Ken Alton

University of British Columbia - Vancouver, Vancouver, British Columbia, Canada

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Publications (6)1.71 Total impact

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    Ken Alton, Ian M. Mitchell
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    ABSTRACT: We define a δ-causal discretization of static convex Hamilton-Jacobi Partial Differential Equations (HJ PDEs) such that the solution value at a grid node is dependent only on solution values that are smaller by at least δ. We develop a Monotone Acceptance Ordered Upwind Method (MAOUM) that first determines a consistent, δ-causal stencil for each grid node and then solves the discrete equation in a single-pass through the nodes. MAOUM is suited to solving HJ PDEs efficiently on highly-nonuniform grids, since the stencil size adjusts to the level of grid refinement. MAOUM is a Dijkstra-like algorithm that computes the solution in increasing value order by using a heap to sort proposed node values. If δ>0, MAOUM can be converted to a Dial-like algorithm that sorts and accepts values using buckets of width δ. We present three hierarchical criteria for δ-causality of a node value update from a simplex of nodes in the stencil. The asymptotic complexity of MAOUM is found to be O(([^(Y)]r)d N logN)\mathcal {O}((\hat{\Psi}\rho )^{d} N \log N), where d is the dimension, [^(Y)]\hat{\Psi} is a measure of anisotropy in the equation, and ρ is a measure of the degree of nonuniformity in the grid. This complexity is a constant factor ([^(Y)]r)d(\hat{\Psi}\rho)^{d} greater than that of the Dijkstra-like Fast Marching Method, but MAOUM solves a much more general class of static HJ PDEs. Although ρ factors into the asymptotic complexity, experiments demonstrate that grid nonuniformity does not have a large effect on the computational cost of MAOUM in practice. Our experiments indicate that, due to the stencil initialization overhead, MAOUM performs similarly or slightly worse than the comparable Ordered Upwind Method presented in (Sethian and Vladimirsky, SIAM J. Numer. Anal. 41:323, 2003) for two examples on uniform meshes, but considerably better for an example with rectangular speed profile and significant grid refinement around nonsmooth parts of the solution. We test MAOUM on a diverse set of examples, including seismic wavefront propagation and robotic navigation with wind and obstacles. KeywordsOrdered upwind methods–Anisotropic optimal control–Anisotropic front propagation–Hamilton-Jacobi equation–Viscosity solution–Dijkstra-like methods–Dial-like methods
    Journal of Scientific Computing 01/2011; 51(2):313-348. · 1.71 Impact Factor
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    K. Alton, I.M. Mitchell
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    ABSTRACT: We present an efficient dynamic programming algorithm to solve the problem of optimal multi-location robot rendezvous. The rendezvous problem considered can be structured as a tree, with each node representing a meeting of robots, and the algorithm computes optimal meeting locations and connecting robot trajectories. The tree structure is exploited by using dynamic programming to compute solutions in two passes through the tree: an upwards pass computing the cost of all potential solutions, and a downwards pass computing optimal trajectories and meeting locations. The correctness and efficiency of the algorithm are analyzed theoretically, while a continuous robot arm problem demonstrates the algorithm¿s practicality.
    Decision and Control, 2008. CDC 2008. 47th IEEE Conference on; 01/2009
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    K. Alton, I.M. Mitchell
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    ABSTRACT: Optimal path planning under full state and map knowledge is often accomplished using some variant of Dijkstra's algorithm, despite the fact that it represents the path domain as a discrete graph rather than as a continuous space. In this paper we compare Dijkstra's discrete algorithm with a variant (often called the fast marching method) which more accurately treats the underlying continuous space. Analytically, both generate a value function free of local minima, so that optimal path generation merely requires gradient descent. We also investigate the use of optimality metrics other than Euclidean distance for both algorithms. These different norms better represent optimal paths for some types of problems, as demonstrated by planning optimal collision-free paths for a multiple robot scenario. When considering approximations consistent with the underlying state space, our conclusion is that fast marching places fewer constraints upon grid connectivity, and that it achieves better accuracy than Dijkstra's discrete algorithm in many but not all cases
    Robotics and Automation, 2006. ICRA 2006. Proceedings 2006 IEEE International Conference on; 02/2006
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    K. Alton, M. van de Panne
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    ABSTRACT: We present a semi-parametric control policy representation and use it to solve a series of nonholonomic control problems with input state spaces of up to 7 dimensions. A nearest-neighbor control policy is represented by a set of nodes that induce a Voronoi partitioning of the input space. The Voronoi cells then define local control actions. Direct policy search is applied to optimize the node locations and actions. The selective addition of nodes allows for progressive refinement of the control representation. We demonstrate this approach on the challenging problem of learning to steer cars and trucks-with-trailers around winding tracks with sharp corners. We consider the steering of both forwards and backwards-moving vehicles with only local sensory information. The steering behaviors for these nonholonomic systems are shown to generalize well to tracks not seen in training.
    Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on; 05/2005
  • Ken Alton, Ian M Mitchell
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    ABSTRACT: We present an efficient algorithm to solve the problem of optimal multi-location robot rendezvous. The rendezvous problem considered can be structured as a tree, with each node representing a meeting of robots, and the algorithm computes optimal meeting locations and connecting robot trajectories. The tree structure is exploited by using dynamic programming to compute solutions in two passes through the tree: an upwards pass computing the cost of all potential solutions, and a downwards pass computing optimal trajectories and meeting locations. The correctness and efficiency of the algorithm are analyzed theoretically, while a discrete robotic clinic problem and a continuous robot arm problem demonstrate the algorithm's practicality. This is an extended version of a paper submitted to ICRA 2008 [1].
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    Ken Alton, Ian M Mitchell
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    ABSTRACT: The Fast Marching Method (FMM) has proved to be a very efficient algorithm for solving the isotropic Eikonal equation. Because it is a minor modification of Dijkstra's algorithm for finding the shortest path through a discrete graph, FMM is also easy to implement. In this paper we describe a new class of Hamilton-Jacobi (HJ) PDEs with axis-aligned anisotropy which satisfy a causality condition for standard finite difference schemes on orthogonal grids and can hence be solved using the FMM; the only modification required to the algorithm is in the local update equation for a node. Since our class of HJ PDEs and grids permit asymmetries, we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and PDEs. This class of HJ PDEs has applications in robotic path planning, and a brief example is included. In support of this and similar applications, we also include explicit update formulas for variations of the Eikonal equation that use the Manhattan, Euclidean and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing.