Differential Kinematics and Statics Ref: 理论力学，洪嘉振，杨长俊，高等教育出版社， 2001

1. Differential Kinematics and Statics Ref: 理论力学，洪嘉振，杨长俊，高等教育出版社，2001

2. Incremental Motion • What small (incremental) motions at the end-effector (Dx, Dy, Dz) result from small motions of the joints (Dq1, Dq2, …, Dqn )? • Alternatively, what velocities at the end-effector (vx, vy, vz) result from velocities at the joints (w1, w2, … wn)?

3. Some Definitions • Linear Velocity: The instantaneous rate-of-change in linear position of a point relative to some frame. v=(vx, vy, vz)T • Angular Velocity: The instantaneous rate-of-change in the orientation of one frame relative to another. • Angular Velocity depends on the way to represent orientation (Euler Angles, Rotation Matrix, etc.) • Angular Velocity Vector and the Angular Velocity Matrix.

4. Some Definitions • Angular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to another and whose magnitude is the rate of rotation about that axis.

5. Free Vector • Linear velocityare insensitive to shifts in origin but are sensitive to orientation. {D} x x

6. {A} {B} Free Vector • Angular velocityare insensitive to shifts in origin but are sensitive to orientation. {D} x x x x

7. Velocity Frames • frame of reference: this is the frame used to measure the object’s velocity • frame of representation.: this is the frame in which the velocity is expressed.

8. Y0 R y0 x2 y2 v a2 Y1 q2 X1 a1 v v q1 0 X0 v 0 x0 Figure 2.13: Two-Link Planar Robot

9. End-effector velocity for w1 Y0 y0 v r0n v v w1 0 X0 v 0 x0

10. End-effector velocity for w2 Y0 y0 v r1n w2 v v 0 X0 v 0 x0

11. Two-Link Planar Robot • Direct kinematics equation

12. Incremental Motion • taking derivatives of the position equation w.r.t. time we have • note that

13. Incremental Motion • written in the more common matrix form, • or in terms of incremental motion,

14. Differential Kinematics • Find the relationship between the joint velocities and the end-effector linear and angular velocities. Linear velocity Angular velocity for a revolute joint for a prismatic joint

15. Differential Kinematics • Differential kinematics equation • Geometric Jacobian

16. Relationship with T(q) • Direct kinematics equation • Linear velocity • Angular velocity?

17. Vector (Cross) Product • Vector product of x and y • Skew-symmetric matrix

18. Vector (Cross) Product • Skew-symmetric matrix

19. Derivative of a Rotation Matrix define S(t) is skew-symmetric

20. Interpretation of S(t)

21. Interpretation of S(t) Given R(t)

22. Example 3.1: Rotation about Z