X.H. Duan
J. Q. Sun, R.H. Allen
Department of Mechanical Engineering
University of Delaware
Newark, Delaware 19176
INTRODUCTION: Recent neurophysiological research has focused on the elastic nature of muscle. Muscle spring like behavior plays an important role in the execution of single and multiple joint posture and movement (1). There is plenty evidence supporting the assertion that joint stiffness is indeed time-varying. The analysis of joint stiffness may be done in two ways. One is a top-down procedure, i.e., from the planing and control of the central nervous system (CNS) to the determination of the joint stiffness at the skeleton level. Another is an inverse procedure, i.e., from the skeleton system performance to deduce the required joint stiffness. So far, most published work has focused on the analysis of the CNS planning and control. The method to determine the joint stiffness from the stability requirement has been studied by Schemer et. al [2]. In this work, the effects of joint stiffness are emulated by proportional- derivative controls. Because of the complexity of the controller in the gait model, this approach cannot deal with the time varying nature of the joint stiffness.
In the present study, we assume that the spring-like property of co-contraction muscles provides inherent stability of the human body during walking. This muscle function is included in our skeleton model and is represented by a nonlinear angular spring and dash pot on each joint. The time varying joint stiffness elements are determined by imposing dynamic stability requirements of the gait and a hypothesis of minimum muscle effort for providing the joint stiffness.
MATERIALS AND METHODS We have developed a three-link model consisting of a stand leg, a stand calf and the upper body and derived equations of motion for joint angles in the sagittal plane:
(1)
Here [A(0)] is the inertial matrix, [B(0)] the corolis matrix, [C(0)] the damping matrix, [K(0)] the stiffness matrix and [G] the gravity force matrix. [T] is a second order virtual moment vector. When linearized about an arbitrary configuration, O,, the equations become:
(2)
Here we assume that the matrix [C(0 )] is positive or semi positive definite. Under this assumption, the stability condition of the system is that the matrix -[A(0 )]-'[K(0 )] is negative or semi negative definite. Hence, all the solutions of the characteristic equation lie either on the left side of the phase plane, or on the imaginary axis. Following Routh's stability criterion, the stability condition can be determined in terms of K, 0, and its derivative. This condition serves as a constraint for determining the joint stiffness.
The joint stiffness stems from the muscle co-contractions and are active in the sense that it is adjustable and it consumes ballistic energy during contraction and elongation. The stiffer the spring, the more muscle co-contraction forces are needed and therefore the more energy is consumed. We hypothesize that the minimal muscle effort is required for maintaining the stability in the normal human gait. A stiffness effort measure can be defined as follows:
(3)
During the walking process, the Joint stiffness is adjusted such that the muscle effort J for providing stiffness is minimal subject to the stability constraint. Mathematically, determining the joint stiffness thus becomes a nonlinear time varying constrained optimization problem.
RESULTS: Figure 1 compares the trajectories obtained via direct dynamic analysis of a model with active joint stiffness (la) to one without active stiffness (lb). As expected, the latter model diverges from its expected trajectory when exposed to perturbations. With joint stiffness, the model can withstand perturbations and can be used to predict trajectories given initial conditions and joint forces. Figure 2 shows the optimal joint stiffness at the ankle, knee and hip for three walking speeds: slow, natural and fast. The different walking trajectories obtained from experiments [3] are used to calculate their respective optimal joint stiffness. Figure 3 shows the optimal joint stiffness for a very slow walking (12.5 steps/min).
DISCUSSION: The results show that for faster walking, larger joint stiffness amplitude and variation range are required. Increasing the stride length and cadence both increases the joint stiffness amplitude and the variation range. For the very slow walking, the joint stiffness are nearly constant, which is in agreement with the previous observations in [4].
For the same walking trajectory, the optimal joint stiffness vary with joints. The ankle provides higher joint stiffness than the knee and the hip in order to stabilize the human body. This means the higher net muscle effort will be required on the ankle joint than on those of the knee and the hip. This is in agreement with the experimental results in [3].
CONCLUSION: Joint stiffness plays an important role in stabilizing human gait. Joint stiffness are dynamically adjusted in the walking process according to the walking speeds and postures. The minimum muscle stiffness criterion has led to the prediction of the time varying joint stiffness that are in agreement with the results in the literature. More clinical evidence is needed to validate the assumptions in the present study.
1. N. Hogan 1988 Can. J. Physiol. Pharmacol. 66, Planning and excution of multijoint movements.
2. A. Scheiner, D. C. Ferencz and H. J. Chizeck 1994 Proceedings of Thirteenth Southern Biomedical Engineering Conference, University of the District of Colubia, Washington, DC. Simulation of the Complete Gait Cycle Using A 23 Degree Of Freedom Model.
3. D. A. Winter 1980 The Biomechanics and Motor Control of Human Gait: Normal, Elderly and Pathological. University of Waterloo Press.
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