If forces and velocities produce zero power then they represent reciprocal screws.

1. The principal screw of inertia

2. Coordinate Transformations
• The skew symmetric vector cross product
operator is
Translating a velocity twist Av and a force wrench
Af from point A to another point B can be done
utilizing position vector ABr between points A and B.
The linear velocity is B A ABv v r    and the
moment is B A ABr f    . The term ABr 
corresponds to the vector cross product which is
defined in terms of 3 3 matrix as
0
0
0
x z y
y z x
z y x
   
     
   
      

3. skew-symmetric
• In mathematics, and in particular linear
algebra, a skew-symmetric (or antisymmetric
or antimetric) matrix is a square matrix whose
transpose is its negation; that is, it satisfies the
condition −A = AT.

4. • Then the spatial form of the translations are
• With the 6x6 spatial transformation
( )B B A Av X v
( ) T
B B A Af X f

1
0 1
AB
B A
r
X
  
  
 
Note that ( )T
r r    and
1
( )B A A BX X
 .
Rotations are actually easier to define since both
linear and angular parts transform the same way.
Using E, an orthogonal 3x3 rotation matrix from
coordinate frames B’ to B, any 'B Be Ee . Thus,

5. A general twist transformation from A
to B
• and its inverse
A general twist transformation from A to B with
translation ABr followed by rotation R is described
by the transformation matrix
( )
0
T
AB
B A
R R r
X
R
 
  
 
0
T T
AB
A B T
R r R
X
R
 
  
 

6. A general wrench transformation from
A to B
• and its inverse
A general wrench transformation from A to B with
translation ABr followed by rotation R is described
by the transformation matrix
0
( )
( )
T
B A T
AB
R
X
R r R
  
   
0
( )
T
T
A B T T
AB
R
X
r R R
  
  
 

7. Spatial accelerations transform
Spatial accelerations transform the same way
velocities do. The transformation of an acceleration
twist from coordinate frame A to coordinate frame
B is defined as ( )B B A Aa X a using the same
transformation matrix as
( )
0
T
AB
B A
R R r
X
R
 
  
 

8. Euler’ Method
• Orientation is given by a 33 rotation matrix.
It is possible to describe an orientation with
fewer than nine numbers  Cayley’s formula
for orthogonal matrices.
)SI()SI(R 3
1
3  
O. Bottema, B. Roth, Theoretical Kinematics, North Holland, Amsterdam, 1979
Skew-symmetric matrix (i.e. T
SS  ) by three parameters














0SS
S0S
SS0
S
XY
XZ
YZ

9. Six constraints on nine elements
• Multiplication of matrices is not
commutative.
• (Finite rotation in 3-D space)  be performed
in a specific order
Nine elements of rotation matrix are not all independent.
 ZˆYˆXˆR 






,0ZˆYˆ,0ZˆXˆ,0YˆXˆ
1Zˆ,1Yˆ,1Xˆ
Six constraints on nine elements

10. X-Y-Z Fixed Angles:
• Start with the frame coincident with a known
frame (Global frame). First rotate L about X by
an angle , then rotate about Y by an angle ,
and then rotate about Z by any angle .
• Rotations are specified about the fixed (i.e.
non-moving) reference frame.

11.    































































333231
232221
131211
XYZXYZ
coscoscos
coscoscos
coscoscos
coscossincossin
sincoscossinsincoscossinsinsincossin
sinsincossincoscossinsinsincoscoscos
cossin0
sincos0
001
cos0sin
010
sin0cos
100
0cossin
0sincos
RRR),,(R

12. atan2
• where


45)0.2,0.2(2tanA
135)0.2,0.2(2tanA
)
cos
cos
,
cos
cos
(2tanA
)
cos
cos
,
cos
cos
(2tanA
)coscos,cos(2tanA
3332
1121
2
21
2
1131








13. Zy’x” Euler Angles:
• Start with the frame coincident with a known
frame (Global frame). First rotate L about Z by an
angle , then rotate about y by an angle , and
then rotate about x by an angle .
• Each rotation is performed about an axis of the
moving system (L), rather than the fixed frame
(G). Such a set of three rotations are called “Euler
Angles”. Note that each rotation takes place
about an axis whose location depend on the
preceding rotations.

14. • SAME AS that obtained for the same three
rotation taken in the reserve order about fixed
axes!!!
   






































cossin0
sincos0
001
cos0sin
010
sin0cos
100
0cossin
0sincos
RRR),,(R "x'yZ"x'Zy

15. Two rotations
• Example: two rotations, Z by 30 and X by 30:
Example: two rotations, Z by 30 and X by 30:









 

000.1000.0000.0
000.0866.0500.0
000.0500.0866.0
)30(RZ












866.0500.0000.0
500.0866.0000.0
000.0000.0000.1
)30(RX



























87.043.025.0
50.075.043.0
00.050.087.0
)30(R)30(R
87.050.000.0
43.075.050.0
25.043.087.0
)30(R)30(R ZXXZ

 Multiplication of matrices is not commutative.
(Finite rotation in 3-D space)  be performed in a specific order

16. Joint Rotation Convention
• To identify the relative attitude of two body segments,
three axes should be specified. In the joint rotation
convention (JRC), two axes are fixed with the body
segments, proximal and distal and one is a floating axis.
• The first axis is the first fixed body axis and is
perpendicular to the sagittal plane of the proximal
segment.
• The second axis is the floating axis (the cross product
of the first and third axes).
• The third, or the second fixed-body axis, is the long axis
of the distal segment.

17. JRC angles are anatomically
meaningfull.

18. Definition of Anatomical Coordinate
System
• The anatomical coordinate system of the shank, considered to be
fixed, is defined by the four landmarks on the shank as follows:
• Origin is at the midpoint of the line joining MM and LM
• Y-axis is orthogonal to the quasi-frontal plane defined by the MM,
LM and HF
• Z-axis is orthogonal to the quasi-sagittal plane defined by the Y-axis
and TT
• X-axis is the cross product of the Y- and Z-axis
• The positive directions of the coordinate axes are defined as for
both the right foot and the left foot plantarflexion about the z-axis,
inversion about the y-axis, and adduction about the x-axis are
positive. Figure 3-6 shows the shank anatomical coordinate system
of both the left foot and the right foot pictorially.
• At the reference position, the local coordinate system of the foot is
assumed to be coincident initially with that of the shank. So the
coordinate system of the foot is not landmark based coordinate
system.

19. Anatomical coordinate system of the
shank
HF
TT
TT
HF
LM
MM
MM
LMY
YZ
Z
X
X
Right Left

20. Momentum Wrench
• A moving rigid body has a linear and an
angular momentum associated with it. The
spatial momentum is defined as
0p
P
r p h
   
       
Which represents the linear momentum p acting
on a line through the center of mass, and the
intrinsic angular momentum h about the center of
mass. The intrinsic angular momentum is
h I 

21. • Assembling the linear and angular parts
together the spatial momentum is defined as
where I is the angular inertia about the center of
mass and  is its angular velocity vector. The linear
momentum is proportional to the linear velocity
vector of the center of mass and thus
( )p m v r   
where v is the linear velocity vector of the
reference point and r is the relative position vector
of the center of mass from the reference point.
( )
( )
m v r
P
I r m v r

 
  
      
The r p term represents the moment of
momentum about the reference point.

22. Momentum wrenches transform
• Momentum wrenches transform the same
way force wrenches transform. The
momentum transformation from coordinate
frame A to coordinate frame B is defined as
( ) T
B B A AP X P

using the transformation matrix ( ) T
B AX 
from
0
( )
( )
T
B A T
AB
R
X
R r R
  
   

23. Spatial Inertia
• The spatial inertia is defined as the tensor that
multiplies a velocity twist to produce a
momentum wrench. Therefore, by factoring
out the velocity terms from momentum the
symmetric 6x6 inertia tensor is defined as
• where
• and
P Iv
1 ( )
( ) ( )( )
T
T
m m r
I
m r I m r r
 
  
    
v
v

 
  
 

24. • which can be interpreted from right to left as
transforming the spatial velocities from B to A,
then evaluating the spatial momentum at A
and transforming back the momentum to B.
Note that ( )( )T
m r r  is the 3x3 angular inertia
matrix of a point mass m with relative position
vector r .
The transformation of the inertia from coordinate
frame A to coordinate frame B is easily defined
from equation P Iv as
( ) ( )T
B A B A A BI X I X

25. Kinetic Energy and Power
• It is easy to show that under the spatial
transformations the kinetic energy of a rigid
body is an invariant quantity. If the kinetic
energy is evaluated at a coordinate system A
as
• Then by transforming all the quantities to a
coordinate system B the kinetic energy
becomes
1
2
T
A A AK v I v
1
( ) ( ) ( )
2
1
2
T T
A B B B A B B A A B B
T
B B B
K X v X I X X v
v I v



26. Reciprocal screws
• Thus the kinetic energy is the same regardless on
where it is evaluated.
• Power is also an invariant defined as the linear form of
a velocity twist with a force wrench.
• Power evaluated on a coordinate system A as
• is transformed to a coordinate system B by
• If forces and velocities produce zero power then they
represent reciprocal screws.
T
A AW v f
( ) ( )T T
A B B A B B
T
B B
W X v X f
v f


