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Guaranteed Collision Avoidance for Autonomous Systems with Acceleration Constraints and Sensing Uncertainties

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Abstract

A set of cooperative and noncooperative collision avoidance strategies for a pair of interacting agents with acceleration constraints, bounded sensing uncertainties, and limited sensing ranges is presented. We explicitly consider the case in which position information from the other agent is unreliable, and develop bounded control inputs using Lyapunov-based analysis, that guarantee collision-free trajectories for both agents. The proposed avoidance control strategies can be appended to any other stable control law (i.e., main control objective) and are active only when the agents are close to each other. As an application, we study in detail the synthesis of the avoidance strategies with a set-point stabilization control law and prove that the agents converge to the desired configurations while avoiding collisions and deadlocks (i.e., unwanted local minima). Simulation results are presented to validate the proposed control formulation.

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Notes

  1. In what follows, we will assume that the control objective for the \(i\hbox {th}\) and \(j\hbox {th}\) vehicles is to converge to a desired configuration \({\mathbf {q}}_i^o\in {\mathbb {R}}^n\) and \({\mathbf {q}}_j^o\in {\mathbb {R}}^n\), respectively. See Sect. 3 for details.

  2. In real applications, it is advised or desirable to enforce a minimum distance between the vehicle and any obstacle larger than \(r^*\). In addition, it is a standard practice whenever position information is not updated continuously [46].

  3. Other bounded stable control laws can be chosen. The main results on collision avoidance (i.e., Theorems 4.1 and 4.2) are independent of the objective control.

  4. For the most part of this paper, we will choose \(\mu _i^o = \bar{\mu }_i^o\). However, in Sect. 5 we will make the distinction between both parameters by choosing \(\mu _i^o = \bar{\mu }_i^o - \bar{\mu }_i^L\), where \(\bar{\mu }_i^L \in ]0,\bar{\mu }_i^o[\).

  5. We can always choose design parameters \(R_i=R_j\) and \(h_i=h_j\) such that the assumption holds.

  6. Herein, we make reference to LaSalle’s Invariance Principle for nonsmooth systems developed in [49], which only requires \(\dot{{\xi }}\) to be Lebesgue measurable, or equivalently, \({\xi }\) to be absolutely continuous. Moreover, note that our two-agent system under no sensing uncertainties is time-invariant.

  7. Note that the results in Sect. 4 (i.e., Lemma 4.1 and Theorems 4.1 and 4.2) still hold since \(\left\| {\mathbf {u}}^o_i+{\mathbf {u}}^L_i\right\| \le \bar{\mu }_i^o\) and, therefore, we can arrive to the same conclusions on collision avoidance by replacing \({\mathbf {u}}^o_i\) with \({\mathbf {u}}^o_i+{\mathbf {u}}^L_i\).

  8. Note that \({\mathbf {d}}_i\) and \({\mathbf {d}}_j\) are piecewise continuous and time-invariant. Therefore, \({\mathbf {u}}_i^a\) and \({\mathbf {u}}_j^a\) are also bounded, time-invariant, piecewise continuous vector functions, which allow us to apply LaSalle’s Invariance Principle for nonsmooth systems [49].

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Acknowledgments

This work was partially supported by The Boeing Company via the Information Trust Institute, University of Illinois at Urbana–Champaign, and by NSF Grant ECCS-0725433.

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Authors and Affiliations

  1. Department of Weapons and Systems Engineering, U.S. Naval Academy, Annapolis, MD, 21402, USA

    Erick J. Rodríguez-Seda

  2. Coordinated Science Laboratory, University of Illinois, Urbana, IL, 61801, USA

    Dušan M. Stipanović

  3. School of Engineering and Computer Science, University of Texas, Dallas, TX, 75080, USA

    Mark W. Spong

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  1. Erick J. Rodríguez-Seda
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  2. Dušan M. Stipanović
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  3. Mark W. Spong
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Correspondence to Erick J. Rodríguez-Seda.

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Communicated by Martin Corless.

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Rodríguez-Seda, E.J., Stipanović, D.M. & Spong, M.W. Guaranteed Collision Avoidance for Autonomous Systems with Acceleration Constraints and Sensing Uncertainties. J Optim Theory Appl 168, 1014–1038 (2016). https://doi.org/10.1007/s10957-015-0824-7

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