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Abstract

This thesis aims at developing mathematical models and numerical methods for establishing a classification and diagnosis scheme for the pathological gait of Cerebral Palsy (CP) patients based on Optimal Control (OC). The pathological gait of CP patients is still an ongoing field of research and not completely understood. Methods to diagnose or classify the gait of CP patients are important for intervention and treatment planning. For this purpose we model the patient’s body by a biomechanical rigid multibody system. The dynamics of this multibody system appear as constraints in an Optimal Control Problem (OCP), which we use to describe the pathological gait. This is based on the common assumption that optimization serves as a fundamental principle of human locomotion, such that the gait can be interpreted as optimal with respect to a well-chosen combination of varying optimization criteria. To achieve a calibrated patient-specific gait model, unknown model parameters and objective weights of various optimization criteria in the OCP have to be identified under consideration of given motion capture data from the HEIDELBERG MOTIONLAB. We formulate a Bilevel Inverse OCP: on the upper level of this bilevel optimization problem we have a Parameter Estimation (PE) Problem, constrained by the lower level parametrized OCP describing the gait of a patient with CP. With given motion capture data the unknowns can be determined, such that the developed gait model fits the given measurements best.

For solving Bilevel Inverse OCPs we consider a direct solution approach by applying the Direct Multiple Shooting Method and replace the resulting lower level Nonlinear Programming Problem (NLP) by its firstorder optimality conditions. This results in a so-called Mathematical Program with Complementarity Constraints (MPCC) - a challenging class of NLPs. We establish a novel mathematical method under consideration of known specific characteristics of the underlying constraints, and propose efficient numerical algorithms. In this so-called Direct Simultaneous Approach with Fixed Active Set (DISIMFAS), we furthermore exploit given structures, which arise as a result of the Direct Multiple Shooting Method and the bilevel optimization problem. In this thesis, we develop the software package PARDYNOPT with an efficient implementation of the proposed DISIMFAS. PARDYNOPT furthermore implements numerical methods for solving OCPs in general based on the Direct Multiple Shooting Method and is designed in a modular way for further extensions and investigations. The performance of the developed numerical method is demonstrated on a set of different Bilevel Inverse OCPs. Among these, we derive and analyse a new Bilevel Inverse OCP for a human-like locomotion model with promising results for future application of the proposed method to identify unknown parameters in the gait model of patients with CP. As an essential step in this direction, in this thesis we develop a rigid multibody system model for a patient with CP, such that its dynamics can capture the main characteristics of the pathological gait. We propose an OCP constrained by the rigid multibody system dynamics with a least-squares objective to investigate the reconstruction of the underlying dynamics of the proposed CP model to given motion capture data. Furthermore, we derive an adequate parametrized OCP with a well-chosen combination of optimization criteria for the gait of a CP patient. This gait model can serve as a lower level in the Bilevel Inverse OCP formulation. The suitability of the CP gait model is evaluated by analyzing solutions of differently weighted OCPs.