Adaptive Control of Human Posture in a Specific Movement

Transcription

1 Journal of Informatics and Computer Engineering (JICE) Vol. 2(5), Oct. 216, pp Adaptive Control of Human Posture in a Specific Movement Seyyed Arash Haghpanah School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran haghpanah@mech.sharif.ir Fatemeh Haghpanah School of Computer and Electrical Engineering Yazd University Yazd, Iran fatemehhaghpanah@yazd.ac.ir Abstract Human posture control is a complex issue in biomechanics. Human body is unstable without any controller. The stabilization of the body is achieved by the activation of the muscles and creating the joint torques. In this paper, human body in upright standing position has been modeled using an inverted double pendulum. Since the body parameters are different among the individuals, it is assumed that these parameters are not known exactly and are uncertain. An adaptive controller based on the inverse dynamics in addition to parameter adaptation law has been designed. The simulation of the system using this controller shows the effectiveness of the proposed method in controlling the human posture. Keywords adaptive control; human body; stability; inverted double pendulum I. INTRODUCTION Biomechanics uses the principles and laws of other scientific branches and it is a multi-disciplinary field of research. Control theory is a topic used to regularize the biomechanical systems. Human body, considered as a biomechanical system, is unstable without using any controller. The purpose of human body control is to compensate the gravitational forces and stabilize the body posture. This matter is conducted by the torques applied at the joints using the muscles. So, the human body can be considered as a dynamical system that has some inputs and outputs. By tuning the inputs, the system would be stable during standing or any special movement. Human can control his body posture in different circumstances. Generally human body position can be divided to two categories in the standing situation: Static posture in standing and dynamic posture during walking. entral nervous system implements various control strategies to maintain the stability of human body in the two mentioned scenarios. This mechanism is an interaction of the musculoskeletal system and motor control processes. To study the underlying basis of these mechanisms the biological control is developed by the engineers. Various models of human body have been created and control methods are used to investigate the physiological foundation of CNS. These findings can be used in clinics for treatment of motor control disorders such as Parkinson disease. Also the results of clinical methods like electrical stimulation can be modeled. The study standing posture has valuable information from clinical point of view. The figures show that the medical cost for treatment of patients experienced falling was about 19 billion dollars in 2 in the USA [1]. Human modeling can be beneficial in designing humanoid robots. Therefore the study of human posture control is an important subject in biomechanics and the related fields of research. The stability problem of human body in standing position has attracted a great deal of attention for four decades [2-4]. In most of these studies, the human body has been modeled as an inverted pendulum [4-7]. Ankle strategy is investigated in some researches. It is assumed that the whole body rotates about the ankle joint, and used the stiffness of this joint to maintain the stability of the body [8, 9]. Qu et al. used a single inverted pendulum model to represent the human body and an optimal control method was used for balancing of the human upright stance [1]. They extended their work to a 3D balance control model to simulate the human body sway in the anterior posterior and medial lateral directions [11].Gunther et. al showed experimentally that all leg joints including hip, knee and the ankle contribute to human balance in quiet stance and calculated their contribution percentage [12]. The stabilization of human body in people suffering from vertiginous is studied using the robust control method [13]. Guelton et. al used T-S observer model as an alternative to inverse dynamics for estimating the joint torques [14]. In model based controllers, an exact knowledge of the parameters of the system is necessary. So in controlling of the human body based in the simplified physical models the anthropometric parameters of the body should be available. These data are different based on the age and gender. Usually these data are extracted based on the measurements on cadavers and used for a specific range of people. Therefore it is not possible to measure the human body parameters for each one and personalize the human body parameters. Hence there exists some error in controlling the human motion due the uncertainty in the parameters of the model. In this paper, an adaptive controller is proposed for controlling the human motion in a specified movement. Adaptation laws should be designed to estimate the parameters Article History: JICE DOI: / , Received Date: 3 Jun. 216, Accepted Date: 28 Aug. 216, Available Online: 3 Oct

2 of the model. So the stabilization of the human body would be fulfilled although the parameters are not known precisely. The human body is considered as an inverted pendulum and the nonlinearities of the model are taken in to account and linearization of the model is avoided which is done in some papers. A nonlinear controller for compensation of the gravitational effects and disturbances in the input is designed. The simulation of the model with the proposed controller would show the efficiency of the proposed method. II. HUMAN BODY MODELING A. Equations of Motion As mentioned above, simple inverted pendulum doesn't show the effects of all joints. So in this study a double inverted pendulum is used for modeling the human body. The model has two degree of freedom including ankle and hip joint. UU = mm 1 ggggll 1 ccccccθθ 1 + mm 2 ggll 1 ccccccθθ 1 + mm 2 ggll 2 ccccccθθ 2 (2) Lagrange equation is as follows dd (3) = uu dddd θθ ii θθ ii ii LL = TT UU (4) by substituting the energy terms in Lagrange equation and simplifying (II 1 + mmll 1 2 kk mm 2 ll 1 2 )θθ 1 + mm 2 ll 1 ll 2 cos(θθ 1 θθ 2 ) θθ 2 +ccθθ 2 2 sin(θθ 1 θθ 2 ) + dd ssssssθθ 1 = uu 1 mm 2 ll 1 ll 2 cos(θθ 1 θθ 2 ) θθ 1 + II 2 + mm 2 ll 2 2 θθ 2 (6) ccθθ 12 sin(θθ 1 θθ 2 ) + ee ssssssθθ 2 = uu 2 Now rearrange the equations of motion in the following form HH(qq)qq + CC(qq, qq )qq + gg(qq) = uu (7) where aa cc cos (θθ HH(qq) = 1 θθ 2 ) cc cos (θθ 1 θθ 2 ) bb (5) ccθθ 2sin (θθ CC = 1 θθ 2 ) ccθθ 1sin (θθ 1 θθ 2 ) gg = dd ssssssθθ 1 ee ssssssθθ 2 Fig. 1. human body model in upright stance The parameters of the model are as follows: also the parameters are aa = II 1 + mm 1 ll 1 2 kk 2 + mm 2 ll 1 2 bb = II 2 + mm 2 ll 2 2 Symbol ll 1 ll 2 II 1 II 2 mm 1 mm 2 kk Table Ι. The parameters of the model Definition Lower extremity length The distance of the hip joint to C.G of the upper extremity Lower extremity moment of inertia Upper extremity moment of inertia Lower extremity mass Upper extremity mass The length ratio θθ 1 and θθ 2 are the absolute angles of the ankle and hip joints. To derive the dynamics equations of motion, Lagrange method is implemented. For this purpose, the kinematic and potential energies of the system are calculated: TT = 1 2 II 1θθ mm 1ll 1 2 kk 1 2 θθ II 2θθ mm 2[ll 1 2 θθ 12 +ll 2 2 θθ ll 1 ll 2 θθ 1 2 θθ 2 2 cos(θθ 1 θθ 2 )] and the potential energy (1) cc = mm 2 ll 1 ll 2 dd = (mm 2 + mm 1 kk)ggll 1 III. ADAPTIVE INVERSE DYNAMICS CONTROL ee = mm 2 ggll 2 In this section adaptive inverse dynamics control is explained [15]. this controller is based on the inverse dynamics but since the parameters of system are uncertain, an approximation of the matrices of the system are used in the control law which are variable with time and some rules are designed to adapt the parameters in each step. Consider the following control command: uu = HH(qq)qq dd KK DD qq KK PP qq + CC (qq, qq )qq + gg(qq) (8) where H C and g are estimates of H C and g and q = q q d where q d is the desired of q. Also suppose that the system can be considered as linear in parameters such that uu = YY(qq, qq, qq )ρρ (9) where ρ is (r 1) consisting of system parameters and Y(q, q, q ) is an (n r) matrix. Since the parameters of the system are unknown so 196

3 uu = YY(qq, qq, qq )ρρ (1) where ρ is an estimation of the parameters. By applying the control law to (7) the closed loop error equation of the system is derived HH(qq)qq + KK DD qq + KK PP qq = YY(qq, qq, qq )ρρ (11) l 1 (m) l 2 (m) I 1 (kg/m 2 ) I 2 (kg/m 2 ) m 1 (kg) m 2 (kg) k And YY(qq, qq, qq )ρρ = HH(qq) HH(qq) qq + CC (qq, qq ) CC(qq, qq ) qq + (gg(qq) gg(qq)) suppose that H(q) is invertible so we have (12) q + K D q + K P q = H 1 (q)y(q, q, q ) = Φ(q, q, q, ρ)ρ (13) T by choosing ξ 1 = q and ξ 2 = q, ξ = (ξ 1 ξ T 2 ) T as state variables. The state space equation is obtained ξξ = AAAA + BBΦρρ (14) AA = KK PP II KK DD BB = II the adaptation law is for the parameters is ρρ = Γ 1 Φ TT BB TT PPPP (15) now we can use the above formulation to control the system to the desired trajectory. The equation of the motion should be written in the linear in parameter form. Since there exists 7 parameters in the model, we should select 7 parameters in the linear in parameters form and adapt them. The equations of motion in the parametric form are YY = [ qq 1 qq 1 qq 2 qq 2 cos(qq 1 qq 2 ) qq 2 + qq 2 2 sin(qq 1 qq 2 ) sinqq 1 cos(qq 1 qq 2 ) qq 1 qq 1 2 ] sin (qq 1 qq 2 ) sinqq 2 ρρ = [II 1 mm 1 ll 2 1 kk mm 2 ll 1 II 2 mm 2 ll 2 mm 2 ll 1 ll 2 (mm 2 + mm 1 kk)ggll 1 mm 2 ggll 2 ] TT (16) (17) IV. SIMULATION According to the method presented in the previous section, the joint torques of hip and ankle for a specific movement would be designed. The nominal parameters of the model are shown in Table ΙΙ [16]. in this simulation the purpose is to do a flexion from (θ 1, θ 2 ) = (,) to (θ 1, θ 2 ) = ( π/6, π/3) according to the below pattern (18) θθ 1dd = tttttt 1 (2(tt 2)) (19) θθ 2dd = tttttt 1 (4(tt 2)) For the estimation of the parameters we assume 1 percent error exists in the anthropometric data and based on the adaptation law they are changed. The other parameters of the control rule and adaptation equation are represented in Table ΙΙΙ. TableΙΙΙ. parameters of the control law KK PP = 5 Γ = 2 II(7) 5 PP = KK PP +.5KK DD. 5 II(2) KK DD = II(2) KK PP +.5KK DD The results of the simulation are shown in the following figures. θ 1 (rad) Fig. 2. variation of θθ 1 θ 2 (rad) Fig. 3. variation of θθ 2 actual desired actual desired -.2 Table ΙΙ. The parameters of the human body 197

4 a (kg.m 2 ) Fig. 4. variation of aa 5.5 e (kg.m 2 /s 2 ) Fig. 8. variation of ee b (kg.m 2 ) u 1 (Nm) Fig. 5. variation of bb -25 Fig. 9. Torque of the ankle c (kg.m 2 ) u 2 (Nm) Fig. 6. variation of cc d (kg.m 2 /s 2 ) Fig. 7. variation of dd -2 Fig. 1. Torque of the hip Fig. 2 and Fig. 3 demonstrate that the controller is designed very well and the tracking problem is accomplished precisely. Although the exact values of the parameters are not known but the controller has tracked the desired trajectory of the joints with a high accuracy. The parameters of the system converge to constant values and remain stable but they are not the actual values of them. Fig. 1 and Fig. 11 show the torques applied at the joints of the body. These reveal that the torque acting at the ankle is greater than the hip during flexion. V. CONCLUSION In this paper an inverted pendulum model for human body in standing position proposed. The equations of motion converted to the robotic form to use the robotic control methods. Since the anthropometric data for each person is not available, the data extracted from cadaver measurements should be used. so they are not accurate and uncertainty exists in the parameters. So an adaptive inverse dynamics was used to control the system for a specified motion and adaptation 198

5 laws were designed for parameters. The simulation of the system with the suggested controller showed the effectiveness of the method during a flexion movement. REFERENCES [1] Stevens, J. A., Corso, P. S., Finkelstein, E. A., & Miller, T. R., The costs of fatal and non-fatal falls among older adults, Injury prevention, 12(5), , 26. [2] Murray, M. P., Seireg, A., & Scholz, R. C., Center of gravity, center of pressure, and supportive forces during human activities, Journal of Applied Physiology, 23(6), , [3] Gurfinkel, V., Osevets, M., Dynamics of the vertical posture in man, Biophysics 17, , [4] Geursen, J., Altena, D., Massen, C., Verduin, M., A model of standing man for the description of his dynamic behaviour, Agressologie 17, 63 69, [5] Winter, D., Patla, A., Ishac, M., Gage, W., Motor mechanisms of balance during quiet standing, Journal of Electromyography and Kinesiology 13 (1) 49 56, 23. Madigan, M., Davidson, B., Nussbaum, M., Postural sway and joint kinematics during quiet standing are affected by lumbar extensor fatigue, Human Movement Science 25 (6), , 26. [6] Bottaro, A., Yasutake, Y., Nomura, T., Casadio, M., Morasso, P., Bounded stability of the quiet standing posture: an intermittent control model, Human Movement Science 27 (3), , 28. [7] Winter, D., Patla, A., Prince, F., Ishac, M., Gielo-Perczak, K., Stiffness control of balance in quiet standing, Journal of Neurophysiology 8 (3), , [8] Loram, I., Maganaris, C., Lakie, M., Human postural sway results from frequent, ballistic bias impulses by soleus and gastrocnemius, The Journal of Physiology 564 (Pt 1), , 25. [9] Xingda, Q., Muray A., A balance control of quiet upright stance based on optimal control strategy, Journal of Biomechanics 4 (16), , 27. [1] Qu, X., & Nussbaum, M. A. (). Modelling 3D control of upright stance using an optimal control strategy, Computer methods in biomechanics and biomedical engineering, 15(1), , 212. [11] Gunther, M., Grimmer, S., Siebert, T., Blickhan, R., All leg joints contribute to quiet human stance: A mechanical analysis, Journal of Biomechanics 42, [12] Li, C. L., Lin, C. L., Chen, C. K., Stabilizing postural control for emulated human balancing systems, International Journal of Engineering Science 46, , 28. [13] Guelton, K., Delprat, S., Guerra, T. M., An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi Sugeno unknown-inputs observer in the descriptor form, Control Engineering Practice 16, , 28. [14] Canudas de Wit, C., Siciliano B., Bastin G., Theory of robot control, London: Springer, [15] Winter, D. A., Biomechanics and motor control of human movement, New York: Wiley,