Exact robot navigation using power diagrams

Abstract

We reconsider the problem of reactive navigation in sphere worlds, i.e., the construction of a vector field over a compact, convex Euclidean subset punctured by Euclidean disks, whose flow brings a Euclidean disk robot from all but a zero measure set of initial conditions to a designated point destination, with the guarantee of no collisions along the way. We use power diagrams, generalized Voronoi diagrams with additive weights, to identify the robot's collision free convex neighborhood, and to generate the value of our proposed candidate solution vector field at any free configuration via evaluation of an associated convex optimization problem. We prove that this scheme generates a continuous flow with the specified properties. We also propose its practical extension to the nonholonomically constrained kinematics of the standard differential drive vehicle. Exact Robot Navigation Using Power DiagramsOmur Arslan and Daniel E. Koditschek Abstract-We reconsider the problem of reactive navigation in sphere worlds, i.e., the construction of a vector field over a compact, convex Euclidean subset punctured by Euclidean disks, whose flow brings a Euclidean disk robot from all but a zero measure set of initial conditions to a designated point destination, with the guarantee of no collisions along the way. We use power diagrams, generalized Voronoi diagrams with additive weights, to identify the robot's collision free convex neighborhood, and to generate the value of our proposed candidate solution vector field at any free configuration via evaluation of an associated convex optimization problem. We prove that this scheme generates a continuous flow with the specified properties. We also propose its practical extension to the nonholonomically constrained kinematics of the standard differential drive vehicle.

Smooth extensions of feedback motion planners via reference governors

“…Given a closed-loop system that performs satisfactorily in the absence of constraints, a reference governor modifies the desired reference command, whenever necessary, to the closed-loop system in order to avoid constraint violation for all future time while ensuring system stability. To demonstrate an application of reference governors in motion planning, we now present a reinterpretation of our recently introduced provably correct reactive robot navigation algorithm [22], for a first-order disk-shaped robot operating in a "sphere world" [6], in the reference governor framework. 4 Later in Section IV-B, we shall also provide its reference governor based smooth extension.…”

“…Accordingly, for the fully-actuated single-integrator robot dynamics in (2), we propose in [22] a simple reactive navigation strategy, called "move-to-projected-goal" law, v : F → R n that drives the robot at location x ∈ F toward a designated global goal x * ∈ F through a safe local goal, x * : = Π LF(x) (x * ), called "projected-goal", as follows:…”

“…In [22], we consider the collision-free navigation problem of a velocity-controlled disk-shaped robot, centered at x ∈ W with radius r ∈ R, in a closed compact convex environment…”

“…1, that continuously modifies the target position to the closed-loop motion planner to avoid collisions while preserving the system stability. Note that, the closed-loop motion planner is (a positive constant times) the negated gradient of the squared Euclidean distance A reference governor interpretation of the "move-to-projected goal" law of [22], where the reference governor ensures collision avoidance and the feedback motion planner guarantees stability.…”

“…Using the power diagram -a generalized Voronoi diagram [25] -of the robot's workspace W, generated by disks representing the robot and obstacles, in [22] we define the robot's local workspace and local free space, respectively, as…”