A quadratic regulator-based heuristic for rapidly exploring state space

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 53-55).Kinodynamic planning algorithms like Rapidly-Exploring Randomized Trees (RRTs) hold the promise of finding feasible trajectories for rich dynamical systems with complex, non-convex constraints. In practice, these algorithms perform very well on configuration space planning, but struggle to grow efficiently in systems with dynamics or differential constraints. This is due in part to the fact that the conventional proximity metric, Euclidean distance, does not take into account system dynamics and constraints when identifying which node in the existing tree is capable of producing children closest to a given point in state space. Here we argue that the RRTs' coverage of state space is maximized by using a proximity psuedometric proportional to the length, in time, of the quickest possible trajectory between two points in state space. We derive this minimum-time metric for the double integrator and show that an affine quadratic regulator (AQR) design can be used to approximate the exact minimum-time proximity pseudometric at a reasonable computational cost. We demonstrate improved exploration of the state spaces of the double integrator and simple pendulum when using this pseudometric within the RRT framework. However, for more complex nonlinear systems, experiments thus far suggest that the AQR-based proximity pseudometric and the conventional metric produce equivalent coverage of the state space, on average. This drop-off in benefit as system complexity and nonlinearity increase may be due to the linearization of system dynamics that is required to calculate the AQR-based pseudometric. Future work includes exploring methods for approximating the exact minimum-time proximity pseudometric that can reason about dynamics with higher-order terms.by Elena Leah Glassman.M.Eng

Joint Exploration of Local Metrics and Geometry in Sampling-based Planning

This thesis addresses how the local geometry of the workspace around a system state can be combined with local metrics describing system dynamics to improve the connectivity of the graph produced by a sampling-based planner over a fixed number of configurations. This development is achieved through generalization of the concept of the local free space to inner products other than the Euclidean inner product. This new structure allows for naturally combining the local free space construction with a locally applicable metric. The combination of the local free space with two specific metrics is explored in this work. The first metric is the quadratic cost-to-go function defined by a linear quadratic regulator, which captures the local behavior of the dynamical system. The second metric is the Mahalanobis distance for a belief state in a belief space planner. Belief space planners reason over distributions of states, called belief states, to include modeled uncertainty in the planning process. The Mahalanobis distances metric for a given belief state can be exploited to include notions of risk in local free space construction. Numerical simulations are provided to demonstrate the improved connectivity of the graph generated by a sampling-based planner using these concepts