Implementation and Control of X--Y Pedestal Using Dual-Drive Technique and Feedback Error Learning...

1646 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 22, NO. 4, JULY 2014

Implementation and Control of X–Y Pedestal Using Dual-Drive Techniqueand Feedback Error Learning for LEO Satellite Tracking

Alireza Taheri, Mahdi Aliyari Shoorehdeli, Hamid Bahrami, and Mohammad Hossein Fatehi

Abstract— This brief presents a X–Y pedestal using thefeedback error learning (FEL) controller with adaptive neuralnetwork for low earth orbit (LEO) satellite tracking applications.The aim of the FEL is to derive the inverse dynamic model ofthe X–Y pedestal. In this brief, the kinematics of X–Y pedestal isobtained. To minimize or eliminate the backlash between gears,an antibacklash gearing system with dual-drive technique isused. The X–Y pedestal is implemented and the experimentalresults are obtained. They verify the obtained kinematics of theX–Y pedestal, its ability to minimize backlash, and the reductionof the tracking error for LEO satellite tracking in the typicalNOAA19 weather satellite. Finally, the experimental results areplotted.

Index Terms— Antibacklash, dual drive, feedback errorlearning (FEL), kinematics, low earth orbit (LEO), satellitetracking, X–Y pedestal.

I. INTRODUCTION

ASATELLITE tracking station (STS) must be able to tracka satellite at any position in the sky above a few degrees

elevation. The best quality of data reception is obtained whenthe telemetry signal is stronger, i.e., where the satellite isclosest to the ground station. In addition, it is most importantthat the station can be capable of continuing a good trackingthrough and near the local zenith to assure quality receptionof the telemetry data [1], [2]. The pedestals are one of theimportant parts of each STS. The most common and well-known type of pedestals are the elevation over azimuth (here-after abbreviated as Az–Elv) pedestals. The keyhole problemin an Az–Elv pedestal occurs when the payload is trackinga satellite near its zenith position. The Az–Elv pedestals areunable to track continuously in cross of high elevation, becausein this point, the difference of azimuth angle after zenith is180° in comparison with prior azimuth angle before zenith andazimuth motor must turn the antenna during a short time [3].

One way to solve the problem of tracking through thezenith is to add a third axis to the Az–Elv mount; however,it increases complexity of the servo and driver system. Theother solution is using the X–Y pedestals because they have a

Manuscript received September 3, 2012; accepted August 27, 2013. Manu-script received in final form September 11, 2013. Date of publication October9, 2013; date of current version June 16, 2014. Recommended by AssociateEditor D. Liu.

A. Taheri, H. Bahrami, and M. H. Fatehi are with the Department of Elec-trical Engineering, Science and Research Branch, Islamic Azad University,Tehran 658815611, Iran (e-mail: [email protected]; [email protected];[email protected]).

M. A. Shoorehdeli is with the Department of Electrical Engineering,K. N. Toosi University of Technology, Tehran 19697, Iran (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2013.2281838

smoother tracking near to zenith. The X–Y pedestals have alsobeen named: X over Y, cross-elevation over elevation, traverseover elevation, cross-level over elevation, dual elevation, andtwo-axis zenith search [3]. The applications and advantagesof the X–Y mounts have been described in [1]–[7]. A three-degree-of-freedom (DOF) mechanism for using in the antennapositioning system has been presented in [8] that in itsconfiguration, the axes of rotation are X–Y over azimuth andthe effect of backlash has been eliminated by an antibacklashgearing system. The challenges of antenna positioning controland antenna performance with proportional-integral, LQG, andH∞ position controllers have been presented in [9] and [10].The predictive controller has been designed and simulated fortracking of the NASA Deep Space Network 70-m antennain [11]. In [12], errors in the telescope drive systems have beenanalyzed and models for the errors have been developed andalgorithms for online adaptive parameter identification havebeen presented. In [19], a control and measurement systemfor an Az–Elv antenna on a Navy Jammer pedestal has beendesigned and implemented.

There is the backlash between the teeth of gears that affectsthe tracking precision, and is an undesirable characteristic inthe X–Y pedestals. Hence, it must be minimized or elimi-nated. By electronic circuits and producing a torque difference(torque bias or counter torque) at [13] and [14], the backlashbetween gears has been eliminated in a 34-m beam waveguideantenna (analog backlash controller). In [15] and [16] usinga digital backlash controller and generating a counter torque,the backlash has been eliminated in an array antenna testbed.In [17] and [18], an antibacklash gearing system has beenprovided, which includes first and second gears that areenmeshed with another gear to provide a resultant torque formaking the rotational motion.

In this brief, the feedback error learning (FEL) controller isimplemented on experimental pedestal with X–Y configurationthat diameter of antenna is 1.2 m. The main advantage of FELis that modification of the previous conventional controlleralgorithm and plant Jacobian calculation is not required.To drive the freedom on X–Y-axes, two ac servo motorsare used separately and the elimination of the backlash iscried out by dual-drive technique. During the control action,the angles of rotation in antenna are necessary, and theyare obtained indirectly via measuring the angles on X- andY- axes by absolute SSI encoders. The control algorithm isimplemented on industrial computer by visual studio C++language.

The structure of this brief is as follows. Section II presentsthe X–Y pedestal kinematics and Section III formulates inverse

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TAHERI et al.: IMPLEMENTATION AND CONTROL OF X–Y PEDESTAL 1647

Fig. 1. Diagram of the X–Y pedestal and coordinate frames.

kinematics of X–Y pedestal. Backlash compensation and dual-drive technique are introduced in Section IV. Sections Vand VI introduce the FEL and neural controller. Experimen-tal setup description is discussed in Section VII and theexperimental results are presented in Section VIII. Finally,Section IX concludes this brief.

II. X–Y PEDESTAL KINEMATIC

Kinematic equation is necessary for analyzing of motion.This analysis is done without regarding the forces that causeit and it is the first stage to control the X–Y pedestal. An X–Ypedestal can be assumed as two-DOF manipulator robot thatplaced on a fixed surface. The reference frame can be fixed atany point on this surface and two coordinate frames are fixedon each link. The dependence of motion between two linkscan be described with a homogeneous transformation matrix.These frames are defined using the Denavit–Hartenberg(D–H) conventional system [20], [21]. The D–H system allowsfinding the relative of coordinates between any frames using acoordinate transformation. Hence, it is possible and simple tofind the position of antenna in any orientation. In this section,we derive the kinematic model of a common X–Y pedestalshown in Fig. 1. The detailed structure, the coordinate system,the joint variables, and parameters of X–Y pedestal are shownin Fig. 1 that XYZ represents the global reference frame [23].As shown in Fig. 1, the X–Y pedestal is an open-chain serialrobot arm with three links that Link1 is fixed to the base andLink2, Link3 are mounted on X and Y rotation axes as theseaxes rotate between ±90°. The D–H parameters are specifiedin Table I [24], [25]. The frames are defined as follows [23].

Frame {0} is pointed on the base. Frame {1} is connectedto the Link1, held on the joint1. Frame {2} is connected tothe Link2, held on the joint2. Frame {3} is connected to theantenna dish and is held on antenna feed and general (global)frame, as shown in Fig. 1.

Using Table I and D–H convention [24], the transformationmatrixes can be obtained in the following forms. In the

TABLE I

D–H PARAMETERS FOR X–Y PEDESTAL

following matrixes Cx = Cosx and Sx = Sinx:

01T =

⎡⎢⎢⎣

C θ1 −Sθ1 0h0Sθ1 Cθ1 0 00 0 1 00 0 0 1

⎤⎥⎥⎦

12T =

⎡⎢⎢⎣

Cθ2 −Sθ20 h10 0 −1 0Sθ2 Cθ2 0 00 0 0 1

⎤⎥⎥⎦

23T =

⎡⎢⎢⎣

1 0 0 h20 1 0 00 0 1 00 0 0 1

⎤⎥⎥⎦. (1)

The rotation matrixes are as follows:

01 R =

⎡⎣

Cθ1 −Sθ1 0Sθ1 Cθ1 00 0 1

⎤⎦

21 R = 1

2 RT =⎡⎣

Cθ2 0 Sθ2−Sθ2 0 Cθ2

0 −10

⎤⎦

32 R = 2

3 RT =⎡⎣

1 0 00 1 00 0 1

⎤⎦. (2)

The transformation between frame {0} and general frame canbe expressed as follows:

G0 T =

⎡⎢⎢⎣

0 1 0 00 0 1 01 0 0 00 0 0 −1

⎤⎥⎥⎦. (3)

The rotation matrix frame {0} at general frame is as follows:

G0 R =

⎡⎣

0 1 00 0 11 0 0

⎤⎦. (4)

After multiplying the matrixes, the total transformation matrixand rotation matrix can be obtained as follows [21]:

03T = 0

1T × 12T × 2

3T

=

⎡⎢⎢⎣Cθ1 ·Cθ2 −Cθ1 ·Sθ2 Sθ1 h2 ·Cθ1 ·Cθ2+h1 ·Cθ1+h0Sθ1 ·Cθ2 −Sθ1 ·Sθ2 −Cθ1 h1 ·Sθ1 + h2 ·Sθ1 ·Cθ2

Sθ2 Cθ2 0 h2 · Sθ20 0 0 1

⎤⎥⎥⎦

(5)


Fig. 2. Orientation of the antenna dish as described by Az–Elv and X–Y [36].

03 R =

⎡⎣

Cθ1 · Cθ2 −Cθ1 · Sθ2 Sθ1Sθ1 · Cθ2 −Sθ1 · Sθ2 −Cθ1

Sθ2 Cθ2 0

⎤⎦. (6)

As shown in (5) and (6), the h0, h1, and h2 parameters makeeffect on the translation vector and they are not importantin the orientation. The rotation matrix that converts into thegeneral coordinate can be calculated as follows:

G3 R = G

0 R × 03 R =

⎡⎣

Sθ1 · Cθ2 − Sθ1 · Sθ2 − Cθ1Sθ2 Cθ2 0Cθ1 · Cθ2 − Cθ1 · Sθ2Sθ1

⎤⎦. (7)

Thus, the translation vector is as follows:

p =⎡⎣

Sθ1.Cθ2.h2 + Sθ1.h1Sθ2.h2Cθ1.Cθ2.h2 + Cθ1.h1 + h0

⎤⎦. (8)

III. INVERSE KINEMATICS OF X–Y PEDESTAL

Robot kinematic equations are divided in two types, namelyforward or direct and inverse kinematics [20], [21].

The aim is to determine the desirable angles in joint space(θ1, θ2) for directing the antenna toward the satellite. As shownin Fig. 2, the �P unit vector is the orientation of satellitethat is expressed by Az–Elv angles. In X–Y pedestal, thejoint variables are (θ1, θ2), hence the Az–Elv angles mustbe described in X–Y coordinates. The �P can be obtained asfollows:

�P =⎡⎣

−Cos (Az) .Cos (Elv)Sin (Az) .Cos (Elv)Sin (Elv)

⎤⎦. (9)

After orientation action, the joint variables (θ1, θ2) must bepointed to satellite. Using the rotation matrix (7), the vector�P can be obtained according to (θ1, θ2) angles

G3

�P = G3 R.

⎡⎣

100

⎤⎦

=⎡⎣

Sinθ1.Cosθ2Sinθ2Cosθ1.Cosθ2

⎤⎦

⇒{

θ1 = −A tan 2 [Cos(El).Sin(Az) , Sin(El)]θ2 = Sin−1 [Sin (Az) .Cos (Elv)]

(10)

where

−π

2< θ1, θ2 < +π

2. (11)

Fig. 3. Mapping between Az–Elv and X–Y.

Fig. 4. Schematic diagram of backlash in a two-gear [38].

Fig. 5. Straight bevel gears [24].

Thus, the mapping that converts Az–Elv to X–Y angles canbe expressed as shown in Fig. 3 and (10). By this mapping, thedesirable joint variables (θ1, θ2) can be obtained from Az–Elvangles of satellite trajectory.

IV. BACKLASH COMPENSATION AND

DUAL-DRIVE TECHNIQUE

In mechanical engineering, clearance between mating com-ponents is called backlash. A schematic diagram of backlashin a pair of gear is shown in Fig. 4. The backlash has a stronginfluence on positioning accuracy. In precision positioningapplications with frequent load reversal such as robotics,satellite tracking, and so on, the effect of backlash should beminimized or eliminated. In this brief, an antibacklash gearingsystem, which includes two motors on each axis, is used.The gears that are used in this brief are bevel gears. Whenthe direction of a shaft’s rotation needs to be changed, thebevel gear is very useful and it usually mounted on shaftsthat are 90° apart. A typical straight bevel gear is shownin Fig. 5. In this brief, the motors have to be driven inopposite direction to control the backlash on the gears. Asa sign change in torque is occurred for one of the motors,there is a correction made to the torque as defined by thetorque curves for each motor so that the commanded torquefor each motor adds up to the total commanded torque for thetwo motors [17]. Fig. 6 shows the applied torques to motorswith a bias torque. As long as the total drive torque for twomotors is less than the bias torque, the backlash does not occur.


Fig. 6. Torque applied to the motor drivers with torque bias δ = 2 andwithout change in slope of the torques curve.

Fig. 7. Torque applied to the motor drivers with torque bias δ = 2 and withchange in slope of the torques curve.

Referring to Fig. 6, the input voltage versus torque can bederived as follows:

torque1 = (V − δ), torque2 = −(V + δ). (12)

When a sign change in torque is occurred for each motor, therate of torque change is reduced to minimize impact loadingon gear teeth during backlash. As shown in Fig. 7 at the signchange in torque for motor2, there is an increase in the slopeof the torque curve for motor1 and a simultaneous decrease inthe slope of the torque curve for motor2. The reduced gain onmotor1 is compensated by increasing the gain on the motor2 inan equal amount with opposite slope, as shown in Fig. 7. Theadvantage of torque bias with change in slope of the torquecurve is that it decreases the effect of backlash; hence, gearsare more preserved against of damage. The combined slopereduction and compensation shown in Fig. 7 is referred to asbacklash softening or backlash minimization [17]. Referringto Fig. 7, the input voltage versus torque can be derivedas [2]

torque1 =[(V−δ) − (V−δ)e−(V−δ)2

]+ (V+δ)e−(V+δ)2

torque2 = −[(V+δ) − (V+δ)e−(V+δ)2

]+ (V−δ)e−(V−δ)2

.

(13)

Fig. 8. FEL architecture [33].

In (12) and (13), V is controller output. In this brief, V isprocessed by an industrial computer and δ is the offset voltage(torque bias). Torque1 and torque2 are output voltages of D/Acard (PCL726) that is applied to motor driver1 and motordriver2, respectively (refer to Fig. 11). Additional equationsmay be derived that can have the same general characteristicsas (13) and eliminate the backlash and allow smooth engage-ment of mating gears.

V. FEEDBACK ERROR LEARNING

FEL technique has been proposed in [26] and [27], and itsgeneral structure is shown in Fig. 8. This approach requires thefeedforward controller to identify the complete plant inversedynamics for compensation and is one of the powerful meth-ods to obtain such an inverse model. Recently, Miyamura et al.have been established a control theoretical validity of the FELmethod in the framework of adaptive control and proved itsstability based on the strictly positive realness [28]–[32]. Ingeneral, FEL algorithm involves two steps [33].

In the first step, input signals are fed in a feedforwardcontroller through the network to produce actual outputs. Inthe second step, the output vector of a conventional feedbackcontroller (CFC) is considered as the error to propagatebackward through the feedforward controller. In the FEL, theCFC is essential to guarantee global asymptotic stability ofthe whole system. The stability proof of this controller wasdiscussed in [28] and [34]. In general, the CFC was realized bya predetermined constant gain controller [PID or proportional-plus-derivative (PD)] in FEL scheme in many applications.The only criterion used to select the gain is the stability ofthe system [35]. The success of training is depended directlyon the proper choice on CFC outputs. Hence, there is alwaysa problem of how to select the gain values of the CFC. Thefeedforward controller acquires a fully nonlinear inverse modelby trying to eliminate the feedback error. In the block diagramof Fig. 8, the UT is actual input vector to the plant, UF is theoutput vector from the feedforward controller, and UCFC isthe CFC output vector. The controller using FEL scheme isalso a hybrid controller. The feedforward controller is realizedby an artificial neural network or fuzzy controller that learnsthe inverse dynamics of controlled object using outputs of theCFC as teacher signals during controlling online. The mainobjective of FEL is to implement the feedforward controller asa function approximation, instead of designing linear controlleron the basis of a plant model [36]. The main advantageof FEL is that it does not require any modification of theprevious conventional controller algorithm and plant Jacobiancalculation is not required because the feedforward controller


Fig. 9. X–Y pedestal FEL controller scheme.

uses the output signal generated by the CFC as its own errorsignal that is back propagated for learning purposes. In thisbrief, a neural network controller that operates in parallel witha CFC based on the FEL architecture is used and implementedfor low earth orbit (LEO) satellite tracking. During trainingneural network, the output of CFC is gradually minimized,whereas the output of neural network becomes the maincontrol effort. The schematic diagram of used control methodis shown in Fig. 9 as the feedforward controller adopts inputsfrom error signals and CFC used two PD controllers in CFCsection. The output signal of PDX and PDY can be generatedby

uPDX = K px (Xd − XOUT) + Kdx(Xd − XOUT)

uPDY = K py(Yd − YOUT) + Kdy(Yd − YOUT). (14)

In (14), Xd and Yd are the desired position of pedestalantenna. K px and Kdx are proportional and derivational feed-back gain belong to PDX , and K py and Kdy are proportionaland derivational feedback gain belong to PDY , respectively.In addition, uPDX and uPDY are the outputs of CFC for twoaxes that are used for updating the unknown parameters of theneural network controller during the learning.

VI. NEURAL NETWORK CONTROLLER

The neural network that is used in this brief is multilayerperceptions (MLPs), at the feedforward path and its structure isshown in Fig. 10 as it consists of three layers. In this structure,R is number of inputs, S1 is number of hidden neurons, andS2 is number of output neurons. The relation between signalsof layers can be expressed by [37]

n1 = W 1 P + b1, a1 = f 1(n1)

n2 = W 2a1 + b2, a2 = f 2(n2) (15)

where P is input vector, f 2 is linear function, W 2 and b2 areweight matrix and bias of output layer, and a1 is the output ofsecond layer. f 1 is nonlinear sigmoid function, W 1 and b1 areweight matrix and bias of the second layer, and a2 is outputof neural network. According to Fig. 9, it is obvious that twoneural network controllers are needed, one for X-axis and the

Fig. 10. Structure of neural controller [27].

other for Y-axis as two signals uNNX and uNNY are outputs ofthem, respectively.

As it is said in Section II, the mechanism that is used in thisbrief is a two-DOF, so the reference inputs relating to themare used for training in neural network. In addition, first- andsecond-order derivatives are used to improve training as inputsof neural network. Hence, input vectors of neural section areselected as follows:

PX = [Xd Xd Xd

], PY = [

Yd Yd Yd].

Briefly, outputs of neural network controllers can be writtenas follows:

uNN = f 2(W 2 f 1(W 1 P + b1) + b2). (16)

A. Earning Algorithm

The error signal is defined as

E = uT − uNN.

The aim of training algorithm is to adjust the network weightsthrough the minimization of these cost functions

JX = 1

2E2

X , JY = 1

2E2

Y

where

EX = uTX − uNNX , EY = uTY − uNNY . (17)

In (17), uTX and uTY are the final control signals and uNNX

and uNNY are the neural network controllers output. It isdesirable that the output of CFC controllers (uPDX and uPDY )


Fig. 11. X–Y pedestal experimental system photograph.

reaches to zero. Using the backpropagation learning algorithm,the neural network parameters are adjusted such that the errorin (17) is less than a desired threshold value after a givennumber of training cycles. The adaptation law for weights andbias of neural network can be written as follows:

�(k + 1) = �(k) − η

(∂ J (k)

∂�(k)

)(18)

where η and � = [W 1, b1, W 2, b2

]are the learning rate

and tuning parameters of neural controller, respectively. Byrecursive applications of the chain rule, the error term foreach layer is first calculated, and then the parameters in thecorresponding layers are adjusted. The neural network actsonline in this method.

VII. EXPERIMENTAL SETUP DESCRIPTION

This section describes the control implementation and theexperimental setup. The experimental setup of the X–Ypedestal is shown in Fig. 11. It consists of a parabolic dishwith 1.2-m antenna diameter that mounted on a platform thatits freedom degrees are on the X and Y axes, as shown inFig. 1. The rotations with respect to each axis actuated withtwo ac servo motors and two gearboxes plus two servo drivesand four same straight bevel gears. The antenna rotations atX- and Y- axes are sensed by two encoders. These axes rotatebetween ±90°. The implementation of the control software isdone using VC6++ (Visual studio C++ 2006). This sectionis composed of two parts: 1) software implementation and2) hardware implementation, which are given in details in thefollowing.

A. Software Implementation

Software implementation includes graphical user interface(GUI) that is created in VC6++ and control algorithms areimplemented on industrial computer. Fig. 12 shows GUI ofcontrol software. To apply this software, the first stage isturning up and setting up the servo drivers by means of drivers

Fig. 12. GUI.

TABLE II

FORMATS OF SATELLITE TRAJECTORY PASS FROM PPP

control buttons in GUI program so drivers will be placed inready mode and then by click on open pass button in GUIprogram, the satellite trajectory that is obtained before canbe uploaded inside the algorithm. In this stage, the desiredtrajectory relating to the target satellite, that are Az–Elv anglesduring the time, is available in a text file such as *.txtfile for GUI program and it has before been generated byprepass planning (PPP) according to Fig. 9 and Table II. Then,according to the (10) and mapping block that is shown inFig. 3, satellite trajectory in Az–Elv angles can be mapped tothe X and Y angles so desired trajectory (mechanism rotationangles) that are named Xd , Yd can be obtained for usingin tracking algorithm. In the next stage, start pass button ispressed and the pedestal move to the start point (on the otherhand, rise point of satellite) and then by pressing satellitetracking button in GUI, satellite tracking will be ready fortracking the satellite. After this time, algorithm wait untilsatellite rises and when satellite begins to rise from risepoint, pedestal begins to track it and received signals are sentto the demodulator and then it draws received informationas meteorological images. Therefore, software can recognizerising time and setting time of satellite that it means beginningand end of tracking are determined for tracking system. Timesetting for industrial PC is done on GMT time according to


Fig. 13. Schematic diagram of X–Y pedestal.

Fig. 14. Weather map received from the NOAA 19 satellite on August 2010.

the GPS that is connected to PC. In addition to GUI program,the neural network and training algorithm is implemented byVC6++ and as it was said before, the initial weights ofneural network choose random and neural network controllerdo control activity online beside PD controller.

B. Hardware Implementation

The schematic diagram of hardware implementation forX–Y pedestal is shown in Fig. 13. It consists of two acservo motors (Mecapion model APM-SC05D) and two gear-boxes (with ratio 90:1) plus two ac servo drives (Mecapionmodel APD-VN04) for each axis and the antenna rotations atX and Y axes are sensed by two encoders (type SSI absolute15 b). Turning the drivers on and off and reading their status

Fig. 15. (a) NOAA 19 satellite trajectory produced by PPP. (b) NOAA 19satellite trajectory after mapping Az–Elv to X–Y.

are done by GUI program and one DI/O card (ICP DI/Omodel P32C32), that it is setup on PC (Advantech industrialcomputer). The number of pulses in encoder is read onlineby Deva01 card and then these data are used in softwareand converted to angle information, then software obtains thetracking error and uses it in control algorithm according toFig. 9. A six-channel Advantech D/A card (model PCL726)makes control voltage between ±10 V and it is sent to theac servo drives (Mecapion model APD-VN04) as setpoint.These ac servo drivers are set on torque mode and according toreceived control signal from control algorithm (voltage fromD/A card), they send required command to motors. Along withthis description, Figs. 9 and 11–13 can show more details.Limit switches are used to restrict range of motion on eachaxis between ±90°. In addition, time setting for industrialcomputer is done on GMT time by the GPS that is connectedto PC. Therefore, software can recognize acquisition of signal(AOS—the time when the satellite rises above the horizon)time and loss of signal time (LOS—the time when the satellitesets below the horizon) of satellite that it means beginning andend of tracking are determined for tracking system.

VIII. EXPERIMENTAL RESULTS

Several controllers on above system are applied and imple-mented to tracking LEO satellites and verify the inversekinematics of X–Y pedestal and dual-drive technique forminimizing the backlash effect. In addition, we consider PD


TABLE III

PD AND PID COEFFICIENTS

TABLE IV

PARAMETERS FOR X-AXIS

TABLE V

PARAMETERS FOR Y-AXIS

TABLE VI

SSE OF TRACKING ERROR

and PID controllers for CFC. The parameters of CFC areso designed that the stability of whole structure must beachieved. For neural network structure, we consider 3-5-1MLP network as Fig. 10. That contains three inputs, fiveneurons in hidden layer, and one output. The number ofhidden neurons and learning rate was determined by trialand error so that the tracking error is minimized and overfitting problem not occurs for neural network. The learningrate for adaptation processes is considered 0.04 and 0.05 forthe X and Y axes, respectively. In addition, the initial valuesof the neural network weights were randomly obtained fora range of ±2. The other method to choose initial valuesfor weights is according to the obtained experience of offlinetracking, so in the first, system must be run with a defaulttrajectory and during this tracking, the weights of neural net-work are saved and then the latest saved weights can be usedas initial weights of neural network in training. To ensure theaccuracy of the obtained inverse kinematics for X–Y pedestal,we have used the NORAD two-line element (TLE) fromwww.NORADcelestrack.com, produced the NOAA19 weathersatellite trajectory, and tracked this satellite via X–Y pedestal(the TLE is a standard mathematical model to describe asatellite’s orbit and TLEs are just one type of format for orbital

Fig. 16. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) PD controller output. (d) Outputtorques applied to X driver1 and X driver2.

elements [38]). The satellite trajectory passes format that isproduced by TLE and PPP is described in Table II. In thistable, the Az–Elv angles are at degrees and they are updatedin each 0.1 s. As can be seen in Table II, the AOS and LOStimes of satellite are 09:57:48.7 and 10:08:38.4, respectively,and the total time of satellite tracking is 6497 s. For satellite


Fig. 17. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) PID controller output. (d) Outputtorques applied to X driver1 and X driver2.

tracking, the Az–Elv angles of satellite (columns 6 and 7 ofTable II) are fed in algorithm—this information is sent rowby row, every 0.1 s, and used in control algorithm—and thenby use mapping (15), desired tracking angles in mechanismthat are named Xd , Yd are obtained. These obtained angles arereference inputs in control algorithm and they are shown inFig. 15 for tracking the NOAA19 satellite. If mapping (15)

Fig. 18. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) FEL outputs. (d) Output torquesapplied to X driver1 and X driver2.

be incorrect, the X–Y pedestal will not receive the data fromsatellite. However, the weather map received from NOAA19,which is shown in Fig. 14, demonstrates that the mappinggot in Fig. 3 is correct. Using the Xd , Yd on experimentalsystem at each 0.1 s and saving the results in the computer, theexperimental results are plotted. In these results, performance


Fig. 19. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) PD controller output. (d) Outputtorques applied to Y driver1 and Y driver2.

of PD controller, PID controller, and FEL technique on the X-and Y- axes and effect of dual-drive technique are illustratedat the following. Coefficients that are considered for PD andPID are shown in Table III. These coefficients are chosen withtrial and error, so that the system is stable and the error isminimized.

Fig. 20. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) PID controller output. (d) Outputtorques applied to Y driver1 and Y driver2.

A. Plot Experimental Results

In this section, the experimental results on the X and Y axesin X–Y pedestal are plotted. The backlash in the experimentalX–Y pedestal system at X- and Y- axes are about 2.0° and 5.0°,respectively. The experimental results by applying PD andPID controllers in X–Y pedestal system are shown in


Fig. 21. (a) Input and output satellite trajectory for tracking NOAA 19.(b) Satellite trajectory tracking error. (c) FEL outputs. (d) Output torquesapplied to Y driver1 and Y driver2.

Figs. 16(a)–(d), 17(a)–(d), 19(a)–(d), and 20(a)–(d) for twoaxes. As shown in Figs. 16(a) and (b), 17(a) and (b), 19(a)and (b), and 20(a) and (b), by applying PD and PID controllersand using dual-drive technique, tracking of satellite has asteady-state error. Figs. 16(d), 17(d), 19(d), and 20(d) showthe torques that are applied to the motor drivers by using PDand PID controllers (torque bias). The experimental results

by applying FEL method are shown in Figs. 18(a)–(d) and21(a)–(d) for X and Y axes, respectively. The performanceresults by FEL technique are shown in Fig. 18(a) and (b)for X-axis and Fig. 21(a) and (b) for Y-axis. It is obviousthat the tracking error in FEL is less than PD and PIDcontrollers. As shown Figs. 18(c) and 21(c) as soon as theneural network controller (UNNX ) is inserted to control activity,the PD controller output (UPDX ) leads to zero. The error leadsto zero. As Figs. 18(c) and 21(c) show, after a short time,neural network control system will be on hand. Figs. 18(d) and21(d) show the torques that are applied to the X and Y motordrivers using FEL and dual-drive technique (torque bias). Theparameters that are used for the X and Y axes are shown inTables IV and V.

In these tables, torque1 and torque2 are torques that areapplied to the ac servo drivers and torque1 and torque2are obtained from (18). Table VI shows that with numericalapproach, using the SSE of tracking error for each strategy.The SSE of Figs. 16(d), 17(d), 18(d), 19(d), 20(d), and 21(d)is calculated and shown in Table VI. In this table, one can seethat the SSE is decreased with FEL and the SSE in FEL isless than PD and PID controllers significantly for LEO satellitetracking.

IX. CONCLUSION

In this brief, a common X–Y pedestal with an antibacklashgearing system that used the FEL controller with adaptiveneural network for LEO satellite tracking has been presented.The used antibacklash gearing system is based on dual-drivetechnique. The main advantage of FEL is that the modificationof the previous conventional controller algorithm and plantJacobian calculation is not required. The kinematics andinverse kinematics of the X–Y pedestal have been analyzed.By the implementation and experimental results, the mappingthat converts Az–Elv to X–Y angles has been validated.The PD, PID, and FEL controllers have been applied to theX–Y pedestal for LEO satellite tracking. In the FEL technique,the output of PD controller is used as a teaching signal forthe neural network controller, hence the plant Jacobian is notneeded for tuning of the controller parameters.

The experimental results illustrate the proposed FEL con-troller provides better control performance compared with PDand PID controllers in satellite tracking. It was also shownthat using the dual-drive technique can minimize the effect ofbacklash.

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Implementation and Control of X--Y Pedestal Using Dual-Drive Technique and Feedback Error Learning...

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