An interesting and challenging problem in robotics is the off-line determination of the minimum cost path along which an end effector should move from a given initial to a given final state. This paper presents a discrete minimum cost path/trajectory planner which provides a general solution and allows for a range of constraints such as bounds on joint coordinates, joint velocities, joint torques and joint jerks. To demonstrate the practicability and feasibility of the planner, simulation results are presented for the Stanford manipulator using three and then the full six of its degrees of freedom. Simulation runs with two-link planar arms are also presented to enable a comparison with previously published results.

A finite transverse shock wave propagates through an unbounded medium consisting of two joined incompressible elastic half-spaces of different material properties, in the direction normal to the plane interface. A semi-inverse method is used to examine the reflection-transmission of this wave at the interface. It is found that, depending on the material properties, the reflected wave is either a simple wave or a shock; the transmitted wave is always a shock.

A centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.

A problem in combustion theory with tenperature-dependent conductivity is considered. It is shown that information regarding criticality dependence on data and parameters can be obtained from a transformed equation in which the conductivity is constant, while the nonlinear source term is modified. Some previous work can then be used in such a study.

We study the large time behaviour of the free boundary for a one-phase Stefan problem with supercooling and a kinetic condition u = −ε|⋅ṡ| at the free boundary x = s(t). The problem is posed on the semi-infinite strip [0,∞) with unit Stefan number and bounded initial temperature ϕ(x) ≤ 0, such that ϕ → −1 − δ as x → ∞, where δ is constant. Special solutions and the asymptotic behaviour of the free boundary are considered for the cases ε ≥ 0 with δ negative, positive and zero, respectively. We show that, for ε > 0, the free boundary is asymptotic to , δt/ε if < δ > 0 respectively, and that when δ = 0 the large time behaviour of the free boundary depends more sensitively on the initial temperature. We also give a brief summary of the corresponding results for a radially symmetric spherical crystal with kinetic undercooling and Gibbs-Thomson conditions at the free boundary.

We consider in this article an evolutionary monotone follower problem in [0,1]. State processes under consideration are controlled diffusion processes , solutions of dyx(t) = g(yx(t), t)dt + σu(yx(t), t) dwt + dυt with yx(0) = x ∈[0, 1], where the control processes υt are increasing, positive, and adapted. The cost functional is of integral type, with certain explicit cost of control action including the cost of jumps. We shall present some analytic results of the value function, mainly its characterisation, by standard dynamic programming arguments.

A number of Kuhn-Tucker type sufficient optimality criteria for a class of variational problems under weaker invexity assumptions are presented. As an application of these optimality results, various Mond-Weir type duality results are proved under a variety of generalised invexity assumptions. These results generalise many well-known duality results of variational problems and also give a dynamic analogue of certain corresponding (static) results relating to duality with generalised invexity in mathematical programming.

A new implementation of the BFGS algorithm for unconstrained optimisation is reported which utilises a conjugate factorisation of the approximating Hessian matrix. The implementation is especially useful when gradient information is estimated by finite difference formulae and it is well suited to machines which are able to exploit parallel processing.