This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Coverage control for mobile sensing networks

A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Variational Principles in Mechanics

H. J. Kushner and P. G. Dupuis. Springer-Verlag, New York/Heidelberg, May 1992. 439 pp., DM 98. ISBN 3-540-97834-8

Numerical methods for stochastic control problems in continuous time

Games and Decisions. By R. Duncan Luce and Howard Raiffa. Pp. xi, 509. 70s. 1957. (J Wiley & Sons)

In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work, we have shown that by utilizing the state feedback control policy parametrization, this stochastic optimal control problem can be associated with a difference of convex functions program. Here, we revisit the same covariance control problem but this time we focus on the analysis of the problem. Specifically, we establish the existence of solutions to the optimization problem and derive the first and second order conditions for optimality. We provide analytic expressions for the gradient and the Hessian of the performance index by utilizing specialized tools from matrix calculus. Subsequently, we prove that the optimization problem always admits a global minimizer, and finally, we provide a sufficient condition for the performance index to be a strictly convex function (under the latter condition, the problem admits a unique global minimizer). In particular, we show that when the terminal state covariance is upper bounded, with respect to the L\"{o}wner partial order, by the covariance matrix of the desired terminal normal distribution, then our problem admits a unique global minimizing state feedback gain. The results of this paper set the stage for the development of specialized control design tools that exploit the structure of the solution to the covariance steering problem with a squared Wasserstein distance terminal cost.

On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost.

In this paper, we propose a novel optimization-based trajectory planner that utilizes spherical harmonics to estimate the collision-free solution space around an agent. The space is estimated using a constrained over-determined least-squares estimator to determine the parameters that define a spherical harmonic approximation at a given time step. Since spherical harmonics produce star-convex shapes, the planner can consider all paths that are in line-of-sight for the agent within a given radius. This contrasts with other state-of-the-art planners that generate trajectories by estimating obstacle boundaries with rough approximations and using heuristic rules to prune a solution space into one that can be easily explored. Those methods cause the trajectory planner to be overly conservative in environments where an agent must get close to obstacles to accomplish a goal. Our method is shown to perform on-par with other path planners and surpass these planners in certain environments. It generates feasible trajectories while still running in real-time and guaranteeing safety when a valid solution exists.

Collision Avoidance Using Spherical Harmonics.

On the effect of manipulating Large Scale Motions in a Boundary Layer

In this paper, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observable functions (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observable functions that can span the lifted space. Recent methods either use tools from machine learning to learn the observable functions or guess a suitable set of observables that best describes the nonlinear dynamics. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observable functions converges pointwise to the zero function, which allows us to select only a finite set of observable functions to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.

Koopman Operator Based Modeling for Quadrotor Control on $SE(3)$

In this paper, we present a game-theoretic feedback terminal guidance law for an autonomous, unpowered hypersonic pursuit vehicle that seeks to intercept an evading ground target whose motion is constrained along an axis. We formulate this problem as a pursuit-evasion game whose saddle point solution is in general difficult to compute onboard the hypersonic vehicle due to its highly nonlinear dynamics. To overcome this difficulty, we linearize the nonlinear hypersonic dynamics around a reference trajectory and subsequently utilize feedback control design techniques from Linear Quadratic Differential Games (LQDGs). In our proposed guidance algorithm, the hypersonic vehicle computes its open-loop optimal state and input trajectories off-line and prior to the commencement of the game. These trajectories are then used to linearize the nonlinear equations of hypersonic motion. Subsequently, using this linearized system model, we formulate an auxiliary two-player zero-sum LQDG which is effective in the neighborhood of the given reference trajectory and derive its feedback saddle point strategy that allows the hypersonic vehicle to modify its trajectory online in accordance with the changes/dispersion to the target's position due to its evasive maneuvers. We provide numerical simulations to verify the performance of our proposed guidance law.

Efstathios Bakolas

Papers

On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost.

Collision Avoidance Using Spherical Harmonics.

On the effect of manipulating Large Scale Motions in a Boundary Layer

Koopman Operator Based Modeling for Quadrotor Control on $SE(3)$

Feedback Strategies for Hypersonic Pursuit of a Ground Evader.