# Mathematical modelling of human motion with below-knee prothesis Other conference contribution, 2002

The problem of modelling of dynamic characteristics of human motion with below knee prosthesis is under the investigation. It is assumed that the angles between main segments of the legs are specified based on the experimental study. The amount of the angular coordinates is not sufficient to restore the kinematics of motion of human locomotor apparatus. Optimization approach is proposed for mathematical modelling of motion of human locomotor apparatus wearing below knee prosthesis. Human locomotor apparatus is modelled by a plane system of nine rigid bodies connected by ideal cylindrical joints. The torques at healthy joints of the apparatus are the active (or powered) control stimuli. There are also passive (or unpowered) torques at the prosthesis structure that are modelled by spring-damper like actuators. It is assumed that the stiffness parameters of the prosthesis are known from experimental study. The following optimal control problem is considered. PROBLEM. Let the length and the time duration of single step, and the hip, the knee and the ankle angles of both legs are given. It is required determining the active controlling stimuli that minimize augmented energy cost function subject to differential equations of motion, boundary conditions and constraints imposed both on motion of the system and on the controlling stimuli. The equations of motion are derived based on the Lagrange formalisms and can be found in /1/. The boundary conditions and the restrictions on the kinematic, dynamic and control characteristics of the motion are written in the way that they reflect the anthropomorphic properties of motion of the system in question. The augmented energy cost function comprises three terms that represent the value of mechanical energy consumption, the discrepancy of angular coordinates at leg joints from the respective experimental data, and the discrepancy of evaluated passive torques from the respective torques acting at the prosthesis joints which available from experiment. The algorithm has been developed to solve numerically the above formulated optimal control problem. The algorithm proposed is based on parameter optimization approach and inverse dynamics technique. The independent variable functions have been parameterised by using the cubic smooth splines. Several simulations of energy optimal motion of human locomotor apparatus with below knee prosthesis have been done by using of the developed algorithm. Analysis of the results obtained demonstrated high efficiency of the developed algorithm and the possibility of realistic prediction of dynamic and energetic characteristics of human motion with below knee prosthesis based on utilization of experimental data corresponding to the angular coordinates of legs.

## Author

### Viktor Berbyuk

Chalmers, Department of Machine and Vehicle Systems, Mechanical Systems

### Myroslav Demydyuk

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine

### Bogdan Lytwyn

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine

### Subject Categories

Other Materials Engineering

4/22/2021