A Simplified Nonlinear Biomechanical Model for FES-cycling Einar S. Idsø1,*, Tor A. Johansen1, Kenneth J. Hunt2 1
Dept. of Eng. Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Centre for Rehabilitation Engineering, University of Glasgow, Glasgow G12 8QQ, Scotland
2
{esi,torj}@itk.ntnu.no
[email protected]
Introduction Cycling by means of functional electrical stimulation has been well established as a useful tool for rehabilitation of spinal cord injured people as well as a means for recreation. For recreational purposes, and often so for rehabilitation, open-loop control is sufficient: by using a predefined stimulation pattern based on the measured crank angle, the patient can directly control the intensity of the stimulation. For many experimental setups, where for instance the effect of training is to be evaluated, it is required that a certain cadence, power or other output criterion is maintained over a period of time. In these cases open-loop control comes short, and closed-loop control must be used. Also, with motor-assisted cycling, feedback control can enable cycling in optimal conditions and thus improve efficiency. Rather few types of feedback controllers have been designed for this system. An early solution by Chen and coworkers [1] utilized a PID-controller on a linear first-order model. The group later went on to design a model-free fuzzy logic controller to overcome the uncertainties of the model [2]. Recently Schauer et.al. synthesized a controller using polynomial design on an identified linear model [5]. Both nonlinear controllers and system identification schemes, which are necessary for adaptive feedback control, require rather thorough understanding of the system, but when properly implemented can yield excellent output performance and further insight into the properties of the system. Unfortunately, due to the complexity of the muscle activation and force generation, coupled with the mathematically complicated nonlinear characteristics of the cycling motion, it has so far been very difficult to implement model-based feedback controllers and system identification schemes. This paper discusses how the mechanical part of the rider-tricycle model can be simplified to a highly accurate, yet simple and understandable, nonlinear model that can be used both to enhance insight into the system, and as a foundation for the design of model-based nonlinear controllers and adaptation schemes. Methods A model for a recumbent rider-tricycle system has been developed [4] using the Euler-Lagrange equations of motion [3] and implemented in MATLAB. The model is in the sagittal plane and has a single degree of freedom, the crank-angle q: M (q )q&& + C ( q)q& 2 + G (q) =
∑ [J
i ={r ,l }
τ
h ,i ( q ) h ,i
]
+ J k ,i (q )τ k ,i + Q0
(1)
M(q) is the total moment of inertia of the system observed at the crank, C(q) contains coriolis- and centripetal terms, and G(q) is the torque due to gravity. The torques generated at the hip and knee are denoted τh and τk. The index i indicates left or right leg. These torques are transformed to the crank via the Jacobians Jk(q) and Jh(q). Q0 is the sum of additional torques, such as friction and air resistance. The advantage of this model lies in it being single-dimensional. Its major disadvantage is the complexity of the expressions for M(q), C(q), G(q), Jk(q) and Jh(q), some of which take up a full printed A4 page. This results in not only high computational complexity, but more importantly also in nonlinear and complex inclusion of the anthropometric parameters. These equations are not fit for an intuitive understanding of the system. Moreover, due to the nonlinearities in parameters, parameter identification and the design of model-based controllers will be difficult. Therefore model simplification is performed, based on curve-fitting. The functions M(q), C(q), G(q), Jk(q) and Jh(q) from equation (1) are computed for 0≤q
