681 - Introduction to Computer Graphics

1
Inverse Kinematics
Set goal configuration of end effector and
calculate joint angles
Analytic approach when linkage is simple enough
Numeric approach complex linkages At each time
slice, determine joint movements that take you in
direction of goal position (and orientation)
2
Forward Kinematics - review
Articulated linkage hierarchy of joint-link
pairs
Pose linkage is a specific configuration
Pose Vector vector of joint angles for linkage
Degrees of Freedom (DoF) of joint or of whole
figure
Types of joints revolute, prismatic
Tree structure arcs nodes Recursive traversal
concatenate arc matrices Push current matrix
leaving node downward Pop current matrix
traversing back up to node
3
Inverse Kinematics
End Effector
L1
q3
q2
L3
L2
q1
Goal
4
Inverse Kinematics
Underconstrained if fewer constraints than
DoFs Many solutions
Overconstrained too many constraints No solution
Reachable workspace volume the end effector can
reach
Dextrous workspace volume end effector can
reach in any orientation
5
Inverse Kinematics - Analytic
Given arm configuration (L1, L2, )
Given desired goal position (and orientation) of
end effector x,y or x,y,z, y1,y2, y3
Analytically compute goal configuration (q1,q2)
Interpolate pose vector from initial to goal
6
Analytic Inverse Kinematics
7
Analytic Inverse Kinematics
8
Analytic Inverse Kinematics
9
Analytic Inverse Kinematics
When linkage is too complex for analytic methods
At each time step, determine changes to joint
angles that take the end effector toward goal
position and orientation
Need to recompute at each time step
10
Inverse Kinematics - Numerically
w2
d2
End Effector
q2
w2 x d2
- Compute instantaneous effect of each joint
- Linear approximation to curvilinear motion
- Find linear combination to take end effector
towards goal position
11
Inverse Kinematics
12
Inverse Kinematics
Solution only valid for an instantaneous step
Angular affect is really curved, not straight
line
Once a step is taken, need to recompute solution
13
Inverse Kinematics - Mathematics
Use chain rule
14
Matrix Form
Y F(X)
15
The Matrices
V linear and angular velocities
J Jacobian Matrix of partials
16
The Matrices
N DoFs
V linear and angular velocities
3x1, 6x1
J Jacobian Matrix of partials
3xN, 6xN
N x 1
17
Pseudo Inverse of the Jacobian
18
Solving using the Pseudo Inverse
LU decomposition
19
Adding a Control Term
A solution of this form
Doesnt affect the desired configuration
But it can be used to bias The solution vector
20
Form of the Control Term
Desired angles and corresponding gains are input
Bias to desired angles (not the same as hard
joint limits)
z is H differentiated
21
Some Algebraic Manipulation
Add this, using previous form to solution
indicated by pseudo-inverse of Jacobian
Rearrange and solve
22
Solving the Equations
LU decomp.
23
Control Term
Use to bias to desired mid-angle
Does not enforce joint angles
Does not address human-like or natural motion
Only kinematic control no forces involved