Motion planning for a multi-robotic system refers to finding trajectories for each robot in a team so that a certain task is performed. In general, this problem is under-determined because of two main reasons. First, the tasks are usually specified in terms of reaching a final position starting from a given initial one. Mathematically, this translates to generating interpolating curves with given boundary conditions in a certain configuration space, problem which usually admits several solutions. One of the main ideas of this dissertation is that a natural way to solve this indeterminacy is to find a solution which is optimal with respect to a performance criterion, e.g., energy consumption. Second, especially in the case when the team is composed of large numbers of robots, the task might be specified in high level terms of the type "the robots should gather in a certain region of the space". Explicitly generating individual trajectories, though feasible, is highly under-determined and computationally unattractive. The second main idea of the dissertation is that, in this case, the motion generation and control problems should be solved in a lower dimensional space which captures the behavior of the group and the nature of the cooperative task.
First, we consider the problem of generating minimum kinetic energy motion for a rigid body in a 3D environment. We develop a computationally efficient method for interpolation on SE(3) that produces nearly optimal trajectories, which are also invariant to changes in the reference frame. Second, we study the rigidity condition and develop a method of optimal motion planning for groups of robots required to maintain a rigid formation. In the third part of this work, we propose a method to control a large number of agents based manifold, which has a product structure of a Lie group, which captures the dependence of the ensemble on the world frame, and a shape manifold, which is an intrinsic description of the team. Illustrative experimental results are included.
