This paper addresses the finite-time formation tracking problem for multiple vehicles with dynamics model on SE(3) (the specific Euclidean group of rigid body motions), under the condition that the tracking time is preassigned according to the task requirements. By using Pontryagin's maximum principle on Lie groups, a class of finite-time optimal tracking control laws are designed for vehicles to track a desired trajectory within a given finite time. Meanwhile, the corresponding cost function is minimized. Furthermore, a tracking-time lower bound is derived for multi-vehicle systems with control constraints. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed control laws.
FORMATION TRACKING CONTROL FOR MULTIPLE VEHICLES
3131In all the aforementioned works, a common feature is the choice of a parameterization, where three-parameter representations of the attitude, such as Euler angles, modified Rodriguez parameters, or unit quaternions (four-parameter), are used to describe the configuration manifold. Parameterization methods convert the configuration space from nonlinear to normal Euclidean by identifying some different velocity spaces to the same Euclidean space. For stability and tracking problems, these methods provide good local approximations. However, for rigid-body systems that cannot be considered locally in any neighborhood, these methods present difficulties for different rigid bodies to keep the rigid formation when the nonlinear trajectory-tracking problem is considered. Furthermore, these methods cause singularities or ambiguities, as there is no three-parameter representation for the Lie group SO(3), which is global without singularities [21]. Quaternions do not have singularities, but they have ambiguities in representing the attitude, as the three-sphere S 3 doubly covers SO(3) [3]. Besides, considering that computations for rigid bodies performed with different choices of coordinates will produce different results, it is desirable to take into account the geometric structure of the configuration space of the rigid bodies and work with it directly.The tracking problem of rigid bodies on a nonlinear manifold, like the group SE(3) of rigid-body motions, has also been studied extensively. Some pioneering works can be found in [3,4,[22][23][24][25], where the asymptotical tracking control laws were designed and in which the obtained results were coordinate-free. The trajectory tracking results were derived from the general Riemannian framework to Lie groups in [4,22]. Note that this approach may fail to fully exploit the additional structure, which is available in Lie groups. Thus, many researchers studied the tracking problem of rigid bodies on Lie groups [3,[23][24][25]. Here, the velocity error is obtained by using coordinate transformations from one velocity to another, which is provided by the algebraic structure of Lie groups and the symmetry of the vector field on Lie groups. Considering that SE(3) is a matrix Lie group, the ordinary operations, such a...