Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. For robust execution, we must take into account uncertainty, which arises due to uncertain localization, modeling errors, and disturbances. Prior work handled the case of set-bounded uncertainty. We present here a chance-constrained approach, which uses instead a probabilistic representation of uncertainty. The new approach plans the future probabilistic distribution of the vehicle state so that the probability of failure is below a specified threshold. Failure occurs when the vehicle collides with an obstacle or leaves an operator-specified region. The key idea behind the approach is to use bounds on the probability of collision to show that, for linear-Gaussian systems, we can approximate the nonconvex chance-constrained optimization problem as a disjunctive convex program. This can be solved to global optimality using branch-and-bound techniques. In order to improve computation time, we introduce a customized solution method that returns almost-optimal solutions along with a hard bound on the level of suboptimality. We present an empirical validation with an aircraft obstacle avoidance example.

Abstract-Robotic systems need to be able to plan control actions that are robust to the inherent uncertainty in the real world. This uncertainty arises due to uncertain state estimation, disturbances and modeling errors, as well as stochastic mode transitions such as component failures. Chance constrained control takes into account uncertainty to ensure that the probability of failure, due to collision with obstacles, for example, is below a given threshold.In this paper we present a novel method for chance constrained predictive stochastic control of dynamic systems. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore the approximation becomes exact as the number of particles tends to infinity. This method applies to arbitrary noise distributions, and for systems with linear or jump Markov linear dynamics we show that the approximate problem can be solved using efficient Mixed Integer Linear Programming techniques. We also introduce an importance weighting extension that enables the method to deal with low probability mode transitions such as failures. We demonstrate in simulation that the new method is able to control an aircraft in turbulence and can control a ground vehicle while being robust to brake failures.

Abstract-Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. Previous approaches that used a constrained optimization approach to solve for finite sequences of optimal control inputs have been highly effective. For robust execution, it is essential to take into account the inherent uncertainty in the problem, which arises due to uncertain localization, modeling errors, and disturbances.Prior work has handled the case of deterministically bounded uncertainty. We present here an alternative approach that uses a probabilistic representation of uncertainty, and plans the future probabilistic distribution of the vehicle state so that the probability of collision with obstacles is below a specified threshold. This approach has two main advantages; first, uncertainty is often modeled more naturally using a probabilistic representation (for example in the case of uncertain localization); second, by specifying the probability of successful execution, the desired level of conservatism in the plan can be specified in a meaningful manner.The key idea behind the approach is that the probabilistic obstacle avoidance problem can be expressed as a Disjunctive Linear Program using linear chance constraints. The resulting Disjunctive Linear Program has the same complexity as that corresponding to the deterministic path planning problem with no representation of uncertainty. Hence the resulting problem can be solved using existing, efficient techniques, such that planning with uncertainty requires minimal additional computation. Finally, we present an empirical validation of the new method with a number of aircraft obstacle avoidance scenarios.

In this paper we consider finite-horizon optimal control of dynamic systems subject to stochastic uncertainty; such uncertainty arises due to exogenous disturbances, modeling errors, and sensor noise. Stochastic robustness is typically defined using chance constraints, which require that the probability of state constraints being violated is below a prescribed value. Prior work showed that in the case of linear system dynamics, Gaussian noise and convex state constraints, optimal chance-constrained finite-horizon control results in a convex optimization problem. Solving this problem in practice, however, requires the evaluation of multivariate Gaussian densities through sampling, which is time-consuming and inaccurate. We propose a new approach to chance-constrained finite-horizon control that does not require the evaluation of multivariate densities. We use a new bounding approach to ensure that chance constraints are satisfied, while showing empirically that the conservatism introduced is small. This is in contrast to prior bounding approaches that are extremely conservative. Furthermore we show that, as long as the prescribed maximum probability of constraint violation is below 0.5, the resulting optimization is convex and hence amenable to online control design.

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