Adaptive robust disturbance compensating control for ...

Original Article

Adaptive robust disturbance compensating control for a servo system in the presence of both friction and deadzone

Proc IMechE Part C: J Mechanical Engineering Science 0(0) 1–14 ! IMechE 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406215616422 pic.sagepub.com

Jinho Jung1, Donghyuk Lee2, Jong Shik Kim1 and Seong Ik Han2

Abstract An adaptive robust control that does not need sophisticated plant modeling work is proposed for precise output positioning of a servo system in the presence of both friction and deadzone nonlinearities. It is difficult to achieve effective motion control by traditional linear control methodology for these types of nonlinearities, without the aid of a proper compensation scheme for nonlinearity. In this study, dynamic friction is modeled by a Tustin friction model, and inverse deadzone method is adopted to compensate deadzone effect. The adaptive laws of the unknown system dynamic parameters, friction and deadzone, are derived. Furthermore, a robust control method with funnel control is proposed to compensate for unmodeled and estimation errors. The boundedness and convergence of the closed-loop system are ensured by a Lyapunov stability analysis. The performance of the proposed control scheme is verified through experiments on the XY table servo system and the robotic manipulator. Keywords Adaptive control, deadzone, friction, funnel control, robust uncertainty compensator, servo system Date received: 2 July 2015; accepted: 19 October 2015

Introduction In recent times, with the increasing demand for high quality production, precise motion in industrial servo machines, such as machine tools and industrial robots, has become an important requirement. A wide variety of control strategies have been investigated for mechanical servo systems including PD and nonlinear feedback control1,2 and the open-loop optimization control.3,4 In most servo machines, besides mechanical elements such as inertia, and coupled dynamics, there are often inherent nonlinear elements such as friction and deadzone in the contact surface and in the transmitting power actuator; it is often difficult to compensate for these parameters by a simple linear controller. Nonlinear friction gives rise to transmission lag between contact surfaces, and deadzone leads to loss of power efficiency. In order to achieve both high motion control performance and convenient applicability of the servo system, a reasonable identification and modeling process for these mechanical parameters, and efficient control algorithm without depending on a complex control structure are prerequisites. Nonlinear friction models include classic Tustin friction model5 and dynamic friction models such as the LuGre model6 and generalized Maxwell-slip

friction model (GMS).7 The classic friction model cannot compensate for pre-sliding friction properties. While the dynamic friction model operates in the presliding friction, the shortcomings of these advanced friction models are that there are a lot of parameters which lead to failures in real-time operation, and require a deep understanding of the machine structure system. Several researchers have developed deadzone compensation methods in the servo and robotic systems.8–10 In most cases, modeling of deadzone is not simple because of irregular and asymmetric width and slope of the deadzone. The approaches for the servo system control that take case of both friction and deadzone are rare; the limitations are on account of the difficulty in obtaining exact compensation for friction and deadzone. Some control approaches for servo mechanical systems without modeling 1

School of Mechanical Engineering, Pusan National University, Busan, Korea Republic 2 Department of Electronic Engineering, Pusan National University, Busan, Korea Republic Corresponding author: Seong Ik Han, Department of Electronic Engineering, Pusan National University, Busan 609-735, Korea Republic. Email: [email protected]

Downloaded from pic.sagepub.com at PUSAN NATL UNIV on December 15, 2015

2

Proc IMechE Part C: J Mechanical Engineering Science 0(0)

mechanical parameters are found in the literatures11–14 wherein the adaptive estimation-based control was employed instead of the model-based control. These adaptive estimation-based controls showed efficient control performance to some extent in applications to robotic manipulator systems without the presence of friction or deadzone, or both. In this paper, the adaptive laws were developed not only for the unknown parameters such as inertia, coupled dynamics, and gravity term, but also for friction and deadzone parameters in order to design a non-model-based controller without introducing any modeling procedure, for ensuring fast controller design and improved tracking motion in industrial applications. However, the adaptive estimation for unknown parameters does not always guarantee satisfactory performance because of its incomplete choice of the adaptive gain and difficulty in determining the optimal adaptive gain. Generally, it is known that the switching law compensates for unknown estimation errors and disturbances in sliding mode control (SMC).15 However, this control causes chattering, generated by the sign function and high frequency oscillation in the control action. Therefore, a robust disturbance compensation law that borrowed its concept from the funnel control16,17 is considered to add robustness to the adaptive controller without generating chattering during tracking. The gains of the funnel-based compensator are obtained from the estimate for the unknown upper bound of disturbance. From the Lyapunov function, all the adaptive laws are derived, and a stability analysis is performed. The designed controller is implemented using Matlab Realtime Toolbox and MF624 interface board18 for conducting experiments on the XY table and the articulate robot manipulator.

Problem formulation Description of a non-smooth nonlinear servo system Consider a servo system in the presence of both deadzone and friction, which is described by the dynamic equation

_ are Property 219. Since MðqÞ and therefore MðqÞ symmetric matrices, the skew-symmetry of the _ _ can be also be seen from the matrix MðqÞ 2Cðq, qÞ _ _ þ CT ðq, qÞ. _ fact MðqÞ ¼ Cðq, qÞ Property 319. Even though the skew-symmetry prop_ _ is guaranteed if Cðq, qÞ _ is erty of MðqÞ 2Cðq, qÞ defined by the Christoffel symbols, it is always true _ _ q_ ¼ 0. that q_ T ½MðqÞ 2Cðq, qÞ Assumption 1. There exist some finite positive constants, ki 4 0, 14i44 and finite nonnegative constants, such that 8q 2 Rn , 8q_ 2 Rn , jMðqÞ j4k1 , _ j4k2 , jGðqÞ j4k4 , and supt50 jFd j4k4 . jCðq, qÞ The deadzone nonlinearity DðuÞ is shown in Figure 1(a) and a mathematical model is described by 8 > < mr ðuðtÞ dr Þ DðuÞ ¼ 0 > : ml ðuðtÞ dl Þ

for uðtÞ5dr for dl 5 uðtÞ 5 dr for uðtÞ4dl

ð2Þ

where mr and ml denote the slope of the deadzone, and dr , and dl stand for the deadzone width parameters. The practical assumptions concerning the deadzone are given below for the control problem: Assumption 2. The deadzone output is not available for measurement. The deadzone parameters, dr and dl , are unknown but their signs are known as dr 50 and dl 40. Assumption 3. The deadzone slopes are assumed as mr ¼ 1 and ml ¼ 1 to simplify the problem. The deadzone inverse technique is a useful method for compensating the deadzone effect.8–10 Setting ud ðtÞ as the control signal from the controller for achieving control of the plant without deadzone, the following control signal uðtÞ is generated according to certainty equivalence deadzone inverse described in Figure 1(b) uðtÞ ¼ D1 ðud ðtÞÞ ¼ ðud ðtÞ þ d^r Þ p þ ðud ðtÞ þ d^l Þð1 pÞ

_ þ GðqÞ þ Ff ðq, qÞ _ þ Fd ðtÞ ¼ DðuÞ MðqÞ þ Cðq, qÞ

ð3Þ ð1Þ

where q ¼ ½q1 , . . . , qn T 2 Rn denotes the vector of generalized coordinates; MðqÞ 2 Rnn represents the symmetrically bounded positive definite matrix; _ 2 Rnn is the centripetal Coriolis torque Cðq, qÞ _ 2 Rn matrix; GðqÞ is the gravity force vector; Ff ðq, qÞ n is the nonlinear friction vector; Fd 2 R is the external disturbance; and u 2 Rn is the control input vector. Property 1. The inertia matrix, MðqÞ, is known to be symmetric and positive definite.

where d^r and d^l are the estimates of dr and dl , respectively and p¼

1 if ud ðtÞ50 0 if ud ðtÞ 5 0

ð4Þ

The resulting error between u and ud is given by DðuðtÞÞ ud ðtÞ ¼ d~r p þ d~l ð1 pÞ

ð5Þ

The resulting control system with deadzone inverse is shown in Figure 2.


Jung et al.

3

Figure 1. Configuration of deadzone and deadzone inverse. (a) Deadzone model (b) Deadzone inverse.

Design of controller and nonlinearity compensator

Figure 2. The block diagram of the controlled system with deadzone inverse.

Tustin friction model The friction force as the classical Tustin model can be expressed as h 2 i Ff ¼ Fc þ ðFs Fc Þ exp vv1 sgnðvÞ þ Fv v s ð6Þ where Fc denotes Coulomb friction, Fs denotes the stiction level, Fv represents the viscous friction, and vs is the Stribeck velocity. The estimation of the friction force can be obtained from h

2 i F^f ¼ F^c þ ðF^s F^c Þ exp vv1 sgnðvÞ þ F^v v s ð7Þ where Fî are estimates of Fi , i ¼ f, c, s, v, respectively. From equations (6) and (7), the following expression is obtained F~f ¼ Ff F^f h 2 i ¼ F~c 1 exp vv1 sgnðvÞ s 2 sgnðvÞ þ F~v v þ F~s exp vv1 s where F~ð Þ ¼ Fð Þ F^ð Þ .

ð8Þ

In this section, three controllers are designed and the adaptive laws for the unknown system parameters are derived from the stability analysis, by using Lyapunov function. First, a model reference controller is designed under the assumption that the dynamic parameters of the servo system are known. However, this controller cannot be applied efficiently to a real system because most of the dynamic parameters are unknown. The second adaptive controller is then designed using estimated parameters from the adaptive laws for the unknown dynamic parameters. In the third step, since the adaptive laws cannot estimate the unknown parameters precisely and some estimate errors exist, a robust control is added in order to ensure robustness of the adaptive controller, by introducing funnel control, without considering relay control of the conventional SMC, which generates undesirable chattering.

Model reference control The control objective for a servo dynamic system is to determine a state feedback control system such that the system output q can track a desired trajectory qd , while ensuring that all the closed loop signals are bounded. Consider the following signals e ¼ q qd

ð9Þ

r_ ¼ q_d e

ð10Þ

where qd ¼ ½qd1 , . . . , qdn T is the desired trajectory, ¼ diagð1 , . . . , n Þ is the constant matrix, and e ¼ ½e1 , . . . , en T . The filtered error surface s and its derivative s_ are defined as s ¼ e_ þ e


4


¼ q_ q_d þ e ¼ q_ r_

ð11Þ

s_ ¼ e€ þ e_ ¼ q€ q€d þ e_

ð12Þ

¼ q€ r€ Based on the terms of the definitions (9) to (12) and if the deadzone is compensated by the inverse deadzone technique precisely, i.e. DðuðtÞÞ ¼ ud ðtÞ in (5), the dynamic equation (1) can be written as MðqÞ_s ¼ MðqÞq€ MðqÞ€r _ q_ MðqÞ€r GðqÞ Ff ðq, qÞ _ ¼ Cðq, qÞ Fd ðtÞ þ ud _ MðqÞ€r Cðq, qÞ_ _ r GðqÞ ¼ Cðq, qÞs _ Ff ðq, qÞ Fd ðtÞ þ ud ð13Þ Define the Lyapunov function candidate as follows 1 V1 ¼ sT MðqÞs 2

ð14Þ

Considering (13) and the obtained result of _ _ ¼ 0 from the property 3, the 2Cðq, qÞs sT ½MðqÞ time derivative of V1 is written as 1 _ V_ 1 ¼ sT MðqÞ_s þ sT MðqÞs 2 1 _ _ s ¼ sT MðqÞ 2Cðq, qÞ 2 _ GðqÞ þ sT ½MðqÞ€r Cðq, qÞr _ þ ud Fd ðtÞ Ff ðq, qÞ

ð15Þ

_ r GðqÞ ¼ sT ½MðqÞ€r Cðq, qÞ_ _ Fd ðtÞ þ ud Ff ðq, qÞ Let us choose the control input as u ¼ ueq þ us

ð16Þ

¼ s Tm m signðsÞ

where ueq ¼ s Tm m , us ¼ signðsÞ, m ¼ ½€r _ GðqÞ Ff ðq, qÞ _ Fd T , r_ 1 1 1T , m ¼ ½MðqÞ Cðq, qÞ 4 0 are constants, signðsÞ are the sign functions. Based on equation (16), equation (15) is written as V_ 1 4 ks k2 jsj

ð17Þ

4 min ks k2 40 Integrating equation (17) leads to Z V1 ðtÞ V1 ð0Þ4 min 0

t

ks k2 d40

ð18Þ

Therefore, s remains to small compact set containing the origin, as time goes to infinity. However, this model-based control requires exact information on mechanical dynamics and disturbances. When the dynamics are complex and disturbances are unknown, this control scheme undergoes limitations in real applications. Therefore, we consider an adaptive control method by adopting estimates of unknown dynamics and disturbances.

Adaptive control In this section, the controller and adaptive laws are derived. Defining the tracking error as e ¼ q qd , the command vector, rðtÞ, and its derivative, r_ðtÞ, are defined as r_ ¼ q_d e

ð19Þ

r€ ¼ q€d e_

ð20Þ

Based on equations (5) and (8), and the terms of the definitions (19) and (20), the dynamic equation (1) can be written as _ MðqÞ€r Cðq, qÞ_ _r MðqÞ_s ¼ Cðq, qÞs _ GðqÞ Ff ðq, qÞ Fd þ ud ðtÞ þ d~mr p þ d~ml ð1 pÞ _ k^1 r€ k^2 r_ k^3 k^4 F^f k~1 r€ ¼ Cðq, qÞs k~2 r_ k~3 k~4 h 2 i F~c 1 exp vv1 sgnðvÞ s 1 2 F~s exp ðvvs Þ sgnðvÞ F~v v þ ud ðtÞ þ d~mr p þ d~ml ð1 pÞ ð21Þ _ where k1 ¼ MðqÞ, k2 ¼ Cðq, qÞ, k3 ¼ GðqÞ, and k4 ¼ Fd ðtÞ, kî are estimates of ki , and k~i ¼ ki kî . As the dynamic parameters ki cannot be known a priori, adaptive laws are considered to estimate the unknown parameters. The Lyapunov function candidate is defined as follows 4 v X 1 1 ~T ~ X 1 ~T ~ ki ki þ V2 ¼ sT MðqÞs þ Fj Fj 2 2 2 i j j¼c i¼1 l X 1 ~T ~ d dk þ 2k k k¼r

ð22Þ

where i 4 0, i ¼ 1, . . . , 4, and j 4 0, j ¼ c, s, v, are constants. Differentiating equation (22) with respect to time, we obtain

h 1 _ T _ _ MðqÞ s þ sT k^1 r€ k^2 r_ k3 V2 ¼ s Cðq, qÞ 2 k^4 F^f k~1 r€ k~2 r_ k~3 k~4


Jung et al.

5 where K^ ¼ ½k^1 k^2 k^3 k^4 T and ¼ ½ k€r k k r_ k 1 1T . 4 0 and 4 0 are constant diagonal matrices, 0 5 5 1 is constant, and 0i 4 0 are constants. Substituting equations (24) to (30) into equation (23), we obtain

h 2 i F~c 1 exp vv1 sgnðvÞ s 2 sgnðvÞ F~v v þ ud ðtÞ F~s exp vv1 s 4 i X 1 ~T _~ þd~mr p þ d~ml ð1 pÞ þ ki ki 2 i¼1 i

þ

V_ 2 4 sT s jsjþ1

v X 1 j¼c

l X 1 ~T _~ _ dk dk F~Tj F~j þ j k¼r k

h i ¼ sT k^1 r€ k^2 r_ k3 k^4 F^f þ ud ðtÞ h i þ sT k~1 r€ k~2 r_ k~3 k~4 h 2 i sT F~c 1 exp vv1 sgnðvÞ s 2 sgnðvÞ sT F~s exp vv1 s h i sT F~v v þ sT d~r p þ d~l ð1 pÞ 4 X 1

þ

þ

4 v 0 FT F X 0ki k2i X j j j þ 2 2 j¼c i¼1

l X 0 dT dk k k

2

4 lmin ðÞ js j2

The control input and adaptive law are chosen as ud ¼ s K^ T þ F^f jsj signðsÞ

ð24Þ

_ kî ¼ i js ji þ 0i kî , k ¼ 1, . . . , 4

ð25Þ

h i 2 _ sgnðvÞÞ þ 0 F^c F^c ¼ c js jð1 exp vv1 s c ð26Þ

_ d^l ¼ l js jð1 pÞ 0l d^l

2

k¼r

l X 1 ~T _^ _ _ k~Ti kî dk dk F~Tj F^j 2 j¼c j i¼1 i k¼r k

h i 1 _ 4sT K^ T þ F^f þ ud k~1 ks k€r k^1 1

1 _ 1 _ k~2 ks k_r k^2 k~3 ks k k^3 2 3

1 _ k~4 ks k k^4 4

2 sgnðvÞÞ þ 1 F_^c F~c ks kð1 exp vv1 s c

1 2 sgnðvÞ þ F_^s F~s ks k exp vv1 s s

1 1 _ _ F~v ks kv þ F^v þ d~Tr ks k p d^r v r

1 _ þ d~Tl ks kð1 pÞ d^l ð23Þ l

_ F^v ¼ v ð js jv þ 0v F^v Þ _ d^r ¼ r js j p 0r d^r

k k

k¼r

v X 1

h i 2 _ Þ sgnðvÞ þ 0 F^s F^s ¼ s js j exp ðvv1 s s

l X 0 d~T d~k

l X 0 d~T d~k k k

k¼r

where and

4 v 0 F~T F~ X 0ki k~2i X j j j 2 2 j¼c i¼1

2

4 v 0 F~ T F~ X 0ki k~2i X j j j 2 2 j¼c i¼1

þ 2 4 &2 V2 þ 2

ð31Þ

&2 ¼ min½lmin ðÞ, 1=2ð0ki , 0c , 0s , 0v , 0r , 0l Þ 4 0 P Pv 0 T Pl 4 0 2 0 2 ¼ 12 i¼1 ki ki þ j¼c j Fj Fj þ k¼r k

dTk dk Þ. Multiplying (31) by e&2 t yields d V2 e&2 t 42 e&2 t dt

ð32Þ

Integrating (32) over ½0, t leads to 0 4 V2 4 ½V2 ð0Þ ð1=&2 Þ2 e&2 t þ ð1=&2 Þ2 . Therefore, this means that all the error signals are semi-globally uniformly ultimately bounded. Remark 1. In the designed controller in equation (24), we use the finite time-based control term20 instead of the relay control term, i.e. sign function given as signðsÞ, which gives chattering in the control action, to alleviate chattering and guarantee finitetime convergence of the filtered error. ^ Remark 2. If the estimation is precise, i.e. k ¼ k, ^ ~ ~ Fi ¼ Fî , and dj ¼ dj , then k ¼ 0, F~i ¼ 0, and dj ¼ 0. Equation (31) can be rewritten as V_ 2 4 sT s jsjþ1 ¼ 20 V2 2V2

0

ð33Þ

0 where 0 ¼ k1 1 and ¼ ð þ 1Þ=2. Therefore, from 20 Slotine and Li, the settling time ts can be given as 0

ð27Þ ts 4

0 V1 ð0Þ þ 1 2 ln 20 ð1 0 Þ

ð34Þ

ð28Þ ð29Þ

Robust adaptive control ð30Þ

The adaptive laws given in the previous section cannot estimate all the unknown parameters precisely


6

Proc IMechE Part C: J Mechanical Engineering Science 0(0) where the uncertainty is defined as _ r þ Ff ðq, qÞ _ þ Fd ðtÞ, ðÞ Fu ðtÞ ¼ MðqÞ€r þ Cðq, qÞ_ are the expected adaptation perturbations of each dynamic term, jFu j4 , is unknown upper bound of Fu . Define the following Lyapunov function candidate as 4 v X 1 1 ~T ~ X 1 ~T ~ ki ki þ F Fj V3 ¼ sT MðqÞs þ 2 2i 2j j j¼c i¼1

Figure 3. Basic concept of the Funnel control.

though robust control is inserted in equation (24). If the estimation errors are large, the control gain is high and larger control inputs appear in the actuator. Therefore, to provide robustness to the incomplete estimation and improve tracking control performance, the following disturbance compensator is included in the controller 1 ^ F ðtÞ jsðtÞ j uu ðtÞ ¼ s

ð35Þ

where ^ and F ðtÞ are defined later. This concept is borrowed from the funnel control16,17 as shown in Figure 3. If the boundary F ðtÞ satisfies the funnel condition in equation (36) with sð0Þ 5 F ð0Þ n o F : t ! s 2 Rn F1 js j 5 1

ð36Þ

The scale factor ^ of equation (35) is adjusted in order to ensure that the filtered error surface sðtÞ evolves inside the prescribed boundary function F ðtÞ. Thus, as the filtered error sðtÞ approaches the boundary F ðtÞ, the control action of uu ðtÞ increases, and as the error sðtÞ becomes small, the control action of uu ðtÞ decreases conversely. A proper boundary to constrain the uncertainty is selected by F ðtÞ ¼ ð"0 "1 ÞexpðatÞ þ "1

ð37Þ

where "0 ¼ F ð0Þ5"1 4 0, "1 ¼ lim inf F ðtÞ, and t!1 a 4 0 is the decay rate of the boundary function. It is known that "0 4 sð0Þ regulates the transient time response, and "1 affects the steady-state time response. Therefore, we adopt the following modified control law that guarantees robust control if the initial uncertainties are located within the boundaries of the prescribed constraint function. Considering the modeling errors and disturbance, equation (21) can be written as _ MðqÞ€r Cðq, qÞ_ _r MðqÞ_s ¼ Cðq, qÞs _ GðqÞ Ff ðq, qÞ Fu ðtÞ þ ud ðtÞ þ d~r p þ d~l ð1 pÞ

ð38Þ

l X 1 ~T ~ 1 T dk dk þ þ ~ ~ 2 2 k k¼r

ð39Þ

^ ^ is estimate of . The time where ~ ¼ , derivative of equation (39) is given similarly as follows

h i 1 _ V_ 3 4sT K^ T þ F^f þ ud k~1 js j€r k^1 1

1 1 _^ _^ ~ ~ K4 k2 js j_r k2 k3 js j k3 2 3

1 _ k~4 js j k^4 4

1 2 1 _^ ~ Fc js j 1 expð vvs sgnðvÞÞ þ Fc c

1 2 1 _ ~ ^ Fs js j exp ð vvs Þ sgnðvÞ þ Fs s

1 _^ 1 _^ T ~ ~ Fv js jv þ Fv þ dr js j p dr v r

1 _^ 1 _ T T ~ þ dl js jð1 pÞ dl þ ~ js j ^ l ð40Þ The controller and adaptive law with consideration of the adaptive laws previously selected in equations (25) to (30) is selected as ud ¼ s K^ T þ F^f jsj signðsÞ uu ðtÞ

ð41Þ

_^ ¼ js j2 ðF js jÞ1 0 ^

ð42Þ

Equation (40) is then written as V_ 3 4 sT s jsjþ1

l X 0 d~T d~k k k

k¼r

2


0 ~ T ~ 2

þ js j js j2 ðF js jÞ1 þ

4 v 0 FT F l X X 0ki k2i X 0k dTk dk 0 T j j j þ þ þ 2 2 2 2 j¼c i¼1 k¼r


ð43Þ

Jung et al.

7

Figure 4. XY table system. (a) XY table and controller. (b) Schematic description of the ball-screw.

Table 1. XY table specification.

4 lmin ð 2 Þ js j2

Component

Specification

Servo Motor Servo Amplifier Lead of ball-screw Resolution of linear encoder Motor power

HP-KP13, HP-KP23 MR-J3-10A, MR-J3-20A 10 mm 0.1 mm resolution

l X 0 d~T d~k k k

k¼r

200 W (X-axis), 100 W(Y-axis)

2


0 ~ T ~ þ 3 4 &3 V3 þ 3 ð46Þ 2

where &3 ¼ min½lmin ð 2 Þ, 1=2ð0ki , 0c , 0s , 0v , 0r , 0l Þ Pl P4 0 2 Pv 0 T 0 4 0 and 3 ¼ 12 i¼1 ki ki þ j¼c j Fj Fj þ k¼r k 1 dTk dk þ 0 T Þþ F þ 2

j j2 . Therefore, all the

error signals are semi-globally uniformly ultimately

The following inequalities are easily proved 1 js j2 F js j 4 ðF þ js jÞ js j 4 js j2 þ

1 j j2 4

bounded for 4 2 and s ! 0, e ! 0, and e_ ! 0, ð44Þ

as t ! 1 by Barbalat’s Lemma.21

ð45Þ

Remark 3. In fact, the robustness to uncertainty obtained from the controllers and the adaptive law in equations (35) and (42) is conditionally guaranteed provided that the prescribed uncertainty boundary function, F ðtÞ, satisfies sð0Þ 5 F ð0Þ or "0 4 sð0Þ. Next, the uncertainty suppression of the control input of uu ðtÞ depends on the value of the scale ^ factor ðtÞ. The proper estimation for uncertainty and selection of F ðtÞ improve the robustness of the proposed controller.

for a constant 4 0. Using the relationships (44) and (45), equation (43) can be rewritten as V_ 3 4 ð 2 Þ js j2 jsjþ1

l X 0 d~T d~k k k

k¼r

þ

2

l X 0 dT dk k k

k¼r

2

4 v 0 F~T F~ X 0ki k~2i X j j j 2 2 j¼c i¼1

4 v 0 FT F 0 ~ T ~ X 0ki k2i X j j j þ þ 2 2 2 j¼c i¼1

0 T 1 þ F þ j j2 þ 2 2

Remark 4. If the condition of sð0Þ 5 F ð0Þ is violated, the proposed uncertainty compensation property disappears and the controller becomes the


8


Figure 5. Experimental results for the XY table system. (a) Estimated dynamic parameter in the X axis. (b) Estimated dynamcis paramters in the Y axis. (c) Estimated friction parameters in the X axis. (d) Estimated friction paramters in the Y axis. (e) ^ x and ^ y . (f) Control inputs of the robust adaptive control system in the XY axis. (g) Tracking errors of each control system in the X axis. (h) Tracking errors of each control system in the Y axis. (i) Tracking errors of the adaptive and robust adaptive control systems in the X axis. (j) Tracking errors of the adaptive and robust adaptive control systems in the Y axis.

conventional terminal sliding mode controller. In this case, the trade-off gain tuning between chattering and robustness is considered to obtain a desirable performance. However, the controller term of jsj signðsÞ in equation (42) does not induce larger chattering compared to the term of signðsÞ adopted

frequently in the conventional first-order SMC scheme. Remark 5. The general selection rule for the constraint boundary function F ðtÞ does not exist, but this depends on the designer’s experience and trial


Jung et al.

9

Figure 5. Continued.


10


and error method according to variations of control system.

Experimental evaluations

0x,4 ¼ 0:002, 0y,1 ¼ 0:005, 0y,2 ¼ 0:0025, 0y,3 ¼ 0:0025, and 0y,4 ¼ 0:001. The performance functions were selected as follows "x ðtÞ ¼ ð0:2 0:001Þe5t þ 0:001ðmmÞ,

In this section, the experimental applications for a XY table and an articulated manipulator are presented to verify the control performance of the proposed control strategy. Several controllers were designed: adaptive controller without deadzone and/or friction compensation and with deadzone and friction compensation, and robust adaptive controller with deadzone and friction compensation.

Experiment for the XY table system The XY table system is shown in Figure 4, where an AC servo motor, an inverter and an universal motor driver produced by Mitsubishi company are equipped in the actuator system. The linear encoders attached in the side of linear motion guides of each axis measure the information regarding the position. The ballscrew transfer mechanism produced by Samick company is described in Figure 2. The complete specification of the XY table system is listed in Table 1. The MF 624 DSP control board combined with Matlab realtime toolbox software is used to transform input/ output signals, and implement the design control algorithm The parameters of the controller and adaptive laws are selected as follows: x ¼ 10, y ¼ 10, x ¼ 10, y ¼ 5, i ¼ 0:5, i ¼ x, y, x,1 ¼ 1:5, x,2 ¼ 1, x,3 ¼ 1:2, x,4 ¼ 1:2, y,1 ¼ 1:5, y,2 ¼ 1, y,3 ¼ 1:5, y,4 ¼ 1:5, 0x,1 ¼ 0:005, 0x,2 ¼ 0:001, 0x,3 ¼ 0:002, Table 2. RMS tracking error of the adaptive and robust adaptive control system in the XY table. Axis

Adaptive

Robust Adaptive

X Y

0.009 mm (100%) 0.027 mm (100%)

0.007 mm (78%) 0.010 mm (37%)

"y ðtÞ ¼ ð0:2 0:002Þe5t þ 0:002ðmmÞ

ð47Þ

The command inputs are selected as qdx ðtÞ ¼ 2 sin 0:2t ðmmÞ and qdy ðtÞ ¼ 2 sin 0:25t ðmmÞ. The estimated dynamic parameters and friction parameters are shown in Figure 5(a) to (d); Figure 5(e) presents the estimated uncertainty. The tracking errors without or with deadzone and friction compensation are presented in Figure 5(g) and (h), where the effect of deadzone compensation is greater than friction compensation effect. The tracking errors for robust control are shown in Figure 5(i) and (j), where the root mean square (RMS) error of the proposed control decreased to maximum 37% of that of the adaptive control system, as shown in Table 2. Therefore, the proposed uncertainty compensator suppresses the uncertainty effectively. Figure 6(f) shows the control inputs of the robust adaptive control system in each axis.

Experiment for the articulated manipulator system As the second application, the experiment for the articulated robot manipulator presented in Figure 6 was carried out to prove the efficacy of the proposed control strategy. In this application, two controllers were designed: the adaptive controller and robust adaptive controller. The designed controllers were implemented through the Matlab realtime toolbox with MF624 control board equipped with D/A and A/D converter. The angles of each link are measured by the rotary encoder attached to each DC servo motor. The parameters of the controller and adaptive laws are selected as follows: 1 ¼ 120, 2 ¼ 90, 1 ¼ 60, 2 ¼ 60, i ¼ 0:5, i ¼ 1, 2, 1,1 ¼ 5, 1,2 ¼ 2:5,

Figure 6. Photograph and diagram of the articulated manipulator control system.


Jung et al.

11

Figure 7. Experimental results for the robotic manipulator system. (a) Estimated dynamic parameter in the link 1. (b) Estimated dynamcis paramters in the link 2. (c) Estimated friction parameters in the link 1. (d) Estimated friction paramters in the link 2. (e) Estimated deadzone parameters in the link 1. (f) Estimated deadzone parameters in the link 2. (g) ^ 1 and ^ 2 . (h) Tracking output of the robust control system in the link 1. (i) Tracking output of the robust control system in the link 2. (j) Control inputs of the robust adaptive control system in the link 1 and 2. (k) Tracking errors of the adaptive and robust adaptive control systems in the link 1. (l) Tracking errors of the adaptive and robust adaptive control systems in the link 2.

2,3 ¼ 2:2, 1,4 ¼ 1:6, 2,1 ¼ 2, 2,2 ¼ 2, 2,3 ¼ 5, 2,4 ¼ 4, 01,1 ¼ 0:01, 01,2 ¼ 0:02, 01,3 ¼ 0:03, 0 0 0 1,4 ¼ 0:035, 2,1 ¼ 0:01, 2,2 ¼ 0:02, 02,3 ¼ 0:025, and 02,4 ¼ 0:01. The performance functions were selected as follows "1 ðtÞ ¼ ð0:1 0:005Þet þ 0:005 ðrad Þ, "2 ðtÞ ¼ ð0:1 0:0025Þet þ 0:0025 ðrad Þ

ð48Þ

The command inputs are selected as qd1 ðtÞ ¼ 0:1 sin 0:4t ðrad Þ and qd2 ðtÞ ¼ 0:15 sin 0:4t ðrad Þ. In Figure 7(a) to (g), the estimated results for the dynamic parameters, friction, deadzone, and uncertainty are shown. The tracking outputs of the

proposed control are presented in Figure 7(h) and (i) and the tracking errors of the adaptive and robust adaptive control systems are shown in Figure 7(j) and (k), and listed in Table 3. The error size in the proposed control system decreased to 38% of that of the adaptive control system. Therefore, it is proved that the proposed control scheme has more advanced control performance than the adaptive control system.

Conclusions In this paper, the adaptive control scheme with funnel robust compensation was developed to provide a high enhanced position tracking performance


12




Jung et al.

13


Link1

Adaptive

Robust Adaptive

position tracking performance of the proposed control scheme was validated from experiments by its effective compensation for the deadzone, friction, and uncertainties.

1 2

0.008 rad (100%) 0.014 rad (100%)

0.003 rad (38%) 0.007 rad (50%)

Declaration of Conflicting Interests

Table 3. RMS tracking error of the adaptive and robust adaptive control system in the manipulator.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

of the nonlinear servo dynamic system in the presence of both deadzone and friction. First, an adaptive controller was designed using the adaptive laws for estimation of the dynamic parameters of the servo system, Tustin friction, and deadzone parameters. Next, a funnel compensator was added to enforce the robustness of the adaptive controller. From the Lyapunov stability theorem, adaptive laws for the controller, friction, and deadzone observes were derived. As design examples, the XY table and the robotic manipulator in the presence of friction and deadzone were chosen. The favorable

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by a 2-Year Research Grant of Pusan National University.

References 1. Kelly R and Salgado R. PD control with computed feedback forward of robot manipulators: a design procedure. IEEE Trans Robot and Autom 1992; 10: 566–571.


14


2. Spong MW, Hutchison S and Vdyasagar M. Robot modeling and control. Vidyasagar, USA: John Wiley and Sons, 2006. 3. Udwadia FE and Schutte AD. Equations of motion for general constrained systems in Lagrangian mechanics. Acta Mech 2010; 213: 111–129. 4. Udwadia FE. A new approach to stable optimal control of complex nonlinear dynamical systems. J Appl Mech ASME 2013; 81: 031001–031006. 5. Armstrong-Helouvry B, Dupont P and Canudas de Wit C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 1994; 30: 1083–1138. 6. Canudas de Wit C, Olsson H and Astrom KJ. A new model for control of systems with friction. IEEE Trans Automat Ctrl 1995; 40: 419–425. 7. Al-Bender F, Lampaert V and Swevers J. The generalized Maxwell-slip model: a novel model for friction simulation and compensation. IEEE Trans Automat Ctrl 2005; 50: 1883–1887. 8. Tao G and Kokotovic PV. Adaptive control of plants with unknown dead-zones. IEEE Trans A C 1994; 39: 59–68. 9. Lewis FL, Tim WK, Wang LZ, et al. Deadzone compensation in motion control systems using adaptive fuzzy logic control. IEEE Trans Contr Sys Tech 1999; 7: 731–742. 10. Hu C, Yao B and Wang Q. Adaptive robust precision motion control of systems with unknown input deadzones: a case study with comparative experiments. IEEE Trans Indust Electr 2011; 58: 2454–2464. 11. Wang Z, Ge SS and Lee TH. Robust motion/force control of uncertain holonomic/nonholonomic mechanical

12.

13.

14.

15. 16.

17. 18. 19.

20.

21.

systems. IEEE/ASME Trans Mechatron 2004; 9: 118–123. Li Z, Ge SS and Ming A. Adaptive robust motion/force control of holonomic-constrained nonholonomic mobile manipulators. IEEE Trans Syst Man CyberPart B: Cyber 2007; 37: 607–616. Mohanty A and Yao B. Integrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband. IEEE/ASME Trans Mechatr 2011; 16: 707–715. Dasmahapatra S, Sarkar BK, Saha R, et al. Design of an adaptive fuzzy-bias SMC and validation for a rugged electrohydraulic system. IEEE/ASME Trans Mechatr 2015. DOI 10.1109/TMECH.2015.2393437. Spurgeon EC. Sliding mode control: theory and applications. London: Taylor & Francis, 1998. Ilchman AE, Ryan P and Trenn S. PI-funnel control for two mass systems. IEEE Trans A C 2009; 54(4): 981–923. Hackl CM. High-gain position control. Int J Ctrl 2011; 84: 1695–1716. Humusoft Comp. MF 624 Multifunction I/O card manual. Czech: Czech Republic, 2006. Ge SS, Lee TH and Harris CJ. Adaptive neural network control of robot manipulators. River Edge, NJ: World Scientific, 1998. Yu S, Yu X, Shirinzadeh B, et al. Continuous finitetime control for robotic manipulators with terminal sliding mode. Automatica 2005; 41: 1957–1964. Slotine JE and Li W. Applied nonlinear control, Englewood Cliffs, New Jersey: Prentice-Hall, International, 1991.
