1. CSCE 441: Computer Graphics Forward/Inverse kinematics
Jinxiang Chai

2. Outline
Animation basics:
Forward kinematics
Inverse kinematics

3. Kinematics
The study of movement without the consideration of the masses or forces that bring about the motion

4. Degrees of Freedom (Dofs)
The set of independent displacements that specify an object’s pose

5. Degrees of Freedom (Dofs)
The set of independent displacements that specify an object’s pose
How many degrees of freedom when flying?

6. Degrees of Freedom (Dofs)
The set of independent displacements that specify an object’s pose
How many degrees of freedom when flying?
So the kinematics of this airplane permit movement anywhere in three dimensions
Six
x, y, and z positions
roll, pitch, and yaw

7. Degrees of Freedom How about this robot arm? Six again
2-base, 1-shoulder, 1-elbow, 2-wrist

8. Work Space vs. Configuration Space
The space in which the object exists
Dimensionality
R3 for most things, R2 for planar arms
Configuration space
The space that defines the possible object configurations
Degrees of Freedom
The number of parameters that are necessary and sufficient to define position in configuration

9. Forward vs. Inverse Kinematics
Forward Kinematics
Compute configuration (pose) given individual DOF values
Inverse Kinematics
Compute individual DOF values that result in specified end effector position

10. Example: Two-link Structure
Two links connected by rotational joints θ2 l2 l1 θ1
X=(x,y)
End Effector
Base

11. Example: Two-link Structure
Animator specifies the joint angles: θ1θ2
Computer finds the position of end-effector: x θ2 l2 l1 θ1
X=(x,y)
End Effector
Base
(0,0)
x=f(θ1, θ2)

12. Example: Two-link Structure
Animator specifies the joint angles: θ1θ2
Computer finds the position of end-effector: x θ2 θ2 l2 l1 θ1
X=(x,y)
End Effector
Base
(0,0)
x = (l1cosθ1+l2cos(θ2+ θ2)
y = l1sinθ1+l2sin(θ2+ θ2))

13. Forward Kinematics
Create an animation by specifying the joint angle trajectories θ2 θ2 l2 l1 θ1
X=(x,y)
End Effector
Base
(0,0) θ1 θ2

14. Inverse Kinematics
What if an animator specifies position of end-effector? θ2 θ2 l2 l1
End Effector θ1
X=(x,y)
Base
(0,0)

15. Inverse Kinematics Animator specifies the position of end-effector: x
Computer finds joint angles: θ1θ2 θ2 θ2 l2 l1
End Effector θ1
X=(x,y)
Base
(0,0)
(θ1, θ2)=f-1(x)

16. Inverse Kinematics Animator specifies the position of end-effector: x
Computer finds joint angles: θ1θ2 θ2 θ2 l2 l1
End Effector θ1
X=(x,y)
Base
(0,0)

17. Why Inverse Kinematics?
Motion capture
Basic tools in character animation
- key frame generation
- animation control
- interactive manipulation
Computer vision (video-based mocap)
Robotics
Bioinfomatics (Protein Inverse Kinematics)
Etc.

18. Inverse Kinematics
Given end effector positions, compute required joint angles
In simple case, analytic solution exists
Use trig, geometry, and algebra to solve

19. Inverse Kinematics
Analytical solution only works for a fairly simple structure
Iterative approach needed for a complex structure

20. Iterative Approaches
Inverse kinematics can be formulated as an optimization problem

21. Function Optimization
Finding the minimum for nonlinear functions

22. Iterative Approaches
Find the joint angles θ that minimizes the distance between the character position and user specified position θ2 θ2 l2 l1 θ1
C=(Cx,Cy)
Base
(0,0)

23. Iterative Approaches
Find the joint angles θ that minimizes the distance between the character position and user specified position θ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

24. Iterative Approaches
Mathematically, we can formulate this as an optimization problem:
The above problem can be solved by many nonlinear optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc

25. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:

26. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
How can we decide the amount of update?

27. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:

28. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Known!

29. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Taylor series expansion

30. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Taylor series expansion
rearrange

31. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Taylor series expansion
rearrange
Can you solve this optimization problem?

32. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Taylor series expansion
rearrange
This is a quadratic function of

33. Gauss-Newton Approach
Optimizing an quadratic function is easy
It has an optimal value when the gradient is zero

34. Gauss-Newton Approach
Optimizing an quadratic function is easy
It has an optimal value when the gradient is zero J b Δθ
Linear equation!

35. Gauss-Newton Approach
Optimizing an quadratic function is easy
It has an optimal value when the gradient is zero J b Δθ

36. Gauss-Newton Approach
Optimizing an quadratic function is easy
It has an optimal value when the gradient is zero J b Δθ

37. Jacobian Matrix
Dofs (N)
Jacobian is a M by N matrix that relates differential changes of θ to changes of C
Jacobian maps the velocity in joint space to velocities in Cartesian space
Jacobian depends on current state
The number of constraints (M)

38. Jacobian Matrix
Jacobian maps the velocity in joint angle space to velocities in Cartesian space
(x,y) θ2 θ1

39. Jacobian Matrix
Jacobian maps the velocity in joint angle space to velocities in Cartesian space
(x,y)
A small change of θ1 and θ2 results in how much change of end-effector position (x,y) θ2 θ1

40. Jacobian Matrix
Jacobian maps the velocity in joint angle space to velocities in Cartesian space
(x,y)
A small change of θ1 and θ2 results in how much change of end-effector position (x,y) θ2 θ1

41. Jacobian Matrix
Jacobian maps the velocity in joint angle space to velocities in Cartesian space
(x,y)
A small change of θ1 and θ2 results in how much change of end-effector position (x,y) θ2 θ1

42. Jacobian Matrix
Jacobian maps the velocity in joint angle space to velocities in Cartesian space
(x,y)
A small change of θ1 results in how much change of end-effector position (x,y) θ2 θ1

43. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

44. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0) f2 f1

45. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

46. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

47. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

48. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

49. Jacobian Matrix: 2D-link Structureθ2 θ2 l2 l1 θ1
C=(c1,c2)
Base
(0,0)

50. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles:
Step size: specified by the user

51. Iterative Approaches for IK
Mathematically, we can formulate this as an optimization problem:
The above problem can be solved by many nonlinear optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc

52. C/C++ Optimization Library
levmar : Levenberg-Marquardt nonlinear least squares algorithms in C/C+
- Works with/without analytical Jacobian matrix
- Speeds up the optimization process with analytical Jacobian matrix.

53. Another Example
A 2D lamp character 53

54. Another Example
A 2D lamp character 54

55. ? Forward Kinematics A 2D lamp character
Given , , how to compute the global position of the point A? 55

56. ? Forward Kinematics A 2D lamp with 6 degrees of freedom lower arm
middle arm ?
Upper arm
base

57. Inverse Kinematics A 2D lamp with 6 degrees of freedom lower arm
middle arm
Upper arm
If you do inverse kinematics with position of this point
What’s the size of Jacobian matrix?
base

58. Inverse Kinematics A 2D lamp with 6 degrees of freedom lower arm
middle arm
Upper arm
If you do inverse kinematics with position of this point
What’s the size of Jacobian matrix?
2-by-6 matrix
base

59. Human Characters 59

60. Inverse Kinematics
Analytical solution only works for a fairly simple structure
Iterative approach needed for a complex structure

61. Inverse Kinematics Is the solution unique?
Is there always a good solution?

62. Ambiguity of IK
Multiple solutions

63. Ambiguity of IK
Infinite solutions

64. Failures of IK
Solution may not exist

65. Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints

66. Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0

67. Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0

68. Minimize the difference between the solution and a reference pose
Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0
Minimize the difference between the solution and a reference pose
Satisfy the constraints

69. Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0
Naturalness g(θ) (particularly for human characters)

70. Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0
Naturalness g(θ) (particularly for human characters)

71. How natural is the solution pose?
Inverse Kinematics
Generally ill-posed problem when the number of Dofs is higher than the number of constraints
Additional objective
Minimal Change from a reference pose θ0
Naturalness g(θ) (particularly for human characters)
Satisfy the constraints
How natural is the solution pose?

72. Interactive Human Character Posing
video