Optimal Control of Isometric Muscle Dynamics

Abstract

Abstract. We use an indirect optimal control approach to calculate the optimal neural stimulation needed to obtain measured isometric muscle forces. The neural stimulation of the nerve system is hereby considered to be a control function (input) of the system 'muscle' that solely determines the muscle force (output). We use a well-established muscle model and experimental data of isometric contractions. The model consists of coupled activation and contraction dynamics described by ordinary differential equations. To validate our results, we perform a comparison with commercial optimal control software.Keywords: biomechanics; inverse dynamics; muscle model; optimal control, stimulation. 0 B IntroductionMathematical models for everyday phenomena often ask for a control or input such that a system reacts in an optimal or at least in a desired way. Whether finding the optimal rotation of a stick for cooking potatoes on the open fire such that the potato has a desired temperature, see [1], or computing the optimal neural stimulation of a muscle such that the force output is as close as possible to experimentally measured data. Typical examples for biomechanical optimal control problems occur in the calculation of goal directed movements, see [2][3][4] and in robotics [5]. Concerning huge musculoskeletal systems, the load sharing problem of muscle force distribution has to be solved using optimal control [6,7]. The most common application for solving the load sharing problem is the inverse dynamics of multi-body systems (MBS) as in [8,9]. The aim is to approximate observed multi-body trajectories by a forward simulation. The problem occurs to find a set of muscle activations such that the muscle forces resulting from the MBS simulation are similar to the measured ones.Considering a general optimal control problem there is a process described by a vector of state variables which has to be influenced by control variable u ∈ within a time interval [t 0 , t 1 ] such that a given objective function ℑ( , u) is minimized subject to the model equations. These model equations can be either ordinary differential equations (ODE), partial differential equations (PDE) or differential algebraic equations (DAE). Additional constraints on the control Optimal Control of Isometric Muscle Dynamics 13 variable as well as the state variable itself can be imposed. The corresponding general optimal control problem reads as follows:For solving this minimization problem we introduce an adjoint (or costate) variable λ which operates as a penalty function, if the ODE or DAE are not fulfilled. The optimized control variable u * is found at the saddle point of the Lagrangian ℒ.Almost exclusively in biomechanical literature, the problem in Eq. (1) is solved by the technique of first discretize then optimize also known as direct method. Therefore ℑ as well as the ODE/DAE constraints are discretized on a given time grid resulting in a huge non-linear program (NLP), see [1,[10][11][12]. For solving such NLPs, several efficient solvers have been desig...

Optimization problems in epidemiology, biomechanics & medicine

“…7 and 8. A detailed analysis of a similar scenario, including objective function values, can be found in [4]. Regarding both figures there is a high congruence of the experimental force curves, the optimal control result and the direct model output using [14].…”

“…In [4], an optimal control of model (11) was performed, using activation dynamics from [19]. It was proposed to exchange the activation dynamics to improve the performance of the model.…”

“…We shortly outline the model structure based on [4]. It is assumed that the muscle-tendon-complex MTC consists of four elements; the contractile element CE that is responsible for the force production, the parallel elastic element PEE that represents the connective tissue around the CE, the serial elastic element SEE that represents the tendon, and the serial damping element SDE that acts parallel to the SEE.…”

“…The upcoming equation system (2)-(4) was solved with Algorithm 1 from [4]. As a starting stimulation we chose u 0 ðtÞ 0:5.…”

“…Examples for such problems are manyfold, e.g. finding the optimal rotation of a stick for cooking potatoes on the open fire such that the potato has a desired temperature, see [1], constructing an optimal strategy against the spread of dengue disease such that maximal reduction of the number of infected cases is achieved with minimal cost [2,3] or computing the optimal neural stimulation of a muscle such that the force output is as close as possible to experimentally measured data [4].…”