Amos, Gideon; (2002) Solving the Hamilton-Jacobi-Bellman Equation for Animation. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis addresses the construction of a practical method for solving the Hamilton-Jacobi-Bellman (HJB) equation for the purpose of computer animation. Solutions of the HJB equation yield optimal feedback controllers which may be applied to dynamic models of simple robots or animals to generate motion sequences. Our implementation has proven effective at producing controllers for models with a phase space of up to four dimensions. An efficient swing-up and balance controller was obtained for the acrobot, a double pendulum with a single actuator at its middle joint. A number of specific techniques are proposed to improve both the efficiency of the method and the confidence we may have in the solutions it produces. These include improvements to the convergence rate of the algorithm and a method to quantify the influence of boundaries upon the solution. An adaptive meshing technique, based upon a kd-tree decomposition of the solution space, is presented. It provides a more efficient solution than regular, non-adaptive, meshes. We are optimistic that our approach may prove effective at generating lifelike motion, which is a key requirement for compelling computer animation. Techniques such as the coupling of simplified dynamic models to kinematic animation methods promise to extend the applicability of our approach to systems of higher dimension, such as animated humanoids.

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