Singularity-Robust Task Priority Redundancy Resolution for Real-time Kinematic Control of Robot Manipulators

Stefano Chiaverini

Outline

1.Redundancy2. Inverse differential kinematic control 3. Robust techniques for kinematic control4. Task-priority redundancy resolution5. Numerical analysis of existing solutions.

Redundancy

N > M

Degrees of Freedom Workspace

Redundancy

Example:7 DOF arm vs. 6 DOF Trajectory

Redundancy

Many solutions per problem.

Redundancy

Which one to pick?

Inverse Differential Kinematic Control:

A redundant system!

Kinematic Control

Gives an end-effector velocity which minimizes joint velocities.

Jacobian Pseudo-Inverse:

Kinematic Control

Generalized:

Min Norm Null space

Arbitrary

x

Null

Span

Kinematic Control

Problem: singularites

Min Norm Null space

Singularities

Kinematic Singularities

•Occur when Jacobian has null singular values•Inherent to all techniques.

Algorithmic Singularities

•Occur when no solution exists which satisfies constraints and provides desired EE motion.

Singularities

How to deal with kinematic singularities?

Robust Techniques

First Technique: Damping

Damping term

From SVD

Robust Techniques***

First Technique: Damping

Damped Norm Null Space

Robust Techniques

Pros•Good performance•Robust to singularities

Cons•Accumulates error

First Technique: Damping

Robust Techniques

How to deal with error?

Filter out singularities.

How to choose q0?

Consequence of redundancy

Task Priority Redundancy Resolution

How to choose q0?

Minimizes an objective function H(q)

Task Priority Redundancy Resolution

Or, define a task-space constraint:

Complicates the update rule:

Task Priority Redundancy Resolution

Task space constraint:Can satisfy (N – M) parameters

Task Priority Redundancy Resolution

Problem: Leads to Algorithmic Singularities

Might approach singular

Task Priority Redundancy Resolution

Condition for algorithmic singularities

Null spaces linearly dependant

Task Priority Redundancy Resolution

Algorithmic Singularities Difficult to predict. Arise because of competing demands. Leads to extreme joint velocities.

How do we get rid of them?

Task Priority Redundancy Resolution

Solve for the constraint instead:

A lot of math later….

Zeros out at singularities…

Numerical Analysis

For a 7-dof manipulator:Exact solution (796 flops):

Chiaverni’s robust simplification (632 flops):

Naïve minimum norm (519 flops):

Numerical Analysis

Experiments:

Numerical Analysis

Constraint:

(damped)

(undamped)

(damped)

(undamped)

Conclusion

Need to avoid two kinds of singularities. Presented approach which handles both. New approach is more computationally efficient.

Limitations

Null space motions only Constraints always prioritized lower