Research Thrusts

  • Convolution Complementarity Problems
Convolution complementarity problems have the form: Find the function u where

0 ≤ u(t) |---- (k * u)(t) + q(t) ≥ 0     for all t
 
where k * u is the convolution of a given kernel function k with the unknown u:

(k * u)(t) = ∫  0t k(t - s) u(s) ds
 
Usually k represents the impulse resonse of a system, and can be square-matrix valued. These are useful for studying dynamic complementarity systems that have additional "memory" such as systems with delays, or in partial differential equations. If k is the impulse response for a linear time-invariant system

dx/dt = Ax + Bu,
y = Cx + Du,
 
then this is equivalent to the Linear Complementarity System as described by Schumacher, Weiland, Heemels, Camlibel, and van der Schaft. But because convolution complementarity problems can incorporate memory effects that cannot be replicated by ODEs, these have found use in studying impacts of elastic bodies. In electrical systems, they can be used for transmission line problems with diodes, for example.
  • Modeling Contacts
  • Impact & Friction :: Forward, initial value problems
  • Inverse and Motion Planning Problems
  • Numerics
Numerical methods are a key part of the work done in this project.  Even for theoretical issues, we use numerical (or at least discrete-time) methods for proving existence of solutions.  We are also working on numerical methods for rigid-body dynamics, convolution complementarity problems, and differential variational inequalities.  For solving the resulting complementarity problems, we can use complementarity solvers such as Lemke's method, non-smooth Newton methods, smoothing methods.
  • Differential variational inequalities
Differential Variational Inequalities (DVIs) are problems of the form: find the functions u(t) and x(t) satisfying the conditions

dx/dt = f(t,x,u),

u(t) solves the Variational Inequality VI(K,F(t,x(t),.));

That is, (v-u(t))TF(t,x(t),u(t)) ≥ 0    for almost all t.
 

We can consider initial value problems of this kind, as well as boundary value problems of this kind.  These are a nonlinear generalization of the Linear Complementarity Systems of Schumacher, van der Schaft et al. Existence of solutions has been established for a number of families of DVIs.

The Pontryagin conditions for optimal control can often be represented as a DVI, as can "Variational Inequalities of Evolution'' which have the form


dx/dt in f(t,x) -- NC(x)
 
where C is a given set and NC (x) is the normal cone to C at x.

  • Optimal design and control

    Application Thrusts

  • Fixture insertion
A fixture is a device that immobilizes a part in a known position and
orientation relative to a known frame of reference. Typically immobilization is accomplished through a number of unilateral contacts between the part and the fixture. However, it is often difficult to manipulate the part to achieve the intended contacts. Furthermore, even when the contacts have been achieved, it can be difficult to determine this fact, and to guarantee their maintenance during subsequent manufacturing operations. These considerations give rise to the following fixture insertion problem:

Given the geometry of a fixture and a part, determine a manipulation strategy to "seat" the part, i.e., insert the part such that all intended contacts are acheived.

This problem was previously solved under the assumptions that the parts and fixture are rigid and the part is at rest infinitesimally near its seated position without knowledge of any contacts. Related open problems under study are those defined by relaxing several constraints. For example, the part and fixture may be flexible, the part may initially be quite distant from its seated configuration, and the geometry of the part and the fixture may be partially free.

  • Manipulation
  • Fixture Design
  • Manipulation Planning and Design
  • Mechanism Analysis
  • Any-time Algortihms