Optimal Impulsive Control of Structure-Variant Rigidbody Mechanical Systems
The optimal control of rigidbody dynamical systems with discontinuous states is a largely non-investigated area. Optimal control of impulsive systems inevitably entails optimal control with discontinuous states. Possible areas of application involve legged locomotion, manipulators with blockable degrees of freedom (DOF), aerospace applications where system parameters such as inertial parameters change discontinuously, robotic applications that involve contacts such as grasping.
In this work, necessary conditions for the optimal control of such rigid-body systems will be stated. In addition, a method is introduced that locates the optimal primal and dual arcs in space of bounded variation functions. Contrary to the approach taken in literature so far, instead of taking an interval opening approach, the instant of discontinuity is reduced to an instant with Lebesgue measure zero.
The approach requires the different system modes and their order to be specified in advance. The necessary conditions obtained, enable the determination the optimal transition time and location. For the underlying non-convex problem the given conditions can only propose the candidates for extremizers, for the
certificate of sufficiency further work has to be conducted.
By the application of subdifferential calculus techniques to lower semicontinuous functionals, Pontryagin’s Maximum Principle (PMP) like conditions are obtained. The considered functional will be a generalized Bolza functional that is evaluated on multiple intervals. The well-known PMP entails the necessary conditions for optimal control problems with differential constraints and end-point constraints with sufficient regularity properties in the space of absolutely continuous arcs. However, impulsive optimal control requires to search extremizing arcs in the space of bounded variation functions. So the obtained necessary conditions will entail PMP conditions under mild hypotheses.
Another issue is the representation of mechanical system dynamics in different modes such that it enables the formulation of necessary conditions for the whole problem. Inevitably this requires addressing the question of modeling the rigidbody mechanical system dynamics. A method that relies on projecting the mechanical dynamics into subspaces without additionally introducing algebraical constraints and generalized coordinates will be introduced. As can be seen, the sources of impacts are manifold. Recent research showed that such rigid-body systems can best be described by variational inequalities which lead to nonlinear and linear complementarity type of systems to be treated in order to obtain the accelerations/velocities and forces. The main concern in the research so far has been to determine the forces and accelerations/velocities for autonomous/uncontrolled mechanical systems. In the modeling framework considered in this work, impulsive forces can arise autonomously, due to contact interactions such as collisions or controlled/nonautonomously, due to actions as blocking of DOF of manipulators impactively. The introduced framework will have the ability to model and control of hybrid mechanical systems with discontinuous transitions among different system modes. Some numerical and simulational results are presented. The applicability of the obtained results to other Lagrangian systems in physics will be indicated.