carrier differential gps for real-time control of large flexible structures

Carrier Differential GPS for Real-Time Control of Large Flexible Structures

E. HARRISON TEAGUE, JONATHAN P. HOW, and BRADFORD W. PARKINSON

Stanford University, Stanford, California Received October 1996 Revised May 1997

ABSTRACT

Current laboratory tests demonstrate vibration and orientation control of a highly flexible vehicle using only the GPS carrier to measure motion. A 30 ft long test structure has been constructed that is suspended from above, and moves in a manner analogous to that of a flexible orbiting platform. Experiments show simultaneous rigid-body orientation and elastic vibration control by closing a feedback loop from the GPS differential carrier phase (DCP) measurements to on- board thrusters.

Use of the subcentimeter-level differential position information in DCP measurements is challenging because of inherent, and arbitrarily large, measurement biases (cycle ambiguities). This paper presents a new bias ambiguity resolution algorithm that combines measurements taken during motion of the structure with a model of the platform dynamics to initialize the unknown biases. The techniques presented are applicable to systems that exhibit relative motions with frequencies (< 10 Hz) and deflections (> 1 cm) that are detectable by current receivers.

NAVIGATION: Journal of The Zmtitute of Nauigation Vol. 44, No. 2, Summer 1997 Printed in U.S.A.

INTRODUCTION

Active structural control has been a topic of much research over the past two decades. Particular attention has been focused on orientation control and vibration suppression of orbiting space platforms, such as the Space Station. Such a structure is large and lightweight by necessity, and thus is very flexible, allowing slow vibrations with significant deflections if excited. Also, because of the vacuum environment in space, vibrations may persist for a long time. These structural vibrations could be counter- productive for humans and experiments on board, and over time could cause weakening of the platform. To control the motion of a structure, it is obviously necessary to measure its motion. This paper investigates using the GPS carrier as a source of information for structural vibration and orientation motion control.

Previous Work

The presence of GPS carrier signals offers an opportunity for subcentimeter relative antenna position measurements. It is challenging to use this high-precision information, however, because of the bias ambiguity (primarily cycle ambiguity) associated with carrier phase measurements. One of the first successful applications of this global information source was in the field of geodetic surveying. Scientists showed accurate measurement of the relative position between two static antennas by measuring the differential carrier phase (DCP) from several satellites in view (see sessions 5-7 in 111). Motion of the GPS satellites over time allowed resolution of the bias ambiguities.

The first application of DCP for sensing the motion of a dynamic platform in real time was aboard ships. In 121, ship heading, pitch, and roll were calculated using measurements at three antennas. The system used the assumption that the antennas were mounted to a rigid platform. Next, DCP was used for real-time aircraft attitude determination 131. Again, the antennas were assumed to be mounted on a rigid platform, but this time, the assumption was less valid. Wing flex can cause significant deflections of wing-tip- mounted antennas.

Aircraft wing flex was addressed in 141 by the addition of a single wing flex dynamic state. Further, a novel gradient search technique was developed for determining bias ambiguities using platform motion alone, without relying on the slow motion of satellites. Unfortunately, platform flexing belongs to a class of motion that is not observable using this gradient search bias initialization algorithm. As a result, wing flex was

215

assumed to be static during initialization of the biases, and then dynamically estimated with the attitude states.

This paper extends previous techniques to platforms whose relative antenna motions due to flexing are on the same order of magnitude as those due to overall platform attitude changes. An important part of this extension is a bias initialization algorithm that can account for antenna motion due to platform flexibility. This paper also shows the results of real-time control experiments using the flexible structure shown in Figure 1.

The paper first presents the description and results of the experiments, followed by details of the DCP measurements and bias ambiguity resolution algorithms. Finally, discussion and conclusions are presented.

THE GPS/FLEXIBLE STRUCTURE EXPERIMENT

Experimental Objectives

The following is a list of the experimental objectives for this research project:

Design an estimator that can observe selected rigid-body and elastic deformation states in real time using only GPS measurements. Evaluate the accuracy of the state estimates by comparing them with estimates based on independent rate gyro measurements.

l Successfully damp elastic vibrations in the test structure with a real-time control system.

l Demonstrate an automatic rigid-body orientation slew maneuver with simultaneous structural vibration control.

Experimental Setup

A diagram of the main elements of the experimental setup and their interconnection is shown in Figure 2. The following subsections describe each of these elements in turn. The hardware descriptions here are brief; see [51 for more details.

Physical Structure

The structure is an aluminum body that hangs on 12 m threads from an overhead crane in the laboratory. A diagram of the structure is shown in Figure 3. The structure consists of three massive rigid bodies (65 kg) that are connected by long, elastic aluminum tubes in a horizontal line. This configuration was selected to achieve relatively large, low-frequency deflections in the laboratory environment. The highly flexible tubes bend and twist as the whole structure vibrates slowly, with significant deflections in three dimensions. Passive structural damping is very small (time constants of several minutes), making the system a good candidate for active vibration control.

Fig. l-Experimental Structure in Motion

216 Navigation Summer 1997

; -stiuiture - - ---__-----__-----~-~-

process I , noise

o/Q /

I __--_____----__---___-----_---- *not used for control

Fig. S-Elements in the Control Loop

The suspension threads are attached at the top to a rigid beam that is mounted to a bearing, allowing the beam, and the structure below, to rotate freely in the horizontal plane. The system provides a platform whose motions are detectable by GPS antennas mounted on the structure.

The estimator and controller make use of a dynamic model of the test structure that was computed using MSCLNASTFUN on computer facilities at NASA/Ames. Suspension-induced modes were predicted in addition to the structure’s elastic modes. The modal frequencies predicted by this model were adjusted to fit Fourier transformed experimental data. The final linear

model contains under 10 modes below 2 Hz. Figure 4 shows the most important suspension, bending, and twist modes.

GPS Sensors

Since the testing was done indoors, the signals from the GPS satellites were not detectable. Therefore, a set of six pseudolite transmitters was constructed and mounted on the ceiling and walls of the laboratory. The transmitters are equally distributed in the “sky” to provide favorable signal geometry. One of the pseudolites was modified to broadcast a data message used for coarse synchronization of the GPS receivers.

Two hemispherical GPS patch antennas are mounted to each rigid body (Figure 51, for a total of six antennas. (A seventh antenna is also on a static stand in the laboratory; it is used for calibration and testing, but is not essential to system operation.) Two 4-antenna Trimble TANS Quadrex@ receivers measure the pseudolite signals and output DCP measurements at 10 Hz. The receiver core code is modified at the signal processing level to suit the experiment requirements. The signal from one of the antennas on the structure is split and runs to both receivers. This common signal at each receiver allows carrier measurements to be differenced only between antennas connected to the same receiver, which eliminates timing errors (see Appendix A).

Fig. 3-The Test Structure

Vol. 44, No. 2 Teague, et al.: GPS for Control of Flexible Structures 217

Suspension Modes Elastic Modes I- 4

,/’ jJJJ ,/ /’

/’ ,/”

1. /’

I ;

,’ lils “-x_.,, * . .

‘1 r --A.~

1. /’

“!i, /’

, ‘J”

5. jl’

4 ,/”

,/ ’ -ir,, ‘.~ J / “...,._

6 /’

/

Fig. I-Structural Modes

Fig. 5-Antennas and Thrusters

The resulting GPS sensing system consists of an array of interconnected antennas that provide enough information to resolve the overall motion, with sufficient bandwidth and deflection sensitivity for control. The DCP measurement equation is discussed further in the next section,

Rate Gyros

Six inexpensive rate gyroscopes were mounted to the test structure, primarily to provide an independent measurement with which to compare the GPS results. It is possible to take advantage of


Fig. 6-Rate Gyros

the good high-frequency performance of the gyros to enhance state estimates [61, but that is not the primary focus of this work. Estimates using a combination of GPS and other sensor types would make the capabilities and limitations of GPS difficult to distinguish. Figure 6 shows the gyroscope mounts (these can also be seen to the left of the block elements in Figure 1).

State Estimator

The states of the experimental system are defined as selected elastic and rigid-body structural modes and their derivatives. These states are estimated in real time at 10 Hz using an extended Kalman filter that has been optimized for execution speed. The original analytical filter development for this work is described in [71. The time updates of the filter are computed using the NASTRAN dynamic model.

The measurement updates are computed from the measurement equation linearized about the current estimated state. Relinearization is performed only when the state perturbations from the previous linearization exceed certain

thresholds, thus minimizing unnecessary processing.

The measurement equation and its gradients are formulated using the symbolic manipulator Autoled [81 to process the vector functions of the state variables. A custom text file manipulation code written in Per1 reads the Autolev output, and generates a C language program that can then be compiled and linked with the real-time application. The automatically generated C code text file is over 100 kb in size, but executes in less than 5 ms on a PentiumO 200.

The state estimator relies on an initialization process that provides an estimate of the biases intrinsic to DCP measurements. Bias estimation is discussed further in a subsequent section. Experimental results of the estimator are shown and discussed later in this section.

Controller

The controller used for the data presented in this paper is a multi-input/multi-output full-state regulator that minimizes a quadratic cost function of weighted states and control outputs. This


“linear quadratic regulator” (or LQR) minimizes the cost, J, where xj and uj, respectively, are the state and control vectors at time j, and

m

J = C x,‘Qxxj + UTQuUj (1) j=O

The weights (QX and Q,) were tuned by processing simulated data. Analysis of the system equations yields open- and closed-loop poles as shown in the z-plane (first quadrant) in Figure 7. Figure 8 shows that the closed-loop poles are

System poles (x open loop, + closed loop)

0 0.2 0.4 0.6 0.6 1

0.25

0.2

0.15

0.1

0.05

00

,

/ :

I-

Fig. 7-Pole Locations

System poles (x open loop, + closed loop)

5 0.8 0.85 0.9 0.95 1

Fig. g--Pole Locations (zoomed)

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indeed stable and damped. The single closed-loop pole that is closest to z = 1 corresponds to the rigid-body motion and has a settling time of approximately 30 s. The other closed-loop poles correspond to vibration modes and have settling times of approximately 5 s. The main limitation on the control performance of this system is the 1.4 N maximum thrust of the gas jet actuators. See [5] for a full description of the control formulation.

Actuators

A system of 24 on/off cold gas thrusters is used to impart forces on the test structure. The system pressure is 100 psi, and each thruster can deliver approximately 1.4 N of force. The thruster positions and orientations were chosen to provide controllability of all modes of interest while minimizing the number of thrusters. A cluster of four jets is shown in Figure 5. The thruster valves are capable of cycling at a maximum rate of 100 Hz.

Continuous control force requests from the controller were realized using pulse-width, pulse- frequency (PWPF) modulation of the on/off valves. For details, see 151.

Real-Time Computer

Real-time data collection, processing, and control implementation were performed on a Pentium Pro 200 running LynxOSO Ver. 2.4.0 from Lynx Real-Time Systems@ [9]. The code was written in a multithreaded, priority-based framework conforming to POSIX real-time standards.

Real- Time Data Processing

Figure 9 illustrates the main components of the real-time code. GPS data is received though two

Q thread of execution

Fig. 9-Real-Time Processing Setup

Summer 1997

serial ports and stored in global space. When the state estimator is signaled of the presence of new data, it processes the data, and control feedback is computed based on the new estimates. This control feedback is passed to the control modulator that commands the thruster valves though a digital I/O card.

Real- Time Es tima tor Results

Figure 10 plots estimates of the rotation angle of the structure’s center rigid body about a laboratory fixed vertical axis. This data was processed in real time following a dynamic bias estimation (described in a later section). The structure was initially rotated (manually) as a whole in the horizontal plane to illustrate state estimation during a large-scale platform slew. Then (at about 140 s), the structure was randomly excited to illustrate estimation during elastic vibration.

GPS- and gyro-based estimates are plotted. The gyro estimate is obtained by integrating the body fixed vertical axis gyro signal, which is equal to the laboratory fixed rotation for small out-of-plane rotations. Figure 11 shows the difference between the two estimates. The first plot shows a linear trend in the difference that is attributable to drift in the gyroscopes. The second plot shows the difference with the linear trend removed. For analysis of GPS estimates, it is fair to remove this trend, which is known to be contributed by the

gyros alone. The second plot shows that the difference is always less than 1 deg and has an RMS value of 0.3 deg. These results show that the estimates based on the independent sensors agree during both the slew maneuver and elastic vibration.

These results are significant in that they show a subdegree, subcentimeter sensor of general structural deformations with zero drift characteristics. This performance is available wherever the GPS signal environment exists, or can be recreated (as in our laboratory).

Real- Time Control Results

Thus far, the GPS sensor system and estimation objectives have been discussed. Now, experimental test results that address the goals for real-time control are presented. The first test illustrates active vibration damping. The structure was perturbed manually such that it vibrated “randomly” with relative antenna deflections on the order of 10 to 20 cm. Because of the low system damping, vibrations typically persist for 5 to 10 min after excitation. Figure 12 shows estimated angles of rotation of each of the three rigid sections about the longitudinal twist axis (coinciding with the elastic beams when the structure is at rest; see Figure 3) during this vibration. The first 15 s shows free motion. At about 15.5 s on the plots, the control loop is closed,

Center body vertical axis rotation


2-

0-

$ -2-. p! B 0 -4-

-6 -

-6-

-lO-

.- 0 I

100 150 200 250 seconds

Fig. lo-Comparison of GPS- and Gyro-Based Estimates

Difference in GPS and Gyro estimate

Linear trend removed 31 I I I I 1

IRMS=0.32 degree$

-v

0 50 100 150 200 seconds I

Fig. II-GPS and Gyro Estimate Difference

and the thrusters begin firing. Vibration damping is achieved with a settling time of approximately 5 s, and the angles are controlled to within 0.15 deg as measured by GPS. This settling time agrees with the prediction based on the computed closed-loop poles.

The second plot in Figure 12 shows two raw integrated gyro measurements. The measurements were not initialized at 0 deg, so the absolute y-axis values are not significant. In contrast with the top plot of estimator outputs, this second plot shows unfiltered structure motion measurements before and after loop closure. Also, as a measure of the open- and closed-loop estimator accuracy, the bottom plot of Figure 12 shows the difference between the GPS- and gyroscope-based state estimates. Agreement to better than 1 deg of rotation angle between the independent sensors is maintained throughout the test.

The next experiment was a test of the feedback system’s ability to perform a slew maneuver. The structure was rotated 16 deg in the horizontal plane and left at rest. At time zero, a command for 0 deg rigid-body angle was issued to the control system. The maneuver settled within 30 s, which is fast given the limited thruster force and large rotational inertia associated with this motion. Figure 13 shows the time response plots. The first plot is the estimated slew angle, which settles to within 0.5 deg as measured by GPS. The second

222 Navigation

plot shows the relative vertical axis rotations of each of the rigid sections of the structure. These plots indicate the ability of the control system to simultaneously regulate internal vibrations (to within 1 deg) during a slew maneuver.

The experimental results presented in this paper depend on the ability to make subcentimeter-level GPS carrier-based relative position measurements, and the ability to resolve the system states given these measurements. The following two sections discuss the basics of DCP and some methods for resolving the unknown biases associated with these measurements.

DCP MEASUREMENT EQUATION

This research uses carrier measurements that are differenced relative to a common clock reference. This is achieved by multiplexing several antennas connected to the same receiver. (The general carrier phase observable is developed in Appendix A.) The resulting DCP measurement equation is illustrated in Figure 14. In our current receivers, a DCP measurement is the difference in phase between a “master” antenna and one of the “slave” antennas. The vector between the antennas is called the “baseline vector.” The measurement consists of the difference in the line-of-sight distance from a transmitter to each antenna, plus a bias that is fixed at the time of carrier lock at

Summer 1997

Horizontal axis twist rotation of each rigid section

- Rigid section 1 - - Rigid section 2 ‘- -. - Rigid section 3

Closed loop : I I I

20 25 30 35

Raw integrated gyro measurements (no gyro on 3rd section)

Difference in GPS and Gyro based estimates (no gyro on 3rd section)

5 10 15 20

Fig. 12-Elastic Vibration Control

25 30 35

each antenna. In some cases, this bias may be written as the sum of the so-called integer ambiguity and line bias, but for this paper, the quantity is left as a general bias. The DCP measurement may be written

PNjPi =

bij =

Vij =

position vector from Nj to Pi bias associated with baseline i and transmit antenna j stochastic noise

Qij = IpNjpiI - lpNjpMl + bij + vij (2)

where A+ij = DCP of baseline i, transmitter j

Pi = point corresponding to the phase center of antenna i (i = M denotes the master antenna)

Nj = point corresponding to the phase center of transmit antenna j

If the transmitter is sufficiently distant from the antennas, the received wavefront is well approximated as planar. This allows the carrier phase measurement to be written as the vector dot product of the baseline vector with the line-of-sight vector to the transmitter. The indoor experiment uses pseudolite transmitters that are too close to the experiment to allow for the planar assumption. See [lo] for further details.


Full structure rigid-body rotation I I t I I I I I

-20 I I , I I I I I 0 10 20 30 40 50 60 70 80

Time (set)

Vertical axis rotation of each rigid section (- 1, -- 2, -. 3) 3 1 I I 1 1 , I f

2_ . ...’ ‘. :, : ..; ,,,,,,., I :..._

-3’ 0

I I

10 20

I 1 I I I I I 30 40 50 60 70 80

Time (set)

Fig. 13-Slew Maneuver Control

‘*

N

** ‘.

‘. ,’ ‘. ,’

,’

Fig. II-DCP Measurement

The next step is to form the vector measurement equation for the whole system by stacking measurements:

A+ ; h(x) + b + v (3)

Symbols without subscripts denote a vector of stacked quantities, and x represents the system state. (A number under the equals sign is a reference to a previous equation). h(x) contains the geometric functional dependence in the

measurements and depends on the vector magnitudes, which in turn depend on x.

Note that for nonplanar wavefronts, h(x) also depends strongly on the transmitter phase center locations, which must be estimated. A phase center estimation algorithm based on GPS measurements was developed and used, but is not presented here. See the appendix of [51 for details of the algorithm.

Equation (3) can be linearized about a nominal state and bias, X and 6, respectively, to form

Ar$ ; H(ii)Gx + h(E) + i; + 6b + v (4)

where

x = x + 6x, b=6+6b

(5)

Since there are always as many unknown biases as measurements, this system of equations is underdetermined. Hence, another method of estimating b is sought.

BIAS ESTIMATION

The purpose of a bias estimator is to find b in equation (3). Three types of estimators (in order of


increasing complexity) are discussed below: static, kinematic, and dynamic. First, the formulation of the mathematics of each estimator type is presented. Then the operational and performance differences of the estimator types are discussed.

Static

A static bias estimator is simple, but is rarely practical for operational use because it requires motionless data collection during which an accurate state estimate is known. The biases then may be computed from equation (3) as b = Z$ - h(x), where @ is the averaged phase measurements, and St is the best guess of the state. If the measurement noise, v, is zero mean, the primary source of bias error is errors in the state estimate, X. The sensitivity of b can be analyzed by computing the maximum singular value of H(E) from equation (5). This value gives an indication of the potential worst-case bias error due to small errors in X.

Kinematic

A kinematic bias estimator uses multiple measurement sets (the P set being a vector of measurements taken from the system that is in some state, ~(~9, and uses the knowledge that the biases are constant to form an overdetermined system. If there are n states and m measurements per set, combining N sets and linearizing yields

0

’ H’N’

I I

_j I

6X”’

6X(2’

s&N' + v (6)

Sb _

or, (7)

where the superscript is the measurement set index, and

tjy(i) = A+(i) _ h($i)) _ 6 (8)

H(i) = H@(i)) (9)

iSy(” is the measurement residual. If the system is observable, a solution can be found by iterating the weighted least-squares equation with weights defined by the diagonal matrix, R (a good choice of R is an estimate of the inverse of the measurement noise covariance):

= (fiT Rfi)-l# R Sy

Given an initial guess of X and b, the iteration proceeds by computing I% from equations (5)~(7)

and Sy from equations (6)~(81, computing updates from equation (lo), updating 5Z and b using &,,, = &Id + 6% and S,,, = bold + 86, and repeating until convergence.

For this system to be observable, $I must have full column rank. First, this requires that enough datasets be collected such that fi is “tall.” This requires N > m/(m - n). Another interesting requirement for observability is that no column of H”’ be constant for all i (see Appendix B for a proof). This indicates that the measurement, h(x), must be a nonlinear function of the state x, and that the data, 6y, must depend on this nonlinearity. Further, the sensitivity of each state variable (column of H”‘) must be a function x.

These observations lead to somewhat detailed requirements for system motion during data collection. An alternative that may provide a better bias estimate, given a sufficiently accurate model, uses knowledge of system dynamics.

Dynamic

If the antennas are mounted to a system for which there is a dynamic model, it is possible to use this information to improve our bias estimate. For example, given a (discrete) dynamic model of the form

Xi+1 = Axi + BWi, i = 1, . . . N (11)

the augmented state can be formed as

b F 1 Xl

X aug = (12)

A linearized equation that incorporates all the data and known system information 1111 is given by or

(14)

The process noise, wi, is not written as a perturbation because it is assumed to be linear and zero mean.

The procedure for performing a batch dynamic solution of the measurement biases is as follows:

1) Collect and store measurements at N equally spaced epochs while the physical system is in motion.

2) Define the diagonal weighting matrix W-l = E[vv&J as the best estimate of the process noise covariance. As in equation (lo), define


3)

4)

Sy”’ Sy’2’

&Gil

0 0

i,

=

L

I H”’ 0 0 0 I Hc2’ A Ht2’ B 0 0 I Hc3’ A2 Hc3’ AB H’3’)3 . . . 0

I Hc4’ A3 Hc4’ A2 B Hc4’ AB 0

1 H’N’ AN-1 H’N’ AN-2 B H’N’ AN-3 B . :. HOi B

R-l = E[vivr] as the best estimate of the measurement noise covariance. Define the diagonal weighting matrices, p and X1. These determine the relative weighting of the b and x1 portions of the state, and they provide a means of placing a cost on very large jumps in b and x1 per estimator iteration. They may be set to zero, or scaled to improve the convergence of the batch estimator. Perform the batch solution by iterating equation (15) until convergence:

I -!

[f] = (CTRC + [” X1 A)hTRSy (15)

The batch dynamic approach provides a method for refining the bias estimates by including additional information from a platform dynamic model. As shown in 151, this dynamic approach eliminates the need for the rigid platform assumption and allows accurate estimation of measurement biases on platforms with significant flexibility. The best bias estimator for a particular application depends on several factors, including (1) required estimate accuracy, (2) motion available during data collection, (3) magnitude of platform flexibility, and (4) model accuracy. All the experimental results presented in this paper used the dynamic algorithm.

Discussion of Bias Estimators

The static estimator has value for certain applications because of its computational simplicity. Given a static bias estimate based on a state guess, X, subsequent (kinematic least- squares) state estimate errors are equal to the error in si, as long as the linearization in equation (4) remains valid. However, the static estimator requires data taken while the system is at rest in a known state. For this reason, the static algorithm is not oRen valuable outside of a laboratory setting.

Finally, the bias estimators presented here use a batch approach, where the estimate update is computed using all the available data simultaneously. It is possible to use iterative recursive smoothing techniques to arrive at the same result 1131. As is well known, the computational efficiency of recursive processing is superior, especially for systems with many states. Also, recursive techniques are more flexible to operational nonidealities, such as temporary signal losses 1121. However, a much better initial guess is needed for the recursive than for the batch algorithm for reliable convergence. Because of this limitation, a batch algorithm was chosen for the experimental work.

Kinematic algorithms have been used exclusively for the research on which this work builds [4, 121. The advantages of this approach over dynamic bias estimation are speed and simplicity. The sparse structure of equation (6) can be exploited for efficient computation. However, we have not yet obtained usable bias estimates from a kinematic algorithm for the flexible structure experimental system. Furthermore, it is shown in El that kinematic bias resolution algorithms that use platform motion to provide bias observability rely on a rigid (or nearly rigid) platform assumption. Since the flexible structure designed for these experiments is not well approximated as rigid, a new technique was needed that could take platform flexibility into account for bias estimation.

For the experimental work described in this paper, the unknown biases were initialized by first storing measurements during system motion, and then applying the dynamic algorithm to the stored measurements (convergence required iteration for approximately 1 min). These estimates were then used in the extended Kalman filter that computed real-time state estimates for control.

CONCLUSIONS

The experimental results presented in this paper are

l

promising for a number of reasons:

Estimation of the elastic and orientation states of a highly flexible structure was demonstrated using only GPS carrier measurements. Rotation estimate accuracies of


Sb 6x1

Wl

w2

WN-1

0.3 deg standard deviation were measured relative to independent gyro sensors.

l The implication is that high-accuracy, zero- drift deformation measurements are available anywhere the GPS signal environment exists, or can be created.

l A constant gain controller based on a simple linear dynamic model was able to control both structural orientation and elastic vibrations simultaneously using pulsed thrusters for actuation. Tests show settling times of approximately 5 s for vibration damping and 30 s for pointing control at subcentimeter/ subdegree positioning accuracy.

l These results show the use of GPS for control of a system with complicated dynamics depending on many parameters (the experiments in this paper use 36 measurements and 18 states). This represents a next step in the sophistication of distributed antenna GPS sensing systems.

Three different methods of resolving the intrinsic bias uncertainty in DCP measurements have been shown: static, kinematic, and dynamic. Kinematic algorithms that use antenna motion to provide bias observability were found to be insufficient for the highly flexible experimental platform. Thus, a dynamic algorithm was developed that combines measurements taken during platform motion with a model of platform dynamics. This algorithm was used successfully to initialize measurement biases prior to the real- time control experiments presented in this paper.

Based on the success of the experiments presented herein, the authors believe that this GPS-based platform attitude and flexibility sensor is directly applicable to real systems that have motions commensurate with the bandwidth and sensitivity limits of current receiver technology. The only requirement is the existence of a GPS signal environment, which could be supplied by the GPS satellites, by pseudolites, or both. Further system performance improvements could be obtained by more accurate model determination (either analytically or by using measurement- based system identification) and a more refined control compensator design.

ACKNOWLEDGMENTS

We would like to thank several individuals and organizations whose participation in this research has been invaluable. We acknowledge London Lawson for designing and constructing the actuation system. Thanks to Michael Boerjes for the hardware interface to the sensors and actuators. Thanks also to Kurt Zimmerman, Konstantin Gromov, Stewart Cobb, Paul

Montgomery, and Clark Cohen for valuable help and suggestions. We would also like to thank Roy Hampton and Mladen Chargin at NASA/Ames for their assistance with MSCYNASTRAN.

We gratefully acknowledge Trimble Navigation for providing receivers and access to the receivers’ core source code. This research is funded by NASA contract NAS8-39225.

Based on a paper presented at The Institute of Navigation ION GPS-96, Kansas City, Missouri, September 1996.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Goad, C. C., ed., Proceedings of the First International Symposium on Precise Positioning with GPS, International Union of Geodesy and Geophysics, International Association of Geodesy, U.S. Department of Defense, U.S. Department of Commerce, 1985. Kruczynski, L. R., Li, P. C., Evans, A. G., and Hermann, B. R., Using GPS to Determine Vehicle Attitude: USS Yorktown Test Results, In Proceedings of the International Technical Meeting of the Satellite Division of The Institute of Navigation, Colorado Springs, CO, September 1989. van Graas, F. and Braasch, M., GPS Znterferometric Attitude and Heading Determination: Initial Flight Test Results, NAVIGATION, Journal of The Institute of Navigation, Vol. 38, No. 4, Winter 1991, pp. 359-78. Cohen, C. E., Attitude Determination Using GPS, Ph.D. dissertation, Stanford University, December 1992. Teague, E. H., Flexible Structure Estimation and Control Using the Global Positioning System, Ph.D. dissertation, Stanford University, May 1997. Montgomery, P. Y., Uematsu, H., and Parkinson, B. W., Analysis of Angular Velocity Determination Using GPS, In Proceedings of The Institute of Navigation GPS-94, Salt Lake City UT, 1994. Teague, E. H., How, J. P., Lawson, L. G., and Parkinson, B. W., GPS as a Structural Deformation Sensor, In Proceedings of the AIAA Guidance, Navigation and Control Conference, Baltimore, MD, August 1995. Autolev, Autolev 3.1 Users Manuals, OnLine Dynamics, Inc., Sunnyvale, CA, 1996. LynxOS, LynxOS 2.4.0 Users Manuals, Lynx Real- Time Systems, San Jose, CA, 1995. Teague, E. H., How, J. P., Lawson, L. G., Boerjes, M., and Parkinson, B. W., Technique for Real-Time Control of Flexible Structures Using GPS, In Proceedings of the AAS Guidance and Control Conference, Breckenridge, CO, February 1996.

11. Bryson, A. E., Class Notes, Optimal Estimation with Uncertainty, Stanford University, 1993.

12. Pervan, B. S., Navigation Integrity for Aircraft Precision Landing Using the Global Positioning System, Ph.D. dissertation, Stanford University, Stanford, CA, March 1996.


13. &lb, A., Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974.

14. Kailath, T., Linear Systems, Prentice-Hall Information and System Sciences Series, Prentice- Hall, Englewood Cliffs, NJ, 1980.

APPENDIX A GENERAL DIFFERENTIAL CARRIER PHASE ANALYSIS

The phase of an incoming, down-converted carrier wave may be written as (ignoring a constant offset)

+ = (0 - o,>t + $x (A-1)

where

o, A = frequency and wavelength of incoming wave

channel specific: 4 = measured phase at the antenna phase

center o, = downconverter reference oscillator

frequency t = time of phase measurement

applicability x = distance from the antenna to the

transmitter

Figure 15 shows a diagram of such a receiver channel. Let AC) = (‘jZ - (-jl. If we assume that there is perfect phase lock in both receivers, and that At is small with respect to the antenna motion bandwidth, the difference in carrier phase between two antennas is

Downconvertcr Correlator I

I t I

I I

I I

I I

I I

I I

I I

I Clock ’ I

I I

___---------se-4

Receiver

Fig. 15-General Phase Reception Diagram

A+ = Kw - w&z - (w - w,,)tll + !f Ax (A-2)

Neglecting second-order terms and letting t1 = t and w,, = oe,

A$ = Ko - o,)At - Aw,t] (A-3)

+ 27FAx A

The term in brackets in equation (A-3) represents error terms related to timing. Our receivers use antenna multiplexing and a common clock reference; thus Aw, = 0:

A@ = (w - w,)At + F Ax (A-4)

Now, At is a function only of multiplexing switching time and unequal antenna cable lengths. These are constant over time after phase lock, and thus constitute a line bias term.

In the absence of a common clock reference, other techniques must be employed to make use of the differential carrier observable, such as double differencing to remove relative clock offsets between differenced channels.

APPENDIX B KINEMATIC ESTIMATOR OBSERVABILITY PROOF

Theorem: Given

r I..$” 0

fi= H(2)

(B-1)

$I is not full column rank if any column of H”’ is constant for all i.

Proof: n is not full column rank if all its columns are not linearly independent 1141, equivalently, if we can find any B # 0 such that

BV = 0 (B-2)

Let hi) equal the kth column of H(“. Define vk as a vector whose length equals the number of columns of Hci), and whose elements are all zero except for the kth element > which is 1.

-0-

&= 1 - kti element

_o_ Suppose

(B-3)


with vI arbitrary. Then I-

(B-4)

Thus, if hi) is constant for all i, we can set the arbitrary vl = -he), and then

ii*=0

! H(l) vk + VI

H@' vk + VI na= :

HcN’ v; + VI

Vol. 44, No. 2

(B-5)

Teague, et al.: GPS for Control of Flexible Structures 229

carrier differential gps for real-time control of large flexible structures

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