Keywords: human movements, temporal finite elements, optimization, contacts, free-time formulation.

Previous work by the authors [1] has shown that temporal finite element approximations can be used for the representation of targeted optimal control problems, and that low order Lagrange interpolation and a weak equilibrium formulation lead to robust and efficient simulations. A free-time formulation is now introduced to increase the degree of freedom in finding optimal movement patterns and also solve the speed of the movement. The formulation deals with contact constraints and unknown boundaries. The intended applications lie within biomechanics, human musculo-skeletal movements in particular, but the method can be applied to a variety of optimal control problems.

The time-scale variable gives the problem some new characteristics. Common objective functions within biomechanics contain integrated measurements, like energy usage or forces along the motion, which are directly affected by the time-scale variable. Other problems contain neutral configurations according to the objective function which makes the optimal time undetermined. Also the interpolation error is dependent on the time-scale variable.

In human musculo-skeletal movements a partial contact during the movement is common. Multiple phases are then included, where each phase may have individual constraints and mechanical properties. The free-time formulation allows these phases to be determined by the optimization.

Many applications in biomechanics are characterised by an optimal performance at the final state, such as any jumping event. Therefore, the use of objective functions, which are directly dependent only on the last state, are investigated here. This gives other demands on the formulation. In order to avoid the nonlinear programming solver being able to introduce force impulses in the unbounded controls corresponding to contact forces, the actual contact was introduced by demanding zero accelerations. The contact was applied as a constraint in weak form. This formulation required more time elements in order to satisfactorily fulfil the contact properties but gave good numerical properties.

The effects of the time-scale parameter are discussed and verified by numerical examples and are seen to be dependent on the objective function. First, a targeted movement representing a rising movement is shown. Minimizing the control forces along the movement gives smooth and well defined optimal solutions. Objective functions dependent only on the last state mainly show bang-bang characteristics in the control. This also raises the problem with non-unique optimal movement and movement time. Secondly, a three-phase two-foot high jump motion was simulated, where the optimization finds a prior motion preparing for the subsequent phases with different mechanical properties.

The conclusion from the present work is that free-time optimization of movements can be an important tool in understanding optimal movements in many important everyday and athletic contexts.

1

A. Eriksson, A. Nordmark, "Temporal finite element formulation of optimal control in mechanisms", Computer Methods in Applied Mechanics and Engineering, 2010. doi:10.1016/j.cma.2010.02.003

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