In this paper, "chance optimization" problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations based on the theory of measures and moments is provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. Indeed, we provide a sequence of semidefinite programs of increasing dimension which can arbitrarily approximate the solution of the original problem. To be able to efficiently solve the resulting large-scale semidefinite relaxations, a first-order augmented Lagrangian algorithm is implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.

In this paper, we address the risk estimation problem where one aims at estimating the probability of violation of safety constraints for a robot in the presence of bounded uncertainties with arbitrary probability distributions. In this problem, an unsafe set is described by level sets of polynomials that is, in general, a non-convex set. Uncertainty arises due to the probabilistic parameters of the unsafe set and probabilistic states of the robot. To solve this problem, we use a moment-based representation of probability distributions. We describe upper and lower bounds of the risk in terms of a linear weighted sum of the moments. Weights are coefficients of a univariate Chebyshev polynomial obtained by solving a sum-of-squares optimization problem in the offline step. Hence, given a finite number of moments of probability distributions, risk can be estimated in real-time. We demonstrate the performance of the provided approach by solving probabilistic collision checking problems where we aim to find the probability of collision of a robot with a non-convex obstacle in the presence of probabilistic uncertainties in the location of the robot and size, location, and geometry of the obstacle.In this paper, we leverage SOS based techniques to provide upper and lower bounds of the probability of violation of safety constraints described by level sets of n-variate polynomials. The proposed method can deal with bounded uncertainties with arbitrary probability distributions and also uncertain nonconvex safety constraints e.g., obstacles with uncertain location, size, and geometry. The provided method relies on a convex optimization that looks for a univariate polynomial indicator function. Using the proposed approach, we describe upper and lower bounds of the risk as a linear weighted sum of the moments of uncertainties. The weights are coefficients of a univariate Chebyshev polynomial obtained by solving a univariate SOS optimization.

In this paper, we introduce "risk contours map" that contains the risk information of different regions in uncertain environments. Risk is defined as the probability of collision of robots with obstacles in presence of probabilistic uncertainties in location, size, and geometry of obstacles. We use risk contours to obtain safe paths for robots with guaranteed bounded risk. We formulate the problem of obtaining risk contours as a chance constrained optimization. We leverage the theory of moments and nonnegative polynomials to provide a convex optimization in the form of sum of squares optimization. Provided approach deals with nonconvex obstacles and probabilistic bounded and unbounded uncertainties. We demonstrate the performance of the provided approach by solving risk bounded motion planning problems.

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This paper introduces Probabilistic Chekov (p-Chekov), a chance-constrained motion planning system that can be applied to high degree-of-freedom (DOF) robots under motion uncertainty and imperfect state information. Given process and observation noise models, it can find feasible trajectories which satisfy a user-specified bound over the probability of collision. Leveraging our previous work in deterministic motion planning which integrated trajectory optimization into a sparse roadmap framework, p-Chekov shows superiority in its planning speed for high-dimensional tasks. P-Chekov incorporates a linear-quadratic Gaussian motion planning approach into the estimation of the robot state probability distribution, applies quadrature theories to waypoint collision risk estimation, and adapts risk allocation approaches to assign allowable probabilities of failure among waypoints. Unlike other existing risk-aware planners, p-Chekov can be applied to high-DOF robotic planning tasks without the convexification of the environment. The experiment results in this paper show that this p-Chekov system can effectively reduce collision risk and satisfy user-specified chance constraints in typical real-world planning scenarios for high-DOF robots.