Single Point of Contact Manipulation of Unknown Objects

1
Single Point of Contact Manipulation of Unknown
Objects

• Stuart Anderson
• Advisor Reid Simmons
• School of Computer Science
• Carnegie Mellon University

2
The Problem

• Given an object with unknown geometric and
  dynamic parameters, how can a robot discover
  these parameters?
• A six degree of freedom fiducial (pose sensor) is
  affixed to the object.
• We can manipulate the object by applying forces
  to its surface.
• Applied forces and fiducial observations have
  white gaussian noise

3
How can we learn about the world?

• Experimentation
• Based on a belief about the state of the world
  make a prediction of what will happen when a
  particular action is taken. Then observe what
  really happens and incorporate this new
  information into the belief.

4
Learning

• Two approaches
• Extensions to the Kalman filter
• Particle Filter
• What do we learn about the system?
• Geometry
• pre-selected set of normals
• learn their offsets, define a convex hull
• Dynamics Parameters
• Mass, Inertial Tensor, Center of Mass,
  Coefficient of Friction

5
Physics

• The motion of a rigid body is described by the
  Newton-Euler equations.
• For simplicity, we use the term wrench to
  describe vector composed of a force and a torque.
• We also introduce a generalize mass matrix. This
  allows us to write

6
Physics

• In order to predict the motion of an object, we
  need to know the forces applied to it.

• If we have some belief about the parameters of
  the object, then we can explicitly compute the
  wrench due to the manipulator and to gravity

7
Physics

• The contact wrench cannot be computed explicitly.

The pressure at any given point is indeterminate
and both the normal force and frictional force
depend on it. This severely limits what can be
known about the contact wrench
8
Physics
So the total contact wrench falls within the
convex hull of the wrenches produced by applying
all pressure at a single point.
9
Physics

• We need to know more about these convex hulls.
• The frictional load is modeled by Coulomb
  friction.
• The upper bound on the magnitude of the
  frictional load is linear in the contact pressure
• The direction of the frictional load at a given
  point is the direction that point is sliding. If
  the point is not sliding then the load direction
  is not known.

Thus, when the object is sliding we have
10
Physics

• If the motion is a pure translation, then the
  direction of motion is constant for all points

and
So the set of w(x) where x is the vertices of
the convex hull of the contact region defines the
convex hull of possible wrenches, and lies within
a two dimensional slice of wrench space.
11
Physics

• If the motion is a generalized rotation then the
  direction of the frictional load is no longer
  linear in position.
• The velocity of points on the surface is linear
  in X though.

12
Physics

• Since distance is linear in polar coordinates, we
  apply a change of coordinates to x.

How do we visualize these equations?
13
Physics

• Developing an exact description of the convex
  wrench hull in this case is difficult.
• Instead, we take samples of w(x) and show bounds
  on the error in the convex hull constructed from
  these samples.
• Because w(x) is linear in r we need only sample
  the minimal and maximal values of r for a given
  theta.
• The number of samples needed for a given maximum
  error bound grows linearly in the radius of a
  circle centered at the rotation center which
  encloses the contact hull.

14
Physics

• Finally, when the object is not moving we take
  additional samples to model the indeterminacy in
  the direction of the frictional load.

15
Particle Filter

• Approximate the posterior distribution using a
  set of particles.
• Whats a particle?
• A description of the system a state vector
• Lets us work with any system, dont need to worry
  about linearization.
• But we have to be careful or dimensionality
  becomes a problem.
• m(1), I(6), mu(1), cm(3), x(3), v(3),geom(250)

16
Particle Filter Approach

• Given a particle b, when we make an observation o
• Then resample the distribution based on the
  relative likelihood of each particle.

17
Observations and Predictions

• We observe accelerations based on the fiducial
  movement.
• Caveat assume we take observations frequently
  enough that the forces applied to the object
  remain about constant. We can stop looking when
  bad stuff (impact) happens.
• So we need to predict the acceleration based on
  state and force applied.
• In general, we cant get an explicit answer to
  this question since the contact pressure
  distribution is undetermined.
• But we can compute bounds on the resulting
  contact force and torque based on the velocity,
  geometry, and applied force.
• We also observe the pose of the object, and use
  that to refine geometric estimates.

18
Particle Filter

• We cant use a standard particle filter with our
  system, since it is not possible to compute
  p(ob) exactly.
• We let each particle have an upper and lower
  bound on its probability, instead of an explicit
  value.
• But how to normalize the upper bound after an
  observation?

19
Particle Filter
20
Particle Filter

• How do we compute a new upper bound for a given
  particle?

21
Particle Filter

• We can solve a simpler case by quadratic
  programming.

But we have no insight into finding the true
normalized upper bound without resorting to
general optimization strategies.
22
Planning and Behaviors

• Planning has a problem with dimensionality too.
• The set of actions is a (potentially unconnected)
  path through the set of all forces at all points.
• Behaviors are a way to deal with this
• Slide (maintaining current contact face)
• Switch contact face (a.k.a roll)

23
Uncertainty

• Given a set of particles, we can simulate a
  behavior many times, assuming a different
  particle to be the truth each time.
• We can look at the statistics of these trials to
  see what effect a given behavior will have on the
  belief state.
• Some behaviors tend to reduce the variance or
  entropy of the distribution. Pick these.

24
The Software

• Dynamics Simulation / Visualization
• Human Interface
• Planning Interface
• Geometric Reasoning
• We can refine the convex hull and find the
  predicted contact hull.
• Wrench Reasoning
• We can compute the probability of an observation
  within known and controllable error bounds for
  all motion cases.
• Particle Filter
• Working implementation of the particle filter
  based on an incorrect assumption about the upper
  bound update.
• Behaviors
• Slide, shift face. Reactive control is a little
  too good because the force can be shifted
  instantaneously.