Daniel Lyons

Chance constrained model predictive control for multi-agent systems with coupling constraints

We consider stochastic model predictive control of a multi-agent systems with constraints on the probabilities of inter-agent collisions. First, we discuss a method based on sample average approximation of the collision probabilities to make the stochastic control problem computationally tractable. Empirical results indicate that the complexity of the resulting optimization problem can be too high to be solved under realtime requirements. To reduce the computational burden we propose a second approach. It employs probabilistic bounds to determine regions of increased probability of presence for each agent and introduce constraints for the control problem prohibiting overlap of these regions. We prove that the resulting problem is conservative for the original problem, i.e., every control strategy that is feasible under our new constraints will automatically be feasible for the true original problem. Furthermore, we present simulations demonstrating improved run-time performance of our second approach and compare our stochastic method to robust control.

Lazy auctions for multi-robot collision avoidance and motion control under uncertainty

We present an auction-flavored multi-robot planning mechanism where coordination is to be achieved on the occupation of atomic resources modeled as binary inter-robot constraints. Introducing virtual obstacles, we show how this approach can be combined with particle-based obstacle avoidance methods, offering a decentralized, auction-based alternative to previously established centralized approaches for multi-robot open-loop control. We illustrate the effectiveness of our new approach by presenting simulations of typical spatially-continuous multi-robot path-planning problems and derive bounds on the collision probability in the presence of uncertainty.

Uncertainty-constrained robot exploration: A mixed-integer linear programming approach

In this paper we consider the situation in which a robot is deployed in an unknown scenario and has to explore the entire environment without possibility of measuring its absolute position. The robot can take relative position measurements (from odometry and from place revisiting episodes) and can then estimate autonomously its trajectory. Therefore, the quality of the resulting estimate depends on the motion strategy adopted by the robot. The problem of uncertainty-constrained exploration is then to explore the environment while satisfying given bounds on the admissible uncertainty in the estimation process. We adopt a moving horizon strategy in which the robot plans its motion T steps ahead. Our formulation leads to a mixed-integer linear problem that has several desirable properties: (i) it guarantees that the robot motion is collision free, (ii) it guarantees that the uncertainty constraints are met, (iii) it enables the design of algorithms that efficiently solve moderately sized instances of the exploration problem. We elucidate on the proposed formulation with numerical experiments.

Nonlinear information filtering for distributed multisensor data fusion

The information filter has evolved into a key tool for distributed and decentralized multisensor estimation and control. Essentially, it is an algebraical reformulation of the Kalman filter and provides estimates on the information about an uncertain state rather than on a state itself. Whereas many practicable Kalman filtering techniques for nonlinear system and sensor models have been developed, approaches towards nonlinear information filtering are still scarce and limited. In order to deal with nonlinear systems and sensors, this paper derives an approximation technique for arbitrary probability densities that provides the same distributable fusion structure as the linear information filter. The presented approach not only constitutes a nonlinear version of the information filter, but it also points the direction to a Hilbert space structure on probability densities, whose vector space operations correspond to the fusion and weighting of information.

On mixed-integer random convex programs

We consider a class of mixed-integer optimization problems subject to N randomly drawn convex constraints. We provide explicit bounds on the tails of the probability that the optimal solution found under these N constraints will become infeasible for the next random constraint. First, we study constraint sets in general mixed-integer optimization problems, whose continuous counterpart is convex. We prove that the number of support constraints (i.e., constraints whose removal strictly improve the optimal objective) is bounded by a number depending geometrically on the dimension of the decision vector. Next, we use these results to show that the tails of the violation probability are bounded by a binomial distribution. Finally, we apply these bounds to an example of robust truss topology design. The findings in this paper are a first step towards an extension of previous results on continuous random convex programs to the case of problems with mixed-integer decision variables that naturally occur in many real-world applications.

Robust model predictive control with least favorable measurements

Closed-loop model predictive control of nonlinear systems, whose internal states are not completely accessible, incorporates the impact of possible future measurements into the planning process. When planning ahead in time, those measurements are not known, so the closed-loop controller accounts for the expected impact of all potential measurements. We propose a novel conservative closed-loop control approach that does not calculate the expected impact of all measurements, but solely considers the single future measurement that has the worst impact on the control objective. In doing so, the model predictive controller guarantees robustness even in the face of high disturbances acting upon the system. Moreover, by considering only a single dedicated measurement, the complexity of closed-loop control is reduced significantly. The capabilities of our approach are evaluated by means of a path planning problem for a mobile robot.

Modellierung von Unsicherheiten und Zustandsschätzung mit Mengen von Wahrscheinlichkeitsdichten

Zusammenfassung Die systematische Behandlung von Unsicherheiten stellt eine wesentliche Herausforderung in der Informationsfusion dar. Einerseits müssen geeignete Darstellungsformen für die Unsicherheiten bestimmt werden und andererseits darauf aufbauend effiziente Schätzverfahren hergeleitet werden. Im Allgemeinen wird zwischen stochastischen und mengenbasierten Unsicherheitsbeschreibungen unterschieden. Dieser Beitrag stellt ein Verfahren zur Zustandsschätzung vor, welches simultan stochastische und mengenbasierte Fehlergrößen berücksichtigen kann, indem unsichere Größen nicht mehr durch eine einzelne Wahrscheinlichkeitsdichte, sondern durch eine Menge von Dichten repräsentiert werden. Besonderes Augenmerk liegt hier auf den Vorteilen und Anwendungsmöglichkeiten dieser Unsicherheitsbeschreibung. Abstract Treating uncertainties properly is a difficult but necessary task in data fusion applications. In order to meet this challenge, appropriate uncertainty models are required and efficient state estimation techniques need to be derived. The most common approaches are to model uncertain quantities as random variables or as bounded sets. This paper presents a state estimation framework that allows for a simultaneous treatment of stochastic and unknown but bounded uncertainties. This is achieved by characterizing an uncertain quantity by a set of probability densities. Finally, promising applications of the derived concept are described.