» APPLICATIONS OF CONTROL

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1 » APPLICATIONS OF CONTROL A PID Model of Human Balance Keeping KIMURA HIDENORI and YIFA JIANG The basic requirement of biped (two-legged) motion is apparatus records what we know today as the center of upright standing, which is inherently a balancekeeping task. The goal of our research is to under- were often discussed in relation to other physiological pressure (COP). In the early studies, sway characteristics stand what enables standing in humans and use this insight to develop diagnostics for medical applications. Children and the elderly often suffer from falls because of an underdeveloped or reduced capacity to maintain balance [1]. In children, the nervous system and physique are in development, and thus the balance-keeping control system is imperfect, while in the elderly, aging is a leading factor in the deterioration of this capacity. The increased properties such as mental development and intelligence [4]. Later, the relationship between aging and body sway motivated studies on balance keeping in the elderly. Recently, the research emphasis has shifted toward the relationship between body sway and balance-keeping ability, including the connection between body sway and vulnerability to falls, using an index that can be calculated from records of body sway [5]. risk of falling, which is a serious problem in elderly populations, is the motivating factor behind intensive research on the neuromechanisms behind 15 1 Eyes Closed Eyes Closed Amplitude ( 1 2 mm) human balance keeping. 5 4 RESEARCH ON BALANCE KEEPING Studies on human posture control during upright standing were popular in the late 19th century [2]. Normal standing was recognized not as a static posture but rather as a continuous movement as shown in Figure 1. Graphical tracings of human sway at the head level, using an apparatus known as the ataxiagraph, shows that the extent of anteroposterior movement is somewhat greater with the eyes open than with the eyes closed [2]. Studies on the projection of the center of gravity onto the support surface led to the development of the balance platform [3]. This Body Sway in Roll Direction (mm) (a) Eyes Open Time (s) (b) FIGURE 1 (a), (b) Records of body sway in the roll direction over time, measured with a CMOS video camera. These traces show the movement of a marker on the subject s back at the level of the fourth thoracic vertebrate. A typical range of angular rotation around the ankle joint is about 1 2 degrees. The average body sway speed is significantly increased when the subject s eyes are closed. (c), (d) Power spectra show that the increased movement is mainly due to high-frequency components between.8.12 Hz (c) Eyes Open Frequency (Hz) (d) 18 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER X/6/$2. 26IEEE

2 Head Psoas Major Glutaeus Medium Lesser Trochanter (a) Pelvic Lumbosacral Joint Sacrum Ilium Pubis Ischium Hip Joint Anterior View of Left Femur Greater Trochanter m 3 θ 2 h1 f 2 A f 1 B h 2 f 3 E D m 2 F f 4 γ l 4 l 3 C φ θ 1 l 1 m 1 m1 l 2 FIGURE 2 (a) The human pelvis and muscles related to balance keeping. The pelvis is composed of four bones, the sacrum, ilium, pubis, and ischium, which are linked with the fifth lumbar spine and femur by the lumbosacral and hip joint, respectively. (b) The femur includes a head that is articulated with the ilium. The two bone structures of the femur are the great trochanter and less trochanter, where two pairs of muscle, the glutaus medius and psoas major, terminate. Experimental observations show that the glutaus medius and psoas major play a key role in balancekeeping control in standing. (c) Structural model of the upright standing body. In normal subjects, the upper body is kept perpendicular to the horizontal. The forces f 1, f 2 are produced by the psoas major, while f 3 and f 4 are produced by the glutaus medius. m 1, m 2, and m 3 represent the masses of the lower leg, pelvis, and upper body, respectively. θ 1 is the sway angle of ankle joint, while θ 2 is the sway angle of the upper trunk. (b) (c) 2d The inverted pendulum provides a simplified model of upright standing in humans since, like the inverted pendulum, humans need feedback to maintain balance. Similar to control systems for artifacts, balance-keeping control consists of sensors, actuators, and a central control system. A minimum of three sensory organs contribute to balance keeping: the vestibular organ (located in the inner ear), eyes, and proprioceptors (nerve endings that detect force and displacement). These sensory organs transmit information representing the current body states to the central nervous system. Actuators, which include the muscles present in various parts of the body, help to sustain body balance. Studies show that two pairs of muscle in the pelvis, namely, the glutaus medius (GM) and psoas major (PM), as shown in Figure 2, are especially important for balance keeping. Although the central nervous system is responsible for body balance control, the mechanisms that underlie body balance control are not as clear as in other motor control functions. Body sway is in some sense an outcome of body balance control and thus can provide a useful clue for investigating balance mechanisms. FIGURE 3 A simplified structural model of static upright standing. m 1 is the mass of the two legs, m 2 is the mass of the pelvis, and m 3 is the mass of the upper trunk. Experimental results show that the lumbosacral joint moves opposite to the direction of the ankle joint, suggesting that the trunk is kept perpendicular to the horizontal. Left Right D m 3 θ l 2 C m 2 m 1 A θ B m 1 l 1 d d l g DECEMBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 19

3 Sensors Neuromuscular System Inverted Pendulum Body K P [2m 1 l g (m 2 m 3 )l 1 ]gsin θ θ ref = _ K D s [2m 1 l 2 g (m 2 m 3 )l 2 1 ]. s 2 com V cop K l /s e st d sin(ωt) Sensory Systems FIGURE 4 Human static upright stance control. Since the aim of standing is to keep the body upright, the reference value of postural sway angle is set to zero. The central nervous system detects the error signal and sends an output signal to the muscles so as to keep the body upright in a state of equilibrium. The controller is modeled as a PID control, and the plant is modeled as an inverted pendulum. Here, com denotes the center of mass. with the control laws CMOS Camera e(t) = r(t) θ(t), (2) de(t t d ) t τ(t) = K P e(t t d ) K D K I e(t t d )dt, (3) dt where τ represents the moment of rotation, t d is the time lag, r(t) =, and K P, K D, and K I are the gains of a PID control law. Combining (1) and (3), and approximating sin θ(t) θ(t) leads to [ 2m 1 l 2 g (m 2 m 3 )l 2 1] θ(t) [2m 1 l g (m 2 m 3 )l 1 ]gθ(t) FIGURE 5 The body-sway measuring system composed of a marker for image reorganization, a CMOS camera for image recording, and a personal computer for data processing. A whiteboard with marker is set 1. m before the subject to act as a visual index at eye level. PID PARAMETER ESTIMATION THROUGH BODY SWAY MEASUREMENT To analyze body balance from a control theory point of view, we construct a multilink model of the human body, as shown in Figure 3, with proportional-integralderivative (PID) control taken as a working model of the body balance control algorithm. The body dynamics are modeled by [ τ = 2m 1 l 2 g (m 2 m 3 )l1] 2 θ [2m 1 l g (m 2 m 3 )l 1 ]g sin θ, (1) = K P e(t t d ) K D de(t t d ) dt t K I e(t t d )dt. (4) The overall system is depicted in Figure 4. Estimating the PID gains based on measurements of body sway is a closed-loop identification problem involving the estimation of control parameters instead of plant parameters. The extent of body sway around the ankle joint is small, often not more than 2 [6]. To measure sway, we have developed an optical measuring system. As shown in Figure 5, the system is composed of three parts: a black marker with a diameter of 3 mm, a CMOS video camera with a recording speed of 15 frames/s, and a computer system for image recognition and data processing. A marker is placed on the subject s back at the level of the fourth thoracic vertebrate, and the recording period is set to 3 s or 6 s. Body sway is recorded for cases in which 2 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 26

4 the subject s eyes are open and when they are closed. Figure 1 shows the body sway in terms of the angle θ defined in Figure 3 in both the eyes-open and eyes-closed states. The body sway is significantly different between the two states, especially for the recorded angular velocity θ(t). In addition, the frequency component around.1 Hz (.8.15 Hz) is present when the subject s eyes are closed, whereas, in the eyes-open state, only a component of.3 Hz (.1.5 Hz) is observed. Using body sway data for PID gain estimation, it follows from (4) that the scalar parameters, K P, K D, K I, and t d can be identified using observations of θ. To use standard system identification techniques, (4) is rewritten in the linearin-the-parameters form τ(t) = K P u(t) K D v(t) K I w(t) e(t), (5) u(t) = e(t t d ), (6) v(t) = de(t t d), dt t (7) w(t) = e(t t d )dt, (8) where τ(t), u(t), v(t), and w(t) are computable when t d is determined. For a collection of measurements, (5) can be rewritten as τ 1... K P e 1... = K D, (9) τ n K I e n Body Sway (rad) Amplitude (rad) Experimental Data Simulated Data.5.1 Time (s) (a) Experimental Simulated Frequency (Hz) (b) FIGURE 6 Simulated and experimental recorded body sway in the eyes-open state. (a) Recorded and simulated body sways over time. (b) Frequency response. The subject is a 37-year-old male. The body height is 1.65 m, and the body weight is 65 kg. The values of K P, K D, K I, and t d estimated from the experimental data are 519. N-m/rad, 72.3 N-m-s/rad, 3. N-m/rad-s, and.11 s, respectively. The simulated frequency response is similar to the experimental data. where is defined by u 1 v 1 w 1 =..., (1) u n v n w n the entries of which are measurements of u, v, and w, and the parameters of K P, K D, and K I are given by the leastsquares solution K P τ 1... K D = ( T ) 1 T K I τ n. (11) Letting S = e T e = (t d ), the value of t d is estimated using the relation S t d =. (12) The partial derivative in (12) is evaluated numerically. After identifying the parameters, body sway can be simulated with the startup function θ(t) =.1 sin t, where t d < t <. Figure 6 shows simulated body sway with parameters estimated from the experimental data DECEMBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 21

5 collected from a healthy 37-year-old male. White noise is added to the input in the simulation since balance-keeping control is regarded as a white-noise-driven system. As shown in Figure 6(a), the time record of the simulated body sway and the actual measurement are similar in both amplitude and frequency, which is further validated by comparing their power spectra as shown in Figure 6(b). The similarity between the simulated and experimental results suggests that the plant-controller model captures the essential mechanisms of body sway and balance control. Furthermore, considering the sensor noise to be as shown in Figure 4, the transfer function from sensor noise to output becomes T(s) = G 1G 2 e st d G 1 G 2 e st d 1, (13) G 1 (s) = K Ds 2 K P s K 1, s (14) 1 G 2 (s) = Is 2 G, (15) where I = 2m 1 l 2 g (m 2 m 3 )l 2 1, G = [2m 1l g (m 2 m 3 )l 1 ]g, and sin θ θ. Gain 1,4 1,2 1, Time Lag.1 (s) 5 Eyes Open (.6 Hz) FIGURE 7 Frequency response of the simulated balance-keeping control system for the subject in Figure 6. The magnitude is a function of ω and t d. The peaks of the magnitude in the eyes-open (.6 Hz) and eyes-closed (.16 Hz) states are different, which is roughly consistent with the experimental data, where peaks appear at.2 Hz in the eyes-open state and.12 Hz in the eyes-closed state. 5 Eyes Closed (.16 Hz) Omega.1 (rad) The magnitude T( jω) as a function of ω and t d is shown in Figure 7. The peaks of the magnitude appearing in both the eyes-open and eyes-closed states are consistent with the recorded experimental data. These results suggest that the PID model is useful for analyzing balance-keeping control. DIFFERENCE BETWEEN NORMAL AND BALANCE-DEFICIENT SUBJECTS Balance-keeping ability can be analyzed based on the speed of body sway. As observed in [7], body sway is significantly increased in a balance-deficient group compared to a normal group. The elderly, who are more prone to falls, exhibit more extensive body sway, suggesting poorer balance-keeping ability. To quantify the influence of aging, we use PID-based gain estimation as described above. Our results show that the value of K D in the PID controller is vision related. In particular, K D is significantly decreased when the subject s eyes are closed. It is also observed that aging causes the value of K D to decrease, as shown in Figure 8, while the damping effect of the controller is mainly related to balance-keeping ability. However, the clinical importance of K P and K I is still unclear and further research is necessary. CONCLUSIONS Standing involves a complex balance-keeping control mechanism in the brain. In addition to the brain, the bone frame and muscular system are developed to make bipedal motion possible. We have developed a balance-keeping control model that can be used for analyzing the balancekeeping ability in humans. PID control model is used to approximate the control process that enables standing. This closedloop model provides insight into balancerelated phenomena such as body sway and falls. Experimental results indicate that the derivative gain K D is significantly reduced when the subject s eyes are closed, and that K D decreases with aging. This observation suggests that K D is a balance- and stabilityrelated parameter. Future research will focus on applications of this approach to disease diagnosis and fall prevention. AUTHOR INFORMATION Hidenori Kimura obtained his Ph.D. from the University of Tokyo. He joined Osaka University in 197 and moved to the University of Tokyo in In 2, he became a member of the Institute of Physical and Chemical Research (RIKEN), where he is a team leader in 22 IEEE CONTROL SYSTEMS MAGAZINE» DECEMBER 26

6 KdI Aging FIGURE 8 Values of K D taken from 12 healthy elderly subjects with average ages 64.1 ± 16.3 years. The value of K D is normalized as KdI = K D /mgh, where m is body weight, g is the acceleration due to gravity, and h is body height. The result shows that K D generally decreases with age along with the ability to maintain equilibrium. the Biological Control Systems Laboratory. He is a recipient of multiple SICE and IFAC paper awards as well as the IEEE Control Systems Society George Axelby Award. He is the president of Asian Control Association and vice-president of the Trans-Disciplinary Federation of Science and Technology. His current research interests lie in biological control theory including neuro-robotics and intracellular regulation. Yifa Jiang obtained his M.D. from Zhejiang Medical University, China, and his Ph.D. from Gifu University of Japan in 199 and 2, respectively. He is currently a research scientist in the Biological Control Systems Laboratory in RIKEN of Japan. Prior to joining RIKEN in 22, he worked five years at Zhejiang Medical University as a teaching assistant, and served two years on the faculty of the Virtual Systems Laboratory in Gifu University. REFERENCES [1] I. Melzer, N. Benjuya, and J. Kaplanski, Postural stability in the elderly: A comparison between fallers and non-fallers, Age Aging, vol. 33, no. 6, pp , 24. [2] W.N. Bullard and E.G. Brackett, Observation on the steadiness of the hand and on static equilibrium, Boston Med. Surg. J., vol. 119, no. 25, pp , [3] F.A. Hellerbrandt, R.H. Tepper, G.L. Braun, and M.C. Elliott, The location of the cardinal anatomical orientation planes passing through the center of weight in young adult women, Amer. J. Physiology, vol. 121, pp , [4] T.L. Bolton, The relation of motor power to intelligence, Amer. J. Physiology,, vol. 14, pp , 193. [5] S. Nagasaki, Y.F. Jiang, S.M. Mirbod, Y. Furuta, T. Matsuoka, and H. Kimura, Fall risk prediction in old women evaluated by trunk sway tests in static upright stance, J. Education Health Sci., vol. 48, pp , 23. [6] Y.F. Jiang, S. Nagasaki, T. Matsuoka, Y. Furuta, and H. Kimura, The relationship between trunk sway and the motion of center of pressure during quite stance, Japan. J. Phys. Fitness Sports Med., vol. 52, pp , 23. [7] J.H. Allum, A.L. Adkin, M.G. Carpenter, M. Held-Ziolkowska, F. Honegger, and K. Pierchala, Trunk sway measures of postural stability during clinical balance tests: Effects of a unilateral vestibular deficit, Gait & Posture, vol. 14, no. 3, pp , 21. Communicated by Shuzhi Sam Ge. DECEMBER 26 «IEEE CONTROL SYSTEMS MAGAZINE 23