Provably Safe Robot Navigation with Obstacle Uncertainty

Abstract

Abstract-As drones and autonomous cars become more widespread it is becoming increasingly important that robots can operate safely under realistic conditions. The noisy information fed into real systems means that robots must use estimates of the environment to plan navigation. Efficiently guaranteeing that the resulting motion plans are safe under these circumstances has proved difficult. We examine how to guarantee that a trajectory or policy is safe with only imperfect observations of the environment. We examine the implications of various mathematical formalisms of safety and arrive at a mathematical notion of safety of a long-term execution, even when conditioned on observational information. We present efficient algorithms that can prove that trajectories or policies are safe with much tighter bounds than in previous work. Notably, the complexity of the environment does not affect our method's ability to evaluate if a trajectory or policy is safe. We then use these safety checking methods to design a safe variant of the RRT planning algorithm.

Convex Risk Bounded Continuous-Time Trajectory Planning in Uncertain Nonconvex Environments

“…at time steps t = 0, 0.5, 1, b) Dynamic ∆-risk contours for ∆ = 0.1 at time steps t = 0, 0.5, 1 described in (12). Dashed line shows the expected value of the given uncertain trajectory, i.e., E[(px 1 (t, ω 2 ), px 2 (t, ω 3 ))].…”

“…Similar to illustrative example 1, we compute the polynomials E[P 2 (x, ω, t)] and E[P(x, ω, t)] using the moments of the uncertain parameters ω i , i = 1, 2, 3 and the polynomial obstacle. We then construct the dynamic ∆-risk contour as a function of time as described in (12). Figure 3 shows the obtained dynamic ∆-risk contours for ∆ = 0.1 at time steps t = 0, 0.5, 1 along the given uncertain trajectory…”

“…Remark 2: We can use (10) and (12) to construct static and dynamic risk contours in real-time. Therefore, standard motion planning algorithms such as RRT * , PRM, and motion primitive-based methods can be used for real-time risk bounded motion planning.…”

Rapid uncertainty propagation and chance‐constrained path planning for small unmanned aerial vehicles

“…There is a wide array of solution strategies for planning a vehicle's path subject to uncertainty, including convex programming, 26 mixed integer linear programming, 27 graph search, 28 fast marching trees, 29 and probabilistic roadmaps. 30 One of the more popular approaches [31][32][33][34][35] utilizes a class of stochastic search algorithms, called rapidly expanding random trees (RRTs). 36 These algorithms are well suited for real-time implementation 32 and are sampling-based, so they scale well with problem size, but only offer a probabilistic guarantee of completeness.…”

Chance Constrained Motion Planning for High-Dimensional Robots

“…Unlike car-like robots, a typical robotic manipulator can have seven degrees-of-freedom (DOFs), and this high-dimensionality makes it extremely difficult to quantify uncertainties into collision risks and to make safe motion plans in real time. Existing systems that tackle the risk-aware motion planning problem (Van Den Berg et al 2012;Luders et al 2013;Ono et al 2013;Sun et al 2015;Chen et al 2017;Axelrod et al 2018;Luo et al 2019) lack the ability of efficiently handling high-dimensional robots and non-convex environments. In order to address these difficulties, we propose probabilistic Chekov (p-Chekov), a combined sampling-based and optimization-based approach that takes advantage of the fact that most obstacles in a lot of practical motion planning tasks are static and only a small number of objects are dynamic during deployment.…”