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A new approach for trajectory optimization of musculoskeletal dynamic models is introduced. The model combines rigid-body and muscle dynamics described with a Hill-type model driven by neural control inputs. The objective is to find input and state trajectories that are optimal with respect to a minimum-effort objective and meet constraints associated with musculoskeletal models. The measure of effort is given by the integral of pairwise average forces of the agonist-antagonist muscles. The concepts of flat parameterization of nonlinear systems and sum-of-squares optimization are combined to yield a method that eliminates the numerous set of dynamic constraints present in collocation methods. With terminal equilibrium, optimization reduces to a feasible linear program, and a recursive feasibility proof is given for more general polynomial optimization cases. The methods of the article can be used as a basis for fast, and efficient solvers for hierarchical, and receding-horizon control schemes. Two simulation examples are included to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

This article presents a computational framework to efficiently generate feasible and safe trajectories for multiple autonomous vehicle operations. We formulate the optimal motion planning problem as a continuous-time optimal control problem, and approximate its solutions in a discretized setting using Bernstein polynomials. The latter possess convenient properties that allow to efficiently compute and enforce constraints along the vehicles’ trajectories, such as maximum speed and angular rates, minimum distance between trajectories and between the vehicles and known obstacles, etc. Thus, the proposed method is particularly suitable for generating trajectories in real-time for safe operations in complex environments and multiple vehicle missions. We show, using a rigorous mathematical framework, that the solution to the discretized optimal motion planning problem converges to that of the continuous-time one. The advantages of the proposed method are investigated through numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses a one-dimensional optimal persistent monitoring problem using second-order agents. The goal is to control the movements of agents to minimize a performance metric associated with the environment (targets) over a finite time horizon. In contrast to earlier results limited to first-order dynamics for agents, we control their accelerations rather than velocities, thus leading to a better approximation of agent behavior in practice and to smoother trajectories. Bounds on both velocities and accelerations are also taken into consideration. Despite these added complications to agent dynamics, we derive a necessary condition for optimality and show that the optimal agent trajectories can be fully characterized by two parameter vectors. A gradient-based algorithm is proposed to optimize these parameters and yield a minimal performance metric. In addition, a collision avoidance algorithm is proposed to solve potential collision and boundary-crossing problems, thus extending the gradient-based algorithm solutions. Finally, simulation examples are included to demonstrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation. [ABSTRACT FROM AUTHOR]

Hybrid optimal control problems are studied for a general class of hybrid systems, where autonomous and controlled state jumps are allowed at the switching instants, and in addition to terminal and running costs, switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework, and it is shown that under certain assumptions, the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

Recent research has established the importance of (strict) dissipativity for proving stability of economic MPC in the case of an optimal steady state. In many cases, though, steady-state operation is not economically optimal and periodic operation of the system yields a better performance. In this technical note, we propose ways of extending the notion of (strict) dissipativity for periodic systems. We prove that optimal $P$-periodic operation and MPC stability directly follow, similarly to the steady-state case, which can be seen as a special case of the proposed framework. Finally, we illustrate the theoretical results with several simple examples. [ABSTRACT FROM AUTHOR]