A model of a human standing. by Richard Denker. January 7, 2013

Transcription

1 A model of a human standing by Richard Denker January 7, 2013 Analytical Mechanics (FYGC04), HT 2012 Faculty of Technology and Science Department of Physics

2 Contents 1 Introduction 3 2 Theory The Model Feedback control The compliance of muscle and tendon Discussion 8 2

3 Figure 1: The basic inverted pendulum model of standing 1 Introduction Human quiet standing is one of the classical problems in the study of biomechanics. Experiments show that when a human stands quietly it can be described approximately as an inverted pendulum, pivoted at the ankle joints, with the center of mass of the body (COM) on average a few centimeters in front of the ankle joints (see Fig. 1). The inverted pendulum sways back and forward somewhat erratically around this position, using muscles and the spring-like Achilles tendon to adjust the angle. An experimental measurement of this kind of erratic motion is shown in Fig. 2. In the relatively simplistic model which will be described here, only motions back and forth are considered. The most dicult part of this model is to nd a way to describe how the body controls the swaying motion. The content of this essay is a humble attempt to summarize the simplest of the models described and discussed by Frank Borg in his paper, created with physic students in mind, called An inverted pendulum with a springy control as a model of human standing [3]. All pictures and reasoning are borrowed from this document. 2 Theory 2.1 The Model As illustrated in Fig. 1, a rectangular force plate of a known length b with transducers in each corner sensing the vertical forces F i, can be used to nd the center of pressure (COP). The origin of the coordinate system (the ankle joints 3

4 Figure 2: An example of the measured forward-backward sway in term of the center of pressure (COP) where the pendulum is pivoted) is located above the center of the force plate, so if we denote the y-coordinate of the COP by u we get Now using the notations in Fig. equations are obtained [2], u = b 2 F3 + F 4 F 1 F 2 F 3 + F 4 + F 1 + F 2. (1) 1 and Newtonian mechanics the following mÿ = F y, (2) m z = F z mg (3) I θ = mglsin(θ) N, (4) N = uf z + ζf y. (5) Here I and m denotes the moment of inertia and mass of the body (minus feet) respectively, L is the distance from the ankle joints to the center of mass (COM), F y and F z are the components of the ground reaction force which contributes to N, the torque that is produced by the plantar exors counteracting the gravitational torque mgy (y-coordinate of COM). For small angles θ the approximation y = L sin(θ) Lθ can be used in Eq. (4) and F z mg (since z 0 for small angles) in Eq. (5) which gives, Iÿ L = mgy N, (6) N = mgu + ζmÿ. (7) 4

5 Inserting (7) in (6) and simplifying we get g ζ + I ml (y u) = ÿ, (8) where we can let the constant in front of (y u) be the square of the characteristic angular velocity, ω c, which gives The corresponding characteristic frequency is f c = ω c 2π = 1 g 2π ζ + ÿ = ω 2 c (y u). (9) I ml. (10) From Eq. (7) it can be argued that that the muscle torque N is approximately proportional to u, since the rst term in general is much larger than the second term. The COP-coordinate u usually varies in the range 2-8 cm, while ÿ may be of the order of about 1 cm/s 2 and ζ less then 10 cm. Since g 981 cm/s 2 the argument above indeed seems valid. Thus we have N mgu. (11) Let's now look at Eq. (9) in the frequency domain. In the frequency domain we replace the functions in Eq. (9) with their corresponding Fouriertransformations, which are denoted by a hat. Without going into details, the rules governing transformation of derivatives gives that the transformed equation becomes û(f) ŷ(f) = ( ) 2, (12) 1 + f f c which means that y is a low-pass ltered version of u(t). 2.2 Feedback control The task of the body's balance control is to vary the function u(t) such that y(t) in Eq. (9) remains bounded in a small interval around an average slightly leaning forward position. Solving for y(t) in term of u(t) can be done by rst introducing a new variable together with q(t) = y(t) + 1 ω c ẏ(t), (13) q(t) = ω c [q(t) u(t)], (14) 5

6 which combined is equivalent to Eq. (9). If we nd q(t) we can then solve for y(t), using Eq. (13) and the method of Laplace-transforms. The solution is y(0) cosh(ω c t) + ẏ(0) ω c t y(t) = y(0) e ωct + 1 ω c e ωc(s t) q(s)ds = 0 t sinh(ω c t) + ω c sinh(ω c (s t))u(s)ds. 0 (15) The rst row of Eq. (15) shows that y(t) stays bounded whenever q(t) < C, for some constant C. Looking at Eq. (14) it is possible to design a threshold controller, which starts to pull on the pendulum whenever q crosses a threshold value q th. We will refer to this as bang-control and it will have the form of u(t) = f(q(t τ)), where τ is a time delay, f(q) = { 0 C + ɛ 1 if q q th + ɛ 2 otherwise. (16) The parameter C determines the strength of the feedback force that occurs a time τ after q surpasses its threshold q th, and the terms ɛ i represents additional stochastic elements (noise). With this, Eq. (14) now becomes q(t) = ω c q(t) ω c f(q(t τ)), (17) which is a system that can sustain an oscillatory motion, veried by a simulation made by Frank Borg (see Fig. 3). 2.3 The compliance of muscle and tendon The muscle-tendon combination in Fig. 1 can be divided into two dierent parts connected in series; the muscle and the Achilles tendon. The compliant (springlike) nature of the Achilles tendon makes it impossible for the pendulum (body) to simply stop rigidly in place by locking the length of the muscle once the pendulum reaches a certain angle. Experimental measurements of the tendon stiness and calculations made by Frank Borg also indicates that the tendon stiness is by itself not enough to counter the gravitational stiness, and a person would fall forward by such an action. In this model the muscle length will therefore instead change dynamically with the tendon elongation once q crosses a threshold q th. This helps to increase the eective stiness of the muscle-tendon combination as will be demonstrated below. Let the muscle be of length x 1 and the tendon of length x 2. Together these two parts have the combined length l. To get a simple model we will here assume that there is a maintained proportionality between an elongation of the tendon x 2 and a contraction (dynamic change) of the muscle x 1, x 2 = γ x 1, (18) 6

7 Figure 3: The saw-line represents motion of the the control data q without the stochastic variables ɛ i. The wavy line represent the COG-coordinate y with the stochastic variables. and that the Achilles tendon acts as a linear spring (in reality it is not linear) with spring constant K. We have which gives Thus, l = x 2 + x 1 = (1 1 γ ) x 2, (19) ( ) γ K x 2 = K l. (20) γ 1 K eff = ( ) γ K, (21) γ 1 which means that if the muscle contracts half as fast as the tendon lengthens (γ=2 ), then the eective combined muscle-tendon stiness will be two times the tendon stiness. The bang of Section (2.2) did not discuss how the feedback force was delivered (such as by a compliant muscle-tendon combination) but the feedback must be related to the spring elongation by f(q(t τ)) = K(x 2 (t) x 0 2) (22) where x 0 2 is the tendon length at the operating point. It is implicated by Eq. (22) that when q(t) crosses its threshold the elongation of the tendon will instantly jump from x 0 2 to x C/K. This is obviously not physiologically 7

8 possible since both the muscle and the tendon would need a nite time to adjust its shape. It's also to be noted that the force of the muscle depends on the number of motor units activated each time q crosses its threshold, which in a more rened model should be a probabilistic function of q. These inconsistencies are partly addressed by the stochastic elements ɛ i of Eq. (16), allowing for mall uctuations of threshold level and the force. 3 Discussion The model described here is a very simplistic model which covers only the basic phenomena regarding human standing. The variables involved to make a more realistic model are numerous, one would have to take into account voluntary movements like shifting the weight between legs, a far more complex balancing control system, movement at all the joints of the body, etc. There exists alternative models of human standing, for example a model of double linked pendulums involving both hip and ankle sway [3], though both the inverted pendulum and double pendulum models are acknowledged merely as approximations. Another possible control system that could have been chosen in this model is the PID-controller (common in engineering applications), which is brought up as an alternative by Frank Borg. References [1] An inverted pendulum with a springy control as a model of human standing by Frank Borg - [2] Karlsson, A., & Lanshammar, H. (1997). Analysis of postural sway strategies using an inverted pendulum model and force plate data. Gait And Posture, 5(3), 198. [3] Yasuyuki S, Taishin N, Maura C, Pietro M. Intermittent control with ankle, hip, and mixed strategies during quiet standing: A theoretical proposal based on a double inverted pendulum model. Journal Of Theoretical Biology [serial online]. n.d.;310: Available from: ScienceDirect, Ipswich, MA. Accessed January 9,