Keywords: structures, dynamics, target states, mechanisms, robotics, biomechanics, finite element approximation.

This paper discusses a temporal finite element approximation in the analysis of dynamics of mechanical systems, with a special emphasis on problems where a targeted control is desired. This is defined as a situation where forces are to be introduced for the movement of a structure from an initial to one or more target states, but where the behavior between these states is arbitrary. The primary applications are related to bio-mechanical simulations of skeleto-muscular systems, or to robotic analyses. By interpolating simultaneously displacements and velocities in the discrete degrees of freedom, a collocation over the time interval can be used to decide the necessary control variations. As a second step, the control can be optimized for chosen criteria on the integrated force components. By the introduced interpolation of control forces and discrete displacements, a degree of continuity is introduced in the obtained results.

The presentation focusses on the similarity in computational formulations between several types of dynamic simulations, and sets them in a common algorithmic context. The temporal descriptions of all discrete displacement components are thereby based on a Hermitian finite element form, where each variable is represented by its value and its time differential at a set of discrete time stations. All displacement variables are thereby represented as piecewise cubic polynomials.

Using the basic equilibrium for the stated problem, and introducing the temporal interpolation of the variables, a finite element form of the problem can be established, with elements in the time dimension, supplementing the discrete or discretized description at each time instance. A set of equations is then established by using a two-point collocation within each time element. This view allows equilibrium equations of any complexity, but is primarily suited for problems of low to moderate numbers of degrees of freedom. The acting forces consist of prescribed external forces and a priori unknown control forces. Prescribed boundary conditions add equations to the system to be solved.

Dependent on the formulation of the problem, the solution method handles three basic classes of problems. These are distinguished by the number of boundary conditions on the displacements and velocities, and the number of free control force values. For the evolution problem, without control forces, the problem formulation must specify two values for each displacement component. For the fixed control problem, where a target state is desired, the number of free control force values is equal to the number of excessive boundary conditions, and their values can be determined. For an optimal control problem, the number of free control force values is higher than the number of excessive displacement conditions, allowing the optimization of their values.

For all the three classes of problems, a set of equation is established. In the optimal control problem, the set will add equations of optimality, increasing the size of the problem; a general algorithm can, however, be easily established, where only the number of prescribed displacement values and the number of free control force components decide the used method.

Performed tests indicate that the developed viewpoint and algorithm can be efficient in the study of complex, but primarily small to moderate size problems, with an improved continuity in the description of motion, and a good stability in dynamic solution. Comparing accuracy and computational effort, the method is efficient for a small problem, compared to Euler and Newmark methods, [1], and comparable to a Runge-Kutta 4th order method.

The method avoids the common shooting procedure to find a target displacement state, by solving for all discrete time stations at once. For a target controlled non-linear mechanism problem, the convergence with discretization is studied, and shown to be quick, given that a reasonably good initial approximation can be introduced. The method also allows full Newton iterations, leading to high accuracy in results. Alternative local optima in control force cost are discussed, starting from a well-known problem, [2].

With a sparse matrix for the established system, the efficiency of the method can in many cases be improved. Ongoing work expands the method to allow redundant force systems, limits in control force values, and interpolation of activation measures in the muscular system, rather than in forces themselves.

1

M. Géradin, D. Rixen, Mechanical vibrations. Theory and application to structural dynamics, 2nd Edition, Wiley, Chichester, 1997.

2

M. L. Kaplan, J. H. Heegaard, "Second-order optimal control algorithm for complex systems", Int. J. Numer. Meth. Eng., 53, 2043-2060, 2002. doi:10.1002/nme.372

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