Adaptive Robust Posture Control of a Pneumatic - Purdue University

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1 Proceedings of the 2007 American Control Conference ThB18.5 Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 Adaptive Robust Posture Control of a Pneumatic Muscles Driven Parallel Manipulator with Redundancy Guoliang Tao, Xiaocong Zhu, Bin Yao and Jian Cao AbstractThis paper presents an adaptive robust posture and high power/volume ratio of pneumatic muscles, which controller for a pneumatic muscles driven parallel manipulator will have promising wide applications in robotics, industrial with a redundant degree-of-freedom (DOF). The symmetric automation, and bionic devices. [2] geometric structure of the parallel manipulator driven by Due to the symmetric geometric structure of the parallel identical pneumatic muscles studied in this paper makes the manipulator and the identical properties of the pneumatic rotation angle of the manipulator along its axial direction negligible and a non-factor in using the manipulator. As such, muscles used, the rotation angle of the moving platform along the axial rotation angle is normally not measured and controlled the axial direction of the parallel manipulator (i.e., z-axis in when these types of manipulator are used in practice, leading to Fig.1) is nearly close to zero during normal operations. Thus, a single DOF redundancy in synthesizing the precise posture this angle is normally not measured and controlled to save controller for rotation angles along other axes. To make full use cost when using these types of parallel manipulators in of this redundancy as well as effectively tackle severe practice, leading to a DOF redundancy in synthesizing uncertainties in the system dynamics, an equivalent average-stiffness-like desired constraint is introduced in the posture controllers for the rotation angles along the other two development of adaptive robust posture controller to achieve axes. Such an added design freedom is not utilized in our precise posture tracking while reducing control chattering due previous study where an adaptive robust posture controller to measurement noise. Experimental results are obtained to was developed for pneumatic muscle driven parallel verify the validity of the proposed controller for the pneumatic manipulator without redundancies. [3] muscles driven redundant parallel manipulator. The utilization of adjustable stiffness concept in the control of the pneumatic muscle systems is not new. Many I. INTRODUCTION researchers have investigated adjusting the average pressure P neumatic muscle is a new kind of flexible actuator similar to human muscle, which is made up of rubber tube and cross-weave sheath material. When the rubber tube is inflated, scalar with respect to the entire stiffness of the single or antagonistic pneumatic muscle system to eliminate the redundant freedom and improve performance. [4-9] However, the cross-weave sheath experiences lateral expansion, especially for the pneumatic muscles driven parallel resulting in axial contractive force and the change of the end manipulator, not only it is very difficult to control the posture point position of pneumatic muscle. Thus, the position or in the presence of various parametric uncertainties and force control of a pneumatic muscle along its axial direction uncompensated model uncertainties that are inherited to the can be realized by regulating the inner pressure of its rubber pneumatic muscle system, but also there are added difficulties tube.[1] The parallel manipulator driven by pneumatic muscles in utilizing the adjustable stiffness since it requires finding a can realize rotation motion, which consists of three pneumatic uniform equivalent average stiffness scalar derived from the muscles connecting the moving platform of the parallel coupling equivalent stiffness matrix associated with the manipulator to its base platform as shown in Fig.1.Such a coupled multi-input-multi-output (MIMO) complex parallel manipulator combines the advantages of compact dynamics of the parallel manipulator. structure of parallel mechanisms with the adjustable stiffness In this paper, the adaptive robust posture control strategy [14,15] is used to effectively deal with large extent of nonlinear Manuscript received March 15, 2007. This work is supported by Festo Inc. uncertainties and parametric uncertainties to achieve accurate through an international cooperation. The third author is supported by the posture tracking control. In addition, with the guaranteed National Natural Science Foundation of China (NSFC) under the Joint Research Fund (grant 50528505) for Overseas Chinese Young Scholars. small tracking errors of the proposed adaptive robust posture Guoliang Tao is with the State Key Laboratory of Fluid Power control, the method of the prior equivalent average stiffness Transmission and Control, Zhejiang University, China. (telephone: adjustment is developed to eliminate the redundant DOF as +86-571-87951318; fax: +86-571-87951491; e-mail: [email protected] cn). Xiaocong Zhu is with the State Key Laboratory of Fluid Power well as to reduce control input chattering due to noise or to Transmission and Control, Zhejiang University, China. (e-mail: save energy. [email protected]) Bin Yao is with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA, and a Kuang-piu Professor at the State Key II. PRINCIPLE OF POSTURE CONTROL WITH REDUNDANCY Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. ([email protected]). A. Dynamics of the parallel manipulator with redundancy Jian Cao is with the State Key Laboratory of Fluid Power Transmission The parallel manipulator driven by three pneumatic and Control, Zhejiang University, China. (e-mail: [email protected] com). muscles is shown in Fig.1. Define the posture vector as =[x, 1-4244-0989-6/07/$25.00 2007 IEEE. 3408

2 ThB18.5 y, z]T and the pressure vector as p=[p1, p2, p3]T, the moving platform and the base platform and the identical dynamics of the parallel manipulator is as Eq.(1)[3] properties of the pneumatic muscles used, z is nearly close to M a ( ) + Ca ( , ) + Ga ( ) + Ffa (, ) + d ta (t ) = J a T ( ) Fm (1) zero in normal operations. For this reason, it is normally not measured and controlled when using these types of where , , R 3 is the posture vector, velocity vector and manipulators in practice. Thus, in the following controller acceleration vector of the parallel manipulator, Ma() is the design, z is assumed to be zero, and subsequently, from 33 rotational inertial matrix, C a (, ) is the 3-vector of Eq.(1), the system dynamics for the rotation angles along the centripetal and Coriolis torques, Ga() is the 3-vector of other two axes can be written as gravitational torques, Ffa (, ) is all kinds of friction forces M(x)x + C(x, x)x + G(x) + Ff (x, x) + dt (t)= = f ( x) p + g ( x) (6) coming from the pneumatic muscles and link joints, dta(t) is p = fm ( x)qm + gm ( x, x, p) + dm (t) the disturbance in task-space and Ja() is Jacobian qm = Kq ( p, sign(u))u transformation matrix, Fm =[Fm1,Fm2,Fm3]T is the muscle force vector with each component Fmi is given by Eq.(2) where x = [ x , y ] R2 , M (x ) , C (x, x ) , G (x ) , Ff (x, x) , T without considering the modeling error. [12,13] d t (t ) and are the corresponding 2-vecor or 22 matrix Fmi ( xmi , pi ) = pi [a (1 k i ) 2 b]+Fri ( xmi ) (2) from M a ( ) , C a ( , ) , Ga ( ) , Ffa ( , ) , d ta (t ) and where a and b are constants related to the structure of a respectively, f ( x ) and g ( x ) are the corresponding 23 pneumatic muscle, k is a factor accounting for the slide effect, components of f a ( ) and ga ( ) respectively. i is the contractive ratio given by i =xmi/L0, L0 is the initial muscle length, and Fri is the compensation for the rubber B. The principle of posture control with redundancy elastic force. With the backstepping design procedure, the desired The torque in task-space is a= JaT()Fm, thus substituting torque d in task-space can be calculated under any control Eq.(2) into Eq.(1), a can be described as the following form. strategy. Then, from Eq.(6), one obtains. a ( , p) = f a ( ) p + ga ( ) (3) d ( x, x, xd , xd , xd ) = f ( x ) pd + g ( x ) (7) [3] The pressure dynamics in muscle-space is However, the solution of desired pressure pd from d along p = f m ( xm )qm + gm ( xm , xm , p) + d m (t ) (4) Eq.(7) is not unique since d is a 21 vector and pd is a 31 where f m ( xm ) , gm ( xm , xm , p) are nonlinear functions vector with the matrix f(x) is a 23 matrix. The system and d m (t ) is disturbance in muscle-space. possesses a redundant freedom; hence an extra constraint related to the pressure p is introduced into the posture control Mass flow rate of air through the fast switching valve is a design to remove the redundancy. function of its duty cycle u given by Considering the manipulator being at the equilibrium state qm = K q ( p, sign(u))u (5) ( x = x = x = [0, 0]T ) and p / 0 , an equivalent stiffness where Kqi is a nonlinear flow gain function referred to [3]. is described as Eq.(8) [11]. z(z1) z f a g y y1 1 K ( p, a ) = p + a (8) B3 2 a a 3 Therefore, the equivalent stiffness is mainly dependent on r O1 B2 the inner pressure of the pneumatic muscle and on the posture B1 x of the manipulator. In addition, the equivalent stiffness is a x1 principle diagonal matrix, whose components have the l3 h 4 feature that magnitudes of the equivalent stiffness along x and l1 l2 y axes are at the same level and much larger than equivalent y 5 A3 stiffness along z-axis. Then, an equivalent average stiffness of the manipulator R A2 Kd=(Kx+Ky)/2 is defined as Eq.(9), which also depends on the O 6 inner pressure of the pneumatic muscle. x Kd = f k ( x) p + gk ( x) (9) A1 Therefore, integrating with Eq.(7) and Eq.(9), the desired 1. Moving platform 2. Ball joint 3.Spherical joint pressure in task-space is calculated by 4. Central pole 5. Pneumatic muscle 6. Base platform pd = 11 ( x )[ 2 ( x )] (10) Fig 1. Structure of the pneumatic muscles driven parallel manipulator f ( x) g ( x ) d where 1 ( x ) = f ( x ) , 2 ( x) = g ( x) , = K Due to the symmetric distribution of the link-joints on the k k d 3409

3 ThB18.5 Hence, any desired torque and equivalent average stiffness where 2 is a positive design parameter referred in [10]. could be achieved in the proposed posture controller of the As analyzed in section 2, according to the desired torque d manipulator with a redundant freedom. and the equivalent average stiffness Kd designed in next subsection, the desired pressure pd can be obtained by III. CONTROLLER DESIGN Eq.(10). Define the positive semi-definite Lyapunov function A. ARC design of posture control V2=z2TMz2T/2 and let the input discrepancy be z3=ppd. The Rather severe parametric uncertainties and nonlinear derivative of V2 along the solution of Eq.(13) is uncertainties exist in the dynamics of pneumatic muscles V2 = z2T K 2 z2 + z2T f z3 + z2T ( ds2 + 2T t d t ) (17) driven parallel manipulator, both due to the hysteresis and time-varying friction forces and the large extent of modeling 2) Step2 errors of pneumatic muscle as well as the strong coupling and A virtual control input qm is synthesized so that z3 inherent nonlinearities of the manipulator. An adaptive robust converges to zero or a small value with a guaranteed transient control is adopted to deal with the parametric uncertainties performance. and unknown nonlinearities[3,10] Let d m = d m0 + d m . With a set of unknown parameter vector 1) Step1 in muscle-space denoted by m, the term of parametric Define a switching-function-like quantity as uncertainties in muscle-space is z2 = z1 + K c z1 (11) fm ( x)qm + gm ( x, x, p) + dm0 =3T ( x, x, p) m + fc3 ( x, x, p) (18) where z1=xyd is the trajectory tracking error vector and Kc is where 3T ( x, x, p) is the regressor for parameter adaptation a positive diagonal feedback matrix. If z2 converges to a small in muscle-space and f c3 ( x , x , p) is the known vector. value or zero, then z1 will converge to a small value or zero since the transfer function from z2 to z1 is globally The derivative of input discrepancy along the solution of asymptotically stable. Thereby, the next objective is to design Eq.(6) and Eq.(18) is the torque to make z2 as small as possible. z3 = p pd = fmqm + gm + dm0 + dm pdc pdu (19) Define xr = yd K c z1 and let d t = d t0 + d t . With a set of pd pd pd pd unknown parameter vector in task-space denoted by t, the where pdc = + x+ x+ t , term due to parametric uncertainties in task-space can be t x x t described as pd p and pdu = ( x x ) + d ( x x ) , in which x and x are Mxr + Cxr + G + Ff + dt0 = ( x, xr xr ) t + fc2 ( x, xr , xr ) (12) 2 T x x obtained from x by an output differential observer [3], pdc where 2T ( x , xr xr ) is the regressor for parameter adaptation represents the calculable part of pd and can be used to design in task-space and f c2 ( x , xr , xr ) is the known vector. And the following equation is obtained along the solution control functions, but pdu can not due to various of Eq.(6) and Eq.(12). uncertainties. The virtual flow input qmd is designed as follows Mz 2 + Cz 2 = 2 T t f c2 + 2 T t d t (13) qmd = qmda + qmds , qmda = fm1 ( fT z2 gm dm0 + pdc ) (20) The virtual control input d consists of two parts given by where qmda is used for adaptive model compensation through d = da + ds , da = 2 T t + f c2 (14) on-line parameter adaptation via m , qmds is a robust control where da is used for adaptive model compensation through law to be synthesized later. m is updated by a discontinuous on-line parameter adaptation via t , ds is a robust control projection-based adaptation law m = Proj ( 33 ) with the law to be synthesized later. t is updated by a discontinuous projection-based adaptation law t = Proj ( 2 2 ) with the parameter adaptation function given by 3 = 3 z3 .[3] The robust control law qmds is given by parameter adaptation function given by 2 = 2 z2 .[3] ds consists of two parts for robust control qmds = qmds1 + qmds2 , qmds1 = fm 1 K 3 z3 (21) ds = ds1 + ds2 , ds1 = K 2 z2 (15) where K3 is a positive feedback gain matrix, qmds2 is a robust control function synthesized to dominate the model where K2 is a positive control gain matrix and ds2 is uncertainties and chosen to satisfy the following conditions. synthesized to dominate the model uncertainties, which is z3 (fm qmds2 3 m + d m pdu ) 3 T T chosen to satisfy the following conditions. (22) z2 ( ds2 + 2 t d t ) 2 T T z3T fm qmds2 0 (16) z2 T ds2 0 where 3 is a positive design parameter. To see how the above control function works, define a 3410

4 ThB18.5 p.s.d. function V3=V2+z3Tz3/2, its derivative along the solution posture obtained through a 2nd-order differential filter, which of Eq.(16), Eq.(17), Eq.(19),and Eq.(22) is may amplify noises and lead to control chattering. Therefore V3 z 2 T K1 z2 z3T K 3 z3 + 2 + 3 an optimal equivalent average stiffness is sought to decrease (23) noise gain and then reduce control chattering. The solution of Eq.(23) is The acceleration in feedback control consists of the v following two parts. V3 (t ) exp(vt )V3 (0) + [1 exp(vt )] (24) v x = x + n(t ) 0 (30) where v=2min{K2, K3}, v=2+3. where x 0 is the noise-free signal and n(t) is the measurement Thus, the tracking errors are ultimately bounded since z2 noise. When the trajectory tracking error is small, pd is nearly and z3 exponentially converge to some balls whose sizes are equal to p. With this approximation, the control input in proportional to 2 and 3. Furthermore asymptotic output Eq.(25) can be rewritten as tracking (or zero final tracking error) is obtained in the p ( K ) presence of parametric uncertainties only. u = K q 1 ( pd ( K d )) fm 1 d d [ x 0 + n(t )] + ( x , x , t ) (31) 3) Step3 x Once the control function qmd is synthesized, the inverse where ( x , x , t ) is a nonlinear function. flow mappings are used to calculate the specific duty cycle In order to obtain the optimal equivalent average stiffness, commands of the fast switching valves. the gain from noise to control inputs should be as small as u = K q 1qmd (25) possible. Thus the objective function to be minimized is pd ( K d ) B. Equivalent average stiffness design J opt = K q 1 ( pd ( K d )) fm 1 (32) x In the above subsection, the equivalent average stiffness Kd in Eq.(10) can be designed to reduce control chattering and The schematic diagram of adaptive robust posture control save energy as follows. with adjustable equivalent average stiffness is shown in Fig.2. 1) Permissible range of equivalent average stiffness The permissible and optimal equivalent average stiffness The equivalent average stiffness Kd is constrained by the along a desired sinusoidal trajectory (amplitude is x=5 and minimum and maximum permissible pressure of the entire y=4, period is 20s) are shown in Fig.3. system. C. Some issues on posture control with redundant DOF Another form of Eq.(2) is 1) The posture controller of the parallel manipulator with Fm ( xmi , pi ) = A( xmi ) pi + Fr ( xmi ) (26) such redundant freedom could be flexibly designed. Besides where A(xmi) is the equivalent area of pneumatic muscle the ARC adopted in this paper, other posture control methods For a single muscle, the minimum pressure is such as SMC or deterministic robust control can also be used pmini=Fr(xmi)/A(xmi) since the pneumatic muscle can only to obtain the desired torque. And then the equivalent generate positive contractive force. And the maximum average-stiffness-like desired constraint according to pressure is pmaxi=max{pmi, ps} in which pmi is the permissible requirements is integrated into the posture controller to maximum inner pressure of the pneumatic muscle and ps is effectively improve precise posture tracking as well as the pressure of source. Let pmin and pmax be the vectors anti-interference and reducing control chattering. consisting of such three pmini and pmaxi respectively. 2) Since it is only the equivalent average stiffness that is According to Eq.(10), the permissible pressure range is designed in controller, the equivalent stiffness along x and y pmin pd = 11 ( 2 ) pmax (27) axes fluctuates respectively due to the posture variations during a period, while the average of the two equivalent 1 Let 1 = a d + ak Kd in which a is a 32 matrix and ak stiffness values is almost invariant. is a 31 matrix from 11 , and substitute it into Eq.(27), then 3) The desired pressure pd can also be obtained by Eq. (6). pmin + 112 a d ak Kd pmax + 112 a d pd = f + ( x )( d g ) + [ I 33 f + ( x ) f ( x )] p0 (33) (28) where p0 is the equivalent stiffness pressure, which is an Hence, the permissible equivalent average stiffness for the arbitrary 31 vector. desired posture is obtained. Substitute the above pd as p into Eq.(10), thus the relation between p0 and Kd becomes Kdmax = min r1 , r2 , r3 , Kdmin = max l1 , l2 , l3 (29) ak1 ak2 ak3 ak1 ak2 ak3 Ak ( x ) p0 = Bk ( x, K d ) (34) where r = pmax + 2 a d = [ r1 , r2 , r3 ] , 1 T 1 where Ak ( x ) = f k ( x )[ I 33 f + ( x ) f ( x )] , l = pmin + 2 a d = [ l1 , l2 , l3 ] 1 1 T Bk ( x , K d ) = K d g k ( x ) f k ( x ) f + ( x )( d g ) 2) Optimal equivalent average stiffness in which f + ( x ) is the pseudo-inverse of f ( x ) . The control inputs contain the acceleration of manipulator 3411

5 ThB18.5 If the three components of vector p0 are set to the same equivalent average stiffness only used in Fig.6 greatly value represented by the average pressure p0 , a unique reduces control chattering and saves energy while the mapping between the equivalent average stiffness Kd and the tracking errors are as small as those of Fig.4 and Fig.5. average pressure p0 will be established. In general, the The experimental results of tracking the same trajectory with time-varying equivalent average stiffness are shown in methods of adjusting equivalent average stiffness and Fig.7. As can be seen from it, the time-varying equivalent regulating average pressure almost have the same effects on average stiffness hardly has influence on the tracking errors control performance, but the method of regulating average during the control process. In other words, the added pressure is simpler. equivalent average-stiffness-like desired constraint almost doesnt influence the posture control. Desired Kd equivalent Pneumatic To sum up, all results prove the original intention that average stiffness pd Pneumatic u muscles posture control with equivalent average-stiffness-like desired 11 i muscle driven Desired torque d ( 2 ) controller parallel constraint for the parallel manipulator with redundancy could using adaptive manipulator not only guarantee small tracking errors but also reduce robust control control chattering and save energy. t m Parameter x, p estimation y d , y d , y d , yd Fig. 2. Schematic diagram of posture control Fig. 4. Sinusoidal tracking results with Kd =800 N.m/rad Fig. 3. The permissible and optimal equivalent average stiffness along a trajectory IV. EXPERIMENTAL RESULTS The experimental setup in [2] is used to evaluate the effectiveness of the proposed posture controller with the constraint of equivalent average stiffness. The controller of the parallel manipulator is firstly tested for tracking a sinusoidal desired posture with the amplitude of x=5,y=4 and the period of 20s under the same conditions except different constant equivalent average stiffness shown in Fig.4, Fig.5 and Fig.6. As seen, larger equivalent average stiffness of the manipulator requires higher inner pressure of pneumatic muscle and larger control efforts while brings smaller control chattering. Due to the larger hysteresis and time-varying friction forces caused by larger average pressure, the tracking errors increase slightly. On the contrary, smaller equivalent average stiffness results in lower inner pressure of pneumatic muscle and smaller control efforts Fig. 5. Sinusoidal tracking results with Kd =1650 N.m/rad while brings much more control chattering due to the systems sensitivity to disturbances and measurement noises under lower pressure. It is especially noted that the optimal 3412

6 ThB18.5 REFERENCES [1] A. Hildebrandt, O. Sawodny, R. Neumann and A. Hartmann, A flatness based design for tracking control of pneumatic muscle actuators, in Seventh International Conference on Control, Automation, Robotics and Vision (ICARCV02), vol. 3, Singapore, Dec. 2002, pp. 11561161. [2] G. Tao, X. Zhu and J. Cao, Modeling and controlling of parallel manipulator joint driven by pneumatic muscles, Chinese Journal of Mechanical Engineering (English Edition), vol.18, no.4, pp. 537541, 2005 [3] X. Zhu, G. Tao, J. Cao and B. Yao, Adaptive robust posture control of a pneumatic muscles driven parallel manipulator, in 4th IFAC Symposium on Mechatronic Systems, Heidelberg, Germany. Sep.12-14, 2006, pp.764769. [4] L. Sui, Z. Wang and G. Bao, Analysis of stiffness characteristics of pneumatic muscle actuators, China Mechanical Engineering, vol. 15, no.3, pp.242244, 2004. [5] B. Tondu, V. Boitier and P. Lopez, Naturally Compliant robot-arms actuated by McKibben Artificial Muscles, 1994 IEEE International Conference on Systems, Man and Cybernetics, vol.3, San Antonio, TX, Oct. 2-5, 1994, pp.26352640. [6] R. Liu, On the compliant control of parallel mechanism and actuating Fig. 6. Sinusoidal tracking results with Kd=1500 N.m/rad with artificial muscle, Beijing University of Aeronautics and Astronautics, PhD dissertation, 1996. [7] G. Tonietti and A. Bicchi, Adaptive simultaneous position and stiffness control for a soft robot arm, in Proc. of the 2002 IEEE/RSJ Conference on Intelligent Robots and Systems EPFL, Lausanne, Switzerland, vol.2, 30 Sept.-5 Oct. 2002, pp. 19921997. [8] R. Q. van der Linde, Design, analysis, and control of a low power joint for walking robots by phasic activation of Makibben muscles, IEEE Transactions on Robotics and Automation, vol. 15, no.4, pp. 599604,1999. [9] A. Hildebrandt, O. Sawodny, R. Neumann, and A. Hartmann, Cascaded control concept of a robot with two degrees of freedom driven by four artificial pneumatic muscle actuators,2005 American Control Conference, Portland, OR, USA. Jun. 8-10, 2005, pp. 680685. [10] B. Yao and M. Tomizuka, Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica, vol. 3, no.5, pp. 893900, 1997. [11] Y. Li, J. Wang and L. Wang, Stiffness analysis of a Stewart platform-based parallel kinematic machine, in Proc. of the 2002 IEEE International Conference on Robotics & Automation, Washington DC, May 2002, pp.3672-3677. [12] N. Tsagarakis and D. G. Caldwell, Improved modelling and assessment of pneumatic muscle actuators, in Proc. of the 2000 IEEE International Conference on Robotics & Actuators, vol.4, San Fig. 7. Sinusoidal tracking results with time-varying Kd Francisco, CA, Apr. 24-28, 2000, pp. 36413646. [13] C. P. Chou and B. Hannaford, Measurement and modeling of McKibben pneumatic artificial muscles, IEEE Transactions on V. CONCLUSIONS Robotics and Automation, vol. 12, no.1, pp. 90102, 1996. [14] B. Yao, Advanced motion control: an adaptive robust control The adaptive robust posture control with adjustable framework, The 8th IEEE International Workshop on Advanced equivalent average stiffness as a constraint is proposed for Motion Control, Kawasaki, Japan, March 25-28, 2004, pp. 565570. precise planar posture tracking control of the pneumatic [15] B. Yao and M. Tomizuka, Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form, Automatica, vol. 3, no.5, pp. muscles driven parallel manipulator with a redundant design 893900, 1997. freedom. An adaptive robust posture controller has been developed to effectively deal with the model uncertainties existing in the complex dynamics of such a system. Meanwhile, to make full use of the added design freedom of not controlling the rotational angle of the pneumatic muscle driven parallel manipulator, the equivalent average stiffness adjustment is introduced to integrate with the ARC controller and its optimal value is determined to achieve secondary objectives such as reducing the control chattering due to measurement noise. Comparative experimental results are presented to illustrate the proposed control strategy as well. 3413

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