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## On the application of mathematical methods in Hill-type muscle modeling: stability, sensitivity and optimal control

• While reading this sentence, you probably gave (more or less deliberately) instructions to approximately 100 to 200 muscles of your body. A sceptical face or a smile, your fingers scrolling through the text or holding a printed version of this work, holding your head, sitting, and much more. All these processes take place almost automatically, so they seem to be no real achievement. In the age of digitalization it is a defined goal to transfer human (psychological and physiological) behavior to machines (robots). However, it turns out that it is indeed laborious to obtain human facial expression or walking from robots. To optimize this transfer, a deeper understanding of a muscle's operating principle is needed (and of course an understanding of the human brain, which will, however, not be part of this thesis). A human skeletal muscle can be shortened willingly, but not lengthened, thereto it takes an antagonist. The muscle's change in length is dependent on the incoming stimulus from the central nervous system, the current length of the muscle itself, and certain muscle--specific quantities (parameters) such as the maximum force. Hence, a muscle can be mathematically described by a differential equation (or more exactly a coupled differential--algebraic system, DAE), whose structure will be revealed in the following chapters. The theory of differential equations is well-elaborated. A multitude of applicable methods exist that may not be known by muscle modelers. The purpose of this work is to link the methods from applied mathematics to the actual application in biomechanics. The first part of this thesis addresses stability theory. Let us remember the prominent example from middle school physics, in which the resting position of a ball was obviously less susceptible towards shoves when lying in a bowl rather than balancing at the tip of a hill. Similarly, a dynamical (musculo-skeletal) system can attain equilibrium states that react differently towards perturbations. We are going to compute and classify these equilibria. In the second part, we investigate the influence of individual parameters on model equations or more exactly their solutions. This method is known as sensitivity analysis. Take for example the system "car" containing a value for the quantity "pressure on the break pedal while approaching a traffic light". A minor deviation of this quantity upward or downward may lead to an uncomfortable, abrupt stop or even to a collision, instead of a smooth stop with a sufficient gap. The considered muscle model contains over 20 parameters that, if changed slightly, have varying effects on the model equation solutions at different instants of time. We will investigate the sensitivity of those parameters regarding different sub--models, as well as the whole model among different dynamical boundary conditions. The third and final part addresses the \textit{optimal control} problem (OCP). The muscle turns a nerve impulse (input or control) into a length change and therefore a force response (output). This forward process is computable by solving the respective DAE. The reverse direction is more difficult to manage. As an everyday example, the OCP is present regarding self-parking cars, where a given path is targeted and the controls are the position of the steering wheel as well as the gas pedal. We present two methods of solving OCPs in muscle modeling: the first is a conjunction of variational calculus and optimization in function spaces, the second is a surrogate-based optimization.

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Author: Robert Rockenfeller urn:nbn:de:kola-12735 Thomas Götz, Jörg Fehr Thomas Götz Doctoral Thesis English 2016/02/15 2016/02/18 Universität Koblenz-Landau, Universitätsbibliothek Universität Koblenz-Landau, Campus Koblenz, Fachbereich 3 2016/02/17 2016/02/18 xiii, 169 Fachbereich 3 / Mathematisches Institut 31 Mathematik / 31.44 Gewöhnliche Differentialgleichungen 31 Mathematik / 31.80 Angewandte Mathematik Es gilt das deutsche Urheberrecht: § 53 UrhG