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1. 1. 2 Link Planar Manipulator  2  1 a 1 a 2 O 2 O 1 O 0 x 1 x 0 x 2 y 1 y 2 y 0 Frame 0 – ground reference Frame 1 – link 1, distal end Frame 2 – link 2, distal end Length of Link 1 = a 1 Length of Link 2 = a 2 Note: coordinate systems are consistent with the Denavit-Hartenburg system
2. 2. Forward Kinematics <ul><li>Find endpoint at 0 d 02 </li></ul><ul><ul><ul><li>(i.e. endpoint w.r.t. ground reference) </li></ul></ul></ul>
3. 3. Homogeneous Transformations in a plane O 1 O 0 p 0 p 1  0 d 01 y 0 x 0 x 1 y 1 P
4. 4. Composition of Homogeneous Transforms <ul><li>Coordinate transformations can be chained </li></ul><ul><ul><li>Forward: 0 T 2 = 0 T 1 1 T 2 </li></ul></ul><ul><ul><li>Inverse: 0 P = ( 0 T 1 ) -1 0 P </li></ul></ul>
5. 5. Frame 1 is displaced from Frame 0 by rotation of 30 degrees and a translation of (1,1). Frame 2 is displaced from Frame 1 by a rotation of 60 degrees and translation of (1/2,  3/2) O 1 O 0 p 2 30 d 01 y 0 x 0 x 1 y 1 1 1 d 12 p 1 p 0 y 2 x 2 60 1/2 Sqrt(3)/2 1 1
6. 6. O 1 O 0 p 2 30 d 01 y 0 x 0 x 1 y 1 1 1 d 12 p 1 p 0 y 2 x 2 60 1/2 Sqrt(3)/2 1 1
7. 7. Homogeneous Coordinates <ul><li>In graphics HC used to represent: scaling, shear, translation, rotation. </li></ul><ul><li>Point denoted by set of coordinates </li></ul><ul><ul><li>A set of points in R 3 whose last coordinate1 is referred to as the standard affine plane in R 3 </li></ul></ul><ul><ul><li>A vector is denoted by 3 coordinate vector whose last component is 0 and lies in the affine plane </li></ul></ul>
8. 8. P1 P2
9. 9. Operators Note: in composing homogeneous transformations Translation matrix, then Rotation Matrix
10. 10. Trans and Rot Operators <ul><li>Translate a point P 1 , represented relative to origin O 0 by p 1 , by displacement d to point P 2 , represented relative to origin O 0 by p 2 </li></ul><ul><li>The rotation operator rotates a point P 1 to P 2 by  1 </li></ul>
11. 11. General Transformation Operator <ul><li>First rotates a point and then translates it </li></ul>
12. 12. Two views of movement P1-> P2 <ul><li>Coordinate Transformation </li></ul><ul><ul><li>Change the position of the origin </li></ul></ul><ul><li>Point Transformation </li></ul><ul><ul><li>Move point relative to a fixed reference frame </li></ul></ul>
13. 13. Composition of operators <ul><li>As a coordinate transformation </li></ul><ul><ul><li>Translate origin of frame 0 to (1,1/2) then rotates the axes by 30 degrees </li></ul></ul><ul><li>As an operator </li></ul><ul><ul><li>Rotate frame 1 by 30 degrees and then translate it to (1,1/2) all relative to frame 0 </li></ul></ul>
14. 14. Poles of planar displacements <ul><li>Pole: a point that does not move under arbitrary planar operators except pure translation </li></ul>
15. 15. Spatial Transformations and Displacements <ul><li>2D to 3D is straightforward for Linear displacements </li></ul><ul><li>Rotation in 3D is not Commutitave </li></ul><ul><ul><li>R 1 R 2  R 2 R 1 </li></ul></ul>