1. Chap 5 Kinematic Linkages
Animation (U), Chap 5 Kinematic Linkages

2. Model Hierarchy vs. Motion Hierarchy
Relative motion
An object’s motion is relative to another object
Object hierarchy + relative motion form motion hierarchy
Linkages
Components of the hierarchy represent
objects that are physically connected
Motion is often restricted
Animation (U), Chap 5 Kinematic Linkages
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3. Kinematics
How to animate the linkages by specifying or determining position parameters over time
The branch of mechanics concerned with the motions of objects without regard to the forces that cause the motion
Animation (U), Chap 5 Kinematic Linkages
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4. Kinematics Forward Kinematics
Animator specifies rotation parameters at joints
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5. Kinematics Inverse Kinematics
Animator specifies the desired position of hand, and system solves for the joint angles that satisfy that desire.
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6. Hierarchical Modeling Articulated Model
Represent an articulated figure as a series of links connected by joints
root
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7. Degrees of Freedom (DOF)
The minimum number of coordinates required to specify completely the motion of an object
yaw
roll
pitch
6 DOF: x, y, z, raw, pitch, yaw
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8. Degrees of Freedom One DOF Joints
Allow motion in one direction
Exam: Revolute joint, Prismatic joint
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9. Degrees of Freedom Complex Joints
2 DOF joint
Planar joint, universal joint
3 DOF joint
Gimbal
Ball and socket
Complex joints of DOF n can be modeled as n one-DOF joints connected by n-1 links of 0 length.
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10. Degrees of Freedom Complex Joints
Animation (U), Chap 5 Kinematic Linkages
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11. Degrees of Freedom in Human Model
Root: 3 translational DOF + 3 rotational DOF
Rotational joints are commonly used
Each joint can have up to 3 DOF
Shoulder: 3 DOF
Wrist: 2 DOF
Knee: 1 DOF
3 DOF
1 DOF
Animation (U), Chap 5 Kinematic Linkages
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12. Hierarchical Modeling Data Structure
Hierarchical linkages can be represented by a tree-like structure
Root node – corresponds to the root object whose position is known in the global coordinate system
Other nodes – located relative to the root node
Leaf nodes –
Mapping between hierarchy and tree
Node – object part
Arc – joint or transformation to apply to all nodes between it in the hierarchy
Animation (U), Chap 5 Kinematic Linkages
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13. Hierarchical Modeling Data Structure
Animation (U), Chap 5 Kinematic Linkages
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14. Hierarchical Modeling Data Structure
Animation (U), Chap 5 Kinematic Linkages
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15. Hierarchical Modeling A Simple Example
Animation (U), Chap 5 Kinematic Linkages
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16. Hierarchical Modeling Tree Structure
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17. Hierarchical Modeling Transformations
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18. Hierarchical Modeling Variable rotations at joints
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19. Hierarchical Modeling Tree Structure
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20. Hierarchical Modeling Transformations
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21. Hierarchical Modeling With Two Appendages
Animation (U), Chap 5 Kinematic Linkages
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22. Hierarchical Modeling With Two Appendages
Animation (U), Chap 5 Kinematic Linkages
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23. Joint Space vs. Cartesian Space
space formed by joint angles
position all joints—fine level control
Cartesian space
3D space
specify environment interactions
Animation (U), Chap 5 Kinematic Linkages
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24. Forward and Inverse Kinematics
Forward kinematics
mapping from joint space to cartesian space
Inverse kinematics
mapping from cartesian space to joint space
Forward Kinematics
Inverse Kinematics
Animation (U), Chap 5 Kinematic Linkages
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25. Forward Kinematics Evaluate the tree
Depth-first traversal from the root to leaf
Repeat the following until all nodes and arcs have been visited
The traversal then backtracks up the tree until an unexplored downward arc is encountered
The downward arc is then traversal
In implementation
Requires a stack of transformations
Animation (U), Chap 5 Kinematic Linkages
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26. Forward Kinematics Tree Traversal
M = I T0
M = T0
T1.1
T 1.2
M = T0*T1.1
M = T0*T1.1*T2.1
T2.1
T2.2
M = T0*T1.1
M = T0
M = T0*T1.2
M = T0*T1.2*T2.2
Animation (U), Chap 5 Kinematic Linkages
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27. Forward Kinematics Example?
End Effector
Base
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28. Inverse Kinematics
The initial pose vector and a desired pose vector are given, come out the joint values required to attain the desired pose
Can have zero, one, or more solutions
Over-constrained system
Under-constrained system
Singularity problems
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29. Inverse Kinematics Example This is not uncommon!
2 equations (constraints)
3 unknowns
Infinitely many solutions exist!
This is not uncommon!
See how you can move your elbow while keeping your finger touching your nose
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30. Other problems in IK Infinite many solutions
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31. Other problems in IK No solutions
Animation (U), Chap 5 Kinematic Linkages
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32. Inverse Kinematics
Once the joint values are calculated, the figure can be animated by interpolating the initial pose vector values to the final pose vector values.
Does not provide a precise or an appropriate control when there is large differences between initial and final pose vectors. Alternatively,
Interpolate pose vectors
Do inverse kinematics for each interpolated pose vector
Animation (U), Chap 5 Kinematic Linkages
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33. Inverse Kinematics Analytic vs. Numerical
Analytic solutions exist only for relatively simple linkages
Inspecting the geometry of linkages
Algebraic manipulation of equations that describe the relationship of the end effector to the base frame.
Numerical incremental approach
Inverse Jacobian method
Other methods
Animation (U), Chap 5 Kinematic Linkages
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34. Inverse Kinematics Analytic Methodq2 L2 L1
Goal q1
(X,Y)
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35. Inverse Kinematics Analytic Method
180- q2 L1
(X,Y) q1 Y qT X
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36. Inverse Kinematics Cosine LawB
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37. Inverse Kinematics Analytic Method
(X,Y)
180- q2 q1 qT Y X
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38. Inverse Kinematics End Effector=P Base More complex joints
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39. Why is IK hard? Redundancy Natural motion control Singularities
joint limits
minimum jerk
style?
Singularities
ill-conditioned
Singular
Animation (U), Chap 5 Kinematic Linkages
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40. Solving Inverse Kinematics Numerically
Inverse-Jacobian method
Optimization-based method
Example-based method
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41. Inverse-Jacobian method
Jacobian is the n by m matrix relating differential changes of q (dq ) to differential changes of P (dP)
So Jacobian maps velocities in joint space to velocities in Cartesian space
Animation (U), Chap 5 Kinematic Linkages
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42. Inverse-Jacobian method
Recall the inverse kinematics problem
f is highly nonlinear
Make the problem linear by localizing about the current operating position and inverting the Jacobian
and iterating towards the desired post over a series of incremental steps.
Animation (U), Chap 5 Kinematic Linkages
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43. Inverse-Jacobian method
Given initial pose and desired pose, iteratively change the joint angles to approach the goal position and orientation
Interpolate the desired pose vectors for some intermediate steps
For each step k, find the angular velocities for joints, and do
Animation (U), Chap 5 Kinematic Linkages
CS, NCTU, J. H. Chuang

44. Inverse-Jacobian method
Animation (U), Chap 5 Kinematic Linkages
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45. Computing Jacobian Compute analytically Geometric approach
Ok for simple joints
Geometric approach
For complex joints
Let’s say we are just concerned with the end effector position for now.
This also implies that the Jacobian with be an 3×N matrix where N is the number of DOFs
For each joint DOF, we analyze how e would change if the DOF changed
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46. Computing Jacobian Analytically An Example
End Effector=P
Base
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47. Computing Jacobian Analytically An Example
Animation (U), Chap 5 Kinematic Linkages
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48. Computing Jacobian Geometrically Jacobian for a 2D arm
Let’s say we have a simple 2D robot arm with two 1-DOF rotational joints:
• e=[ex ey] φ2 φ1
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49. Computing Jacobian Geometrically Jacobian for a 2D arm
The Jacobian matrix J(Φ) shows how each component of e varies with respect to each joint angle
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50. Computing Jacobian Geometrically Jacobian for a 2D arm
Consider what would happen if we increased φ1 by a small amount. What would happen to e ? • φ1
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51. Computing Jacobian Geometrically Jacobian for a 2D arm
What if we increased φ2 by a small amount? • φ2
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52. Computing Jacobian Geometrically Jacobian for a 2D arm• φ2 φ1
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53. Computing Jacobian Geometrically Jacobian for a 2D arm
Angular and linear velocities induced by joint axis rotation.
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54. Computing Jacobian Geometrically Jacobian for 2D arm
Let’s consider a 1-DOF rotational joint first
We want to know how the global position e will change if we rotate around the axis. • φ1
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55. Computing Jacobian Geometrically Jacobian for 2D arm•
a’i: unit length rotation axis in world space
r’i: position of joint pivot in world space
e: end effector position in world space •
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56. Computing Jacobian Geometrically 3-DOF Rotational Joints
Once we have each axis in world space, each one will get a column in the Jacobian matrix
At this point, it is essentially handled as three 1-DOF joints, so we can use the same formula for computing the derivative as we did earlier:
We repeat this for each of the three axes
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57. Computing Jacobian Geometrically Jacobian for another 2D arm
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58. Computing Jacobian Geometrically Jacobian for another 2D arm
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59. Computing Jacobian Geometrically Jacobian for another 2D arm
The problem is to determine the best linear combination of velocities induced by the various joints that would result in the desired velocities of the end effector.
Jacobian is formed by posing the problem in matrix form.
Animation (U), Chap 5 Kinematic Linkages
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60. Computing Jacobian Geometrically Jacobian for another 2D arm
Animation (U), Chap 5 Kinematic Linkages
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61. Inverse-Jacobian method
Linearize about qk locally
Jacobian depends on current configuration
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62. Inverse-Jacobian method Problems
Problems in localizing the P = f (θ)
Problems in solving the system
Non-square matrix, but independent rows
Use pseudo inverse
Singularity
When the inverse of Jacobian does not exist
occurs when any cannot achieve a given
Near singularity
ill-conditioned
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63. Inverse-Jacobian method Iterative approach
Given initial pose and desired pose, iteratively change the joint angles to approach the goal position and orientation
Interpolate the desired pose vectors for some intermediate steps
For each step k, compute V, and find the angular velocities for joints, and do
Animation (U), Chap 5 Kinematic Linkages
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64. Inverse-Jacobian method Iterative approach
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65. Inverse-Jacobian method Iterative approach
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66. Inverse-Jacobian method Iterative approach
No linear combination of vectors gi could produce the desired goal.
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67. Inverse-Jacobian method Iterative approach
Frame:
Total: 21 frames
No linear combination of vectors gi could produce the desired goal.
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68. Inverse-Jacobian method Problems
Problems in localizing the P = f (θ)
Problems in solving the system
Non-square matrix, but independent rows
Use pseudo inverse
Singularity
When the inverse of Jacobian does not exist
occurs when any cannot achieve a given
Near singularity
ill-conditioned
Animation (U), Chap 5 Kinematic Linkages
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69. Inverse-Jacobian method Remedy to Singularity Problems
Problems with singularities can be reduced if the manipulator is redundancy – when there are more DOF than there are constraints to be satisfied
Jacobian is not square
If rows of J are linearly independent (J has full row rank), then exists and instead, pseudo inverse of Jacobian can be used!
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70. Inverse-Jacobian method Pseudo Inverse of the Jacobian
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71. Inverse-Jacobian method Pseudo Inverse of the Jacobian
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72. Inverse-Jacobian method Dealing with singularities
Physically the singularities usually occur when the linkage id fully extended or when the axes of separate links align themselves
V1 and V2 are
in parallel
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73. Inverse-Jacobian method Dealing with singularities
Two approaches to the case of full extension
Simply not allow the linkage to become fully extended
Adopt a different iterative process with the region of singularity to avoid the case of singularity
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74. Inverse-Jacobian method Dealing with near singularities
Small V implies very large
Two cases
of near
singularity
Near singularity occurs
When 3 links all aligned
And the end is positioned
over the base.
Small V
implies
very large
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75. Inverse-Jacobian method Dealing with near singularities
Damped least square approach
a user-supplied parameter is used to add in a term that reduces the sensitivity of the pseudoinverse,
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76. Adding more control to IK
Pseudoinverse computes one of many possible solutions that minimizes joint angle velocities
IK using pseudoinverse Jacobian may not provide natural poses
A control term can be added to the pseudoinverse Jacobian solution
The control term should not add anything to the velocities, that is
control term
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77. Adding more control to IK Control Term Adds Zero Linear Velocities
A solution of this form
When put into this formula
Like this
After some manipulation, you can show that it…
…doesn’t affect the desired configuration
But it can be used to bias
the solution vector
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78. Adding more control to IK
To bias the solution toward specific joint angles, such as the middle joint angle between joint limits, z is defined as
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79. Adding more control to IK
Joint gain
indicates the relative importance of the associated desired angle
The higher the gain, the stiff the joint
High: the solution will be such that the joint angle quickly approach the desired joint angle
All gians are zero: reduced to the conventional pseudoinverse of Jacobian.
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80. Adding more control to IK How to solve the system?
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81. Adding more control to IK
Gain s
Top: {0.1, 0.5, 0.1}
Bottom: {0.1, 0.1, 0.5}
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82. Null space of Jacobian The control term is in the null space of J
The null space of J is the set of vectors which have no influence on the constraints
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83. Utility of Null Space
The null space can be used to reach secondary goals
Or to find natural pose / control stiffness of joints
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84. Optimization-based Method
Formulate IK as an nonlinear optimization problem
Example
Objective function
Constraint
Iterative algorithm
Nonlinear programming method by Zhao & Badler, TOG 1994
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85. Objective Function
“distance” from the end effector to the goal position/orientation
Function of joint angles : G(q)
Goal
distance
Base
End Effector
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86. Objective Function Position Goal Orientation Goal
Position/Orientation Goal
weighted sum
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87. Nonlinear Optimization
Constrained nonlinear optimization problem
Solution
standard numerical techniques
MATLAB or other optimization packages
usually a local minimum
depends on initial condition
limb coordination
joint limits
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88. Example-based Method Utilize motion database to assist IK solving
IK using interpolation
Rose et al., “Artist-directed IK using radial basis function interpolation,” Eurographics’01
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89. Example-based Method (cont.)
IK using constructed statistical model
Grochow et al., “Style-based inverse kinematics,” SIGGRAPH’04
Provide the most likely pose based on given constraints
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90. Example-based Method (cont.)
Constructed pose space (training, extrapolated)
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91. Videos “Style-based inverse kinematics”
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92. Forward Kinematics
A series of transformations on an object can be applied as a series of matrix multiplications
: position in the global coordinate
: position in the local coordinate
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93. Hierarchical Articulated Model
Represents a character’s skeleton as a series of links connected by joints
connectivity constraints is enforced in a tree-like structure
family of parent-child spatial relationships
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