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Register a new userMarkov Chain-Based Stochastic Strategies for Robotic Surveillance
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Added by
Xiaoming Duan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This article surveys recent advancements of strategy designs for persistent robotic surveillance tasks with the focus on stochastic approaches. The problem describes how mobile robots stochastically patrol a graph in an efficient way where the efficiency is defined with respect to relevant underlying performance metrics. We first start by reviewing the basics of Markov chains, which is the primary motion model for stochastic robotic surveillance. Then two main criteria regarding the speed and unpredictability of surveillance strategies are discussed. The central objects that appear throughout the treatment is the hitting times of Markov chains, their distributions and expectations. We formulate various optimization problems based on the concerned metrics in different scenarios and establish their respective properties.
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This paper studies a stochastic robotic surveillance problem where a mobile robot moves randomly on a graph to capture a potential intruder that strategically attacks a location on the graph. The intruder is assumed to be omniscient: it knows the current location of the mobile agent and can learn the surveillance strategy. The goal for the mobile robot is to design a stochastic strategy so as to maximize the probability of capturing the intruder. We model the strategic interactions between the surveillance robot and the intruder as a Stackelberg game, and optimal and suboptimal Markov chain based surveillance strategies in star, complete and line graphs are studied. We first derive a universal upper bound on the capture probability, i.e., the performance limit for the surveillance agent. We show that this upper bound is tight in the complete graph and further provide suboptimality guarantees for a natural design. For the star and line graphs, we first characterize dominant strategies for the surveillance agent and the intruder. Then, we rigorously prove the optimal strategy for the surveillance agent.
Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy is the weighted average, over all graph nodes, of the entropy of the first return times of the Markov chain; this objective function is a function series that does not admit in general a closed form. The paper features theoretical and computational contributions. First, we obtain a discrete-time delayed linear system for the return time probability distribution and establish its convergence properties. We show that the objective function is continuous over a compact set and therefore admits a global maximum; a unique globally-optimal solution is known only for complete graphs with unitary travel times. We then establish upper and lower bounds between the return time entropy and the well-known entropy rate of the Markov chain. To compute the optimal Markov chain numerically, we establish the asymptotic equality between entropy, conditional entropy and truncated entropy, and propose an iteration to compute the gradient of the truncated entropy. Finally, we apply these results to the robotic surveillance problem. Our numerical results show that, for a model of rational intruder over prototypical graph topologies and test cases, the maximum return time entropy chain performs better than several existing Markov chains.
The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for minimizing Lipschitz differentiable functions, which can be nonconvex. When the functions are convex and strongly convex, we establish both sublinear and linear convergence rates, respectively. We also present a method of Markov chain inertial BCD. Finally, we discuss potential applications.
This paper investigates the visual servoing problem for robotic systems with uncertain kinematic, dynamic, and camera parameters. We first present the passivity properties associated with the overall kinematics of the system, and then propose two passivity-based adaptive control schemes to resolve the visual tracking problem. One scheme employs the adaptive inverse-Jacobian-like feedback, and the other employs the adaptive transpose Jacobian feedback. With the Lyapunov analysis approach, it is shown that under either of the proposed control schemes, the image-space tracking errors converge to zero without relying on the assumption of the invertibility of the estimated depth. Numerical simulations are performed to show the tracking performance of the proposed adaptive controllers.
This paper analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate optimization problems and design robotic strategies. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide the first closed-form expression for the expected meeting time in terms of the transition matrices of the moving agents. This novel expression leads to necessary and sufficient conditions for the meeting time to be finite and to insightful graph-theoretic interpretations. Second, based on the closed-form expression, we setup and study the minimization problem for the expected capture time for a pursuer/evader pair. We report theoretical and numerical results on basic case studies to show the effectiveness of the design.
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