COMP790-072 Robotics: An Introduction

1. COMP790-072Robotics: An Introduction • Kinematics & Inverse Kinematics M. C. Lin

2. Forward Kinematics M. C. Lin

3. What is f ? M. C. Lin

4. What is f ? M. C. Lin

5. Other Representations • Separate Rotation + Translation: T(x) = R(x) + d • Rotation as a 3x3 matrix • Rotation as quaternion • Rotation as Euler Angles • Homogeneous TXF: T=H(R,d) M. C. Lin

6. Forward Kinematics • As DoF increases, there are more transformation to control and thus become more complicated to control the motion. • Motion capture can simplify the process for well-defined motions and pre-determined tasks. M. C. Lin

7. Forward vs. Inverse Kinematics M. C. Lin

8. Inverse Kinematics (IK) • As DoF increases, the solution to the problem may become undefined and the system is said to be redundant. By adding more constraints reduces the dimensions of the solution. • It’s simple to use, when it works. But, it gives less control. • Some common problems: • Existence of solutions • Multiple solutions • Methods used M. C. Lin

9. Numerical Methods for IK • Analytical solutions not usually possible • Large solution space (redundancy) • Empty solution space (unreachable goal) • f is nonlinear due to sin’s and cos’s in the rotations. • Find linear approximation to f-1 • Numerical solutions necessary • Fast • Reasonably accurate • Yet Robust M. C. Lin

10. The Jacobian M. C. Lin

11. The Jacobian M. C. Lin

12. The Jacobian M. C. Lin

13. Computing the Jacobian • To compute the Jacobian, we must compute the derivatives of the forward kinematics equation • The forward kinematics is composed of some matrices or quaternions M. C. Lin

14. Matrix Derivatives M. C. Lin

15. Rotation Matrix Derivatives M. C. Lin

16. Angular Velocity Matrix M. C. Lin

17. M. C. Lin

18. M. C. Lin

19. Computing J+ • Fairly slow to compute • Breville’s method: J+(JJT)-1 • Complexity: O(m2n) • ~ 57 multiply per DOF with m = 6 • Instability around singularities • Jacobian loses rank in certain configur. M. C. Lin

20. Jacobian Transpose • Use JT rather thanJ+ • Avoid excessive inversion • Avoid singularity problem M. C. Lin

21. Principles of Virtual Work • Work = force x distance • Work = torque x angle M. C. Lin

22. Jacobian Transpose • Essentially we’re taking the distance to the goal to be a force pulling the end-effector. • With J-1, the solution was exact to the linearized problem, but this is no longer so. M. C. Lin

23. Jacobian Transpose M. C. Lin

24. Jacobian Transpose • In effect this JT method solves the IK problem by setting up a dynamical system that obeys the Aristotilean laws of physics: F = m v ;  = I and the steepest descent method. • The J+ method is equivalent to solving by Newtonian method M. C. Lin

25. Pros & Cons of Using JT + Cheaper evaluation + No singularities - Scaling Problems • J+has minimal norm at every step and JTdoesn’t have this property. Thus joint far from end-effector experience larger torque, thereby taking disproportionately large time steps • Use a constant matrix to counteract - Slower Convergence than J+ • Roughly 2x slower [Das, Slotine & Sheridan] M. C. Lin

26. Cyclic Coordinate Descend (CCD) • Just solve 1-DOF IK-problem repeatedly up the chain • 1-DOF problems are simple & have analytical solutions M. C. Lin

27. CCD Math - Prismatic M. C. Lin

28. CCD Math - Revolute M. C. Lin

29. CCD Math - Revolute • You can optimize orientation too, but need to derive orientation error and minimize the combination of two • You can derive expression to minimize other goals too. • Shown here is for point goals, but you can define the goal to be a line or plane. M. C. Lin

30. Pros and Cons of CCD + Simple to implement + Often effective + Stable around singular configuration + Computationally cheap + Can combine with other more accurate optimizations - Can lead to odd solutions if per step not limited, making method slower - Doesn’t necessarily lead to smooth motion M. C. Lin

31. References M. C. Lin