Theory 

The methodology of control theory is to begin with a practical problem; to abstract the central issues and formulate an idealized, hypothetical problem; to develop, if necessary, new mathematical methods for its solution; and to work out a rigorous solution. Then one has a framework on which to do real applications. 

We (my students, co-supervisors, and I) began by formulating the hypothetical problem of rendezvous for point robots. That is, the robots are modelled as kinematic points in the plane, and the objective is for them to meet at a common location. This is entirely analogous to birds flying in a flock; the heading angles are all equal. How do birds do this, and how do fish swim in schools? The rendezvous problem is a challenge because there are no leaders, the robots do not have a common map and preferably should have identical stored programs, and there can be no human intervention. The robots are assumed to have onboard cameras, motors, and microcomputers, but no GPS or other global sensor.  

We began with the simple strategy of cyclic pursuit: The robots are numbered 1 to n, and robot i pursues robot i+1 (robot n pursues robot 1). Using the theory of circulant matrices, we proved that cyclic pursuit solves the rendezvous problem. The theory of robot formations relies heavily on graph theory. The graph is defined by having a node for each robot and a link from node i to node j if robot i can see robot j. In general the graph is directed. Beautiful mathematics by Perron and Frobenius (1907, 1912) is relevant here. We derived a new control-theoretic result: a necessary and sufficient condition on the graph for rendezvous to be achievable.

Of course, a kinematic point in the plane is not close to a real wheeled rover. Closer models are the unicycle and the bicycle. We studied the cyclic pursuit strategy applied to unicycles. We found that stable formations could be achieved, where the unicycles are moving in a circle in the same direction. We confirmed this experimentally using lab rovers made in-house.

The rendezvous problem itself has limited application; its purpose is to clarify the visibility requirements, i.e., who needs to see whom, in a hypothetical case. (But there actually are a few practical applications. When underwater robotic rovers that are mapping temperature gradients need energy charging, the most efficient way is to have them rendezvous.) We moved to the problem of formation control, for example getting four rovers to form a square, derived control laws, proved stability of the formation, and verified by experiments on real rovers.

In our major theoretical work we studied the rendezvous problem when the robots can see only a fixed distance away. We gave the first mathematical proof of an algorithm for this case. This and the preceding theoretical results were published in 10 journal papers, including 8 in the top journals in control (IEEE Trans. Auto. Control, Automatica, SIAM J. Control and Opt).

My most recent work, on the stability of formations of vehicles, is with Abie Feintuch. In studying the formation of a very large number of vehicles, one approach is instead to model an infinite number of vehicles. The question then arises as to what mathematical framework to take so that the latter model correctly describes the behaviour of the former. We argue that the usual Hilbert space framework is not the right one, because an infinite chain does not behave like a finite but large one. For example, formulating the rendezvous problem in Hilbert space results in a rendezvous at the origin, whereas a finite chain would not do so. The Banach space formulation seems more appropriate in our opinion. However, the problems are harder. Designing controllers that are optimal is an open problem.

Application 

During 2007-2009 we conducted a major application of our theory in collaboration with Professor Tim Barfoot of the University of Toronto Aerospace Institute and Jared Giesbrecht of Defense Research and Development Canada (DRDC), Suffield, Alberta. Motivating this research is a military situation in which a manned vehicle convoy traverses hostile territory to deliver supplies. Naturally, protecting the soldiers is important, but equipping every vehicle in the convoy with heavy armour is expensive. To reduce the cost and the number of soldiers required, autonomous unarmoured supply vehicles may be used, whereby each supply vehicle would autonomously follow the trajectory of its immediate leader. With this setup, the vehicle convoy would be comprised of autonomous unarmoured vehicles and manually-driven armoured vehicles. To follow its immediate leader, an autonomous vehicle can sometimes take advantage of a global positioning system (GPS), inter-vehicle communications, and/or lane markers/magnets. However, since the vehicle convoy is in hostile territory, GPS signals may be jammed, inter-vehicle communications may be intercepted, and the roads may be unstructured.

Based on this motivating example, we designed and tested a vehicle-following system to allow a convoy of full-sized autonomous vehicles with large inter-vehicle spacing to follow a manually-driven lead vehicle's trajectory without cutting corners on turns. Our testing was done on MultiAgent Tactical Sentry (MATS) vehicles that were provided by DRDC. Since there are no inter-vehicle communications to relay the lead vehicle's position, the goal of an autonomous follower is to track the trajectory of its immediate leader. The resulting design and field trials were successful enough to appear in the premier journal on field robotics.