SOLVING DYNAMICS OF QUIET STANDING AS NONLINEAR POLYNOMIAL SYSTEMS Zhiming Ji Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey 070 ji@njit.edu Abstract Many problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations, as is succinctly pointed out by Raghavan and Roth (995. Developing methods for solving sets of nonlinear polynomial equations represents one of Bernard Roth s many significant contributions towards establishment of theoretical basis for the mechanisms analysis and synthesis and the mechanical aspects of robotics. In this paper, results from studying the human postural stability during quiet standing are discussed to show that some problems in biomechanical systems may also lead to systems of nonlinear polynomial equations. Keywords: Systems of polynomial equations, Postural stability, Biomechanical systems.. Introduction Postural control is coordinated by the central nervous system with input from three systems: visual, vestibular, and somatosensory (collectively known as the proprioceptive system. Body sway is used as an indicator of postural stability. Falls due to impaired postural control present a serious health hazard to elderly as well as to persons with balance disorder (for examples, deficits of the proprioceptic system or muscle weakness, and diminish a person's ability to perform activities of daily living. The Sensory Organization Test (SOT, Motor Control Test (MCT and Adaptation Test (ADT form the core battery of tests, recognized as Computerized Dynamic Posturography (CDP, for diagnosing the functional impairments underlying balance disorders. Most Computerized Dynamic Posturography systems qualify postural stability based on force plate technology. These systems measure the ground reaction force with transducers on force plate to determine the center of pressure (COP. Then they use the upward projection of the COP as an estimate for the body center of mass (COM. Different low-pass filters are used on the COP time series to remove the high frequency content in the resulted COP (Benda, Riley and Krebs, 994; Caron, Faure and Breniere, 997, based on the assumption that postural sway is quasi-static.
Another approach is to estimate COM with the second integral of horizontal acceleration, which is assumed to be proportional to the horizontal ground reaction force (Shimba, 984. This method requires the initial constants of integration and several techniques were suggested for estimating these integration constants (Zatsiorsky and King, 998. Winter et al (998 estimate COM based on their 4-segment model and the measurements of markers. This approach works well for research purpose but less practical for clinical applications. Most of the studies on quiet standing use the single-joint inverted pendulum model (Winter, Patla, Rietdyk and Ishac, 00; Morasso and Schieppati, 999 and only the moment equilibrium is considered as the system equation. By including the force equilibrium into the system equation, we can determine the COM directly from the measured ground reaction force, without using integration or filtering. We show that the resulted nonlinear differential equation can be solved as a set of nonlinear polynomial equations, even for quiet standing on inclined surface.. Equations of Motion for Quiet Standing A complete set of dynamic equilibrium equations can be easily derived to establish the relationship between sway movement and the ground reaction forces. Free body diagrams in Figures and illustrate the human inverted pendulum for sway in the sagittal plane. Figures shows the entire body excluding feet as inverted pendulum rotating about the ankle joint A. M is the mass of body above ankle, F, and F V are horizontal and vertical force acting H A at ankle joint, τ is moment acting at ankle joint by muscles and θ is absolute sway angle with respect to fixed vertical reference. Figure. Free body diagram of body (above ankle
Figures shows the feet together with the force plate. m is the total mass of the feet and the force plate; F F and front and rear transducers respectively; F R are ground reaction forces perpendicular to the force plate, measured with F H is ground reaction force parallel to the force plate, measured with transducer at pin join; d is the distance between to the pin axis and transducers that measures forces perpendicular to the force plate; e is the distance between ankle join and transducer that measuring force parallel to force plate; a is distance from the line through ankle and pin joints to the center of mass of the feet, θ m is sway angle of the center of mass relative to the line perpendicular to force plate, and φ is incline angle of the force plate. Figure. Free body diagram of feet with force plate If h is the distance between the ankle joint and the center of mass of the body and I is the moment of inertia of the body about the ankle joint, then the equation of motion for the body can be written as F ( &, θ cosθ & H A = M h θ sinθ ( F ( & θ sinθ & V = Mg Mh + θ cosθ ( τ + Mgh sin θ = I & θ (3 The equilibrium of the feet can be written as 3 ( F F φ F = F cosφ + + sin (4 H, A H F R
F V = ( F + F cosφ F sinφ mg (5 F R H τ = ( F F d + F e mga cosφ (6 F R H After eliminating all the internal forces and moments F,, F V, and τ from eqs. (-6, we H A have the following three differential equations for the postural system: Mh ( & θ cosθ & θ sinθ = F cosφ + ( F + F sinφ (7 H F Mh ( & θ sinθ + & θ cosθ = ( M + m g ( F + F cosφ + F sinφ (8 F I & θ = Mghsinθ ( F F d F e mga cosφ (9 F R H + R R H The ground reaction forces F F, F R and F H are induced by the sway motion. Once they are measured by the transducers embedded in the force plate, the unknown state θ, & θ, and & θ of the sway for a fixed inclination φ can be obtained by solving eqs. (7-9. Although & θ and & θ are derivatives of the sway θ (t, they can be treated as independent to each other. Let x = sinθ, x = cosθ, & x 3 = θ, and x = & θ 4, we convert the set of differential eqs. (7-9 into a set of polynomial equations a ( = a x x4 xx3 xx4 x x3 4 x a5x4 a6 + x = (0 a ( = a a = + ( 3 + ( x (3 where a = Mh, a = FH cosφ + ( FF + FR sinφ, a 3 = ( M + m g ( FF + FR cosφ + FH sinφ, a 4 = Mgh, a5 = I, a6 = ( FF FR d + FH e mga cosφ. The resulted polynomial system can be solved with one of the three well-known methods (Raghavan and Roth, 995: Dialytic elimination, Polynomial Continuation, and Grobner bases. Applying Sylvester's Dialytic Elimination, we obtain a quadratic equation of x as: [( a + a a + a a ] x a a ( a a + a a x + a a a a 0 a (4 4 3 5 5 6 4 3 5 6 5 = 4
From the solution of equation (4, we can obtain the rest by formulas [ aa6 ( aa4 a3a5 x ] aa5 x = + (5 x 3 a3x a x = ( a (6 4 a3x a x x = ( + a (7 The sway angle can now be solved as θ = ATAN ( x, x. Note that the process only yields & x 3 = θ. Thus only the magnitude θ &, not the direction, of the angular velocity θ & is found. Therefore, this direction information must be derived from the time history of angle θ. For small sway, an approximation sin θ θ and cosθ may be used to simplify the solution. The ankle moment can be evaluated with eq. (6. With the computed sway angle and ankle moment, muscle stiffness during quiet standing can now be evaluated. 3. Application Examples We have been using EquiTest system (NeuroCom International, 00 to evaluate patients and subjects for balance and postural stability. This system consists of a movable dual platform, which is surrounded by a visual scene that can also rotate about the ankle joints. It quantifies the ground reaction force using five force transducers. The two force plates are connected by a pin joint and supported by four force transducers mounted symmetrically on a supporting plate. These four transducers measure forces perpendicular to the force plate. A fifth transducer is mounted to the supporting plate directly beneath the pin joint for measuring shear force. The force transducers are sampled at 00Hz, with a detection threshold about one Newton. The system estimates COM from COP using a moving average filter. We applied the derived formulas with measured ground reaction forces from several groups of subjects. The subjects body segment lengths and inertial parameters are calculated using anthropometric data taken from the literature (Dempster, 955; Miller and Nelson, 973; Le Veau, William and Lissner, 977, based on the subjects height and weight. The results for one of the subjects are presented here. The subject s height and weight are H =. 67 m and W = 75. 500 kgf, respectively. We find M = 73. 35 kg, m =. 65 kg, I = 85.0 kg m, h = 0. 933 m, d = 0. 07 m, e = 0. 065 m and a = 0. 035 m. The following plots are generated based on one of the trials (each trial lasts 0 seconds. 5
Figure 3 shows the computed COM ( y = h sinθ and the moving average reported by the Equitest system for the same trial. We can see clearly the smoothing effect of the moving average by comparing the computed and the reported COM. The system estimates h as 0.557H or h = 0. 93m, which is slightly smaller than the value we used. As a result, the reported COM curve is lower than our computed COM. Figure 3. Example of COM displacement Fig. 4 shows the result of a linear regression of ankle moment versus sway angle for the trial presented in Fig. 3. The resulted line is τ = 655.73 θ + 0. 978 ( N m with the coefficient of determination R = 0. 993. This indicates that the ankle stiffness closely resembles an ideal spring, which is in agreement with (Winter, Patla, Rietdyk and Ishac, 00. This computed ankle stiffness is very close to the value of Mgh, which is 670. 30. In this particular trial, the difference between the stabilizing ankle moment and the destabilizing gravity moment is less than % of the ankle moment, as shown in Fig. 5. The discrete nature of the force measurement can be seen from the plot of the net moment Mgh θ τ. 6
Figure 4. Example of linear regression between ankle moment and sway angle Figure 5. Example of net moment at the ankle 4. Conclusions The equations of motion for human quiet standing, when modeled as a single-joint inverted pendulum, form a set of nonlinear differential equations. By using the measured ground reaction force, this set of equations can be solved as a set of nonlinear polynomial equations. The resulted polynomial equations can be solved based on the work of Raghavan and Roth to obtain the COM 7
displacement during quiet standing. Such a solution can be used to evaluate ankle muscle stiffness. 5. Acknowledgements The work reported here was inspired by the scholarship of Bernard Roth and discussions with Thomas Findley and Hans Chaudhry. References Benda B. J., Riley P. O., and Krebs D. E., 994, Biomechanical relationship between the center of gravity and center of pressure during standing, IEEE Transactions on Rehabilitation Engineering, Vol., pp. 3 0. Caron O., Faure B. and Breniere Y., 997, Estimating the centre of gravity of the body on the basis of the centre of pressure in standing posture, Journal of Biomechanics, Vol. 30(/, pp.69-7. Dempster W. T., 955, Space Requirements of the seated operator, Aero Medical Lab., Wright- Patterson Air Force Base, WADC TR 55-59. Miller D., and Nelson R., 973, Biomechanics of sport, Lea & Febiger, Philadelphia. NeuroCom International Inc., 00, EquiTest System operator s manual, ver. 8.0. B. Le Veau, William and Lissner, 977, Biomechanics of human motion, nd. Ed., W. B. Saunders Co., Philadelphia. Morasso P.G., and Schieppati M., 999, Can muscle stiffness alone stabilize upright standing? J. Neurophysiol. Vol. 8(3, pp. 6-66. Raghavan M., and Roth B., 995, Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators, Special 50th Anniversary Design Issue, Trans. ASME 7, pp. 7-79. Shimba T., 984, An estimation of center of gravity from force platform data, Journal of Biomechanics, Vol. 7, pp. 53 60. Winter D. A., Patla A. E., Prince F., Ishac M., and Gielo-Perczak K., 998, Stiffness control of balance in quiet standing, J. Neurophysiol. Vol. 80 (3, pp.-. Winter D. A., Patla A.E., Rietdyk, S., and Ishac, M. G., 00, Ankle muscle stiffness in the control of balance during quiet standing, J. Neurophysiol. Vol. 85 (6, pp. 630-633. Zatsiorsky V. M. And King D. L.,998, An algorithm for determining gravity line location from posturographic recordings, Journal of Biomechanics, Vol. 3(, pp. 6-64. 8