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. Animation Robot arm animation (click here)
Pixar lamp animation (click here)

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

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

7. 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

8. Degrees of Freedom How about this robot arm? Click (here) Six
2-shoulder, 1-elbow, 3-wrist

9. Configuration Space vs. Work 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
Work space
The space in which the object exists
Dimensionality
R3 for most things, R2 for planar arms

10. 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’s position

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

12. 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)

13. 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))

14. 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

15. Forward Kinematics A 2D lamp with 6 degrees of freedom lower arm
middle arm
Upper arm
base 15

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

17. 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)

18. 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)

19. 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.

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

21. Inverse Kinematics
Analytical solution only works for a fairly simple structure
Numerical/iterative solution needed for a complex structure

22. Numerical Approaches
Inverse kinematics can be formulated as an optimization problem

23. Function Optimization
Finding the minimum for nonlinear functions
F(x) x

24. Optimization As a Tool
How to use optimization to solve the following linear system?
3x + 2y = 5;
4x – 5y = 6

25. Optimization As a Tool
How to use optimization to solve the following linear system?
3x + 2y = 5;
4x – 5y = 6
Define a function:
F(x,y) = (3x+2y-5)2+(4x-5y-6)2

26. Optimization As a Tool
How to use optimization to solve the following linear system?
3x + 2y = 5;
4x – 5y = 6
Could be nonlinear equations!
Define a function:
F(x,y) = (3x+2y-5)2+(4x-5y-6)2
Finding the minimum for F(x,y) :
x+,y+= argminx,y (3x+2y-5)2+(4x-5y-6)2

27. Optimization As a Tool
How to use optimization to solve the following linear system?
f(x, y) = 0;
g(x,y) = 0
Could be nonlinear equations!
Define a function:
F(x,y) = f(x,y)2+g(x,y)2
Finding the minimum for F(x,y) :
x+,y+= argminx,y f(x,y)2+g(x,y)2

28. Formulation
So how to convert the IK process into an optimization function?
Find the pose satisfying: θ2 θ2 l2 l1 θ1
C=(Cx,Cy)
Base
(0,0)

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

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

31. 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

32. Gradient-based Optimization

33. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles until the solution converges:

34. Gauss-Newton Approach
Step 1: initialize the joint angles with
Step 2: update the joint angles until the solution converges:
How can we decide the amount of update? 34

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

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

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

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

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

40. 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

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

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

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

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

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

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

47. 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

48. 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’s position (x,y) θ2 θ1

49. 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

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

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

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

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

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

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

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

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

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

59. When to stop
Option #1: When the change of solution from previous iteration to the current one is below a user-specified threshold.
Option #2: When the number of iteration reaches a user-specified threshold.
Option #3: either #1 or #2 is satisfied. 59

60. 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

61. 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.

62. Another Example
A 2D lamp character 62

63. Another Example
A 2D lamp character 63

64. ? Forward Kinematics A 2D lamp character
Given , , how to compute the global position of the point A? 64

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

66. 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

67. 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

68. Human Characters 68

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

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

71. Ambiguity of IK
Multiple solutions

72. Ambiguity of IK
Infinite solutions

73. Failures of IK
Solution may not exist

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

75. 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

76. 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

77. 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

78. 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)

79. 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)

80. 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?

81. Interactive Human Character Posing
Video (click here)

82. Summary of IK Very simple structure allows an analytic solution
Most of complex articulated figures requires a numerical solution
May not always get the “right” solution
- need additional objectives