Karthik Elamvazhuthi

An Optimal Control Approach to Mapping GPS-Denied Environments Using a Stochastic Robotic Swarm

This paper presents an approach to mapping a region of interest using observations from a robotic swarm without localization. The robots have local sensing capabilities and no communication, and they exhibit stochasticity in their motion. We model the swarm population dynamics with a set of advection-diffusion-reaction partial differential equations (PDEs). The map of the environment is incorporated into this model using a spatially-dependent indicator function that marks the presence or absence of the region of interest throughout the domain. To estimate this indicator function, we define it as the solution of an optimization problem in which we minimize an objective functional that is based on temporal robot data. The optimization is performed numerically offline using a standard gradient descent algorithm. Simulations show that our approach can produce fairly accurate estimates of the positions and geometries of different types of regions in an unknown environment.

Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms

This paper presents a novel procedure for computing parameters of a robotic swarm that guarantee coverage performance by the swarm within a specified error from a target spatial distribution. The main contribution of this paper is the analysis of the dependence of this error on two key parameters: the number of robots in the swarm and the robot sensing radius. The robots cannot localize or communicate with one another, and they exhibit stochasticity in their motion and task-switching policies. We model the population dynamics of the swarm as an advection-diffusion-reaction partial differential equation (PDE) with time-dependent advection and reaction terms. We derive rigorous bounds on the discrepancies between the target distribution and the coverage achieved by individual-based and PDE models of the swarm. We use these bounds to select the swarm size that will achieve coverage performance within a given error and the corresponding robot sensing radius that will minimize this error. We also apply the optimal control approach from our prior work in [13] to compute the robots’ velocity field and task-switching rates. We validate our procedure through simulations of a scenario, in which a robotic swarm must achieve a specified density of pollination activity over a crop field.

Decentralized stochastic control of robotic swarm density: Theory, simulation, and experiment

This paper explores a stochastic approach for controlling swarms of independent robots toward a target distribution in a bounded domain. The robot swarm has no central controller, and individual robots lack both communication and localization capabilities. Robots can only measure a scalar field (e.g. concentration of a chemical) from the environment and from this deduce the desired local swarm density. Based on this value, each robot follows a simple control law that causes the swarm as a whole to diffuse toward the target distribution. Using a new holonomic drive robot, we present the first confirmation of this control law with physical experiment. Despite deviations from assumptions underpinning the theory, the swarm achieves the theorized convergence to the target distribution in both simulation and experiment. In fact, simulated and experimental performance agree with one another and with our hypothesis that the error from the target distribution is inversely proportional to the square root of the number of robots. This is evidence that the algorithm is both practical and easily scalable to large swarms.

Optimal transport over nonlinear systems via infinitesimal generators on graphs

We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.

Optimal perturbations for nonlinear systems using graph-based optimal transport

Abstract We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge–Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

Scalable Formation Control of Multi-robot Chain Networks Using a PDE Abstraction

This work investigates the application of boundary control of the wave equation to achieve leader-induced formation control of a multi-robot network with a chain topology. In contrast to previous related work on controlling formations of single integrator agents, we consider a model for double integrator agents. For trajectory planning, we use the flatness based method for assigning trajectories to leader agents so that the agents’ trajectories and control inputs are computed in a decentralized way. We show how the approximation greatly simplifies the planning problem and the resulting synthesized controls are bounded and independent of the number of agents in the network. We validate our formation control approach with simulations of 100 and 1000 agents that converge to configurations on three different type of target curves.

Confinement control of double integrators using partially periodic leader trajectories

We consider a multi-agent confinement control problem in which a single leader has a purely repulsive effect on follower agents with double-integrator dynamics. By decomposing the leaders control inputs into periodic and aperiodic components, we show that the leader can be driven so as to guarantee confinement of the followers about a time-dependent trajectory in the plane. We use tools from averaging theory and an input-to-state stability type argument to derive conditions on the model parameters that guarantee confinement of the followers about the trajectory. For the case of a single follower, we show that if the follower starts at the origin, then the error in trajectory tracking can be made arbitrarily small depending on the frequency of the periodic control components and the rate of change of the trajectory. We validate our approach using simulations and experiments with a small mobile robot.