Keywords: musculo-skeletal, tensegrity structure, cost function, Hill's model, biomechanics, configuration space, visco-elastic.

The musculoskeletal system can be viewed as an indeterminate structure at one level and an unstable mechanism at another. In this paper a sub-structured inverse dynamics approach has been adopted to determine the muscle forces. Individual muscle forces are found through an optimization procedure using joint equilibrium equations as constraints. Thus a special set of cost functions must be specified and the problem has to be handled as a mathematical programming problem. This paper emphasizes on the different choices of `cost functions' for different types of activities. The cost function is not unique and depends on the magnitude of external load ( ), its duration (), its sense (push-pull), dynamic nature (cyclic and impact) of the load. It is necessary to account for the viscous nature of muscles and the dissipation of energy. To take these factors into account different cost functions are being proposed for different types of loads. Equations of equilibrium form the equality constraints for all the models and are given as follows:

where, vectors , and represent vector containing muscle forces, configuration parameters and external force resultant respectively.

Selection of the cost function:

In a living and self-organizing biological system cost function is not unique. It is dependent on the external load , configuration and also on the type of the particular activity. In this section four distinct criteria are considered.

a)

Equal stress model
In inverse problems, the classical equally stressed criterion is a rational choice. However, in a given configuration there may not be enough flexibility to manipulate the muscle forces. Hence the optimal criterion is slightly modified to the following form:

(35)

where, is the force in -th muscles, is the average force in the muscles and is the total number of muscles at any joint.

b)

Energy cost function
In case of nonlinear elasticity or visco elasticity energy functional may not be quadratic. In such cases energy functional in terms of displacement and forces will also have dissipative terms.

(36)

where, is an appropriate real number. Since equations of equilibrium are already solutions to the optimization problem it is practically the same conditions in a different garb.

c)

Dissipative energy cost function
For the case of sustainable repeated operations a minimum energy dissipation criteria is more appropriate. Considering Hill-type three element model the dissipation function may be written as

(37)

The subscript denotes dissipation in the viscous dashpot element in each muscle and denotes the volume of each muscles.

d)

Minimum total pain criteria
For a non-repeated sustained load the criteria should more likely be the total visco-elastic strain as developed in Hill-type model.

for

(38)

Conclusion:

In the present work some of the probable cost functions are presented. These are based on total energy, minimum dissipation and maximum viscous strain. It also shows that depending on the sense of loading, that is active or passive, group of muscle involved are different. In cyclic loading this phenomenon is considered. Consequences of different cost functions involved are demonstrated through numerical example.

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