Robust-tracking control for robot manipulator with deadzone and ...

www.ietdl.org Published in IET Control Theory and Applications Received on 8th August 2010 Revised on 16th December 2010 doi: 10.1049/iet-cta.2010.0460

ISSN 1751-8644

Robust-tracking control for robot manipulator with deadzone and friction using backstepping and RFNN controller S.H. Park1 S.I. Han2 1

Department of Mechatronics Engineering, Dongseo University, Busan, Republic of Korea School of Electrical Engineering, Pusan National University, Busan, Republic of Korea E-mail: [email protected]

2

Abstract: This study deals with a robust non-smooth non-linearity compensation scheme for the direct-drive robot manipulator with an asymmetric deadzone, dynamic friction in joints and between the environmental contact space and end-effector and uncertainty. A model-free recurrent fuzzy neural network (RFNN) control system to approximate the ideal backstepping control law is designed to replace the traditional model-based adaptive controller, which requires information on the robots dynamics in advance. The simple dead-zone estimator and friction compensator based on the elasto-plastic friction model are developed in order to estimate unknown dead-zone width and friction parameters. The Lyapunov stability analysis yields the adaptive laws of the RFNN controller as well as the estimators of a dead-zone width and an elasto-plastic friction parameter. The validity of the proposed control scheme is confirmed from simulated results for free and constrained direct-drive robots with a deadzone in joint actuator, dynamic friction in joints and contact surfaces of the end-effector and uncertainty.

1

Introduction

Recently, compensation of non-smooth non-linearities in the actuator such as deadzone, saturation, backlash, as well as joint and contact friction is essential for maintenance of precise motion of a robot manipulator and adaptability of the control scheme for several advanced applications, from industrial robots to medical robots. For the control researchers, the study of non-smooth non-linearities has been of great interest for a long time because of its nonanalytic nature in actuators and joints and the fact that the exact parameters of it are unknown. Tao and Kokotovic [1] initiated the pioneered approach of dead-zone compensation. The dead-zone pre-compensator based on fuzzy and neural networks were proposed by Lewis et al. [2] and Selmic and Lewis [3] in non-linear industrial motion system. Selmic and Lewis [3] constructed a dead-zone pre-compensator using the dead-zone inverse technique as a dead-zone compensation tool. A dead-zone compensation method without the inverse deadzone was proposed by Wang et al. [4] and Zhonghua et al. [5]. In this study, the dead-zone estimation law will be constructed without an inverse deadzone, which has a simpler structure than the previous methods. While most of these reaches on the deadzone are only focused to offset the deadzone itself; however, other non-smooth non-linear friction included in the same plant or environmental contact surface was not dealt with simultaneously. Thus, this non-linear friction should also be compensated for in order to achieve advanced control performance because frictional contacts IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

are often related to the complex tasks of manufacturing equipments or robots such as grinding, polishing, scribing, material handling and spot welding. Friction modelling and compensation are particularly important issues in a robotic manipulator with direct-drive actuators and arm-workpiece frictional contact. Hence, without depending on high control gains, friction needs to be adequately modelled and compensated for to improve the transient performance and reduce the steady-state tracking errors. As endeavours to develop further advanced friction model corresponding to real physical friction phenomena have proceeded, several dynamic friction models have been proposed. Leuven et al. [6] and GMS [7] models claim more accurate reproducibility of friction behaviour, especially in the pre-sliding region than the LuGre model [8] and elastoplastic model [9]; however, these models pay considerable identification effort owing to their complexity. Furthermore, the control schemes based on these models are even more complicated at the design level and during implementation. On the other hand, the LuGre friction model is an extension of Dahl’s model [10] and can provide representations of Stribeck friction, viscous friction, rising static friction and friction memory during slip, and has a single-state mathematical structure that can be easily implemented for more applications than the Leuven and GMS models. A significant shortcoming of this model, however, is that it exhibits the drift phenomenon, that is, systems subjected to an arbitrarily small bias force and arbitrarily small vibration. Practical experimental result shows that this drift is spurious. It is shown that the drift is 1397

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www.ietdl.org because of the fact that pre-sliding displacement in the LuGre model always includes a plastic component. To minimise the drift, a class of single-state models are defined in which pre-sliding is elasto-plastic, that is, under loading the displacement is first purely elastic before transitioning to plastic. In the elasto-plastic friction model, an important term that achieves elasto-plastic property is the piecewise continuous function [9]. For many practical applications, it is not easy to predict this function value accurately and laborious and repetitive experiments for its identification may be required. In the field of the friction control, the friction estimation based on the elasto-plastic model has almost never touched except the discrete-time estimation method [11]. Thus, in this paper, we suggest the estimation laws that estimate the value of the piecewise continuous function as well as other friction state and parameters based on the elastoplastic model. For the LuGre friction model, several friction observers [12 – 15] are developed to estimate the friction parameters, but these observers have some limitations. A developed friction parameter observer in this paper has no limitations when compared with the previous observers proposed by Canudas de Wit and Lischinsky [12] and Xie [15]. In general, the robotic behaviour is divided into free and constrained motions. When a robot moves in space, it is unconstrained and its end-effector is free of external forces. In this case, compensation of friction in joints and transmissions of robotic actuator is important to achieve accurate tracking performance of the robot manipulator. The joint friction compensation of robots has been reviewed in [16 – 20]. If the robot contacts its environments, the endeffector moves into environmental condition, and a reaction force is generated between the end-effector and environment. Although various aspects for constrained robots appear, the control of both position and contact force is the most important problem. Thus, in most constrained robot applications, the frictional interference of the armworkpiece contact should also be considered. The Karnopp model [21] and the reset integrator method [22] are used to compensate contact friction. The reset integrator method accurately represents the bonding effect of slip-stick friction when sticking, but these models give inaccurate results because of incomplete description of complex dynamic friction phenomena. The friction estimation using the LuGre model was employed by Khayati et al. [23]. Most of friction compensation schemes for dynamic friction in the joints and surfaces are based on a model-based method or the LuGre model, which is weak against the uncertainties of robots because of their incomplete modelling of manipulator dynamics, friction and external unknown disturbances as well as the drift of the LuGre friction model. For numerous works to control the constrained robot system [23 – 29], in this paper, a robotic manipulator with the joint or the end-effector that is holonomically constrained on a frictional surface is considered as a general extension of friction compensation technique for robotic tasks. The traditional force control using the Lagrange multiplier is replaced into friction force compensation under assumption of a constant normal force. Until now, the fuzzy logic and neural networks has been much researched in identification and complex non-linear dynamic systems control. While the fuzzy logic has characteristic of linguistic information and high-level logic control, the neural networks contain characteristics of parallelism and low-level learning. The fuzzy neural 1398 & The Institution of Engineering and Technology 2011

network (FNN) has the advantages of both the fuzzy and neural networks. The FNN combines the fuzzy reasoning in handling uncertain information and artificial neural networks in learning from process [30 – 32]. However, the major drawback of the existing FNN is that it has a static feedforward network structure, so that it requires a large number of neuron or membership functions. Furthermore, the weight updates of the feedforward neural network do not utilise the neural network’s internal information and the function approximation is sensitive to the training data. To improve these drawbacks, the recurrent FNN (RFNN) [33, 34] has been introduced to involve dynamic structures in the form of feedback connections used as internal memories. Thus, since the RFNN has a dynamic mapping structure, it provides fast and good approximation performance in the presence of uncertainty such as parameter variations of the system, external load and unmodelled dynamics compared with the feedforward FNN. Therefore in this paper, we will develop a backstepping control-based RFNN system combined with a dead-zone observer and an elasto-plastic model-based friction observer for precise tracking performance and robustness to the deadzone of joint actuator, friction of joint or contact surface and uncertainty of the direct-drive robot system. Numerical simulations for both free and constrained tasks of a direct-drive robot manipulator were carried out to evaluate the validity of a proposed control scheme.

2 Unconstrained robot manipulator with joint friction and deadzone 2.1 Model of unconstrained robot manipulator dynamics The non-linear dynamic model of an n-rigid-link robot system with direct-drive joints moving free space is given in the following Lagrange form M(q)¨q + C(q, q˙ )˙q + G(q) + T f (q, q˙ ) + T L = t

(1)

where q, q˙ , q¨ [ Rn represent the joint position, velocity and acceleration vectors, respectively; the moment of inertia matrix M(q) [ R n×n is a positive definite symmetric matrix; C(q, q˙ ) [ Rn×n is the centripetal Coriolis matrix; ˙ M(q) − 2C(q, q˙ ) is a skew-symmetric matrix; G(q) [ R n is the gravity vector; T f (q, q˙ ) [ Rn is the non-linear friction torque vector; TL [ R n is the external disturbance vector. The actuator output t [ R n is related to the control input td [ R n through the deadzone. When the modelling uncertainties exist and considering preceded deadzone, the robot system in (1) can be formulated as M n (q)¨q + f n (q, q˙ ) + T fn (q, q˙ ) + T u = t

(2)

where the subscript n represents the system parameters in the nominal condition: f n (q, q˙ ) = C n (q, q˙ )˙q + G n (q); Tu is a lumped uncertainty defined as T u = DM(q)¨q + DC(q, q˙ )˙q + DG(q) + DT f + T L ; DM(q), DC(q, q˙ ), DG(q) and DT f denote the unknown uncertainties of M(q), C(q, q˙ ), G(q) and Tf , respectively. It is assumed that the uncertainties of DM(q), DC(q, q˙ ), DG(q) and DTf are bounded by some positive constants ri(i ¼ m, c, g, f) such that DM ≤ rm , DC ≤ rc , DG ≤ rg and DTf ≤ rf . The disturbance is assumed such that TL [ L2[0, T ], ∀T [ [0, 1), and TL is bounded by some positive constant rd: TL ≤ rd . IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

www.ietdl.org Thus, the lumped uncertainty can be assumed to be bounded with a finite value. Moreover, modelling of the non-linear friction in (2) often is difficult owing to its complex dynamic behaviour. Thus, to guarantee more improved control performance, an elaborate non-linear friction model should be chosen. First, we consider the following LuGre friction model as follows ⎡

⎤ q˙ 1 − s01 h1 (˙q1 )z1 ⎢ q˙ 2 − s02 h2 (˙q2 )z2 ⎥ ⎢ ⎥ z˙ = ⎢ ⎥ .. ⎣ ⎦ . q˙ n − s0n hn (˙qn )zn

(3)

where zi is the average defection vector of the elastic bristle, s0i . 0 is the stiffness of elastic bristle and |˙qi | hi (˙qi ) = [Tci + (Tsi − Tci )exp(−(˙qi /˙qsi )2 )]

where zbai (0 , zbai , zssi (˙qi )) is the breakaway displacement below which the pre-sliding is purely elastic. ami (˙zi , q˙ i ) is indeed a function of q˙ i and zi , which describes the transition between elastic and plastic behaviour, as

ami (zi , q˙ i ) =

1 z − (zssi (˙qi ) + zbai )/2 1 + sin p i 2 zssi (˙qi ) − zbai

2 sgn(˙qi ) [Tci + (Tsi − Tci )e−(˙qi /qsi ) ] s0i

(8)

(5)

(6)

(9)

Since the friction torque of the elasto-plastic model is equal to that of the LuGre model, from (5) and (6), Tfn can be rewritten as Tfni = s0i zi − ai s3i hi (˙qi )zi + s4i q˙ i

The main difference with the LuGre friction model is represented by the function ai (zi , q˙ i ): this is an adhesion map that controls the rate of change of zi; especially, when the bristle displacement zi is smaller than a given breakaway displacement zbai , then ai (zi , q˙ i ) is set to zero and consequently (6) reduces to z˙i = q˙ i . This means that in the range |zi| ≤ zbai , the model shows purely elastic presliding regime, and therefore does not exhibit drift as shown in Figs. 1c and e. These results correspond to the experimental result as shown in Fig. 1f. Note that the LuGre friction model is a special case of (6) where IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

sgn(˙qi ) = sgn(zi ) sgn(˙qi ) = sgn(zi ) sgn(˙qi ) = sgn(zi ) sgn(˙qi ) = sgn(zi ) (7)

zssi (˙qi ) =

where, s1i . 0 is the damping coefficient in elastic range and s2i . 0 is the viscous friction coefficient. The LuGre friction model captures many aspects of friction, but allows no further modelling of the pre-sliding domain. In Figs. 1b and d, it is shown that, when the LuGre friction model, a drift is always produced when arbitrarily small forces or torques are applied as shown in Fig. 1a. This behaviour is not physically consistent, since in reality an object remains situated in the presence of small loads as shown in Fig. 1f, where this is the previous experimental result [35] for the input torque shown in Fig. 1a. While the LuGre friction model assumes that plastic and elastic displacements are always combined, the elasto-plastic friction model enables precise modelling of the pre-sliding domain and the dynamic transition to sliding domain, where the following formulation is used for the bristle displacement (i = 1, 2, . . . , n)

|zi | , zbai , zbai , |zi | , zssi (˙qi ), |zi | , zssi (˙qi ),

(4)

⎡

z˙i = q˙ i − ai (zi , q˙ i )s0i hi (˙qi )zi

⎧ 0, ⎪ ⎪ ⎨ ami , ai (zi , q˙ i ) ⎪ 1, ⎪ ⎩ 0,

Following [9], zssi (˙qi ) is defined as

The function hi (˙qi ) . 0 depends on many factors such as the material properties, lubrication and temperature and Tci is the Coulomb friction, Tsi is the stiction level, q˙ si is the Stribeck angular velocity. The dynamic friction torque vector is described by ⎤ s01 z1 + s11 z˙1 + s21 q˙ 1 ⎢ s02 z2 + s12 z˙2 + s22 q˙ 2 ⎥ ⎢ ⎥ Tf n = ⎢ ⎥ .. ⎣ ⎦ . s0n zn + s1n z˙n + s2n q˙ n

ai (zi , q˙ i ) = 1. The function ai (zi , q˙ i ) is parameterised as

(10)

where s3i ¼ s0i . s1i , s4i ¼ s1i + s2i . Moreover, if we assume that the input torque deadzone on each link does not affect to other links, we can describe input uncertainty as ⎡

⎤ t1 ⎢ t1 ⎥ ⎢ ⎥ t = ⎢ . ⎥ = D D (t d ) ⎣ .. ⎦ tn

(11)

In order to model the effect of the input torque deadzone, we employ the asymmetric dead-zone model as shown in Fig. 2 as follows ⎧ ⎨ tdi − di+ , ti = Dd (tdi ) = 0, ⎩ tdi + di− ,

for for for

di+ ≤ tdi di− ≤ tdi ≤ di+ tdi ≤ −di−

(12)

where i ¼ 1, 2, . . . , n. In practical motion control systems, the width of the deadzone is unknown, so that compensation of it is difficult. Most compensation covers only the case of symmetric deadzone, where di+ ¼ di2 . The asymmetric deadzone may be written as

t = DD (td ) = td − satD (td )

(13)

where the asymmetric saturation function is defined as ⎧ ⎨ di+ , satdi (tdi ) = tdi , ⎩ −di− ,

for di+ ≤ tdi for di− ≤ tdi ≤ di+ for tdi ≤ −di−

(14)

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Fig. 1 Simulated results of the LuGre and the elasto-plastic friction model and real experimental result a b c d e f

Applied input torque Angular displacement predicted by the LuGre model Angular displacement predicted by the elasto-plastic model Torque against displacement for the LuGre model Torque against displacement for the elasto-plastic model Experimental result in ball-screw system: torque against displacement

and the block diagonal matrix of dead-zone width is T D = diag{d 1 , d 2 , . . . , d n } [ R2n×n = d + d − . 2.2

Fig. 2 Dead-zone model 1400 & The Institution of Engineering and Technology 2011

Backstepping control

The control objective for a direct-drive manipulator is to determine an intelligent adaptive control system such that the system output vector q can track a desired joint trajectory qd , which is a given continuously differentiable IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

www.ietdl.org and uniformly bounded in joint or task space, and the tracking error y1 ¼ qd 2 q should be kept as small as possible. Then the derivative of the tracking error can be represented by y˙ 1 = q˙ d − v

(15)

where v = q˙ , which is the speed vector of the robot joints, can be viewed as a virtual control. Define the following stabilising function

a1 = −L1 y1 − q˙ d

(16)

where L1 ¼ diag(L11 , L12 , . . . , L1n) ≥ 0 is a matrix as design parameter. Moreover, define y2 = v + a1

(17)

In the elasto-plastic friction model, the state variable zi cannot be measured directly and ai (zi , q˙ i ), (0 ≤ ai (zi , q˙ i ) ≤ 1) is an important factor in the elasto-plastic model, we suggest the friction state observer to estimate zi as follows z˙î = q˙ i − aˆ i sˆ 0i hi (˙qi )ˆzi

y˙ 2 = v˙ + a˙ 1

where zî is the estimation of zi , aˆ i (zi , q˙ i ) is the estimation of ai (zi , q˙ i ) and sˆ 0i is the estimation of s0i . Then, the estimation error of the friction state variable is given from (6) and (24) as follows z˙˜i = −ai s0i hi (˙qi )˜zi − ai hi (˙qi )ˆzi s˜ 0i − sˆ 0i hi (˙qi )ˆzi a˜ i

Tˆ f ni = sˆ 0i zî − aˆ i sˆ 3i hi (˙qi )ˆzi + sˆ 4i q˙ i

M −1 n (qi )(−f n (q,

q˙ ) − T u − T f n + t) + L1 y˙ 1 − q¨ d (18)

Assuming all system parameters as well as friction and deadzone are well known, an ideal control is given as

tid = M n (q)(¨qd − L1 z˙ 1 − L2 z2 + z1 ) + f n (q, q˙ ) + T u + T f n = uw + uf (19) where L2 ¼ diag(L21 , L22 , . . ., L2n) ≥ 0 is a matrix as design parameter and uw = M n (q)[¨qd − L1 y˙ 1 − L2 y2 + y1 ] + f n (q, q˙ ) + T u (20) uf = T f n

(21)

Define the following Lyapunov function 1 1 V1 = yT1 y1 + yT2 y2 2 2

(24)

(25)

where z˜i = zi − zî , a˜ i =ai − aˆ i and s˜ 0i = s0i − sˆ 0i . Thus, the estimation of friction Tfn is given as

Then the derivative of z2 can be represented as

=

2.3 Estimators of the friction state and dead-zone width

(22)

(26)

where sˆ 3i and sˆ 4i are estimations of s3i and s4i, respectively. Thus T˜ f ni = Tf ni − Tˆ f ni = Gi (˙qi )˜zi + zî s˜ 0i − ai hi (˙qi )ˆzi s˜ 3 + q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i (27) where Gi (˙qi ) = s0i − ai s3i hi (˙qi ). In the elasto-plastic model, ai (zi , q˙ i ) is a very important factor, but it is difficult to determine it experimentally. Therefore the estimation procedure of it as well as other friction parameters should be considered. To compensate the deleterious effects of deadzone, a precompensator can be placed as illustrated in Fig. 3. The desired function of the pre-compensator acts a role of the composite throughout from u to t to be unity. As the width [ d + d − ]T of deadzone is not known, the learning or adaptation method is considered intuitively. An estimate ˆ = [ dˆ ˆ T D + d− ] of the dead-zone width parameter matrix D is as the pre-compensator to offset the effects of deadzone. The function from u to t with the precompensator can be expressed by Selmic and Lewis [3]

Taking the time derivative of (22) with (15) and (18) yields

td = u + jdˆ + + (I − j)dˆ −

V˙ 1 = yT1 y˙ 1 + yT2 y˙ 2

(28)

˙) = yT1 (˙qd − v) + yT2 [M −1 n (qi )(−f n (q, q − T u − T f n + t) + Ly˙ 1 − q¨ d ]

where j ¼ I, (u ≥ 0) and j ¼ 0, (u ≤ 0). The composite throughout from u to t of the adaptive compensator plus the deadzone is

= yT1 ( − y2 − L1 y1 ) + yT2 (y1 − L2 y2 )

t = DD (td ) = u + jdˆ + + (I − j)dˆ − − ED [u + jdˆ + + (I − j)dˆ − ]

=

−yT1 L1 y1

−

yT2 L2 y2

≤0

(23)

From the Lyapunov stability theory [36], since y1 0 and y2 0 as t 1, then the ideal control system is asymptotically stable. However, the ideal controller in (19) cannot be implemented for practical applications since f n (q, q˙ ) cannot be exactly obtained in general and the lumped uncertainty Tu is unknown in advance. Also, the estimation of the friction and deadzone in (19) should be provided to guarantee the stability of (23). IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

˜ T j + D ˜ Td =u−D

(29)

Fig. 3 Adaptive dead-zone compensator 1401


www.ietdl.org where the matrix of dead-zone width estimation errors is ˜ =D−D ˆ = diag{d˜ 1 , d˜ 2 , . . . , d˜ n } and j = [ j I − j ]T D and the modelling mismatch d satisfies the bound [2] |d| ≤

√ n

(30)

When considering preceded deadzone, the robot system in (2) can be formulated as

to the input layer through a time delay. The signal propagation and the basic function in each layer of the RFNN are introduced as follows. Layer 1 – Input layer: Each node i in this layer is denoted by P, which multiplies the input signals and outputs the result of the product. The net inputs and the net output are represented as

M n (q)¨q + fn (q, q˙ ) + T fn (q, q˙ ) + T u T

T

˜ j+D ˜ d =u−D

OIi = uIi wi OIi (t − T ), (31)

3 RFNN controller and adaptive laws of the unconstrained robotic system In this paper, we propose the intelligent controller system to achieve the desired tracking performance robust to uncertainty. For an unconstrained case, this type of controller consists of an RFNN with estimation laws of friction, deadzone and reconstruction suggested by ubrfnn = urfnn + uˆ f + ud + ur

Layer 2 – Membership layer: In this layer, each node performs a membership function. The Gaussian function is adopted as the membership function. The jth node input and output are represented as

OIIj

(OIi + wij OIIj (t − T) − mij )2 = exp − , s2ij

j = 1, 2, . . . , nb

(34)

where mij and sij are, respectively, the mean and the standard deviation of the Gaussian function in the jth term of the ith term input linguistic variable and nb is the total number of the linguistic variables with respect to the input nodes. Layer 3 – Rule layer: Each node k in this layer is also denotes by P. The kth rule node is represented as II OIII k = P Oj , j

3.1

(33)

where uIi represent the inputs; wi are the recurrent weights of the input layer; and na is the number of input.

(32)

where urfnn is used to imitate controller uw , ˜ Tm ˜ T d and the robust compensator ur −D uˆ f = Tˆ fn , ud = D is designed to compensate the approximated error. The laws of the intelligent approximation and estimation of the deadzone and friction parameters will be derived on the basis of the approximation theory and the Lyapunov stability theory to achieve the robust position-tracking performance in the closed system.

i = 1, 2, . . . , na

k = 1, 2, . . . , nc

(35)

RFNN system

A four-layer RFNN, as shown in Fig. 4, which comprises the input (the i layer), membership (the j layer), rule (the k layer), and output layer (the o layer), is adopted to implement the function of online gain tuning. Moreover, T represents a tapped time delay, and the output of the RFNN is recurrent

where nc is the number of rules with complete rule connection if the same linguistic variables are selected for each input node. Layer 4 – Output layer: The single node o in this layer is labelled as S, which computes the overall output as the summation of all input signals OIV o =

wko OIII k +

k

wf i uIi ,

o = 1, 2, . . . , n

(36)

i

where the connecting weight wfi is the output action strength of the ith input associated with the output of the fuzzy neural network; wko is the output action strength of the oth output associated with the kth rule; and OIV o is the output of the fuzzy neural network. The output of the RFNN can be expressed in a vector form as IV IV T T T T Oo = [OIV i . . . , Oi , . . . , On ] = W w Q + wf x = W o Qo

(37) where W o = [W Tw wTf ]T [ Rnd ,

W w = [wk1 , . . . , wkn ]T

wf = [wf 1 , . . . , wf p ]T [ Rna , Fig. 4 Proposed RFNN architecture 1402 & The Institution of Engineering and Technology 2011

Qo = [Q x]T [ Rnd

and nd ¼ na + nc . IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

www.ietdl.org 3.2

Design of controller and estimator

s˙˜ 4i = −s˙ˆ 4i = −h4 q˙ i h(˙qi )T˜ f ni

(49)

a˙˜ i = −a˙ˆ i = ha zî sˆ 3 h(˙qi )T˜ f ni

(50)

A RFNN-based controller is the output of the RFNN that can be represented in a vector form urfnn = Oo = W To Qo

(38)

The optimal RFNN-based controller u∗rfnn can be designed to mimic the controller uw in (20) such that uw = u∗rfnn = W ∗T o Qo + 1 where W ∗o is the optimal parameter vectors of approximation error vector 1 is assumed to be 1 ≤ r, in which r is a positive constant control law of the RFNN system is assumed following form ubrfnn = urfnn + uˆ f + ur + ud

(39) Wo and the bounded by vector. The to take the

(40)

where urfnn utilises to approximate the ideal approximator u∗rfnn , and the robust compensator ur is designed to reduce residual approximation error. Substituting (40) into (31) yields (see (41) and (42))

where hi ≥ 0, (i ¼ o, f, 0, 3, 4, a)are the design parameters and the diagonal matrix, Ji . 0, is a design parameter. Note that the proposed dead-zone estimation law of (46) has more simple structure than the estimation laws proposed by Lewis et al. [2] and Zhonghua and Lewis [3]. Proof: Define the following Lyapunov function V2 =

n

V2i

where 1 1 1 T 1 2 1 ˜ T −1 ˜ ˜ w ˜ + ˜ + d J d V2i = y21i + y22i + w w 2 2 2ho ki ki 2hf f i 2 i i i 1 1 2 1 2 1 2 1 2 + z˜2i + s˜ 0i + s˜ 3i + s˜ 4i + a˜ 2 2h0 2h3 2h4 2ha i (52) Differentiating with respect to time of (52) yields

˜ ki = wki − w ˜ ∗ki and w˜ f i = wf i − w∗f i . where w Theorem 1: Consider the robot dynamic system with friction, deadzone and uncertainty represented in (31). If the control law is designed as (40), in which the adaptation laws of the RFNN controller are chosen as (43) and (44) with the adaptive bound, friction and dead-zone estimation laws shown in (45) – (50), then the stability of the proposed control system can be guaranteed.

1 T˙ 1 ˜˙ ˙˜ + d˜ Ti J−1 ˜ w˜ + w ˜ w V˙ 2i = y1i y˙ 1i + y2i y˙ 2i + w i di ho ki ki hf f i f i 1 1 1 1 s˜ 0i s˙˜ 0i + s˜ 3i s˙˜ 3i + s˜ 4i s˙˜ 4i + a˜ i a˙˜ i h0 h3 h4 ha

1 = −L1i y21i − L2i y22i + w˜ Tki y2i Q − wˆ˙ ki ho

1 ˙ ˆ + w˜ f i y2i xi − w hi fi + z˜i z˙˜i +

wˆ˙ ki = ho y2i Q

(43)

˙ˆ f i = hf y2i xi w

(44)

T ˆ˙ ˜T i y2i − J−1 + d˜ i (m i d i ) − y2i 1i − d i di y2i + y2i uri

(45)

− M −1 qi )˜zi + zî s˜ 0i − ai hi (˙qi )ˆzi s˜ 3i ni (qi )y2i [Gi (˙ − q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i ]

(r2 + 1) uri = i 2 y2i 2ri

(51)

i=1

i y2i dˆ˙ i = Ji m

(46)

s˙˜ 0i = −s˙ˆ 0i = −h0 zî T˜ f ni

(47)

s˙˜ 3i = −s˙ˆ 3i = h3 hi (˙qi )ˆzi T˜ f ni

(48)

+ z˜i z˙˜i +

1 1 1 1 s˜ 0i s˙˜ 0i + s˜ 3i s˙˜ 3i + s˜ 4i s˙˜ 4i + a˜ i a˙˜ i h0 h3 h4 ha (53)

We choose the following learning and adaptive laws in order to achieve V˙ 2i ≤ 0. Thus, considering (43) – (50), (53) can be

T ∗T ˜ y˙ 2 = (y1 − L2 y2 ) + M −1 n (q)[− T f n + W o Qo − W o Qo − 1 + ur + ud ] ⎡ ⎤ T T ˜ ˜ Tk1 Q + w ˜ f 1 x1 − 11 + ur1 + d˜ 1 m 1 − d˜ 1 d1 ) (y11 − L21 y21 ) + M −1 n1 (q1 )(− T f n1 + w ⎢ ⎥ T T ⎢ ⎥ ˜ ˜ Tk2 Q + w ˜ f 2 x2 − 12 + ur2 + d˜ 2 m 2 − d˜ 2 d2 ) ⎥ ⎢ (y12 − L22 y22 ) + M −1 n2 (q2 )(− T f n2 + w ⎢ ⎥ =⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦

(41)

(42)

T T ˜ ˜ Tkn Q + w˜ f n xn − 1n + urn + d˜ n m n − d˜ n dn ) (y1n − L2n y2n ) + M −1 nn (qn )(− T f nn + w

IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

1403


www.ietdl.org From di ≤ 1, (57) can be given by

rearranged as (r + 1) T V˙ 2i = −L1i y21i − L2i y22i − y2i 1i − i 2 y22i − d˜ i di y2i 2ri 2

− M −1 qi )˜zi + zî s˜ 0i − ai hi (˙qi )ˆzi s˜ 3i ni (qi )y2i [Gi (˙ − q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i ] − zî Gi (˙qi )s˜ 0i z˜i − zˆ2i s˜ 20i + ai hi (˙qi )ˆz2i s˜ 0i s˜ 3i − q˙ i zî s˜ 0i sˆ 4i + sˆ 3i h(˙qi )ˆz2 s˜ 0i a˜ i + zˆ2i hi (˙qi )s˜ 3i s˜ 0i − ai h2i (˙qi )ˆz2i s˜ 23i + q˙ i hi (˙qi )ˆzi s˜ 3i s˜ 4i + hi (˙qi )ˆzi Gi (˙qi )s˜ 3i z˜i − h2i (˙qi )ˆz2i s˜ 3i a˜ i − zî q˙ i hi (˙qi )s˜ 4i s˜ 0i ˜ 24i − hi (˙qi )˙qi Gi (˙qi )s˜ 4i z˜i + ai zî hi (˙qi )s˜ 4i s˜ 3i − q˙ 2i hi (˙q)s + sˆ 3i hi (˙qi )ˆzi s˜ 4i ai + zˆ2i sˆ 3i hi (˙qi )a˜ i s˜ 0i − ai h2i (˙qi )ˆz2i sˆ 3i a˜ i s˜ 3i + zî sˆ 3i hi (˙qi )˙qi a˜ i s˜ 4i + zî sˆ 3i hi (˙qi )Gi (˙qi )a˜ i z˜i − zˆ2 sˆ 3i2 h2i (˙qi )a˜ 22 − ai hi (˙qi )ˆzi z˜i s˜ 0i − ai s0i hi (˙qi )˜z2i − aˆ i hi (˙qi )ˆzi a˜ i z˜i

= −VTi Fi Vi − L1i y21i − L2i y22i − y2i 1i −

(54)

(r2i + 1) 2 y2i 2r2i

T − d˜ i di y2i − M −1 qi )˜zi + zî s˜ 0i − ai hi (˙qi )ˆzi s˜ 3i ni (qi )y2i [Gi (˙ − q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i ] (55)

where Vi = s˜ 0i

s˜ 3i

s˜ 4i

z˜i

a˜ i

T

1 ˜ 2i V˙ 2i ≤ r2i 12i − L21i y22i − L22i y22i − dy 2 − M −1 qi )˜zi + zî s˜ 0i ni (qi )y2i [Gi (˙ − ai hi (˙qi )ˆzi s˜ 3i − q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i ]

2 1 2 2 L M −1 qi )˜zi 2 22i ni (qi )Gi (˙ y2i + 3 = ri 1i − L21i y2i − 6 L22i 2

2 M −1 zi s˜ 0i ni (qi )ˆ + y2i + 3 L22i

2 L22i M −1 qi )ˆzi s˜ 3i ni (qi )ai hi (˙ − y2i − 3 6 L22i

2 M −1 qi s˜ 4i ni (qi )˙ + y2i − 3 L22i

2 L M −1 ˆ 3i hi (˙qi )ˆzi a˜ i ni (qi )s − 22i y2i − 3 6 L22i

2 ˜ M −1 (q ) d ni i i + y2i + 3 L22i 2 9M −1 ni (qi ) [Gi (˙qi )2 z˜2i + zˆ2i s˜ 20i + a2i h2i (˙qi )ˆz2i s˜ 23i L22i + q˙ 2i s˜ 24i + sˆ 3i2 h2i (˙qi )ˆz2i a˜ 2i + d˜ i 2 ]

+

(see (56))

Note that

≤ −L21i y22i +

+ a2i h2i (˙qi )ˆz2i s˜ 23i + q˙ 2i s˜ 24i + sˆ 3i2 h2i (˙qi )ˆz2i a˜ 2i 1 + d˜ i 2 ] + r2i 12i 2

F + FTi Fi − FTi + = Fi1 + Fi2 Fi = i 2 2 where Fi1 = FTi1 = (Fi + FTi )/2 is a symmetric matrix and Fi2 = −FTi2 = (Fi − FTi )/2 is a skew-symmetric matrix. Thus, Mi1 is positive semi-definite by the fact that det Fi1 (1, 1) = zˆ2i . 0,

where L2i ¼ L21i + L22i , L21i and L22i are the positive constants. Integrating both sides of (58), we have 2 9M −1 ni (qi ) L22i 0 T T × Gi (˙qi )2 z˜2i dt + zˆ2i s˜ 20i dt

V2i (T ) − V2i (0) ≤ −L21i

(j = 2, 3, 4, 5) Therefore, since the matrix Fi is positive semi-definite, (55) is represented as

−ai hi (˙qi )ˆz2i ai h2i (˙qi )ˆz2i ai hi (˙qi )ˆzi 0 ai h2i (˙qi )ˆz2i sˆ 3i

1404 & The Institution of Engineering and Technology 2011

0

T

T

s˜ 23i dt + q˙ 2i

+a2i h2i (˙qi )ˆz2i

s˜ 24i dt 0 0 T T 2 2 2 2 2 ˜ a˜ i dt + d i dt +sˆ 3i hi (˙qi )ˆzi

1 T + r2i 12i − L2i y22i − d˜ i di y2i 2

zˆ2i 2 ⎢ −ˆz h (˙q ) i i i ⎢ Fi = ⎢ ⎢ zî q˙ i hi (˙qi ) ⎣ a h (˙q )ˆz i i i i −ˆz2i sˆ 3i hi (˙qi )

y22i dt +

0

1 1 y22i y2i 2 2 + 2 r 1 + r 1 V˙ 2i ≤ − y22i − i i ri i i 2 2 r2i

⎡

(58)

T

det Fi1 (1:j, 1:j) = 0

− M −1 qi )˜zi + zî s˜ 0i ni (qi )y2i [Gi (˙ − ai hi (˙qi )ˆzi s˜ 3i − q˙ i s˜ 4i − sˆ 3i hi (˙qi )ˆzi a˜ i ]

2 9M −1 ni (qi ) [Gi (˙qi )2 z˜2i + zˆ2i s˜ 20i L22i

1 + r2i 2 (57)

0

T 12i dt

0

(59)

0

Since V2i(t) ≥ 0, the inequality in (59) implies the following

q˙ i zî −˙qi zî hi (˙q) q˙ 2i hi (˙q) 0 zî sˆ 3i hi (˙qi )˙qi

⎤ zî Gi (˙qi ) −sˆ 3i hi (˙qi )ˆz2i −hi (˙qi )ˆzi Gi (˙qi ) h2i (˙qi )ˆz2i ⎥ ⎥ hi (˙q)˙qi Gi (˙qi ) −sˆ 3i hi (˙qi )ˆzi ⎥ ⎥ ai s0i hi (˙qi ) aˆ i hi (˙qi )ˆzi ⎦ −ˆzi sˆ 3i hi (˙qi )Gi (˙qi ) zˆ2i sˆ 3i2 h2i (˙qi )

(56)

IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

www.ietdl.org inequality T

1 1 1 T ˜ (0)w˜ wi (0) w y22i dt ≤ z21i (0) + z22i (0) + 2 2 2ho wi 0 1 1 + z˜2i (0) + s˜ 20i (0) + s˜ 23i (0) h0 h3 1 2 1 2 + s˜ 4i (0) + a˜ i (0) h4 ha T T −1 2 9|M ni (qi )| + Gi (˙qi )2 z˜2i dt + zˆ2i s˜ 20i dt L22i 0 0 T T s˜ 20i dt + a2i h2i (˙qi )ˆz2i s˜ 23i dt + zˆ2i

L21i

0

0

T

T

s˜ 24i dt + sˆ 3i2 h2i (˙qi )ˆz2i a˜ 2i dt 0 0 T T 1 12i dt + d˜ i 2 dt + r2i 2 0 0 + q˙ 2i

z˜i 0, s˜ ji 0, a˜ i 0, d˜ i 0 and 1˜ i 0 as t 1 by Barbalat’s Lemma [36]. We have also ti ui , y1i 0 and y2i 0 as t 1. Therefore we have completed the proof. A 3.3

On-line learning of the RFNN parameters

The performance of the RFNN will be significantly affected by selection of the connective weight, recurrent weight, mean and variance. For achieving effective on-line learning, the gradient descent method is introduced. From the adaptation law of (43), it follows that ∂V ∂V2i ∂OIV ∂V2i i wˆ˙ ki = −ho 2i = −ho IV = −ho IV Q ∂wki ∂Oi ∂wki ∂Oi = ho y2i Q

(60)

(68)

In the above equation, the Jacobian of the controlled system is obtained as

˜ wi (0) = 0, If the system starts with initial conditions w wf i (0) = 0, d˜ i (0) = 0, z˜i (0) = 0, s˜ ji (0) = 0, ( j ¼ 0, 3, 4) and a˜ i (0) = 0, (60) is rewritten as T

T y22i dt

L21i

T

≤ az

0

z˜2i dt 0

+ a0

T

s˜ 20i dt 0

T

a˜ 2i dt

0

0

T

1 2 d˜ i dt + r2i 2 0

+ ad

II

˙ ij = −hm m

T 12i dt

(61)

0

=

where (62)

2 9M −1 ni (qi ) 2 zî a0 = L22i

aa =

(64)

2 9M −1 ni (qi ) 2 q˙ i L22i

(65)

2 9M −1 ni (qi ) sˆ 3i2 h2i (˙qi )ˆz2i L22i

(66)

2 9M −1 ni (qi ) L22i

(67)

ad =

1 z˜2i dt

, 1,

0

1

s˜ 2ji dt

, 1,

0

1˜ 2i dt

, 1,

(70)

= hs y2i wki OIIj

2[OIi + wij OIIj (t − T ) − mjk ] s3ij

(71)

w˙ ij = −hij

III ∂O ∂V2i ∂V2i ∂OIV j i ∂Ok = −hij IV II ∂w ∂wij ∂O ∂Oi ∂OIII ij j k

= hij y2i wki OIIj

2[OIi + wij OIIj (t − T ) − mij ] II Oj (t − T ) s3ij (72)

1 ||d˜ i ||2 dt

a˜ i dt , 1, 0

1

2[mjk − OIi − wij OIIj (t − T )] s2ij

II

Since z˜i , s˜ ji (j = 0, 3, 4), a˜ , d˜ i and 1˜ i , are square integrable, that is 1

hm y2i wki OIIj

III ∂O ∂V ∂V2i ∂OIV j i ∂Ok s˙ ij = −hs 2i = −hs IV III ∂sij ∂Oi ∂Ok ∂OIIj ∂sij

(63)

2 9M −1 ni (qi ) a2i h2i (˙qi )ˆz2i L22i

a4 =

III ∂O ∂V2i ∂V2i ∂OIV j i ∂Ok = −hm IV III II ∂mij ∂Oi ∂Ok ∂Oj ∂mij

II

2 9M −1 ni (qi ) G2i (˙qi ) az = L22i

a3 =

(69)

The learning algorithms based on the gradient descent method for the membership functions and recurrent weights in membership layer and input layer are derived as follows

s˜ 23i dt 0

T

s˜ 24i dt + aa

+ a4

+ a3

∂V2i = −y2i ∂OIV i

0

,1

0

respectively and ai ,(i ¼ z, 0, 3, 4, a, d ) and ri are finite, then, IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

III ∂OII ∂V2i ∂V2i ∂OIV j i ∂Ok = −hi IV II ∂w ∂wi ∂O ∂Oi ∂OIII i j k I II II 2(mij − Oi − wij Oj (t − T )) = hi y2i wki Oj uIi OIi (t − T) 2 s ij j

w˙ i = −hi

(73) where hm , hs , hij and hi are the learning rates related to the mean, variation and recurrent weights, respectively. The configuration of the proposed backstepping control-based RFNN combined with deadzone and friction observer is depicted in Fig. 5. 1405


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Fig. 5 Block diagram of the proposed RFNN_WDFO control system

4 Constrained robot manipulator dynamics with contact friction and deadzone

subspace of the contact point and H(q) [ R(n2m) × n satisfies K(q) · H(q)T = 0

4.1 Model of constrained robot manipulator dynamics

(78)

The constrained dynamic model of an n-rigid-link robot system can be represented in the following Lagrange form [27] M(q)¨q + C(q, q˙ )˙q + G(q) + J (q)T (Fn + F t ) + T L = t

(74)

where J(q) [ R n×ndenotes the Jacobian matrix of the system; Fn [ R n a vector of the normal force of contact described in the task space; Ft [ R n a vector of the tangential force exerted on the end-effector described in the task space. In the constrained case, we assume that the joint friction torque in (1) is very small to make the problem simple. The stiffness of the environmental contact leads the holonomically constrained motion of the manipulator described by m algebraic equations [26] K(q)J (q)˙q = 0

(75)

where K(q) [ R m×n denotes the normal direction of the constrained surface with the normal force defined as F n = K(q)T · le

(76)

where le [ R m are the Lagrangian multipliers. The tangential friction force is defined by F t = H(q) F f e T

(77)

where Ffe [ R n2m represents the frictional force in tangential Table 1

Value of the link parameter of the robot system

Parameter

Value, unit

m1 m2 L1 L2 g s01n s02n

3.5 kg 0.75 kg 1m 0.8 m 9.806 m/s2 775 Nm/rad 523 Nm/rad

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Fig. 6 Sketch of the unconstrained two-link manipulator with deadzone in joint actuator and joint friction

Table 2

Value of the friction and dead-zone parameters of the unconstrained robot system Parameter

s01n s02n s11n s12n s21n s22n Ts1 Ts2 Tc1 Tc2 q˙ s1 q˙ s2 a1 a2 d1n+ d1n2

Value, unit 775 Nm/rad 523 Nm/rad 20.8 Nms/rad 12 Nms/rad 32 Nms/rad 5 Nms/rad 25.1 Nm 24 Nm 20.7 Nm 17.6 N 0.13 rad/s 0.08 rad/s 0.3 0.5 60.5 Nm 55.7 Nm


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Fig. 7 Simulated results of the unconstrained RFNN_WDFO system a Position-tracking results of joint1. b Position-tracking results of joint2 c Position-tracking errors of each joint d Cartesian reference-tracking result of endpoint in the X– Y space e sˆ 01 and sˆ 02 f sˆ 31 and sˆ 32 . g sˆ 41 and sˆ 42 h aˆ 1 and aˆ 2 i zi and zî j Tfi and Tˆ f i k dˆ +1 l u1 and t1


1407


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Fig. 7 Continued

Furthermore, the relative velocity of the end-effector on the contact surface is defined by q˙ c = H(q)J (q)˙q

q = [ qT1 qTm ]T (79)

Using adequate column permutations, it follows that K(q)J (q) = [ K 1 K m ] 1408 & The Institution of Engineering and Technology 2011

where K1 [ R m×(n 2 m) and Km [ R m×m. q is represented as

(80)

(81)

with qT1 [ Rn−m and qTm [ Rm . The joint space of the robot has only n 2 m independent variables during constrained motion [26]. Thus, the new coordinates is chosen as q˙ = J 1 (q)˙q1

(82)


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Fig. 8 Simulated results of the unconstrained systems a b c d

Position-tracking errors of joint1 Position-tracking errors of joint2 u1 t1

In addition, from (78), (80) and (85), H(q) can be selected as

and q¨ = J˙ 1 (q)˙q1 + J 1 (q)¨q1

(83)

where J1(q) [ R n×(n 2 m) is described by T J 1 (q) = [ I n−m −K −1 m K 1]

(84)

From (80) and (84), it follows that J T1 (q) · J T · K(q)T = 0 Table 3

(85)

H(q) = [J (q) · J 1 (q)]T

(86)

The elasto-plastic friction model instead of LuGre friction model is chosen to represent the contact sliding (not rolling) friction between the end-effector and external surface. We assume the friction to be directly proportional to normal force Fn between the end-effector and the surface. Also without loss of generality, 1D dynamic friction model of the end-effector is selected under assumption that the friction force coefficients are considered unidirectional motion as

RMS joint-tracking error for the unconstrained robot system

Control system RFNN RFNN_WFO RFNN_WDO RFNN_WDFO

Link1, rad

Reduction rate, %

Link2, rad

Reduction rate, %

0.003 0.0028 3.72e-04 1.73e-04

100 93 12 6

2.62e-04 8.97e-06 2.62e-04 2.93e-06

100 3 100 1


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Fig. 9 Simulated results of the unconstrained RFNN_WDFO system for Case II a b c d e

Cartesian reference tracking result of endpoint in the X– Y space zi and zî . Tfi and Tˆ f i . dˆ +1 u1 and t1

F f e = le m

(87)

where le is the one-dimensional normal contact force previously given by (76). Thus, the tangential friction force of (77) can be given by F t = H(q)T le m = H(q)T F f 1410 & The Institution of Engineering and Technology 2011

(88)

0z − s (z, q˙ 1 ) 3 z¯˙ + s 4 q˙ 1 and z¯˙ = q˙ 1 − a where F f = le m = s q1 )z. In (88), the traditional force control scheme using 0 h(˙ s Lagrange multiplier in (87) is replaced with the surface friction estimation under the assumption that the applied nominal contact force are maintained as unchanged state in tasks space to limit the content of the paper. More developments for the force control are beyond this paper. The other friction properties are all same as an unconstrained case. IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

www.ietdl.org ˙ˆ f i = hf y2i xi w

For the constrained robot dynamics, the dynamic equation (74) can be represented in terms of reduced one by taking into account of the above results as follows 1 (q)¨q1 + f1 (˙q1 , q) + H 1 Ff + T L1 = t1 M

uri =

(89)

where

(94)

(r2i + 1) y2i 2r2i

(95)

˙ d¯ˆ i = Ji jiy2i

(96)

˙ˆ 0i = −h0zî F˜ ni s ˙˜ 0i = −s

(97)

L1 = J T1 T L and t1 = J T1 1 = HH T , T H

˙ˆ 3i = h3 hi (˙q1i )zî F˜ ni s ˙˜ 3i = −s

(98)

Considering the modelling uncertainty and dead-zone input, (89) can be described by

s ˙ˆ 4i = −h4 q˙ 1i h(˙q1i )F˜ ni ˙˜ 4i = −s

(99)

ˆ 3 h(˙q1i )F˜ ni a ˙ˆ i = hazî s ˙˜ i = −a

(100)

1 (q) = J T1 M(q)J 1 f 1 (˙q1 , q) M = J T1 [M(q)J˙ 1 + C(q, q˙ )J 1 ]˙q1 + J T1 G(q)

1 Ff n + T u1 = u ˜ T1 j + D ˜ 1d 1n (q)¨q1 + f1n (˙q1 , q) + H 1 − D M (90) 1 (q)¨q1 + Df1 (˙q1 , q) + H 1 DF f + T L1 and u1 = DM where T T T ˜T ˜ D1 = J 1 D . It is assumed that the uncertainty of the constrained robot is also bounded like the unconstrained robotic system. To avoid the complexity and excessive expansion of this paper, the detail procedures of the control system design and proofs on the constrained robot system is simplified since the most process of constrained case are similar to those of unconstrained ones. 4.2

Design the controller and estimator

For a constrained robot, the control objective is to let the motion of end-effector q1 move along a desired trajectory q1d . Thus, the tracking error y1 = q1d − q1 should be kept as small as possible by compensating both the dead-zone non-linearity of the joint input and the contact friction between the end-effector and task surface. The controller can be given by brfnn = u rfnn + u¯ˆ f + u d + u r u

(91)

with

T

M(q) = C(q, q˙ )˙q =

3i i

1i

i i

= diag(L ,L , ..., L L 1 11 12 1(n−m) ) ≥ 0 Since the remaining results and proof procedures for the constrained control system are almost same as the unconstrained case, the detail processes are omitted.

5 Simulation for the robot manipulator with deadzone, friction and uncertainty This section describes the simulated evaluation of the proposed control scheme applied to two-degrees-of-freedom (DOF) robot manipulator direct-drive manipulator with two vertical links. The values of links are illustrated in Table 1. 5.1

Unconstrained robotic system

M(q)¨q + C(q, q˙ )˙q + G(q) + T f (q, q˙ ) = t

(92)

rfnn is same as the unconstrained case. The adaptive In (91), u laws are derived as follows: w¯ˆ˙ ki = hoy2i Q

y − q˙ , F˜ y2 = q˙ 1 − L 1 1 1d ni ˜ ˜ ˆ 0i − a ˜ 3 − a ˜ 3 + q˙ 1i s ˜ 4i = Gi (˙q1i )zi + zi s i hi (˙q1i )zî s i hi (˙q1i )zî s ˜ −s ˆ h (˙q )zˆ a

The dynamic equation of the unconstrained robot manipulator shown in Fig. 6 is represented by (1)

1 Fˆ f n f = H u ˜ 1 j − D ˜ 1d ud = D

where

(93)

where (see equation at the bottom of the page) The parameters of the friction and dead-zone parameters of the unconstrained robot system are given in Table 2. The direct kinematics for a circle trajectory in task space is

(m1 + m2 )L21 + m2 L22 + 2m2 L1 L2 cos(q2 ) m2 L22 + L1 L2 m2 cos(q2 ) m2 L22 + L1 L2 m2 cos(q2 ) −m2 L1 L2 sin(q2 )˙q22 − 2m2 L1 L2 sin(q2 )˙q1 q˙ 2 m2 L1 L2 sin(q2 )˙q21

m2 L2 cos(q1 + q2 ) + (m1 + m2 )L1 g cos(q1 ) G(q) = m2 L2 cos(q1 + q2 ) s01 z1 + s11 z˙1 + s21 q˙ 1 T f (q, q˙ ) = s02 z2 + s12 z˙2 + s22 q˙ 2 IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

(101)

m2 L22

1411


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Fig. 10 Simulated results for Case III a b c d e f

Position-tracking errors of the joint1 for Position-tracking errors of the joint2 for Position-tracking errors of the joint1 for Position-tracking errors of the joint2 for ur1 ur2

the impulse disturbance the impulse disturbance the sinusoidal disturbance the sinusoidal disturbance

given by

Then, the desired end-effector trajectory of the manipulator is

L1 sin(q1 ) + L2 sin(q1 + q2 ) h(q) = L1 cos(q1 ) + L2 cos(q1 + q2 )

1412 & The Institution of Engineering and Technology 2011

(102)

x Y d (t) = d yd

x + R cos(v × t) = c yc + R sin(v × t)

(103)


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Fig. 11 Sketch of the constrained two-link manipulator with deadzone in joint actuator and contact surface friction

where xc ¼ yc ¼ 0.5 m, R ¼ 0.01 m and v ¼ 1.45 rad/s. This trajectory makes the manipulator tip trace a circle in the x0 2 y0 plane having a radius R ¼ 0.01 m. The desired trajectory Yd was translated to the corresponding joint space desired position trajectory qd via the inverse kinematics of the simulated two-DOF links manipulator qd = h−1 (Y d ) = ⎡

qd1 qd2

⎤

yd /xd − tan−1 (L2 sin(qd2 )) ⎢ tan ⎥ L1 + L2 cos (qd2 ) ⎢ ⎥ =⎢

⎥ ⎣ −1 1 − (x2d + y2d − L21 − L22 )2 /(2L1 L2 )2 ⎦ tan (x2d + y2d − L21 − L22 )/(2L1 L2 ) (104) −1

All simulated tests were carried out under the initial conditions q1(0) ¼ 20.1403 rad, q2(0) ¼ 2.3549 rad, q˙ 1 (0) = 0 rad/s and q˙ 2 (0) = 0 rad/s. The parameters of the backstepping control are chosen as L11 ¼ 500, L12 ¼ 500. The learning rates of the RFNN network are selected as low values, ho ¼ hf ¼ hm ¼ hs ¼ hij ¼ hi ¼ 0.5, respectively, to guarantee stability of the RFNN network. High learning rates may cause the RFNN network to produce unstable output, although the convergence speed would become faster. The RFNN had four, five, five and two neurons at the input, membership, rule and output layers, respectively. The Gaussian function and the connecting weights were initialised within the maximum value of each input and the random number via some trials. uI1 = y11 = q1d − q1 , uI2 = y˙ 11 , uI3 = y12 = q2d − q2 , uI4 = y˙ 12 are selected as the inputs of the RFNN network. The design gains of the approximated error compensator are r1 ¼ 0.5 and r2 ¼ 0.5. In the simulation, we designed four controllers to evaluate the performance of the proposed control system: backstepping control-based RFNN system (RFNN), backstepping control-based RFNN with friction estimator (RFNN_WFO), backstepping control-based RFNN system with the dead-zone estimator (RFNN_WDO), backstepping control-based RFNN system with the deadzone and friction observer (RFNN_WDFO). Moreover, the following three cases IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

were considered in the simulation process to illustrate the effectiveness of the proposed control systems: Case I: All parameters are nominal condition, and TL ¼ 0; Case II: s0i ¼ 1.5 × s0in , s1i ¼ 1.5 × s1in , s2i ¼ 1.5 × s2in , ai ¼ 1.5 × ain and d1+ ¼ 1.5 × d1n+; Case III: i) TL1 ¼ 0, TL2 ¼ 2500 Nm and m2 ¼ 20 × m2n act on link2 at 1 s and 3 s, respectively, ii) TL1 ¼ 0, TL2 ¼ 10 sin(2.5t) Nm act on link2 during all simulated time. where the subscript n means a nominal value of each parameter. The simulated results of the RFNN_WDFO for Case I is represented in Fig. 7. The introduced three friction parameter observers well estimate the given friction parameters s0 , s3 , s4 and a as shown in Figs. 7e– h. Thus, the friction state z and torque Tf of each joint are also well estimated, as shown in Figs. 7i and j, respectively. In Fig. 7k, the estimated dead-zone width of joint1, dˆ 1+ , approaches to its real dead-zone width of joint1 and thus, as shown in Fig. 7l, the estimated dead-zone output is almost the same as the real input torque. Therefore we can see that the proposed dead-zone compensation scheme gives a satisfactory tracking performance for the joints and Cartesian motion as shown in Figs. 7a – d. Next, for the RFNN, RFNN_WFO and RFNN_WDO systems, simulations were carried out to compare the control Table 4

Value of the friction parameter of the constrained robot

system Parameter

Value, unit

01n s 11n s 21n s F s1 F C 1 ˙ s1 q

12 797 N/m 74.6 N/m 18 Ns/m 4.2 N 3.6 N 0.005 m/s 0.8

1 a

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Fig. 12 Simulated results of the constrained RFNN_WDFO system a b c d e f g h i

Position-tracking results s ˆ 0 s ˆ 3 s ˆ 4 a ˆ z and zˆ Ff and Fˆ f dˆ +1 u1 and t1 (dashed)

performance of the proposed system with them. In Figs. 8a and b, the tracking errors of two links are represented, in which the tracking performance of link1 and link2 mainly depends on the 1414


deadzone and friction. In Figs. 8c and d, u1 and t1 of the RFNN and RFNN_WFO systems are significantly different because of uncompensation of deadzone. Table 3 represents the numerical IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397–1417 doi: 10.1049/iet-cta.2010.0460

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Fig. 12 Continued

results of the control performances of each control system as the root-mean-square value of the joint tracking error. To evaluate the robustness of the proposed estimators to modelling error, simulated results for 50% increased case of the nominal value of friction and dead-zone parameters are represented in Fig. 9, where the control performances are almost unchanged like nominal case. Finally, simulation was carried out to test robustness to mass variation and external disturbance of Case III. As shown in Figs. 10a and b, when the impact disturbance and mass variation are applied into joint2, the tracking errors of the RFNN and RFNN_WDFO system of each joint were rapidly stabilised in certain small range. For sinusoidal external disturbance applied to link2, the tracking errors of the RFNN_WDFO system are almost unchanged and chattering phenomena are not found in of each robust control input ur . However, the tracking error and robust control input of the RFNN are slowly increased as shown in Figs. 10d and f. Therefore from the simulated results, it can be verified that the proposed RFNN_WDFO system is robust to the uncertainty of the robot manipulator as well as the deadzone of input torque and non-linear friction of each joint. 5.2

Constrained robot system

A two-DOF robot manipulator as shown in Fig. 11 is considered for simulation of the constrained motion. The IET Control Theory Appl., 2011, Vol. 5, Iss. 12, pp. 1397– 1417 doi: 10.1049/iet-cta.2010.0460

end-effector moves along the frictional straight line on the upper horizontal plane. The Jacobian matrix is given by [37] J=

−L1 sin q1 − L2 sin(q1 + q2 ) L1 cosq1 + L2 cos(q1 + q2 )

−L2 sin(q1 + q2 ) L2 cos(q1 + q2 )

(105) J1(q) is selected from the constraint equation L1 sin q1 + L2 sin(q1 + q2) ¼ 1 as follows J 1 (q) = 1

L1 cos q1 −1 − L2 cos(q1 + q2 )

T (106)

The normal and tangential transformation matrices are computed as follows K(q) = 0

1 ,

H(q) =

L1 sin q2 0 cos(q1 + q2 )

(107)

The parameters of the backstepping control are chosen as = 10. The parameters of the friction and = 250, L L 11 12 dead-zone parameters of the constrained robot system are given in Table 4. The learning rates of the RFNN are selected as low values, ho ¼ hf ¼ hm ¼ hs ¼ hij ¼ hi ¼ 0.5, respectively, to guarantee stability of the RFNN. The design gains of the approximated error compensator are r1 ¼ 0.5 1415


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Fig. 13 Simulated results of the constrained PD, RFNN and RFNN_WFDO systems a Position-tracking errors b uli

and r2 ¼ 0.05. The sinusoidal desired input q1d ¼ 0.2 sin 2.5t (m) under the initial conditions q1(0) ¼0.45 m is selected. Simulated results of the RFNN_WDFO system are given in Fig. 12. For the desired tracking input, the actual output is represented in Fig. 12a. The estimated results of the contact surface friction are shown in Figs. 12b – f. Thus, the surface friction force is well estimated in Fig. 12g. The reflected deadzone in task space is given in Figs. 12g and h. The control input u1 is similar to t1 although it is a little fluctuation comparing with the unconstrained case. In Fig. 13, the position-tracking error and control input of three control systems are represented, where it is shown that the proposed RFNN_WDFO system has a more outstanding performance than the PD and RFNN systems.

6

Conclusion

In this paper, the deadzone and friction compensation schemes with the RFNN controller as robust model-free control have been studied to establish much enhanced position-tracking performance of the uncertain robot manipulator containing both joint input deadzone and nonlinear dynamic friction in joint or task contact surface. The adaptive estimation laws for both dead-zone and friction parameters are derived from the Lyapunov stability theorem. To improve the drawback of the LuGre friction model, the elasto-plastic model-based friction parameter estimator is proposed to estimate the friction parameters of the elasto-plastic friction model in robot system. The validation of the proposed control scheme can be evaluated from some simulations for both unconstrained and constrained robotic tasks in the presence of deadzone in joint actuator, both joint friction and contact friction between the end-effector and external surface and uncertainty.

7

Acknowledgment

This work was supported by Dongseo Frontier Project (Research Fund of 2009) of Dongseo University.

8

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