### Singularity-Robust

```Singularity-Robust
Task Priority Redundancy Resolution
for Real-time Kinematic Control
of Robot Manipulators
Stefano Chiaverini
Outline
1.Redundancy
2. Inverse differential kinematic control
3. Robust techniques for kinematic control
4. Task-priority redundancy resolution
5. 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
Jacobian Pseudo-Inverse:
Gives an end-effector velocity which
minimizes joint velocities.
Kinematic Control
Generalized:
Span
Min Norm
x
Null
Arbitrary
Null space
Kinematic Control
Min Norm
Null space
Problem: singularites
Singularities
Kinematic
Singularities
Algorithmic
Singularities
• Occur when
Jacobian has null
singular values
• Inherent to all
techniques.
• 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
From SVD
Damping term
Robust Techniques
***
First Technique: Damping
Damped Norm
Null Space
Robust Techniques
First Technique: Damping
Pros
Cons
• Good
performance
• Robust to
singularities
• Accumulates
error
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
```