Kendra Lesser

Path-guided artificial potential fields with stochastic reachable sets for motion planning in highly dynamic environments

Highly dynamic environments pose a particular challenge for motion planning due to the need for constant evaluation or validation of plans. However, due to the wide range of applications, an algorithm to safely plan in the presence of moving obstacles is required. In this paper, we propose a novel technique that provides computationally efficient planning solutions in environments with static obstacles and several dynamic obstacles with stochastic motions. Path-Guided APF-SR works by first applying a sampling-based technique to identify a valid, collision-free path in the presence of static obstacles. Then, an artificial potential field planning method is used to safely navigate through the moving obstacles using the path as an attractive intermediate goal bias. In order to improve the safety of the artificial potential field, repulsive potential fields around moving obstacles are calculated with stochastic reachable sets, a method previously shown to significantly improve planning success in highly dynamic environments. We show that Path-Guided APF-SR outperforms other methods that have high planning success in environments with 300 stochastically moving obstacles. Furthermore, planning is achievable in environments in which previously developed methods have failed.

Reachability for partially observable discrete time stochastic hybrid systems

Abstract When designing optimal controllers for any system, it is often the case that the true state of the system is unknown to the controller. Imperfect state information must be taken into account in the controller’s design in order to preserve its optimality. The same is true when performing reachability calculations. To estimate the probability that the state of a stochastic system reaches, or stays within, some set of interest in a given time horizon, it is necessary to find a controller that drives the system to that set with maximum probability, given the controller’s knowledge of the true state of the system. To date, little work has been done on stochastic reachability calculations with partially observable states. The work that has been done relies on converting the reachability optimization problem to one with an additive cost function, for which theoretical results are well known. Our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information. Our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value. We develop a dynamic programming recursion for the solution of the equivalent perfect information problem, proving that the recursion is valid, an optimal solution exists, and results in the same solution as to the original problem. We also show that our results are equivalent to those for the reformulated additive cost problem, and so such a reformulation is not required.

Aggressive Moving Obstacle Avoidance Using a Stochastic Reachable Set Based Potential Field

Identifying collision-free trajectories in environments with dynamic obstacles is a significant challenge. However, many pertinent problems occur in dynamic environments, e.g., flight coordination, satellite navigation, autonomous driving, and household robotics. Stochastic reachable (SR) sets assure collision-free trajectories with a certain likelihood in dynamic environments , but are infeasible for multiple moving obstacles as the computation scales exponentially in the number of Degrees of Freedom (DoF) of the relative robot-obstacle state space. Other methods, such as artificial potential fields (APF), roadmap-based methods, and tree-based techniques can scale well with the number of obstacles. However, these methods usually have low success rates in environments with a large number of obstacles. In this paper, we propose a method to integrate formal SR sets with ad-hoc APFs for multiple moving obstacles. The success rate of this method is 30 % higher than two related methods for moving obstacle avoidance, a roadmap-based technique that uses a SR bias and an APF technique without a SR bias, reaching over 86 % success in an enclosed space with 100 moving obstacles that ricochet off the walls.

Stochastic reachability based motion planning for multiple moving obstacle avoidance

One of the many challenges in designing autonomy for operation in uncertain and dynamic environments is the planning of collision-free paths. Roadmap-based motion planning is a popular technique for identifying collision-free paths, since it approximates the often infeasible space of all possible motions with a networked structure of valid configurations. We use stochastic reachable sets to identify regions of low collision probability, and to create roadmaps which incorporate likelihood of collision. We complete a small number of stochastic reachability calculations with individual obstacles a priori. This information is then associated with the weight, or preference for traversal, given to a transition in the roadmap structure. Our method is novel, and scales well with the number of obstacles, maintaining a relatively high probability of reaching the goal in a finite time horizon without collision, as compared to other methods. We demonstrate our method on systems with up to 50 dynamic obstacles.

Finite state approximation for verification of partially observable stochastic hybrid systems

We consider the problem of verification of safety specifications for stochastic hybrid systems with a controller that has access to partial observations of the state. We address this problem through a finite state approximation of the stochastic hybrid system, which enables the use of existing solution techniques for partially observable Markov decision processes. First, we review a dynamic programming formulation of the safety (viability) problem over an equivalent information state. We then solve a dynamic program over the finite state approximation to generate a lower bound to the viability probability, using a point-based method that generates samples of the information state. Our approach produces approximate probabilistic viable sets and synthesizes a controller to satisfy safety specifications. We provide error bounds and convergence results, assuming additive Gaussian noise in the continuous state dynamics and observations. Finally, we demonstrate performance of the approximation on a simple temperature regulation problem.

Stochastic reachability for control of spacecraft relative motion

The concept of stochastic reachability allows for the assessment, before any maneuvers are initiated, of the probability of successfully implementing a rendezvous or docking procedure for spacecraft. The so-called reach-avoid problem lets us find the probability of reaching a target set while avoiding some unsafe or undesired set, despite uncertainty due to nonlinearity and disturbances. This paper examines two novel methods for the calculation of stochastic reachable sets, and specifically for rendezvous and docking problems. In particular, we examine a) particle (or scenario) approximations to expected values, and b) conversion of the reach-avoid probability to a chance-constrained convex optimization problem. Both methods allow for computation of the reach-avoid set in higher dimensions, as compared to other existing methods for computing stochastic reachable sets. We describe in detail both of these methods, and then apply them to spacecraft relative motion, a four-dimensional problem.

Hybrid Dynamic Moving Obstacle Avoidance Using a Stochastic Reachable Set-Based Potential Field

One of the primary challenges for autonomous robotics in uncertain and dynamic environments is planning and executing a collision-free path. Hybrid dynamic obstacles present an even greater challenge as the obstacles can change dynamics without warning and potentially invalidate paths. Artificial potential field (APF)-based techniques have shown great promise in successful path planning in highly dynamic environments due to their low cost at runtime. We utilize the APF framework for runtime planning but leverage a formal validation method, Stochastic Reachable (SR) sets, to generate accurate potential fields for moving obstacles. A small number of SR sets are computed a priori, then used to generate a potential field that represents the obstacles stochastic motion for online path planning. Our method is novel and scales well with the number of obstacles, maintaining a relatively high probability of reaching the goal without collision, as compared to other traditional Gaussian APF methods. Here, we demonstrate our method with up to 900 hybrid dynamic obstacles and show that it outperforms the traditional Gaussian APF method by up to 60% in the holonomic case and up to 20% in the unicycle case.

Multiobjective Optimal Control With Safety as a Priority

This work develops a lexicographic approach to multi-objective optimal control on models for cyber-physical systems, encompassing in particular stochasticity, limited access to model variables (partial observations), and possibly hybrid (continuous and discrete) dynamics (with the finite state POMDP framework as a known special instance). The technique is showcased in two new case studies in the area of building automation systems. Technically, the main achievements of this work are: The application of the lexicographic framework to multi-objective optimization including quantitative probabilistic safety requirements, thus leading to a principled and scalable integration of correct-by-design synthesis for safety and optimal synthesis for performance; the novel extension of the lexicographic framework to partially-observed stochastic models with continuous (possibly hybrid) dynamics; the emphasis on computational aspects, including the use of compact and approximate representations of value functions combined with the quantification of error bounds on model abstractions.

Approximate Safety Verification and Control of Partially Observable Stochastic Hybrid Systems

Assuring safety in stochastic hybrid systems is particularly difficult when only noisy or partial observations of the state are available. We first review a formulation of the probabilistic safety problem under partial hybrid observations as a dynamic program over an equivalent information state. Two methods for approximately solving the dynamic program are presented. The first approximates the hybrid system as a finite state Markov decision process, so that the information state is a probability mass function. The second method approximates an indicator function over the safe region using radial basis functions, to represent the information state as a Gaussian mixture. In both cases we discretize the hybrid observation process, then use point-based value iteration to under-approximate the safety probability and synthesize a safety-preserving control policy. We obtain error bounds and convergence results in both cases, assuming switched affine dynamics and additive Gaussian noise on the continuous states and observations. We compare the performance of the finite state and Gaussian mixture approaches on a simple numerical example.

Safety verification of output feedback controllers for nonlinear systems

A high-gain observer is used for a class of feedback linearisable nonlinear systems to synthesize safety-preserving controllers over the observer output. A bound on the distance between trajectories under state and output feedback is derived, and shown to converge to zero as a function of the gain parameter of an observer. We can therefore recover safety properties under output feedback and control saturation constraints by synthesizing a controller as if the full state were available. We specifically design feedback linearising controllers that satisfy certain properties, such as stability, and then construct the associated maximal safety-invariant set, namely the largest set of all initial states that are guaranteed to produce safe trajectories over a given (possibly infinite) time horizon.