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英文原文Adaptive robust posture control of a parallel manipulator driven by pneumatic musclesKeywords:Pneumatic muscleParallel manipulatorAdaptive controlNonlinear robust controlAbstractRather severe parametric uncertainties and uncertain nonlinearities exist in the dynamic modeling of a parallel manipulator driven by pneumatic muscles. Those uncertainties not only come from the time-varying friction forces and the static force modeling errors of pneumatic muscles but also from the inherent complex nonlinearities and unknown disturbances of the parallel manipulator. In this paper, a discontinuous projection-based adaptive robust control strategy is adopted to compensate for both the parametric uncertainties and uncertain nonlinearities of a three-pneumatic-muscles-driven parallel manipulator to achieve precise posture trajectory tracking control. The resulting controller effectively handles the effects of various parameter variations and the hard-to-model nonlinearities such as the friction forces of the pneumatic muscles. Simulation and experimental results are obtained to illustrate the effectiveness of the proposed adaptive robust controller. 2008 Elsevier Ltd. All rights reserved.1. IntroductionPneumatic muscle is a new type of flexible actuator similar to human muscle. It is usually made up of a rubber tube and crossweave sheath material. Pneumatic muscles have the advantages of cleanness, cheapness, light-weight, easy maintenance, and the higher power/weight and power/volume ratios when compared with pneumatic cylinders (Ahn, Thanh, & Yang, 2004). The basic working principle of a pneumatic muscle is as follows: when the rubber tube is inflated, the cross-weave sheath experiences lateral expansion, resulting in axial contractive force and the movement of the end point position of the pneumatic muscle. Thus, the position or force control of a pneumatic muscle along its axial direction can be realized by regulating the inner pressure of its rubber tube. The parallel manipulator driven by pneumatic muscles (PMDPM) studied in this paper is a new application of pneumatic muscles.It consists of three pneumatic muscles connecting the moving arm of the parallel manipulator to its base platform as shown in Fig. 1. By controlling the lengths of three pneumatic muscles,three degrees-of-freedom (DOF) rotation motion of the parallel manipulator can be realized. Such a parallel manipulator combines the advantages of compact structure of parallel mechanisms with the adjustable stiffness and high power/volume ratio of pneumatic muscles, which will have promising wide applications in robotics,industrial automation, and bionic devices.Many researchers have investigated the precise position control of pneumatic muscles during the past several years.Most of them dealt with the control of single or antagonistic pneumatic muscles. Specifically, Bowler, Caldwell, and Medrano-Cerda (1996), employed an adaptive pole-placement scheme to control a bi-directional pneumatic muscle actuator system, for use on a 7-DOF anthropomorphic robot arm. Cai and Yamaura (1996)presented a sliding mode controller for a manipulator driven by artificial muscle actuators. Kimura, Hara, Fujita, and Kagawa(1997), applied the feedback linearization method to the position control of a single-input pneumatic system with a third-order dynamics including the pressure dynamics. Kimoto and Ito (2003)added nonlinear robust compensations to a linear controller in order to stabilize the system globally and achieve robustness to uncertain nonlinearities. Carbonell, Jiang, and Repperger (2001),Chan, Lilly, Repperger, and Berlin (2003), Repperger, Johnson, andPhillips (1998) and Repperger, Phillips, and Krier (1999), proposed several methods such as fuzzy backstepping, gain-scheduling,variable structure and fuzzy PD+I for a SISO pneumatic muscle system with a second-order dynamics to achieve asymptotic position tracking. Lilly (2003), Lilly and Quesada (2004) and Lilly and Yang (2005), applied the sliding mode control technique with boundary layer to control pneumatic muscle actuators arranged in bicep and tricep configurations. Tian, Ding, Yang, and Lin(2004), adopted the RPE algorithm to train neural networks for modeling and controlling an artificial muscle system. Hildebrandt,Sawodny, Neumann, and Hartmann (2002); Sawodny, Neumann,and Hartmann (2005), presented a cascade controller for a twoaxis planar articulated robotic arm driven by four pneumatic muscles.As reviewed above, some of the researchers designed robust controllers without considering the pressure dynamics, while the effect of pressure dynamics is essential for the precise control of pneumatic muscles (Carbonell et al., 2001; Chan et al., 2003; Lilly, 2003; Lilly & Quesada, 2004; Lilly & Yang, 2005; Repperger et al., 1998,1999). Some of the researchers developed controllers with the assumption that the system model is accurate, or that model uncertainties satisfy matching condition only, while those assumptions are hard to be satisfied in practice (Hildebrandt et al., 2002, 2005; Kimura et al., 1997). For the PMDPM shown in Fig. 1, it not only has all the control difficulties associated with the pneumatic muscles, but also the added difficulties of the coupled multi-input-multi-output (MIMO) complex dynamics of the parallel manipulator and the large extent of unmatched model uncertainties of the combined overall system. In other words,there exist rather severe parametric uncertainties and uncertain nonlinearities, which are caused not only by the time-varying friction forces and static force modeling errors of pneumatic muscles but also by the inherent complex nonlinearities and unknown disturbances of the parallel manipulator. Therefore, it is very difficult to control precisely the posture of the PMDPM.The recently proposed adaptive robust control (ARC) has been shown to be a very effective control strategy for systems with both parametric uncertainties and uncertain nonlinearities (Xu & Yao,2001; Yao, 2003; Yao, Bu, Reedy, & Chiu, 2000; Yao & Tomizuka,2001). This approach effectively integrates adaptive control with robust control through utilizing on-line parameter adaptation to reduce the extent of parametric uncertainties and employing certain robust control laws to attenuate the effects of various uncertainties. In ARC, a projection-type parameter estimation algorithm is used to solve the design conflict between adaptive control and robust control. Thus, high final tracking accuracy is achieved while guaranteeing excellent transient performance.In this paper, the posture control of a PMDPM shown in Fig. 1 is considered in which each pneumatic muscle is controlled by two fast switching valves. The adaptive robust control strategy is applied to reduce the lumped uncertain nonlinearities and parametric uncertainties while using certain robust feedback to attenuate the effects of uncompensated model uncertainties.In addition, pressure dynamics are explicitly considered in the proposed controller. Consequently, good tracking performance is achieved in practice as demonstrated by simulation and experimental results.The paper is organized as follows: Section 2 gives the dynamic models of the PMDPM controlled by fast switching valves. Section3 presents the proposed adaptive robust controller, along with proofs of the stability and asymptotic output tracking of the resulting closed-loop system. Section 4 shows the advantages of the proposed adaptive robust controller over the traditional deterministic robust controller via simulation results. Section 5 details the obtained experimental results to verify the effectiveness of the proposed adaptive robust posture controller and Section 6 draws the conclusions.2. Dynamic modelsThe PMDPM shown in Fig. 1 consists of a moving platform,a base platform, a central pole, and three pneumatic muscles.Pneumatic muscles are linked with the moving platform and the base platform by spherical joints Bi and Ai (i = 1, 2, 3) respectively.The joints are evenly distributed along a circle on the moving platform and the base platform respectively. The central pole is rigidly fixed with the base platform and is connected with the moving platform by a ball joint. Consider two frames, the first one,reference frame Oxyz fixed to the base platform, and the second one, moving frame O1x1y1z1 attached to the moving platform at the center (Tao, Zhu,&Cao, 2005). The posture of thePMDPMis defined through the standard RollPitchYaw (RPY) angles: first rotate the moving frame around the fixed x-axis by the yaw angle _x, then rotate the moving frame around the fixed y-axis by the pitch angle_y, and finally rotate the moving frame around the fixed z-axis by the roll angle _z. Two fast switching valves are used to regulate the pressure inside each pneumatic muscle, and this combination is subsequently referred to as a driving unit.Some realistic assumptions are made as follows to simplify the analysis: (a) The working medium of the pneumatic muscles satisfies the ideal gas equation, (b) Resistance and dynamics of various pipes of the pneumatic muscles are neglected, (c) The gas leakage from the pipe is neglected, and (d) The opening and closing time of the fast switching valves are also neglected.2.1. Dynamic model of parallel manipulator2.1.1. Inverse kinematics of parallel manipulatorLet the posture vector of the moving platform be=. Define as the position vector of the ith spherical joint of the base platform in frame Oxyz and as the position vector of the ith spherical joint of the moving platform in frame Oxyz. Let be the position vector of the center Oin frame Oxyz andthe position vector of the ith spherical joint of he moving platform in frame Oxyz. Then,+ （1）where is the rotational matrix from frame Oxyzto frame Oxyz and was uniquely determined using the posture vector by Huang, Kou, and Fang (1997) （2）where c and s are short-hand notations for cosand sinrespectively. The contractive length of the ith pneumatic muscle can then be determined as （3）where is the initial length of pneumatic muscle, is the ineffective length between the base platform and the moving platform excluding the effective length of the ith pneumatic muscle, and | | represents the Euclidean norm of a vector. Eq.(3) solves the inverse kinematics of the moving platform from the posture vector _ to the contractive length vector of three pneumatic muscles.2.1.2. Dynamics of parallel manipulatorLet be the value of the force acting on the moving platform by the ith pneumatic muscle along its axial direction, with positive value for the contractive force. Denote a force vector as F=. Let the angular velocity of the moving platform expressed in the base frame Oxyz be. Then, according to the dynamic equation of a rigid body, the dynamic model of the moving platform can be written as Tao et al. (2005) (4)where I（） is the rotational inertia matrix and Crepresents the Coriolis terms, represents the coefficient matrix of viscous frictions of the spherical joints, represents the external disturbance vector, and J()is the Jacobian transformation matrix between the contractive velocity vector of pneumatic muscles and angular velocity vector of the moving platform.The dynamic equations of the driving units can be written as （5）Where M=diag（）is the equivalent mass matrix of three pneumatic muscles and their spherical joints, F= is the friction force vector of pneumatic muscles,and is the static force vector of pneumatic muscles detailed in Section 2.2. Merging Eqs. (4) and (5) while noting and in which （G（）is the transformation matrix from the angular velocity vector to the posture velocity vector of the PMDPM via three RPY angles, the dynamic model of PMDPM can be obtained as follows in terms ofposture vector. （6）where+.2.2. Models of actuator2.2.1. Characteristics of pneumatic muscleFor each driving unit i, the static force of pneumatic muscle is(Chou & Hannaford, 1996; Tondu & Lopez, 2000; Yang, Li, & Liu,2002) （7） （8）where pis the pressure inside the pneumatic muscle, a and b are constants related to the structure of pneumatic muscle, kis a factor accounting for the side effect, is the contractive ratio given by=x/L, F is the rubber elastic force, D is the initial diameter of the pneumatic muscle, is the thickness of shell and bladder, E is the bulk modulus of elasticity of rubber tube with cross-weave sheath, is the initial braided angle of the pneumatic muscle, is the current braided angle of the pneumatic muscle given by and is the modeling error.2.2.2. Actuator dynamicsThe pressure dynamics that generates the pressure inside the pneumatic muscle will be given. Suppose the relationship between volume and pressure inside the pneumatic muscle can be described by polytropic gas law （9）where is the pneumatic muscles inner volume, which is a function of , is the air mass inside the pneumatic muscle, and is the polytropic exponent. The ideal gas equation describes the dependency of the air mass. （10）where R is the gas constant and Ti is the thermodynamic temperature inside the pneumatic muscle.Differentiate Eq. (9) while noting Eq. (10), one obtains （11）where is the air mass flow rate through the valve.Suppose in which qmi is the calculable air mass flow rate given in Section 2.2.3, and are the coefficient and disturbance considering model errors respectively.Then the following general pressure dynamic equation instead ofEq. (11) will be used (Richer & Hurmuzlu, 2000) （12）where 2.2.3. Valve modelThe relationship between the calculable air mass flow rate into the ith pneumatic muscle and control input (the duty cycle of two fast switching valves in the ith driving unit) can be put into the following concise form (Tao et al., 2005) （13）where is a nonlinear flow gain function given by =within which is the effective orifice area of fast switching valve, is the upstream pressure, pdi is the downstream pressure, is the upstream temperature and k is the ratio of specific heat.2.3. Dynamics in state-space formLet be the calculable and differentiable part of Fm. And define the drive moment of the PMDPM in task-space as （14）Differentiate Eq. (14) while noting Eqs. (12) and (13), the dynamics of actuators are described as （15）Where-=Define a set of state variables as x=.Then the entire system can be expressed in a state-space form as （16）where is the inverse function of Eq. (14) from to p,and. Note that both and are positive definite matrices. For the simplicity of notation, the variables such as and are expressed by and when no confusions on the function variables exist.3. Adaptive robust controller3.1. Design issues to be addressedGenerally the system is subjected to parametric uncertainties due to the variation of etc., and uncertain nonlinearities represented by and that come from uncompensated model terms due to complex computation, unknown model errors and disturbances etc. Practically, dp and d_ may be decomposed into two parts, the constant or slowly changing part denoted by and, and the fast changing part denoted by i.e., The main unknown parameters will be updated on-line through adaptation law to improve performance while the effects of other minor parameters will be lumped into uncertain nonlinearities. For simplicity, the following notations are used throughout the paper: i is used for the ith component of the vector , the operation for two vectors is performed in terms of the corresponding elements of the vectors, and denotes the uncertain parameter vector. Then, the following assumption is made as in Yao et al. (2000).Assumption: The extent of parametric uncertainties and uncertain nonlinearities are known, i.e. , whereis the maximum parameter vector, is the minimum parameter vector, and are known vectors.It can be seen that the major difficulties in controlling are:(a) The entire system is a highly nonlinear MIMO coupling dynamic system, due to either unknown disturbances, and nonlinear flow gains or nonlinear and time-varying terms such as. Therefore, it is more suitable to design a nonlinear controller based on the MIMO model instead of a linear controller based on linearization.(b) The system has severe parametric uncertainties represented by the lack of knowledge of damping coefficients Cs and the polytropic exponents and . Hence on-line parameter adaptation method should be adopted to reduce parametric uncertainties.(c) The system has large extent of lumped modeling errors like unknown disturbances and unmodeled friction forces, which are contained in and .So both on-line adaptation and effective robust attenuation to these terms should be used for good control performance.(d) The model uncertainties are unmatched, i.e., both parametric uncertainties and uncertain nonlinearities appear in the dynamic equations of the PMDPM that are not directly related to the control input u. Therefore, the backstepping design technology should be employed to overcome design difficulties for achieving nasymptotic stability.3.2. Projection mappingLet denote the estimate of and the estimation error. In view of Assumption, the following projection-type parameter adaptation law is used to guarantee that the parameter estimates remain in the known bounded region all the time (Yao et al., 2000). （17） （18）where is a positive definite diagonal matrix of adaptation rates and is a parameter adaptation function to be synthesized later.It can be shown that for any adaptation function, such a parameter adaptation law guarantees （19）3.3. ARC controller designThe design is carried out using the recursive backstepping design procedure via ARC Lyapunov functions in task-space and muscle-space as follows (Yao et al., 2000).(1) Step 1Define a switching-function-like quantity as （20）where is the trajectory tracking error vector and Kc is a positive definite diagonal feedback matrix. If converges to a small value or zero, then will converge to a small value or zero since the transfer function from to is stable. Thus the next objective is to design the drive moment for making as small as possible. For this purpose, define the unknown parameter vector in task-space as and denote the effects of parametric uncertainties in task-space as （21）where is a regressor given by ,with .The desired drive moment consists of two terms （22）where functions as the adaptive control part used to achieve an improved model compensation through on-line parameter adaptation via , and is a robust control law to be synthesized later. is updated by an adaptation law of the form Eq. (17), i.e., （23）with the parameter adaptation function given by .And consists of two terms （24）where is a positive definite feedback gain matrix. is synthesized to dominate the model uncertainties coming from both parametric uncertainties and uncertain nonlinearities, which is chosen to satisfy the following conditions. （25）where
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