Structure preserving simulation of non-smooth dynamics and optimal control - Koch, Michael (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2016)
This work deals with so-called structure preserving integrators which are applied to systems with non-smooth dynamics. In addition to forward dynamic simulations of simple mechanical systems, herein the focus particularly lies on the optimal control of multibody systems. The aim is to provide a biomechanical modelling of the human lower extremities and the analysis of human jumping movements and of the upright gait. In order to do this, the solutions of the investigated optimal control problems in consideration of different physiologically motivated objective functions are compared with human movements in reality. A method like this is especially useful to improve prostheses, for the performance enhancement of athletes or the accompanying analysis of rehabilitation patients as well as to clarify questions of ergonomic aspects. The presented approaches enable to determine the trajectory and control sequence from the infinite number of different possibilities between two given states, which minimise the chosen objective function. An advantage thereby is that the underlying integration schemes have structure preserving properties and consequently certain characteristics of the continuous systems are inherited to the approximated solutions. The two used structure preserving methods can be derived from the Lagrangian or the Hamiltonian mechanics, whereby for example the momentum or the angular momentum consistency are inherited from real dynamics to the integration schemes. Before the optimal control simulation can be described in detail, the questions regarding to model the bodies’ rigidity and the assembly by joints to multibody systems is answered. In order to actuate multibody systems, corresponding structure preserving force formulations are introduced in the discrete equations of motion. Thus, we achieve two possibilities to solve optimal control problems, thereby the integrator which based on the discrete variational principle is also known as DMOCC and serves as a reference solution for the herein introduced energy and momentum consistent scheme. Both approaches have in common that the optimal control problem is transcribed to an optimisation problem. A comparison between both optimisation methods is done by using the following examples: first, one examines a satellite reorientation manoeuvre inspired by the Hubble Space Telescope, while the second example is focussed on the bipedal compass gait. The results of both integrators have many similarities and consequently, with both schemes it is possible to handle complex systems. In the following sections we restrict ourselves to the variational integrator, because its derivation shows an elegant way to treat impacts and frictional contacts. The resulting event-driven integration schemes are verified using simple but very expressive examples. Using the example of a slider-crank mechanism, the results from the forward dynamics simulation are compared with the results of the widely used Moreau time-stepping scheme. This enable us to evaluate the performance of the event-driven integrator in comparison to a standard method. Based on the results so far achieved, the simulation of the human locomotor system in context of the optimisation problems comes to the fore again. At first, rigid body models are required iii which involves the implementation of the human hip, knee and ankle joint and subsequently the human anatomy and the movement abilities are implemented to the multibody systems. In a next step, physiologically motivated objective functions are formulated and these are minimised subject to the equations of motion with respect to the different occurring contact sequences. To enhance the modelling optimisation complexity, we are interested primarily in monopedal jumper models. The resulting behaviour of different jumping movements is analysed and compared with literature values. Afterwards, the obtained knowledge is transferred to a two-legged walker with feet and the upright gait cycle is simulated by a skilful mathematical formulation, whereby the choice of the optimal step length and the required optimal movement intervals in accordance to the objective functions are left to the optimiser.<br