A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. Finally, we show that the discrete equations are a meaningful discretisation of the continuous ones.Comment: Key words: Variational analysis, Mechanical systems, Lagrangian mechanics, Damping, Fractional derivatives, Discretisation, Variational integrators. 13 pages, no figures. Contributed paper to 6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Contro

Learning discrete Lagrangians for variationalPDEs from data and detection of travelling waves

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE

Optimal Control for a Pitcher's Motion Modeled as Constrained Mechanical System

In this contribution, a recently developed optimal control method for constrained mechanical systems is applied to determine optimal motions and muscle force evolutions for a pitcher's arm. The method is based on a discrete constrained version of the Lagrange-d'Alembert principle leading to structure preserving time-stepping equations. A reduction technique is used to derive the nonlinear equality constraints for the minimization of a given objective function. Different multi-body models for the pitcher's arm are investigated and compared with respect to the motion itself, the control effort, the pitch velocity, and the pitch duration time. In particular, the use of a muscle model allows for an identification of limits on the maximal forces that ensure more realistic optimal pitch motions

Discrete mechanics and optimal control: An analysis

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated

Optimal Reconfiguration of Formation Flying Spacecraft--a Decentralized Approach

This paper introduces a hierarchical, decentralized, and parallelizable method for dealing with optimization problems with many agents. It is theoretically based on a hierarchical optimization theorem that establishes the equivalence of two forms of the problem, and this idea is implemented using DMOC (Discrete Mechanics and Optimal Control). The result is a method that is scalable to certain optimization problems for large numbers of agents, whereas the usual “monolithic” approach can only deal with systems with a rather small number of degrees of freedom. The method is illustrated with the example of deployment of spacecraft, motivated by the Darwin (ESA) and Terrestrial Planet Finder (NASA) missions

High order variational integrators in the optimal control of mechanical systems

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.Comment: 25 pages, 9 figures, 1 table, submitted to DCDS-