Preview

PDF, English Download (5MB) | Terms of use

Abstract

In this thesis, we develop new mathematical models and methods for the Optimal Control of constrained biomechanical Multi-Body Systems (MBSs) for problems appearing in therapy design of Cerebral Palsy (CP). We model the human body while walking as a constrained rigid MBS, and the gait as a solution of an Optimal Control Problem (OCP) which is constrained by the dynamics of this MBS. Here, changing foot-ground contact configurations lead to jumps in the differential states. Assuming that it is possible to provide a patient-specifically calibrated OCP whose (selected) solution models the gait of a patient, such kind of Optimal Control model can be employed to predict the effect of medical treatments on the gait pattern. In this setting, we focus on three aspects: possibly changing sequences of foot-ground contact configurations due to medical interventions, worst-case scenarios in presence of uncertainties, e. g., in the applied medical treatments, and a suitable translation of interventions into changes of the employed Optimal Control model. For the case that the sequence of foot-ground contact configurations after a medical treatment is unknown, we develop an approach for the numerical solution of OCPs with switches, switching costs, and jumps in the differential states, which can occur at switching. For this, we consider a Mixed-Integer Optimal Control Problem and extend the Partial Outer Convexification approach. We develop two types of so-called switching indicators which are utilized on the one hand as a trigger for events that are associated with certain types of switches, and on the other hand for the computation of switching costs. In the considered setting, medical interventions can be seen as changes of parameters that enter the gait modeling OCP. However, in medical practice unavoidable inaccuracies can occur in the implementation of an intervention. Therefore, we study worst-case scenarios for parametric OCPs with parameter uncertainties. We develop and examine an approach for the determination of worst-possible parameter realizations and the according OCP solutions which is suited for model-based treatment planning of CP. Here, we deal with a bilevel optimization problem with an OCP on the lower level. In order to apply our approach for worst-case treatment planning, we provide a suitable model for medical treatments. Since many interventions in CP management eventually aim at extending the ranges of motion of joints, we present a modeling approach that translates treatments of this kind into changes of parameters which enter the dynamics of the gait modeling OCP. The usefulness of the developed approaches is demonstrated in two case studies.