Dynamic modeling and active control of a cable-suspended parallel

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Mechatronics 18 (2008) 1–12

Dynamic modeling and active control of a cable-suspended parallel robot B. Zi *, B.Y. Duan, J.L. Du, H. Bao School of Electromechanical Engineering, Xidian University, Xi’an 710071, China Received 20 July 2006; accepted 12 September 2007

Abstract Cable-suspended parallel robot (CPR), in which cables are utilized to replace links to manipulate objects, is developed from parallel and serial cable-driven robot. Compared with the parallel robot, this kind of robot has more advantages. The cooperative variation of lengths of six cables pulls the feed cabin to track radio source with six degrees of freedom (DOFs). Similar to a parallel robot, the cablecabin mechanism for 500-m aperture spherical radio telescope (FAST) can be viewed as a CPR. According to its 5-m scaled model, based on the inverse kinematics analysis the inverse dynamic formulation of CPR with non-negligible cable mass is established using Lagrangian dynamic formulation. Then, considering random wind forces acted on the cabin are simulated based on the characters of the mechanism, a fuzzy plus proportional–integral control (FPPIC) method, which can enhance the control performance for steady-state errors, is utilized to control the wind-induced vibration of the trajectory tracking of the feed cabin. Finally, we provide the examples of simulation and experiment to justify the dynamic modeling for control and to test the proposed method.  2007 Elsevier Ltd. All rights reserved. Keywords: Cable-suspended parallel robot; Dynamic modeling; Active control; Fuzzy plus proportional–integral control

1. Introduction With the development of radio astronomy, it becomes necessary to build a new-generation large radio telescope. So far the largest antenna is the Arecibo spherical radio telescope with the diameter of 305 m, which was built in the 1970s and located in Puerto Rico, USA. As for the Arecibo-type telescope, there exist three problems, namely: (1) high cost; (2) narrow frequency band; and (3) spherical error. Considering the above drawbacks, one plan proposed by Chinese astronomers and engineers is to build a set of large spherical reflectors in the extensively existing karst landform in Guizhou province, southwest China. As an engineering demonstrator, a 500-m aperture spherical telescope (FAST) is proposed, which will be over twice as large as the Arecibo radio telescope. Technically, the

*

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FAST has a number of innovations. Firstly, a cable support system for the feed cabin, which integrates optical, mechanical, and electronic technologies, will reduce the structural weight considerably [1,2]. Secondly, for the sake of having a good observing precision, the proposed two level control systems [3] are utilized in the design of FAST. One is the feed cable-suspended system with centimeter positioning precision by varying the cables’ lengths. The other is Stewart platform mounted in the cabin. The millimeter positioning precision of the feed fixed in the moving platform of Stewart platform can be realized by varying the six legs’ lengths. The feed cable-suspended system has been investigated with some useful results [4,5]; however, the cables were modeled as a pure time delay or a massless cable element. In addition, researchers in Canada have recently studied a multi-tethered aerostat positioning system that is at a similar scale to our system, which consists of a platform supported by a helium-filled aerostat and attached to three anchored ground tethers actuated using computer-controlled winches [6]. To conduct a systematic

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study of the key technology of FAST, an inner FAST model of 5 m was built in Xidian University (see Fig. 1). The cooperative variation of lengths of six flexible cables pulls the feed cabin to track the given radio source with six degrees of freedom (DOFs). Similar to a parallel robot, the cable-cabin structure of FAST can be viewed as a cablesuspended parallel robot (CPR). Conventional robots with serial or parallel structures are impractical for some applications since the workspace requirements are higher than what the conventional robots can provide. CPR uses cables instead of links to manipulate objects. The cables are so light that actuators of CPR have only to drive their loads. Furthermore, CPR gives a wide range of motion, because drums of the mechanism can wind long cables. For the above reasons, cable-driven mechanisms have received attention and have been recently studied since 1980s [7]. Some reported research on CPR are NIST Robocrane [8], ultra-high speed robot, tendon-driven Stewart platform, parallel wire mechanism for measuring a robot pose, controller designs for CPR [9], and wrench-feasible workspace analysis for CPR [10]. Particularly, dynamical characteristics of this type of mechanism has been well studied, such as the manipulation problem of load by multiple wires [11,12], and the cables are considered to be massless. However, the cable will extend when stretched and an error in length of 0.1% can give rise to an error in force of about 50%, thus accurate modeling of the cables is crucial [13]. The static analysis of cable-driven robotic manipulators with non-negligible cable mass is presented in [14], and the results show that the cable sag can have a significant effect on both the inverse kinematics and stiffness of such manipulators. Now, most methods of analysis of the cable structure are based on the discretisation of FEM, and many relative cable elements have been developed, which can be divided into two types – elements based on polynomial interpolation functions and on analytical functions [15,16]. Based on the preceding analysis, the cables of CPR in this paper act as the elastic catenary. In view of a control aspect, a conventional proportional–integral-derivative (PID) control has been employed

in industry but it does not always guarantee a high performance for the CPR. Thus, a sophisticated control scheme is required for an enhanced control performance. The idea of fuzzy set and fuzzy control is introduced by Zadeh in an attempt to control systems that are structurally difficult to model [17]. In [17], fuzzy sets and fuzzy logic have been considered as effective tools to deal with uncertainties in terms of vagueness, ignorance, and imprecision. Since Mamdani did the first fuzzy control application [18], fuzzy control is being increasingly applied to many systems. In particular, fuzzy control has emerged as one of the most active and fruitful areas for research in the application of fuzzy set theory [19]. For example, Lin et al. [20] developed an integrated fuzzy-logic-based missile guidance law against high speed target. Hwang and Kim [21] employed an adaptive fuzzy-logic method to provide the solution to the control of an electrically-driven robot. In addition, in many applications, the fuzzy-logic-control-based systems have proved to be superior in performance to conventional systems. Recently, fuzzy-logic and conventional-technique are combined to design fuzzy-logic controller, and the control technology based on fuzzy-logic is being increasingly applied to systems with nonlinearity and uncertainty; see e.g., [22–25]. In [22], the genetic algorithms can be used effectively for generating fuzzy rules in fuzzy-PID hybrid control structures. In [23], a new hybrid fuzzy P+ID control scheme is developed to control a direct drive two-link manipulator under inertial parameters’ changes. A fuzzy plus integral control approach is applied to the control of the effluent turbidity in direct filtration in [24]. In [25], a fuzzy PI+PD controller, which includes a dynamic switching fuzzy system, is applied to swing-up an underactuated mechanism: the pendubot. In this paper, first, according to its 5 m scaled model, based on the inverse kinematics analysis the inverse dynamic formulation of CPR with non-negligible cable mass is established using Lagrange’s equations. Then, a fuzzy plus proportional–integral control (FPPIC) method of CPR is proposed to control the wind-induced vibration of the trajectory tracking of the feed cabin. The random wind forces acted on the cabin is simulated based on the characters of the structure. Finally, we verify the performance of the proposed method through simulations and experiments. 2. System dynamic model 2.1. Model definition and kinematics

Fig. 1. The model of feed support for 5 m FAST.

A simple schematic of the CPR for FAST 5-m representing the coordinate systems is shown in Fig. 2. The CPR is considered that is kinematically and statically determined. In the design, there is a cable tower/winch pair at each vertex of a regular hexagon of radius a, which actuate six cables that are linked to a cabin. Therefore, the cabin can translate and rotate in the inertial frame. Let the origin of the inertial frame OXYZ be the bottom of tower A1.

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A5

A6 A1

A4

A2 B6

B1

B2

Z1

o1 X1

A3

B5

tor rBi ¼ ½ xBi y Bi zBi T , which contains the Cartesian coordinates x, y, z of the origin of frame O1X1Y1Z1 with respect to frame OXYZ. The length vector li expressed with respect to frame OXYZ can be computed by l i ¼ rBi  rAi

B4

Y1

ði ¼ 1; 2; . . . ; 6Þ

Z Y X

Fig. 2. Schematic diagram of CPR for FAST.

The origin of O1X1Y1Z1 (cabin coordinate system) is in the center of the bottom of the cabin as shown in Fig. 2. In Fig. 2, towers A1–A6 are distributed evenly on a circle with the diameter of D. The cables A1B1, A3B3, and A5B5 are connected from the bottom of the cabin to the towers A1, A3, and A5, respectively. The other three cables A2B2, A4B4, and A6B6 are connected from the towers A2, A4, and A6 to the top points B2, B4, and B6 of the cabin as shown in Fig. 3. Angle g between O1B2 and O1B1 indicates the relative position relationship of spherical joint B1 and spherical joint B2. It can be noted from Fig. 3 that the coordinates ofB2,B4, and B6 have relationships with g. The coordinates of the joint Ai on the tower i with respect to the frame OXYZ can be described as 9 T T > rA1 ¼ ð 0 0 h Þ ; rA2 ¼ ð a 0 h Þ > >  T pffiffi =  T > p ffiffi ffi 3 3 rA3 ¼ 2 a 2 a h ; rA4 ¼ a 3a h ð1Þ >  T > pffiffi > pffiffiffi > T rA5 ¼ ð 0 3a h Þ ; rA6 ¼  12 a 23 a h ; where h is the height of tower and a is the side length between the tower A1 and tower A2 in regular hexagon. The length of the cable can be changed by rotating the winch to reel the cable in or let it out. The cabin is assumed to be a rigid body. We proceed to consider the vector diagram for an ith cable. The position of frame OXYZ is represented by vec-

Y1 B3

B4

B2 B6

O 2 (O1 )

ð2Þ

in which,

B3

rBi ¼ rO1 þ R  rBi1

o

3

η B1

ði ¼ 1; 2; . . . ; 6Þ

rO1 is the position vector of the origin of O1X1Y1Z1 with respect to OXYZ; rBi1 is the position vector of the point Bi related to O1X1Y1Z1. rBi1 can be expressed as 9 T > rB11 ¼ ½r; 0; 0 > = T ð3Þ rB31 ¼ ½r cosðhÞ; r sinðhÞ; 0 > > ; T rB51 ¼ ½r cosð2hÞ; r sinð2hÞ; 0 9  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT > 2 > 2 rB21 ¼ r1 cosðgÞ; r1 sinðgÞ; r  ðr1 Þ > > > > > >   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T > = 2 ð4Þ rB41 ¼ r1 cosðg þ hÞ; r1 sinðg þ hÞ; r2  ðr1 Þ > > > >  > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT > > > ; rB61 ¼ r1 cosðg þ 2hÞ; r1 sinðg þ 2hÞ; r2  ðr1 Þ2 > In which, r and r1 represent the radii of the points B1, B3, B5 and the points B2, B4, B6 of the cabin, respectively. h = 120 as shown in Fig. 3. R is the orientation matrix of the cabin with respect to OXYZ. By introducing the roll, pitch, and yaw angles into R, the matrix becomes 3 2 cacb casbsc  sacc casbcc þ sasc 6 7 ð5Þ R ¼ 4 sacb sasbsc þ cacc sasbcc  casc 5 sb

cbsc

cbcc

where roll is the rotation about the fixed axis Z by a, pitch is the rotation about the fixed axis Y by b, and yaw is the rotation about the fixed axis X by c; c represents cosine function; and s represents sine function. The Jacobian matrix of CPR is defined as the relation between the velocity of the cabin and the velocity of the driven cable. We specify the velocity of the T cable-driven as l_ ¼ ½ l_ 1 l_ 2 l_ 3 l_ 4 l_ 5 l_ 6  (m/s), define _ z_ ÞT (m/ the velocity of the center of the cabin as v ¼ ð_x; y; s), and express the angular velocity of the cabin with _ c_ ÞT (rad/s), _ b; respect to three axes Z, Y and X as w ¼ ða; so the velocity of the cabin can be described as r_ O1 ¼ ½ v w T ¼ ½ x_ y_ z_ a_ b_ c_ T . The relation between l_ i and r_ O1 can be defined as [26] l_ ¼ J  r_ O1

X1

B5 Fig. 3. Connection between the cables and the cabin.

where Jacobian matrix J is h i oli oli i J ¼ oloxi oloyi olozi ol oa ob oc

ð6Þ

ði ¼ 1; 2; . . . ; 6Þ

ð7Þ

For simplicity, we use the symbolism to develop the Jacobian matrix. The formula can be obtained [3].

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Thus, we can compute the cable-driven lengths, i.e., norms of li, from the given positions and orientations of the cabin. This problem is called an inverse kinematics problem of a CPR. The forward kinematics problem is the opposite of the inverse kinematics, i.e., to obtain the positions and orientations from the given cable-driven lengths. 2.2. Dynamic equations of motion The general dynamic equations of motion can be obtained from the Lagrangian formulation. The generalized coordinates is q = [x, y, z, a, b, c]T. Lagrange’s equation can be written in the form of potential energy V, kinetic energy T, and generalized forces or torques in the following form:

_ _ d oT ðq; qÞ oT ðq; qÞ oV ðqÞ þ ¼s ð8Þ  dt oq_ oq oq This results in a kinetic energy of the cabin, which can be written in Cartesian coordinates as 1 1 T ¼ mvT v þ wT RI O1 RT w 2 2

ð9Þ

or, 1 1h _ _ T ¼ mð_x2 þ y_ 2 þ z_ 2 Þ þ I xx ðacacb þ bsacb  c_ sbÞ2 2 2 2 þI yy ððcasbsc  saccÞa_ þ ðcacc þ sasbscÞb_ þ c_ cbscÞ 2 þI zz ððcasbcc þ sascÞa_ þ ðsasbcc  cascÞb_ þ c_ cbccÞ T

i

T

where v ¼ ½ x_ y_ z_  ; w ¼ ½ a_ b_ c_  ; m and IO1 is the mass and the moment of inertia of the cabin, respectively. IO1 is expressed as 2 3 I xx I xy I xz 6 7 ð10Þ I O1 ¼ 4 I yx I yy I yz 5 I zx

I zy

I zz

In this study, each cable is assumed to be a force element. Therefore, the potential energy of the system is due only to gravitational forces. The potential energy takes the form V ¼ mgz

ð11Þ

where g is the acceleration due to gravity. It is noted that there is a relation between externally applied wrench on the cabin and the cable tensions required to keep the system in equilibrium, this relationship [27] is ½F x ; F y ; F z ; M o ; M b ; M c T ¼ J T u

equations of motion in terms of q generalized coordinates as the following general form: _ q_ þ GðqÞ þ sd ¼ JT u MðqÞ€ q þ Cðq; qÞ

_ is the where M(q) is the inertia matrix of the system, Cðq; qÞ vector of Coriolis and centripetal terms, G(q) is the vector of gravity terms, sd is the vector of external disturbance terms (e.g., random wind, etc.). Here, u is the vector of cable-driven tensions TJi for i = 1, 2, . . . , 6. In the following discussion, we will briefly describe how TJi is derived. 2.3. Mechanics analysis of the cable Six flexible cables are used to drive the cabin to implement high-precision trajectory tracking. Due to the eigenfrequency of the cable-cabin system is very low, the response of the cabin is very slow in the presence of external disturbances. The quasi-dynamics analysis is made by substituting static tension of cable at a static position for the dynamic force of the cable at the same position. When the cabin operates at the practically observed speed (0.5– 2 cm/s), the difference between static tension and the dynamic force is very small [28]. However, this difference has little influence on the calculation of the cable length, so the dynamic characteristic of cables can be neglected. Owing to six identical cables, we only perform mechanics analysis of a cable. The first challenge for analyzing a cable-suspended system is to mathematically describe the static displacement (or the shape) of the cables under the influence of gravity [14]. To compute the static displacement of a cable, we will assume for the moment that the unstressed length of the cable, Lu, are known. The procedure in this section determines the complete geometry of the cable, end forces from a given strained or unstrained cable length and the positions of cable ends by using Newton–Raphson method. Considering an elastic cable element stretched in vertical plane, with unstressed length Lu, corresponding stressed length L, elastic modulus E, cross-sectional area A, and self-weight per unit length q0, as shown in Fig. 4, the rigorous relationship between cable projection and end force of the cable is subjected to [16] L2 ¼ h2 þ

l2 sin h2 g g2

ð14Þ

F4 F3

J

z

ð12Þ

where the external force on the cabin at the reference point is Fx, Fy, Fz and the external moment on the cabin body is given by Ma, Mb, Mc; u is the vector of cable-driven tensions. Through a series of transformations, substitutions and simplifying the resulting expression, here, we write the

ð13Þ

h

F2 F1

x

E, A, q 0

I l Fig. 4. Elastic catenary cable element.

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where g¼

q0 l 2F 1

ð15Þ

Lu ¼ L 

1 F4 þ TJ F 4 T J þ F 2 T I þ F 21 ln 2EAq0 TI  F2

ð16Þ

and   q0 cosh g F2 ¼ Lh sinh g 2   Lu 1 F4 þ TJ þ ln l ¼ F 1 EA q0 T I  F 2 1 TJ  TI ðT 2J  T 2I Þ þ h¼ 2EAq0 q0

ð17Þ ð18Þ ð19Þ

The variables F1, F2, F3, F4, TIi and TJi (for i = 1, 2, . . . , 6) are subjected to the following equations: ) F 3 ¼ F 1 ; F 2 þ F 4 ¼ q0 Lu ð20Þ T Ii ¼ ðF 21 þ F 22 Þ1=2 ; T Ji ¼ ðF 23 þ F 24 Þ1=2 There exists an unique solution for the cable-driven tension using (13), for a given motion of the cabin, as long as J is not singular. In this case, the solution of (13) can be written as T 1

_ q_ þ GðqÞ þ sd  u ¼ ðJ Þ ½MðqÞ€q þ Cðq; qÞ

ð21Þ

3. Controller design A robot control scheme stems from two frameworks. One is to design a control on the workspace and the other is on the linkspace. The linkspace control is a conventional control, and is a kind of tracking control to follow the desired link length computed from the position command of the mobile plate by inverse kinematics. Normally, most controllers in applications are based on the linkspace coordinates, which they consider only an approximated manipulator model [29]. From an engineering viewpoint, considering a control aspect, a PID control has been employed in industry due to its simple control structure, ease of design, and inexpensive cost. But it does not always guarantee a high performance for a parallel robot. Therefore, we would like to keep the advantages of the PID controller while designing a new controller for an enhanced control performance. Fuzzy-logic has been considered to be an efficient and effective tool to manage system uncertainties since the seminal paper of Zadeh. Among the many applications of fuzzy-logic, control design appears to be the one that has attracted the most attention in the past two decades [30]. In this paper, a fuzzy plus proportional–integral controller combining PI control with fuzzy-logic control is developed for more effective and robust performance in the presence of random wind and other disturbances. Its viability for practical applications has been demonstrated by simulation and experiments.

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The CPR control system of FAST consists of six cabledriven alternating current (AC) subservosystem, and the control mode of the subservosystem is Insulated Gate Bipolar Transistor (IGBT) Modem Pulse-Width Modulated (PWM) sine wave control. For the high requirement of real-time and synchronization of CPR, we adopt a centralized control scheme, which consists of a main computer (M-C), a measure computer (Measure-C) and the multi-movement-node based on servodrive. In real-time motion,the real position and orientation of the center of the feed cabin can be detected by three laser ranging equipments under the control of Measure-C, together with the desired trajectory are fed into M-C, and M-C calculates the lengths of six cables in each time slice based on the inverse dynamic model, and then converts the length change value of cable into the number of pulse. Thus each servo control system can realize corresponding movement via multi-axis movement controller. That is, the length of each cable at every sampling time can be adjusted by high power servomotor through a favorable control method to guarantee the high-precision trajectory tracking. 3.1. Fuzzy plus proportional–integral control Fuzzy-logic systems belong to the category of computational intelligence techniques. One advantage of the fuzzylogic over other forms of knowledge-based controllers lies in the interpolative nature of the fuzzy control rules. In a fuzzy-logic controller, the dynamic behavior of a fuzzy system is characterized by a set of linguistic description rules based on expert knowledge usually of the form: IF (a set of conditions are satisfied) THEN (a set of consequence can be inferred). Fig. 5 shows a basic configuration of a fuzzy-logic system. The inputs, which are the state variables, characterize the key features of the system, and the output control variable influence the states of the system. The basic consideration is aimed at stabilizing the rotor angles of the machines and also damping the oscillation of the generator speed to zero [31]. The CPR for FAST is a nonlinear system with a large envelope of control space. One common task in controlling the CPR system is to demand the cabin to follow a given reference trajectory. System dynamics in tracking control may change significantly. Therefore, the fuzzy control is a convenient choice because it is possible to finely tune the control actions in a nonlinear fashion for different tracking error ranges [24]. Hence, in order to improve its control

Fuzzy rule Input

Fuzzification

Defuzzification Fuzzy inference

Fig. 5. Basic configuration of a fuzzy-logic system.

Output

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performance, in some nonlinear systems and in the case of strong disturbances, fuzzy and PID controllers are used in parallel [24,32]. In view of the characteristics of nonlinearity for CPR, a proportional–integral controller needs to be added in parallel to the fuzzy controller for the tracking control of the CPR in order to improve the steady-state error performance. The proposed controller consists of fuzzy-logic control and proportional–integral control algorithms. A proportional–integral controller is added in parallel to enhance the control performance for steady-state errors, a block diagram as shown in Fig. 6. Here, KP and KI are called proportional gain and integral gain, respectively. sd is the external disturbance. The two inputs are the error e(k) and the change of the error De(k). The output is the change in the control action Du(k) eðkÞ ¼ y d ðkÞ  yðkÞ DeðkÞ ¼ eðkÞ  eðk  1Þ

ð22Þ ð23Þ

where yd(k) is the desired output at time instant k, y(k) is the actual output, e(k) is the error and De(k) is the change of the error. The change of control action Du(k) is calculated based on the two inputs using a fuzzy inference mechanism. uf(k) is the proportional–integral control signal, and the total control signal is ut(k) = uf(k) + Du(k). The fuzzy controller is implemented in a discrete-time form using the zero-order-hold (ZOH), which holds its input signal for a specified sampling period to get a continuous signal out. The inputs to the controller system, e(k), and De(k), are actual values in form of ‘‘crisp’’ numbers. Fuzzification converts the numerical value into a linguistic variable which can be understood by the fuzzy control system. Singleton fuzzification, which transforms a crisp value into a fuzzy singleton value, is adopted in this fuzzy control scheme. Triangular type membership functions are defined over the range of input and output space, which linguistically describe the variable’s universe of discourse, due to their ease in real-time hardware implementation [33]. All membership functions of the fuzzy-logic control inputs, e(k), De(k), and the output, Du(k), are defined on the common normalized domain [1, 1] as shown in Fig. 7. Each of the fuzzy-logic reasoning inputs, e(k) and De(k), and the output, Du(k), is assumed to take seven linguistic sets defined as negative big (NB), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive

Fig. 7. Membership functions of e, De and Du.

medium (PM) and positive big (PB). Thus, a complete set of 49 fuzzy rules is given in Table 1 based on expert knowledge and testing. For example, if the output is falling far away from the desired trajectory, a large control signal that pulls the output toward the desired trajectory is expected, whereas a small control signal is required when the output is near and approaching the desired trajectory. For both the input and output variables membership functions are defined to be symmetric, equi-spaced with an equal area so that each set can be described by its central value. The max–min (Mamdani type) inference is used to generate the best possible conclusion. This type of inference is computationally easy and effective; thus it is appropriate for real-time control applications. The crisp control command is calculated here using the center-of-gravity (COG) defuzzification [34]. The criterion provides the defuzzified output with better continuity. 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 fuzzy controller. It can be expressed as  DU ðkÞ ¼ haEðkÞ þ ð1  aÞDEðkÞi a ¼ ð0; 1Þ ð24Þ DuðkÞ ¼ bDU ðkÞ 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(k) is the fuzzy value of the error of input variable, DE(k) is the fuzzy value of the change in error, a is the weighting factor, b is the proportional factor, and

Fig. 6. Block diagram of the closed-loop fuzzy plus proportional–integral control system.

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B. Zi et al. / Mechatronics 18 (2008) 1–12 Table 1 Rules for the fuzzy tracking system Du

e

De

NB NM NS ZO PS PM PB

NB

NM

NS

ZO

PS

PM

PB

NB NB NB NB NM NS ZO

NB NB NM NM NS ZO PS

NB NB NM NS ZO PS PM

NB NM NS ZO PS PM PB

NM NS ZO PS PM PB PB

NS ZO PS PM PB PB PB

ZO PS PM PB PB PB PB

DU(k) and Du(k) are the fuzzy and non-fuzzy outputs of the fuzzy controller. 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 [35]. 4. Simulation studies and experimental results In this section, we provide the examples of simulation and experiment to justify the dynamic modeling for control and to test our proposed method. The model parameters of FAST 5 m used in the simulation and experiment are given in Table 2. Moment of inertia of CPR in cabin frame (in kg m2 unit): 2 3 0:782 0 0 6 7 0:782 0 5 I O1 ¼ 4 0 0 0 0:293 4.1. Simulation example studies Example 1 aims to justify the dynamic modeling for control, while Example 2 demonstrates the effectiveness of the FPPIC active control method through simulations on the comparison between the conventional PID and the proposed FPPIC method. Example 1. The algorithm for the inverse dynamics of CPR without sd described in the foregoing statement has been implemented in a program written in MATLAB and C, separately. The program has been developed to plan straight-line and circular paths. We use the simulation programs for solving the inverse dynamics. That is, the given data values representing the task space are passed through the mathematical equation representing the

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inverse dynamics problem solution [i.e., f : (x, y, z, a, b, c) ! (l1, l2, l3, l4, l5, l6)] to obtain the corresponding data values for the cable space. By inverting these data values [i.e., f : (l1, l2, l3, l4, l5, l6) ! (x, y, z, a, b, c)], the data values corresponding the cable space become the input values, and task-space values become the output to the forward dynamics model. The trajectory planning scheme described above has been selected just for simplicity. In fact, any scheme for planning the trajectory in the Cartesian space can be used with equal ease based on the established dynamic formulation. Considering the length of the paper, the simulation result of a circle case is presented only. In the simulation of tracking a moving target with constant velocity v = 0.01 m/s and constant pose (a, b and c are all assumed to be 0), the initial point is (2.45, 2.5 · cos(p/ 6), 1.3) (m). The geometric path is a circle with the radius of 0.6 m, parallel to the x–y plane 8 > < x ¼ 1:85 þ 0:6 sinð0:0167tÞ y ¼ 2:5 cosðp=6Þ þ 0:6 cosð0:0167tÞ ð25Þ > : z ¼ 1:3 The trajectory tracking of the center of the cabin is shown in Fig. 8. From Fig. 8, the formulation follows the planned trajectory relatively well. Fig. 9 illustrates the changes in length of the cables. Li, for i = 1, 2, . . . , 6, denotes the length of cable between Ai and Bi. Owing to the coordinates of the joint A1, A2 and A3 is just symmetric to that of the point A4, A5 and A6, as shown in Fig. 2, the length of cable vary symmetrically. Obviously, the plots shown in Fig. 9 indicate the changes in length of the cables are valid. From Fig. 10, it is known that the tracking displacements of the center of cabin in X, Y and Z directions accord with the desired motion trajectory. Fig. 11 represents the cabin upon which the forces are exerted in all directions. The forces actuating on the cabin in the direction of X and Y are equal to the centripetal forces in the direction of X and Y as the cabin moves along the circle, respectively. On the other hand, the force actuating on the cabin in the direction of Z equals the gravity of the cabin. From the above simulation results, it may be concluded that we justify the dynamic modeling for control. Example 2. Because the CPR is suspended in the space, the wind disturbance has significant effect on the tracking precision. Therefore, the simulation is carried out based on the trajectory tracking with the random wind sd. According to the historical maximum wind speed and the simulation of the maximum wind-induced displacement of the CPR, random wind sd, which is a disturbance at the velocity of 17 m/s, is modeled in a program written in MATLAB.

Table 2 Parameters of CPR m (kg) 20

a (m) 2.5

D (m) 5

h (m) 2.5

r (m) 0.21

r1 (m) 0.015

g () 8

q0 (N/m) 2.57

E (GPa) 28

A (m2)

g (m/s2) 5

2.788 · 10

9.8

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Fig. 8. Following trajectory of the circle motion.

velocity spectrum, respectively; Dx is the frequency increment; wjm(xl) refers to phase angle related to two points at different heights; hml is random variable ranging from 0 to 2p. In this paper, the wind-induced vibration caused by the along-wind velocity is considered especially. Further, the wind pressure can be simulated as follows: 1 1 2 wðx; y; z; tÞ ¼ qV 2 ðx; y; z; tÞ ¼ q½V ðzÞ þ vðx; y; z; tÞ 2 2

ð27Þ

where v(x, y, z, t) is the fluctuating velocity and V ðzÞ is the mean velocity at height z satisfying the exponential law, V ðzÞ ¼ V 10 ðz=10Þ Fig. 9. Change in length of cable.

a

ð28Þ

where V 10 indicates the mean velocity at 10 m high; a is coefficient related to the ground roughness. As a comparison, a conventional PID controller is also integrated to the system in the form of uðkÞ ¼ k p eðkÞ þ k i

k X j¼0

Fig. 10. Tracking displacement of the center of cabin in X-, Y- and Z-directions.

Along-wind velocity is considered as an ergodic stationary random process and samples of the wind velocity related to time are generated by the following equation [36]: V j ðx; y; z; tÞ ¼ V j ðzÞ þ

j X N X

pffiffiffiffiffiffiffiffiffiffi jH jm ðxl Þj 2Dx cos½xl  t

m¼1 l¼1

þ wjm ðxl Þ þ hml 

ðj ¼ 1; 2; . . . ; nÞ

ð26Þ

In which V j ðzÞ is the mean wind velocity at jth height; n and N denote the numbers of divided segments uniformly along the structural height and in frequency band of the wind

eðjÞT þ k d

eðkÞ  eðk  1Þ T

ð29Þ

where kp, ki and kd represent the proportional, integral and derivative gains, respectively, and T is the sampling period. Because of the same selected servomotor systems, the parameters for all the six control subsystem are determined as identical primarily, and then make the necessary small tuning according to the respective simulation results. After several simulations, the sampling period is determined as T = 0.02 s; the parameters for the PID and the FPPIC can be determined as, kp = 21.5, ki = 1.4 and kd = 0.46; a = 0.34, b = 1.7, KP = 12.5, KI = 0.23. To see the performance of the proposed fuzzy plus proportional–integral controller, Fig. 12 shows the actual motions in the x–y plane tracked by the conventional PID and FPPIC methods, and the tracking errors in the x- and y-direction. The arrow in the Fig. 12 indicates switching state of controllers and disturbance. Fig. 12 shows that the active control strategy can effectively attenuate the wind-induced vibration, and improve the tracking accuracy of the feed cabin. We observe that the maximum and average errors of the FPPIC method are smaller than those of the conventional PID controller. From the above simulation results, it may be concluded that the FPPIC

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Fig. 11. The forces actuating on the cabin in X-, Y- and Z-directions.

Fig. 12. The tracking trajectory and tracking errors in the direction of X and Y.

strategy can achieve a favorable control performance and has high robustness. 4.2. Experimental results The proposed algorithm is verified by experiments using the CPR control system in Fig. 1. Two different control schemes are implemented: conventional PID control and FPPIC. Fig. 13 shows the hardware configuration of the experimental system. The extendable actuator is comprised of an AC servomotor and drive unit, worm-gear speed

EVOC industrial computer

(Main-C)

reducer (reduction ratio: 1:60), and bobbin. The servomotor and driver used is YASKAWA R–L series (low inertial moment) SGML-04AF14 AC servomotor with an incremental encoder (1024 P/R) and the matching SGDL04AP driver. The motor parameters are: rated power 0.4 kW, rated voltage 200 V, rated torque 1.27 N m, rated speed 3000 r/min, motor moment of inertia 1.91 · 105 kg m2. The servo drive was set in the position mode. The FPPIC algorithm and the PID method were implemented in C on an EVOC industrial computer with a Pentium 4-2.4 GHz, i.e., M-C. The position and orientation of

Servomotor Control case 1#-6# Drive

FPPIC

• • •

Servomotor

EVOC industrial computer ( Measure-C)

Encoder

Reducer

Bobbin

Cable

• Feed cabin

• •

Encoder

Reducer

Bobbin

three Leica laser ranging equipments

Fig. 13. The hardware configuration of experimental system.

Cable

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the center of the feed cabin is measured using three Leica TCA1800 laser ranging equipments (made in Switzerland) under the control of a Measure-C. A series of experiments with FAST 5-m was performed with the same reference trajectory as proposed in Section 4.1. During experiment, the cabin is asked to move along the given trajectory. The artificial wind forces, through a fan with 5 kW, on the structure with a mean velocity of 17 m/s is produced. For real-time, the parameters of the controller, obtained by simulation, are manually adjusted on the FAST 5-m experimental setup to minimize the following error. Experimental results are presented for three

cases: an open-loop case; a FPPIC case; and a conventional PID control case. We begin with the open-loop case to justify the dynamic modeling. In this case, the number of pulse inputs is calculated using inverse dynamics. Figs. 14 and 15 show the desired trajectory and actual trajectory of the circle motion and tracking errors in X-, Y- and Z-directions under openloop. Figs. 16 and 17 illustrate the tracking performance and errors using conventional PID and FPPIC methods under closed-loop. From the results of the proposed method as shown in Fig. 17, it is clear that the tracking error is

Fig. 14. Desired trajectory and actual trajectory of the circle motion under open-loop.

Fig. 15. Tracking errors in X-, Y- and Z-directions under open-loop.

Fig. 16. Desired trajectory and actual trajectory of the circle motion under closed-loop.

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Fig. 17. Tracking errors in X (a), Y (b) and Z (c) directions under closed-loop.

improved and is smaller than the PID control. In addition, it is also shown that the computational and experimental performance values are in good agreement. The simulation and experimental results in Sections 4.1 and 4.2 together highlight the effectiveness of the proposed control framework. However, the trajectory following errors in the experiments are poor in contrast to in the simulation. During the experimental tests, there are mostly due to problems in practical implementation such as nonlinear effects (backlash and friction, etc.), (dynamic) uncertainties, cross-coupling and other imperfections of such a nonlinear system. On the other hand, the other major assumption we have made in our model is the gearing, the shafts, and the bearings are not flexible. In reality, all of these elements have finite stiffness. However, for purposes of control-system analysis and design, we neglect these effects and use a simpler dynamic model in the simulation.

4.1 and 4.2 together highlight the effectiveness of the proposed control framework. This design is carried on directly in discrete-time and the structure chosen for the realization permits fast computation. Furthermore, the FPPIC controller can be applied to other servo system requiring high tracking precision. Above all, the structure of the controller is simple and directly intuitive to the practitioners. Acknowledgments This work was supported by the National Natural Science Foundation under Grant No. 10433020 and Foundation of physical study on electromechanical coupling of large antenna structure under Grant No. 50475171. The authors would like to appreciate the reviewers for their valuable comments and suggestions. References

5. Conclusions This paper deals with the theoretical and experimental studies on the CPR of FAST model of 5 m. To begin with, the inverse dynamics problem of the CPR with non-negligible cable mass has been formulated by the Lagrange formulation. Then, in view of random wind forces acted on the cabin are simulated based on the characters of the structure, a fuzzy plus proportional–integral controller is designed to control the wind-induced vibration of the trajectory tracking of the feed. In the end we provide the examples of simulation and experiment to justify the dynamic modeling for control and to test the proposed method. The simulation and experimental results in Section

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