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Efficient algorithms for globally optimal trajectories

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1 Author(s)
Tsitsiklis, J.N. ; Lab. for Inf. & Decision Syst., MIT, Cambridge, MA, USA

We present serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point xo and follows a unit speed trajectory x(t) inside a region in ℛm until an unspecified time T that the region is exited. A trajectory minimizing a cost function of the form ∫0T r(x(t))dt+q(x(T)) is sought. The discretized Hamilton-Jacobi equation corresponding to this problem is usually solved using iterative methods. Nevertheless, assuming that the function r is positive, we are able to exploit the problem structure and develop one-pass algorithms for the discretized problem. The first algorithm resembles Dijkstra's shortest path algorithm and runs in time O(n log n), where n is the number of grid points. The second algorithm uses a somewhat different discretization and borrows some ideas from a variation of Dial's shortest path algorithm (1969) that we develop here; it runs in time O(n), which is the best possible, under some fairly mild assumptions. Finally, we show that the latter algorithm can be efficiently parallelized: for two-dimensional problems and with p processors, its running time becomes O(n/p), provided that p=O(√n/log n)

Published in:

Automatic Control, IEEE Transactions on  (Volume:40 ,  Issue: 9 )