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A dynamic recurrent neural network-based controller for a rigid–flexible manipulator system
Mechatronics 14 (2004) 471–490
A dynamic recurrent neural network-based controller for a rigid–flexible manipulator system Lianfang Tian *, Curtis Collins Department of Mechanical Engineering, University of California at Riverside, Riverside, CA 92521, USA
Abstract This paper deals with the tracking control problem of a manipulator system with unknown and changing dynamics. In this study, a fuzzy logic controller (FLC) in the feedback configuration and an efficient dynamic recurrent neural network (DRNN) in the feedforward configuration are proposed. The DRNN, which possesses the ability of approaching arbitrary nonlinear function, is applied to approximate the inverse dynamics of the robotic manipulator system. Based on the outputs of the FLC, parameter updating equations are derived for the adaptive DRNN model. The analysis of the stability of the system is also carried out. Finally, extensive simulations are conducted under different conditions. Results demonstrate the remarkable performance of the proposed controller. It can successfully identify the inverse dynamics of the flexible manipulator system and perform accurate tracking for a given trajectory. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Robotic system; Dynamic recurrent neural network; Fuzzy logic control; Adaptive control
1. Introduction It is well known that the demand for increased productivity by robots can be partly met by the use of lighter robots operating at high speed and consuming less energy, which may lead to a reduction in the stiffness of the manipulator structure. This would result in an increase in robot deflection and poor performance due to the effect of mechanical vibration in the links and make it hard to control. Thus,
*
Corresponding author. E-mail address: [email protected] (L. Tian).
0957-4158/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2003.10.002
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L. Tian, C. Collins / Mechatronics 14 (2004) 471–490
trajectory following control of a robotic manipulator system has been an important research area in the last decade [1–3]. On the other hand, a robotic system is a highly nonlinear and heavily coupled mechanical system. The mathematical model of such a system often consists of a set of linear or nonlinear differential/difference equations derived using some form of approximation and simplification. However, when an exact robot model is difficult to obtain due to uncertainty and complexity, the traditional model-based control techniques will break down [4,5]. From the aforementioned, it can be summarized that a control method for the robotic manipulator system must be able to effectively deal with the adverse effects of transient vibration and complexity of the system. It is apparent that the conventional model-based control cannot meet the requirement of system performance. Because of this reason, much attention has recently been directed towards intelligent control methods such as fuzzy logic control and neural networks [6,7]. Those techniques have the following advantages: a fuzzy logic controller (FLC), based on fuzzy set theory, can provide a means of converting a linguistic control strategy into control action and thus offer a high level computation. In addition, it also shows more excellent robustness over PID. A neural network can provide low-level computation and demonstrate many other salient features such as learning, default-toleration, parallelism and generalization [8–11]. Therefore, those novel control techniques have drawn much attention to many researchers and have been practically applied in manipulator control systems. Hu et al. [12] proposed a neural network (NN) controller for a constrained robot manipulator. The NN is used as a feed-forward loop to adaptively compensate for the uncertainties in the robot dynamics. The weights of the NN are trained on-line. Gao et al. [13,14] introduced a robust adaptive fuzzy neural network (AFNC) for the motion control of a multi-link robot manipulator system. The proposed AFNC is employed to compensate the modeling errors and handle external disturbance. Sun et al. [15] developed a discrete time tracking control approach based on dynamic neural networks (DNN) for robotic manipulators. The inverse dynamics established by the DNN and the NN variable structure control are used for the design of an adaptive tracking controller. Zhang et al. [16,17] presented a one-layer dual neural network model that provides a new parallel distributed computational approach to online kinematic control of physically constrained redundant manipulators, and the dynamic equation of the dual neural network is piecewise linear and does not contain any high-order nonlinear term, and thus, the architecture is very simple. Moreover, it can guarantee fast convergence due to the exponential convergence. In this paper, a novel FLC, a learning controller and a feedforward controller are designed for a two-link rigid–flexible manipulator system. In the feedback loop, an FLC is used to provide control signals for the manipulator system. In the learning controller, a dynamic recurrent neural network (DRNN) contains a state feedback and provides more computational advantages than a back-propagation neural network and models the inverse dynamics of the manipulator system. The weights of the DRNN are updated on-line based on the outputs of the FLC during the control process. It provides the manipulator system the ability to adapt to the changes of
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uncertainties. A feedforward controller, which is a complete copy of the learning controller, is used to generate the joint torques and enhance the performance of the control system. The remarkable characteristics of the proposed controller include: (1) a dynamic multi-layer neural network is instead of the back-propagation neural network. It has the memory ability. (2) the weights of the network are trained on-line using the realtime data from the actual manipulator movement instead of off-line training. The model is first trained for the identification of the dynamics of the robotic system and then is placed in the feedforward configuration to be used to generate control signals. The layout of this paper is organized as follows: Section 2 gives the dynamic modeling of the two-link flexible manipulator system. Section 3 presents the fuzzy logic controller. Section 4 describes the DRNN. The analysis of the stability of the control system is carried out in Section 5. Simulations are conducted in Section 6. Finally, conclusions are drawn in Section 7.
2. Dynamic modeling of the flexible robot system 2.1. The robot arm dynamics The system considered here consists of two links connected with a revolute joint moving in a horizontal plane as shown in Fig. 1. The first link is assumed to be rigid, while the second one is flexible, which is composed of a slender flexible beam cantilevered onto a rigid rotating hub. The longitudinal deformation of the second link is neglected. It is assumed that the second link can be bent freely in the horizontal plane but is stiff in the vertical bending and torsion. Thus, the Euler–Bernoulli beam theory is sufficient to describe the flexural motion of the flexible link. The Lagrange’s
Fig. 1. The schematic diagram of a two-link rigid–flexible manipulator system.
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equation and model expansion method can be utilized to develop the dynamic modeling of the robotic manipulator system [18,19]. As shown in Fig. 1, X0 O0 Y0 represents the inertial coordinate frame, X1 O1 Y1 and X2 O2 Y2 are the moving coordinate frames with origin at the hubs of links 1 and 2, respectively. From Fig. 1, it shows that h1 and h2 are the revolving angles of the two links with respect to their frames, respectively. R1 and R2 denote position vectors of an arbitrary point on the links 1 and 2 relative to the inertial frame, respectively. wðx; tÞ is the transverse elastic displacement. Note that the axial deformations are not considered in this study, because the flexural motion of the beams does not induce significant axial vibrations. Two pairs of orthogonal unit vectors (i1 ; j1 ) and (i2 ; j2 ), which are fixed at the moving coordinate frames of the links 1 and 2, are shown in Fig. 1. For links 1 and 2, the coordinate of the general points R1 and R2 can be expressed as follows:
x1 r1 cos h1 R1 ¼ ¼ y1 r1 sin h1 x L1 cos h1 þ r2 cosðh1 þ h2 Þ w sinðh1 þ h2 Þ R2 ¼ 2 ¼ L1 sin h1 þ r2 sinðh1 þ h2 Þ þ w cosðh1 þ h2 Þ y2
ð1Þ ð2Þ
where r1 and r2 are the spatial coordinates relative to X1 O1 Y1 and X2 O2 Y2 coordinate frames, respectively. L1 and L2 are the lengths of links 1 and 2, respectively. The time derivatives of R1 and R2 are: _ _R1 ¼ x_ 1 ¼ r1 h1 sin h1 ð3Þ y_ 2 r2 h_1 cos h1 "
R_ 2 ¼ " ¼
x_ 2
#
y_ 2 L1 h_1 sin h1 r2 ðh_1 þ h_2 Þ sinðh1 þ h2 Þ wðh_1 þ h_2 Þ cosðh1 þ h2 Þ w_ sinðh1 þ h2 Þ L1 h_1 cos h1 þ r2 ðh_1 þ h_2 Þ cosðh1 þ h2 Þ wðh_1 þ h_2 Þ sinðh1 þ h2 Þ þ w_ cosðh1 þ h2 Þ
#
ð4Þ
Now, the total kinetic energy T can be written as Z 1 1 1 1 L2 _ T _ 2 R2 R2 q2 dr T ¼ J1 h_21 þ Jh ðh_1 þ h_2 Þ þ Mh L21 h_21 þ 2 2 2 2 0
ð5Þ
where J1 is the moment of inertia of link 1, Jh and Mh are the moment of inertia and mass of hub at O2 and q2 is the unit mass density of link 2. Note that the first term on the right-hand side in Eq. (5) is due to the inertia of link 1, the second and the third terms are due to the inertia and mass of the hub at O2 , respectively. The fourth term is due to the rotation of the flexible link 2. The potential energy can be written as Z 1 L2 2 U¼ EIðw00 ðt; xÞÞ dx ð6Þ 2 0
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where EI is the flexural rigidity of the flexible link 2 and x is a spatial vector along with O2 X2 . A prime donates the derivative with respect to the spatial variable x. However, there exist other energies, such as the potential energy due to shear, potential energy from the centrifugal stiffening, tension–compression energy, and so on. The most important and dominant term in the potential energy is the strain energy described by Eq. (6) in this study. Thus, the dynamic motion equation of the two-link rigid–flexible manipulator system can be derived in terms of the Lagrange–Euler formulation d oL oL ð7Þ ¼ si ði ¼ 1; 2Þ dt oh_i ohi d oL oL ¼0 ð8Þ dt ow_ ow where L is the Lagrangian function and L ¼ T U . si are the generalized torques applied to the system at joint i. Substituting Eqs. (5) and (6) into Eqs. (7) and (8) yields the following dynamic equations: Z L2 1 2 2 3 J1 þ Jh þ Mh L1 þ q2 L1 L2 þ q2 L2 þ q2 w2 ðx; tÞ dx þ q2 L1 L22 cos h2 3 0 Z L2 Z L2 1 3 € 2q2 L1 sin h2 wðx; tÞ dx h1 þ Jh þ q2 L2 þ q2 w2 ðx; tÞ dx 3 0 0 Z L2 1 2 þ q2 L1 L2 cos h2 q2 L1 sin h2 wðx; tÞ dx h€2 2 0 Z L2 1 q2 L1 L22 sin h2 þ q2 L1 cos h2 wðx; tÞ dx h_22 2 0 Z L2 q L1 L2 sin h2 þ 2q L1 cos h2 wðx; tÞ dx h_1 h_2 2
2q2 L1 sin h2 þ q2
2
2
Z 0
Z
0 L2
_ tÞ dx 2q2 wðx;
0 L2
x€ wðx; tÞ dx þ q2 L1 cos h2
Z 0
Z
L2
_ tÞ dx ðh_1 þ h_2 Þ wðx; tÞwðx;
0 L2
€ ðx; tÞ dx ¼ s1 w
ð9Þ
Z L2 Z L2 1 1 w2 ðx; tÞ dx þ q2 L1 L22 cos h2 q2 L1 sin h2 wðx; tÞ dx h€1 Jh þ q2 L32 þ q2 3 2 0 0 Z L2 1 þ Jh þ q2 L32 þ q2 w2 ðx; tÞ dx h€2 3 0 Z L2 Z L2 _ tÞ dxðh_1 þ h_2 Þ þ q2 x€ wðx; tÞ dx þ 2q2 wðx; tÞwðx; 0 0 Z L2 1 2 q L1 L2 sin h2 þ q2 L1 cos h2 wðx; tÞ dx h_21 ¼ s2 ð10Þ þ 2 2 0
476
L. Tian, C. Collins / Mechatronics 14 (2004) 471–490 0000 € ðx; tÞ wðx; tÞðh_1 þ h_2 Þ2 þ L1 h_21 sin h2 þ EI w ¼ 0 ðx þ L1 cos h2 Þh€1 þ xh€2 þ w
ð11Þ with the following boundary conditions: wð0; tÞ ¼ 0
ð12aÞ
o wð0; tÞ ¼ 0 ox o2 wðL2 ; tÞ ¼ 0 ox2 o3 wðL2 ; tÞ ¼ 0 ox3
ð12bÞ ð12cÞ ð12dÞ
and the initial conditions wðx; 0Þ ¼ w0 ðx; 0Þ x 2 ½0; L2 o wðx; 0Þ ¼ w_ 0 ðx; 0Þ ot _ hð0Þ ¼ h_0
x 2 ½0; L2
ð13aÞ ð13bÞ ð13cÞ ð13dÞ
hð0Þ ¼ h0 T
where h ¼ ½h1 ; h2 . Note that Eqs. (12c) and (12d) are due to the absence of bending moment and shearing forces at the tip of the link 2. 2.2. Finite-dimensional approximation The flexible-link manipulator system is a distributed-parameter (continuous) system. Due to actuator/sensor bandwidth limits, it is reasonable to construct the system as a truncated model at lower frequency. To this end, the system can be approximated by a finite-dimensional model, which ignores the high-frequency modes. There exist different techniques to discretize a continuous system. The basic idea is to approximate the distributed parameters by a finite set of trial functions. In our case, the flexure variable wðx; tÞ can be approximated as follows: wðx; tÞ ¼
n X
gi ðxÞqi ðtÞ
ð14Þ
i¼1
where gi ðxÞ can be obtained by solving the eigenfunctions that arise from the linearized problem, and qi ðtÞ is the position vector in joint space. The selection of function gi ðxÞ is rather crucial. The best way to choose gi ðxÞ is to use the eigenfunctions. If so, the expansion of Eq. (14) is known as unconstrained expansion. However, for practical purpose, solving the eigenvalue problem may not be easy, sometimes impossible. Therefore, to discretize the system, a set of functions needs to be chosen. The approximation, known as ‘‘assumed mode’’ method or constrained mode expansion, employs ‘‘a complete set of functions’’. The trial
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functions in this case are derived by fixing the rigid link and setting the external inputs as zero. Therefore, the dynamics described in Eq. (11) can be reduced to the dynamics of an Euler–Bernoulli beam, i.e. 0000
q2 g€ðxÞ þ EIg ðxÞ ¼ 0
ð15Þ
It is well known that the solution of the eigenvalue problem of Eq. (15) is gi ðxÞ ¼ ½coshðbi xÞ cosðbi xÞ
coshðbi L2 Þ þ cosðbi L2 Þ ½sinhðbi xÞ sinðbi xÞ sinhðbi L2 Þ þ sinðbi L2 Þ ð16Þ
and bi satisfy coshðbi L2 Þ cosðbi L2 Þ ¼ 0
ð17Þ
Note that the eigenfunctions given by Eq. (16) are pairwisely orthogonal. If choosing the first n eigenfunctions (mode-shapes) as the assumed modes for the discretized process, the following lumped representation of the system is obtained M X€ þ KX ¼ F ðX ; X_ Þ þ Bu
ð18Þ
where X ¼ ½h1 ; h2 ; q1 ; q2 ; . . . ; qn , M, K, F and B are the inertia, stiffness, vector of nonlinearities due to Coriolis and centrifugal forces, and input matrix, respectively. Details are given in Appendix A.
3. Fuzzy logic controller (FLC) The concept of fuzzy theory was first introduced by Zadeh in 1965 and is used to control dynamic systems that are too complex and/or too ill-defined. Since Mamdam [20] developed the first FLC based on fuzzy set theory, many kinds of FLCs for various applications have been proposed. Generally, an FLC consists of a set of linguistic conditional statements derived from human operators and represent the experts’ knowledge about the controlled system. These statements define a set of control actions by ‘‘if–then’’ rules. The FLC can be considered as a fuzzy reasoning process to mimic the control actions of human operator. Usually, the FLC has four components: fuzzification, knowledge base, inference, and defuzzification. Since inputs to the fuzzy controller are not vague in nature, each input should be first fuzzified. This can be done through associating each input with a set of fuzzy variables. In order to give a fuzzy variable a numerical sense, a membership function is assigned with each variable. The form of a membership function can be discrete or continuous. The range of the membership function is varied from zero to one. Two types of continuous membership functions commonly used in the FLC are exponential and triangle shape. Knowledge base contains information about the boundaries, possible transformations of the domains and the fuzzy sets with their corresponding linguistic terms. It is usually represented by a set of control rules, each of those control rules has
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L. Tian, C. Collins / Mechatronics 14 (2004) 471–490 Knowledge base
e
r
+ -
ec
E Fuzzification
EC
Inference
U
Defuzzification
u
Manipulator system
y
Fig. 2. Diagram of fuzzy logic controller.
‘‘if–then’’ statement. Matching fuzzified inputs with each control rule will activate a set of control actions. Inference represents the processing unit. It determines the corresponding output value from the measured inputs according to the knowledge base. Defuzzification completes the task of determining a crisp output value by taking the information about the control variable provided by the decision logic into account. Finally, if necessary, it carries out a transformation of the output value into the appropriate domain. The most commonly used defuzzification methods are center of gravity and maximum value methods [21,22]. In fuzzy control systems, the performance of the fuzzy controller, which is dependent on the decision of fuzzy control rules, influences that of the controlled system greatly. In order to adjust the control rules conveniently, a weighting factor is introduced to the FLC. It can be expressed as Uf ¼ haE þ ð1 aÞECi
a ¼ ð0; 1Þ
uf ¼ kUf
ð19Þ ð20Þ
where h i is the operator, denoting a minimum integer which has the same sign as its content, but the absolute value is required. E is the fuzzy value of the error of input variable, EC the fuzzy value of the change in error, a is the weighting factor, k is the proportional factor, and Uf and uf are the fuzzy and non-fuzzy outputs of the FLC. Through adjusting the weighting factor the weighted degree to error and change in error can be changed. If the order of the controlled system is low, the weighted value to error should be larger than that to change in error. On the contrary, if the order of the controlled system is high, the weighted value to change in error should be larger than that to error. Fig. 2 shows the diagram of the fuzzy controller.
4. Dynamic recurrent neural network (DRNN) Usually, an NN consists of many interconnected nonlinear processing elements called neurons. The conventional NN is that individual neurons sum their weighted inputs and yield an output through a nonlinear activation function, which is called a static model. Currently, multi-layered neural networks based on the static model are widely used in various applications [23]. As the extension of the static neural net-
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works, DRNNs constructed by the static neurons through adding state feedback were developed [8–11,24,25]. In this paper, a kind of DRNN shown in Fig. 3 is introduced, which includes one hidden layer with h nonlinear neurons and an output layer with n2 linear neurons. The nonlinear neurons serve to capture the nonlinearities of the approximated system (namely, inverse plant dynamics), while the linear neurons are used to allow the approximation in arbitrary range. Fig. 3 shows the structure of the DRNN, which is a three-layer network. The neurons of the hidden layer have the dynamic self-recurrent connection relationship. The mapping relationship of the DRNN can be described as follows: Input units: Hidden units:
Ii ðkÞ ¼ xi ðkÞ ð2Þ Oj ðkÞ ð2Þ
¼
Ij ðkÞ ¼
ði ¼ 1; 2; . . . ; n1Þ
ð2Þ f ðIJ ðkÞÞ
n1 X
ðj ¼ 1; 2; . . . ; hÞ
ð1Þ
wji xi ðkÞ þ
i¼1
Output units:
ð3Þ uðmÞ n ðkÞ ¼ Om ðkÞ ¼
ð21Þ
h X
ð2Þ
ð2Þ
wjn Oj ðk 1Þ
ð22Þ
n¼1 n2 X
ð3Þ
ð2Þ
wmj Oj ðkÞ
ðm ¼ 1; 2; . . . ; n2Þ
ð23Þ
m¼1 ð1Þ ð2Þ ð3Þ T where xi ðkÞ are input units, xðkÞ ¼ ðh1 ; h_1 ; h€1 ; h2 ; h_2 ; h€2 Þ, wji , wjn and wmj are the connection weights from input layer to hidden layer, carried layer to hidden layer, ð2Þ ð2Þ and hidden layer to output layer, respectively, Ij ðkÞ and Oj ðkÞ are the inputs and outputs of the hidden layer at time k, Oð3Þ m ðkÞ are the outputs of the neural network at time k. f ðÞ is the activation function, generally expressed as the sigmoid function f ðxÞ ¼ 1:0=ð1:0 þ ex Þ. n1; h and n2 are the numbers of units of input layer, hidden (carried) layer, and output layer, respectively. The error function is written as n2 1X J¼ e2 ðkÞ ð24Þ 2 m¼1 m ðmÞ
where em ðkÞ ¼ uf ðkÞ uðmÞ n ðkÞ, n2 ¼ 2.
Fig. 3. Architecture of the DRNN.
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Then, the connection weights are updated as follows using the gradient descent method oJ ð3Þ owmj
oJ ð2Þ
owjn
¼ em ðkÞ
¼
n2 X
oem ðkÞ
em ðkÞ
m¼1
¼
ð3Þ owmj
n2 X
¼ em ðkÞ
oem ðkÞ ð2Þ
owjn
"
em ðkÞ
m¼1
oJ ð1Þ
owji
¼
n2 X
em ðkÞ
m¼1
¼
n2 X
ð1Þ
em ðkÞ
m¼1
n2 X
em ðkÞ
#
ð2Þ
¼
n2 X
em ðkÞ
m¼1
oOð3Þ m ðkÞ
ð2Þ
¼ em ðkÞOJ ðkÞ
ð25Þ
oOmð3Þ ðkÞ ð2Þ
owjn
ð2Þ
f 0 ðIj ðkÞÞOj ðk 1Þ
ð2Þ oOJ ðkÞ
owji
ð3Þ owmj
m¼1
oOð3Þ m ðkÞ
oem ðkÞ
"
¼
oOð3Þ m ðkÞ
#
ð2Þ oOJ ðkÞ
ð26Þ
oOmð3Þ ðkÞ ð1Þ
owji
ð2Þ
f 0 ðIj ðkÞÞxi ðkÞ
ð27Þ
oJ wðk þ 1Þ ¼ wðkÞ þ a ow ð2Þ
ð2Þ
ð2Þ
ð28Þ ð1Þ
ð2Þ
ð3Þ
where f 0 ðIj ðkÞÞ ¼ OJ ðkÞ½1 OJ ðkÞ, w are the weights wji , wjn and wmj of the DRNN and a is the learning rate.
5. Local stability and convergence analysis For all dynamic systems, the stability of the control system is an important property that has to be addressed. In our control system, the control signal to the plant is made of two components shown in Fig. 4. One is contributed by the FLC denoted by uf ðtÞ and the other is the DRNN represented by un ðtÞ. The stability of the overall control system is governed by the stability characteristics of the FLC and DRNN, which can be established and analyzed utilizing the energy function of the
Fig. 4. Block diagram of the control system.
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individual controller. It is well known that a dynamic system is stable if its energy decreases monotonically until an equilibrium state is reached. According to [7,9], the energy of a fuzzy relation R is expressed as EðRÞ ¼
n X m 1 X uðxi ; yj Þf ðlR ðxi ; yj ÞÞ n m i¼1 j¼1
ð29Þ
where uðxi ; yj Þ is the function responsible for the position of the fuzzy relation in the Cartesian product, f ðlR ðxi ; yj ÞÞ is the function responsible for the shape and the spread of the fuzzy relation, and n and m are cardinals of the Cartesian product. For further proof, readers can refer to [26]. Usually, the stability of the neural network control system mainly depends on the selection of the learning rate and the initial values of the weights. In order to maintain real-time convergence of the learning rate, Lyapunov theorem is used to define the range of the learning rate, which ensures that the control system is stable during the whole control process. Theorem. if an input series of internal dynamic neural network can be activated in the whole control process subject to X ðkÞ 2 Rn , then learning rate satisfies 0 < a < ð2=r2 Þ where r ¼
ð30Þ
h iT oeðkÞ ð1Þ ð2Þ ð3Þ , e ¼ ½e1 ; e2 T , w ¼ wji ; wjn ; wmj . ow
Then, Eq. (30) can ensure the control system is exponentially convergent in the whole process. Proof. The Lyapunov function is defined as LðkÞ ¼ 12e2 ðkÞ DLðkÞ ¼ Lðk þ 1Þ LðkÞ ¼
ð31Þ 1 2 ½e ðk 2
2
þ 1Þ e ðkÞ
ð32Þ
Learning error rate is described as eðk þ 1Þ ¼ eðkÞ þ DeðkÞ DeðkÞ ¼
oeðkÞ Dw ow
ð33Þ ð34Þ
The updated expression of the weights of the DRNN is written as Dw ¼ a
oJ oeðkÞ ¼ aeðkÞ ow ow
# 2 " 2 1 2 1 oeðkÞ oeðkÞ 2 eðkÞ DLðkÞ ¼ ½e ðk þ 1Þ e ðkÞ ¼ a 2a 2 2 ow ow
ð35Þ ð36Þ
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L. Tian, C. Collins / Mechatronics 14 (2004) 471–490
In order to ensure the convergence of the system, DLðkÞ must satisfy DLðkÞ < 0. Thus, the solution of Eq. (36) can be obtained 0
2 oeðkÞ ow
ð37Þ
2
According to the theorem, the ranges of weights of all layers can be obtained.
h
6. Simulation remarks 6.1. System performance An all-purpose software for the two-link rigid–flexible manipulator system with multi-mode was developed. For the sake of simplification of computing, one mode of vibration is applied to describe the dynamics of the system in this study. The physical parameters of the system are given in Table 1. Fig. 4 shows the structure of the control system. First, the comparison of the performance between the FLC and the proposed neuro-fuzzy controller is conducted. Three groups of inputs (0.7, 1.0 and 1.3 rad) are given to the control system, respectively. The parameters of the two controllers are listed in Table 2 (Set 1 for the FLC and Set 2 for the proposed neuro-fuzzy controller) and the simulation results are illustrated in Figs. 5 and 6, respectively. In Figs. 5 and 6, (a) and (b) represent the outputs of the two joints of the system when it responds to the three groups of step inputs separately, and (c) and (d) are the corresponding steady-state error curves of the two joints. For the sake of clearance of observation, the time coordinate is selected from the 2nd second to the 4th second, the first two seconds are omitted; (e) demonstrates the tip deflection of the second flexible link. From the above figures, the overshoot does not appear for both cases, which is necessary for practical applications. In the meantime, the difference between these two controllers can obviously be seen. The performance of the control system is improved greatly by employing the proposed neuro-fuzzy controller, especially in the tip deflection of the second flexible link. This can be verified from both Table 2 and Figs. 5 and 6. The tip deflection decreases more than 10 times compared with the FLC. Moreover, the steady-state error of the system is also decreased and the absolute steady-state errors of both joints are less than 0.002 rad for all three groups of inputs. Table 1 Parameters of the rigid–flexible manipulator system Physical parameters
Link 1
Link 2
Length (m) Moment of inertia (kg m2 ) Density (kg m2 ) Elastic modulus (N m2 ) Mass (kg)
L1 ¼ 0:61 J1 ¼ 0:32
L2 ¼ 0:52 Jh ¼ 0:02 q2 ¼ 0:17 EI ¼ 1:0 Mh ¼ 0:23
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Table 2 Parameters of the proposed controller and simulation results
Set 1
Input (rad)
Joint
a
k
0.7
1 2 1 2 1 2
0.12 0.11 0.16 0.14 0.20 0.17
2.5 1.5 2.5 1.5 2.5 1.5
1 2 1 2 1 2
0.31 0.33 0.37 0.39 0.42 0.40
0.90 0.09 0.90 0.04 0.90 0.40
1.0 1.3 Set 2
0.7 1.0 1.3
Steady-state error (rad)
Tip deflection (mm)
Max
Min
Max
Min
0.0032 0.0012 0.0028 0.0018 0.0024 0.0016
)0.0030 )0.0017 )0.0026 )0.0017 )0.0022 )0.0020
42.7
)43.1
54.9
)53.8
56.2
)57.5
0.0006 0.0008 0.0005 0.0006 0.0013 0.0015
)0.0006 )0.0008 )0.0005 )0.0007 )0.0010 )0.0018
4.00
)3.94
5.71
)5.83
7.55
)7.87
a is the weighting factor expressed in Eq. (19) and k is the proportional factor expressed in Eq. (20).
Next, the performance of the DRNN is evaluated. Two sets of learning rates (0.05 and 0.1) are given to the control system separately, while the moment factor is kept unchanged. The results are demonstrated in Table 3 (Set 1 for k ¼ 0:05, Set 2 for k ¼ 0:1) and Fig. 7, in which (a) and (c) are the responses of the two joints to the two cases, respectively, and (b) and (d) are the tip deflections of the second flexible link. From both Table 3 and Fig. 7, it can be seen that when the learning rate is increased the performance of the control system becomes worse. The absolute steady-state errors are increased more than one time for both joints. The tip deflections are also increased. Therefore, a conclusion can be drawn that it is favorable to improve the performance of the control system by choosing a smaller learning rate. This is in agreement with the analysis in Section 5. 6.2. System robustness The robustness of the control system is also investigated under different conditions. Fig. 8 demonstrates the results when the moment of inertia of the link 1 is increased to 0.64 kg m2 , twice as the previous case. Based on the data from Fig. 8(a) and (b), it can be seen that the steady-state errors for both joints are less than 0.001 rad. The deflection of the second flexible link, shown in Fig. 8(c), is within ±0.0035 m. It is obvious that the performance of the control system is hardly affected by the increased moment of inertia of link 1. Fig. 9 illustrates the results when the mass of hub fixed at the end of the link 1 is increased to 0.46 kg, twice as the previous case. The absolute steady error for link 1 is not more than 0.001 rad. While for link 2, the absolute steady-state error becomes smaller gradually with the increasing of time, the tip deflection of the second flexible link 2 increases a little, less than 0.0045 m.
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Fig. 5. Step responses of robot system with the FLC.
Fig. 10 shows the results when the density of link 2 is increased to 0.34 kg/m, twice as the previous case. It can be seen that the steady-state errors for both links can be controlled within ±0.002 rad and the tip deflection is within ±0.006 m. From the three cases (Figs. 8–10), we can conclude that when the parameters of the system change in the control process, the performance of the control system is hardly affected. Thus, this can be used to explain that the proposed controller has better robustness in resisting against the changes of the parameters of the control system.
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Fig. 6. Step responses of robot system with the proposed neuro-fuzzy controller.
Table 3 Parameters of the proposed controller and simulation results Joint
a
k
Steady-state error (rad)
Tip deflection (mm)
Max
Min
Max
Min
Set 1
1 2
0.39 0.27
0.9 0.08
0.56 1.3
)0.56 )1.2
3.7
)3.6
Set 2
1 2
0.38 0.25
0.9 0.6
1.4 2.8
)1.3 )2.9
4.4
)4.2
a is the weighting factor expressed in Eq. (19) and k is the proportional factor expressed in Eq. (20).
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Fig. 7. Step responses of robot system under different learning rates.
Fig. 8. Step responses of robot system under different inertia.
L. Tian, C. Collins / Mechatronics 14 (2004) 471–490
Fig. 9. Step responses of robot system under different masses.
Fig. 10. Step responses of robot system under different densities.
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7. Conclusion A complete description for the dynamic modeling of a two-link rigid–flexible manipulator system is given. The finite-dimension approximation is discussed. A neuro-fuzzy controller for the system is designed, which includes two parts: the FLC serves as feedback controller; the DRNN is employed as a feedforward controller to enhance the performance of the control system. Furthermore, the analysis of the stability of the control system is also carried out. Simulations on the comparison between the FLC and the proposed neuro-fuzzy controller are conducted. Results show that the performance of the control system is improved greatly with the proposed controller, greatly decreasing the tip deflection of the second flexible link and also enhancing the steady-state accuracy for both joints. Simulation on the performance of the DRNN is carried out, demonstrating that it is favorable for the improvement of the performance of the control system by choosing a smaller learning rate. Finally, the robustness of the control system is investigated. The control system performs well in resisting against the parameter changes of the system, such as moment of inertia of link 1, mass of hub, and density of the link 2.
Acknowledgements This research was supported by Natural Science Foundation of South China University of Technology, grant no. B8-106-016 and Doctor Start Foundation of Guangdong, grant no. B9-108-061. The authors would like to thank the anonymous reviewers for their precious suggestions for this paper and Tricia Butera for the help on this paper.
Appendix A A finite-dimensional approximation of the dynamic flexible-link manipulator 2 m11 m12 m13 6 m m m23 21 22 6 6 q2 S1 ðxÞ þ q2 L1 cos h2 S1 ð1Þ q2 S1 ðxÞ q2 L2 M ¼6 6 .. .. 4 . 0 . q2 Sn ðxÞ þ q2 L1 cos h2 Sn ð1Þ q2 Sn ðxÞ 3 2 0 0 0 0 60 0 0 0 7 7 6 6 0 0 EI L2 b1 0 7 K¼6 7 7 6 .. .. .. 5 4. . 0 . 0
0
EI L2 bn
equation of the two rigid– ..
.
3 m1ð2þnÞ m2ð2þnÞ 7 7 0 7 7 7 5 q2 L2
ðA:1Þ
ðA:2Þ
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489
3
2
f1 7 6 f2 7 6 6 q2 L2 ðh_1 þ h_2 Þ2 q1 q2 L1 sin h2 h_21 S1 ð1Þ 7 F ¼6 7 7 6 .. 5 4 . 2 q2 L2 ðh_1 þ h_2 Þ qn q2 L1 sin h2 h_21 Sn ð1Þ B ¼ ½1; 1; 0; . . . ; 0
ðA:3Þ
T
ðA:4Þ
where, m11 , m12 , m22 , f1 and f2 are as follows: n X 1 m11 ¼ J1 þ Jh þ Mh L21 þ q2 L21 L2 þ q2 L32 þ q2 L2 q2j ðtÞ 3 j¼1
þ q2 L1 L22 cos h2 2q2 L1 sin h2
n X
Sj ð1Þqj ðtÞ
ðA:5Þ
j¼1 n X 1 1 m12 ¼ Jh þ q2 L32 þ q2 L2 q2j ðtÞ þ q2 L1 L22 cos h2 3 2 j¼1
q2 L1 sin h2
n X
Sj ð1Þqj ðtÞ
ðA:6Þ
j¼1 n X 1 m22 ¼ Jh þ q2 L32 þ q2 L2 q2j ðtÞ 3 j¼1 # " n X 1 q L1 L22 sin h2 þ q2 L1 cos h2 f1 ¼ qj ðtÞSJ ð1Þ h_22 2 2 j¼1 # " n X 2 þ q L1 L sin h2 þ q L1 cos h2 qj ðtÞSj ð1Þ h_1 h_2 2
ðA:7Þ
2
2
j¼1
" þ 2q2 L1 sin h2
n X
Sj ð1Þq_ j ðtÞ 2q3 L2
j¼1
n X
#
qj ðtÞq_ j ðtÞ ðh_1 þ h_2 Þ
ðA:8Þ
j¼1
# n X 1 2 f2 ¼ q2 L1 L2 sin h2 þ q2 L1 cos h2 Sj ð1Þqj ðtÞ h_21 2 j¼1 "
2q2 L2 ðh_1 þ h_2 Þ
n X
qj ðtÞq_ j ðtÞ
j¼1
where M ¼ M T S2þj ð1Þ ¼ S2þj ðxÞ ¼
Z Z
L2
gj ðxÞ dx ðj ¼ 1; 2; . . . ; nÞ
0
0
L2
xgj ðxÞ dx ðj ¼ 1; 2; . . . ; nÞ
ðA:9Þ
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References [1] Sun FC, Sun ZQ, Woo P-Y. Neural network-based adaptive controller design of robotic manipulators with an observer. IEEE Trans Neural Networks 2001;12(1):54–67. [2] Behera L, Gopal M, Chandhury S. Trajectory tracking of robot manipulator using Gaussian networks. Robot Autonom Syst 1994;13:107–15. [3] Kim YH, Lewis FI. Neural network output feedback control of robot manipulators. IEEE Trans Robot Automat 1999;15(2):301–9. [4] Jung S, Hsia TC. Robust neural force control scheme under uncertainties in robot dynamics and unknown environment. IEEE Trans Ind Electron 2000;47(2):403–12. [5] Benati M, Morro A. Formulation of equations of motion for a chain of flexible links using Hamilton’s principle. Trans ASME, J Dynam Syst, Measure, Control 1994;116:81–8. [6] Homaifar A, Blkdash M, Gopalan V. Design using genetic algorithm of hierarchical hybrid fuzzyPID controllers of two-link robotic arms. J Robot Syst 1997;14(6):449–63. [7] Ciliz MK, Isik C. On-line learning control of manipulator based on artificial neural network methods. Robotica 1997;13:293–304. [8] Yan LL, Li CJ. Robot learning control based on recurrent neural network inverse model. J Robot Syst 1997;14(3):199–212. [9] Arcinegas JI, Eltimsahy AH, Cios KJ. Neural-networks-based adaptive control of flexible robotic arms. Neurocomputing 1997;17:141–57. [10] Chien CJ, Fu LC. A neural network based learning controller for robot manipulators. In: Proceedings of the 39th IEEE Conference on Decision and Control; 2000. p. 1748–53. [11] Karakasoglu A, Sundareshan MK. A recurrent neural network-based adaptive variable structure model-following control of robotic manipulator. Automatica 1995;31(10):1495–507. [12] Hu SH, Ang Jr. MH, Krishnan H. Neural network controller for constrained robot manipulators. In: Proceedings of the 2000 IEEE International Conference on Robotics and Automation, San Francisco, USA; May 2000. [13] Gao Y, Er MJ, Yang S. Adaptive control of robot manipulation using fuzzy neural network. IEEE Trans Ind Electron 2001;48(6):1274–8. [14] Gao Y, Er MJ, Leithead WE, Leith DJ. On-line adaptive control of robot manipulators using dynamic fuzzy neural networks. In: Proceedings of the American Control Conference, Arlington, USA; June 2001. [15] Sun FC, Sun ZQ, Zhang RJ, Chen YB. Discrete-time tracking control of robotic manipulators based on dynamic inversion using dynamic neural networks. In: Proceedings of the 15th IEEE International Symposium on Intelligent Control, Patras, Greece; 17–9 July 2000. [16] Zhang Y, Wang J, Xu Y. A dual neural network for bi-criteria kinematic control of redundant manipulators. IEEE Trans Robot Automat 2002;18(6):923–31. [17] Zhang Y, Wang J. A dual neural network for constrained torque optimization of kinematically redundant manipulators. IEEE Trans Syst, Man Cybernet B 2002;32(5):654–62. [18] Yigit AS. On the stability of PD control for a two-link rigid–flexible manipulator. Trans ASME, J Dynam Syst, Meas Control 1994;116:208–15. [19] Xi F, Fenton RG. On flexible link manipulators: modelling and analysis using the algebra of rotations. Robotica 1994;12:371–81. [20] Mumdani EH. Application of fuzzy algorithm for control of dynamic plant. Proc IEE 1974;121:1585–8. [21] Lin Y-J, Lee T-S. An investigation of fuzzy logic control of flexible robotics. Robotica 1993;11:363–71. [22] Sugeno M. An introduction survey of fuzzy control. Inform Sci 1985;36:59–83. [23] Jung S, Hsia TC. A study of neural network control of robot manipulators. Robotica 1996;14:7–15. [24] Tang WS, Wang J. A recurrent neural network for minimum infinity-norm kinematic control of redundant manipulators with an improved problem formulation and reduced architecture complexity. IEEE Trans Syst, Man, Cybernet B 2001;31(1):98–105. [25] Jung S, Hsia TC. A study of neural network control of robot manipulators. Robotica 1996;14:7–15. [26] Kiszku JB, Gupta MM, Nikutorus PN. Energetic stability of fuzzy dynamic systems. IEEE Trans SMC 1995;15:786–92.
A dynamic recurrent neural network-based controller for a rigid–flexible manipulator system Lianfang Tian *, Curtis Collins Department of Mechanical Engineering, University of California at Riverside, Riverside, CA 92521, USA
Abstract This paper deals with the tracking control problem of a manipulator system with unknown and changing dynamics. In this study, a fuzzy logic controller (FLC) in the feedback configuration and an efficient dynamic recurrent neural network (DRNN) in the feedforward configuration are proposed. The DRNN, which possesses the ability of approaching arbitrary nonlinear function, is applied to approximate the inverse dynamics of the robotic manipulator system. Based on the outputs of the FLC, parameter updating equations are derived for the adaptive DRNN model. The analysis of the stability of the system is also carried out. Finally, extensive simulations are conducted under different conditions. Results demonstrate the remarkable performance of the proposed controller. It can successfully identify the inverse dynamics of the flexible manipulator system and perform accurate tracking for a given trajectory. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Robotic system; Dynamic recurrent neural network; Fuzzy logic control; Adaptive control
1. Introduction It is well known that the demand for increased productivity by robots can be partly met by the use of lighter robots operating at high speed and consuming less energy, which may lead to a reduction in the stiffness of the manipulator structure. This would result in an increase in robot deflection and poor performance due to the effect of mechanical vibration in the links and make it hard to control. Thus,
*
Corresponding author. E-mail address: [email protected] (L. Tian).
0957-4158/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2003.10.002
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trajectory following control of a robotic manipulator system has been an important research area in the last decade [1–3]. On the other hand, a robotic system is a highly nonlinear and heavily coupled mechanical system. The mathematical model of such a system often consists of a set of linear or nonlinear differential/difference equations derived using some form of approximation and simplification. However, when an exact robot model is difficult to obtain due to uncertainty and complexity, the traditional model-based control techniques will break down [4,5]. From the aforementioned, it can be summarized that a control method for the robotic manipulator system must be able to effectively deal with the adverse effects of transient vibration and complexity of the system. It is apparent that the conventional model-based control cannot meet the requirement of system performance. Because of this reason, much attention has recently been directed towards intelligent control methods such as fuzzy logic control and neural networks [6,7]. Those techniques have the following advantages: a fuzzy logic controller (FLC), based on fuzzy set theory, can provide a means of converting a linguistic control strategy into control action and thus offer a high level computation. In addition, it also shows more excellent robustness over PID. A neural network can provide low-level computation and demonstrate many other salient features such as learning, default-toleration, parallelism and generalization [8–11]. Therefore, those novel control techniques have drawn much attention to many researchers and have been practically applied in manipulator control systems. Hu et al. [12] proposed a neural network (NN) controller for a constrained robot manipulator. The NN is used as a feed-forward loop to adaptively compensate for the uncertainties in the robot dynamics. The weights of the NN are trained on-line. Gao et al. [13,14] introduced a robust adaptive fuzzy neural network (AFNC) for the motion control of a multi-link robot manipulator system. The proposed AFNC is employed to compensate the modeling errors and handle external disturbance. Sun et al. [15] developed a discrete time tracking control approach based on dynamic neural networks (DNN) for robotic manipulators. The inverse dynamics established by the DNN and the NN variable structure control are used for the design of an adaptive tracking controller. Zhang et al. [16,17] presented a one-layer dual neural network model that provides a new parallel distributed computational approach to online kinematic control of physically constrained redundant manipulators, and the dynamic equation of the dual neural network is piecewise linear and does not contain any high-order nonlinear term, and thus, the architecture is very simple. Moreover, it can guarantee fast convergence due to the exponential convergence. In this paper, a novel FLC, a learning controller and a feedforward controller are designed for a two-link rigid–flexible manipulator system. In the feedback loop, an FLC is used to provide control signals for the manipulator system. In the learning controller, a dynamic recurrent neural network (DRNN) contains a state feedback and provides more computational advantages than a back-propagation neural network and models the inverse dynamics of the manipulator system. The weights of the DRNN are updated on-line based on the outputs of the FLC during the control process. It provides the manipulator system the ability to adapt to the changes of
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uncertainties. A feedforward controller, which is a complete copy of the learning controller, is used to generate the joint torques and enhance the performance of the control system. The remarkable characteristics of the proposed controller include: (1) a dynamic multi-layer neural network is instead of the back-propagation neural network. It has the memory ability. (2) the weights of the network are trained on-line using the realtime data from the actual manipulator movement instead of off-line training. The model is first trained for the identification of the dynamics of the robotic system and then is placed in the feedforward configuration to be used to generate control signals. The layout of this paper is organized as follows: Section 2 gives the dynamic modeling of the two-link flexible manipulator system. Section 3 presents the fuzzy logic controller. Section 4 describes the DRNN. The analysis of the stability of the control system is carried out in Section 5. Simulations are conducted in Section 6. Finally, conclusions are drawn in Section 7.
2. Dynamic modeling of the flexible robot system 2.1. The robot arm dynamics The system considered here consists of two links connected with a revolute joint moving in a horizontal plane as shown in Fig. 1. The first link is assumed to be rigid, while the second one is flexible, which is composed of a slender flexible beam cantilevered onto a rigid rotating hub. The longitudinal deformation of the second link is neglected. It is assumed that the second link can be bent freely in the horizontal plane but is stiff in the vertical bending and torsion. Thus, the Euler–Bernoulli beam theory is sufficient to describe the flexural motion of the flexible link. The Lagrange’s
Fig. 1. The schematic diagram of a two-link rigid–flexible manipulator system.
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L. Tian, C. Collins / Mechatronics 14 (2004) 471–490
equation and model expansion method can be utilized to develop the dynamic modeling of the robotic manipulator system [18,19]. As shown in Fig. 1, X0 O0 Y0 represents the inertial coordinate frame, X1 O1 Y1 and X2 O2 Y2 are the moving coordinate frames with origin at the hubs of links 1 and 2, respectively. From Fig. 1, it shows that h1 and h2 are the revolving angles of the two links with respect to their frames, respectively. R1 and R2 denote position vectors of an arbitrary point on the links 1 and 2 relative to the inertial frame, respectively. wðx; tÞ is the transverse elastic displacement. Note that the axial deformations are not considered in this study, because the flexural motion of the beams does not induce significant axial vibrations. Two pairs of orthogonal unit vectors (i1 ; j1 ) and (i2 ; j2 ), which are fixed at the moving coordinate frames of the links 1 and 2, are shown in Fig. 1. For links 1 and 2, the coordinate of the general points R1 and R2 can be expressed as follows:
x1 r1 cos h1 R1 ¼ ¼ y1 r1 sin h1 x L1 cos h1 þ r2 cosðh1 þ h2 Þ w sinðh1 þ h2 Þ R2 ¼ 2 ¼ L1 sin h1 þ r2 sinðh1 þ h2 Þ þ w cosðh1 þ h2 Þ y2
ð1Þ ð2Þ
where r1 and r2 are the spatial coordinates relative to X1 O1 Y1 and X2 O2 Y2 coordinate frames, respectively. L1 and L2 are the lengths of links 1 and 2, respectively. The time derivatives of R1 and R2 are: _ _R1 ¼ x_ 1 ¼ r1 h1 sin h1 ð3Þ y_ 2 r2 h_1 cos h1 "
R_ 2 ¼ " ¼
x_ 2
#
y_ 2 L1 h_1 sin h1 r2 ðh_1 þ h_2 Þ sinðh1 þ h2 Þ wðh_1 þ h_2 Þ cosðh1 þ h2 Þ w_ sinðh1 þ h2 Þ L1 h_1 cos h1 þ r2 ðh_1 þ h_2 Þ cosðh1 þ h2 Þ wðh_1 þ h_2 Þ sinðh1 þ h2 Þ þ w_ cosðh1 þ h2 Þ
#
ð4Þ
Now, the total kinetic energy T can be written as Z 1 1 1 1 L2 _ T _ 2 R2 R2 q2 dr T ¼ J1 h_21 þ Jh ðh_1 þ h_2 Þ þ Mh L21 h_21 þ 2 2 2 2 0
ð5Þ
where J1 is the moment of inertia of link 1, Jh and Mh are the moment of inertia and mass of hub at O2 and q2 is the unit mass density of link 2. Note that the first term on the right-hand side in Eq. (5) is due to the inertia of link 1, the second and the third terms are due to the inertia and mass of the hub at O2 , respectively. The fourth term is due to the rotation of the flexible link 2. The potential energy can be written as Z 1 L2 2 U¼ EIðw00 ðt; xÞÞ dx ð6Þ 2 0
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475
where EI is the flexural rigidity of the flexible link 2 and x is a spatial vector along with O2 X2 . A prime donates the derivative with respect to the spatial variable x. However, there exist other energies, such as the potential energy due to shear, potential energy from the centrifugal stiffening, tension–compression energy, and so on. The most important and dominant term in the potential energy is the strain energy described by Eq. (6) in this study. Thus, the dynamic motion equation of the two-link rigid–flexible manipulator system can be derived in terms of the Lagrange–Euler formulation d oL oL ð7Þ ¼ si ði ¼ 1; 2Þ dt oh_i ohi d oL oL ¼0 ð8Þ dt ow_ ow where L is the Lagrangian function and L ¼ T U . si are the generalized torques applied to the system at joint i. Substituting Eqs. (5) and (6) into Eqs. (7) and (8) yields the following dynamic equations: Z L2 1 2 2 3 J1 þ Jh þ Mh L1 þ q2 L1 L2 þ q2 L2 þ q2 w2 ðx; tÞ dx þ q2 L1 L22 cos h2 3 0 Z L2 Z L2 1 3 € 2q2 L1 sin h2 wðx; tÞ dx h1 þ Jh þ q2 L2 þ q2 w2 ðx; tÞ dx 3 0 0 Z L2 1 2 þ q2 L1 L2 cos h2 q2 L1 sin h2 wðx; tÞ dx h€2 2 0 Z L2 1 q2 L1 L22 sin h2 þ q2 L1 cos h2 wðx; tÞ dx h_22 2 0 Z L2 q L1 L2 sin h2 þ 2q L1 cos h2 wðx; tÞ dx h_1 h_2 2
2q2 L1 sin h2 þ q2
2
2
Z 0
Z
0 L2
_ tÞ dx 2q2 wðx;
0 L2
x€ wðx; tÞ dx þ q2 L1 cos h2
Z 0
Z
L2
_ tÞ dx ðh_1 þ h_2 Þ wðx; tÞwðx;
0 L2
€ ðx; tÞ dx ¼ s1 w
ð9Þ
Z L2 Z L2 1 1 w2 ðx; tÞ dx þ q2 L1 L22 cos h2 q2 L1 sin h2 wðx; tÞ dx h€1 Jh þ q2 L32 þ q2 3 2 0 0 Z L2 1 þ Jh þ q2 L32 þ q2 w2 ðx; tÞ dx h€2 3 0 Z L2 Z L2 _ tÞ dxðh_1 þ h_2 Þ þ q2 x€ wðx; tÞ dx þ 2q2 wðx; tÞwðx; 0 0 Z L2 1 2 q L1 L2 sin h2 þ q2 L1 cos h2 wðx; tÞ dx h_21 ¼ s2 ð10Þ þ 2 2 0
476
L. Tian, C. Collins / Mechatronics 14 (2004) 471–490 0000 € ðx; tÞ wðx; tÞðh_1 þ h_2 Þ2 þ L1 h_21 sin h2 þ EI w ¼ 0 ðx þ L1 cos h2 Þh€1 þ xh€2 þ w
ð11Þ with the following boundary conditions: wð0; tÞ ¼ 0
ð12aÞ
o wð0; tÞ ¼ 0 ox o2 wðL2 ; tÞ ¼ 0 ox2 o3 wðL2 ; tÞ ¼ 0 ox3
ð12bÞ ð12cÞ ð12dÞ
and the initial conditions wðx; 0Þ ¼ w0 ðx; 0Þ x 2 ½0; L2 o wðx; 0Þ ¼ w_ 0 ðx; 0Þ ot _ hð0Þ ¼ h_0
x 2 ½0; L2
ð13aÞ ð13bÞ ð13cÞ ð13dÞ
hð0Þ ¼ h0 T
where h ¼ ½h1 ; h2 . Note that Eqs. (12c) and (12d) are due to the absence of bending moment and shearing forces at the tip of the link 2. 2.2. Finite-dimensional approximation The flexible-link manipulator system is a distributed-parameter (continuous) system. Due to actuator/sensor bandwidth limits, it is reasonable to construct the system as a truncated model at lower frequency. To this end, the system can be approximated by a finite-dimensional model, which ignores the high-frequency modes. There exist different techniques to discretize a continuous system. The basic idea is to approximate the distributed parameters by a finite set of trial functions. In our case, the flexure variable wðx; tÞ can be approximated as follows: wðx; tÞ ¼
n X
gi ðxÞqi ðtÞ
ð14Þ
i¼1
where gi ðxÞ can be obtained by solving the eigenfunctions that arise from the linearized problem, and qi ðtÞ is the position vector in joint space. The selection of function gi ðxÞ is rather crucial. The best way to choose gi ðxÞ is to use the eigenfunctions. If so, the expansion of Eq. (14) is known as unconstrained expansion. However, for practical purpose, solving the eigenvalue problem may not be easy, sometimes impossible. Therefore, to discretize the system, a set of functions needs to be chosen. The approximation, known as ‘‘assumed mode’’ method or constrained mode expansion, employs ‘‘a complete set of functions’’. The trial
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functions in this case are derived by fixing the rigid link and setting the external inputs as zero. Therefore, the dynamics described in Eq. (11) can be reduced to the dynamics of an Euler–Bernoulli beam, i.e. 0000
q2 g€ðxÞ þ EIg ðxÞ ¼ 0
ð15Þ
It is well known that the solution of the eigenvalue problem of Eq. (15) is gi ðxÞ ¼ ½coshðbi xÞ cosðbi xÞ
coshðbi L2 Þ þ cosðbi L2 Þ ½sinhðbi xÞ sinðbi xÞ sinhðbi L2 Þ þ sinðbi L2 Þ ð16Þ
and bi satisfy coshðbi L2 Þ cosðbi L2 Þ ¼ 0
ð17Þ
Note that the eigenfunctions given by Eq. (16) are pairwisely orthogonal. If choosing the first n eigenfunctions (mode-shapes) as the assumed modes for the discretized process, the following lumped representation of the system is obtained M X€ þ KX ¼ F ðX ; X_ Þ þ Bu
ð18Þ
where X ¼ ½h1 ; h2 ; q1 ; q2 ; . . . ; qn , M, K, F and B are the inertia, stiffness, vector of nonlinearities due to Coriolis and centrifugal forces, and input matrix, respectively. Details are given in Appendix A.
3. Fuzzy logic controller (FLC) The concept of fuzzy theory was first introduced by Zadeh in 1965 and is used to control dynamic systems that are too complex and/or too ill-defined. Since Mamdam [20] developed the first FLC based on fuzzy set theory, many kinds of FLCs for various applications have been proposed. Generally, an FLC consists of a set of linguistic conditional statements derived from human operators and represent the experts’ knowledge about the controlled system. These statements define a set of control actions by ‘‘if–then’’ rules. The FLC can be considered as a fuzzy reasoning process to mimic the control actions of human operator. Usually, the FLC has four components: fuzzification, knowledge base, inference, and defuzzification. Since inputs to the fuzzy controller are not vague in nature, each input should be first fuzzified. This can be done through associating each input with a set of fuzzy variables. In order to give a fuzzy variable a numerical sense, a membership function is assigned with each variable. The form of a membership function can be discrete or continuous. The range of the membership function is varied from zero to one. Two types of continuous membership functions commonly used in the FLC are exponential and triangle shape. Knowledge base contains information about the boundaries, possible transformations of the domains and the fuzzy sets with their corresponding linguistic terms. It is usually represented by a set of control rules, each of those control rules has
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L. Tian, C. Collins / Mechatronics 14 (2004) 471–490 Knowledge base
e
r
+ -
ec
E Fuzzification
EC
Inference
U
Defuzzification
u
Manipulator system
y
Fig. 2. Diagram of fuzzy logic controller.
‘‘if–then’’ statement. Matching fuzzified inputs with each control rule will activate a set of control actions. Inference represents the processing unit. It determines the corresponding output value from the measured inputs according to the knowledge base. Defuzzification completes the task of determining a crisp output value by taking the information about the control variable provided by the decision logic into account. Finally, if necessary, it carries out a transformation of the output value into the appropriate domain. The most commonly used defuzzification methods are center of gravity and maximum value methods [21,22]. In fuzzy control systems, the performance of the fuzzy controller, which is dependent on the decision of fuzzy control rules, influences that of the controlled system greatly. In order to adjust the control rules conveniently, a weighting factor is introduced to the FLC. It can be expressed as Uf ¼ haE þ ð1 aÞECi
a ¼ ð0; 1Þ
uf ¼ kUf
ð19Þ ð20Þ
where h i is the operator, denoting a minimum integer which has the same sign as its content, but the absolute value is required. E is the fuzzy value of the error of input variable, EC the fuzzy value of the change in error, a is the weighting factor, k is the proportional factor, and Uf and uf are the fuzzy and non-fuzzy outputs of the FLC. Through adjusting the weighting factor the weighted degree to error and change in error can be changed. If the order of the controlled system is low, the weighted value to error should be larger than that to change in error. On the contrary, if the order of the controlled system is high, the weighted value to change in error should be larger than that to error. Fig. 2 shows the diagram of the fuzzy controller.
4. Dynamic recurrent neural network (DRNN) Usually, an NN consists of many interconnected nonlinear processing elements called neurons. The conventional NN is that individual neurons sum their weighted inputs and yield an output through a nonlinear activation function, which is called a static model. Currently, multi-layered neural networks based on the static model are widely used in various applications [23]. As the extension of the static neural net-
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works, DRNNs constructed by the static neurons through adding state feedback were developed [8–11,24,25]. In this paper, a kind of DRNN shown in Fig. 3 is introduced, which includes one hidden layer with h nonlinear neurons and an output layer with n2 linear neurons. The nonlinear neurons serve to capture the nonlinearities of the approximated system (namely, inverse plant dynamics), while the linear neurons are used to allow the approximation in arbitrary range. Fig. 3 shows the structure of the DRNN, which is a three-layer network. The neurons of the hidden layer have the dynamic self-recurrent connection relationship. The mapping relationship of the DRNN can be described as follows: Input units: Hidden units:
Ii ðkÞ ¼ xi ðkÞ ð2Þ Oj ðkÞ ð2Þ
¼
Ij ðkÞ ¼
ði ¼ 1; 2; . . . ; n1Þ
ð2Þ f ðIJ ðkÞÞ
n1 X
ðj ¼ 1; 2; . . . ; hÞ
ð1Þ
wji xi ðkÞ þ
i¼1
Output units:
ð3Þ uðmÞ n ðkÞ ¼ Om ðkÞ ¼
ð21Þ
h X
ð2Þ
ð2Þ
wjn Oj ðk 1Þ
ð22Þ
n¼1 n2 X
ð3Þ
ð2Þ
wmj Oj ðkÞ
ðm ¼ 1; 2; . . . ; n2Þ
ð23Þ
m¼1 ð1Þ ð2Þ ð3Þ T where xi ðkÞ are input units, xðkÞ ¼ ðh1 ; h_1 ; h€1 ; h2 ; h_2 ; h€2 Þ, wji , wjn and wmj are the connection weights from input layer to hidden layer, carried layer to hidden layer, ð2Þ ð2Þ and hidden layer to output layer, respectively, Ij ðkÞ and Oj ðkÞ are the inputs and outputs of the hidden layer at time k, Oð3Þ m ðkÞ are the outputs of the neural network at time k. f ðÞ is the activation function, generally expressed as the sigmoid function f ðxÞ ¼ 1:0=ð1:0 þ ex Þ. n1; h and n2 are the numbers of units of input layer, hidden (carried) layer, and output layer, respectively. The error function is written as n2 1X J¼ e2 ðkÞ ð24Þ 2 m¼1 m ðmÞ
where em ðkÞ ¼ uf ðkÞ uðmÞ n ðkÞ, n2 ¼ 2.
Fig. 3. Architecture of the DRNN.
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Then, the connection weights are updated as follows using the gradient descent method oJ ð3Þ owmj
oJ ð2Þ
owjn
¼ em ðkÞ
¼
n2 X
oem ðkÞ
em ðkÞ
m¼1
¼
ð3Þ owmj
n2 X
¼ em ðkÞ
oem ðkÞ ð2Þ
owjn
"
em ðkÞ
m¼1
oJ ð1Þ
owji
¼
n2 X
em ðkÞ
m¼1
¼
n2 X
ð1Þ
em ðkÞ
m¼1
n2 X
em ðkÞ
#
ð2Þ
¼
n2 X
em ðkÞ
m¼1
oOð3Þ m ðkÞ
ð2Þ
¼ em ðkÞOJ ðkÞ
ð25Þ
oOmð3Þ ðkÞ ð2Þ
owjn
ð2Þ
f 0 ðIj ðkÞÞOj ðk 1Þ
ð2Þ oOJ ðkÞ
owji
ð3Þ owmj
m¼1
oOð3Þ m ðkÞ
oem ðkÞ
"
¼
oOð3Þ m ðkÞ
#
ð2Þ oOJ ðkÞ
ð26Þ
oOmð3Þ ðkÞ ð1Þ
owji
ð2Þ
f 0 ðIj ðkÞÞxi ðkÞ
ð27Þ
oJ wðk þ 1Þ ¼ wðkÞ þ a ow ð2Þ
ð2Þ
ð2Þ
ð28Þ ð1Þ
ð2Þ
ð3Þ
where f 0 ðIj ðkÞÞ ¼ OJ ðkÞ½1 OJ ðkÞ, w are the weights wji , wjn and wmj of the DRNN and a is the learning rate.
5. Local stability and convergence analysis For all dynamic systems, the stability of the control system is an important property that has to be addressed. In our control system, the control signal to the plant is made of two components shown in Fig. 4. One is contributed by the FLC denoted by uf ðtÞ and the other is the DRNN represented by un ðtÞ. The stability of the overall control system is governed by the stability characteristics of the FLC and DRNN, which can be established and analyzed utilizing the energy function of the
Fig. 4. Block diagram of the control system.
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individual controller. It is well known that a dynamic system is stable if its energy decreases monotonically until an equilibrium state is reached. According to [7,9], the energy of a fuzzy relation R is expressed as EðRÞ ¼
n X m 1 X uðxi ; yj Þf ðlR ðxi ; yj ÞÞ n m i¼1 j¼1
ð29Þ
where uðxi ; yj Þ is the function responsible for the position of the fuzzy relation in the Cartesian product, f ðlR ðxi ; yj ÞÞ is the function responsible for the shape and the spread of the fuzzy relation, and n and m are cardinals of the Cartesian product. For further proof, readers can refer to [26]. Usually, the stability of the neural network control system mainly depends on the selection of the learning rate and the initial values of the weights. In order to maintain real-time convergence of the learning rate, Lyapunov theorem is used to define the range of the learning rate, which ensures that the control system is stable during the whole control process. Theorem. if an input series of internal dynamic neural network can be activated in the whole control process subject to X ðkÞ 2 Rn , then learning rate satisfies 0 < a < ð2=r2 Þ where r ¼
ð30Þ
h iT oeðkÞ ð1Þ ð2Þ ð3Þ , e ¼ ½e1 ; e2 T , w ¼ wji ; wjn ; wmj . ow
Then, Eq. (30) can ensure the control system is exponentially convergent in the whole process. Proof. The Lyapunov function is defined as LðkÞ ¼ 12e2 ðkÞ DLðkÞ ¼ Lðk þ 1Þ LðkÞ ¼
ð31Þ 1 2 ½e ðk 2
2
þ 1Þ e ðkÞ
ð32Þ
Learning error rate is described as eðk þ 1Þ ¼ eðkÞ þ DeðkÞ DeðkÞ ¼
oeðkÞ Dw ow
ð33Þ ð34Þ
The updated expression of the weights of the DRNN is written as Dw ¼ a
oJ oeðkÞ ¼ aeðkÞ ow ow
# 2 " 2 1 2 1 oeðkÞ oeðkÞ 2 eðkÞ DLðkÞ ¼ ½e ðk þ 1Þ e ðkÞ ¼ a 2a 2 2 ow ow
ð35Þ ð36Þ
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In order to ensure the convergence of the system, DLðkÞ must satisfy DLðkÞ < 0. Thus, the solution of Eq. (36) can be obtained 0
2 oeðkÞ ow
ð37Þ
2
According to the theorem, the ranges of weights of all layers can be obtained.
h
6. Simulation remarks 6.1. System performance An all-purpose software for the two-link rigid–flexible manipulator system with multi-mode was developed. For the sake of simplification of computing, one mode of vibration is applied to describe the dynamics of the system in this study. The physical parameters of the system are given in Table 1. Fig. 4 shows the structure of the control system. First, the comparison of the performance between the FLC and the proposed neuro-fuzzy controller is conducted. Three groups of inputs (0.7, 1.0 and 1.3 rad) are given to the control system, respectively. The parameters of the two controllers are listed in Table 2 (Set 1 for the FLC and Set 2 for the proposed neuro-fuzzy controller) and the simulation results are illustrated in Figs. 5 and 6, respectively. In Figs. 5 and 6, (a) and (b) represent the outputs of the two joints of the system when it responds to the three groups of step inputs separately, and (c) and (d) are the corresponding steady-state error curves of the two joints. For the sake of clearance of observation, the time coordinate is selected from the 2nd second to the 4th second, the first two seconds are omitted; (e) demonstrates the tip deflection of the second flexible link. From the above figures, the overshoot does not appear for both cases, which is necessary for practical applications. In the meantime, the difference between these two controllers can obviously be seen. The performance of the control system is improved greatly by employing the proposed neuro-fuzzy controller, especially in the tip deflection of the second flexible link. This can be verified from both Table 2 and Figs. 5 and 6. The tip deflection decreases more than 10 times compared with the FLC. Moreover, the steady-state error of the system is also decreased and the absolute steady-state errors of both joints are less than 0.002 rad for all three groups of inputs. Table 1 Parameters of the rigid–flexible manipulator system Physical parameters
Link 1
Link 2
Length (m) Moment of inertia (kg m2 ) Density (kg m2 ) Elastic modulus (N m2 ) Mass (kg)
L1 ¼ 0:61 J1 ¼ 0:32
L2 ¼ 0:52 Jh ¼ 0:02 q2 ¼ 0:17 EI ¼ 1:0 Mh ¼ 0:23
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Table 2 Parameters of the proposed controller and simulation results
Set 1
Input (rad)
Joint
a
k
0.7
1 2 1 2 1 2
0.12 0.11 0.16 0.14 0.20 0.17
2.5 1.5 2.5 1.5 2.5 1.5
1 2 1 2 1 2
0.31 0.33 0.37 0.39 0.42 0.40
0.90 0.09 0.90 0.04 0.90 0.40
1.0 1.3 Set 2
0.7 1.0 1.3
Steady-state error (rad)
Tip deflection (mm)
Max
Min
Max
Min
0.0032 0.0012 0.0028 0.0018 0.0024 0.0016
)0.0030 )0.0017 )0.0026 )0.0017 )0.0022 )0.0020
42.7
)43.1
54.9
)53.8
56.2
)57.5
0.0006 0.0008 0.0005 0.0006 0.0013 0.0015
)0.0006 )0.0008 )0.0005 )0.0007 )0.0010 )0.0018
4.00
)3.94
5.71
)5.83
7.55
)7.87
a is the weighting factor expressed in Eq. (19) and k is the proportional factor expressed in Eq. (20).
Next, the performance of the DRNN is evaluated. Two sets of learning rates (0.05 and 0.1) are given to the control system separately, while the moment factor is kept unchanged. The results are demonstrated in Table 3 (Set 1 for k ¼ 0:05, Set 2 for k ¼ 0:1) and Fig. 7, in which (a) and (c) are the responses of the two joints to the two cases, respectively, and (b) and (d) are the tip deflections of the second flexible link. From both Table 3 and Fig. 7, it can be seen that when the learning rate is increased the performance of the control system becomes worse. The absolute steady-state errors are increased more than one time for both joints. The tip deflections are also increased. Therefore, a conclusion can be drawn that it is favorable to improve the performance of the control system by choosing a smaller learning rate. This is in agreement with the analysis in Section 5. 6.2. System robustness The robustness of the control system is also investigated under different conditions. Fig. 8 demonstrates the results when the moment of inertia of the link 1 is increased to 0.64 kg m2 , twice as the previous case. Based on the data from Fig. 8(a) and (b), it can be seen that the steady-state errors for both joints are less than 0.001 rad. The deflection of the second flexible link, shown in Fig. 8(c), is within ±0.0035 m. It is obvious that the performance of the control system is hardly affected by the increased moment of inertia of link 1. Fig. 9 illustrates the results when the mass of hub fixed at the end of the link 1 is increased to 0.46 kg, twice as the previous case. The absolute steady error for link 1 is not more than 0.001 rad. While for link 2, the absolute steady-state error becomes smaller gradually with the increasing of time, the tip deflection of the second flexible link 2 increases a little, less than 0.0045 m.
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Fig. 5. Step responses of robot system with the FLC.
Fig. 10 shows the results when the density of link 2 is increased to 0.34 kg/m, twice as the previous case. It can be seen that the steady-state errors for both links can be controlled within ±0.002 rad and the tip deflection is within ±0.006 m. From the three cases (Figs. 8–10), we can conclude that when the parameters of the system change in the control process, the performance of the control system is hardly affected. Thus, this can be used to explain that the proposed controller has better robustness in resisting against the changes of the parameters of the control system.
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Fig. 6. Step responses of robot system with the proposed neuro-fuzzy controller.
Table 3 Parameters of the proposed controller and simulation results Joint
a
k
Steady-state error (rad)
Tip deflection (mm)
Max
Min
Max
Min
Set 1
1 2
0.39 0.27
0.9 0.08
0.56 1.3
)0.56 )1.2
3.7
)3.6
Set 2
1 2
0.38 0.25
0.9 0.6
1.4 2.8
)1.3 )2.9
4.4
)4.2
a is the weighting factor expressed in Eq. (19) and k is the proportional factor expressed in Eq. (20).
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Fig. 7. Step responses of robot system under different learning rates.
Fig. 8. Step responses of robot system under different inertia.
L. Tian, C. Collins / Mechatronics 14 (2004) 471–490
Fig. 9. Step responses of robot system under different masses.
Fig. 10. Step responses of robot system under different densities.
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7. Conclusion A complete description for the dynamic modeling of a two-link rigid–flexible manipulator system is given. The finite-dimension approximation is discussed. A neuro-fuzzy controller for the system is designed, which includes two parts: the FLC serves as feedback controller; the DRNN is employed as a feedforward controller to enhance the performance of the control system. Furthermore, the analysis of the stability of the control system is also carried out. Simulations on the comparison between the FLC and the proposed neuro-fuzzy controller are conducted. Results show that the performance of the control system is improved greatly with the proposed controller, greatly decreasing the tip deflection of the second flexible link and also enhancing the steady-state accuracy for both joints. Simulation on the performance of the DRNN is carried out, demonstrating that it is favorable for the improvement of the performance of the control system by choosing a smaller learning rate. Finally, the robustness of the control system is investigated. The control system performs well in resisting against the parameter changes of the system, such as moment of inertia of link 1, mass of hub, and density of the link 2.
Acknowledgements This research was supported by Natural Science Foundation of South China University of Technology, grant no. B8-106-016 and Doctor Start Foundation of Guangdong, grant no. B9-108-061. The authors would like to thank the anonymous reviewers for their precious suggestions for this paper and Tricia Butera for the help on this paper.
Appendix A A finite-dimensional approximation of the dynamic flexible-link manipulator 2 m11 m12 m13 6 m m m23 21 22 6 6 q2 S1 ðxÞ þ q2 L1 cos h2 S1 ð1Þ q2 S1 ðxÞ q2 L2 M ¼6 6 .. .. 4 . 0 . q2 Sn ðxÞ þ q2 L1 cos h2 Sn ð1Þ q2 Sn ðxÞ 3 2 0 0 0 0 60 0 0 0 7 7 6 6 0 0 EI L2 b1 0 7 K¼6 7 7 6 .. .. .. 5 4. . 0 . 0
0
EI L2 bn
equation of the two rigid– ..
.
3 m1ð2þnÞ m2ð2þnÞ 7 7 0 7 7 7 5 q2 L2
ðA:1Þ
ðA:2Þ
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3
2
f1 7 6 f2 7 6 6 q2 L2 ðh_1 þ h_2 Þ2 q1 q2 L1 sin h2 h_21 S1 ð1Þ 7 F ¼6 7 7 6 .. 5 4 . 2 q2 L2 ðh_1 þ h_2 Þ qn q2 L1 sin h2 h_21 Sn ð1Þ B ¼ ½1; 1; 0; . . . ; 0
ðA:3Þ
T
ðA:4Þ
where, m11 , m12 , m22 , f1 and f2 are as follows: n X 1 m11 ¼ J1 þ Jh þ Mh L21 þ q2 L21 L2 þ q2 L32 þ q2 L2 q2j ðtÞ 3 j¼1
þ q2 L1 L22 cos h2 2q2 L1 sin h2
n X
Sj ð1Þqj ðtÞ
ðA:5Þ
j¼1 n X 1 1 m12 ¼ Jh þ q2 L32 þ q2 L2 q2j ðtÞ þ q2 L1 L22 cos h2 3 2 j¼1
q2 L1 sin h2
n X
Sj ð1Þqj ðtÞ
ðA:6Þ
j¼1 n X 1 m22 ¼ Jh þ q2 L32 þ q2 L2 q2j ðtÞ 3 j¼1 # " n X 1 q L1 L22 sin h2 þ q2 L1 cos h2 f1 ¼ qj ðtÞSJ ð1Þ h_22 2 2 j¼1 # " n X 2 þ q L1 L sin h2 þ q L1 cos h2 qj ðtÞSj ð1Þ h_1 h_2 2
ðA:7Þ
2
2
j¼1
" þ 2q2 L1 sin h2
n X
Sj ð1Þq_ j ðtÞ 2q3 L2
j¼1
n X
#
qj ðtÞq_ j ðtÞ ðh_1 þ h_2 Þ
ðA:8Þ
j¼1
# n X 1 2 f2 ¼ q2 L1 L2 sin h2 þ q2 L1 cos h2 Sj ð1Þqj ðtÞ h_21 2 j¼1 "
2q2 L2 ðh_1 þ h_2 Þ
n X
qj ðtÞq_ j ðtÞ
j¼1
where M ¼ M T S2þj ð1Þ ¼ S2þj ðxÞ ¼
Z Z
L2
gj ðxÞ dx ðj ¼ 1; 2; . . . ; nÞ
0
0
L2
xgj ðxÞ dx ðj ¼ 1; 2; . . . ; nÞ
ðA:9Þ
490
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