Dynamic analysis and control of a simple nonlinear limb model

Motivation and Aim: Even the simplest limb model exhibits strongly nonlinear dynamic behavior that calls for applying the results of nonlinear systems and control theory. The analysis and control of limb models are important in the fields of designing and controlling artificial limbs, muscle prosthesis and in neuro-physiological investigations. The aim of this study is to investigate the possibility to applying input-output linearization [11] for nonlinear control of a simple limb model. Material and Methods: A nonlinear input-affine state-space model has been developed for a simple onejoint system with a flexor and an extensor muscle (see figure 1) which is suitable for nonlinear systems analysis and control. The model takes the nonlinear properties of the force-length relation and the force-contraction velocity relation into account. Exerted forces depend linearly on the activation state of muscles, and a viscoelestic tendon is considered following the principles in [28], [24] and [22]. This model has been extended with a simple model of the gamma-loop mechanism, but only the non-extended model is used for the control studies. The inputs of the model are the normalized activation signal of muscles, the output is the joint angle, and the number of state variables is 8. As preliminary model analysis we performed stability, controllability and observability analysis of the linearized model around steady-state points. Moreover, the relative degree of the model and the stability of its zero-dynamics [3] were also determined. Both regulating and servo controllers were designed for the simple limb model and were compared to the standard reference case being an LQcontroller designed for the locally linearized system. A pole-placement control was designed for the input-output linearized system, and also a fuzzy controller was designed. Result: The model was verified against engineering intuition and proved to be suitable for controller design purposes. The model analysis showed that the nonlinear limb model was controllable and was in the edge of stability because of the Hamlitonian properties of the model. The relative degree of the model is 3 for both of the inputs with a stable zero dynamics. Therefore input-output linearization was applicable and a 3rd order linear system was obtained in the new coordinates. A simple poleplacement controller was designed for this input-output linearized model. Both the pole-placement, the fuzzy, and the reference LQ-controller were suitable for control purposes but the fuzzy and the LQ-controller were more sensitive to the disturbances. 4