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Stable Gain Adaptation for Time-Delay Control of Robot Manipulators ⁎
8th 8th IFAC IFAC Symposium Symposium on on Mechatronic Mechatronic Systems Systems Vienna, Austria, Sept. on 4-6,Mechatronic 2019 8th IFACAustria, Symposium Systems online at www.sciencedirect.com Vienna, Sept. 4-6, 2019 Available 8th IFACAustria, Symposium Systems Vienna, Sept. on 4-6,Mechatronic 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-15 (2019) 217–222
Stable Stable Gain Gain Adaptation Adaptation for for Time-Delay Time-Delay Stable Gain Adaptation for Time-Delay Control Robot Manipulators Stable Gainof Adaptation for Time-Delay Control of Robot Manipulators Control of Robot Manipulators Control of Robot Manipulators ∗∗∗ ∗∗ Kap-Ho Seo ∗∗∗ Junyoung Lee ∗∗ Pyung Hun Chang ∗∗
Junyoung Lee ∗ Pyung Hun Chang Kap-Ho Seo ∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗∗ Maolin Junyoung Lee ∗ Pyung HunJin Chang Kap-Ho Seo Maolin Jin ∗∗ ∗∗∗∗ Junyoung Lee Pyung HunJin Chang Kap-Ho Seo ∗∗∗ Maolin Maolin Jin ∗∗∗∗ ∗ ∗ Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) ∗∗ Institute of 37666, Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗Korea Daegu Gyeongbuk Institute of and Technology Namgu, Pohang, Korea (e-mail: [email protected]) Daegu Gyeongbuk Institute of Science Science and Technology (DGIST), (DGIST), ∗∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Daegu 711-873, Korea (e-mail: [email protected]) Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 711-873, Korea (e-mail: [email protected]) ∗∗ ∗∗∗ Gyeongbuk Institute of Convergence Science and Technology (DGIST), Daegu 711-873, Korea (e-mail: [email protected]) ∗∗∗ Daegu Korea Institute of and (KIRO), Korea Institute of Robot Robot and Convergence (KIRO), Jigok-Ro Jigok-Ro 39, 39, ∗∗∗ Namgu, Daegu 711-873, Korea (e-mail: [email protected]) Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, Namgu, Pohang, 37666, Korea (e-mail: [email protected]) ∗∗∗ ∗∗∗∗Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗∗∗ Namgu, Korea Institute of Robot and Convergence (KIRO), Jigok-Ro Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute Robot Korea and Convergence (KIRO), Jigok-Ro 39, 39, ∗∗∗∗Namgu, Namgu, Pohang,of 37666, (e-mail: [email protected]) [email protected]) Pohang, 37666, Korea (e-mail: Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗∗∗Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute RobotKorea and Convergence (KIRO), Jigok-Ro 39, Namgu, Pohang, of 37666, (e-mail: [email protected]) Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Abstract: Abstract: This This paper paper proposes proposes adaptive adaptive gain gain dynamics dynamics for for time-delay time-delay control control (TDC) (TDC) of of robot robot Abstract: This paper proposes adaptive gain dynamics for time-delay control (TDC) of robot manipulators. The TDC is a widely employed approach for control of robot manipulators because manipulators. The TDC is a widely employed approach for control of robot manipulators because Abstract: This paper proposes adaptive gain dynamics for time-delay control (TDC) of robot it is model-independent, simple, and robust. Recently, however, it is reported that TDC with manipulators. The TDC is a widely employed approach for control of robot manipulators because it is model-independent, and robust.approach Recently,for however, it robot is reported that TDC with manipulators. The TDC issimple, a widely employed control of manipulators because excessively high gains causes unstable or oscillated responses. To overcome this limitation, this it is model-independent, simple, and robust. Recently, however, it is reported that TDC with excessively high gains causes unstable or oscillated responses. To overcome this limitation, this it is model-independent, simple, and robust. Recently, however, it is reported that TDC with study designs adaptive gain dynamics for the TDC using a sliding variable and an acceptance excessively high gains causes unstable or oscillated responses. To overcome this limitation, this study designs adaptive gain dynamics for the TDC using a sliding variable and an acceptance excessively high gains causes unstable or To overcome this limitation, this study designs adaptive gain dynamics foroscillated the adjusts TDCresponses. using a sliding variable and an acceptance layer. When the gain dynamics automatically aa control gain according to system states, layer. When the gain dynamics automatically adjusts control gain according to system states, study designsthe adaptive gain dynamics for therange. TDC Therefore, using a sliding variable and an acceptance the adaptive control gain moves into a stable the TDC with the proposed gain layer. When gain dynamics automatically adjusts a control gain according to system states, the adaptive control gain movesautomatically into a stable range. Therefore, the TDC with the proposed gain layer. When the gain dynamics adjusts a control gain according to system states, dynamics becomes adaptive and stable. The effectiveness of the proposed dynamics has the adaptive control gain moves into a stable range. Therefore, the TDC withgain the proposed gain dynamics becomes adaptive and stable. The effectiveness of the proposed gain dynamics has the adaptive control gain moves into a stable range. Therefore, the proposed TDC withgain the proposed gain been verified by simulations and experiments. dynamics becomes adaptive and stable. The effectiveness of the dynamics has been verified by simulations and experiments. dynamics becomes adaptive and and experiments. stable. The effectiveness of the proposed gain dynamics has been verified by simulations © 2019, IFAC by (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. been verified simulations and experiments. Keywords: Keywords: Adaptive Adaptive control, control, sliding sliding mode mode control, control, time-delay time-delay control, control, robot robot manipulator. manipulator. Keywords: Adaptive control, sliding mode control, time-delay control, robot manipulator. Keywords: Adaptive control, sliding mode control, time-delay control, robot manipulator. 1. Youcef-Toumi 1. INTRODUCTION INTRODUCTION Youcef-Toumi and and Ito, Ito, 1990; 1990; Hsia Hsia et et al., al., 1991). 1991). Since Since the the 1. INTRODUCTION TDC does not require the calculation of robot dynamics Youcef-Toumi and Ito, 1990; Hsia et al., 1991). Since the TDC does not require the calculation of robot dynamics 1. INTRODUCTION Youcef-Toumi and Ito, 1990; Hsia et al., 1991). Since the owing to technique, the becomes simple, does notTDE require the calculation robot dynamics Robot manipulators manipulators are employed employed in in various various fields owing to the the TDE technique, the TDC TDCof becomes simple, fields and and TDC Robot are TDC does not require the calculation of robot dynamics owing to the TDE technique, the TDC becomes simple, efficient, and yet robust (Hsia, 1989; Youcef-Toumi and Robot manipulators aretasks employed in various fieldset operated with advanced advanced tasks (Severinson-Eklundh et and al., efficient, and yet robust (Hsia, 1989; Youcef-Toumi and operated with (Severinson-Eklundh al., to the TDE technique, the TDCthe becomeshas simple, RobotVircikova manipulators are employed in various fields Ito, Hsia et 1991). As result, efficient, robust (Hsia, and operated with advanced tasks (Severinson-Eklundh et and al., owing 2003; et al., 2015). To conduct various works, Ito, 1990; 1990; and Hsiayet et al., al., 1991). As aa 1989; result,Youcef-Toumi the TDC TDC has been been 2003; Vircikova et al., 2015). To conduct various works, and yet robust (Hsia, 1989; Youcef-Toumi and operated with advanced tasks (Severinson-Eklundh etconal., efficient, widely applied to various areas such as robot manipulators Ito, 1990; Hsia et al., 1991). As a result, the TDC has been 2003; Vircikova et al., 2015). To conduct various works, accurate control is required. However, in model-based applied to various areas such as robot manipulators accurate control is required. However, in model-based con- widely Ito, 1990; Hsia et al., 1991). As a result, the TDC has been 2003; Vircikova et al., 2015). To conduct various works, (Jung et al., Jin al., 2008; Jin et 2017), applied to various robot manipulators accurate However, in dynamic model-based con- widely trol, it it is is control difficultisto torequired. obtain the the complex dynamic equation (Jung et al., 2004; 2004; Jin et etareas al., such 2008;as Jin et al., al., 2017), an an trol, difficult obtain complex equation widely applied tosystem various asChen, robot manipulators accurate control istorequired. However, in dynamic model-based con(Jung et al., 2004; Jin (Cheng etareas al., such 2008; Jin et 1996), al., 2017), an overhead crane and a wind trol, it is difficult obtain the complex equation of a robot manipulator, which includes uncertainty and overhead crane system (Cheng and Chen, 1996), a wind of a robot manipulator, which includes uncertainty and (Jung et(Kim al., 2004; Jin et al., 2008; Jinreactor et 1996), al., (Gonz´ 2017), an trol, it is difficult to obtain the complex dynamic equation turbine et al., 2015), a chemical a lez overhead crane system (Cheng and Chen, a wind of a robot manipulator, which includes uncertainty and nonlinearity (Baghli and El Bakkali, 2016; Li et al., 2013). turbine (Kim etsystem al., 2015), a chemical reactor (Gonz´ alez nonlinearity (Baghli and El Bakkali, 2016;uncertainty Li et al., 2013). crane (Cheng and (Kim Chen, 1996), a wind of acope robot manipulator, which includes and overhead et al., 2002), an underwater vehicle et al., 2016), a turbine (Kim et al., 2015), a chemical reactor (Gonz´ a lez nonlinearity (Baghli and El Bakkali, 2016; Li et al., 2013). To with this problem, a lot of attempts have been al., 2002), an vehicle (Kim et al.,(Gonz´ 2016), a To cope with(Baghli this problem, a lot of2016; attempts been et turbine (Kim et underwater al., 2015), a2012), chemical reactor aand lez nonlinearity and El Bakkali, Li et have al., 2013). tilt-rotor aircraft (Lee et al., excavators (Chang et al., 2002), an underwater vehicle (Kim et al., 2016), a To cope with this problem, a lot of attempts have been tried: for example, sliding-mode control (SMC) (Fallaha tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and tried: for example, sliding-mode control (SMC) (Fallaha al.,2002; 2002), anetunderwater vehicle (Kim et al., 2016), a Toal., cope with thisand problem, a lotcontrol of attempts have been et tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and Lee, Kim al., 2019), chaotic systems (Kim et al., tried: for example, sliding-mode (SMC) (Fallaha et 2011; Islam Liu, 2011), and fuzzy control (Lian, Lee, 2002; Kim et al., 2019), chaotic systems (Kim et al., et al., 2011; Islam and Liu, 2011), control and fuzzy control (Lian, tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and tried: for example, sliding-mode (SMC) (Fallaha 2017; Jin and Chang, 2009), and a shape memory alloy Lee, 2002; Kim et al., 2019), chaotic systems (Kim et al., et al., 2011; Islam and Liu, 2011), and fuzzy control (Lian, 2011; Biglarbegian et al., 2011) to list a few. Nevertheless, 2017;2002; Jin and Chang, 2009), chaotic and a shape memory 2011; et al., 2011) to and list afuzzy few. control Nevertheless, Kim et al., 2019), systems (Kim etalloy al., et al., Biglarbegian 2011; Liu, 2011), (Lian, Lee, actuator et al., 2015). 2017; Jin(Jin and Chang, 2009), and a shape memory alloy 2011; Biglarbegian etthe al., 2011) to list of a few. Nevertheless, SMCs still Islam requireand the information of robot dynamics, actuator (Jin et al., 2015). SMCs still require information robot dynamics, 2017; Jin and Chang, 2009), and a shape memory alloy 2011; Biglarbegian et al., 2011) to list a few. Nevertheless, A been (Jin et have al., 2015). SMCs still requireis information of control robot dynamics, whose calculation calculation is the complicated. Fuzzy control is aa good good actuator A lot lot of of studies studies been conducted conducted on on the the TDC TDC (Hsia, (Hsia, whose complicated. Fuzzy is (Jin et have al., 2015). SMCs calculation still require information of control robot dynamics, 1989; and Ito, 1990; Hsia al., 1991; Jung A lot Youcef-Toumi of studies have been conducted onet the TDC whose is the complicated. Fuzzy is a good actuator solution for the the estimation of nonlinearity nonlinearity and uncertainty uncertainty 1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991;(Hsia, Jung solution for estimation of and A lot of studies have been conducted on the TDC (Hsia, whose calculation is complicated. Fuzzy control is a comgood 1989; et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim Youcef-Toumi and Ito, 1990; Hsia et al., 1991; Jung solution for the estimation of nonlinearity and uncertainty of the robot dynamics, however, it suffers from the et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim of the robot dynamics, however, it suffers from the com1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991; Jung solution for the estimation of nonlinearity and uncertainty 2015; Gonz´ a lez et al., 2002; Kim et al., 2016; Lee et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim of the robot dynamics, however, it suffers from the complexity and difficulty of implementation, due to a lot of et al., 2015; Gonz´ a lez et al., 2002; Kim et al., 2016; Lee of implementation, due to a plexity and difficulty lot of et al., 2004; Jin etaal., al., 2008; Cheng and Chen, 1996; Kim of therules robot dynamics, however, it suffers due fromtothe comal., 2012; Jin et 2015) with a constant constant gain which is et al., 2015; Gonz´ lez et al., 2002; Kim et al., 2016; Lee plexity and difficulty of implementation, a lot of fuzzy and parameters. et al., 2012; Jin et al., 2015) with a gain which is fuzzy rules and parameters. al., 2012; 2015; Gonz´ aal., leztrial et al., 2002; Kim et al., Lee plexityis and difficulty of implementation, due to a lot of et manually tuned with and error (Jin et al., 2009; Lee et al., Jin et 2015) with a constant gain2016; which is fuzzy rules and parameters. TDC a well-established well-established robust control control algorithm algorithm (Hsia manually tuned with trial and error (Jin et al., 2009; Lee TDC is a robust (Hsia et al., 2012; Jin et al., 2015) with aTDC, constant gain2009; which is fuzzyGao, rules and parameters. et al., 2014). In the aspect of the a relatively small manually tuned with trial and error (Jin et al., Lee TDC is a well-established robust control algorithm (Hsia and 1990; Youcef-Toumi and Wu, 1992). It employs et al., 2014). In the aspect of the TDC, aetrelatively small and Gao, 1990; Youcef-Toumi andcontrol Wu, 1992). It employs manually tuned with trial and error (Jin al., 2009; Lee TDC is a well-established robust algorithm (Hsia gain larger tracking and relatively large al.,provides 2014). In the aspect of errors, the TDC, and Gao, 1990; estimation Youcef-Toumi and technique Wu, 1992). It employs the time-delay time-delay estimation (TDE) technique which inten- et gain provides larger tracking and aaaa relatively relatively small large the (TDE) which intenet al.,induces 2014). instability In the aspect of errors, the TDC, relatively and time-delay Gao,utilizes 1990; estimation Youcef-Toumi and technique Wu, 1992). It previous employs gain provides larger tracking errors, and relatively large or oscillation oscillation of aaasystem. system. In small addithe (TDE) which intentionally time-delayed information in the gain induces instability or of In additionally utilizes time-delayed information in the previous gain provides larger tracking errors, and a relatively large the time-delay estimation (TDE) technique which intention, the constant gain cannot always provide applicable gain induces instability or oscillation of a system. In additionally utilizes time-delayed information in the previous sampling instant instant to to estimate estimate robot robot dynamics dynamics (Hsia, (Hsia, 1989; 1989; tion, the constant gain cannot always provide applicable sampling induces instability or oscillation of provide a system. In additionally utilizes information in the previous performance (Jin et al., 2017). Notice that this constant tion, the constant gain cannot always applicable sampling instanttime-delayed to estimate robot dynamics (Hsia, 1989; gain performance (Jin etgain al.,cannot 2017). always Notice provide that thisapplicable constant tion, the constant sampling instant to estimate robot dynamics (Hsia, 1989; This gain is theoretically to the stability (Hsia performance (Jin etrelated al., 2017). Notice thatcondition this constant This work work was was supported supported by by the the Ministry Ministry of of Trade, Trade, Industry Industry and and gain is theoretically related to the stability condition (Hsia performance (Jin Youcef-Toumi etrelated al., 2017). thatcondition this constant Energy (MOTIE, Korea) under Innovation gain is theoretically to theNotice stability (Hsia This work was supported by theIndustrial Ministry Technology of Trade, Industry and and Gao, 1990; and Wu, 1992). In this Energy (MOTIE, Korea) under Industrial Technology Innovation and Gao, 1990; Youcef-Toumi and Wu, 1992). In this This work was supported by the Ministry of Trade, Industry and gain is theoretically related to theand stability condition (Hsia Program.(MOTIE, No.10080355, “Development of series series elastic actuator actuator Energy Korea) under Industrial Technology Innovation context, some studies have recently focused on achieving and Gao, 1990; Youcef-Toumi Wu, 1992). In this Program. No.10080355, “Development of elastic and context, some studies have recently focused on achieving Energy (MOTIE, Korea) under Industrial Technology Innovation manipulator with control collision and Gao,some 1990; Youcef-Toumi and focused Wu, 1992). In this Program. No.10080355, “Development series elastic actuator manipulator with compliance compliance control for forofcorresponding corresponding collision and and an auto-tuned gain for the TDC. context, studies have recently on achieving an auto-tuned gain forhave the TDC. Program. No.10080355, “Development series elastic actuator and minimizing impulse” and 20001184, of manipulator with compliance forof“Development corresponding collision and context, some studies recentlyfor focused on achieving minimizing impulse” and No. No. control 20001184, “Development of ultra-thin ultra-thin A study on automatic an auto-tuned gain for gain-tuning the TDC. A study on automatic gain-tuning for the the TDC TDC exploited exploited manipulator with compliance control for “Development corresponding collision and and short precision reducers and high torque to weight ratio precision minimizing impulse” and No. 20001184, of ultra-thin anNussbaum auto-tuned gain for gain-tuning the TDC. and short precision reducers and high torque to weight ratio precision A study on automatic for the TDC exploited a technique (Cho et al., 2014); however, this minimizing impulse” and No. 20001184, “Development of ultra-thin aA Nussbaum technique (Cho et al., 2014); however, this reducers HRC (Corresponding Author: Maolin Jin.) and shortfor precision reducers and high torque to weight ratio precision reducers for HRC robots”. robots”. (Corresponding Author: Maolin Jin.) study on automatic gain-tuning for the TDC exploited a Nussbaum technique (Cho et al., 2014); however, this and shortfor precision reducers(Corresponding and high torqueAuthor: to weight ratio precision reducers HRC robots”. Maolin Jin.) a Nussbaum technique (Cho et al., 2014); however, this reducers for HRC robots”. (Corresponding Author: Maolin Jin.)
2405-8963 © © 2019 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 620 Copyright © under 2019 IFAC IFAC 620 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 620 10.1016/j.ifacol.2019.11.677 Copyright © 2019 IFAC 620
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algorithm requires acceleration, which is vulnerable to signal noise. Moreover, the Nussbaum gain technique induces the complexity owing to a number of tuning parameters. Consequently, the simplicity of TDC is not maintained. In another study (Jin et al., 2017), an adaptive gain dynamics for TDC using sliding mode variable was proposed. Satisfactory tracking performances and appropriate gains had been achieved from the adaptive gain dynamics, however, the searching region for the adaptive gain is intentionally bounded by a saturation function. The initial value of the gain should be a small value in this approach. When the initial value of the gain is selected as an excessively large value, the system may become oscillated or unstable. This paper proposes an adaptive gain dynamics for designing adaptive TDC using a sliding variable and an acceptance layer. Using the proposed gain dynamics, the control gain is automatically regulated as a stable value satisfying the stability condition of the TDC even if an initial value of the gain is set as an unstable one. The acceptance layer is used to find the proper gain and is related to the concept of a boundary layer (Chen et al., 2002; Fuh et al., 2008) in the aspect of tracking accuracy. As a result, the TDC with the proposed adaptive gain dynamics becomes simple, robust, adaptive, and stable, because the TDE technique effectively compensates for the system dynamics, and the proposed gain dynamics adjusts the magnitude of the control gain that satisfies the stability condition. The structure of the TDC with proposed gain dynamics is still simple and easy for implementation. The effectiveness of the proposed gain dynamics has been verified by the simulations and experiments.
robot dynamics. The control law of TDC is designed by using (4) as ¯ qt−L + M(¨ ¯ qd + KD e˙ + KP e), (5) τ = τt−L − M¨ where e stands for the joint displacement error, given as e = qd − q; qd is the desired joint displacement; and KD , KP are the positive diagonal feedback gains. Combining (2)–(5), the closed-loop dynamics is represented as ¯ −1 (N − Nt−L ). ¨ e + KD e˙ + KP e = M (6) ¯ Since the constant gain M affects the control performance ¯ is important. In addition, we in (6), the selection of M ¯ need to select M such that the stability of a system is guaranteed. The well-established stability condition for the TDC is denoted (Hsia and Gao, 1990; Youcef-Toumi and Wu, 1992) as I−M−1 (q)M ¯ < 1. (7) When satisfying the stability condition (7), the right-hand side N − Nt−L of (6) is bounded. The bounded TDE error vector ε is defined as follows: ∆
˙ q ¨) − Nt−L . (8) ε = N(q, q, ¯ In lots of the TDE-based controllers, M is a constant gain (Hsia and Gao, 1990; Youcef-Toumi and Wu, 1992; Cho ¯ depends et al., 2009; Jin et al., 2011) and tuning of M on the inertia M(q) and the system noise level (Cho et al., 2009; Jin et al., 2017). The constant gain has been manually tuned by trial and error, although the inertia varies according to the posture of a robot manipulator during operation (Jin et al., 2009; Lee et al., 2014). The manually tuned constant gain may cause either poor tracking performances or unstable responses.
2. REVIEW OF TDC The dynamic model of an n-degree-of-freefom (n-DOF) rigid robot manipulator can be written as follows: M(q)¨ q + C(q, q) ˙ + G(q) ˙ + F(q, q) ˙ + τd = τ, (1) where q denotes a vector of the joint displacement; M(q) the inertia matrix; C(q, q) ˙ the Coriolis and centripetal torques; G(q) the gravitational vector; F(q, q) ˙ the friction torques; τd the disturbances; τ the applied joint torques. The time variable t is omitted for simplicity. ¯ the robot When using the positive diagonal gain matrix M, dynamics (1) can be reformulated as ¯ q + N(q, q, M¨ ˙ q ¨) = τ, (2) where N(q, q, ˙ q ¨) includes nonlinear and/or uncertain terms such as the Coriolis/centripetal torques, the gravitational torques, the friction torques, and disturbances. It is written as ¯ q + C + G + F + τd . (3) N = [M − M]¨ N can be estimated with the TDE technique (Hsia, 1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991) as follows: ¯ qt−L , ˆ = Nt−L = τt−L − M¨ (4) N≈N
where •t−L is an intentional time-delayed value in the previous sampling instant t − L. As L deceases, the ˆ is close to N in (4). Therefore, L is selected estimation N as the sampling period for the accurate estimation in the digital implementation. The TDE technique represents the main idea for reducing the computational complexity of 621
3. DESIGN OF GAIN DYNAMICS 3.1 Gain Dynamics To obtain the proper gain satisfying the stability condition (7), the gain dynamics is proposed by using a linear sliding variable s as ∆ s = e˙ + λe, (9) where λ is the positive diagonal matrix. The control objective is for q to follow qd . Thus, the characteristic equation is designed as s˙ + λs = 0, (10) where λ has strict eigenvalues in the right-half complex plane. Since substituting (9) into (10) yields ¨ e + 2λe˙ + λ2 e = 0 (KD = 2λ and KP = λ2 ), the TDC (5) can be rewritten as ¯ qt−L + M(s)(¨ ¯ τ = τt−L − M(s)¨ qd + 2λe˙ + λ2 e). (11) ¯ is not a constant. The adaptive From now on, the gain M ¯ gain M(s) is utilized in (11) and the gain dynamics for ¯ M(s) is proposed as follows: ¯2 ¯˙ ii = αii M ¯ ii2 |si | sgn(|si | − Mii ); M (12) βi where •i is the i-th element of a vector and •ii are the ii-th diagonal element of a diagonal matrix, respectively; αii is the positive diagonal matrix; and βi is the normalizing ¯ 2 and si . A closed-loop system with the factor for M ii
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Fig. 1. The block diagram with the proposed gain dynamics. proposed gain dynamics is shown in Fig. 1. ¯ 2 /βi , M ¯ ii inIn the gain dynamics (12), when |si | > M ii creases to compensate for large tracking errors affected by large si which makes sgn() positive in (12). In contrast, ¯ 2 /βi , M ¯ ii decreases to avoid excessively when |si | < M ii ¯ 2 /βi determines tracking aclarge gain. Since the term M ii ¯ 2 /βi is named curacy according to (9) and (12), the term M ii as the acceptance layer. In this paper, the acceptance layer ¯ 2 /βi of the proposed gain dynamics is used to obtain a M ii proper gain which is satisfied with the stability condition by using relatively small βi .
4. SIMULATION 4.1 Simulation Setup The simulation with a one-link arm is implemented to verify the effectiveness of the TDC with the proposed gain dynamics. The model of the one-link arm is written as follows: τ = I q¨ + G(q) + F (q), ˙ (13) where I = ml2 , G(q) = mlgcos(q), and F (q) ˙ = fV q˙ + fC sgn(q). ˙ fV and fC denote the viscous friction coefficient and the Coulomb friction coefficient, respectively. The simulation parameters are set to be m = 1.0 kg, l = 1.0 m, fV = 5.0 Nms, fC = 5.0 Nm, and g = 9.8 m/s2 , respectively. The desired trajectory is selected as qd = 45 · (1 − e−0.4πt ) · sin(0.4πt) deg. The desired error dynamics is selected as e¨ + 20 e˙ + 100 e = 0. The parameters α and β are tuned as α = 1000.0 and 622
¯ is set as β = 120.0, respectively. The initial value of M 2 ¯ M = 4 kg · m which is unstable. Notice that the stable ¯ is derived as 0 < M ¯ < 2 by (7) (Hsia and Gao, range of M 1990), which is shown in Fig. 2 (a). 4.2 Simulation results The simulation results are shown in Fig. 2. The control ¯ becomes adaptive by the gain dynamics, as shown gain M ¯. in Fig. 2 (a), and it stays in the stable range of M ¯ is the unstable one, the Although the initial value of M TDC using the proposed adaptive gain dynamics (12) ¯ stable. The control input and tracking makes the gain M error are shown in Fig. 2 (b) and (c), respectively. When the significant large tracking errors occur, these errors are compensated because of the increase of the adaptive gain ¯. M 5. EXPERIMENT 5.1 Experimental Setup The control performance of the TDC with the proposed gain dynamics is validated using a PUMA-type robot in Fig. 3 (a). At AC servo motors, harmonic drives have gear ratios of 120:1, 120:1, and 100:1, and the maximum continuous torques are 0.637, 0.319, and 0.319 Nm, for joints 1, 2, and 3 respectively. Encoders have a resolution of 2048 pulses/rev and the resolution is 3.66 × 10−4 deg at each joint. The control system is operated under Linux RTAI,
2019 IFAC MECHATRONICS 220 Vienna, Austria, Sept. 4-6, 2019
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that the adaptive gain using (12) converges on the stable ¯ is selected range, an excessively large constant gain M ¯ diag(1.298, 1.145, 0.574) through trial and error as M= because it is difficult to calculate the exact inertia matrix in a practical system.
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Fig. 2. (a) Adaptive gain (solid) and the upper bound from the stability criterion (7) (dashed). (b) Control input. (c) Tracking error. a real-time operating system environment. The desired trajectory is given in Fig. 3 (b) for all joints. The sampling period L is set as 0.002 s. When the damping ratio and angular frequency are selected as ξ = 1 and ω = 10 rad/s respectively, the desired error dynamics is derived as ¨ e +20e+100e ˙ = 0. Therefore, the slope gain λ is obtained as λ = diag(10.0, 10.0, 10.0). The control parameters, α and β, of the proposed method are tuned as α = diag(2000.0, 1800.0, 1800.0) and β = [0.04,0.01,0.01], respectively. ¯ is intentionally set to be M ¯ int = The initial value of M diag(1.553, 1.4, 0.701) which is the unstable one. To verify 623
The experimental results are shown in Fig. 4. The adaptive ¯ gain M(s) is adjusted by proposed gain dynamics (12). In ¯ ¯ reveals Figs. 4(a)–(c), comparison between M(s) and M that the proposed adaptive gain approaches the stable ¯ by the gain dynamics (12). The large gains range of M in Figs. 4 (a)–(c) can be regarded as the upper bound ¯ values of the gain M(s) which were found with trial and error. In Figs. 4 (d)–(f) and (g)–(i), proper tracking errors are shown and chatters are not discovered in the proposed method. On the other hand, the adaptive gains increase to compensate for significant pulse-type tracking errors caused by the discontinuity such as Coulomb friction and stiction at velocity reversal (Jin et al., 2009), as shown in Figs. 4 (a)–(f). To obtain more insight into the gain dynamics, the phase ¯ ii and si are shown in Fig. 5. portraits with respect to M The proposed method using the gain dynamics converges ¯ ii even though it starts at an on the stable region of M unstable initial gain. The adaptive TDC with the proposed gain dynamics provides stable gains in spite of unstable initial values. 6. CONCLUSION This study has proposed an adaptive gain dynamics for the TDC. The adaptive gain increases when the sliding variable is larger than the acceptance layer (tracking error
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¯ ii and si of joints 1, 2, and 3, respectively. Fig. 5. Phase portraits with respect to M is large), and decreases when the sliding variable becomes smaller than the acceptance layer to avoid an unnecessary large control gain. The resultant adaptive TDC makes the control gain stable despite unstable initial values. Simulations and experiments have demonstrated the effectiveness of the proposed algorithm: the gains are appropriate and in a stable region range, and the control performance with the proposed gain dynamics has been improved compared with the conventional TDC. REFERENCES Baghli, F. and El Bakkali, L. (2016). Design and simulation of robot manipulator position control system 624
based on adaptive fuzzy PID controller. In Robotics and Mechatronics, volume 37, chapter 4, 243–250. Zeghloul et al., Eds. Cham, Switzerland: Springer. Biglarbegian, M., Melek, W., and Mendel, J.M. (2011). Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: theory and experiments. IEEE Transactions on Industrial Electronics, 58(4), 1371–1384. Chang, P.H. and Lee, S.J. (2002). A straight-line motion tracking control of hydraulic excavator system. Mechatronics, 12(1), 119 – 138. Chen, M.S., Hwang, Y.R., and Tomizuka, M. (2002). A state-dependent boundary layer design for sliding mode control. IEEE Transactions on Automatic Control,
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IFAC PapersOnLine 52-15 (2019) 217–222
Stable Stable Gain Gain Adaptation Adaptation for for Time-Delay Time-Delay Stable Gain Adaptation for Time-Delay Control Robot Manipulators Stable Gainof Adaptation for Time-Delay Control of Robot Manipulators Control of Robot Manipulators Control of Robot Manipulators ∗∗∗ ∗∗ Kap-Ho Seo ∗∗∗ Junyoung Lee ∗∗ Pyung Hun Chang ∗∗
Junyoung Lee ∗ Pyung Hun Chang Kap-Ho Seo ∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗∗ Maolin Junyoung Lee ∗ Pyung HunJin Chang Kap-Ho Seo Maolin Jin ∗∗ ∗∗∗∗ Junyoung Lee Pyung HunJin Chang Kap-Ho Seo ∗∗∗ Maolin Maolin Jin ∗∗∗∗ ∗ ∗ Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) ∗∗ Institute of 37666, Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗Korea Daegu Gyeongbuk Institute of and Technology Namgu, Pohang, Korea (e-mail: [email protected]) Daegu Gyeongbuk Institute of Science Science and Technology (DGIST), (DGIST), ∗∗ Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Daegu 711-873, Korea (e-mail: [email protected]) Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 711-873, Korea (e-mail: [email protected]) ∗∗ ∗∗∗ Gyeongbuk Institute of Convergence Science and Technology (DGIST), Daegu 711-873, Korea (e-mail: [email protected]) ∗∗∗ Daegu Korea Institute of and (KIRO), Korea Institute of Robot Robot and Convergence (KIRO), Jigok-Ro Jigok-Ro 39, 39, ∗∗∗ Namgu, Daegu 711-873, Korea (e-mail: [email protected]) Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, Namgu, Pohang, 37666, Korea (e-mail: [email protected]) ∗∗∗ ∗∗∗∗Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗∗∗ Namgu, Korea Institute of Robot and Convergence (KIRO), Jigok-Ro Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute Robot Korea and Convergence (KIRO), Jigok-Ro 39, 39, ∗∗∗∗Namgu, Namgu, Pohang,of 37666, (e-mail: [email protected]) [email protected]) Pohang, 37666, Korea (e-mail: Korea Institute of Robot and Convergence (KIRO), Jigok-Ro 39, ∗∗∗∗Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Korea Institute RobotKorea and Convergence (KIRO), Jigok-Ro 39, Namgu, Pohang, of 37666, (e-mail: [email protected]) Namgu, Pohang, 37666, Korea (e-mail: [email protected]) Abstract: Abstract: This This paper paper proposes proposes adaptive adaptive gain gain dynamics dynamics for for time-delay time-delay control control (TDC) (TDC) of of robot robot Abstract: This paper proposes adaptive gain dynamics for time-delay control (TDC) of robot manipulators. The TDC is a widely employed approach for control of robot manipulators because manipulators. The TDC is a widely employed approach for control of robot manipulators because Abstract: This paper proposes adaptive gain dynamics for time-delay control (TDC) of robot it is model-independent, simple, and robust. Recently, however, it is reported that TDC with manipulators. The TDC is a widely employed approach for control of robot manipulators because it is model-independent, and robust.approach Recently,for however, it robot is reported that TDC with manipulators. The TDC issimple, a widely employed control of manipulators because excessively high gains causes unstable or oscillated responses. To overcome this limitation, this it is model-independent, simple, and robust. Recently, however, it is reported that TDC with excessively high gains causes unstable or oscillated responses. To overcome this limitation, this it is model-independent, simple, and robust. Recently, however, it is reported that TDC with study designs adaptive gain dynamics for the TDC using a sliding variable and an acceptance excessively high gains causes unstable or oscillated responses. To overcome this limitation, this study designs adaptive gain dynamics for the TDC using a sliding variable and an acceptance excessively high gains causes unstable or To overcome this limitation, this study designs adaptive gain dynamics foroscillated the adjusts TDCresponses. using a sliding variable and an acceptance layer. When the gain dynamics automatically aa control gain according to system states, layer. When the gain dynamics automatically adjusts control gain according to system states, study designsthe adaptive gain dynamics for therange. TDC Therefore, using a sliding variable and an acceptance the adaptive control gain moves into a stable the TDC with the proposed gain layer. When gain dynamics automatically adjusts a control gain according to system states, the adaptive control gain movesautomatically into a stable range. Therefore, the TDC with the proposed gain layer. When the gain dynamics adjusts a control gain according to system states, dynamics becomes adaptive and stable. The effectiveness of the proposed dynamics has the adaptive control gain moves into a stable range. Therefore, the TDC withgain the proposed gain dynamics becomes adaptive and stable. The effectiveness of the proposed gain dynamics has the adaptive control gain moves into a stable range. Therefore, the proposed TDC withgain the proposed gain been verified by simulations and experiments. dynamics becomes adaptive and stable. The effectiveness of the dynamics has been verified by simulations and experiments. dynamics becomes adaptive and and experiments. stable. The effectiveness of the proposed gain dynamics has been verified by simulations © 2019, IFAC by (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. been verified simulations and experiments. Keywords: Keywords: Adaptive Adaptive control, control, sliding sliding mode mode control, control, time-delay time-delay control, control, robot robot manipulator. manipulator. Keywords: Adaptive control, sliding mode control, time-delay control, robot manipulator. Keywords: Adaptive control, sliding mode control, time-delay control, robot manipulator. 1. Youcef-Toumi 1. INTRODUCTION INTRODUCTION Youcef-Toumi and and Ito, Ito, 1990; 1990; Hsia Hsia et et al., al., 1991). 1991). Since Since the the 1. INTRODUCTION TDC does not require the calculation of robot dynamics Youcef-Toumi and Ito, 1990; Hsia et al., 1991). Since the TDC does not require the calculation of robot dynamics 1. INTRODUCTION Youcef-Toumi and Ito, 1990; Hsia et al., 1991). Since the owing to technique, the becomes simple, does notTDE require the calculation robot dynamics Robot manipulators manipulators are employed employed in in various various fields owing to the the TDE technique, the TDC TDCof becomes simple, fields and and TDC Robot are TDC does not require the calculation of robot dynamics owing to the TDE technique, the TDC becomes simple, efficient, and yet robust (Hsia, 1989; Youcef-Toumi and Robot manipulators aretasks employed in various fieldset operated with advanced advanced tasks (Severinson-Eklundh et and al., efficient, and yet robust (Hsia, 1989; Youcef-Toumi and operated with (Severinson-Eklundh al., to the TDE technique, the TDCthe becomeshas simple, RobotVircikova manipulators are employed in various fields Ito, Hsia et 1991). As result, efficient, robust (Hsia, and operated with advanced tasks (Severinson-Eklundh et and al., owing 2003; et al., 2015). To conduct various works, Ito, 1990; 1990; and Hsiayet et al., al., 1991). As aa 1989; result,Youcef-Toumi the TDC TDC has been been 2003; Vircikova et al., 2015). To conduct various works, and yet robust (Hsia, 1989; Youcef-Toumi and operated with advanced tasks (Severinson-Eklundh etconal., efficient, widely applied to various areas such as robot manipulators Ito, 1990; Hsia et al., 1991). As a result, the TDC has been 2003; Vircikova et al., 2015). To conduct various works, accurate control is required. However, in model-based applied to various areas such as robot manipulators accurate control is required. However, in model-based con- widely Ito, 1990; Hsia et al., 1991). As a result, the TDC has been 2003; Vircikova et al., 2015). To conduct various works, (Jung et al., Jin al., 2008; Jin et 2017), applied to various robot manipulators accurate However, in dynamic model-based con- widely trol, it it is is control difficultisto torequired. obtain the the complex dynamic equation (Jung et al., 2004; 2004; Jin et etareas al., such 2008;as Jin et al., al., 2017), an an trol, difficult obtain complex equation widely applied tosystem various asChen, robot manipulators accurate control istorequired. However, in dynamic model-based con(Jung et al., 2004; Jin (Cheng etareas al., such 2008; Jin et 1996), al., 2017), an overhead crane and a wind trol, it is difficult obtain the complex equation of a robot manipulator, which includes uncertainty and overhead crane system (Cheng and Chen, 1996), a wind of a robot manipulator, which includes uncertainty and (Jung et(Kim al., 2004; Jin et al., 2008; Jinreactor et 1996), al., (Gonz´ 2017), an trol, it is difficult to obtain the complex dynamic equation turbine et al., 2015), a chemical a lez overhead crane system (Cheng and Chen, a wind of a robot manipulator, which includes uncertainty and nonlinearity (Baghli and El Bakkali, 2016; Li et al., 2013). turbine (Kim etsystem al., 2015), a chemical reactor (Gonz´ alez nonlinearity (Baghli and El Bakkali, 2016;uncertainty Li et al., 2013). crane (Cheng and (Kim Chen, 1996), a wind of acope robot manipulator, which includes and overhead et al., 2002), an underwater vehicle et al., 2016), a turbine (Kim et al., 2015), a chemical reactor (Gonz´ a lez nonlinearity (Baghli and El Bakkali, 2016; Li et al., 2013). To with this problem, a lot of attempts have been al., 2002), an vehicle (Kim et al.,(Gonz´ 2016), a To cope with(Baghli this problem, a lot of2016; attempts been et turbine (Kim et underwater al., 2015), a2012), chemical reactor aand lez nonlinearity and El Bakkali, Li et have al., 2013). tilt-rotor aircraft (Lee et al., excavators (Chang et al., 2002), an underwater vehicle (Kim et al., 2016), a To cope with this problem, a lot of attempts have been tried: for example, sliding-mode control (SMC) (Fallaha tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and tried: for example, sliding-mode control (SMC) (Fallaha al.,2002; 2002), anetunderwater vehicle (Kim et al., 2016), a Toal., cope with thisand problem, a lotcontrol of attempts have been et tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and Lee, Kim al., 2019), chaotic systems (Kim et al., tried: for example, sliding-mode (SMC) (Fallaha et 2011; Islam Liu, 2011), and fuzzy control (Lian, Lee, 2002; Kim et al., 2019), chaotic systems (Kim et al., et al., 2011; Islam and Liu, 2011), control and fuzzy control (Lian, tilt-rotor aircraft (Lee et al., 2012), excavators (Chang and tried: for example, sliding-mode (SMC) (Fallaha 2017; Jin and Chang, 2009), and a shape memory alloy Lee, 2002; Kim et al., 2019), chaotic systems (Kim et al., et al., 2011; Islam and Liu, 2011), and fuzzy control (Lian, 2011; Biglarbegian et al., 2011) to list a few. Nevertheless, 2017;2002; Jin and Chang, 2009), chaotic and a shape memory 2011; et al., 2011) to and list afuzzy few. control Nevertheless, Kim et al., 2019), systems (Kim etalloy al., et al., Biglarbegian 2011; Liu, 2011), (Lian, Lee, actuator et al., 2015). 2017; Jin(Jin and Chang, 2009), and a shape memory alloy 2011; Biglarbegian etthe al., 2011) to list of a few. Nevertheless, SMCs still Islam requireand the information of robot dynamics, actuator (Jin et al., 2015). SMCs still require information robot dynamics, 2017; Jin and Chang, 2009), and a shape memory alloy 2011; Biglarbegian et al., 2011) to list a few. Nevertheless, A been (Jin et have al., 2015). SMCs still requireis information of control robot dynamics, whose calculation calculation is the complicated. Fuzzy control is aa good good actuator A lot lot of of studies studies been conducted conducted on on the the TDC TDC (Hsia, (Hsia, whose complicated. Fuzzy is (Jin et have al., 2015). SMCs calculation still require information of control robot dynamics, 1989; and Ito, 1990; Hsia al., 1991; Jung A lot Youcef-Toumi of studies have been conducted onet the TDC whose is the complicated. Fuzzy is a good actuator solution for the the estimation of nonlinearity nonlinearity and uncertainty uncertainty 1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991;(Hsia, Jung solution for estimation of and A lot of studies have been conducted on the TDC (Hsia, whose calculation is complicated. Fuzzy control is a comgood 1989; et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim Youcef-Toumi and Ito, 1990; Hsia et al., 1991; Jung solution for the estimation of nonlinearity and uncertainty of the robot dynamics, however, it suffers from the et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim of the robot dynamics, however, it suffers from the com1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991; Jung solution for the estimation of nonlinearity and uncertainty 2015; Gonz´ a lez et al., 2002; Kim et al., 2016; Lee et al., 2004; Jin et al., 2008; Cheng and Chen, 1996; Kim of the robot dynamics, however, it suffers from the complexity and difficulty of implementation, due to a lot of et al., 2015; Gonz´ a lez et al., 2002; Kim et al., 2016; Lee of implementation, due to a plexity and difficulty lot of et al., 2004; Jin etaal., al., 2008; Cheng and Chen, 1996; Kim of therules robot dynamics, however, it suffers due fromtothe comal., 2012; Jin et 2015) with a constant constant gain which is et al., 2015; Gonz´ lez et al., 2002; Kim et al., 2016; Lee plexity and difficulty of implementation, a lot of fuzzy and parameters. et al., 2012; Jin et al., 2015) with a gain which is fuzzy rules and parameters. al., 2012; 2015; Gonz´ aal., leztrial et al., 2002; Kim et al., Lee plexityis and difficulty of implementation, due to a lot of et manually tuned with and error (Jin et al., 2009; Lee et al., Jin et 2015) with a constant gain2016; which is fuzzy rules and parameters. TDC a well-established well-established robust control control algorithm algorithm (Hsia manually tuned with trial and error (Jin et al., 2009; Lee TDC is a robust (Hsia et al., 2012; Jin et al., 2015) with aTDC, constant gain2009; which is fuzzyGao, rules and parameters. et al., 2014). In the aspect of the a relatively small manually tuned with trial and error (Jin et al., Lee TDC is a well-established robust control algorithm (Hsia and 1990; Youcef-Toumi and Wu, 1992). It employs et al., 2014). In the aspect of the TDC, aetrelatively small and Gao, 1990; Youcef-Toumi andcontrol Wu, 1992). It employs manually tuned with trial and error (Jin al., 2009; Lee TDC is a well-established robust algorithm (Hsia gain larger tracking and relatively large al.,provides 2014). In the aspect of errors, the TDC, and Gao, 1990; estimation Youcef-Toumi and technique Wu, 1992). It employs the time-delay time-delay estimation (TDE) technique which inten- et gain provides larger tracking and aaaa relatively relatively small large the (TDE) which intenet al.,induces 2014). instability In the aspect of errors, the TDC, relatively and time-delay Gao,utilizes 1990; estimation Youcef-Toumi and technique Wu, 1992). It previous employs gain provides larger tracking errors, and relatively large or oscillation oscillation of aaasystem. system. In small addithe (TDE) which intentionally time-delayed information in the gain induces instability or of In additionally utilizes time-delayed information in the previous gain provides larger tracking errors, and a relatively large the time-delay estimation (TDE) technique which intention, the constant gain cannot always provide applicable gain induces instability or oscillation of a system. In additionally utilizes time-delayed information in the previous sampling instant instant to to estimate estimate robot robot dynamics dynamics (Hsia, (Hsia, 1989; 1989; tion, the constant gain cannot always provide applicable sampling induces instability or oscillation of provide a system. In additionally utilizes information in the previous performance (Jin et al., 2017). Notice that this constant tion, the constant gain cannot always applicable sampling instanttime-delayed to estimate robot dynamics (Hsia, 1989; gain performance (Jin etgain al.,cannot 2017). always Notice provide that thisapplicable constant tion, the constant sampling instant to estimate robot dynamics (Hsia, 1989; This gain is theoretically to the stability (Hsia performance (Jin etrelated al., 2017). Notice thatcondition this constant This work work was was supported supported by by the the Ministry Ministry of of Trade, Trade, Industry Industry and and gain is theoretically related to the stability condition (Hsia performance (Jin Youcef-Toumi etrelated al., 2017). thatcondition this constant Energy (MOTIE, Korea) under Innovation gain is theoretically to theNotice stability (Hsia This work was supported by theIndustrial Ministry Technology of Trade, Industry and and Gao, 1990; and Wu, 1992). In this Energy (MOTIE, Korea) under Industrial Technology Innovation and Gao, 1990; Youcef-Toumi and Wu, 1992). In this This work was supported by the Ministry of Trade, Industry and gain is theoretically related to theand stability condition (Hsia Program.(MOTIE, No.10080355, “Development of series series elastic actuator actuator Energy Korea) under Industrial Technology Innovation context, some studies have recently focused on achieving and Gao, 1990; Youcef-Toumi Wu, 1992). In this Program. No.10080355, “Development of elastic and context, some studies have recently focused on achieving Energy (MOTIE, Korea) under Industrial Technology Innovation manipulator with control collision and Gao,some 1990; Youcef-Toumi and focused Wu, 1992). In this Program. No.10080355, “Development series elastic actuator manipulator with compliance compliance control for forofcorresponding corresponding collision and and an auto-tuned gain for the TDC. context, studies have recently on achieving an auto-tuned gain forhave the TDC. Program. No.10080355, “Development series elastic actuator and minimizing impulse” and 20001184, of manipulator with compliance forof“Development corresponding collision and context, some studies recentlyfor focused on achieving minimizing impulse” and No. No. control 20001184, “Development of ultra-thin ultra-thin A study on automatic an auto-tuned gain for gain-tuning the TDC. A study on automatic gain-tuning for the the TDC TDC exploited exploited manipulator with compliance control for “Development corresponding collision and and short precision reducers and high torque to weight ratio precision minimizing impulse” and No. 20001184, of ultra-thin anNussbaum auto-tuned gain for gain-tuning the TDC. and short precision reducers and high torque to weight ratio precision A study on automatic for the TDC exploited a technique (Cho et al., 2014); however, this minimizing impulse” and No. 20001184, “Development of ultra-thin aA Nussbaum technique (Cho et al., 2014); however, this reducers HRC (Corresponding Author: Maolin Jin.) and shortfor precision reducers and high torque to weight ratio precision reducers for HRC robots”. robots”. (Corresponding Author: Maolin Jin.) study on automatic gain-tuning for the TDC exploited a Nussbaum technique (Cho et al., 2014); however, this and shortfor precision reducers(Corresponding and high torqueAuthor: to weight ratio precision reducers HRC robots”. Maolin Jin.) a Nussbaum technique (Cho et al., 2014); however, this reducers for HRC robots”. (Corresponding Author: Maolin Jin.)
2405-8963 © © 2019 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 620 Copyright © under 2019 IFAC IFAC 620 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 620 10.1016/j.ifacol.2019.11.677 Copyright © 2019 IFAC 620
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algorithm requires acceleration, which is vulnerable to signal noise. Moreover, the Nussbaum gain technique induces the complexity owing to a number of tuning parameters. Consequently, the simplicity of TDC is not maintained. In another study (Jin et al., 2017), an adaptive gain dynamics for TDC using sliding mode variable was proposed. Satisfactory tracking performances and appropriate gains had been achieved from the adaptive gain dynamics, however, the searching region for the adaptive gain is intentionally bounded by a saturation function. The initial value of the gain should be a small value in this approach. When the initial value of the gain is selected as an excessively large value, the system may become oscillated or unstable. This paper proposes an adaptive gain dynamics for designing adaptive TDC using a sliding variable and an acceptance layer. Using the proposed gain dynamics, the control gain is automatically regulated as a stable value satisfying the stability condition of the TDC even if an initial value of the gain is set as an unstable one. The acceptance layer is used to find the proper gain and is related to the concept of a boundary layer (Chen et al., 2002; Fuh et al., 2008) in the aspect of tracking accuracy. As a result, the TDC with the proposed adaptive gain dynamics becomes simple, robust, adaptive, and stable, because the TDE technique effectively compensates for the system dynamics, and the proposed gain dynamics adjusts the magnitude of the control gain that satisfies the stability condition. The structure of the TDC with proposed gain dynamics is still simple and easy for implementation. The effectiveness of the proposed gain dynamics has been verified by the simulations and experiments.
robot dynamics. The control law of TDC is designed by using (4) as ¯ qt−L + M(¨ ¯ qd + KD e˙ + KP e), (5) τ = τt−L − M¨ where e stands for the joint displacement error, given as e = qd − q; qd is the desired joint displacement; and KD , KP are the positive diagonal feedback gains. Combining (2)–(5), the closed-loop dynamics is represented as ¯ −1 (N − Nt−L ). ¨ e + KD e˙ + KP e = M (6) ¯ Since the constant gain M affects the control performance ¯ is important. In addition, we in (6), the selection of M ¯ need to select M such that the stability of a system is guaranteed. The well-established stability condition for the TDC is denoted (Hsia and Gao, 1990; Youcef-Toumi and Wu, 1992) as I−M−1 (q)M ¯ < 1. (7) When satisfying the stability condition (7), the right-hand side N − Nt−L of (6) is bounded. The bounded TDE error vector ε is defined as follows: ∆
˙ q ¨) − Nt−L . (8) ε = N(q, q, ¯ In lots of the TDE-based controllers, M is a constant gain (Hsia and Gao, 1990; Youcef-Toumi and Wu, 1992; Cho ¯ depends et al., 2009; Jin et al., 2011) and tuning of M on the inertia M(q) and the system noise level (Cho et al., 2009; Jin et al., 2017). The constant gain has been manually tuned by trial and error, although the inertia varies according to the posture of a robot manipulator during operation (Jin et al., 2009; Lee et al., 2014). The manually tuned constant gain may cause either poor tracking performances or unstable responses.
2. REVIEW OF TDC The dynamic model of an n-degree-of-freefom (n-DOF) rigid robot manipulator can be written as follows: M(q)¨ q + C(q, q) ˙ + G(q) ˙ + F(q, q) ˙ + τd = τ, (1) where q denotes a vector of the joint displacement; M(q) the inertia matrix; C(q, q) ˙ the Coriolis and centripetal torques; G(q) the gravitational vector; F(q, q) ˙ the friction torques; τd the disturbances; τ the applied joint torques. The time variable t is omitted for simplicity. ¯ the robot When using the positive diagonal gain matrix M, dynamics (1) can be reformulated as ¯ q + N(q, q, M¨ ˙ q ¨) = τ, (2) where N(q, q, ˙ q ¨) includes nonlinear and/or uncertain terms such as the Coriolis/centripetal torques, the gravitational torques, the friction torques, and disturbances. It is written as ¯ q + C + G + F + τd . (3) N = [M − M]¨ N can be estimated with the TDE technique (Hsia, 1989; Youcef-Toumi and Ito, 1990; Hsia et al., 1991) as follows: ¯ qt−L , ˆ = Nt−L = τt−L − M¨ (4) N≈N
where •t−L is an intentional time-delayed value in the previous sampling instant t − L. As L deceases, the ˆ is close to N in (4). Therefore, L is selected estimation N as the sampling period for the accurate estimation in the digital implementation. The TDE technique represents the main idea for reducing the computational complexity of 621
3. DESIGN OF GAIN DYNAMICS 3.1 Gain Dynamics To obtain the proper gain satisfying the stability condition (7), the gain dynamics is proposed by using a linear sliding variable s as ∆ s = e˙ + λe, (9) where λ is the positive diagonal matrix. The control objective is for q to follow qd . Thus, the characteristic equation is designed as s˙ + λs = 0, (10) where λ has strict eigenvalues in the right-half complex plane. Since substituting (9) into (10) yields ¨ e + 2λe˙ + λ2 e = 0 (KD = 2λ and KP = λ2 ), the TDC (5) can be rewritten as ¯ qt−L + M(s)(¨ ¯ τ = τt−L − M(s)¨ qd + 2λe˙ + λ2 e). (11) ¯ is not a constant. The adaptive From now on, the gain M ¯ gain M(s) is utilized in (11) and the gain dynamics for ¯ M(s) is proposed as follows: ¯2 ¯˙ ii = αii M ¯ ii2 |si | sgn(|si | − Mii ); M (12) βi where •i is the i-th element of a vector and •ii are the ii-th diagonal element of a diagonal matrix, respectively; αii is the positive diagonal matrix; and βi is the normalizing ¯ 2 and si . A closed-loop system with the factor for M ii
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4. SIMULATION 4.1 Simulation Setup The simulation with a one-link arm is implemented to verify the effectiveness of the TDC with the proposed gain dynamics. The model of the one-link arm is written as follows: τ = I q¨ + G(q) + F (q), ˙ (13) where I = ml2 , G(q) = mlgcos(q), and F (q) ˙ = fV q˙ + fC sgn(q). ˙ fV and fC denote the viscous friction coefficient and the Coulomb friction coefficient, respectively. The simulation parameters are set to be m = 1.0 kg, l = 1.0 m, fV = 5.0 Nms, fC = 5.0 Nm, and g = 9.8 m/s2 , respectively. The desired trajectory is selected as qd = 45 · (1 − e−0.4πt ) · sin(0.4πt) deg. The desired error dynamics is selected as e¨ + 20 e˙ + 100 e = 0. The parameters α and β are tuned as α = 1000.0 and 622
¯ is set as β = 120.0, respectively. The initial value of M 2 ¯ M = 4 kg · m which is unstable. Notice that the stable ¯ is derived as 0 < M ¯ < 2 by (7) (Hsia and Gao, range of M 1990), which is shown in Fig. 2 (a). 4.2 Simulation results The simulation results are shown in Fig. 2. The control ¯ becomes adaptive by the gain dynamics, as shown gain M ¯. in Fig. 2 (a), and it stays in the stable range of M ¯ is the unstable one, the Although the initial value of M TDC using the proposed adaptive gain dynamics (12) ¯ stable. The control input and tracking makes the gain M error are shown in Fig. 2 (b) and (c), respectively. When the significant large tracking errors occur, these errors are compensated because of the increase of the adaptive gain ¯. M 5. EXPERIMENT 5.1 Experimental Setup The control performance of the TDC with the proposed gain dynamics is validated using a PUMA-type robot in Fig. 3 (a). At AC servo motors, harmonic drives have gear ratios of 120:1, 120:1, and 100:1, and the maximum continuous torques are 0.637, 0.319, and 0.319 Nm, for joints 1, 2, and 3 respectively. Encoders have a resolution of 2048 pulses/rev and the resolution is 3.66 × 10−4 deg at each joint. The control system is operated under Linux RTAI,
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that the adaptive gain using (12) converges on the stable ¯ is selected range, an excessively large constant gain M ¯ diag(1.298, 1.145, 0.574) through trial and error as M= because it is difficult to calculate the exact inertia matrix in a practical system.
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Fig. 2. (a) Adaptive gain (solid) and the upper bound from the stability criterion (7) (dashed). (b) Control input. (c) Tracking error. a real-time operating system environment. The desired trajectory is given in Fig. 3 (b) for all joints. The sampling period L is set as 0.002 s. When the damping ratio and angular frequency are selected as ξ = 1 and ω = 10 rad/s respectively, the desired error dynamics is derived as ¨ e +20e+100e ˙ = 0. Therefore, the slope gain λ is obtained as λ = diag(10.0, 10.0, 10.0). The control parameters, α and β, of the proposed method are tuned as α = diag(2000.0, 1800.0, 1800.0) and β = [0.04,0.01,0.01], respectively. ¯ is intentionally set to be M ¯ int = The initial value of M diag(1.553, 1.4, 0.701) which is the unstable one. To verify 623
The experimental results are shown in Fig. 4. The adaptive ¯ gain M(s) is adjusted by proposed gain dynamics (12). In ¯ ¯ reveals Figs. 4(a)–(c), comparison between M(s) and M that the proposed adaptive gain approaches the stable ¯ by the gain dynamics (12). The large gains range of M in Figs. 4 (a)–(c) can be regarded as the upper bound ¯ values of the gain M(s) which were found with trial and error. In Figs. 4 (d)–(f) and (g)–(i), proper tracking errors are shown and chatters are not discovered in the proposed method. On the other hand, the adaptive gains increase to compensate for significant pulse-type tracking errors caused by the discontinuity such as Coulomb friction and stiction at velocity reversal (Jin et al., 2009), as shown in Figs. 4 (a)–(f). To obtain more insight into the gain dynamics, the phase ¯ ii and si are shown in Fig. 5. portraits with respect to M The proposed method using the gain dynamics converges ¯ ii even though it starts at an on the stable region of M unstable initial gain. The adaptive TDC with the proposed gain dynamics provides stable gains in spite of unstable initial values. 6. CONCLUSION This study has proposed an adaptive gain dynamics for the TDC. The adaptive gain increases when the sliding variable is larger than the acceptance layer (tracking error
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¯ ii and si of joints 1, 2, and 3, respectively. Fig. 5. Phase portraits with respect to M is large), and decreases when the sliding variable becomes smaller than the acceptance layer to avoid an unnecessary large control gain. The resultant adaptive TDC makes the control gain stable despite unstable initial values. Simulations and experiments have demonstrated the effectiveness of the proposed algorithm: the gains are appropriate and in a stable region range, and the control performance with the proposed gain dynamics has been improved compared with the conventional TDC. REFERENCES Baghli, F. and El Bakkali, L. (2016). Design and simulation of robot manipulator position control system 624
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