Lyapunov-Based Cooperative Avoidance Control for Multiple Lagrangian Systems with Bounded Sensing Uncertainties Erick J. Rodr´ ıguez-Seda, Duˇ san M. Stipanovi´ c, and Mark W. Spong Abstract— We present a decentralized, real-time, cooperative avoidance control law for a group of nonlinear Lagrangian systems with bounded control inputs, limited sensing ranges, and bounded sensing errors. The control formulation builds on the concept of avoidance control and uses Lyapunov-based analysis to guarantee collision-free trajectories for a group of N vehicles with sensing uncertainties. Advantages of the cooperative avoidance strategy include its easy synthesis with other stable control laws and its null effect on the agent’s main task when other vehicles and obstacles are sufficiently away. Two numerical examples are finally presented that illustrate the performance of the proposed control framework. I. I NTRODUCTION One of the most critical challenges in multi-vehicle sys- tems is to guarantee collision avoidance between neighboring agents and obstacles at all times independently of sensing errors. Unmanned vehicles and mobile robots typically rely on navigation and localization sensors to estimate the dis- tance to nearby agents and obstacles or on wireless commu- nication networks for the broadcast of position coordinates among agents. These sensing mechanisms, in which we include communication networks, may inaccurately estimate the position of obstacles and agents as a result of process delays, interferences, noise, and quantization. For instance, obstacle’s position measurements sampled by vision-based sensing mechanisms on many mobile robotic systems are easily affected by weather conditions and light variations [1]. Similarly, underwater localization equipment on board of unmanned vehicles, such as sonar radars and inertial measurements units, may also experience substantial delays, slow sampling rates, and dead reckoning errors [2]. If these estimation errors are not carefully considered when con- trolling the motion of the vehicle, the system may become vulnerable to collisions. Therefore, avoidance strategies for autonomous navigation must provide robustness to sensing uncertainties. Collision avoidance strategies coping with sensing uncer- tainties have been predominantly studied within the field of path planning, where a complete obstacle-free path from the agent’s current location to the next target is developed based on estimates of the initial position of obstacles. Examples include the certainty [3] and occupancy grid [4], where the This research was partially supported by the National Science Foundation Grant ECCS 07-25433 and by the University of Texas at Dallas. E. J. Rodr´ ıguez-Seda and M. W. Spong are with the School of Engineer- ing and Computer Science, University of Texas, Dallas, TX 75080, USA. erodriguez@utdallas.edu, mspong@utdallas.edu D. M. Stipanovi´ c is with the Coordinated Science Laboratory, Uni- versity of Illinois, 1308 W. Main St., Urbana, IL 61801, USA. dusan@illinois.edu robot’s environment is divided into an array of cells with each cell containing a probability of having an obstacle. Then, a safe path, which the robot is meant to follow, is traced according to this probability map. Although these control strategies have been shown to be robust to common sensor uncertainties, they require other agents and obstacles to be static or to move at low speeds such that the agent’s initial sensing observation remains true throughout the entire trajectory. An alternate approach with a similar drawback is proposed in [5], where a noncooperative collision avoidance strategy based on the concept of reachable sets [6] is de- scribed for zero-velocity obstacles. In contrast to path planning algorithms, real-time collision avoidance strategies compute the avoidance control inputs online as obstacles are detected, therefore, facilitating (in most cases) the treatment of fast-moving obstacles. Real- time collision avoidance algorithms considering sensing un- certainties have been introduced in [7] and [8] based on a variation of the occupancy grid [4] that incorporates estimates of the obstacles’ velocities. Yet, these previous control approaches do not fully investigate the case of time- varying speed obstacles and assume the worst case scenario in which other agents do not try to avoid a collision (i.e., a noncooperative strategy). In [9], a decentralized real-time avoidance strategy for the case of two agents with double integrator dynamics and bounded control inputs is presented using Lyapunov-based analysis. However, the theoretical results are not extended to the general case of multiple nonlinear agents. In this paper, we now introduce a decentralized, real- time, cooperative avoidance control strategy for a group of heterogeneous nonlinear Lagrangian systems with bounded control inputs and limited sensing. The collision avoidance control formulation is based on the concept of avoidance control [10], [11], yet the avoidance functions and control inputs proposed herein are bounded. The overall control framework is able to cope with bounded sensing errors (including those caused by delays, noise, and quantization) by treating the effect of uncertainties as a disturbance in the control input, similar to [12]. However, the control for- mulation in [12] does not guarantee robustness with respect to sensing uncertainties and assumes unbounded control inputs. Advantages of the proposed controller also include the activation of the avoidance control only when the vehicle is close to another agent and the relative easy synthesis with other stable control laws. By using Lyapunov-based analysis we are able to present sufficient conditions that guarantee collision-free transit for a group of N nonlinear agents.