FINAL Consecutive Rotations of a DiceRotation Matrix (continue – 7)
17 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates SOLO We have two frames of coordinate systems A and B, with the same origin O. We can reach B from A by performing a rotation. Let describe the vector in both frames.v  BzBByBBxBAzAAyAAxA zvyvxvzvyvxvv  111111 ++=++= ( )           = zA yA xA A v v v v  ( )           = zB yB xB B v v v v  & ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) BBABBABBAA BBABBABBAA BBABBABBAA zzzyyzxxzz zzyyyyxxyy zzxyyxxxxx ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆˆ ⋅+⋅+⋅= ⋅+⋅+⋅= ⋅+⋅+⋅=  ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] zABBABBABBA yABBABBABBA xABBABBABBA vzzzyyzxxz vzzyyyyxxy vzzxyyxxxxv ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆ ⋅+⋅+⋅+ ⋅+⋅+⋅+ ⋅+⋅+⋅=  from which Rotation Matrix (continue – 8)
18 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 1) SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                     ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ =           zA yA xA BABABA BABABA BABABA zB yB xB v v v zzzyzx yzyyyx xzxyxx v v v ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ ( ) ( )AB A B vCv  = where is the Transformation Matrix (or Direction Cosine Matrix – DCM) from frame A to frame B. B AC ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ==↑ BABABA BABABA BABABA B A B A zzzyzx yzyyyx xzxyxx CC ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ : In the same way ( ) ( ) ( ) ( )BA B BB A A vCvCv  == −1 therefore ( ) 1− = B A A B CC Rotation Matrix (continue – 9)
19 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 2) SOLO ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( )ATAAB A TB A TAAB A TAB A BTB vvvCCvvCvCvvv  ====2 Since the scalar product is independent of the frame of coordinates, we have [ ] [ ] [ ] 1− =→= B A TB A B A TB A CCICC [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           =                         = 100 010 001 3,33,23,1 2,32,22,1 1,31,21,1 3,32,31,3 3,22,21,2 3,12,11,1 B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A TB A CCC CCC CCC CCC CCC CCC CC or ( ) ( )    =≠ == ==∑ = 3,2,10 3,2,11 ,, 3 1 jji iji kjCkiC ij k B A B A δ Those are 9 equations in , but by interchanging i with j we get the same conditions, therefore we have only 6 independent equations. ( ) 3,2,1,, =jijiC B A We see that the Rotation Matrix is ortho-normal (having real coefficients and the rows/columns are orthogonal to each other and of unit absolute value. Rotation Matrix (continue – 10) This means that the relation between the two coordinate systems is defined by 9 – 6 = 3 independent parameters.
20 ROTATIONS Differential Equations of the Rotation Matrices SOLO We want to develop the differential equation of the Rotation Matrix as a function of the Angular Velocity of the Rotation. Let define by: -the Rotation Matrix that defines a frame of coordinates B at the time t relative to some frame A. ( )tC B A -the Rotation Matrix that defines the frame of coordinates B at the time t+Δt relative to some frame A. ( )ttC B A ∆+ ( )φω ∆−,ˆ33xR -the Rotation Matrix from the frame of coordinates B at the time t to B at time t+Δt relative to some frame A. ( ) ( ) ( )tCRttC B Ax B A φω ∆−=∆+ ,ˆ33 and ( ) [ ] [ ] [ ] ( ) [ ]{ } [ ] [ ] [ ] [ ]             ∆       ∆ ×−      ∆ ××+= ∆×−∆−××+=∆− 2 cos 2 sinˆ2 2 sinˆˆ2 sinˆcos1ˆˆ,ˆ 2 33 3333 φφ ω φ ωω φωφωωφω x xx I IR Rotation Matrix (continue – 11)
21 ROTATIONS Differential Equations of the Rotation Matrices (continue – 1) SOLO Let differentiate the Rotation Matrix ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tC dt dIR tC t IR tC t IR t tCtCR t tCttC t C dt dC B A xxB A xx t B A xx t B A B Ax t B A B A t B A t B A φ φ φωφ φ φω φωφω θ       ∆ −∆− =      ∆ ∆ ∆ −∆− =       ∆ −∆− = ∆ −∆− = ∆ −∆+ = ∆ ∆ = →∆→∆ →∆→∆ →∆→∆ 3333 0 3333 0 3333 0 33 0 00 ,ˆ lim ,ˆ lim ,ˆ lim ,ˆ lim limlim ( ) [ ][ ] [ ] [ ]×−=                     ∆       ∆       ∆ ×−      ∆       ∆       ∆ ××= ∆ −∆− →∆→∆ ω φ φ φ ω φ φ φ ωω φ φω θθ ˆ 2 cos 2 2 sin ˆ 2 2 2 sin ˆˆlim ,ˆ lim 2 2 0 3333 0 xx IR and Therefore ( ) [ ] ( )tC dt d dt tdC B A B A φ ω ×−= ˆ Rotation Matrix (continue – 12)
22 ROTATIONS Differential Equations of the Rotation Matrices (continue – 2) SOLO The final result of the Rotation Matrix differentiation is: Since defines the unit vector of rotation and the rotation rate from B at time t to B at time t+Δt, relative to A, then is the angular velocity vector of the frame B relative to A, at the time t ωˆ dt dφ ω φ ˆ dt d ( ) ω φ ω ˆ dt dB AB =←  ( ) ( )[ ]( ) ( )tCt dt tdC B A B AB B A ×−= ←ω  By changing indixes A and B we obtain ( ) ( )[ ] ( ) ( )tCt dt tdC A B A BA A B ×−= ←ω  Rotation Matrix (continue – 13)
23 ROTATIONS Differential Equations of the Rotation Matrices (continue – 3) SOLO Let find the relation between and[ ]( )B AB ×←ω  [ ]( )A AB ×←ω  For any vector let perform the following computationsv  [ ]( ) ( ) [ ]( ) [ ]( )A AB B A B AB BB AB vCvv  ×=×=× ←←← ωωω [ ]( ) ( ) [ ]( ) ( ) [ ]( ) ( )BA B A AB B A AB A A B A AB B A AA AB B A vCCvCCCvC  ×=×=×= ←←← ωωω Since this is true for any vector we havev  [ ]( ) [ ]( ) A B A AB B A B AB CC ×=× ←← ωω  Pre-multiplying by and post-multiplying by we get: A BC B AC [ ]( ) [ ]( ) B A B AB A B A AB CC ×=× ←← ωω  Rotation Matrix (continue – 14)
24 ROTATIONS Differential Equations of the Rotation Matrices (continue – 4) SOLO Let differentiate the equation 33x A B B A ICC = to obtain [ ] ( ) [ ] ( ) 0=+×−=+×−=+ ←← dt dC C dt dC CCC dt dC CC dt dC A BB A B AB A BB A A B B A B AB A BB A A B B A ωω  Post-multiplying by we get A BC [ ] ( ) [ ] ( ) [ ] ( ) A B A AB A B B A B AB A B B AB A B A B CCCCC dt dC ×=×=×= ←←← ωωω  We obtained for the differentiation of the Rotation Matrix ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )B AB A B A B A AB A B A BA A B ttCtCttCt dt tdC ×=×=×−= ←←← ωωω  Note We can see that ( )[ ] ( ) ( )[ ] ( ) ( ) ( )tttt ABBA A AB A BA ←←←← =−⇒×=×− ωωωω  End Note Rotation Matrix (continue – 15)
25 ROTATIONS Differential Equations of the Rotation Matrices (continue – 5) SOLO Suppose that we have a third frame of coordinates I (for example inertial) and we have the angular velocity vectors of frames A and B relative to I. We have [ ]( ) B I B IB B I C dt dC ×−= ←ω  [ ]( ) A I A IA A I C dt dC ×−= ←ω  A I B A B I CCC = dt dC CC dt dC dt dC A IB A A I B A B I += [ ]( ) [ ]( ) I A A I A IA B A I A B I B IB I A A IB A I A B I B A CCCCCC dt dC CC dt dC dt dC ×+×−=−= ←← ωω  or From which we get: [ ]( ) [ ]( )A IA B A B A B IB B A CC dt dC ×+×−= ←← ωω  Rotation Matrix (continue – 16)
26 ROTATIONSSOLO From the equation Computation of the Angular Velocity Vector from .AB←ω  ( ) ( )nRtC x B A ˆ,33 θ−= ( ) ( )[ ]( ) ( )tCt dt tdC B A B AB B A ×−= ←ω  we obtain ( )[ ]( ) ( ) ( )[ ]TB A B AB AB tC dt tdC t −=×←ω  Since the Rotation Matrix is defined also by and ( ) [ ] ( ) [ ]{ }θθθθ sinˆcos1ˆˆcosˆ, 3333 ×−−+=−= × nnnInRC T x B A ( )tC B A nˆθ we can compute as function of and their derivativesnˆθAB←ω  td dθ θ = td nd n ˆ ˆ = • (this is a long procedure described in the complementary work “Notes on Rotations”, and a simpler derivation will be given later, we give here the final result) [ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  Rotation Matrix (continue – 17)
27 ROTATIONSSOLO Computation of and as functions of .AB←ω  td dθ θ = td nd n ˆ ˆ = • Let pre-multiply the equation by and use T nˆ[ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  [ ] 0ˆˆ,0ˆˆ,1ˆˆ ==×= • nnnnnn TTT to obtain [ ] ( ) AB TTTT AB T nnnnnnnnn ← •• ← =→+−×−= ωθθθθω  ˆsinˆˆcos1ˆˆˆˆˆˆ Let pre-multiply the equation by and use[ ]×nˆ[ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  [ ] [ ][ ] ( ) ••• −=−=××=× nnInnnnnnn x T ˆˆˆˆˆˆˆ,0ˆˆ 33 to obtain [ ] [ ] [ ][ ] ( ) [ ] ( ) [ ] θθθθθω sinˆˆcos1ˆsinˆˆcos1ˆˆˆˆˆˆ •••• ← ×+−=×+−××−×=× nnnnnnnnnnn AB  Let pre-multiply the equation by[ ] ( ) [ ] θθω sinˆˆcos1ˆˆ •• ← ×+−=× nnnn AB  [ ]×nˆ [ ][ ] [ ] ( ) [ ][ ] [ ] ( )θθθθω cos1ˆˆsinˆsinˆˆˆcos1ˆˆˆˆ −×+−=××+−×=×× •••• ← nnnnnnnnnn AB  Rotation Matrix (continue – 18)
28 ROTATIONS Computation of and as functions of (continue – 1) SOLO AB←ω  td dθ θ = td nd n ˆ ˆ = • We have two equations: ( ) [ ] [ ] ABnnnn ← •• ×=×+− ωθθ  ˆsinˆˆcos1ˆ [ ] ( ) [ ][ ] ABnnnnn ← •• ××=−×+− ωθθ  ˆˆcos1ˆˆsinˆ with two unknowns and • nˆ [ ] • × nn ˆˆ From those equations we get: ( )[ ] [ ] ( ) [ ][ ] θωθωθθ sinˆˆcos1ˆsincos1ˆ 22 ABAB nnnn ←← • ××−−×=+−  or ( ) [ ] ( ) [ ][ ] θωθωθ sinˆˆcos1ˆcos1ˆ2 ABAB nnnn ←← • ××−−×=−  Finally we obtain: AB T n ←= ωθ  ˆ [ ] [ ][ ] ABnnnn ← •             ××−×= ω θ  2 cotˆˆˆ 2 1 ˆ Rotation Matrix (continue – 19)
29 ROTATIONS Quaternions SOLO The quaternions method was introduced by Hamilton in 1843. It is based on Euler Theorem (1775) that states: The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Therefore every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters • The angle of rotation , defines the third parameter nˆ θ William Rowan Hamilton 1805 - 1865 ( ) ( ) ( ) θθ sinˆcos1ˆˆ1 vnvnnvv  ×+−××+= The rotation of around by angle is given by:nˆ θv  A B C O θ φφ nˆ v  1v  that can be writen ( )[ ] ( ) ( ) θθ sinˆcos1ˆˆ1 vnvvnnvv  ×+−−⋅+= or ( ) ( ) ( ) θθθ sinˆcos1ˆˆcos1 vnvnnvv  ×+−⋅+=
30 ROTATIONS Quaternions (continue – 1) SOLO Computation of the Rotation Matrix We found the Rotation Matrix that describes this rotation from A to B. ( )θ,ˆ33 nRC x B A = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθ θθ θθ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx ×+−××+= ×+−××+= ×+−××+= ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ or from which [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )θθθ ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B =×+−××+==↑
31 ROTATIONS Quaternions (continue – 2) SOLO Definition of the Quaternions The quaternions (4 parameters) were defined by Hamilton as a generalization of the complex numbers ( ) 32100 , qkqjqiqqq  +++== ρ ( )2/cos0 θ=q ( ) nˆ2/sin θρ =  ( ) ( ) ( ) zyx nqnqnq 2/sin&2/sin&2/sin 111 θθθ === where satisfy the relations:kji  ,, 1−=⋅=⋅=⋅ kkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ 1−=⋅⋅ kji  the complex conjugate of is defined asq ( ) 32100 * , qkqjqiqqq  −−−=−= ρ
32 ROTATIONS Quaternions (continue – 3) SOLO Product of Quaternions Product of two quaternions andAq Bq ( )( ) ( )( )3210321000 ,, BBBBAAAABBAABA qkqjqiqqkqjqiqqqqq  ++++++== ρρ ( ) ( ) ( )3210321033221100 AAABBBBABABABABA qkqjqiqqkqjqiqqqqqqqqq  ++++++−−−= ( ) ( ) ( )122131132332 BABABABABABA qqqqkqqqqjqqqqi −+−+−+  therefore ( )( ) ( ) ( )[ ]BAABBABABABBAABA qqqqqqqq ρρρρρρρρ  ×++⋅−== 000000 ,,, Let use this expression to find ( )( ) ( )( ) 2 3 2 2 2 1 2 0 222 000 * 00 * 1ˆˆ 2 sin 2 cos,,,, qqqqnnqqqqqqqqq +++==⋅      +      =⋅+=−==−= θθ ρρρρρρ  The quaternion product can be writen in matrix form as: [ ] [ ]               ×− − =              ×+ − ==      = A A BxBB T BB B B AxAA T AA BA q Iq qq Iq q qq q q ρρρ ρ ρρρ ρ ρ      0 330 00 330 00 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅
33 ROTATIONS Quaternions (continue – 4) SOLO Rotation Description Using the Quaternions Let compute the expression: ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]ρρρρρρρρρρ ρρρρρ   ××−×+×−+⋅⋅×+⋅−⋅= ×−⋅=−= AAAAAAAA AAAAA vvqvqvqvvvqqv qvvqvqvqqvq 00 2 000 0000 * , ,,,,0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ){ } ( ) [ ] [ ][ ]{ } ( ) ( )A AAAA AAAAAAA vqq vvqqvv vvqvqvqvvv    ××+×−⋅+= ××+×−+⋅+⋅××= ××−×+×−+⋅+⋅−⋅= ρρρρρ ρρρρρρρ ρρρρρρρρρρ 22,0 2,0 ,0 0 2 0 0 2 0 00 2 0 Using the relations: ( ) ( ) [ ][ ] ( )[ ][ ] ( )[ ][ ] [ ] ( ) ( )[ ] [ ]      ×=×=× ××−=××=×× =⋅+ →    = = nnq nnnn q n q ˆsinˆ2/sin2/cos22 ˆˆcos1ˆˆ2/sin22 1 ˆ2/sin 2/cos 0 2 2 0 0 θθθρ θθρρ ρρ θρ θ     and ( ) ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )AAAA x AB A B vnnnIvCv  θθ sinˆcos1ˆˆ33 ×−−××+== we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) [ ] [ ][ ]{ } ( ) ( )AAABB vqqvqqvqvv  ××+×−=−=== ρρρρρ 221,0,,0,,0 000 *
34 ROTATIONS Quaternions (continue – 5) SOLO Rotation Description Using the Quaternions (continue – 1) Using the fact that we obtain: [ ] [ ] [ ][ ]××+×−= ρρρ  22 033 qIC x B A           − − −           − − − +           − − − −           = 0 0 0 0 0 0 2 0 0 0 2 100 010 001 12 13 23 12 13 23 12 13 23 0 qq qq qq qq qq qq qq qq qq q             −− −− −− +           − − − +           = 2 2 2 13231 32 2 1 2 321 3121 2 2 2 3 1020 1030 2030 2222 2222 2222 022 202 220 100 010 001 qqqqqq qqqqqq qqqqqq qqqq qqqq qqqq             −−−+ +−−− −+−− = 2 2 2 110323120 3210 2 1 2 33021 20312130 2 2 2 3 2212222 2222122 2222221 qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq 1 2 3 2 2 2 1 2 0 =+++ qqqq ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ = 2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 22 22 22 qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq C B A
35 ROTATIONS Quaternions (continue – 6) SOLO Rotation as a Multiplication of Two Matrices [ ] [ ] [ ][ ]××+×−= ρρρ  22 033 qIC x B A ( )[ ] [ ] [ ][ ]××+×−+= ρρρρρ  22 033 2 0 qIq x T [ ] [ ] [ ][ ] [ ] [ ][ ]××++××+×−= ρρρρρρρ  33033 2 0 2 x T x IqIq [ ] [ ]( ) [ ] [ ]( ) [ ] [ ][ ]××++×−×−= ρρρρρρ  33330330 x T xx IIqIq For any vector we can write ( ) ( ) ( )ρρρρρρ  ⋅−⋅=×× aaaa  or in matrix notation [ ][ ] [ ]( ) [ ] [ ] T x T x T x TT IIaIa ρρρρρρρρρρρρ  ≡+⇒−=×× 333333 Therefore we have [ ] [ ]( ) [ ] [ ]( ) [ ] [ ][ ]××++×−×−= ρρρρρρ  33330330 x T xx B A IIqIqC [ ] [ ]( ) [ ] [ ]( ) =+×−×−= T xx IqIq ρρρρ  330330 [ ]321 3 2 1 012 103 230 012 103 230 qqq q q q qqq qqq qqq qqq qqq qqq           +           − − −           − − − =
36 ROTATIONS Quaternions (continue – 7) SOLO Rotation as a Multiplication of Two Matrices (continue – 1) [ ] [ ][ ] [ ] [ ]           ×− ×−=                 − − −           − − − = ρ ρ ρρ           330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C [ ] [ ][ ] [ ] [ ]           ×− − ×−−=                 − − − −−−           −− −− −− = ρ ρ ρρ           330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C [ ] [ ][ ] [ ] [ ]           ×− ×−=                 − − −           − − − = T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C ρ ρ ρρ          330 330 321 012 103 230 3012 2103 1230 [ ] [ ][ ] [ ] [ ]           − ×− −×−=                 −−− − − −           −− −− −− = T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C ρ ρ ρρ          330 330 321 012 103 230 3012 2103 1230
37 ROTATIONS Quaternions (continue – 8) SOLO Relation Between Quaternions and Euler Angles
38 ROTATIONS Quaternions (continue – 9) SOLO Description of Successive Rotations Using Quaternions Let describe two consecutive rotations: - First rotation defined by the quaternion ( )                   == 1 11 1101 ˆ 2 sin, 2 cos, nqq θθ ρ  - Folowed by the second rotation defined by the quaternion ( )                   == 2 22 2202 ˆ 2 sin, 2 cos, nqq θθ ρ  After the first rotation the quaternion of the vector is transferred to 1 * 1 qvq After the second rotation we obtain ( ) ( ) ( )21 * 2121 * 1 * 221 * 1 * 2 qqvqqqqvqqqqvqq == Therefore the quaternion representing those two rotation is: ( ) ( )( ) ( )       ×            +      +      ⋅            −            = =×++⋅−==== 21 21 1 2 2 1 21 2121 21120210212010220110210 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cos,ˆˆ 2 sin 2 sin 2 cos 2 cos ,,,, nnnnnn qqqqqqqqqq θθθθθθθθ ρρρρρρρρρ  ( ) ( )210 , qqqq == ρ  21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq ⋅            −            =      = θθθθθ 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn ×            +      +      =      = θθθθθ ρ 
39 ROTATIONS Quaternions (continue – 10) SOLO Description of Successive Rotations Using Quaternions (continue – 1) ( ) ( )210 , qqqq == ρ  21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq ⋅            −            =      = θθθθθ 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn ×            +      +      =      = θθθθθ ρ  Two consecutive rotations, followed by , are given by:1q 2q From those equations we can see that: 0ˆˆˆˆˆˆˆˆˆˆ 21212112211221  =×→×−=×→×=×= nnnnnnnnnnifonlyandifqqqq The rotations are commutative if and only if are collinear.21 ˆ&ˆ nn In matrix form those two rotations are given by: First Rotation: ( ) [ ] ( ) [ ]{ }111111331133 sinˆcos1ˆˆcosˆ, θθθθ ×−−+=− nnnInR T xx Second Rotation: ( ) [ ] ( ) [ ]{ }222222332233 sinˆcos1ˆˆcosˆ, θθθθ ×−−+=− nnnInR T xx Total Rotation: ( ) ( ) ( ) [ ] ( ) [ ]{ }θθθθθθ sinˆcos1ˆˆcosˆ,ˆ,ˆ, 331133223333 ×−−+=−−=− nnnInRnRnR T xxxx
40 ROTATIONS Quaternions (continue – 11) SOLO Description of Successive Rotations Using Quaternions (continue – 2) Let find the quaternion that describes the Euler Rotations through the angles respectively. Let write the rotations according to their order 123 →→ ϕθψ ,,             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕθθψψ ijkqqqq xyz B A                    −            +            +                        +      = 2 sin 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos ϕθϕθθϕϕθψψ kjik                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos ϕθψϕθψ                         −                  + 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos ϕθψϕθψ i                          +                  + 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin ϕθψϕθψ j                          −                  + 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin ϕθψϕθψ k 
41 ROTATIONS Quaternions (continue – 12) SOLO Differential Equation of the Quaternions Let define ( ) ( )ρ  ,0qtq B A = - the quaternion that defines the position of B frame relative to frame A at time t. ( ) ( )tqqttq B A ∆+∆+=∆+ ρ  ,00 - the quaternion that defines the position of B frame relative to frame A at time t+Δt. ( )             ∆       ∆ =∆ ∆t B A ntq ˆ 2 sin, 2 cos θθ - the quaternion that defines the position of B frame at time t+Δt relative to frame B at time t. We have the relation: ( ) ( ) ( )tqtqttq B A B A B A ∆=∆+ or ( ) ( ) ( ) ( )( ) ( ) ( ) ( )[ ] ( ) ( )( )ρρρρρρρρ θθ  ∆∆−+=∆∆+−=∆+∆+−=∆+=            ∆       ∆ =∆ ∆ ,,0,1,,,,,ˆ 2 sin, 2 cos 00000000 * qqqqqqqqttqtqntq B A B At B A therefore ( )( )             ∆ −      ∆ =∆∆− ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  ( ) ( )             ∆ −      ∆ =∆∆ ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  or
42 ROTATIONS Quaternions (continue – 13) SOLO Differential Equation of the Quaternions (continue – 1) ( ) ( )             ∆ −      ∆ =∆∆ ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  Let divide both sides by and take the limit .0→∆ tt∆ ( ) ( ) ( )       =      =                   ∆ ∆       ∆       ∆       ∆ ∆       ∆ −      ∆ ==      ∆ ∆ ∆ ∆ ∆∆∆ →∆ tBttB t ntqnqn tt tq td d tt q ˆ 2 1 ,0ˆ 2 1 ,0,ˆ 2 2 sin 2 1 , 2 1 2 cos 2 1 ,lim 0 0 0 θθρ θ θ θ θ θ θ ρ   But is the instant angular velocity vector of frame B relative to frame A.tn∆ ˆθ ( ) ( ) t B AB nt ∆← = ˆθω  ( ) ( ) ( )( ) ( ) ( )ttn B AB B ABt ←←∆ == ωωθ  ,0ˆ,0 So we can write ( ) ( ) ( ) ( )ttqtq td d B AB B A B A ←= ω 2 1 This is the Differential equation of the quaternion that defines the position of B relative to A, at the time t as a function of the angular velocity vector of frame B relative to frame A, . ( )tq B A ( ) ( )t B AB←ω 
43 ROTATIONS Quaternions (continue – 14) SOLO Differential Equation of the Quaternions (continue – 2) Developing this equation, we get ( ) ( ) ( ) ( )ttqtq td d B AB B A B A ←= ω 2 1 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )B AB B AB B AB B AB qtq dt d dt dq ←←←← ×+⋅−==      ωρωωρωρ ρ   00 0 , 2 1 ,0, 2 1 , from which ( )B AB dt dq ←⋅−= ωρ  2 10 ( ) ( ) ( )B AB B ABq dt d ←← ×+= ωρω ρ   0 2 1 or in matrix form [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1
44 ROTATIONS Quaternions (continue – 15) SOLO Differential Equation of the Quaternions (continue – 3)                           − − − −−− =                 ← ← ← zBAB yBAB xBAB qqq qqq qqq qqq q q q q dt d ω ω ω 012 103 230 321 3 2 1 0  [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1 B AAB xBAByBABzBAB xBABzBAByBAB yBABzBABxBAB zBAByBABxBAB q q q q q q q q q dt d ← ←←← ←←← ←←← ←←← Ω=                                   − − − −−− =                 2 1 0 0 0 0 2 1 3 2 1 0 3 2 1 0        ωωω ωωω ωωω ωωω After rearranging or ( )zBAByBABxBAB qqq dt dq ←←← ++−= ωωω 321 0 2 1 ( )zBAByBABxBAB qqq dt dq ←←← +−= ωωω 230 1 2 1 ( )zBAByBABxBAB qqq dt dq ←←← −+= ωωω 103 2 2 1 ( )zBAByBABxBAB qqq dt dq ←←← ++−= ωωω 012 3 2 1
45 ROTATIONS Quaternions (continue – 16) SOLO Pre-multiply the equation Computation of as a Function of the Quaternion and its Derivatives ( ) ( )t B AB←ω  [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1 by [ ] [ ][ ]×−− ρρ    330 xIq [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] ( ) ( ) =           ×+ − ×−−=             ×−− ← • • t Iq Iq q Iq B AB x T xx ω ρ ρ ρρ ρ ρρ             330 330 0 330 2 1 [ ] [ ][ ][ ] ( ) ( ) [ ] [ ]( )[ ] ( ) ( ) ( ) ( )ttIIq tIq B BA B BAx TT x T B BAx T →→ → =−−+= =××−+= ωωρρρρρρ ωρρρρ   2 1 2 1 2 1 3333 2 0 33 2 0 Therefore ( ) ( ) [ ] [ ][ ]             ×−−= • • ← ρ ρρω      0 3302 q Iqt x B AB
46 ROTATIONS Quaternions (continue – 17) SOLO Computation of as a Function of the Quaternion and its Derivatives (continue – 1) But and are related. Differentiating the equation ( ) ( )t B AB←ω  we obtain • 0q • ρ  1 2 0 =+ ρρ T q ( ) ( ) [ ] [ ][ ] [ ] [ ]( ) =       ×−+−=             ×−−= •• • • ← ρρρ ρ ρρω       3300 0 330 22 xx B AB Iqq q Iqt [ ] [ ]( ) [ ] [ ] ••• +×− =       ×−+= ρ ρρρ ρρρρρ    0 033 2 0 330 0 2 1 2 q qIq Iq q T x x T From the equation •••• −=→=+ ρρρρ  TT q qqq 0 000 1 0 we obtain ( ) [ ] [ ]( ) • ← +×−= ρρρρω  T x B AB qIq q 033 2 0 0 2
47 ROTATIONS Quaternions (continue – 18) SOLO Computation of as a Function of , and their Derivatives( ) ( )t B AB←ω  θ nˆ Differentiate the quaternion ( )                   == nqq ˆ 2 sin, 2 cos,0 θθ ρ  to obtain             +            −=      = ••• nnqq ˆ 2 sinˆ 2 cos 2 , 2 sin 2 ,0 θθθθθ ρ   Substitute this in the equation ( ) ( ) [ ] [ ][ ]             ×−−= • • ← ρ ρρω      0 3302 q Iqt x B AB [ ]                       +            −       ×      −            −= • nn nIn x ˆ 2 sinˆ 2 cos 2 2 sin 2 ˆ 2 sin 2 cosˆ 2 sin2 33 θθθ θθ θθθ     [ ] [ ] •• ×      −×            −            +      +      = nnnnnnn ˆˆ 2 sin2ˆˆ 2 cos 2 sinˆ 2 cos 2 sin2ˆ 2 cosˆ 2 sin 222 θθθ θ θθθ θ θ θ  ( ) [ ] •• ← ×−−+= nnnnAB ˆˆcos1ˆsinˆ θθθω  Finally we obtain We recovered a result obtained before.
48 ROTATIONS Quaternions (continue – 18) SOLO Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I The relations between the components of a vector in the frames A, B and I arev  ( ) AB A IA I B A IB I B vCvCCvCv  === Using quaternions the same relations are given by ( ) B A A I IA I B A B I IB I B qqvqqqvqv *** == Therefore B A A I B I qqq = B I A I B A qqq * = Let perform the following calculations B A A I B A A I B I q dt d qqq dt d q dt d += &( )B IB B I B I qq dt d ←= ω 2 1 ( )A IA A I A I qq dt d ←= ω 2 1 and use ( ) ( ) B A A I B A A IA A I B IB B I q dt d qqqq += ←← ωω 2 1 2 1 ( ) ( ) B A A IA A I A I B IB B I A I B A qqqqqq dt d ←← −= ωω  1 ** 2 1 2 1 to obtain ( ) ( ) B A A IA B IB B A B A qqq dt d ←← −= ωω 2 1 2 1
49 ROTATIONS Quaternions (continue – 19) SOLO Differential Equatio of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I (continue – 1) Using the relations ABIAIB ←←← += ωωω  and ( ) ( ) B A A IA B A B IA qq ←← = ωω * ( ) ( ) B A A IA B IA B A qq ←← = ωω we have ( ) ( ) ( ) ( ) ( ) ( ) ( )    0 2 1 2 1 2 1 2 1 2 1 B A A IA B IA B A B AB B A B A A IA B IA B AB B A B A qqqqqq dt d ←←←←←← −+=−+= ωωωωωω from which ( ) ( ) B A A AB B AB B A B A qqq dt d ←← == ωω 2 1 2 1 Since BAAB ←← −= ωω  we get ( ) ( ) ( ) ( ) B A A BA B BA B A B A A AB B AB B A B A qqqqq dt d ←←←← −=−=== ωωωω 2 1 2 1 2 1 2 1 From we get1 * == B A A B B A B A qqqq A B B A qq = * Therefore       +      = B A A B B A A B q dt d qqq dt d 0 *B A B A A B A B qq dt d qq dt d       −=     
50 ROTATIONS Euler Angles SOLO The orientation of the Body Frame relative to the Inertial Frame has three degrees of freedom. We will use 3 Euler Angles that define the orientation by three consecutive rotations around the consecutive frame axes. [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc The three basic Euler rotations around axes are described by the rotation matrices: ,3ˆ,2ˆ,1ˆ
51 ROTATIONS Euler Angles (continue – 1) SOLO Introduce the Piogram that represents the following notation: (from Pio R.L. “Symbolic Representations of Coordinate Transformations”, IEEE on Aerospace and Navigation Electronics, Vol. ANE-11,June 1964, pp.128-134) The Piogram
52 ROTATIONSEuler Angles (continue – 2) SOLO Rotation Around x Axis by an Angle .ϕ [ ]           − == ϕϕ ϕϕϕ cossin0 sincos0 001 x B AC BAAB xx      11 ϕϕω ==← [ ]           −=−= ϕϕ ϕϕϕ cossin0 sincos0 001 x A BC The Piogram (continue – 1)
53 ROTATIONSEuler Angles (continue – 3) SOLO Rotation Around y Axis by an Angle .θ [ ]           − == θθ θθ θ cos0sin 010 sin0cos y B AC BAAB yy  11 θθω ==← [ ]           − =−= θθ θθ θ cos0sin 010 sin0cos y A BC The Piogram (continue – 2)
54 ROTATIONSEuler Angles (continue – 4)SOLO Rotation Around z Axis by an Angle .ψ [ ]           −== 100 0cossin 0sincos ψψ ψψ ψ x B AC BAAB zz      11 ψψω ==← [ ]           − =−= 100 0cossin 0sincos ψψ ψψ ψ x A BC The Piogram (continue – 3)
55 ROTATIONS Euler Angles (continue – 5) SOLO Using the basic Euler Angles we can define the following 12 different rotations: (a) six rotations around three different axes: 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ (b) six rotations such that the first and third are around the sam axes, but the second is different: 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→ Suppose that the Transfer Matrix from A to B is defined by three consecutive Euler Angles: around (unit vector in A Frame), around (unit vector in intermediate frame), around (unit vector in B Frame). B AC iθ Iiˆ jθ Interjˆ kθ Bkˆ [ ] [ ] [ ] [ ] [ ]TB A B A A B A B B Akkjjii B A CCCICCC ==→== −1 &θθθ
56 ROTATIONSEuler Angles (continue – 6) SOLO 123 →→Euler Angles rotations: using the Piogram 1.Rotation from A to A’ around the third axis by the angle ψ 2. Rotation from A’ to B’ around the third axis by the angle θ . 3. Rotation from B’ to B around the third axis by the angle ϕ . xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv The Piogram (continue – 4)
57 ROTATIONSEuler Angles (continue – 7) SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: Example 1: Computation of the relation between to using the PiogramyBv zAyAxA vvv ,, From the Piogram: ( ) ( ) θϕψθϕψϕψθϕψϕ cossinsinsinsincoscoscossinsinsincos zAyAxAyB vvvv ++++−= using the Piogram (continue – 1) The Piogram (continue – 5)
58 ROTATIONSEuler Angles (continue – 8)SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: using the Piogram (continue – 2) Example 2: Computation of the matrix (1st column) using the Piogram B AC 1=xAv 0=yAv 0=zAv ψcos' =xAv ψsin' −=yAv 0' =zAv ψsin' −=yBv ψθcossin' =zBv ψθcoscos=xBv ψθϕ ψϕ cossincos sinsin + +=zBv ψθϕ ψϕ cossinsin sincos + +−=yBv ψcos ψcos ψsin ψsin θcos θsin θcos θsin ϕcos ϕsin ϕcos ϕsin ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           +−+ ++− − =             θϕψθϕψϕψθϕψϕ θϕψθϕψϕψθϕψϕ θψθψθ ccssccscscss csssscccsssc ssccc CCC CCC CCC B A B A B A B A B A B A B A B A B A 3,32,31,3 3,22,21,2 3,12,11,1 The Piogram (continue – 6)
59 ROTATIONSEuler Angles (continue – 9) SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: Example 3: Computation of the angular velocity using the Piogram using the Piogram (continue – 3) AB←ω  xAAB←ω yAAB←ω zAAB←ω 0 0 0 B ψ ψ− A θ θ 'A ϕ− ϕ 'B ( ) ( ) ( ) ( ) [ ] [ ] [ ]                     − − =           −−+           −+           =++=← ψ θ ϕ θ ψψθ ψψθ ϕθψθψψϕθψω        10 0 0 0 0 1 0 1 0 1 0 0 111 233 ' '' ' '' s csc scc xCyCz B B A B A A A A A A A AB The Piogram (continue – 7)
60 ROTATIONSEuler Angles (continue – 10)SOLO 123 →→Euler Angles rotations: Piogram: Example 4: Computation of the angular velocities and using the Piogram using the Piogram (continue – 4) IB←ω  IA←ω  ψ xBIB←ω yBIB←ω zBIB←ω ϕ θ− ψ θ ϕ A 'A 'B B xAIA←ω yAIA←ω zAIA←ω xBIB←ω yBIB←ω zBIB←ω B ϕ− ϕ− ψ− ψ− A'A xAIA←ω yAIA←ω zAIA←ω θ θ− 'B Piogram: ( ) [ ] [ ] [ ]                                               +           −−+           −−+           −−=           = ← ← ← ← ← ← ← zBIB yBIB xBIB zAIA yAIA xAIA A IA ω ω ωϕ ϕθθ ψ ψ ω ω ω ω 0 0 0 0 0 0 123     ( ) [ ] [ ] [ ]                                               +           +           +           =           = ← ← ← ← ← ← ← zAIA yAIA xAIA zBIB yBIB xBIB B IB ω ω ω ψ ψθθ ϕ ϕ ω ω ω ω     0 0 0 0 0 0 321 IAIBBAAAB xyz ←←← −=++= ωωϕθψω      '' 111 The Piogram (continue – 8)
61 ROTATIONS Euler Angles (continue – 9) SOLO 321 →→Euler Angles rotations: [ ] [ ] [ ]           − ++−− +−+ == φθφθθ φψφθψφψφθψθψ φψφθψφψφθψθψ ϕθψ ccscs sccssccssscs sscsccsssccc CB A 123                     − + − =           +− + − =           ← ← ← ψ θ ϕ θ ψϕθ ψϕθ ψθϕ ψθψθϕ ψθψθϕ ω ω ω       10 0 0 s csc scc s csc scc zBAB yBAB xBAB                     −=           ← ← ← zBAB yBAB xBAB csscs ccsc sc c ω ω ω θψθψθ ψθψθ ψψ θ ψ θ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ψψθθϕϕ kjiqqqq zyx B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q Piogram
62 ROTATIONS Euler Angles (continue – 10) SOLO 231 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +− − −+ == ϕθϕθψϕθϕθψθψ ϕθϕψψ ϕθϕθψϕθϕθψθψ ϕψθ ccssssccsssc scccs csscsssccscc C B A 132                     − − − =           − +− − =           ← ← ← θ ψ ϕ θθψ ψ θθψ θψθψϕ θψϕ θψθψϕ ω ω ω       0 10 0 csc s scc csc s scc zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB ssccs ccsc sc c ω ω ω θψψθψ θψθψ θθ ψ θ ψ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos θθψψϕϕ jkiqqqq yzx B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
63 ROTATIONS Euler Angles (continue – 11) SOLO 312 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           − ++− +−+ == ϕθϕϕθ θϕψθψϕψϕθψφψ θϕψθψϕψϕθψθψ θϕψ ccscs cscssccssccs csssccsssscc C B A 213                     − − + =           +− − + =           ← ← ← ψ ϕ θ ϕ ψψϕ ψψϕ ψϕθ ψϕψϕθ ψϕψϕθ ω ω ω       10 0 0 s scc csc s scc csc zBAB yBAB xBAB                     − −=           ← ← ← zBAB yBAB xBAB ccsss sccc cs c ω ω ω ϕψϕψϕ ψϕψϕ ψψ ϕ ψ ϕ θ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ψψϕϕθθ kijqqqq zxy B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
64 ROTATIONS Euler Angles (continue – 12) SOLO 132 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−+ ++− − == ϕθϕθψϕψϕθϕθψ ϕθϕθψϕψϕθϕθψ θψψψθ θψϕ ccssssccsscs sccssccssccs scscc CB A 231                     − =           +− + + =           ← ← ← ϕ ψ θ ϕϕψ ϕϕψ ψ ϕψϕψθ ϕψϕψθ ϕψθ ω ω ω       0 0 10 csc scc s csc scc s zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB sscsc ccsc sc c ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ θ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕψψθθ ikjqqqq xzy B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
65 ROTATIONS Euler Angles (continue – 13) SOLO 213 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           −+− − −+− == ϕθϕθψθψϕθψθψ ϕϕψϕψ ϕθϕθψθψϕθψθψ ψϕθ ccsccssscssc scccs csssccsssscc C B A 312                    − =           + + +− =           ← ← ← θ ϕ ψ θϕθ ϕ θϕθ θϕϕθψ θϕψ θϕϕθψ ω ω ω       0 10 0 scc s ccs scc s ccs zBAB yBAB xBAB           =           −− −− − =           ← ← ← zBAB yBAB xBAB sccss cscc cs c ω ω ω ϕθϕϕθ ϕθϕθ θθ ϕ θ ϕ ψ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos θθϕϕψψ jikqqqq yxz B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
66 ROTATIONS Euler Angles (continue – 14) SOLO 123 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−+ ++− − == θϕψθϕψϕψθϕψϕ θϕψθϕψϕψθϕψϕ θψθψθ ψθϕ ccssccscscss csssscccsssc ssccc CB A 321                     − − =           +− + − =           ← ← ← ψ θ ϕ ϕθϕ ϕθϕ θ ϕθψϕθ ϕθψϕθ θψϕ ω ω ω       ccs scc s ccs scc s zBAB yBAB xBAB 0 0 01                     −=           ← ← ← zBAB yBAB xBAB cs sccc csssc c ω ω ω ϕϕ ϕθϕθ ϕθϕθθ θ ψ θ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕθθψψ ijkqqqq xyz B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
67 ROTATIONS Euler Angles (continue – 15) SOLO 121 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           −−− ++− − == 212121212 212121212 11 11212 ϕϕϕθϕϕϕϕθϕϕθ ϕϕϕθϕϕϕϕθϕϕθ θϕθϕθ ϕθϕ sscccscccscs cssccccscsss scssc C B A                     − =           − + + =           ← ← ← 2 1 22 22 221 221 21 0 0 10 ϕ θ ϕ ϕϕθ ϕϕθ θ ϕθϕθϕ ϕθϕθϕ ϕθϕ ω ω ω       scs css c scs css c zBAB yBAB xBAB                     −− −=           ← ← ← zBAB yBAB xBAB scscs sscs cs s ω ω ω ϕθϕθθ ϕθϕθ ϕϕ θ ϕ θ ϕ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ϕϕθθϕϕ ϕϕ ijiqqqq xyx B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ϕϕθϕθϕϕθϕ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 1 ϕϕθϕθϕϕθϕ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 2 ϕϕθϕθϕϕθϕ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 3 ϕϕθϕθϕϕθϕ q
68 ROTATIONS Euler Angles (continue – 16) SOLO 131 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−− +−−== 212121212 212121212 11 11312 ϕϕϕψϕϕϕϕψϕϕψ ϕϕϕψϕϕϕϕψϕϕψ ψϕψϕψ ϕψϕ ccscscssccss scccsssccccs ssscc C B A                     − − =           ← ← ← zBAB yBAB xBAB scccs csss sc s ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ ϕ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ϕϕψψϕϕ ϕϕ ikiqqqq xzx B A                      − − =           ← ← ← zBAB yBAB xBAB scccs csss sc s ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ ϕ 22 22 22 2 1 0 0 1          +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ϕϕψϕψϕϕψϕ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 1 ϕϕψϕψϕϕψϕ q       −       =                        −                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 122121 2 ϕϕψϕψϕϕψϕ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 3 ϕϕψϕψϕϕψϕ q
69 ROTATIONS Euler Angles (continue – 17) SOLO 212 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−+ −−− == 212122121 11 212122121 21122 θϕθθθθϕθϕθθθ ϕθϕφθ θϕθθθθϕθϕθθθ θϕθ cccsscsccssc sccss ccccsssscscc C B A                     −− =           −− + + =           ← ← ← 2 1 22 22 221 21 221 0 10 0 θ ϕ θ θθϕ ϕ θθϕ θϕθϕθ θϕθ θϕθϕθ ω ω ω       scs c css scs c css zBAB yBAB xBAB                     −− −−=           ← ← ← zBAB yBAB xBAB ccssc sscs cs s ω ω ω θϕϕθϕ θϕθϕ θθ ϕ θ ϕ θ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 θθϕϕθθ θθ jijqqqq yxy B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 θθϕθϕθθϕθ q       +       =                        −                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 1 θθϕθϕθθϕθ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 2 θθϕθϕθθϕθ q       +       =                        +                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 212121 3 θθϕθϕθθϕθ q
70 ROTATIONS Euler Angles (continue – 18) SOLO 232 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−+ − −−− == 212122121 11 212122121 21322 θθθψθθψθθθψθ ϕθψψθ θθθψθθψθθθψθ θψθ ccscssscsscc sscsc scccscsssccc C B A                     + − =           + + − =           ← ← ← 2 1 22 22 221 21 221 0 10 0 θ ψ θ θθψ ψ θθψ θψθψθ θψθ θψθψθ ω ω ω       css c scs css c scs zBAB yBAB xBAB                     −− =           ← ← ← zBAB yBAB xBAB scscc csss sc s ω ω ω θψψθψ θψθψ θθ ψ θ ψ θ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 θθψψθθ θθ jkjqqqq yzy B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 θθψθψθθψθ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 1 θθψθψθθψθ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 2 θθψθψθθψθ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 3 θθψθψθθψθ q
71 ROTATIONS Euler Angles (continue – 19) SOLO 313 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           − +−−− +− == ϕϕψϕψ ψϕψϕψψψψϕψψψ ψϕψϕψψψψϕψψψ ψϕψ cscss cscccssccssc ssscccsscscc C B A 11 221212121 221212121 31132                     − + =           + − + =           ← ← ← 2 1 22 22 21 221 221 10 0 0 ψ ϕ ψ ϕ ψψϕ ψψϕ ψϕψ ψϕψϕψ ψϕψϕψ ω ω ω       c scs css c scs css zBAB yBAB xBAB                     −− −=           ← ← ← zBAB yBAB xBAB sccsc sscs cs s ω ω ω ϕψϕψϕ ψϕψϕ ψψ ϕ ψ ϕ ψ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ψψϕϕψψ ψψ kikqqqq zxz B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ψψϕψϕψψϕψ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 1 ψψϕψϕψψϕψ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 2 ψψϕψϕψψϕψ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 3 ψψϕψϕψψϕψ q
72 ROTATIONS Euler Angles (continue – 20) SOLO 323 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−− −+− == θθψθψ ψθψψψθψψψϕθψ ψθψψψθψψψψθψ ψθψ csssc ssccscscsscc csscccsssccc C B A 11 221212121 221212121 31232                    − =           + + +− =           ← ← ← 2 1 22 22 21 221 221 10 0 0 ψ θ ψ θ ψψθ ψψθ ψθψ ψθψθψ ψθψθψ ω ω ω       c css scs c css scs zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB ssccc csss sc s ω ω ω θψθψθ ψθψθ ψψ θ ψ θ ψ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ψψθθψψ ψψ kjkqqqq zyz B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ψψθψθψψθψ q       −       −=                        −                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 212121 1 ψψθψθψψθψ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 2 ψψθψθψψθψ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 3 ψψθψθψψθψ q
73 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities SOLO Rotation Matrix in Three Dimensional Space We saw that the rotation of a vector is given byx  [ ] [ ] [ ] ( )θθ cos1ˆˆsinˆ' −××+×+= xnnxnxx  [ ] [ ] [ ] ( ){ } ( ) xnRxnnnI x  ˆ,cos1ˆˆsinˆ33 θθθ =−××+×+= The Rotation Matrix has the properties( ) [ ] [ ] [ ] ( ){ }θθθ cos1ˆˆsinˆˆ, 3333 −××+×+= ∆ nnnInR xx ( ) ( )[ ]( )333333 ˆ,ˆ, x T xx InRnR =θθOrtho-normal Unitary ( ) ( )[ ]( )conjugatecomplexInRnR x T xx == * 33 * 3333 ˆ,ˆ, θθ A Theorem from Matrix Algebra states: Every unitary matrix U can be expressed as an exponential matrix where H is hermitian (iH is skew-symmetric) ( )iHU exp= Let find the hermitian matrix that corresponds to the Unitary Rotation Matrix.
74 We found that the matrix has the following properties:[ ]×nˆ [ ] [ ] T x nnInn ˆˆˆˆ 33 +−=×× [ ] [ ] [ ] [ ]×−=××× nnnn ˆˆˆˆ [ ] [ ] [ ] [ ] ( )T x nnInnnn ˆˆˆˆˆˆ 33 +−−=×××× [ ] [ ]×−=× nn T ˆˆ skew-symmetric ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 1) SOLO Rotation Matrix in Three Dimensional Space (continue – 1) Define: [ ]×= ∆ nM ˆθ Therefore [ ][ ] ( )T x nnInnM ˆˆˆˆ 33 222 +−=××= θθ [ ] MnM 333 ˆ θθ −=×−= ( ) [ ]( ) [ ] [ ] [ ] [ ][ ] ( ) [ ] θθ θ θ θθ θ θ θ θθ θθ θθ θ sinˆcos1ˆˆ ˆ !3 ˆ) !4!2 1(ˆ !3 1 ) !4!2 1( 11 !3 1) !4!2 1 ( !4 1 !3 1 !2 1 ˆexpexp 33 3 2 42 2 33 3 2 42 2 2 233 2 2 2 33 432 33 ×+−××+= ×      +−+×++−−×+=       +−+++−−+=       +−++−+= +++++=×= nnnI nnnI MMMI MMI MMMMInM x x x x x     ( ) [ ] [ ] [ ] ( ){ } [ ]( )×=−××+×+= ∆ nnnnInR xx ˆexpcos1ˆˆsinˆˆ, 3333 θθθθ
75 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 2) SOLO Rotation Matrix in Three Dimensional Space (continue – 2) One other way to write this is by using the following matrices:           −= ∆ 010 100 000 1E           − = ∆ 001 000 100 2E           − = ∆ 000 001 010 3E [ ] EnEnEnEn nnn nn nn nn n zyx zyx xy xz yz  ⋅=++=           − +           − +           −=           − − − =× ˆ 000 001 010 001 000 100 010 100 000 0 0 0 ˆ 321 Therefore ( ) [ ] [ ] [ ] ( ){ } [ ]( ) ( )EnnnnnInR xx  ⋅=×=−××+×+= ∆ ˆexpˆexpcos1ˆˆsinˆˆ, 3333 θθθθθ
76 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 3) SOLO Rotation Matrix in Three Dimensional Space (continue – 3) One other way to obtain the same result is the following: compute first [ ] ( ) [ ] [ ] [ ] [ ] ( ){ } [ ] [ ] [ ] [ ][ ] [ ] ( ) [ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] θθ θθ θθ θθθ sinˆˆcosˆ cos1ˆsinˆˆˆ cos1ˆˆˆsinˆˆˆ cos1ˆˆsinˆˆˆ,ˆ 3333 ××+×= −×−××+×= −×××+××+×= −××+×+×=× nnn nnnn nnnnnn nnnInnRn xx ( ) [ ] [ ] [ ]{ }θθ θ θ sinˆˆcosˆ ˆ,33 ××+×= nnn d ndR x Therefore ( ) [ ] ( )nRn d ndR x x ˆ,ˆ ˆ, 33 33 θ θ θ ×= Since is independent of , we can integrate this equation to obtain again:[ ]×nˆ θ ( ) [ ]( ) ( )EnnnR x  ⋅=×= ˆexpˆexpˆ,33 θθθ Secondly compute
77 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 4) SOLO Euler Parameters The quaternion that describes the rotation was found to be: ( ) 32100 , qkqjqiqqq  +++== ρ ( )2/cos0 θ=q ( ) nˆ2/sin θρ =  ( ) ( ) ( ) zyx nqnqnq 2/sin&2/sin&2/sin 111 θθθ === where satisfy the relationskji  ,, 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ The quaternions representing the vector in frames A and B arev  ( ) ( ) ( ) ( ) ( ) ( )BBAA vvvv  ,0&,0 == The relation between those quaternions is given by: ( ) ( ) ( ) ( ) ( )( ) ( ) [ ] [ ][ ]{ } ( ) ( )BBBA vqqqvqqvqv  ××+×+⋅+=−== ρρρρρρρ 0 2 000 * 2,0,,0, We want to perform the same operations using 2x2 matrices with complex entries.       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Pauli Spinor Matrices For this let introduce the following definitions:
78 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 5) SOLO Euler Parameters (continue – 1) Define the operator as the transpose complex conjugate of a matrix A; i.e.:H T * ( ) ( )TTH AAA ** == We can see that 222211 && σσσσσσ === HHH Matrices having the property are called hermitian.AAH = Pauli Spinor Matrices are hermitian with zero trace. UnitaryII x H x H 122112211 11 10 01 01 10 01 10 σσσσσ σσ ⇒=⇒=      =            = = They have the following properties: UnitaryII i i i i x H x H 222222222 22 10 01 0 0 0 0 σσσσσ σσ ⇒=⇒=      =      −       − = = UnitaryII x H x H 322332233 33 10 01 10 01 10 01 σσσσσ σσ ⇒=⇒=      =      −      − = =       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i W. PAULI
79 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 6) SOLO Euler Parameters (continue – 2) Pauli Spinor Matrices properties (continue – 1): 321 0 0 0 0 01 10 σσσ i i i i i =      − =      −       = 312 0 0 01 10 0 0 σσσ i i i i i −=     − =            − = 132 0 0 10 01 0 0 σσσ i i i i i =      =      −      − = 123 0 0 0 0 10 01 σσσ i i i i i −=      − − =      −       − = 213 0 0 01 10 01 10 10 01 σσσ i i i i =      − =      − =            − = 231 0 0 01 10 10 01 01 10 σσσ i i i i −=      − −=      − =      −      = 2233321 xiIi == σσσσσ From those expressions we found that the relations between Pauli Matrices and the quaternions are: or kijiii  === 321 && σσσ 321 && σσσ ikijii −=−=−=        − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i
80 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 7) SOLO Euler Parameters (continue – 3) By similarity with quaternions definition:       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i ( ) ( ) ( )( )knjninqq zyx  +++== 2/sin2/cos,0 θθρ the 2x2 Rotation Matrix is given by: ( ) ( ) ( )( ) ( ) ( )( ) ( )       − =      +− −−− =               − +      − +      −      = ++−= ⋅−= ++−= ** 3012 1230 3210 332211220 22 3212222 10 01 0 0 01 10 10 01 ˆ2/sin2/cos 2/sin2/cosˆ, αβ βα σσσ σθθ σσσθθθ iqqiqq iqqiqq q i i qqiq qqqiIq niI nnniInR x x zyxxx  where 1230 & qiqqiq −−=−= ∆∆ βα Cayley-Klein Parameters
81 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 8) SOLO Euler Parameters (continue – 4)       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i 1230 & qiqqiq −−=−= ∆∆ βα Cayley-Klein Parameters Arthur Cayley (1821-1895) Felix C. Klein (1849-1925) Cayley-Klein Parameters are constrained by 1 2 3 2 2 2 1 2 0 ** =+++=+ qqqqββαα The quaternions representing the vector in frames A and B are v  ( ) ( ) ( ) ( ) ( ) ( )BBAA vvvv  ,0&,0 == Equivalently we define the 2x2 Matrix: ( )       − +      − +      =++= ∆ 10 01 0 0 01 10 32122 zAyAxAzAyAxA A x v i i vvvvvV σσσ We have: ( ) ( )         −+ − =⋅= zAyAxA yAxAzAAA x vivv ivvv vV σ  22 ( ) ( )         −+ − =⋅= zByBxB yBxBzBBB x vivv ivvv vV σ  22
82 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 9) SOLO Euler Parameters (continue – 5)       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i The relation between those matrices is ( ) ( )         −+ − =⋅= zAyAxA yAxAzAAA x vivv ivvv vV σ  22 ( ) ( )         −+ − =⋅= zByBxB yBxBzBBB x vivv ivvv vV σ  22 ( )       − =      +− −−− = ** 3012 1230 22 ˆ, αβ βα θ iqqiqq iqqiqq nR x       +− −−−         −+ −       ++− ++ =         −+ − 3012 1230 3012 1230 iqqiqq iqqiqq vivv ivvv qqiqq iqqiqq vivv ivvv zAyAxA yAxAzA zByBxB yBxBzB ( ) ( ) ( ) ( )nRVnRV x A xx B x ˆ,ˆ, 2222 * 2222 θθ= or
83 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 10) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix ( ) ( ) 222222 ˆ,ˆ, xxx InRnR =−θθ(1) ( )[ ] ( )nRnR xx ˆ,ˆ, 22 1 22 θθ −= − (2) ( )[ ] ( ) 222222 ˆ,ˆ, xx H x InRnR =θθ ( )nR x ˆ,22 θ Unitary ( )[ ] ( ) ( )( )[ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )[ ] 1 222222 22 2222 ˆ,ˆ,ˆ2/sin2/cos ˆ2/sin2/cos ˆ2/sin2/cosˆ, − = =−=⋅+= =⋅+= =⋅−= nRnRniI niI niInR xxx H x H x H x H θθσθθ σθθ σθθθ σσ     Proof (3) ( ) 22 2 ˆ xIn =⋅σ  Proof ( ) ( ) ( ) ( )( )[ ] ( ) ( )( )[ ] ( ) ( )( ) 22 22 22 2 2222 222222 ˆ2/sin2/cos ˆ2/sin2/cosˆ2/sin2/cos ˆ,ˆ, xx xx xxx InI niIniI nRnRI =⋅+= =⋅+⋅−= =−= σθθ σθθσθθ θθ   q.e.d. q.e.d.
84 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 11) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 1) (4) ( ) 0ˆ =⋅σ  ntrace (5) ( ) ( )[ ] ( ) ( ) ( ) ( ) 0ˆ,ˆˆˆ,:ˆ,ˆ, 222222 =⋅−⋅=⋅ nRnnnRnnR xxx θσσθσθ  (6) ( ) ( ) ( ) ( )nRni d nndR x x ˆ,ˆ 2 1ˆˆ, 22 22 θσ θ σθ   ⋅−= ⋅ ( ) ( ) [ ] [ ] [ ] 0ˆ 321321 =++=++=⋅ σσσσσσσ tracentracentracennnntracentrace zyxzyx  Proof q.e.d. ( ) ( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) 222222 2/sinˆ2/cosˆˆ2/sin2/cosˆˆ, xxx IinnniInnR θσθσσθθσθ −⋅=⋅⋅−=⋅  Proof ( ) ( ) ( ) ( ) ( )( )[ ] ( )( ) ( ) 222222 2/sinˆ2/cosˆ2/sin2/cosˆˆ,ˆ xxx IinniInnRn θσθσθθσθσ −⋅=⋅−⋅=⋅  q.e.d. Proof ( ) ( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) ( ) ( )( )[ ]σθθσσθθ σθθ θθ σθ    ⋅−⋅−=⋅−−= ⋅−= ⋅ niIniniI niI d d d nndR xx x x ˆ2/sin2/cosˆ 2 1 ˆ2/cos 2 1 2/sin 2 1 ˆ2/sin2/cos ˆˆ, 2222 22 22 q.e.d.
85 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 12) SOLO Integrate the equation:       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 2) ( ) ( ) ( ) ( )nRni d nndR x x ˆ,ˆ 2 1ˆˆ, 22 22 θσ θ σθ   ⋅−= ⋅ ( ) ( ) ( ) ( ) ( ) θσ σθ σθ dni nnR nndR x x    ⋅−= ⋅ ⋅ ˆ 2 1 ˆˆ, ˆˆ, 22 22 ( ) ( ) ( )    ⋅=    ⋅−= σ θ σ θ θ  n i ninR x ˆ 2 expˆ 2 expˆ,22 ( ) ( ) ( ) ( )    ⋅−−=    ⋅=    ⋅=    ⋅− = σ θ σ θ σ θ σ θ σσ   nininini H H H ˆ 2 ˆ 2 ˆ 2 ˆ 2 skew-hermitian This is in accordance to the Theorem from Matrix Algebra that states: Every unitary matrix U can be expressed as an exponential matrix where H is hermitian (iH is skew-symmetric) ( )iHU exp=
86 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 13) SOLO Let show that we get back the Rotation Matrix       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )σ θθ θ σ θθ σ θ σ θ σ θ σ θ σ θ σ θ θ       ⋅      −      = +      +⋅      −      −⋅      −= +⋅      −+⋅      −+⋅      −+⋅      −+= =    ⋅−= niI In i IniI ninininiI ninR x xxx x x ˆ 2 sin 2 cos 2!4 1 ˆ 2!32!2 1 ˆ 2 ˆ 2!4 1 ˆ 2!3 1 ˆ 2!2 1 ˆ 2 ˆ 2 expˆ, 22 22 43 22 2 22 4 4 3 3 2 2 22 22 Therefore ( ) ( ) ( )σ θθ σ θ θ  ⋅      −      =    ⋅−= niIninR xx ˆ 2 sin 2 cosˆ 2 expˆ, 2222 This is a generalization of the de Moivre expression for complex numbers:       −      =      − 2 sin 2 cos 2 exp θθθ ii
87 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 14) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 4) (7) Proof q.e.d. ( )( ) ( ) ( )21222121 ˆˆˆˆˆˆ nniInnnn x ×⋅+⋅=⋅⋅ σσσ  ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         ×−×+× ×−×× +      ⋅=         ×−⋅×+×− ×+××+⋅ =         −−++−+−− −+−−+++ =         −+ −         −+ − =⋅⋅ zyx yxz zxy xyz xyyxyyxxzzyzzyzxxz yzzyzxxzxyyxyyxxzz zyx yxz zyx yxz nnnninn nninnnn inn nninnnninn nninnnninn nnnninnnnnninnnninnnn innnninnnnnnnninnnnnn ninn innn ninn innn nn 212121 212121 21 21212121 21212121 212121212121212121 212121212121212121 222 222 111 111 21 ˆˆˆˆˆˆ ˆˆˆˆˆˆ 10 01 ˆˆ ˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆ ˆˆ σσ 
88 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 15) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 5) (8) Proof q.e.d. ( ) ( ) ( ) ( ) ( ) ( )      ×            +            +            ⋅−       ⋅            −            =     ⋅−    ⋅−= 2121 2221 21222122 ˆˆ 2 sin 2 sinˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 expˆ 2 expˆ,ˆ, nnnni Inn nininRnR x xx φθφθφθ σ φθφθ σ φ σ θ φθ   ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ] ( ) ( )σ φθ σ φθ σ φθφθ σ φθ σ φθ σσ φθφθ σ φφ σ θθ σ φ σ θ φθ       ⋅            −⋅            − ×⋅+⋅            −            = ⋅            −⋅            − ⋅⋅            −            =       ⋅      −            ⋅      −      =       ⋅−      ⋅−= 21 21222122 21 2122 222122 21222122 ˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 sin 2 cosˆ 2 sin 2 cos ˆ 2 expˆ 2 expˆ,ˆ, nini nniInnI nini nnI niIniI nininRnR xx x xx xx
89 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 16) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 6) (9a) ( )                         −       −      =            −            =      −      =             − − =              −=    −= 2 cos 2 sin 2 sin 2 cos 01 10 2 sin 10 01 2 cos 2 sin 2 cos 0 2 2 0 exp 01 10 2 exp 2 expˆ, 122 122 ϕϕ ϕϕ ϕϕ σ ϕϕ ϕ ϕ ϕ σ ϕ ϕ i i iiI i i iixR x x ( )                               −      =      −       −            =      −      =             − =              − −=    −= 2 cos 2 sin 2 sin 2 cos 0 0 2 sin 10 01 2 cos 2 sin 2 cos 0 2 2 0 exp 0 0 2 exp 2 expˆ, 222 222 θθ θθ θθ σ θθ θ θ θ σ θ θ i i iiI i i iiyR x x
90 ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 17) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 7) (9b) ( )                         − =                   +            −      =       −       −            =      −      =             − =              − −=      −= 2 exp0 0 2 exp 2 sin 2 cos0 0 2 sin 2 cos 10 01 2 sin 10 01 2 cos 2 sin 2 cos 2 0 0 2exp 10 01 2 exp 2 expˆ, 322 322 ψ ψ ψψ ψψ ψψ σ ψϕ ψ ψ ψ σ ψ ψ i i i i iiI i i iizR x x
91 ROTATIONS Gibbs Vector SOLO Using the quaternion notation we can define: ( ) ( ) ( )           == == == ∆ ∆ ∆ z y x n q q g n q q g n q q g 2/tan 2/tan 2/tan 0 3 3 0 2 2 0 1 1 θ θ θ 0 321 ˆˆˆ q zgygxgg ρ   =++= ( )       =++      =++==⋅ 2 tanˆˆˆ 2 tan 222222 3 2 2 2 1 θθ zyx T nnnggggggg  Gibbs in 1901 named this vector “vector semi-tangent of version”. Cayley used the three quantities in 1843 (before the introduction of vector notation), and credits their discovery to Rodriguez. Therefore they also called Euler-Rodriguez Parameters. 321 ,, ggg Josiah Willard Gibbs (1839-1903)
92 ROTATIONS Gibbs Vector (continue – 1) SOLO ( ) [ ] [ ] [ ] ( ){ }θθθ cos1ˆˆsinˆˆ, 3333 −××+×−=−= nnnInRC xx B A [ ] ( ){ }θθθ cos1ˆˆsinˆcos33 −+×−= T x nnnI                   +      ×            −= 2 sinˆ 2 sinˆ2 2 sinˆ 2 cos2cos33 θθθθ θ nnnI x                                       +      ×            −            −      = T x nnnI 2 tanˆ 2 tanˆ 2 cos2 2 tanˆ 2 cos2 2 tan1 2 cos 2222 33 θθθθθθθ                                 +      ×      −            −      = T x nnnI 2 tanˆ 2 tanˆ2 2 tanˆ2 2 tan1 2 cos 2 33 2 θθθθθ ( ) [ ]{ }TT xT gggggI gg   221 1 1 33 +×−− + = ( ) [ ]                     +           − − − −−−−           +++ = 321 3 2 1 12 13 23 2 3 2 2 2 12 3 2 2 2 1 2 0 0 0 21 100 010 001 1 1 ggg g g g gg gg gg ggg ggg Therefore ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ +++ = 2 3 2 2 2 1132231 132 2 3 2 2 2 1321 231321 2 3 2 2 2 1 2 3 2 2 2 1 122 212 221 1 1 ggggggggg ggggggggg ggggggggg ggg CB A
93 ROTATIONS Gibbs Vector (continue – 1) SOLO Let perform the following computations [ ]( ) ( ) [ ]{ } [ ]( )×++×−− + =×+ gIgggggI gg gIC x TT xTx B A    333333 221 1 1 ( ) [ ] ( )[ ] [ ] [ ] [ ]{ }×+××−×−++×−− + = gggggggggggggI gg TTTT xT   221221 1 1 33 ( ) ( )[ ]{ } [ ]{ }×−=×+−+ + = gIgggggI gg x TT xT   3333 11 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] 02 33 =×−− ggfIggedcba T x T  ( )( ) ( ) ( )cbea We obtained ( ) [ ]{ } [ ]{ } [ ]{ } 1 333333 221 1 1 − ×+×−=+×−− + = gIgIgggggI gg C xx TT xT B A   We can see that ( ) ( ) ( ) 2 3 2 2 2 1 2 3 2 2 2 1 1 3 3,32,21,1 ggg ggg CCCCtrace B A B A B A B A +++ −−− =++= 2 3 2 2 2 11 4 1 ggg Ctrace B A +++ =+
94 ROTATIONS Gibbs Vector (continue – 2) SOLO We found ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ +++ = 2 3 2 2 2 1132231 132 2 3 2 2 2 1321 231321 2 3 2 2 2 1 2 3 2 2 2 1 122 212 221 1 1 ggggggggg ggggggggg ggggggggg ggg CB A 2 3 2 2 2 11 4 1 ggg Ctrace B A +++ =+ Therefore ( ) ( ) B A B A B A Ctrace CC g + − = 1 2,33,2 1 ( ) ( ) B A B A B A Ctrace CC g + − = 1 3,11,3 2 ( ) ( ) B A B A B A Ctrace CC g + − = 1 1,22,1 3 Gibbs vector is not widely used because it becomes infinite when the rotation angle is an odd multiply of 180 degrees.
95 ROTATIONS Gibbs Vector (continue – 3) SOLO Differential of Gibbs Vector Let differentiate the equation: 0q g ρ   = We have: ρ ρ    2 0 0 0 q q q g −= • • Using the equation of quaternion differentiation: [ ] [ ] BA x T Iq q →•       ×+ − =         ω ρ ρ ρ      330 0 2 1 we obtain: [ ] [ ] AB T x qq I q q q g ← • •         + × +=−= ωρ ρρ ρ ρ      2 00 332 0 0 0 2 1 or [ ] [ ][ ] AB T x gggIg ← • +×+= ω  33 2 1
96 ROTATIONS Gibbs Vector (continue – 4) SOLO Differential of Gibbs Vector (continue – 1) [ ] [ ][ ] AB T x gggIg ← • +×+= ω  33 2 1 Let multiply this equation by and use:[ ] [ ][ ]×− gI x  33 [ ][ ] [ ] 33x TT Igggggg  −=×× and [ ] 0  =× gg [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] AB T xxx gggIgIggI ← • +×+×−=×− ω  333333 2 1 [ ] [ ] [ ] [ ][ ] [ ] AB TT x gggggggggI ←         ×−××−×−+×+= ω    0 33 2 1 [ ] [ ]( )[ ] ( ) AB T BAx TTT x ggIggggggI ←→ +=−−+= ωω  1 2 1 2 1 3333 We obtain: [ ] [ ][ ] gg ggI T x AB    + ×− = • ← 1 2 33 ω
97 ROTATIONS References SOLO [G] H. Goldstein, “Classical Mechanics”, Adison Wesley, 1980, 2nd Ed. [H] Peter C. Hughes, “Spacecraft Attitude Dynamics”, John Wiley & Sons, 1986 [JT] John L. Junkins, James D. Turner, “Optimal Spacecraft Rotational Maneuvers”, Elsevier, 1986 [McT] http://www-history.mcs.st-andrews.ac.uk/history [MTW] Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, “Gravitation”, Freeman, 1973 [W] James E. Wertz, “Spacecraft Attitude Determination and Control”, Reidel Publishing Company, 1985 [J] Peter Michael Jack’ site “Hypercomplex” http://www.hypercomplex.com [S] Malcom D. Shuster, “A Survey of Attitude Representations”, The Journal of Aeronautical Science, Vol. 41, Oct.-Dec. 1993, pp. 439-517
January 6, 2015 98 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

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Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

  • 1.
    1 NOTES ON ROTATIONS SOLO HERMELIN INITIALINTERMEDIATE FINAL Updated: 3.03.07 Run This http://www.solohermelin.com
  • 2.
    2 ROTATIONS TABLE OF CONTENT SOLO Rotationof a Rigid Body Mathematical Computation of a Rotation Rotation Matrix Computation of the Rotation Matrix Consecutive Rotations Decomposition of a Vector in Two Different Frames of Coordinates Differential Equation of the Rotation Matrices Computation of the Angular Velocity Vector from .AB←ω  ( ) ( )nRtC x B A ˆ,33 θ−= Computation of and as functions of .AB←ω  td dθ θ = td nd n ˆ ˆ = • Quaternions Computation of the Rotation Matrix Definition of the Quaternions Product of Quaternions Rotation Description Using the Quaternions
  • 3.
    3 ROTATIONS TABLE OF CONTENT(continue – 1) SOLO Rotation as a Multiplication of Two Matrices Relations Between Quaternions and Euler Angles Description of Successive Rotations Using Quaternions Differential Equation of the Quaternions Computation of as a Function of the Quaternion and its Derivatives ( ) ( )t B AB←ω  Computation of as a Function of , and their Derivatives( ) ( )t B AB←ω  θ nˆ Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I Euler Angles The Piogram Successive Euler Rotations 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→
  • 4.
    4 ROTATIONS TABLE OF CONTENT(continue – 2) SOLO Cayley-Klein (or Euler) Parameters and Related Quantities Rotation Matrix in Three Dimensional Space Euler Parameters Elementary Features of the 2x2 Rotation Matrix Gibbs Vector Differential Equation of Gibbs Vector References
  • 5.
    5 ROTATIONS Rotation of aRigid Body SOLO 23r 31r 12r1 3 2 P P 1 2 331r 23r 12r A rigid body in mechanics is defined as a system of mass points subject to the constraint that the distance between all pair of points remains constant through the motion. To define a point P in a rigid body it is enough to specify the distance of this point to three non-collinear points. This means that a rigid body is completely defined by three of its non-collinear points. Since each point, in a three dimensional space is defined by three coordinates, those three points are defined by 9 coordinates. But the three points are constrained by the three distances between them: 313123231212 && constrconstrconstr === Therefore a rigid body is completely defined by 9 – 3 = 6 degrees of freedom.
  • 6.
    6 ROTATIONS Rotation of aRigid Body (continue – 1) SOLO We have the following theorems about a rigid body: Euler’s Theorem (1775) The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Chasles’ Theorem (1839) The most general displacement of a rigid body is a translation plus a rotation. Leonhard Euler 1707-1783 Michel Chasles 1793-1880
  • 7.
    7 ROTATIONS Rotation of aRigid Body (continue – 2) SOLO Proof of Euler’s Theorem O – Fixed point in the rigid body A,B – Two point in the rigid body at equal distance r from O.       == rOBOA __________ A’,B’ – The new position of A,B respectively. Since the body is rigid rOBOA == __________ '' Therefore A,B, A’,B’ are one a sphere with center O. α – plane passing through O such that A and A’ are at the same distance from it. β – plane passing through O such that B and B’ are at the same distance from it. PP’ – Intersection of the planes andα β The two spherical triangles APB and A’PB’ are equal. The arcs AA’ and BB’ are equal. That means that rotation around PP’ that moves A to A’ will move B to B’. q.e.d.
  • 8.
    8 ROTATIONS Mathematical Computation ofa Rotation SOLO A B C O θ φφ nˆ v  1v  We saw that every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters.nˆ • The angle of rotation , defines the third parameter.θ Let rotate the vector around by a large angle , to obtain the new vector → = OAv  nˆ θ→ =OBv1  From the drawing we have: →→→→ ++== CBACOAOBv1  vOA  = → ( ) ( )θcos1ˆˆ −××= → vnnAC  Since direction of is: ( ) ( ) φν sinˆˆ&ˆˆ =×××× vnnvnn  and it’s length is: AC → ( )θφ cos1sin −v ( ) θsinˆ vnCB  ×= → Since has the direction and the absolute value CB → vn  ׈ θφsinsinv ( ) ( ) ( ) θθ sinˆcos1ˆˆ1 vnvnnvv  ×+−××+=
  • 9.
    9 ROTATIONS Computation of theRotation Matrix SOLO We have two frames of coordinates A and B defined by the orthogonal unit vectors and{ }AAA zyx ˆ,ˆ,ˆ { }BBB zyx ˆ,ˆ,ˆ The frame B can be reached by rotating the A frame around some direction by an angle .nˆ θ We want to find the Rotation Matrix that describes this rotation from A to B. ( )θ,ˆ33 nRC x B A = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθ θθ θθ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx ×+−××+= ×+−××+= ×+−××+= Let write those equations in matrix form. ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ θθ AAAA B nnnx [ ]( )           − − − =× 0 0 0 ˆ xy xz yz A nn nn nn n [ ] 0ˆ =×ntrace Rotation Matrix
  • 10.
    10 ROTATIONS Computation of theRotation Matrix (continue – 1) SOLO ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ θθ AAAA B nnnx ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 0 1 0 sinˆ 0 1 0 cos1ˆˆ 0 1 0 ˆ θθ AAAA B nnny ( ) [ ]( ) [ ]( ) ( ) [ ]( )           ×+           −××+           = 1 0 0 sinˆ 1 0 0 cos1ˆˆ 1 0 0 ˆ θθ AAAA B nnnz ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ Rotation Matrix (continue – 1)
  • 11.
    11 ROTATIONS Computation of theRotation Matrix (continue – 2) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )θθθ ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B =×+−××+==↑ The matrix has the following properties:[ ]( )A n׈ [ ]( ) { } [ ]( )ATA nn ×−=× ˆˆ [ ]( ) [ ]( ) =             −− −− −− =           − − −           − − − =×× 22 22 22 0 0 0 0 0 0 ˆˆ yxzyzx zyzxyx zxyxyz xy xz yz xy xz yz AA nnnnnn nnnnnn nnnnnn nn nn nn nn nn nn nn T x zzyzx zyyyx zxyxx nnI nnnnn nnnnn nnnnn ˆˆ 000 010 001 33 2 2 2 +−=             +           −= [ ]( ) [ ]( ) ( ) 213ˆˆ −=+−=×× AA nntrace [ ]( ) [ ] [ ] nn nn nn nn nnnnn xy xz yz zyx AT ˆˆ000 0 0 0 ˆˆ ×==           − − − =× [ ]( ) [ ]( ) [ ]( ) ( )[ ]( ) [ ]( ) [ ]( ) [ ]( )AATAAT x AAA nnnnnnnnInnn ×−=×+×−=×+−=××× ˆˆˆˆˆˆˆˆˆˆˆ 22 [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) ( )T x AAAAAA nnInnnnnn ˆˆˆˆˆˆˆˆ 33 +−−=××−=×××× skew-symmetric Rotation Matrix (continue – 2)
  • 12.
    12 ROTATIONS Computation of theRotation Matrix (continue – 3) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( ) ( ) B Axx AAA x TATATA x TA B CnRnR nnnI nnnIC =−=−= ×−−××+= ×+−××+= θθ θθ θθ ,ˆ,ˆ sinˆcos1ˆˆ sinˆcos1ˆˆ 3333 33 33 Note The last term can be writen in matrix form as Therefore In the same way End Note In fact is the matrix representation of the vector product:[ ][ ]vnn  ×× ˆˆ ( ) ( )vInnvvnn x T  33ˆˆˆˆ −→−⋅ ( ) ( ) ( ) ( ) vvnnnnvvnnvnn  −⋅=⋅−⋅=×× ˆˆˆˆˆˆˆˆ [ ][ ] T x nnInn ˆˆˆˆ 33 +−=×× ( )[ ] ( )[ ] [ ][ ][ ] [ ]×−=×××→×−=−⋅×=××× nnnnvnvvnnnvnnn ˆˆˆˆˆˆˆˆˆˆˆ  ( )[ ]{ } ( ) [ ][ ][ ][ ] [ ][ ]××−=××××→××−=×××× nnnnnnvnnvnnnn ˆˆˆˆˆˆˆˆˆˆˆˆ  Rotation Matrix (continue – 3)
  • 13.
    13 ROTATIONS Computation of theRotation Matrix (continue – 4) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( ) [ ]( ) ( )( ) [ ]( ) ( ) θθ θθ θθ sin0cos123 sinˆcos1ˆˆ sinˆcos1ˆˆ 33 33 −−−= =×−−××+= =×+−××+= AAA x TATATA x B A ntracenntraceItrace nnnItracetraceC Therefore θcos21+= B ACtrace Let compute the trace (sum of the diagonal components of a matrix) of B AC Also we have [ ] [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ){ } [ ] [ ]( )( ) [ ]( ) { } [ ] ( ) [ ]( ) { }=×−−+= =×−−+−+= =×+−××+= θθθ θθ θθ sinˆcos1ˆˆcos sinˆcos1ˆˆ sinˆcos1ˆˆ 33 3333 33 AT x AT xx TATATA x B A nnnI nnnII nnnIC ( ) θθθ sin 0 0 0 cos1cos 000 010 001 2 2 2           − − − −−             +           = xy xz yz zzyzx zyyyx zxyxx nn nn nn nnnnn nnnnn nnnnn Rotation Matrix (continue – 4)
  • 14.
    14 ROTATIONS Computation of theRotation Matrix (continue – 5) SOLO Ax Az Ay Bz By Bx  O nˆ θ θ θ θ Therefore we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             −+−−+− +−−+−− −−+−−+ = θθθθθθ θθθθθθ θθθθθθ cos1cossincos1sincos1 sincos1cos1cossincos1 sincos1sincos1cos1cos 2 2 2 zxzyyzx xzyyzyx yzxzyxx B A nnnnnnn nnnnnnn nnnnnnn C We get ( )1 2 1 cos −= B AtraceCθ two solutions for θ If ; i.e. we obtain0sin ≠θ πθ ,0≠ ( ) ( )[ ] ( )θsin2/2,33,2 B A B Ax CCn −= ( ) ( )[ ] ( )θsin2/3,11,3 B A B Ay CCn −= ( ) ( )[ ] ( )θsin2/1,22,1 B A B Az CCn −= Rotation Matrix (continue – 5)
  • 15.
    15 ROTATIONS Consecutive Rotations SOLO - Performfirst a rotation of the vector , according to the Rotation Matrix to the vector . v  ( )1133 ,ˆ θnR x 1v  - Perform a second a rotation of the vector , according to the Rotation Matrix to the vector . 1v  ( )2233 ,ˆ θnR x 2v  ( )vnRv x  11331 ,ˆ θ= ( ) ( ) ( ) ( )vnRvnRnRvnRv xxxx  θθθθ ,ˆ,ˆ,ˆ,ˆ 3311332233122332 === The result of those two consecutive rotation is a rotation defined as: ( ) ( ) ( )1133223333 ,ˆ,ˆ,ˆ θθθ nRnRnR xxx = Let interchange the order of rotations, first according to the Rotation Matrix and after that according to the Rotation Matrix . ( )2233 ,ˆ θnR x ( )1133 ,ˆ θnR x The result of those two consecutive rotation is a rotation defined as: ( ) ( )22331133 ,ˆ,ˆ θθ nRnR xx Since in general, the matrix product is not commutative ( ) ( ) ( ) ( )2233113311332233 ,ˆ,ˆ,ˆ,ˆ θθθθ nRnRnRnR xxxx ≠ Therefore, in general, the consecutive rotations are not commutative. Rotation Matrix (continue – 6)
  • 16.
    16 ROTATIONSSOLO INITIAL INTERMEDIATE FINAL ConsecutiveRotations of a DiceRotation Matrix (continue – 7)
  • 17.
    17 ROTATIONS Decomposition of aVector in Two Different Frames of Coordinates SOLO We have two frames of coordinate systems A and B, with the same origin O. We can reach B from A by performing a rotation. Let describe the vector in both frames.v  BzBByBBxBAzAAyAAxA zvyvxvzvyvxvv  111111 ++=++= ( )           = zA yA xA A v v v v  ( )           = zB yB xB B v v v v  & ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) BBABBABBAA BBABBABBAA BBABBABBAA zzzyyzxxzz zzyyyyxxyy zzxyyxxxxx ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆˆ ⋅+⋅+⋅= ⋅+⋅+⋅= ⋅+⋅+⋅=  ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] zABBABBABBA yABBABBABBA xABBABBABBA vzzzyyzxxz vzzyyyyxxy vzzxyyxxxxv ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆ ⋅+⋅+⋅+ ⋅+⋅+⋅+ ⋅+⋅+⋅=  from which Rotation Matrix (continue – 8)
  • 18.
    18 ROTATIONS Decomposition of aVector in Two Different Frames of Coordinates (continue – 1) SOLO ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                     ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ =           zA yA xA BABABA BABABA BABABA zB yB xB v v v zzzyzx yzyyyx xzxyxx v v v ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ ( ) ( )AB A B vCv  = where is the Transformation Matrix (or Direction Cosine Matrix – DCM) from frame A to frame B. B AC ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )          ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ==↑ BABABA BABABA BABABA B A B A zzzyzx yzyyyx xzxyxx CC ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ : In the same way ( ) ( ) ( ) ( )BA B BB A A vCvCv  == −1 therefore ( ) 1− = B A A B CC Rotation Matrix (continue – 9)
  • 19.
    19 ROTATIONS Decomposition of aVector in Two Different Frames of Coordinates (continue – 2) SOLO ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( )ATAAB A TB A TAAB A TAB A BTB vvvCCvvCvCvvv  ====2 Since the scalar product is independent of the frame of coordinates, we have [ ] [ ] [ ] 1− =→= B A TB A B A TB A CCICC [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           =                         = 100 010 001 3,33,23,1 2,32,22,1 1,31,21,1 3,32,31,3 3,22,21,2 3,12,11,1 B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A TB A CCC CCC CCC CCC CCC CCC CC or ( ) ( )    =≠ == ==∑ = 3,2,10 3,2,11 ,, 3 1 jji iji kjCkiC ij k B A B A δ Those are 9 equations in , but by interchanging i with j we get the same conditions, therefore we have only 6 independent equations. ( ) 3,2,1,, =jijiC B A We see that the Rotation Matrix is ortho-normal (having real coefficients and the rows/columns are orthogonal to each other and of unit absolute value. Rotation Matrix (continue – 10) This means that the relation between the two coordinate systems is defined by 9 – 6 = 3 independent parameters.
  • 20.
    20 ROTATIONS Differential Equations ofthe Rotation Matrices SOLO We want to develop the differential equation of the Rotation Matrix as a function of the Angular Velocity of the Rotation. Let define by: -the Rotation Matrix that defines a frame of coordinates B at the time t relative to some frame A. ( )tC B A -the Rotation Matrix that defines the frame of coordinates B at the time t+Δt relative to some frame A. ( )ttC B A ∆+ ( )φω ∆−,ˆ33xR -the Rotation Matrix from the frame of coordinates B at the time t to B at time t+Δt relative to some frame A. ( ) ( ) ( )tCRttC B Ax B A φω ∆−=∆+ ,ˆ33 and ( ) [ ] [ ] [ ] ( ) [ ]{ } [ ] [ ] [ ] [ ]             ∆       ∆ ×−      ∆ ××+= ∆×−∆−××+=∆− 2 cos 2 sinˆ2 2 sinˆˆ2 sinˆcos1ˆˆ,ˆ 2 33 3333 φφ ω φ ωω φωφωωφω x xx I IR Rotation Matrix (continue – 11)
  • 21.
    21 ROTATIONS Differential Equations ofthe Rotation Matrices (continue – 1) SOLO Let differentiate the Rotation Matrix ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tC dt dIR tC t IR tC t IR t tCtCR t tCttC t C dt dC B A xxB A xx t B A xx t B A B Ax t B A B A t B A t B A φ φ φωφ φ φω φωφω θ       ∆ −∆− =      ∆ ∆ ∆ −∆− =       ∆ −∆− = ∆ −∆− = ∆ −∆+ = ∆ ∆ = →∆→∆ →∆→∆ →∆→∆ 3333 0 3333 0 3333 0 33 0 00 ,ˆ lim ,ˆ lim ,ˆ lim ,ˆ lim limlim ( ) [ ][ ] [ ] [ ]×−=                     ∆       ∆       ∆ ×−      ∆       ∆       ∆ ××= ∆ −∆− →∆→∆ ω φ φ φ ω φ φ φ ωω φ φω θθ ˆ 2 cos 2 2 sin ˆ 2 2 2 sin ˆˆlim ,ˆ lim 2 2 0 3333 0 xx IR and Therefore ( ) [ ] ( )tC dt d dt tdC B A B A φ ω ×−= ˆ Rotation Matrix (continue – 12)
  • 22.
    22 ROTATIONS Differential Equations ofthe Rotation Matrices (continue – 2) SOLO The final result of the Rotation Matrix differentiation is: Since defines the unit vector of rotation and the rotation rate from B at time t to B at time t+Δt, relative to A, then is the angular velocity vector of the frame B relative to A, at the time t ωˆ dt dφ ω φ ˆ dt d ( ) ω φ ω ˆ dt dB AB =←  ( ) ( )[ ]( ) ( )tCt dt tdC B A B AB B A ×−= ←ω  By changing indixes A and B we obtain ( ) ( )[ ] ( ) ( )tCt dt tdC A B A BA A B ×−= ←ω  Rotation Matrix (continue – 13)
  • 23.
    23 ROTATIONS Differential Equations ofthe Rotation Matrices (continue – 3) SOLO Let find the relation between and[ ]( )B AB ×←ω  [ ]( )A AB ×←ω  For any vector let perform the following computationsv  [ ]( ) ( ) [ ]( ) [ ]( )A AB B A B AB BB AB vCvv  ×=×=× ←←← ωωω [ ]( ) ( ) [ ]( ) ( ) [ ]( ) ( )BA B A AB B A AB A A B A AB B A AA AB B A vCCvCCCvC  ×=×=×= ←←← ωωω Since this is true for any vector we havev  [ ]( ) [ ]( ) A B A AB B A B AB CC ×=× ←← ωω  Pre-multiplying by and post-multiplying by we get: A BC B AC [ ]( ) [ ]( ) B A B AB A B A AB CC ×=× ←← ωω  Rotation Matrix (continue – 14)
  • 24.
    24 ROTATIONS Differential Equations ofthe Rotation Matrices (continue – 4) SOLO Let differentiate the equation 33x A B B A ICC = to obtain [ ] ( ) [ ] ( ) 0=+×−=+×−=+ ←← dt dC C dt dC CCC dt dC CC dt dC A BB A B AB A BB A A B B A B AB A BB A A B B A ωω  Post-multiplying by we get A BC [ ] ( ) [ ] ( ) [ ] ( ) A B A AB A B B A B AB A B B AB A B A B CCCCC dt dC ×=×=×= ←←← ωωω  We obtained for the differentiation of the Rotation Matrix ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( )B AB A B A B A AB A B A BA A B ttCtCttCt dt tdC ×=×=×−= ←←← ωωω  Note We can see that ( )[ ] ( ) ( )[ ] ( ) ( ) ( )tttt ABBA A AB A BA ←←←← =−⇒×=×− ωωωω  End Note Rotation Matrix (continue – 15)
  • 25.
    25 ROTATIONS Differential Equations ofthe Rotation Matrices (continue – 5) SOLO Suppose that we have a third frame of coordinates I (for example inertial) and we have the angular velocity vectors of frames A and B relative to I. We have [ ]( ) B I B IB B I C dt dC ×−= ←ω  [ ]( ) A I A IA A I C dt dC ×−= ←ω  A I B A B I CCC = dt dC CC dt dC dt dC A IB A A I B A B I += [ ]( ) [ ]( ) I A A I A IA B A I A B I B IB I A A IB A I A B I B A CCCCCC dt dC CC dt dC dt dC ×+×−=−= ←← ωω  or From which we get: [ ]( ) [ ]( )A IA B A B A B IB B A CC dt dC ×+×−= ←← ωω  Rotation Matrix (continue – 16)
  • 26.
    26 ROTATIONSSOLO From the equation Computationof the Angular Velocity Vector from .AB←ω  ( ) ( )nRtC x B A ˆ,33 θ−= ( ) ( )[ ]( ) ( )tCt dt tdC B A B AB B A ×−= ←ω  we obtain ( )[ ]( ) ( ) ( )[ ]TB A B AB AB tC dt tdC t −=×←ω  Since the Rotation Matrix is defined also by and ( ) [ ] ( ) [ ]{ }θθθθ sinˆcos1ˆˆcosˆ, 3333 ×−−+=−= × nnnInRC T x B A ( )tC B A nˆθ we can compute as function of and their derivativesnˆθAB←ω  td dθ θ = td nd n ˆ ˆ = • (this is a long procedure described in the complementary work “Notes on Rotations”, and a simpler derivation will be given later, we give here the final result) [ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  Rotation Matrix (continue – 17)
  • 27.
    27 ROTATIONSSOLO Computation of andas functions of .AB←ω  td dθ θ = td nd n ˆ ˆ = • Let pre-multiply the equation by and use T nˆ[ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  [ ] 0ˆˆ,0ˆˆ,1ˆˆ ==×= • nnnnnn TTT to obtain [ ] ( ) AB TTTT AB T nnnnnnnnn ← •• ← =→+−×−= ωθθθθω  ˆsinˆˆcos1ˆˆˆˆˆˆ Let pre-multiply the equation by and use[ ]×nˆ[ ] ( ) θθθω sinˆcos1ˆˆˆ •• ← +−×−= nnnnAB  [ ] [ ][ ] ( ) ••• −=−=××=× nnInnnnnnn x T ˆˆˆˆˆˆˆ,0ˆˆ 33 to obtain [ ] [ ] [ ][ ] ( ) [ ] ( ) [ ] θθθθθω sinˆˆcos1ˆsinˆˆcos1ˆˆˆˆˆˆ •••• ← ×+−=×+−××−×=× nnnnnnnnnnn AB  Let pre-multiply the equation by[ ] ( ) [ ] θθω sinˆˆcos1ˆˆ •• ← ×+−=× nnnn AB  [ ]×nˆ [ ][ ] [ ] ( ) [ ][ ] [ ] ( )θθθθω cos1ˆˆsinˆsinˆˆˆcos1ˆˆˆˆ −×+−=××+−×=×× •••• ← nnnnnnnnnn AB  Rotation Matrix (continue – 18)
  • 28.
    28 ROTATIONS Computation of andas functions of (continue – 1) SOLO AB←ω  td dθ θ = td nd n ˆ ˆ = • We have two equations: ( ) [ ] [ ] ABnnnn ← •• ×=×+− ωθθ  ˆsinˆˆcos1ˆ [ ] ( ) [ ][ ] ABnnnnn ← •• ××=−×+− ωθθ  ˆˆcos1ˆˆsinˆ with two unknowns and • nˆ [ ] • × nn ˆˆ From those equations we get: ( )[ ] [ ] ( ) [ ][ ] θωθωθθ sinˆˆcos1ˆsincos1ˆ 22 ABAB nnnn ←← • ××−−×=+−  or ( ) [ ] ( ) [ ][ ] θωθωθ sinˆˆcos1ˆcos1ˆ2 ABAB nnnn ←← • ××−−×=−  Finally we obtain: AB T n ←= ωθ  ˆ [ ] [ ][ ] ABnnnn ← •             ××−×= ω θ  2 cotˆˆˆ 2 1 ˆ Rotation Matrix (continue – 19)
  • 29.
    29 ROTATIONS Quaternions SOLO The quaternions methodwas introduced by Hamilton in 1843. It is based on Euler Theorem (1775) that states: The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Therefore every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters • The angle of rotation , defines the third parameter nˆ θ William Rowan Hamilton 1805 - 1865 ( ) ( ) ( ) θθ sinˆcos1ˆˆ1 vnvnnvv  ×+−××+= The rotation of around by angle is given by:nˆ θv  A B C O θ φφ nˆ v  1v  that can be writen ( )[ ] ( ) ( ) θθ sinˆcos1ˆˆ1 vnvvnnvv  ×+−−⋅+= or ( ) ( ) ( ) θθθ sinˆcos1ˆˆcos1 vnvnnvv  ×+−⋅+=
  • 30.
    30 ROTATIONS Quaternions (continue –1) SOLO Computation of the Rotation Matrix We found the Rotation Matrix that describes this rotation from A to B. ( )θ,ˆ33 nRC x B A = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θθ θθ θθ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx ×+−××+= ×+−××+= ×+−××+= ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33 =           ×+−××+= θθ or from which [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )θθθ ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B =×+−××+==↑
  • 31.
    31 ROTATIONS Quaternions (continue –2) SOLO Definition of the Quaternions The quaternions (4 parameters) were defined by Hamilton as a generalization of the complex numbers ( ) 32100 , qkqjqiqqq  +++== ρ ( )2/cos0 θ=q ( ) nˆ2/sin θρ =  ( ) ( ) ( ) zyx nqnqnq 2/sin&2/sin&2/sin 111 θθθ === where satisfy the relations:kji  ,, 1−=⋅=⋅=⋅ kkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ 1−=⋅⋅ kji  the complex conjugate of is defined asq ( ) 32100 * , qkqjqiqqq  −−−=−= ρ
  • 32.
    32 ROTATIONS Quaternions (continue –3) SOLO Product of Quaternions Product of two quaternions andAq Bq ( )( ) ( )( )3210321000 ,, BBBBAAAABBAABA qkqjqiqqkqjqiqqqqq  ++++++== ρρ ( ) ( ) ( )3210321033221100 AAABBBBABABABABA qkqjqiqqkqjqiqqqqqqqqq  ++++++−−−= ( ) ( ) ( )122131132332 BABABABABABA qqqqkqqqqjqqqqi −+−+−+  therefore ( )( ) ( ) ( )[ ]BAABBABABABBAABA qqqqqqqq ρρρρρρρρ  ×++⋅−== 000000 ,,, Let use this expression to find ( )( ) ( )( ) 2 3 2 2 2 1 2 0 222 000 * 00 * 1ˆˆ 2 sin 2 cos,,,, qqqqnnqqqqqqqqq +++==⋅      +      =⋅+=−==−= θθ ρρρρρρ  The quaternion product can be writen in matrix form as: [ ] [ ]               ×− − =              ×+ − ==      = A A BxBB T BB B B AxAA T AA BA q Iq qq Iq q qq q q ρρρ ρ ρρρ ρ ρ      0 330 00 330 00 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅
  • 33.
    33 ROTATIONS Quaternions (continue –4) SOLO Rotation Description Using the Quaternions Let compute the expression: ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]ρρρρρρρρρρ ρρρρρ   ××−×+×−+⋅⋅×+⋅−⋅= ×−⋅=−= AAAAAAAA AAAAA vvqvqvqvvvqqv qvvqvqvqqvq 00 2 000 0000 * , ,,,,0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ){ } ( ) [ ] [ ][ ]{ } ( ) ( )A AAAA AAAAAAA vqq vvqqvv vvqvqvqvvv    ××+×−⋅+= ××+×−+⋅+⋅××= ××−×+×−+⋅+⋅−⋅= ρρρρρ ρρρρρρρ ρρρρρρρρρρ 22,0 2,0 ,0 0 2 0 0 2 0 00 2 0 Using the relations: ( ) ( ) [ ][ ] ( )[ ][ ] ( )[ ][ ] [ ] ( ) ( )[ ] [ ]      ×=×=× ××−=××=×× =⋅+ →    = = nnq nnnn q n q ˆsinˆ2/sin2/cos22 ˆˆcos1ˆˆ2/sin22 1 ˆ2/sin 2/cos 0 2 2 0 0 θθθρ θθρρ ρρ θρ θ     and ( ) ( ) [ ] [ ]( ) [ ]( ) ( ) [ ]( ) { } ( )AAAA x AB A B vnnnIvCv  θθ sinˆcos1ˆˆ33 ×−−××+== we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) [ ] [ ][ ]{ } ( ) ( )AAABB vqqvqqvqvv  ××+×−=−=== ρρρρρ 221,0,,0,,0 000 *
  • 34.
    34 ROTATIONS Quaternions (continue –5) SOLO Rotation Description Using the Quaternions (continue – 1) Using the fact that we obtain: [ ] [ ] [ ][ ]××+×−= ρρρ  22 033 qIC x B A           − − −           − − − +           − − − −           = 0 0 0 0 0 0 2 0 0 0 2 100 010 001 12 13 23 12 13 23 12 13 23 0 qq qq qq qq qq qq qq qq qq q             −− −− −− +           − − − +           = 2 2 2 13231 32 2 1 2 321 3121 2 2 2 3 1020 1030 2030 2222 2222 2222 022 202 220 100 010 001 qqqqqq qqqqqq qqqqqq qqqq qqqq qqqq             −−−+ +−−− −+−− = 2 2 2 110323120 3210 2 1 2 33021 20312130 2 2 2 3 2212222 2222122 2222221 qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq 1 2 3 2 2 2 1 2 0 =+++ qqqq ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ = 2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 22 22 22 qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq C B A
  • 35.
    35 ROTATIONS Quaternions (continue –6) SOLO Rotation as a Multiplication of Two Matrices [ ] [ ] [ ][ ]××+×−= ρρρ  22 033 qIC x B A ( )[ ] [ ] [ ][ ]××+×−+= ρρρρρ  22 033 2 0 qIq x T [ ] [ ] [ ][ ] [ ] [ ][ ]××++××+×−= ρρρρρρρ  33033 2 0 2 x T x IqIq [ ] [ ]( ) [ ] [ ]( ) [ ] [ ][ ]××++×−×−= ρρρρρρ  33330330 x T xx IIqIq For any vector we can write ( ) ( ) ( )ρρρρρρ  ⋅−⋅=×× aaaa  or in matrix notation [ ][ ] [ ]( ) [ ] [ ] T x T x T x TT IIaIa ρρρρρρρρρρρρ  ≡+⇒−=×× 333333 Therefore we have [ ] [ ]( ) [ ] [ ]( ) [ ] [ ][ ]××++×−×−= ρρρρρρ  33330330 x T xx B A IIqIqC [ ] [ ]( ) [ ] [ ]( ) =+×−×−= T xx IqIq ρρρρ  330330 [ ]321 3 2 1 012 103 230 012 103 230 qqq q q q qqq qqq qqq qqq qqq qqq           +           − − −           − − − =
  • 36.
    36 ROTATIONS Quaternions (continue –7) SOLO Rotation as a Multiplication of Two Matrices (continue – 1) [ ] [ ][ ] [ ] [ ]           ×− ×−=                 − − −           − − − = ρ ρ ρρ           330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C [ ] [ ][ ] [ ] [ ]           ×− − ×−−=                 − − − −−−           −− −− −− = ρ ρ ρρ           330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C [ ] [ ][ ] [ ] [ ]           ×− ×−=                 − − −           − − − = T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C ρ ρ ρρ          330 330 321 012 103 230 3012 2103 1230 [ ] [ ][ ] [ ] [ ]           − ×− −×−=                 −−− − − −           −− −− −− = T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C ρ ρ ρρ          330 330 321 012 103 230 3012 2103 1230
  • 37.
    37 ROTATIONS Quaternions (continue –8) SOLO Relation Between Quaternions and Euler Angles
  • 38.
    38 ROTATIONS Quaternions (continue –9) SOLO Description of Successive Rotations Using Quaternions Let describe two consecutive rotations: - First rotation defined by the quaternion ( )                   == 1 11 1101 ˆ 2 sin, 2 cos, nqq θθ ρ  - Folowed by the second rotation defined by the quaternion ( )                   == 2 22 2202 ˆ 2 sin, 2 cos, nqq θθ ρ  After the first rotation the quaternion of the vector is transferred to 1 * 1 qvq After the second rotation we obtain ( ) ( ) ( )21 * 2121 * 1 * 221 * 1 * 2 qqvqqqqvqqqqvqq == Therefore the quaternion representing those two rotation is: ( ) ( )( ) ( )       ×            +      +      ⋅            −            = =×++⋅−==== 21 21 1 2 2 1 21 2121 21120210212010220110210 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cos,ˆˆ 2 sin 2 sin 2 cos 2 cos ,,,, nnnnnn qqqqqqqqqq θθθθθθθθ ρρρρρρρρρ  ( ) ( )210 , qqqq == ρ  21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq ⋅            −            =      = θθθθθ 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn ×            +      +      =      = θθθθθ ρ 
  • 39.
    39 ROTATIONS Quaternions (continue –10) SOLO Description of Successive Rotations Using Quaternions (continue – 1) ( ) ( )210 , qqqq == ρ  21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq ⋅            −            =      = θθθθθ 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn ×            +      +      =      = θθθθθ ρ  Two consecutive rotations, followed by , are given by:1q 2q From those equations we can see that: 0ˆˆˆˆˆˆˆˆˆˆ 21212112211221  =×→×−=×→×=×= nnnnnnnnnnifonlyandifqqqq The rotations are commutative if and only if are collinear.21 ˆ&ˆ nn In matrix form those two rotations are given by: First Rotation: ( ) [ ] ( ) [ ]{ }111111331133 sinˆcos1ˆˆcosˆ, θθθθ ×−−+=− nnnInR T xx Second Rotation: ( ) [ ] ( ) [ ]{ }222222332233 sinˆcos1ˆˆcosˆ, θθθθ ×−−+=− nnnInR T xx Total Rotation: ( ) ( ) ( ) [ ] ( ) [ ]{ }θθθθθθ sinˆcos1ˆˆcosˆ,ˆ,ˆ, 331133223333 ×−−+=−−=− nnnInRnRnR T xxxx
  • 40.
    40 ROTATIONS Quaternions (continue –11) SOLO Description of Successive Rotations Using Quaternions (continue – 2) Let find the quaternion that describes the Euler Rotations through the angles respectively. Let write the rotations according to their order 123 →→ ϕθψ ,,             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕθθψψ ijkqqqq xyz B A                    −            +            +                        +      = 2 sin 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos ϕθϕθθϕϕθψψ kjik                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos ϕθψϕθψ                         −                  + 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos ϕθψϕθψ i                          +                  + 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin ϕθψϕθψ j                          −                  + 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin ϕθψϕθψ k 
  • 41.
    41 ROTATIONS Quaternions (continue –12) SOLO Differential Equation of the Quaternions Let define ( ) ( )ρ  ,0qtq B A = - the quaternion that defines the position of B frame relative to frame A at time t. ( ) ( )tqqttq B A ∆+∆+=∆+ ρ  ,00 - the quaternion that defines the position of B frame relative to frame A at time t+Δt. ( )             ∆       ∆ =∆ ∆t B A ntq ˆ 2 sin, 2 cos θθ - the quaternion that defines the position of B frame at time t+Δt relative to frame B at time t. We have the relation: ( ) ( ) ( )tqtqttq B A B A B A ∆=∆+ or ( ) ( ) ( ) ( )( ) ( ) ( ) ( )[ ] ( ) ( )( )ρρρρρρρρ θθ  ∆∆−+=∆∆+−=∆+∆+−=∆+=            ∆       ∆ =∆ ∆ ,,0,1,,,,,ˆ 2 sin, 2 cos 00000000 * qqqqqqqqttqtqntq B A B At B A therefore ( )( )             ∆ −      ∆ =∆∆− ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  ( ) ( )             ∆ −      ∆ =∆∆ ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  or
  • 42.
    42 ROTATIONS Quaternions (continue –13) SOLO Differential Equation of the Quaternions (continue – 1) ( ) ( )             ∆ −      ∆ =∆∆ ∆tnqq ˆ 2 sin,1 2 cos,, 00 θθ ρρ  Let divide both sides by and take the limit .0→∆ tt∆ ( ) ( ) ( )       =      =                   ∆ ∆       ∆       ∆       ∆ ∆       ∆ −      ∆ ==      ∆ ∆ ∆ ∆ ∆∆∆ →∆ tBttB t ntqnqn tt tq td d tt q ˆ 2 1 ,0ˆ 2 1 ,0,ˆ 2 2 sin 2 1 , 2 1 2 cos 2 1 ,lim 0 0 0 θθρ θ θ θ θ θ θ ρ   But is the instant angular velocity vector of frame B relative to frame A.tn∆ ˆθ ( ) ( ) t B AB nt ∆← = ˆθω  ( ) ( ) ( )( ) ( ) ( )ttn B AB B ABt ←←∆ == ωωθ  ,0ˆ,0 So we can write ( ) ( ) ( ) ( )ttqtq td d B AB B A B A ←= ω 2 1 This is the Differential equation of the quaternion that defines the position of B relative to A, at the time t as a function of the angular velocity vector of frame B relative to frame A, . ( )tq B A ( ) ( )t B AB←ω 
  • 43.
    43 ROTATIONS Quaternions (continue –14) SOLO Differential Equation of the Quaternions (continue – 2) Developing this equation, we get ( ) ( ) ( ) ( )ttqtq td d B AB B A B A ←= ω 2 1 ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )B AB B AB B AB B AB qtq dt d dt dq ←←←← ×+⋅−==      ωρωωρωρ ρ   00 0 , 2 1 ,0, 2 1 , from which ( )B AB dt dq ←⋅−= ωρ  2 10 ( ) ( ) ( )B AB B ABq dt d ←← ×+= ωρω ρ   0 2 1 or in matrix form [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1
  • 44.
    44 ROTATIONS Quaternions (continue –15) SOLO Differential Equation of the Quaternions (continue – 3)                           − − − −−− =                 ← ← ← zBAB yBAB xBAB qqq qqq qqq qqq q q q q dt d ω ω ω 012 103 230 321 3 2 1 0  [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1 B AAB xBAByBABzBAB xBABzBAByBAB yBABzBABxBAB zBAByBABxBAB q q q q q q q q q dt d ← ←←← ←←← ←←← ←←← Ω=                                   − − − −−− =                 2 1 0 0 0 0 2 1 3 2 1 0 3 2 1 0        ωωω ωωω ωωω ωωω After rearranging or ( )zBAByBABxBAB qqq dt dq ←←← ++−= ωωω 321 0 2 1 ( )zBAByBABxBAB qqq dt dq ←←← +−= ωωω 230 1 2 1 ( )zBAByBABxBAB qqq dt dq ←←← −+= ωωω 103 2 2 1 ( )zBAByBABxBAB qqq dt dq ←←← ++−= ωωω 012 3 2 1
  • 45.
    45 ROTATIONS Quaternions (continue –16) SOLO Pre-multiply the equation Computation of as a Function of the Quaternion and its Derivatives ( ) ( )t B AB←ω  [ ] [ ] ( ) ( )t Iq q dt d B AB x T ←           ×+ − =           ω ρ ρ ρ       330 0 2 1 by [ ] [ ][ ]×−− ρρ    330 xIq [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] ( ) ( ) =           ×+ − ×−−=             ×−− ← • • t Iq Iq q Iq B AB x T xx ω ρ ρ ρρ ρ ρρ             330 330 0 330 2 1 [ ] [ ][ ][ ] ( ) ( ) [ ] [ ]( )[ ] ( ) ( ) ( ) ( )ttIIq tIq B BA B BAx TT x T B BAx T →→ → =−−+= =××−+= ωωρρρρρρ ωρρρρ   2 1 2 1 2 1 3333 2 0 33 2 0 Therefore ( ) ( ) [ ] [ ][ ]             ×−−= • • ← ρ ρρω      0 3302 q Iqt x B AB
  • 46.
    46 ROTATIONS Quaternions (continue –17) SOLO Computation of as a Function of the Quaternion and its Derivatives (continue – 1) But and are related. Differentiating the equation ( ) ( )t B AB←ω  we obtain • 0q • ρ  1 2 0 =+ ρρ T q ( ) ( ) [ ] [ ][ ] [ ] [ ]( ) =       ×−+−=             ×−−= •• • • ← ρρρ ρ ρρω       3300 0 330 22 xx B AB Iqq q Iqt [ ] [ ]( ) [ ] [ ] ••• +×− =       ×−+= ρ ρρρ ρρρρρ    0 033 2 0 330 0 2 1 2 q qIq Iq q T x x T From the equation •••• −=→=+ ρρρρ  TT q qqq 0 000 1 0 we obtain ( ) [ ] [ ]( ) • ← +×−= ρρρρω  T x B AB qIq q 033 2 0 0 2
  • 47.
    47 ROTATIONS Quaternions (continue –18) SOLO Computation of as a Function of , and their Derivatives( ) ( )t B AB←ω  θ nˆ Differentiate the quaternion ( )                   == nqq ˆ 2 sin, 2 cos,0 θθ ρ  to obtain             +            −=      = ••• nnqq ˆ 2 sinˆ 2 cos 2 , 2 sin 2 ,0 θθθθθ ρ   Substitute this in the equation ( ) ( ) [ ] [ ][ ]             ×−−= • • ← ρ ρρω      0 3302 q Iqt x B AB [ ]                       +            −       ×      −            −= • nn nIn x ˆ 2 sinˆ 2 cos 2 2 sin 2 ˆ 2 sin 2 cosˆ 2 sin2 33 θθθ θθ θθθ     [ ] [ ] •• ×      −×            −            +      +      = nnnnnnn ˆˆ 2 sin2ˆˆ 2 cos 2 sinˆ 2 cos 2 sin2ˆ 2 cosˆ 2 sin 222 θθθ θ θθθ θ θ θ  ( ) [ ] •• ← ×−−+= nnnnAB ˆˆcos1ˆsinˆ θθθω  Finally we obtain We recovered a result obtained before.
  • 48.
    48 ROTATIONS Quaternions (continue –18) SOLO Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I The relations between the components of a vector in the frames A, B and I arev  ( ) AB A IA I B A IB I B vCvCCvCv  === Using quaternions the same relations are given by ( ) B A A I IA I B A B I IB I B qqvqqqvqv *** == Therefore B A A I B I qqq = B I A I B A qqq * = Let perform the following calculations B A A I B A A I B I q dt d qqq dt d q dt d += &( )B IB B I B I qq dt d ←= ω 2 1 ( )A IA A I A I qq dt d ←= ω 2 1 and use ( ) ( ) B A A I B A A IA A I B IB B I q dt d qqqq += ←← ωω 2 1 2 1 ( ) ( ) B A A IA A I A I B IB B I A I B A qqqqqq dt d ←← −= ωω  1 ** 2 1 2 1 to obtain ( ) ( ) B A A IA B IB B A B A qqq dt d ←← −= ωω 2 1 2 1
  • 49.
    49 ROTATIONS Quaternions (continue –19) SOLO Differential Equatio of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I (continue – 1) Using the relations ABIAIB ←←← += ωωω  and ( ) ( ) B A A IA B A B IA qq ←← = ωω * ( ) ( ) B A A IA B IA B A qq ←← = ωω we have ( ) ( ) ( ) ( ) ( ) ( ) ( )    0 2 1 2 1 2 1 2 1 2 1 B A A IA B IA B A B AB B A B A A IA B IA B AB B A B A qqqqqq dt d ←←←←←← −+=−+= ωωωωωω from which ( ) ( ) B A A AB B AB B A B A qqq dt d ←← == ωω 2 1 2 1 Since BAAB ←← −= ωω  we get ( ) ( ) ( ) ( ) B A A BA B BA B A B A A AB B AB B A B A qqqqq dt d ←←←← −=−=== ωωωω 2 1 2 1 2 1 2 1 From we get1 * == B A A B B A B A qqqq A B B A qq = * Therefore       +      = B A A B B A A B q dt d qqq dt d 0 *B A B A A B A B qq dt d qq dt d       −=     
  • 50.
    50 ROTATIONS Euler Angles SOLO The orientationof the Body Frame relative to the Inertial Frame has three degrees of freedom. We will use 3 Euler Angles that define the orientation by three consecutive rotations around the consecutive frame axes. [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc The three basic Euler rotations around axes are described by the rotation matrices: ,3ˆ,2ˆ,1ˆ
  • 51.
    51 ROTATIONS Euler Angles (continue– 1) SOLO Introduce the Piogram that represents the following notation: (from Pio R.L. “Symbolic Representations of Coordinate Transformations”, IEEE on Aerospace and Navigation Electronics, Vol. ANE-11,June 1964, pp.128-134) The Piogram
  • 52.
    52 ROTATIONSEuler Angles (continue– 2) SOLO Rotation Around x Axis by an Angle .ϕ [ ]           − == ϕϕ ϕϕϕ cossin0 sincos0 001 x B AC BAAB xx      11 ϕϕω ==← [ ]           −=−= ϕϕ ϕϕϕ cossin0 sincos0 001 x A BC The Piogram (continue – 1)
  • 53.
    53 ROTATIONSEuler Angles (continue– 3) SOLO Rotation Around y Axis by an Angle .θ [ ]           − == θθ θθ θ cos0sin 010 sin0cos y B AC BAAB yy  11 θθω ==← [ ]           − =−= θθ θθ θ cos0sin 010 sin0cos y A BC The Piogram (continue – 2)
  • 54.
    54 ROTATIONSEuler Angles (continue– 4)SOLO Rotation Around z Axis by an Angle .ψ [ ]           −== 100 0cossin 0sincos ψψ ψψ ψ x B AC BAAB zz      11 ψψω ==← [ ]           − =−= 100 0cossin 0sincos ψψ ψψ ψ x A BC The Piogram (continue – 3)
  • 55.
    55 ROTATIONS Euler Angles (continue– 5) SOLO Using the basic Euler Angles we can define the following 12 different rotations: (a) six rotations around three different axes: 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ (b) six rotations such that the first and third are around the sam axes, but the second is different: 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→ Suppose that the Transfer Matrix from A to B is defined by three consecutive Euler Angles: around (unit vector in A Frame), around (unit vector in intermediate frame), around (unit vector in B Frame). B AC iθ Iiˆ jθ Interjˆ kθ Bkˆ [ ] [ ] [ ] [ ] [ ]TB A B A A B A B B Akkjjii B A CCCICCC ==→== −1 &θθθ
  • 56.
    56 ROTATIONSEuler Angles (continue– 6) SOLO 123 →→Euler Angles rotations: using the Piogram 1.Rotation from A to A’ around the third axis by the angle ψ 2. Rotation from A’ to B’ around the third axis by the angle θ . 3. Rotation from B’ to B around the third axis by the angle ϕ . xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv The Piogram (continue – 4)
  • 57.
    57 ROTATIONSEuler Angles (continue– 7) SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: Example 1: Computation of the relation between to using the PiogramyBv zAyAxA vvv ,, From the Piogram: ( ) ( ) θϕψθϕψϕψθϕψϕ cossinsinsinsincoscoscossinsinsincos zAyAxAyB vvvv ++++−= using the Piogram (continue – 1) The Piogram (continue – 5)
  • 58.
    58 ROTATIONSEuler Angles (continue– 8)SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: using the Piogram (continue – 2) Example 2: Computation of the matrix (1st column) using the Piogram B AC 1=xAv 0=yAv 0=zAv ψcos' =xAv ψsin' −=yAv 0' =zAv ψsin' −=yBv ψθcossin' =zBv ψθcoscos=xBv ψθϕ ψϕ cossincos sinsin + +=zBv ψθϕ ψϕ cossinsin sincos + +−=yBv ψcos ψcos ψsin ψsin θcos θsin θcos θsin ϕcos ϕsin ϕcos ϕsin ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           +−+ ++− − =             θϕψθϕψϕψθϕψϕ θϕψθϕψϕψθϕψϕ θψθψθ ccssccscscss csssscccsssc ssccc CCC CCC CCC B A B A B A B A B A B A B A B A B A 3,32,31,3 3,22,21,2 3,12,11,1 The Piogram (continue – 6)
  • 59.
    59 ROTATIONSEuler Angles (continue– 9) SOLO 123 →→Euler Angles rotations: xAv yAv zAv yBv zBv xBv ϕ θ− ψ 'xAv 'yAv 'zAv 'xBv 'yBv 'zBv Piogram: Example 3: Computation of the angular velocity using the Piogram using the Piogram (continue – 3) AB←ω  xAAB←ω yAAB←ω zAAB←ω 0 0 0 B ψ ψ− A θ θ 'A ϕ− ϕ 'B ( ) ( ) ( ) ( ) [ ] [ ] [ ]                     − − =           −−+           −+           =++=← ψ θ ϕ θ ψψθ ψψθ ϕθψθψψϕθψω        10 0 0 0 0 1 0 1 0 1 0 0 111 233 ' '' ' '' s csc scc xCyCz B B A B A A A A A A A AB The Piogram (continue – 7)
  • 60.
    60 ROTATIONSEuler Angles (continue– 10)SOLO 123 →→Euler Angles rotations: Piogram: Example 4: Computation of the angular velocities and using the Piogram using the Piogram (continue – 4) IB←ω  IA←ω  ψ xBIB←ω yBIB←ω zBIB←ω ϕ θ− ψ θ ϕ A 'A 'B B xAIA←ω yAIA←ω zAIA←ω xBIB←ω yBIB←ω zBIB←ω B ϕ− ϕ− ψ− ψ− A'A xAIA←ω yAIA←ω zAIA←ω θ θ− 'B Piogram: ( ) [ ] [ ] [ ]                                               +           −−+           −−+           −−=           = ← ← ← ← ← ← ← zBIB yBIB xBIB zAIA yAIA xAIA A IA ω ω ωϕ ϕθθ ψ ψ ω ω ω ω 0 0 0 0 0 0 123     ( ) [ ] [ ] [ ]                                               +           +           +           =           = ← ← ← ← ← ← ← zAIA yAIA xAIA zBIB yBIB xBIB B IB ω ω ω ψ ψθθ ϕ ϕ ω ω ω ω     0 0 0 0 0 0 321 IAIBBAAAB xyz ←←← −=++= ωωϕθψω      '' 111 The Piogram (continue – 8)
  • 61.
    61 ROTATIONS Euler Angles (continue– 9) SOLO 321 →→Euler Angles rotations: [ ] [ ] [ ]           − ++−− +−+ == φθφθθ φψφθψφψφθψθψ φψφθψφψφθψθψ ϕθψ ccscs sccssccssscs sscsccsssccc CB A 123                     − + − =           +− + − =           ← ← ← ψ θ ϕ θ ψϕθ ψϕθ ψθϕ ψθψθϕ ψθψθϕ ω ω ω       10 0 0 s csc scc s csc scc zBAB yBAB xBAB                     −=           ← ← ← zBAB yBAB xBAB csscs ccsc sc c ω ω ω θψθψθ ψθψθ ψψ θ ψ θ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ψψθθϕϕ kjiqqqq zyx B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q Piogram
  • 62.
    62 ROTATIONS Euler Angles (continue– 10) SOLO 231 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +− − −+ == ϕθϕθψϕθϕθψθψ ϕθϕψψ ϕθϕθψϕθϕθψθψ ϕψθ ccssssccsssc scccs csscsssccscc C B A 132                     − − − =           − +− − =           ← ← ← θ ψ ϕ θθψ ψ θθψ θψθψϕ θψϕ θψθψϕ ω ω ω       0 10 0 csc s scc csc s scc zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB ssccs ccsc sc c ω ω ω θψψθψ θψθψ θθ ψ θ ψ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos θθψψϕϕ jkiqqqq yzx B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         −                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
  • 63.
    63 ROTATIONS Euler Angles (continue– 11) SOLO 312 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           − ++− +−+ == ϕθϕϕθ θϕψθψϕψϕθψφψ θϕψθψϕψϕθψθψ θϕψ ccscs cscssccssccs csssccsssscc C B A 213                     − − + =           +− − + =           ← ← ← ψ ϕ θ ϕ ψψϕ ψψϕ ψϕθ ψϕψϕθ ψϕψϕθ ω ω ω       10 0 0 s scc csc s scc csc zBAB yBAB xBAB                     − −=           ← ← ← zBAB yBAB xBAB ccsss sccc cs c ω ω ω ϕψϕψϕ ψϕψϕ ψψ ϕ ψ ϕ θ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ψψϕϕθθ kijqqqq zxy B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
  • 64.
    64 ROTATIONS Euler Angles (continue– 12) SOLO 132 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−+ ++− − == ϕθϕθψϕψϕθϕθψ ϕθϕθψϕψϕθϕθψ θψψψθ θψϕ ccssssccsscs sccssccssccs scscc CB A 231                     − =           +− + + =           ← ← ← ϕ ψ θ ϕϕψ ϕϕψ ψ ϕψϕψθ ϕψϕψθ ϕψθ ω ω ω       0 0 10 csc scc s csc scc s zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB sscsc ccsc sc c ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ θ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕψψθθ ikjqqqq xzy B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
  • 65.
    65 ROTATIONS Euler Angles (continue– 13) SOLO 213 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           −+− − −+− == ϕθϕθψθψϕθψθψ ϕϕψϕψ ϕθϕθψθψϕθψθψ ψϕθ ccsccssscssc scccs csssccsssscc C B A 312                    − =           + + +− =           ← ← ← θ ϕ ψ θϕθ ϕ θϕθ θϕϕθψ θϕψ θϕϕθψ ω ω ω       0 10 0 scc s ccs scc s ccs zBAB yBAB xBAB           =           −− −− − =           ← ← ← zBAB yBAB xBAB sccss cscc cs c ω ω ω ϕθϕϕθ ϕθϕθ θθ ϕ θ ϕ ψ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos θθϕϕψψ jikqqqq yxz B A                          −                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos2 ϕθψϕθψ q                         +                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
  • 66.
    66 ROTATIONS Euler Angles (continue– 14) SOLO 123 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−+ ++− − == θϕψθϕψϕψθϕψϕ θϕψθϕψϕψθϕψϕ θψθψθ ψθϕ ccssccscscss csssscccsssc ssccc CB A 321                     − − =           +− + − =           ← ← ← ψ θ ϕ ϕθϕ ϕθϕ θ ϕθψϕθ ϕθψϕθ θψϕ ω ω ω       ccs scc s ccs scc s zBAB yBAB xBAB 0 0 01                     −=           ← ← ← zBAB yBAB xBAB cs sccc csssc c ω ω ω ϕϕ ϕθϕθ ϕθϕθθ θ ψ θ ϕ 0 0 1                +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos ϕϕθθψψ ijkqqqq xyz B A                          +                  = 2 sin 2 sin 2 sin 2 cos 2 cos 2 cos0 ϕθψϕθψ q                         −                  = 2 cos 2 sin 2 sin 2 sin 2 cos 2 cos1 ϕθψϕθψ q                         +                  = 2 cos 2 sin 2 cos 2 sin 2 cos 2 sin2 ϕθψϕθψ q                         −                  = 2 sin 2 sin 2 cos 2 cos 2 cos 2 sin3 ϕθψϕθψ q
  • 67.
    67 ROTATIONS Euler Angles (continue– 15) SOLO 121 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           −−− ++− − == 212121212 212121212 11 11212 ϕϕϕθϕϕϕϕθϕϕθ ϕϕϕθϕϕϕϕθϕϕθ θϕθϕθ ϕθϕ sscccscccscs cssccccscsss scssc C B A                     − =           − + + =           ← ← ← 2 1 22 22 221 221 21 0 0 10 ϕ θ ϕ ϕϕθ ϕϕθ θ ϕθϕθϕ ϕθϕθϕ ϕθϕ ω ω ω       scs css c scs css c zBAB yBAB xBAB                     −− −=           ← ← ← zBAB yBAB xBAB scscs sscs cs s ω ω ω ϕθϕθθ ϕθϕθ ϕϕ θ ϕ θ ϕ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ϕϕθθϕϕ ϕϕ ijiqqqq xyx B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ϕϕθϕθϕϕθϕ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 1 ϕϕθϕθϕϕθϕ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 2 ϕϕθϕθϕϕθϕ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 3 ϕϕθϕθϕϕθϕ q
  • 68.
    68 ROTATIONS Euler Angles (continue– 16) SOLO 131 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−− +−−== 212121212 212121212 11 11312 ϕϕϕψϕϕϕϕψϕϕψ ϕϕϕψϕϕϕϕψϕϕψ ψϕψϕψ ϕψϕ ccscscssccss scccsssccccs ssscc C B A                     − − =           ← ← ← zBAB yBAB xBAB scccs csss sc s ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ ϕ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ϕϕψψϕϕ ϕϕ ikiqqqq xzx B A                      − − =           ← ← ← zBAB yBAB xBAB scccs csss sc s ω ω ω ϕψϕψψ ϕψϕψ ϕϕ ψ ϕ ψ ϕ 22 22 22 2 1 0 0 1          +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ϕϕψϕψϕϕψϕ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 1 ϕϕψϕψϕϕψϕ q       −       =                        −                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 122121 2 ϕϕψϕψϕϕψϕ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 3 ϕϕψϕψϕϕψϕ q
  • 69.
    69 ROTATIONS Euler Angles (continue– 17) SOLO 212 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−+ −−− == 212122121 11 212122121 21122 θϕθθθθϕθϕθθθ ϕθϕφθ θϕθθθθϕθϕθθθ θϕθ cccsscsccssc sccss ccccsssscscc C B A                     −− =           −− + + =           ← ← ← 2 1 22 22 221 21 221 0 10 0 θ ϕ θ θθϕ ϕ θθϕ θϕθϕθ θϕθ θϕθϕθ ω ω ω       scs c css scs c css zBAB yBAB xBAB                     −− −−=           ← ← ← zBAB yBAB xBAB ccssc sscs cs s ω ω ω θϕϕθϕ θϕθϕ θθ ϕ θ ϕ θ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 θθϕϕθθ θθ jijqqqq yxy B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 θθϕθϕθθϕθ q       +       =                        −                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 1 θθϕθϕθθϕθ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 2 θθϕθϕθθϕθ q       +       =                        +                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 212121 3 θθϕθϕθθϕθ q
  • 70.
    70 ROTATIONS Euler Angles (continue– 18) SOLO 232 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−+ − −−− == 212122121 11 212122121 21322 θθθψθθψθθθψθ ϕθψψθ θθθψθθψθθθψθ θψθ ccscssscsscc sscsc scccscsssccc C B A                     + − =           + + − =           ← ← ← 2 1 22 22 221 21 221 0 10 0 θ ψ θ θθψ ψ θθψ θψθψθ θψθ θψθψθ ω ω ω       css c scs css c scs zBAB yBAB xBAB                     −− =           ← ← ← zBAB yBAB xBAB scscc csss sc s ω ω ω θψψθψ θψθψ θθ ψ θ ψ θ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 θθψψθθ θθ jkjqqqq yzy B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 θθψθψθθψθ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 1 θθψθψθθψθ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 2 θθψθψθθψθ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 3 θθψθψθθψθ q
  • 71.
    71 ROTATIONS Euler Angles (continue– 19) SOLO 313 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           − +−−− +− == ϕϕψϕψ ψϕψϕψψψψϕψψψ ψϕψϕψψψψϕψψψ ψϕψ cscss cscccssccssc ssscccsscscc C B A 11 221212121 221212121 31132                     − + =           + − + =           ← ← ← 2 1 22 22 21 221 221 10 0 0 ψ ϕ ψ ϕ ψψϕ ψψϕ ψϕψ ψϕψϕψ ψϕψϕψ ω ω ω       c scs css c scs css zBAB yBAB xBAB                     −− −=           ← ← ← zBAB yBAB xBAB sccsc sscs cs s ω ω ω ϕψϕψϕ ψϕψϕ ψψ ϕ ψ ϕ ψ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ψψϕϕψψ ψψ kikqqqq zxz B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ψψϕψϕψψϕψ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 1 ψψϕψϕψψϕψ q       −       =                        −                  = 2 sin 2 sin 2 sin 2 sin 2 cos 2 cos 2 sin 2 sin 212121 2 ψψϕψϕψψϕψ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 3 ψψϕψϕψψϕψ q
  • 72.
    72 ROTATIONS Euler Angles (continue– 20) SOLO 323 →→Euler Angles rotations: Piogram [ ] [ ] [ ]           +−−− −+− == θθψθψ ψθψψψθψψψϕθψ ψθψψψθψψψψθψ ψθψ csssc ssccscscsscc csscccsssccc C B A 11 221212121 221212121 31232                    − =           + + +− =           ← ← ← 2 1 22 22 21 221 221 10 0 0 ψ θ ψ θ ψψθ ψψθ ψθψ ψθψθψ ψθψθψ ω ω ω       c css scs c css scs zBAB yBAB xBAB                     − − =           ← ← ← zBAB yBAB xBAB ssccc csss sc s ω ω ω θψθψθ ψθψθ ψψ θ ψ θ ψ 22 22 22 2 1 0 0 1    ( ) ( )             +                  +                  +      == 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2211 21 ψψθθψψ ψψ kjkqqqq zyz B A        +       =                        −                  = 2 cos 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 212121 0 ψψθψθψψθψ q       −       −=                        −                  = 2 sin 2 sin 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 212121 1 ψψθψθψψθψ q       −       =                        +                  = 2 cos 2 sin 2 sin 2 sin 2 sin 2 cos 2 sin 2 cos 212121 2 ψψθψθψψθψ q       +       =                        +                  = 2 sin 2 cos 2 sin 2 cos 2 cos 2 cos 2 cos 2 sin 212121 3 ψψθψθψψθψ q
  • 73.
    73 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities SOLO Rotation Matrix in Three Dimensional Space We saw that the rotation of a vector is given byx  [ ] [ ] [ ] ( )θθ cos1ˆˆsinˆ' −××+×+= xnnxnxx  [ ] [ ] [ ] ( ){ } ( ) xnRxnnnI x  ˆ,cos1ˆˆsinˆ33 θθθ =−××+×+= The Rotation Matrix has the properties( ) [ ] [ ] [ ] ( ){ }θθθ cos1ˆˆsinˆˆ, 3333 −××+×+= ∆ nnnInR xx ( ) ( )[ ]( )333333 ˆ,ˆ, x T xx InRnR =θθOrtho-normal Unitary ( ) ( )[ ]( )conjugatecomplexInRnR x T xx == * 33 * 3333 ˆ,ˆ, θθ A Theorem from Matrix Algebra states: Every unitary matrix U can be expressed as an exponential matrix where H is hermitian (iH is skew-symmetric) ( )iHU exp= Let find the hermitian matrix that corresponds to the Unitary Rotation Matrix.
  • 74.
    74 We found thatthe matrix has the following properties:[ ]×nˆ [ ] [ ] T x nnInn ˆˆˆˆ 33 +−=×× [ ] [ ] [ ] [ ]×−=××× nnnn ˆˆˆˆ [ ] [ ] [ ] [ ] ( )T x nnInnnn ˆˆˆˆˆˆ 33 +−−=×××× [ ] [ ]×−=× nn T ˆˆ skew-symmetric ROTATIONS Cayley-Klein (or Euler) Parameters and Related Quantities (continue – 1) SOLO Rotation Matrix in Three Dimensional Space (continue – 1) Define: [ ]×= ∆ nM ˆθ Therefore [ ][ ] ( )T x nnInnM ˆˆˆˆ 33 222 +−=××= θθ [ ] MnM 333 ˆ θθ −=×−= ( ) [ ]( ) [ ] [ ] [ ] [ ][ ] ( ) [ ] θθ θ θ θθ θ θ θ θθ θθ θθ θ sinˆcos1ˆˆ ˆ !3 ˆ) !4!2 1(ˆ !3 1 ) !4!2 1( 11 !3 1) !4!2 1 ( !4 1 !3 1 !2 1 ˆexpexp 33 3 2 42 2 33 3 2 42 2 2 233 2 2 2 33 432 33 ×+−××+= ×      +−+×++−−×+=       +−+++−−+=       +−++−+= +++++=×= nnnI nnnI MMMI MMI MMMMInM x x x x x     ( ) [ ] [ ] [ ] ( ){ } [ ]( )×=−××+×+= ∆ nnnnInR xx ˆexpcos1ˆˆsinˆˆ, 3333 θθθθ
  • 75.
    75 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 2) SOLO Rotation Matrix in Three Dimensional Space (continue – 2) One other way to write this is by using the following matrices:           −= ∆ 010 100 000 1E           − = ∆ 001 000 100 2E           − = ∆ 000 001 010 3E [ ] EnEnEnEn nnn nn nn nn n zyx zyx xy xz yz  ⋅=++=           − +           − +           −=           − − − =× ˆ 000 001 010 001 000 100 010 100 000 0 0 0 ˆ 321 Therefore ( ) [ ] [ ] [ ] ( ){ } [ ]( ) ( )EnnnnnInR xx  ⋅=×=−××+×+= ∆ ˆexpˆexpcos1ˆˆsinˆˆ, 3333 θθθθθ
  • 76.
    76 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 3) SOLO Rotation Matrix in Three Dimensional Space (continue – 3) One other way to obtain the same result is the following: compute first [ ] ( ) [ ] [ ] [ ] [ ] ( ){ } [ ] [ ] [ ] [ ][ ] [ ] ( ) [ ] [ ] [ ] [ ] ( ) [ ] [ ] [ ] θθ θθ θθ θθθ sinˆˆcosˆ cos1ˆsinˆˆˆ cos1ˆˆˆsinˆˆˆ cos1ˆˆsinˆˆˆ,ˆ 3333 ××+×= −×−××+×= −×××+××+×= −××+×+×=× nnn nnnn nnnnnn nnnInnRn xx ( ) [ ] [ ] [ ]{ }θθ θ θ sinˆˆcosˆ ˆ,33 ××+×= nnn d ndR x Therefore ( ) [ ] ( )nRn d ndR x x ˆ,ˆ ˆ, 33 33 θ θ θ ×= Since is independent of , we can integrate this equation to obtain again:[ ]×nˆ θ ( ) [ ]( ) ( )EnnnR x  ⋅=×= ˆexpˆexpˆ,33 θθθ Secondly compute
  • 77.
    77 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 4) SOLO Euler Parameters The quaternion that describes the rotation was found to be: ( ) 32100 , qkqjqiqqq  +++== ρ ( )2/cos0 θ=q ( ) nˆ2/sin θρ =  ( ) ( ) ( ) zyx nqnqnq 2/sin&2/sin&2/sin 111 θθθ === where satisfy the relationskji  ,, 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ The quaternions representing the vector in frames A and B arev  ( ) ( ) ( ) ( ) ( ) ( )BBAA vvvv  ,0&,0 == The relation between those quaternions is given by: ( ) ( ) ( ) ( ) ( )( ) ( ) [ ] [ ][ ]{ } ( ) ( )BBBA vqqqvqqvqv  ××+×+⋅+=−== ρρρρρρρ 0 2 000 * 2,0,,0, We want to perform the same operations using 2x2 matrices with complex entries.       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Pauli Spinor Matrices For this let introduce the following definitions:
  • 78.
    78 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 5) SOLO Euler Parameters (continue – 1) Define the operator as the transpose complex conjugate of a matrix A; i.e.:H T * ( ) ( )TTH AAA ** == We can see that 222211 && σσσσσσ === HHH Matrices having the property are called hermitian.AAH = Pauli Spinor Matrices are hermitian with zero trace. UnitaryII x H x H 122112211 11 10 01 01 10 01 10 σσσσσ σσ ⇒=⇒=      =            = = They have the following properties: UnitaryII i i i i x H x H 222222222 22 10 01 0 0 0 0 σσσσσ σσ ⇒=⇒=      =      −       − = = UnitaryII x H x H 322332233 33 10 01 10 01 10 01 σσσσσ σσ ⇒=⇒=      =      −      − = =       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i W. PAULI
  • 79.
    79 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 6) SOLO Euler Parameters (continue – 2) Pauli Spinor Matrices properties (continue – 1): 321 0 0 0 0 01 10 σσσ i i i i i =      − =      −       = 312 0 0 01 10 0 0 σσσ i i i i i −=     − =            − = 132 0 0 10 01 0 0 σσσ i i i i i =      =      −      − = 123 0 0 0 0 10 01 σσσ i i i i i −=      − − =      −       − = 213 0 0 01 10 01 10 10 01 σσσ i i i i =      − =      − =            − = 231 0 0 01 10 10 01 01 10 σσσ i i i i −=      − −=      − =      −      = 2233321 xiIi == σσσσσ From those expressions we found that the relations between Pauli Matrices and the quaternions are: or kijiii  === 321 && σσσ 321 && σσσ ikijii −=−=−=        − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i
  • 80.
    80 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 7) SOLO Euler Parameters (continue – 3) By similarity with quaternions definition:       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i ( ) ( ) ( )( )knjninqq zyx  +++== 2/sin2/cos,0 θθρ the 2x2 Rotation Matrix is given by: ( ) ( ) ( )( ) ( ) ( )( ) ( )       − =      +− −−− =               − +      − +      −      = ++−= ⋅−= ++−= ** 3012 1230 3210 332211220 22 3212222 10 01 0 0 01 10 10 01 ˆ2/sin2/cos 2/sin2/cosˆ, αβ βα σσσ σθθ σσσθθθ iqqiqq iqqiqq q i i qqiq qqqiIq niI nnniInR x x zyxxx  where 1230 & qiqqiq −−=−= ∆∆ βα Cayley-Klein Parameters
  • 81.
    81 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 8) SOLO Euler Parameters (continue – 4)       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i 1230 & qiqqiq −−=−= ∆∆ βα Cayley-Klein Parameters Arthur Cayley (1821-1895) Felix C. Klein (1849-1925) Cayley-Klein Parameters are constrained by 1 2 3 2 2 2 1 2 0 ** =+++=+ qqqqββαα The quaternions representing the vector in frames A and B are v  ( ) ( ) ( ) ( ) ( ) ( )BBAA vvvv  ,0&,0 == Equivalently we define the 2x2 Matrix: ( )       − +      − +      =++= ∆ 10 01 0 0 01 10 32122 zAyAxAzAyAxA A x v i i vvvvvV σσσ We have: ( ) ( )         −+ − =⋅= zAyAxA yAxAzAAA x vivv ivvv vV σ  22 ( ) ( )         −+ − =⋅= zByBxB yBxBzBBB x vivv ivvv vV σ  22
  • 82.
    82 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 9) SOLO Euler Parameters (continue – 5)       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i The relation between those matrices is ( ) ( )         −+ − =⋅= zAyAxA yAxAzAAA x vivv ivvv vV σ  22 ( ) ( )         −+ − =⋅= zByBxB yBxBzBBB x vivv ivvv vV σ  22 ( )       − =      +− −−− = ** 3012 1230 22 ˆ, αβ βα θ iqqiqq iqqiqq nR x       +− −−−         −+ −       ++− ++ =         −+ − 3012 1230 3012 1230 iqqiqq iqqiqq vivv ivvv qqiqq iqqiqq vivv ivvv zAyAxA yAxAzA zByBxB yBxBzB ( ) ( ) ( ) ( )nRVnRV x A xx B x ˆ,ˆ, 2222 * 2222 θθ= or
  • 83.
    83 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 10) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix ( ) ( ) 222222 ˆ,ˆ, xxx InRnR =−θθ(1) ( )[ ] ( )nRnR xx ˆ,ˆ, 22 1 22 θθ −= − (2) ( )[ ] ( ) 222222 ˆ,ˆ, xx H x InRnR =θθ ( )nR x ˆ,22 θ Unitary ( )[ ] ( ) ( )( )[ ] ( ) ( )( ) ( ) ( )( ) ( ) ( )[ ] 1 222222 22 2222 ˆ,ˆ,ˆ2/sin2/cos ˆ2/sin2/cos ˆ2/sin2/cosˆ, − = =−=⋅+= =⋅+= =⋅−= nRnRniI niI niInR xxx H x H x H x H θθσθθ σθθ σθθθ σσ     Proof (3) ( ) 22 2 ˆ xIn =⋅σ  Proof ( ) ( ) ( ) ( )( )[ ] ( ) ( )( )[ ] ( ) ( )( ) 22 22 22 2 2222 222222 ˆ2/sin2/cos ˆ2/sin2/cosˆ2/sin2/cos ˆ,ˆ, xx xx xxx InI niIniI nRnRI =⋅+= =⋅+⋅−= =−= σθθ σθθσθθ θθ   q.e.d. q.e.d.
  • 84.
    84 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 11) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 1) (4) ( ) 0ˆ =⋅σ  ntrace (5) ( ) ( )[ ] ( ) ( ) ( ) ( ) 0ˆ,ˆˆˆ,:ˆ,ˆ, 222222 =⋅−⋅=⋅ nRnnnRnnR xxx θσσθσθ  (6) ( ) ( ) ( ) ( )nRni d nndR x x ˆ,ˆ 2 1ˆˆ, 22 22 θσ θ σθ   ⋅−= ⋅ ( ) ( ) [ ] [ ] [ ] 0ˆ 321321 =++=++=⋅ σσσσσσσ tracentracentracennnntracentrace zyxzyx  Proof q.e.d. ( ) ( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) 222222 2/sinˆ2/cosˆˆ2/sin2/cosˆˆ, xxx IinnniInnR θσθσσθθσθ −⋅=⋅⋅−=⋅  Proof ( ) ( ) ( ) ( ) ( )( )[ ] ( )( ) ( ) 222222 2/sinˆ2/cosˆ2/sin2/cosˆˆ,ˆ xxx IinniInnRn θσθσθθσθσ −⋅=⋅−⋅=⋅  q.e.d. Proof ( ) ( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) ( ) ( )( )[ ]σθθσσθθ σθθ θθ σθ    ⋅−⋅−=⋅−−= ⋅−= ⋅ niIniniI niI d d d nndR xx x x ˆ2/sin2/cosˆ 2 1 ˆ2/cos 2 1 2/sin 2 1 ˆ2/sin2/cos ˆˆ, 2222 22 22 q.e.d.
  • 85.
    85 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 12) SOLO Integrate the equation:       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 2) ( ) ( ) ( ) ( )nRni d nndR x x ˆ,ˆ 2 1ˆˆ, 22 22 θσ θ σθ   ⋅−= ⋅ ( ) ( ) ( ) ( ) ( ) θσ σθ σθ dni nnR nndR x x    ⋅−= ⋅ ⋅ ˆ 2 1 ˆˆ, ˆˆ, 22 22 ( ) ( ) ( )    ⋅=    ⋅−= σ θ σ θ θ  n i ninR x ˆ 2 expˆ 2 expˆ,22 ( ) ( ) ( ) ( )    ⋅−−=    ⋅=    ⋅=    ⋅− = σ θ σ θ σ θ σ θ σσ   nininini H H H ˆ 2 ˆ 2 ˆ 2 ˆ 2 skew-hermitian This is in accordance to the Theorem from Matrix Algebra that states: Every unitary matrix U can be expressed as an exponential matrix where H is hermitian (iH is skew-symmetric) ( )iHU exp=
  • 86.
    86 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 13) SOLO Let show that we get back the Rotation Matrix       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 3) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )σ θθ θ σ θθ σ θ σ θ σ θ σ θ σ θ σ θ θ       ⋅      −      = +      +⋅      −      −⋅      −= +⋅      −+⋅      −+⋅      −+⋅      −+= =    ⋅−= niI In i IniI ninininiI ninR x xxx x x ˆ 2 sin 2 cos 2!4 1 ˆ 2!32!2 1 ˆ 2 ˆ 2!4 1 ˆ 2!3 1 ˆ 2!2 1 ˆ 2 ˆ 2 expˆ, 22 22 43 22 2 22 4 4 3 3 2 2 22 22 Therefore ( ) ( ) ( )σ θθ σ θ θ  ⋅      −      =    ⋅−= niIninR xx ˆ 2 sin 2 cosˆ 2 expˆ, 2222 This is a generalization of the de Moivre expression for complex numbers:       −      =      − 2 sin 2 cos 2 exp θθθ ii
  • 87.
    87 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 14) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 4) (7) Proof q.e.d. ( )( ) ( ) ( )21222121 ˆˆˆˆˆˆ nniInnnn x ×⋅+⋅=⋅⋅ σσσ  ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         ×−×+× ×−×× +      ⋅=         ×−⋅×+×− ×+××+⋅ =         −−++−+−− −+−−+++ =         −+ −         −+ − =⋅⋅ zyx yxz zxy xyz xyyxyyxxzzyzzyzxxz yzzyzxxzxyyxyyxxzz zyx yxz zyx yxz nnnninn nninnnn inn nninnnninn nninnnninn nnnninnnnnninnnninnnn innnninnnnnnnninnnnnn ninn innn ninn innn nn 212121 212121 21 21212121 21212121 212121212121212121 212121212121212121 222 222 111 111 21 ˆˆˆˆˆˆ ˆˆˆˆˆˆ 10 01 ˆˆ ˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆ ˆˆ σσ 
  • 88.
    88 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 15) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 5) (8) Proof q.e.d. ( ) ( ) ( ) ( ) ( ) ( )      ×            +            +            ⋅−       ⋅            −            =     ⋅−    ⋅−= 2121 2221 21222122 ˆˆ 2 sin 2 sinˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 expˆ 2 expˆ,ˆ, nnnni Inn nininRnR x xx φθφθφθ σ φθφθ σ φ σ θ φθ   ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )[ ] ( ) ( )σ φθ σ φθ σ φθφθ σ φθ σ φθ σσ φθφθ σ φφ σ θθ σ φ σ θ φθ       ⋅            −⋅            − ×⋅+⋅            −            = ⋅            −⋅            − ⋅⋅            −            =       ⋅      −            ⋅      −      =       ⋅−      ⋅−= 21 21222122 21 2122 222122 21222122 ˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 sin 2 cosˆ 2 cos 2 sin ˆˆ 2 sin 2 sin 2 cos 2 cos ˆ 2 sin 2 cosˆ 2 sin 2 cos ˆ 2 expˆ 2 expˆ,ˆ, nini nniInnI nini nnI niIniI nininRnR xx x xx xx
  • 89.
    89 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 16) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 6) (9a) ( )                         −       −      =            −            =      −      =             − − =              −=    −= 2 cos 2 sin 2 sin 2 cos 01 10 2 sin 10 01 2 cos 2 sin 2 cos 0 2 2 0 exp 01 10 2 exp 2 expˆ, 122 122 ϕϕ ϕϕ ϕϕ σ ϕϕ ϕ ϕ ϕ σ ϕ ϕ i i iiI i i iixR x x ( )                               −      =      −       −            =      −      =             − =              − −=    −= 2 cos 2 sin 2 sin 2 cos 0 0 2 sin 10 01 2 cos 2 sin 2 cos 0 2 2 0 exp 0 0 2 exp 2 expˆ, 222 222 θθ θθ θθ σ θθ θ θ θ σ θ θ i i iiI i i iiyR x x
  • 90.
    90 ROTATIONS Cayley-Klein (or Euler)Parameters and Related Quantities (continue – 17) SOLO       − =      − =      = 10 01 , 0 0 , 01 10 321 σσσ i i Elementary Features of the 2x2 Rotation Matrix (continue – 7) (9b) ( )                         − =                   +            −      =       −       −            =      −      =             − =              − −=      −= 2 exp0 0 2 exp 2 sin 2 cos0 0 2 sin 2 cos 10 01 2 sin 10 01 2 cos 2 sin 2 cos 2 0 0 2exp 10 01 2 exp 2 expˆ, 322 322 ψ ψ ψψ ψψ ψψ σ ψϕ ψ ψ ψ σ ψ ψ i i i i iiI i i iizR x x
  • 91.
    91 ROTATIONS Gibbs Vector SOLO Using thequaternion notation we can define: ( ) ( ) ( )           == == == ∆ ∆ ∆ z y x n q q g n q q g n q q g 2/tan 2/tan 2/tan 0 3 3 0 2 2 0 1 1 θ θ θ 0 321 ˆˆˆ q zgygxgg ρ   =++= ( )       =++      =++==⋅ 2 tanˆˆˆ 2 tan 222222 3 2 2 2 1 θθ zyx T nnnggggggg  Gibbs in 1901 named this vector “vector semi-tangent of version”. Cayley used the three quantities in 1843 (before the introduction of vector notation), and credits their discovery to Rodriguez. Therefore they also called Euler-Rodriguez Parameters. 321 ,, ggg Josiah Willard Gibbs (1839-1903)
  • 92.
    92 ROTATIONS Gibbs Vector (continue– 1) SOLO ( ) [ ] [ ] [ ] ( ){ }θθθ cos1ˆˆsinˆˆ, 3333 −××+×−=−= nnnInRC xx B A [ ] ( ){ }θθθ cos1ˆˆsinˆcos33 −+×−= T x nnnI                   +      ×            −= 2 sinˆ 2 sinˆ2 2 sinˆ 2 cos2cos33 θθθθ θ nnnI x                                       +      ×            −            −      = T x nnnI 2 tanˆ 2 tanˆ 2 cos2 2 tanˆ 2 cos2 2 tan1 2 cos 2222 33 θθθθθθθ                                 +      ×      −            −      = T x nnnI 2 tanˆ 2 tanˆ2 2 tanˆ2 2 tan1 2 cos 2 33 2 θθθθθ ( ) [ ]{ }TT xT gggggI gg   221 1 1 33 +×−− + = ( ) [ ]                     +           − − − −−−−           +++ = 321 3 2 1 12 13 23 2 3 2 2 2 12 3 2 2 2 1 2 0 0 0 21 100 010 001 1 1 ggg g g g gg gg gg ggg ggg Therefore ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ +++ = 2 3 2 2 2 1132231 132 2 3 2 2 2 1321 231321 2 3 2 2 2 1 2 3 2 2 2 1 122 212 221 1 1 ggggggggg ggggggggg ggggggggg ggg CB A
  • 93.
    93 ROTATIONS Gibbs Vector (continue– 1) SOLO Let perform the following computations [ ]( ) ( ) [ ]{ } [ ]( )×++×−− + =×+ gIgggggI gg gIC x TT xTx B A    333333 221 1 1 ( ) [ ] ( )[ ] [ ] [ ] [ ]{ }×+××−×−++×−− + = gggggggggggggI gg TTTT xT   221221 1 1 33 ( ) ( )[ ]{ } [ ]{ }×−=×+−+ + = gIgggggI gg x TT xT   3333 11 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] 02 33 =×−− ggfIggedcba T x T  ( )( ) ( ) ( )cbea We obtained ( ) [ ]{ } [ ]{ } [ ]{ } 1 333333 221 1 1 − ×+×−=+×−− + = gIgIgggggI gg C xx TT xT B A   We can see that ( ) ( ) ( ) 2 3 2 2 2 1 2 3 2 2 2 1 1 3 3,32,21,1 ggg ggg CCCCtrace B A B A B A B A +++ −−− =++= 2 3 2 2 2 11 4 1 ggg Ctrace B A +++ =+
  • 94.
    94 ROTATIONS Gibbs Vector (continue– 2) SOLO We found ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ +++ = 2 3 2 2 2 1132231 132 2 3 2 2 2 1321 231321 2 3 2 2 2 1 2 3 2 2 2 1 122 212 221 1 1 ggggggggg ggggggggg ggggggggg ggg CB A 2 3 2 2 2 11 4 1 ggg Ctrace B A +++ =+ Therefore ( ) ( ) B A B A B A Ctrace CC g + − = 1 2,33,2 1 ( ) ( ) B A B A B A Ctrace CC g + − = 1 3,11,3 2 ( ) ( ) B A B A B A Ctrace CC g + − = 1 1,22,1 3 Gibbs vector is not widely used because it becomes infinite when the rotation angle is an odd multiply of 180 degrees.
  • 95.
    95 ROTATIONS Gibbs Vector (continue– 3) SOLO Differential of Gibbs Vector Let differentiate the equation: 0q g ρ   = We have: ρ ρ    2 0 0 0 q q q g −= • • Using the equation of quaternion differentiation: [ ] [ ] BA x T Iq q →•       ×+ − =         ω ρ ρ ρ      330 0 2 1 we obtain: [ ] [ ] AB T x qq I q q q g ← • •         + × +=−= ωρ ρρ ρ ρ      2 00 332 0 0 0 2 1 or [ ] [ ][ ] AB T x gggIg ← • +×+= ω  33 2 1
  • 96.
    96 ROTATIONS Gibbs Vector (continue– 4) SOLO Differential of Gibbs Vector (continue – 1) [ ] [ ][ ] AB T x gggIg ← • +×+= ω  33 2 1 Let multiply this equation by and use:[ ] [ ][ ]×− gI x  33 [ ][ ] [ ] 33x TT Igggggg  −=×× and [ ] 0  =× gg [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] AB T xxx gggIgIggI ← • +×+×−=×− ω  333333 2 1 [ ] [ ] [ ] [ ][ ] [ ] AB TT x gggggggggI ←         ×−××−×−+×+= ω    0 33 2 1 [ ] [ ]( )[ ] ( ) AB T BAx TTT x ggIggggggI ←→ +=−−+= ωω  1 2 1 2 1 3333 We obtain: [ ] [ ][ ] gg ggI T x AB    + ×− = • ← 1 2 33 ω
  • 97.
    97 ROTATIONS References SOLO [G] H. Goldstein,“Classical Mechanics”, Adison Wesley, 1980, 2nd Ed. [H] Peter C. Hughes, “Spacecraft Attitude Dynamics”, John Wiley & Sons, 1986 [JT] John L. Junkins, James D. Turner, “Optimal Spacecraft Rotational Maneuvers”, Elsevier, 1986 [McT] http://www-history.mcs.st-andrews.ac.uk/history [MTW] Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, “Gravitation”, Freeman, 1973 [W] James E. Wertz, “Spacecraft Attitude Determination and Control”, Reidel Publishing Company, 1985 [J] Peter Michael Jack’ site “Hypercomplex” http://www.hypercomplex.com [S] Malcom D. Shuster, “A Survey of Attitude Representations”, The Journal of Aeronautical Science, Vol. 41, Oct.-Dec. 1993, pp. 439-517
  • 98.
    January 6, 201598 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA