Adaptive Proxy-based Robust Control Integrated With .
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2020.2997041, IEEE/ASMETransactions on MechatronicsIEEE TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, MARCH 20201Adaptive Proxy-based Robust ControlIntegrated with Nonlinear Disturbance Observerfor Pneumatic Muscle ActuatorsYu Cao, Student Member, IEEE, Jian Huang, Senior Member, IEEE, Caihua Xiong, Member, IEEE,Dongrui Wu, Senior Member, IEEE, Mengshi Zhang, Student Member, IEEE, Zhijun Li, Senior Member, IEEE,Yasuhisa Hasegawa, Member, IEEEAbstract—In Pneumatic Muscle Actuators (PMAs)-drivenrobotic applications, there might exist unpredictable shockswhich lead to the sudden change of desired trajectories andlarge tracking errors. This is dangerous for physical systems.In this paper, we propose a novel adaptive proxy-based robustcontroller (APRC) for PMAs, which is effective in realizing adamped response and regulating the behaviors of the PMA viaa virtual proxy. Moreover, the integration of the APRC and thenonlinear disturbance observer (NDO) further handles the systemuncertainties/disturbances and improves the system robustness.According to the Lyapunov theorem, the tracking states of theclosed-loop PMA control system are proven to be globally uniformly ultimately bounded through two motion phases. Extensiveexperiments are conducted to verify the superior performance ofour approach, in multiple tracking scenarios.Index Terms—Pneumatic muscle actuator, adaptive proxybased robust control, two-phase stability analysis.I. I NTRODUCTIONDUE to the attractive characteristics, i.e., high power/weight ratio, no mechanical parts, low cost, etc [1],the Pneumatic Muscle Actuator (PMA) has been widely usedin a variety of fields, especially exoskeletons that are effectivein power augmentation and rehabilitation training [2]–[4]. Itsdriving force is converted from the air pressure of the innerbladder, which has the features of nonlinearity, hysteresis, andtime-varying parameters [5], making its modeling and controlvery challenging. Different control strategies have been proposed for the PMA, including PID-based control [6], nonlinearmodel predictive control [7], [8], sliding mode control (SMC)This work was supported by the International Science and TechnologyCooperation Program of China (No. 2017YFE0128300), the FundamentalResearch Funds for the Central Universities (HUST: 2019kfyRCPY014), andthe Research Fund of PLA of China (BWS17J024). (Corresponding author:Jian Huang and Caihua Xiong)Y. Cao, J. Huang, D. Wu and M. Zhang are with the Key Laboratory of Image Processing and Intelligent Control, School of Artificial Intelligence and Automation, Huazhong University of Scienceand Technology, Wuhan 430074, China (e-mail: cao yu@mail.hust.edu.cn;huang jan@mail.hust.edu.cn; drwu@hust.edu.cn; dream poem@hust.edu.cn).C.-H. Xiong is with the School of Mechanical Science and Engineeringand the State Key Laboratory of Digital Manufacturing Equipment andTechnology, Huazhong University of Science and Technology, Wuhan 430074,China (e-mail: chxiong@hust.edu.cn).Z. Li is with the Department of Automation, University of Science andTechnology, Hefei 230026, China. (e-mail: zjli@ieee.org).Y. Hasegawa is with the Department of Micro-Nano Systems Engineering,Nagoya University, Nagoya 464-8603, Japan (email: hasegawa@mein.nagoyau.ac.jp).[9], fuzzy control [10], adaptive control [11], dynamic surfacecontrol [12], etc. Unfortunately, an accurate mathematic modelof the PMA is very difficult to obtain in practice which causesdifficulties in precise control. Meanwhile, the traditional PIDcontrol, a typical model-free strategy, works in the positioncontrol of the PMA. However, some significant issues shouldbe taken into account. First, the high-gain PID controllermay cause oscillation and can hardly realize satisfactoryperformance in the physical PMA applications, due to the slowresponse of the PMA and limited sampling rate, etc. Second,the PMA is widely used in the field of robot actuation andindustry, in which the load, running amplitude and frequencymay change within a certain task. The traditional PID controller with a set of fixed control parameters may not meet therequirements of these applications. Next, from a theoreticalviewpoint, it is difficult to theoretically prove the stability ofthe closed-loop system when no theoretical model is involved.Thus, there is still a strong demand for robust PMA control.In robotic applications, the idea of using a proxy is commonbecause a proxy enables robots to track the reference witha damped response to unexpected impacts, which results inthe improvement of the system security and performance[13]. Whereas the physical proxy requires a light-weight andcompact mechanism that leads to difficulties for designation.The virtual proxy is a remedy to fulfill the requirement ofrobot control. A typical strategy called proxy-based slidingmode control (PSMC) [14], which assumes that a zero-qualityvirtual proxy exists between the controlled object and the desired trajectory, is significantly a model-free strategy. Damme[15] presented a PSMC for a two-degree-of-freedom planarmanipulator actuated by Pleated Pneumatic Artificial Muscles,and such a strategy of position control was developed forpiezoelectric-actuated nanopositioning stages in [16]. Anotherapproach supposed that there was a free space around theproxy for the impedance control of a cable-driven system [17].However, most proxy-based strategies lack stability analysis ordepend on a strong conjecture (e.g., see Conjecture 1 in [14]).Therefore, this kind of strategies demands further investigationto establish a sound theoretical foundation.The robustness of the control strategy is another significantissue for robotic systems. Although the proxy-based strategieshave been used in various applications, most of them rarelyconsider the improvement of system robustness. NonlinearDisturbance Observer (NDO) based control is a common1083-4435 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.Authorized licensed use limited to: Huazhong University of Science and Technology. Downloaded on July 06,2020 at 15:05:57 UTC from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2020.2997041, IEEE/ASMETransactions on MechatronicsIEEE TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, MARCH 2020method for improving control performance. The basic ideais to estimate the disturbances/uncertainties from measurablevariables before a control action is taken. Consequently, theinfluence of the disturbances/uncertainties can be suppressed,and the system becomes more robust [18]–[20]. MultipleNDO-based control strategies have been proposed to compensate for the influence of disturbances/uncertainties [21]–[25].However, to our best knowledge, there are very few researcheson the proxy-based control strategy integrated with NDO.This may be due to two challenges. First, most of the proxybased strategies are model-free control approaches, whereas atypical NDO-based controller requires a mathematical modelof the control system. Therefore, the integration of proxybased strategy and NDO is not straightforward. Second, a morerigorous analysis is needed to guarantee the stability of thesystem, which should not be based on a strong conjecture.This paper proposes an adaptive proxy-based robust controlintegrated with nonlinear disturbance observer for the positioncontrol of PMAs. Our main contributions are:1) The proposed adaptive proxy-based robust control extends proxy-based sliding mode control from a model-freestrategy to a model-based strategy by defining the motion behaviors of the proxy. Accompanied by a nonlinear disturbanceobserver, the proposed control method retains the originalcharacteristics of smooth and damped motions and greatlyimproves the robustness of the algorithm.2) The proposed controller ensures the global stability of theclosed-loop system through two stages, in which the controlledobject tracks the proxy, and the proxy tracks the referencetrajectory, simultaneously. Furthermore, this paper elaboratelystudies the case when the proxy is not zero and finds that thenon-zero proxy mass is capable of regulating the behaviors ofthe controlled object.3) Real-world experiments are conducted based on a physical PMA platform for validating the effectiveness of theproposed controller, and the results present better trackingaccuracy and robustness under various reference trajectories.Note that a study presents an extended proxy-based slidingmode control [26]. Compared with [26], this paper proposesa new theoretical proxy-based method by constructing themotion behaviors of the proxy. Integrating with a NDO, thismethod can strictly guarantee the global stability of the systemwhile improving the robustness and retaining the originalcharacteristics of smooth and damped motions. Meanwhile,this paper quantitatively analyzes the effect of proxy on controlperformance. It turns out that as the proxy mass increases,the system’s tracking errors will gradually approach a boundassociated with estimation errors of the system’s uncertainties/disturbances. To the best of our knowledge, this is thefirst study to investigate the effect of the virtual proxy on thephysical plant.The rest of this paper is organized as follows. Section IIintroduces the three-element model of the PMA with thelumped disturbances. Section III first proposes the APRCand then extends the APRC to APRC-NDO to improve thesystem robustness. Section IV presents real-world experimentsto demonstrate the effectiveness and robustness of the APRCNDO. Finally, Section V draws conclusions.2Fig. 1.The PMA and its three-element model.II. T HE T HREE -E LEMENT M ODEL OF THE PMAThe generalized three-element model of the PMA is shownin Fig. 1 [27]. The contractile length varies with the airpressure of inner bladder. The dynamics of the PMA is: mẍ b(P )ẋ k(P )x f (P ) mg bi (P ) bi0 bi1 P (inf lation)bd (P ) bd0 bd1 P (def lation)(1) k(P) k kP 01 f (P ) f0 f1 Pwhere m, x, P are the mass of load, the contractile length ofPMA, and the air pressure, respectively. b(P ), f (P ), k(P ) arethe damping coefficient, the contractile force, and the springcoefficient, respectively.Let τ (t) denote the sum of unmodeled uncertainties, including unmodeled dynamics, friction, inaccurate parameters,and changing loads, etc. The dynamics of the PMA can berewritten as a typical second-order nonlinear model: ẍ f (x, ẋ) b(x, ẋ)u τ (t)1f (x, ẋ) m(f0 mg b0 ẋ k0 x)(2) 1b(x, ẋ) m (f1 b1 ẋ k1 x)where u is the air pressure, and f (x, ẋ) and b(x, ẋ) arenonlinear terms related to the system states.Lemma 1 [28]: Given a differentiable continuous functionΨ(t), t [t0 , t1 ] satisfying σ1 Ψ(t) σ2 with positiveconstant σ1 and σ2 . The derivative Ψ̇(t) is also bounded.Assumption 1 [29]: For the system unknown lumped disturbance τ (t):R R, there exists an unknown positiveconstant ε such that t R satisfy τ (t) ε.III. A DAPTIVE P ROXY- BASED ROBUST C ONTROLI NTEGRATED WITH N ONLINEAR D ISTURBANCE O BSERVERA. Adaptive Proxy-based Robust ControlThe objective of this study is to drive the trajectory of thePMA to track the desired trajectory. In our proxy-based robustcontroller, an imaginary object called “proxy”, assumed to be1083-4435 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.Authorized licensed use limited to: Huazhong University of Science and Technology. Downloaded on July 06,2020 at 15:05:57 UTC from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2020.2997041, IEEE/ASMETransactions on MechatronicsIEEE TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, MARCH 2020connected to the physical actuator, is presented. Before introducing the APRC, we define the following sliding manifolds:ZSq ẋd ẋ c1 (xd x) c2 (xd x) dt(3)ZSp ẋd ẋp c1 (xd xp ) c2 (xd xp ) dt (4)where c1 and c2 are positive constants, xd the desired trajectory, and xp and x the proxy position and the PMA’sdisplacement, respectively.Fig. 2.The principle of proxy-based robust control.First of all, we design a relationship between the proxy andthe controlled object to satisfy:ZṠq Kp (xp x) Ki (xp x) dt Kd (ẋp ẋ) τ 0(5)where Kp , Ki and Kd are positive constants.Remark 1. Traditionally, once the sliding manifold Sq is defined, the controller can be designed using Ṡq k · sgn(Sq ),which is known as the sliding mode control and may causesevere chattering. Hence, our idea of introducing the proxyis to replace k · sgn(Sq ) by a PID controller to establisha connection between the controlled object and the proxy, asshown in Fig. 2. Note that (5) can also be rewritten as: 0001001 X 0 ul 0 ρ (6)Ẋ 0110 c2 c1Rwhere ρ ẍd c1 ẋd c2 xd τ , X [ xdt, x, ẋ], andZul Kp (xp x) Ki (xp x) dt Kd (ẋp ẋ). (7)It is clear that (6) can be regarded as a local relationbetween the controlled object and the proxy. This is a linearsystem with PID control, where X is the system’s states,xp regarded as the desired trajectory, and ẍd , ẋd and xdvarying parameters unrelated to the system’s states. This PIDcontroller drives the PMA’s trajectory x to track the proxy’strajectory xp , when the controller parameters are properlytuned based on the following stability condition.Hence, bringing (2) and (3) into (5), the control signal fedinto the PMA can be computed:1[ẍd c1 (ẋd ẋ) c2 (xd x) f (x, ẋ)b(x, ẋ)Z Kp (xp x) Ki (xp x) dt Kd (ẋp ẋ)]. (8)u However, xp is unknown, and xp should be driven toapproach the desired trajectory xd to fulfill the tracking tasks.3A common idea is to use a sign function to ensure the manifoldSp 0. Hence, we generate the control signal of proxy urbetween the desired trajectory and the proxy, i.e.,ur Γ̂ · sgn(Sp )(9)where sgn(Sp ) is the signum function. Γ̂ is the adaptive gain ofthe sliding surface Sp , and the corresponding optimal constantof Γ̂ is Γ .The adaptive law is described as: γ Sp , Sp δ Γ̂ , Γ̂(0) 0(10)0, Sp δwhere γ is a positive constant that regulates the adaptive rate. δis a boundary layer. When the system achieves a steady-state, Sp is small enough, so that Γ̂ will reach an upper boundinstead of monotonically increasing.Remarkably, the proxy is affected by ur and ul , simultaneously, as shown in Fig. 2, and they are not force signals in thetraditional sense. Hence, we cannot directly use Newton’s lawto establish the relationship between the motion behaviors ofthe proxy and ur , ul . Besides, it is necessary to define suchproperty to ensure the realization of tracking and the system’sstability. Similar to Newton’s law, we define the behavior ofthe proxy under the effects of ur and ul . Let mp 0 be theso-called proxy mass. Then,mp Ṡp ur ul .(11)The effect of ur ul is similar to the resultant force onthe proxy while mp Ṡp can be seen as the motion principle ofthe proxy. Note that this property can be arbitrarily definedaccording to the specific situation, as long as the stability ofthe closed-loop system can be ensured.Combining (4), (7), (9), and (11), the trajectory of the proxyis presented as:Z1[Γ̂sgn(Sp ) Kp (xp x) Ki (xp x) dtẍp mp Kd (ẋp ẋ)] ẍd c1 (ẋd ẋp ) c2 (xd xp ). (12)Once xp is determined, the control signal of the PMA canthen be computed from (4), (8) and (12).For the convenience of presentation, we first define Km diag{Ki c2 , , Kd } with Kp c1 Ki Kd c2 .Theorem 1. Theerror between the proxy R norm of tracking Tstates Xp xp dt, xp , ẋpand the system states X R Txdt, x, ẋ is uniformly ultimately bounded, and a slidingmotion on the surface (4) can be guaranteed when the APRCsatisfies:mp 0, λ(Kc ) 0, Γ λ2 (Kp Ki Kd ), 0where λ(·) and λmin (·) denote the eigenvalues and the minimum eigenvalue of the matrix, respectively. (c1 c2 1)εKp c2 Ki c1 Ki Kd c2, Kc .λ2 Ki Kd c2 Kp Kd c1λmin (Km )Proof: Due to λ(Kc ) 0, a Lyapunov candidate isdefined as V V1 V2 V3 0 with11V1 mp Sp2 Sq2(13)221083-4435 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.Authorized licensed use limited to: Huazhong University of Science and Technology. Downloaded on July 06,2020 at 15:05:57 UTC from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2020.2997041, IEEE/ASMETransactions on MechatronicsIEEE TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, MARCH 2020 1 epep ėp Kcėp21 (Kp Kd c1 Ki Kd c2 )ė2p21 (Kp c2 Ki c1 Ki Kd c2 )e2p21 (Ki Kd c2 )(ep ėp )221 2V3 Γ̃2γZwhere ep (xp x) dt and Γ̃ Γ̂ Γ .V2 (14)1 Γ̃Γ̂γ Γ Sp (Kp ėp Ki ep Kd ëp )Sp Γ Sp λ2 (Kp Ki Kd )Sp 0.(15)(16)According to (3)-(5), we haveṠq Kp ėp Ki ep Kd ëp τSp Sq (ëp c1 ėp c2 ep ).(17)(18)Integrating (10)-(18), the derivatives of V1 , V2 and V3 are:V̇1 Sp ( Γ̂sgn(Sp ) Kp ėp Ki ep Kd ëp ) Sq ( Kp ėp Ki ep Kd ëp τ )It follows from (16) thatV̇p mp Sp Ṡp From (7)-(11), it follows thatmp Ṡp Γ̂sgn(Sp ) Kp ėp Ki ep Kd ëp .4(19) Γ̂ Sp τ Sq (Kp ėp Ki ep Kd ëp )(Sp Sq )When ep is uniformly ultimately bounded, the achievementof a sliding motion on the surface (4) is guaranteed.This completes the proof.Remark 2. The stability analysis of the system has two motionphases. First, the norm of the tracking error between the proxystates Xp and the system states X is uniformly ultimatelybounded. This indicates that the system states converge tothe proxy states. Then, the achievement of sliding motion onthe surface (4) means that the proxy tracks the referencetrajectory, theoretically. In summary, the system states arecapable of indirectly tracking the reference, and the stabilityof the closed-loop system is guaranteed.Corollary 1. If inequality (25) holds, and initially xp xd ,then, as the proxy mass mp increases, Sq will graduallyapproach a bound associated with the upper bound of thelumped disturbances. Γ̂ Sp τ Sq Kd ë2p Kp c1 ė2p Ki c2 e2plim Sq λ2 (c1 c2 1).mp (Kp Kd c1 )ėp ëp (Ki Kd c2 )ep ëp (Kp c2 Ki c1 )ep ėp(20)Kd c2 )ė2p (Ki (Ki Kd c2 )ep ëp .1 V̇3 Γ̃Γ̂ Γ̃ Sp Γ̂ Sp Γ Sp .γThen, it follows thatV̇1 V̇2 Γ̂ Sp τ Sq ė2p (21) Ṡp 1 Γ sgn(Sp ) Kp ėp Ki ep Kd ëp .mplim Ṡp 0.(22)(Kp Ki Kd )(1 c1 c2 )ε ε.min(Ki c2 , , Kd )(23)(30)The proxy mass mp is a fixed value in each experiment. Lettf be the finite duration of the experiment. Then,Z tfSp Ṡp dt υ(31)0From (17)-(23), we havewhere υ is the initial value of xd xp , which equals zero.Hence, it follows thatZ tfZ tf Sp Ṡp dt Ṡp dt.(32)V̇ V̇1 V̇2 V̇3 Γ Sp τ Sq Kd ë2p ė2p Ki c2 e2p ε Sq ε ëp c1 ėp c2 ep (29)Since the system is globally uniformly ultimat
1) The proposed adaptive proxy-based robust control ex-tends proxy-based sliding mode control from a model-free strategy to a model-based strategy by defining the motion be-haviors of the proxy. Accompanied by a nonlinear disturbance observer, the proposed control method retains the original characteristics of smooth and damped motions and greatly
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a fully-actuated subsystem and an under-actuated subsys-tem [9]. Then, he controlled them with a PID controller and a sliding mode controller, respectively. As a result, . Section 2 so that the adaptive robust control for the alti-tude control of the helicopter can be designed in Section 3. Then, Section 3 will discuss an adaptive robust .
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