Many problems in robotics seek to simultaneously optimize several competing objectives under constraints. A conventional approach to solving such multi-objective optimization problems is to create a single cost function comprised of the weighted sum of the individual objectives. Solutions to this scalarized optimization problem are Pareto optimal solutions to the original multi-objective problem. However, finding an accurate representation of a Pareto front remains an important challenge. Using uniformly spaced weight vectors is often inefficient and does not provide error bounds. Thus, we address the problem of computing a finite set of weight vectors such that for any other weight vector, there exists an element in the set whose error compared to optimal is minimized. To this end, we prove fundamental properties of the optimal cost as a function of the weight vector, including its continuity and concavity. Using these, we propose an algorithm that greedily adds the weight vector least-represented by the current set, and provide bounds on the error. Finally, we illustrate that the proposed approach significantly outperforms uniformly distributed weights for different robot planning problems with varying numbers of objective functions.
In the field of motion planning the use of precomputed, feasible, locally optimal motions called motion primitives can not only increase the quality of motions, but decrease the computation time required to develop these motions. In this work we extend the results of our earlier work by developing a technique for computing primitives for a lattice that admits higher-order states like curvature, velocity, and acceleration. The technique involves computing a minimal set of motion primitives that t-span a configuration space lattice. A set of motion primitives t-span a lattice if, given a real number t greater or equal to one, any configuration in the lattice can be reached via a sequence of motion primitives whose cost is no more than t times the cost of the optimal path to that configuration. While motion primitives computed in this way could be used with any graph search algorithm to develop a motion, this paper also proposes one such algorithm which works well in practice in both short complex maneuvers and longer maneuvers. Finally, this paper proposes a shortcut-based smoothing algorithm based on shortest path planning in directed acyclic graphs.
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