Nonholonomic motion planning for a free-falling cat using spline approximation

An optimal motion planning of a free-falling cat based on the spline approximation is investigated.Nonholonomicity arises in a free-falling cat subjected to nonintegrable velocity constraints or nonintegrable conservation laws.The equation of dynamics of a free-falling cat is obtained by using the model of two symmetric rigid bodies.The control of the system can be converted to the motion planning problem for a driftless system.A cost function is used to incorporate the final errors and control energy.The motion planning is to determine control inputs to minimize the cost function and is formulated as an infinite dimensional optimal control problem.By using the control parameterization,the infinite dimensional optimal control problem can be transformed to a finite dimensional one.The particle swarm optimization(PSO) algorithm with the cubic spline approximation is proposed to solve the finite dimension optimal control problem.The cubic spline approximation is introduced to realize the control parameterization.The resulting controls are smooth and the initial and terminal values of the control inputs are zeros,so they can be easily generated by experiment.Simulations are also performed for the nonholonomic motion planning of a free-falling cat.Simulated experimental results show that the proposed algorithm is more effective than the Newtoian algorithm.

APPROXIMATION OF SINGULAR INTEGRALS BY INTERPOLATING SPLINES

In this paper we are mainly concerned with the approximation of the following type of singular Integrals: where w(t)≥0 is a weight function, f(x) a real continuous function on [a, b], satisfying certain smooth conditions, and the integral is of Canchy principal value.

Smooth interpolation on homogeneous matrix groups for computer animation

Homogeneous matrices are widely used to represent geometric transformations in computer graphics, with interpo- lation between those matrices being of high interest for computer animation. Many approaches have been proposed to address this problem, including computing matrix curves from curves in Euclidean space by registration, representing one-parameter curves on manifold by rational representations, changing subdivisional methods generating curves in Euclidean space to corresponding methods working for matrix curve generation, and variational methods. In this paper, we propose a scheme to generate rational one-parameter matrix curves based on exponential map for interpolation, and demonstrate how to obtain higher smoothness from existing curves. We also give an iterative technique for rapid computing of these curves. We take the computation as solving an ordinary differential equation on manifold numerically by a generalized Euler method. Furthermore, we give this algorithm’s bound of the error and prove that the bound is proportional to the shift length when the shift length is sufficiently small. Compared to direct computation of the matrix functions, our Euler solution is faster.