Singularities, Stable Surfaces, and Repeatable Behavior

Rodney G. RobertsAnthony Maciejewski

Presenter: Karthik Sheshadri

Introduction

• =JSolution is of the form, with JG=I.For a repeatable strategy, no end effector movement => no joint movement.

Necessary and sufficient condition for repeatability:LB of any two columns of G is in the column space of G.

Stable surfaces

• Stable surface: m dimensional hypersurface in which the manipulator is repeatable.

• From Shamir and Yomdin 1988: An integral surface for any m dimensional distribution is such that for any point on S, the tangent space to S is exactly the M dimensional space assigned by the distribution.

• Implication: Any movement on the stable surface keeps the manipulator on the stable surface. It cant leave.

Stable surfaces

• When a singularity is not encountered, a manipulator cannot reach a stable surface,otherwise it could leave.

• Example: 3R manipulator, initial config: [0 π k]

With sin k not zero, J is nonsingular.

Stable surfaces

Assume we want an =>joint angle velocities of Let for instance k=, and let the manipulator traverse an anti-clockwise quarter of a unit circle, then only k changes until k= π.Here, J is singular, the needed joint velocities are K=The manipulator is on the stable surface

Range Singularities

• Consider the case when the manipulator is completely extended: K=[k k k].

Which gives J=Range{If L’s are equal, the manipulator cannot escape the singularity.

Likelihood of Stable surfaces

Likelihood of Stable surfaces

Must be of the same dimension as the work space, i.e.,6. This is not the case when

The only configurations that satisfy the above are singularities, and the ,manipulator cannot remain in the singularity.

Likelihood of Stable surfaces

• r=n-m eqns specify a stable surface,• No. of constraint eqns = rWhen m is large, it is unlikely that a stable surface exists.

Existence of stable surfaces

• Let s(θ)=0 be a candidate surface, then=0 has to be in the null space of J, characterised by a null vector for one degree of redundency manipulators.s(θ)=0

Existence of stable surfaces

• Either the null space vector and are proportional or =0.

• In the first case

n-link planar manipulator

n-link planar manipulator

Stable surface exists iff there are no more than two distinct lengths.

Minimum Arc length trajectories

• Suggestion:A trajectory which minimises its arc length in the js is repeatable and has zero torsion

• MinimiseSubject to

Minimum Arc length trajectories

Conclusions

• Severe link length constraint for existence of stable surfaces

• The Lie bracket condition is not in general satisfied for repeatable trajectories.

• Minimum arc length trajectories need not have zero torsion.