Abstract
We present methods for the visualization of the numerical solution of optimal control problems. The solution is based on dynamic programming techniques where the corresponding optimal value function is approximated on an adaptively refined grid. This approximation is then used in order to compute approximately optimal solution trajectories. We discuss requirements for the efficient visualization of both the optimal value functions and the optimal trajectories and develop graphic routines that in particular support adaptive, hierarchical grid structures, interactivity and animation. Several implementational aspects using the Graphics Programming Environment ‘GRAPE’ are discussed.
Highlights
The numerical solution of optimal control problems using dynamic programming techniques or Hamilton-Jacobi-Bellman PDEs has been an active eld of research for the last few decades, cf. e.g. 3] for an overview
We will formulate the properties an e cient interactive visualization should satisfy based on the structure of the optimal control problem and on its numerical approximation
According to the general idea of a hierarchical and procedural visualization interface, which we described in the previous subsection we want to explain some more details of the implementation of this concept
Summary
The numerical solution of optimal control problems using dynamic programming techniques or Hamilton-Jacobi-Bellman PDEs has been an active eld of research for the last few decades, cf. e.g. 3] for an overview. If in addition one demands both interactivity (i.e. a direct coupling with the numerical routine) and animation no standard solutions are available It is the aim of the present paper to provide concepts and implementations to close this gap. For the visualization of optimally controlled as well as uncontrolled trajectories existing concepts for tracing on time dependent vectorelds 16] have been extended and combined with the numerical optimal control routines All in all this results in a comprehensive visualization package for this kind of numerical problems.
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