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Stability Analysis

The subject of multiple cooperative autonomous vehicles has generated a great deal of interest in recent years due to the vision of these vehicles being able to perform tasks faster and more efficiently than an individual vehicle.  Such tasks can include operation in hazardous or remote environments performing repetitive, dangerous, or information gathering duties.  Recent work has taken many different approaches.  The strategies employed are based on diverse fields such as artificial intelligence, game theory, biology, distributed control, and genetic algorithms.  Because we are interested in proven convergence and stability of algorithms, we tend to take a more theoretical approach to their control.  In particular, we have been investigating using large-scale system control theory to develop provably convergent control algorithms.  The example below shows some of the progress made in understanding how these techniques can be used in the design of large-scale distributed cooperative robotic vehicular systems.

Consider a simple one-dimensional problem where a linear chain of interdependent vehicles are to spread out along a line as shown in Figure 1.  The objective is to spread out evenly along the line using only information from the nearest neighbor.  Sandia has previously developed a robotic perimeter detection system that spread the vehicles around a perimeter using one-half the distance between the neighboring two vehicles are the goal point for each vehicle.   In this stability analysis, we were interested in finding out if one-half was a magic number and if we could prove that it provides a stable solution.

Figure 1.  One-dimensional control problem. The top line is the initial state. The second line is the desired final state. The vehicles can only use their neighbors' position to reach the final goal state.

Figure 2.  N vehicle interaction problem.

Using a discrete time model of the individual vehicles with the output of each vehicle being broadcast to it's two closest neighboring vehicles (see Figure 2), we can show that the overall system stability depends on

1. the number of vehicles
2. the interaction gain between the vehicles
3. the responsiveness of each vehicle, and
4. the communication sample time between vehicles. 

Using vector Lyapunov techniques, we can show that the entire group of vehicles is stable if these parameters are within the red regions shown in Figures 3 and 4. Because of the simplicity of this problem, we have also show that the stability region in Figure 4 is approximately the same for an infinite number of vehicles.

Figure 3. Stability region for the N=2 vehicle case. The x-axis is the interaction gain between the vehicles. The y-axis is the product of vehicle's responsiveness and the communication sample period.

Figure 4.  Stability region for the N=10000 vehicle case. The x-axis is the interaction gain between the vehicles.  he y-axis is the product of vehicle's responsiveness and the communication sample period.