1. COMP790-072 Robotics: An Introduction
Kinematics & Inverse Kinematics
UNC Chapel Hill
M. C. Lin

2. Forward Kinematics
UNC Chapel Hill
M. C. Lin

3. What is f ?
UNC Chapel Hill
M. C. Lin

4. What is f ?
UNC Chapel Hill
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)
UNC Chapel Hill
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.
UNC Chapel Hill
M. C. Lin

7. Forward vs. Inverse Kinematics
UNC Chapel Hill
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
UNC Chapel Hill
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
UNC Chapel Hill
M. C. Lin

10. The Jacobian
UNC Chapel Hill
M. C. Lin

11. The Jacobian
UNC Chapel Hill
M. C. Lin

12. The Jacobian
UNC Chapel Hill
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
UNC Chapel Hill
M. C. Lin

14. Matrix Derivatives
UNC Chapel Hill
M. C. Lin

15. Rotation Matrix Derivatives
UNC Chapel Hill
M. C. Lin

16. Angular Velocity Matrix
UNC Chapel Hill
M. C. Lin

17. UNC Chapel Hill
M. C. Lin

18. UNC Chapel Hill
M. C. Lin

19. Computing J+ Fairly slow to compute Instability around singularities
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.
UNC Chapel Hill
M. C. Lin

20. Jacobian Transpose Use JT rather than J+ Avoid excessive inversion
Avoid singularity problem
UNC Chapel Hill
M. C. Lin

21. Principles of Virtual Work
Work = force x distance
Work = torque x angle
UNC Chapel Hill
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.
UNC Chapel Hill
M. C. Lin

23. Jacobian Transpose
UNC Chapel Hill
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
UNC Chapel Hill
M. C. Lin

25. Pros & Cons of Using JT + Cheaper evaluation + No singularities
- Scaling Problems
J+ has minimal norm at every step and JT doesn’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]
UNC Chapel Hill
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
UNC Chapel Hill
M. C. Lin

27. CCD Math - Prismatic
UNC Chapel Hill
M. C. Lin

28. CCD Math - Revolute
UNC Chapel Hill
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.
UNC Chapel Hill
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
UNC Chapel Hill
M. C. Lin

31. References
UNC Chapel Hill
M. C. Lin