Int. J. Systems, Control and Communications, Vol. x, No. x, xxxx 1

Adaptive control of a parallel robot via

backstepping technique

Li Wang

Department of Electrical Engineering, AQ1Lakehead UniversityE-mail: lwang2@lakeheadu.ca

Zongtao Lu

Department of Electrial Engineering and Computer Science,Case Western Reserve UniversityE-mail: zongtao.lu@case.edu

Xiaoping Liu

Department of Electrical Engineering,Lakehead UniversityE-mail: xliu2@lakeheadu.ca

Kefu Liu

Department of Mechanical Engineering,Lakehead UniversityE-mail: kliu@lakeheadu.ca

Dan Zhang

Faculty of Engineering and Applied Science,University of Ontario Institute of TechnologyE-mail: dan.zhang@uoit.ca

Abstract: The adaptive backstepping technique is applied to set point controlof a planar parallel robot. The dynamic model of the robot is characterisedby a set of differential algebraic equations. The model is reduced into aset of Ordinary Differential Equations (ODEs). A model-based adaptivebackstepping controller is designed. The proposed controller is tested byexperiments. Comparison among the adaptive backstepping controller,backstepping controller, adaptive PD controller and PD controller is madebased on experimental results. Experimental results show that the adaptivebackstepping controller outperforms all the other controllers in terms ofsteady-state errors.

Keywords: parallel robot; adaptive backstepping; nonlinear control;DAE; differential algebraic equation; systems.

Reference to this paper should be made as follows: Wang, L., Lu, Z., Liu, X.,Liu, K. and Zhang, D. (xxxx) ‘Adaptive control of a parallel robot viabackstepping technique’, Int. J. Systems, Control and Communications,Vol. x, No. x, pp.xxx–xxx.

Copyright © 2008 Inderscience Enterprises Ltd.

2 L. Wang et al.

Biographical notes: AUTHORPLEASESUPPLYCAREERHISTORYOFNOMORE THAN 100 WORDS FOR EACH AUTHOR.

1 Introduction

Recently, parallel robots have been receiving growing attentions from both academiaand industries due to their advantages over serial robots, such as high stiffness, highspeeds, low inertia and large payload capacity (Stewart, 1965; Hunt, 1978; Fichter,1986; Gosselin, 1988; Boer et al., 1999; Zhang, 2000). In general, the dynamic modelsof parallel robots are much more complex than those of serial robots. Serial robotsare described by a set of Ordinary Diffential Equations (ODEs). Unlike serial robots,parallel robots are oftenmodelled by a set ofDifferential Algebraic Equations (DAEs),so dynamic equations for parallel robots are not in an explicit form of the independentgeneralised coordinates or actuated joints. Calculation of these implicit relations inreal time imposes a severe constraint on application of many well-established controlmethods for serial robots to parallel robots. Therefore, some early attempt in controlof parallel robots focused on the use of non-model based control methods, suchas Proportional Integral Derivative (PID) control (Amiral et al., 1996; Kokkinisand Stoughton, 1991) and artificial intelligence-based algorithms (Begon et al., 1995;Geng and Haynes, 1993). However, as pointed out in Ghorbel et al. (2000), these

Adaptive control of a parallel robot via backstepping technique 3

methods have no guarantee of stability and performance. Some efforts have beenmade to extend model-based control algorithms for serial robots to parallel robots.The study reported in Gosselin (1996) proposed a parallel computational algorithmto speed up on-line computation. In Codourey (1998), the mass and inertia of thelinks were neglected in the dynamic model in order to implement the computed-torquecontrol. A PD plus simple gravity compensation control law was proposed in Ghorbelet al. (2000) for set point control for a planar 2-DOF parallel robot. With the provedskew symmetry property, this controller guarantees a local asymptotic stability. Thesimple gravity compensation is a constant term which can be computed offline to anydegree of accuracy. InLi andWu (2004), the design for control approachwas employedin the design stage of a parallel robot to find an appropriate mechanical structurewith a simple dynamic model. This way, a simple control algorithm is sufficient toachieve a satisfactory control performance. In Vivas and Poignet (2005), a predictivefunctional control strategy is implemented for tracking control of a H4 parallel robot.The dynamic model is simplified by neglecting the effect of arm mass, which greatlyfacilitates the implementation of the controller.

Backstepping is a recursive design method for constructing both feedback controllaws and associated Lyapunov functions in a systematic manner. During the lastten years, backstepping-based designs have emerged as powerful tools for stabilisingnonlinear systems (Kanellakopoulos et al., 1991; Kristic et al., 1995; Marino andTomei, 1995). Adaptive backstepping is able to handle nonlinear systems withunknown parameters (Kristic et al., 1995) whereas robust backstepping design is usedto control nonlinear systems with uncertainties (Freeman and Kokotovic, 1996). Thebacksteppingdesignmethodhasbeenapplied to control of serial robots (Lotfazar et al.,2003; Su and Stepaneko, 1997) and trajectory planning for mobile robots (Ma et al.,2005). To the authors’ best knowledge, there has been no report on application of thebackstepping technique to control of parallel robots. In general, the model governinga parallel robot is highly nonlinear and a precise knowledge of its parameters isnot readily available. Adaptive backstepping appears to be a suitable control designmethodology for parallel robots.

In this paper, a backstepping-based control scheme is applied to set point controlfor a planar 2-DOF parallel robot. By assuming that inertia parameters and somegeometric dimensions of the robot are not known precisely, an adaptive backsteppingcontroller is designed. For the purpose of comparison, an adaptive Proportional andDerivative (PD) controller is designed as well. The performance of each controller istested by experiments.

The rest of the paper is organised as follows. In Section 2, the dynamic modelof the planar 2-DOF parallel robot under study is presented. In Section 3, anadaptive backstepping-based controller and an adaptive PD controller are designed,respectively. In Section 4, the experimental results with the controllers are presented.Finally the study conclusions are drawn in Section 5.

2 Dynamic model

A schematic of a planar 2-DOF parallel robot is shown in Figure 1 where mi, ai

and li are the mass, length and distance to the centre of mass from the lower jointof link i, respectively, Ii denotes the mass moment of inertia of link i. Joints q1 and

4 L. Wang et al.

q2 are actuated while joints q3 and q4 are passive. The dynamical model of the robot,presented in Ghorbel et al. (2000), is described as follows:

D′(q′)q′ + C ′(q′, q′)q′ + g′(q′) = u′ (1)

φ(q′) = 0 (2)

where q′ = [q1, q2, q3, q4]T is the vector of dependent generalised coordinates,u′ = [u1, u2, 0, 0]T with u1 and u2 are the torque applied on joints q1 and q2,respectively, D′(q′) ∈ R4×4 is the inertia matrix, C ′(q′, q′)q′ ∈ R4 represents thecentrifugal andCoriolis termsand g′(q′) ∈ R4 is the gravity vector, andφ(q′) representsthe constraints defined by

φ(q′) =[φ1(q′)φ2(q′)

]=

a1 cos(q1) + a3 cos(q1 + q3) − c − a2 cos(q2)

−a4 cos(q2 + q4)a1 sin(q1)+a3 sin(q1 +q3) − a2 sin(q2)− a4 sin(q2 + q4)

. (3)

Figure 1 The schematic of a 2DOF parallel robot

Assume that the parameters mi, li and Ii are unknown and defineΘ = [θ1, θ2, . . . , θ10]T with θ1 = m1l

21 + m3a

21 + I1, θ2 = m2l

22 + m4a

22 + I2,

θ3 = m3l23 + I3, θ4 = m4l

24 + I4, θ5 = m3a1l3, θ6 = m4a2l4, θ7 = 9.81(m1l1 + m3a1),

θ8 = 9.81(m2l2 + m4a2), θ9 = 9.81m3l3 and θ10 = 9.81m4l4 as unknown parameters.Then,D

′(q′),C

′(q′, q′) and g′(q′) canbe expressed in terms of the unknownparameters

as follows:

D′(q′) =

θ1 +θ3 +2θ5 cos(q3) 0 θ3 +θ5 cos(q3) 00 θ2 +θ4 +2θ6 cos(q4) 0 θ4 +θ6 cos(q3)θ3 + θ5 cos(q3) 0 θ3 00 θ4 + θ6 cos(q3) 0 θ4

C′(q′, q′) =

−θ5 sin(q3)q3 0 −θ5 sin(q3)(q1 + q3) 00 −θ6 sin(q4)q4 0 −θ6 sin(q4)(q2 + q4)θ5 sin(q3)q1 0 0 00 θ6 sin(q4)q2 0 0

Adaptive control of a parallel robot via backstepping technique 5

g′(q′) =

θ7 cos(q1) + θ9 cos(q1 + q3)θ8 cos(q2) + θ10 cos(q2 + q4)θ9 cos(q1 + q3)θ9 cos(q2 + q4)

.

Equations (1) and (2) are a set of DAEs in the dependent generalised coordinates q′.The independent generalised coordinates q or active joints are related to q′ by:

q =[q1

q2

]=

[1 0 0 00 1 0 0

]q′. (4)

In order to obtain a formulation that is suitable for model-based control, a reducedmodel in the independent generalised coordinates, which is derived in Ghorbel et al.(2000), is given below:

D(q′)q + C(q′, q′)q + g(q′) = u (5)

q′ = ρ(q′)q (6)

q′ = σ(q) (7)

where D(q′), C(q′, q′), g(q′), ρ(q′) and σ(q) can be found from Ghorbel et al. (2000).For adaptive controller design, the elements Djk, Cjk and gj with j, k = 1, 2

in D(q′), C(q′, q′) and g(q′) can be expressed as DojkΘ, CojkΘ and gojΘ where

Dojk = [Dojk1 Dojk2 · · · Dojk10] (8)

Cojk = [Cojk1 Cojk2 · · · Cojk10] (9)

goj = [goj1 goj2 · · · goj10] (10)

with Do111 = 1, Do113 = (1 + ρ31)2, Do114 = ρ241, Do115 = 2(1 + ρ31) cos(q3),

Do123 = Do213 = (1 + ρ31)ρ32, Do124 = Do214 = (1 + ρ42)ρ41, Do125 = Do215 =ρ32 cos(q3), Do126 = Do216 = ρ41 cos(q4), Do222 = 1, Do223 = ρ2

32, Do224 = (1+ρ42)2,Do226 = 2(1 + ρ42) cos(q4), Co113 = (1 + ρ31)ρ31, Co114 = ρ41ρ41, Co115 =ρ31 cos(q3) − (1 + ρ31)q3 sin(q3), Co123 = (1 + ρ31)ρ32, Co124 = ρ41ρ42, Co125 =ρ32 cos(q3) − (q1 + q3)ρ32 sin(q3), Co126 = ρ41q2 sin(q4), Co213 = ρ32ρ31, Co214 =(1 + ρ42)ρ41, Co215 = ρ32q1 sin(q3), Co216 = ρ41 cos(q4) − (q2 + q4)ρ41 sin(q4),Co223 = ρ32ρ32, Co224 = (1 + ρ42)ρ42, Co226 = ρ42 cos(q4) − (1 + ρ42)q4 sin(q4),go17 = cos(q1), go19 = (1 + ρ31) cos(q1 + q3), go110 = ρ41 cos(q2 + q4), go28 = cos(q2),go29 = ρ32 cos(q1 + q3), go210 = (1 + ρ42) cos(q2 + q4) and all the other elements arezero, where ρij and ρij are the ij-element of ρ and its time derivative (Ghorbel et al.,2000).

3 Controller design

In the following, two adaptive controllers: adaptive backstepping controller andadaptive PD controller, will be designed to achieve the set-point control. Eachcontroller consists of a control law and an update law for the parameter estimation.For brevity, we denote D(q′) as D, C(q′, q′) as C and g(q′) as g.

6 L. Wang et al.

3.1 Adaptive controller based on backstepping

In order to formulate equation (5) into a form suitable for set point control usingthe adaptive backstepping technique, assign x1 = q1 − qd

1 , x2 = q2 − qd2 , x3 = q1,

x4 = q2 with qd1 and qd

2 being the desired angles for q1 and q2, respectively. Let Θ bethe estimation of Θ. A lower triangular form is obtained as:

x1 = x3 (11)

x2 = x4 (12)

D

[x3

x4

]= u − Cq − g. (13)

Following the backstepping design procedure, first, choose the Lyapunov functioncandidate:

V1 =12x2

1 +12x2

2. (14)

By introducing virtual controllers: α1 = −c1x1, α2 = −c2x2, where c1 and c2 arepositive numbers, V1 can be rewritten into:

V1 = −c1x21 − c2x

22 + x1(x3 − α1) + x2(x4 − α2). (15)

Now, choose the Lyapunov function candidate:

V2 = V1 +12

[x3 − α1

x4 − α2

]T

D

[x3 − α1

x4 − α2

]+

12(Θ − Θ)T Γ(Θ − Θ) (16)

where Γ = diag[γ1, γ2, . . . , γ10] is a positive definite matrix. Note that D is positivedefinite. Differentiating V2 with respect to time yields:

V2 = −c1x21 − c2x

22 + x1(x3 − α1) + x2(x4 − α2) +

[x3 − α1

x4 − α2

]T

D

[x3 − α1

x4 − α2

]

+12

[x3 − α1

x4 − α2

]T

D

[x3 − α1

x4 − α2

]− ˙Θ

T

Γ(Θ − Θ). (17)

As pointed out in Ghorbel et al. (2000), the matrix D − 2C is skew symmetric.As a result, we can get

12

[x3 − α1

x4 − α2

]T

(D − 2C)[x3 − α1

x4 − α2

]= 0. (18)

With the parameterisations in equations (8)–(10), substituting equations (13) and (18)into (17) yields:

V2 = −c1x21 − c2x

22 +

[x3 − α1

x4 − α2

]T (u +

[x1

x2

]+ Λ

)− ˙Θ

T

Γ(Θ − Θ) (19)

Adaptive control of a parallel robot via backstepping technique 7

where:

Λ = ΛoΘ = −[α1Do11 + α2Do12 + α1Co11 + α2Co12 + go1

α1Do21 + α2Do22 + α1Co21 + α2Co22 + go2

]Θ. (20)

Apparently, if the controller is chosen to be:

u =[u1

u2

]= −

[c3(x3 − α1) + x1

c4(x4 − α2) + x2

]− ΛoΘ (21)

and the unknown parameters’ updating law is chosen to be:

˙Θ = Γ−1ΛTo

[x3 − α1

x4 − α2

](22)

where c3 and c4 are positive numbers, the derivative of V2 is negative, i.e.,

V2 = −c1x21 − c2x

22 − c3(x3 − α1)2 − c4(x4 − α2)2 (23)

which means that the corresponding closed-loop system is stable.If Θ is known, the control effort of a corresponding non-adaptive backstepping

controller is given as:

u =[u1u2

]= −

[c3(x3 − α1) + x1

c4(x4 − α2) + x2

]− Λ0Θ. (24)

3.2 Adaptive PD controller

It is worthwhile comparing the controller of equation (21) with an adaptive PDcontroller. Choose the Lyapunov function candidate:

V =12(q − qd)T Kp(q − qd) +

12qT Dq +

12(Θ − Θ)T Γpd(Θ − Θ) (25)

where Γpd = diag[γpd1, γpd2, . . . , γpd8] andKp = diag[kp1, kp2]with γpdi > 0 for 1 ≤i ≤ 8 and kpi > 0 for i = 1, 2. Differentiating V with respect to time yields:

V = (q − qd)T Kpq +12qT Dq + qT Dq − ˙Θ

T

Γpd(Θ − Θ). (26)

According to Ghorbel et al. (2000) D − 2C is skew symmetric and D is symmetric.So we can get:

D = C + CT . (27)

With the parameterisations in equations (8)–(10), substituting equations (13) and (27)into (26) gives

V = (q − qd)T Kpq + qT [u + Λpd] − ˙ΘT

Γpd(Θ − Θ) (28)

8 L. Wang et al.

where

Λpd = ΛpdoΘ =12

[q2(Co21 − Co12) − go1

q1(Co12 − Co21) − go2

]Θ.

Apparently, if the controller is set to be:

u =[u1u2

]= −Kp(q − qd) − Kv q − ΛpdoΘ (29)

and the unknown parameters’ updating law is selected as

˙Θ = Γ−1pd ΛT

pdoq (30)

where Kv = diag[kv1, kv2] is positive definite matrix, the derivative of V2 is

V2 = −qT Kv q (31)

which means that the corresponding closed-loop system is stable.If Θ is known, then the adaptive PD controller becomes an non-adaptive PD

controller with Coriolis and centrifugal terms and gravity compensation:

u =(

u1u2

)= −Kp(q − qd) − Kv q − Λpd0Θ. (32)

4 Experiment

Figure 2 shows a photo of the planar 2-DOF parallel robot developed for the purposeof this study. Links 1 and 2 are driven by two Direct Current (DC) gearhead motors,respectively. The motors are fromKollmorgenMotion Technologies Group. The gearratio is 99 to 1 and the peak torque is 17.1 N-m. The optical encoders are built inthe motors with the resolution of 1000 pulses per revolution. The values of the linkparameters are given in Table 1. The distance between the shafts of the motors isc = 0.4240 m.

Figure 2 The photo of a 2-DOF robot

Adaptive control of a parallel robot via backstepping technique 9

Table 1 Link parameters

Link i mi (kg) ai (m) li (m) Ii (kg·m2)

1 0.1950 0.4600 0.3367 4.567 × 10−3

2 0.1950 0.4600 0.3367 4.567 × 10−3

3 0.2538 0.4600 0.2400 8.626 × 10−3

4 0.2538 0.4600 0.2400 8.626 × 10−3

The computer control system has three main parts: a Pentium III Personal Computer(PC) is used in which two Data Acquisition (DAQ) boards (PCI-6024E andPCI-MIO-16E, National Instruments) are installed; these two DAQ boards areconnected with two SCB-68 connector blocks through shielded cables; the motors aredriven by two H-Bridge circuits, which are controlled by Pulse Width Modulation(PWM) signals from the computer. The controller is implemented by usingVisual C++.Angular positions of Links 1 and 2 are provided by the optical encoders. The encoderreadings are passed through an analog low pass filters before sampled by the DAQboards.Angular velocities ofLinks 1 and2are calculateddigitally basedon thepositionmeasurements. A digital low pass filter is used for the velocity calculation, which isgiven by:

vk+1 = (pk+1 − pk + τvk)/(τ + T ) (33)

where vk and vk+1 are the angular velocity at the sampling instants k and k + 1, pk

and pk+1 are the position measurements of the links at the sampling instants k andk + 1, respectively, T is the sampling period and τ is the time constant set as 0.1.

As for experiments, the control inputs are not the torques applied to the joints. Thedirect control inputs are the armature voltages of the DC motors. Therefore, in orderto implement the designed controllers in terms of torque of the motors, the computedtorque is converted into the armature voltages of the DC motors. The conversionformula is given as follows:

ui =GKt

R(Vai − KeGωi), i = 1, 2 (34)

where ui is the torque applied at the motor, Vai is the armature voltage supplied,G = 99 is the gear ratio of the motor, Kt is the torque constant with the value of2.28N·cm/Amp,Ke is the backElectromotive Force (EMF) constant with the value of0.02282 volts/rad/s, R is the armature resistance with the value of 0.640 Ohms and ωi

is the angular velocity of the gear shaft. The maximum voltage of the driver board is15 volts.

In the experiment, a sampling period of 1 ms was used. In each sampling period,the computer obtains the current positions and velocities of Links 1 and 2, calculatesthe armature voltages in terms of duty cycles of the PWM signals and sends the PWMsignals to the driver boards to control the DC motors.

To compensate the effect of back-lash between gears in the two motors, a voltagecompensation is applied in the experiment. When the computed armature voltageis larger than 0.01 volts, the armature voltage used in experiment is increased by0.05 for motor 1 and 0.35 for motor 2; when the computed armature voltage is

10 L. Wang et al.

less than −0.01 volts, the armature voltage used in experiment is increased by −0.65for both motors.

Four controllers, namely backstepping, adaptive backstepping, PD and adaptivePD, were tested on the parallel robot. The controller gains are c1 = 3.18, c2 = 3.7,c3 = 14 and c4 = 14 for the backstepping; c1 = 3.18, c2 = 3.7, c3 = 14, c4 = 14,γ1 = 0.6, γ2 = 1.5, γ3 = 0.8, γ4 = 2, γ5 = 50.0, γ6 = 60.0, γ7 = 60.0, γ8 = 10.0,γ9 = 60.0 and γ10 = 110.0 for the adaptive backstepping; kp1 = 45, kp2 = 14, kv1 = 52and kv2 = 14 for PD; and kp1 = 45, kp2 = 14, kv1 = 52, kv2 = 14, γ1 = 0.1, γ2 = 1.5,γ3 = 0.13, γ4 = 2, γ5 = 50.0, γ6 = 60.0, γ7 = 60.0 and γ8 = 10.0 for the adaptive PD,which are tuned by trial and error.

Experiments were conducted on the 2DOF parallel robot as shown in Figure 2.In the experiments, the initial angles for the four links are chosen as q1d(0) = 90◦,q2d(0) = 90◦, q3d(0) = −27◦ and q4d(0) = 27◦, while the desired angles are set to beq1d = 150◦, q2d = 160◦, q3d = −96◦ and q4d = 55◦.

Figure 3 shows the errors for no-load tests while Figure 4 displays the errors for aload of 0.4869 kg. It is observed that the transient processes from the four controllersare similar for both no-load tests and load-tests. However, the adaptive backsteppingcontroller outperforms all the other controllers in terms of steady-state errors.

Figure 3 Experimental results: errors with no load

Adaptive control of a parallel robot via backstepping technique 11

Figure 4 Experimental results: errors with a load of 486.9 g

Figure 5 Experimental results: errors against loads

12 L. Wang et al.

Figure 5 demonstrates curves of the steady-state errors vs. the load. It is easilyseen that the adaptive backstepping controller is least sensitive to the load changes.It can be also concluded that the adaptive backstepping controllers are more robustto the load changes than their non-adaptive counterparts.

5 Conclusion

An adaptive backstepping controller has been developed for set point control of aplanar 2-DOF parallel robot. The proposed controller guarantees the stability of theclosed-loop system and is able to handle parameter uncertainties. For comparison,an adaptive PD controller has been also developed. The backstepping controller isnonlinear while the PD controller is a linear controller with Coriolis and centrifugalterms and gravity compensation. Compared with the adaptive PD controller, thecontrol algorithm is more complicated and the update law contains more unknownparameters for the adaptive backstepping controller, so the adaptive backsteppingcontroller demands more computation time. The experiments have been conducted tocompare four controllers: adaptive backstepping, backstepping, adaptive PD and PD.The results have shown that all the controllers perform similarly in terms of transientresponses. However, the adaptive controllers are more robust to load changes thantheir non-adaptive counterparts. The adaptive backstepping controller outperformsall the other controllers in terms of steady-state errors and sensitivity to load changes.

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Query

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