Gimbal_modeling.pdf

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003 3

Direct Versus Indirect Line ofSight (LOS) Stabilization

Peter J. Kennedy, Member, IEEE, and Rhonda L. Kennedy, Member, IEEE

AbstractTwo methods are analyzed for inertially stabilizingthe pointing vector defining the line of sight (LOS) of a two-axisgimbaled laser tracker. Mounting the angular rate and accelera-tion sensors directly on the LOS axes is often used for precisionpointing applications. This configuration impacts gimbal size, andthe sensors must be capable of withstanding high angular slewrates. With the other stabilization method, sensors are mountedon the gimbal base, which alleviates some issues with the directapproach but may be less efficient, since disturbances are not mea-sured in the LOS coordinate frame. This paper investigates the im-pact of LOS disturbances and sensor noise on the performance ofeach stabilization control loop configuration. It provides a detailedanalysis of the mechanisms by which disturbances are coupled tothe LOS track vector for each approach, and describes the advan-tages and disadvantages of each. It concludes with a performancecomparison based upon simulated sensor noise and three sets ofplatform disturbance inputs ranging from mild to harsh distur-bance environments.

Index TermsAngular rate sensor, gimbal, rate gyro, stabiliza-tion.

I. INTRODUCTION

THE REQUIREMENT to point a laser from a dynamic plat-form continues to surface in both defense and communi-cation applications. The laser and track sensor (i.e., camera, de-tector array, etc.) are mounted on the inner axis of a multiaxismechanical gimbal; or below the gimbal, coupled to the line ofsight (LOS) via a stabilized pointing mirror. Pointing controlis implemented via two servo loops, the outer track or pointingloop and an inner stabilization or rate loop. The track sensordetects the laser returns from the target location. The track pro-cessor uses this information to generate rate commands that di-rect the gimbal bore-sight toward the target LOS. The stabiliza-tion loop isolates the laser and sensor from platform motion anddisturbances that would otherwise perturb the aim-point. Thetrack loop must have sufficient bandwidth to track the LOS kine-matics. The stabilization loop bandwidth must be high enoughto reject the platform disturbance spectrum. A typical configu-ration is illustrated in Fig. 1.

Disturbances that affect the pointing vector arise fromplatform angular motion or maneuvers, and external loadssuch as wind and air-stream induced torque. Platform linearmotion and vibration generate disturbance torques due to mass

Manuscript received November 30, 2000; revised March 15, 2002. Manu-script received in final form October 1, 2002. Recommended by Associate Ed-itor D. W. Repperger.

The authors are with David H. Pollock Consultants, Inc., Sensor TechnologySpecialists, Westwood, NJ 07675 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCST.2002.806443

imbalance and gimbal geometry. Conventional stabilizationtechniques employ rate gyros, rate integrating gyros, or ratesensors to sense rate disturbances about the LOS. They areoften mounted on the inner gimbal axis, as close as possible tothe gimbal bore-sight (i.e., in Fig. 1), or in some applicationat the gimbal base. The mounting configuration can have asignificant impact on pointing performance.

This paper focuses on two methods of implementing the sta-bilization servo loop design. A two-axis gimbal ( over ),as shown in Fig. 1, is considered. Direct LOS stabilization, withthe angular rate sensors mounted on the LOS axes, is normallyrecommended for precision pointing applications [1], [2]. Thetechnique is simple to implement, requiring only two rate sen-sors and the associated rate loop compensation. Gimbal size isimpacted, since a larger payload volume is required to mountthe sensors on the inner gimbal. For high-performance aircraftapplications, the penalty in drag and turbulence induced by alarge gimbal penetrating the air-stream may be severe. In ad-dition, the sensors must be capable of functioning, or at leastwithstanding, high angular rates generated during slew.

With indirect LOS stabilization, currently being pursued forseveral applications, the rate sensors are mounted on the gimbalbase. This can alleviate some of the problems associated withthe direct stabilization approach. The disturbances, however, arenot measured in the LOS coordinate frame, which can degradeperformance. The base mounted sensors must accurately mea-sure the rates at the gimbal base. A processor transforms themto an equivalent disturbance about the LOS. A rate loop feed-forward compensator applies a control signal proportional to theestimated rates, attempting to cancel the disturbances. The re-lationship between the actual LOS disturbances and those mea-sured at the base of the gimbal is dependent upon gimbal ge-ometry, structural rigidity, resolver accuracy, tachometer and/orencoder accuracy, and the processor sampling rate (if the con-trol signal is processor generated). This method is more complexthan the direct approach.

II. GIMBAL KINEMATIC RELATIONSHIPS

Three coordinate frames are used to define gimbal motion,starting with the gimbal base frame ( ), followed by the gimbalouter frame ( ), and finally the gimbal inner frame ( ). LOS el-evation control is exercised via the inner gimbal -axis, whileazimuth control is applied via the outer gimbal -axis. It isimportant to note that stabilization is required about the innergimbal -axis, called cross elevation. Therefore, the cross-eleva-tion axis is actually controlled indirectly. The coordinate trans-

1063-6536/03$17.00 2003 IEEE

4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 1, JANUARY 2003

Fig. 1. Two-axis tracker configuration.

formation between the base and outer coordinate frame is (notea bar denotes a matrix, and a hat denotes a vector)

relative azimuth angle between the outerand base coordinate frames.

The base coordinate frame rate vector is related to theouter coordinate frame rate vector by

(1)

where

The coordinate transformation between the outer and innergimbal coordinate frame is given by

relative elevation angle between innerand outer gimbal coordinate frames.

The outer coordinate frame rate vector is related to the innercoordinate frame rate vector, , by

(2)

The relationship between the angular acceleration of the innerand outer gimbal coordinate frames is obtained by differenti-ating (2) or

(3)where

III. GIMBAL DYNAMICS MODELThe gimbal dynamics model can be derived from the torque

relationships about the inner and outer gimbal body axes basedon rigid body dynamics. The expression for the inner gimbalangular dynamics is given by ( denotes the cross productoperation)

(4)where sum of the kinematic torques about the inner gimbal,

, is

withFriction and cable restrainttorques.Inner gimbal inertia matrix.Reaction torques exerted by innergimbal on outer gimbal.Mass imbalance torques abouteach gimbal axis.Elevation stabilization controltorque.

KENNEDY AND KENNEDY: DIRECT VERSUS INDIRECT LINE OF SIGHT (LOS) STABILIZATION 5

The friction and cable restraint torques generate both linearand nonlinear disturbances, [3]. To simplify the analysis, it isassumed that the gimbal rotation axes are aligned with the prin-cipal axes of inertia so that the inertia matrix is diagonal and theinner gimbal dynamics equation can then be expanded as

(5)

(6)(7)

whereviscous friction coefficient;cable restraint coefficient;

;;

nonlinear friction torques;nonlinear cable restraint torques.

In (5)(7), and those equations that follow, the explicit depen-dency on time, , has been suppressed in the expanded expres-sions where no confusion will arise. The reaction torques on theouter gimbal are determined from (5) and (7). The elevation con-trol axis is the inner gimbal -axis, whose dynamics are gener-ated by (6). The inputs to this equation are the base rates and ac-celerations, while the controlled variables are and . Sub-stituting (1) into (2) and then into (6), the elevation axis dynamicscan then be expressed in terms of the base inputs as

(8)In a similar manner, the rigid body torque dynamics for the

outer gimbal body are derived from

(9)

where sum of the kinematic torques about the outer gimbal is

withouter gimbal reaction torques on base;outer gimbal mass unbalance torques;azimuth axis stabilization control torque;friction and cable restraint torque;

(10)

As with the inner gimbal, alignment is assumed between the prin-cipal inertia axes and the axes of rotation, so that the outer inertiamatrix is also diagonal. To express the dynamics in terms of theinner body -axis (cross elevation) one can solve (3) for andsubstitute the resulting expression and (10) into (9) to obtain

(11)The cross elevation axis dynamics are generated by the third

element of vector (11). Expanding the equation for this compo-nent results in

(12)

where denotes the third element of the vector. Also the lastterm on the left-hand side of (11) is zero. From (3), the angularacceleration about the inner -axis can be obtained. Substitutinginto (12), expanding the cross product terms and substituting thekinematic torques leads to

(13)where

outer gimbal viscous friction coefficient;outer gimbal cable restraint coefficient;

;;

;nonlinear friction torque;nonlinear cable restraint torque.

The cross elevation axis dynamics are described by (13). Aswith the elevation axis, this equation needs to be expressed interms of the controlled variables , , the base disturbances

, and the elevation and azimuth angles and . The modifiedexpression is derived in Appendix A as

(14)The impact of the elevation angle on the cross elevation dy-

namics is evident from (A2)(A4). Control of this axis dimin-ishes as the elevation angle increases. A secant gain functioncan compensate to some degree, however, in practice the gaingets very high for angles beyond 70 . The inability of the twoaxis system to track through the cone of occlusion [4] is com-monly termed the NADIR problem.


Fig. 2. Control configuration for direct stabilization.

Fig. 3. Control configuration for indirect stabilization.

IV. STABILIZATION LOOP SERVO CONTROLThe elevation axis control torque and the cross elevation

control torque can be generated by various algorithms. Re-gardless of the method, the primary objective is to suppress theeffect of any base motion, (or its derivative). This can beachieved by direct feedback (direct LOS stabilization) and/ordisturbance reconstruction and cancellation (indirect LOS sta-bilization). Significant differences exist between the two ap-proaches, as will be discussed. The control algorithms presentedare common practical implementations, derived from classicalcontrol methods and bandwidth considerations.

With direct LOS stabilization, a rate sensor measures the ratedisturbances and about the LOS. These are used to gen-erate a feedback compensating torque , which attempts tonull the measured disturbance. The typical control configura-tion is shown functionally in Fig. 2 for either axis. With indirectstabilization, rate sensors mounted on the base of the gimbal

measure platform motion and are used to generate a compen-sating torque , which attempts disturbance cancellationby matching the measured base rates, transformed into the LOScoordinates, to the LOS disturbance rates. In this method, dis-turbances induced by relative motion are compensated for by atachometer feedback loop, which generates a torque , asin the direct method. The typical control configuration is shownfunctionally in Fig. 3 for either axis.

A. LOS Stabilization of the Elevation AxisThe elevation axis dynamics were described by (8). With the

direct approach a rate sensor mounted on the LOS axes mea-sures . To shape the loop response a proportional plus inte-gral (PI) control feedback compensator is often used, or

(15)


.

(measured elevation LOS rate).Elevation rate sensor measurement error.

The sensor measurement error includes sensor noise. The PIcompensator can also be designed to operate on the rate error,which is the difference between the commanded rate from thetrack loop and the measured rate. In this case the compensatoris in the feedforward path, however the results obtained are es-sentially the same. Substituting the control expression into (8),one obtains

(16)

where(inner axis related

disturbances);

(base disturbances);(sensor

noise).The compensator coefficients are often determined as

whereelevation rate loop bandwidth;elevation rate loop lead frequency [ to

].Using differential operator notation , (16) can thenbe expressed as

(17)

The elevation axis response is coupled to the cross elevation axisrate and the outer axis roll rate by the fourth term on the left sideof (17). In addition, the base rate inputs are also coupled to theazimuth axis response by the relative azimuth angle . Equation(17) can be rewritten as

(18)

where

It can be seen that disturbance rejection is derived from the fre-quency dependent gain (or attenuation factor) and theinertia . In a point design, and are chosen such thatthe magnitude of the denominator is sufficient to attenuate ameasured torque or acceleration disturbance at a specific fre-quency(s). The base disturbances are coupled via friction and acable restraint, but are attenuated by the loop gain. Normally,the desired rate loop bandwidth is such that

and

This essentially results in a high gain feed back approach. It issomewhat robust to parameter variations in the sense that exactknowledge of , , and is not required as long as theabove inequality is satisfied. Sensor noise, however, is only mar-ginally affected by the gain, so that it will still have a significantimpact. Sensor noise is coupled via the compensator and will beattenuated (or amplified) roughly as . Sincedisturbance attenuation is inversely proportional to the servoloop bandwidth, it is limited by stability and noise. If the re-quired bandwidth is too high, an alternative algorithm is needed.Identifying the dominant disturbance sources, such as nonlinearfriction torque [3], [5], is necessary. In a severe platform motionenvironment, base sensors can be used to measure andand construct a feedback control torque to compensate for thebase disturbance input, (often termed Strong Stabiliza-tion [4]). The basic servo algorithm can be modified to includeterms that cancel , (or ), and the nonlinearterm in . This requires the additional measurement of theouter axis roll rate .

With the indirect stabilization method, a tachometer is usedto measure the relative elevation angular rate . A tri-axis ratesensor is mounted at the gimbal base to measure . The controltorques for this configuration are

(19a)and

(19b)

where

andmeasured relative elevationangular rate;elevation tachometer noise;measured body , rates;

;

(measured re-solver angle);

; ;azimuth resolver noise.

It is also likely that the command for the compensationtorque, , will be computer generated and contain sam-pling errors associated with the base rate sensor and resolvermeasurements. For a fast enough sampling rate, this error com-ponent will be proportional to the sampling interval, Thisanalysis, ignores this source of error and treats the command


torque as an analog signal. Expanding the expression forand assuming the resolver error is small results in

where

The combined inner gimbal compensation torques are thenobtained as

(20)Substituting (19) into (8), and using the same algorithm to

choose the servo loop compensator coefficients as described inSection IV-A, one has

(21)

Comparison of (18) with (21), indicates that the loop gain forboth approaches will be equivalent. The major difference is inthe noise terms. Whereas the direct approach contains only theelevation rate sensor noise, the indirect approach is perturbedby both tachometer noise as well as base rate sensor noise. Thedifference in the noise terms arises from a fundamental differ-ence in the control technique. The direct approach suppressesthe disturbances via gain attenuation. The disturbance is nevertotally applied to the rate loop since it is always working to nullit. Its magnitude as seen by the loop, is dependent on the loopbandwidth. The indirect approach requires a cancellation of therate so that it must actually be measured. To estimate per-formance of the two approaches, one can compare the root meansquare (rss) rate sensor error of the direct LOS methodwith an estimate of the indirect LOS method noise as given bythe root sum square (rss) of the rms noise of the tachometer, re-solver, and base rate sensors, or

It is important to note that a noise contribution from the az-imuth resolver, dependent on the outer gimbal axis roll rate, nowimpacts the elevation axis performance. Indirect method perfor-mance could be improved by mounting a rate sensor on the outergimbal to measure directly.

B. LOS Stabilization of the Cross-Elevation AxisThe cross elevation axis dynamics were described by (14).

With the direct approach the cross elevation rate is measureddirectly and fed back to suppress disturbances about this axis.

Again, a simple proportional plus integral control compensatoris used, or

(22)where

measured cross-elevation rate.cross-elevation rate sensor measurement error.

As with the elevation axis, the sensor measurement error in-cludes noise. Substituting (22) into (14), and restricting the anal-ysis to small elevation angles, then

(23)where

.

sensor

noise.The PI coefficients are chosen using the same algorithm as

for the elevation axis ( and are again the loop bandwidthand lead frequency), or

and

The cross-elevation response is then approximated as

(24)

where

and . With these approximations, rate loopperformance depends upon loop gain and axis inertia, as withthe elevation axis. Again, coupling of this axis with the elevationaxis is seen to be dependent upon the outer axis roll rate.

The indirect stabilization control configuration again includesa tachometer to measure the relative azimuth angular rate, ,and the tri-axis rate sensor, specified previously, mounted onthe gimbal base. The control torques for this case are given by

(25a)

(25b)


wheremeasured relative elevation axis angle;

; .Again, issues regarding the base torque being computer gener-ated and having an associated sampling error will be ignored.The expression for the rate can be expanded (again as-suming the resolver errors are small) as

where

For small elevation angles, in (25b) can be approxi-mated as and the base control torquecan be expressed as

(26)

The combined outer gimbal azimuth compensation torque canfinally be generated as

(27)

where .Substituting (27) into (14), one obtains

where

(28)

With the small elevation angle assumption, the response can beapproximated as

(29)

The transfer functions and are similar to those definedfor the elevation axis. The primary difference between the di-rect and indirect approaches is again the noise term. The ratesensor noise, , can be compared with an upper boundof the rss of the rms noise associated with the indirect approach,or where therms of is approximated as shown in the equation atthe bottom of the page.

Again the dependence of the rms noise on the resolver anglenoise contribution and the outer axis roll and pitch rates can beobserved. As with the elevation axis, in the presence of band-width limitations, enhanced performance may be possible bycanceling the terms associated with and as well asdirect compensation of friction and cable restraint torques.

C. General Stabilization Servo Loop RequirementsThe preceding stabilization loop analysis focused on specific

control algorithms based on known design techniques. Moreinsight into stabilization loop design requirements may be ob-tained by formulating the problem in a general structure. Themodels describing the gimbal dynamics for the elevation andcross elevation axes, given by (8) (EL) and (14) (Cross EL), canbe expressed in a differential vector form as

(30)

where


and

The direct LOS stabilization feedback design produces a controltorque given by

(31)

where is given in the equation at the bottom of the page.Substituting (31) into (30), one obtains

(32)

where

or

Equation (32) represents the gimbal response to a disturbanceinput vector for the direct LOS stabilization approach. The servoloop gains, which attenuate the input disturbance, are now in-herent in the matrix . A similar expression can be derived

for the indirect LOS stabilization case. As suggested earlier toimprove performance, a strong stabilization approach [4] mightbe utilized in which case the feedback and base torques couldbe modified as ;where is an estimate of the nonlinear feedback vectorand the estimated base disturbance input. With these con-trol inputs, the response becomes

where

and

The difference vectors are the residual errors resulting from thecancellation of the respective terms. Implementation of the basedisturbance cancellation torque requires knowledge of the baserates, gimbal inertia, linear friction and cable restraint constants,and gimbal angles as well as a model of the nonlinear frictiontorque [5], [6].

Another stability measure for the feedback design can be ob-tained by examining the linear part of the response to determineconditions under which the matrix eigenvalues have neg-ative real parts. A first-order estimate of the feedback gain re-quirements can be obtained by treating the time varying termsof as constants. The characteristic equation is then givenby

Using the Routh criterion to establish stability conditions, aquick estimate can be had by assuming that the bandwidths andlead frequencies of each loop are equal, or

With this assumption, it can be shown that the following stabilitycondition results:


This lower bound on loop bandwidth indicates it must exceeda value dependent upon the inertia terms and the maximum ex-pected roll rate, assuming is small and the term (see Ap-pendix A) is bounded. Conditions under which the solution tothe homogenous part of (32) approaches zero as time tends toinfinity can also be derived directly from the state-space rep-resentation. The matrix can be expressed as:

where contains only the constant elementsand eigenvalues with negative real parts and containsthe two time varying roll elements (element 2,4 and element4,2). Equation (32) can then be written as

(33)

This vector equation has the solution

(34)

where is the state transition matrix. The followingpositive bounds are assumed:

The nonlinear term is a product of the state variables. Forsmall, it will satisfy the following condition [ is a max-imum value of the inertia terms associated with ]

since

Therefore, for and the following conditionwill be met: . This bound on the statevector norm is dependent on the initial state vector. Taking thenorm of both sides of (34), one has

(35)

where assuming that

for

It is shown in Appendix B, that this bound can be expressedas: . If the roll rate isbounded such that , then the state

TABLE I

vector norm is exponentially decaying. To satisfy the conditionthat [ for [ [ and all , it is necessaryto select such that

or

for

The bound is a function of the initial state, due to the nonlinearfeedback term. Although the practical significance of this re-sult may be limited, it does provide some insight into the lowerbound on the loop bandwidth and gain required for a reasonablystable response.

V. SIMULATIONThe gimbal model derived in previous sections and defined

by (8) and (14), was simulated in order to provide a quantita-tive comparison of the two stabilization methods. The gimbalparameters assumed were the following.

Inner Axis Inertia:lb-in-s lb-in-s lb-in-s

Inner and Outer Axis Friction:(viscous) lb-in/rad/s(Coulomb) lb-in/rad/s

inner and outer axis cable restraint:lb-in/rad

outer axis inertia:lb-in-s lb-in-s lb-in-s

A 30-Hz servo bandwidth was obtained on the inner (elevation)axis and an approximate 20 Hz bandwidth on the outer (crosselevation) axis using a PI compensator given by

Three cases were simulated, representative of different plat-form types, with the following sinusoidal disturbance rates [i.e.,

] as listed in Table I.The rate sensor noise was assumed to be normally distributed

with a zero mean and a standard deviation of 0.005 rad/s. Theresolver noise was also assumed zero mean normal with a stan-dard deviation of 0.001 rad. Sampling of the resolver angles wasnot modeled so that this source of error is not included.


Fig. 4. Direct LOS stabilization Case One.

Fig. 5. Indirect LOS stabilization Case One.

The simulation results for Case One, representing a severedisturbance environment (with a significant platform roll rate),are shown in Figs. 46. Fig. 4 is a plot of the LOS angular andrate disturbance about the inner axis for the direct approach.Fig. 5 is the same plot, but for the indirect approach. Comparingthe rate disturbances for each approach, it is evident that thedensity of the variations observed with the indirect approach ishigher than for the direct approach. The disturbance rate distri-bution is plotted in Fig. 6 to quantify this observation. It can beseen that for the indirect approach, the distribution has a muchbroader spectrum than for the direct approach. This supportsthe analysis that indicated more error sources were associatedwith the indirect approach. Similar results were obtained forthe outer axis. Figs. 4 and 5 also show the LOS angular devi-

ation. With the indirect approach, the drift in the deviation dueto the low frequency rate bias resulting from the composite ratedisturbances, such as the integrated roll/resolver error distur-bance product [see (28)], is much higher. The inner axis dis-turbance rate distribution plot for Case Two is shown in Fig. 7.The average amplitude of the rate disturbances for this case islower than for Case One, however, the same trends can be ob-served. The disturbance distribution resulting from the indirectcontrol method is again significantly greater than with the directmethod. Again similar results were obtained for the outer axis.The final case represents very low disturbance conditions witha correspondingly low roll rate. The rate distribution plot for theinner axis is shown in Fig. 8 and again the trend is similar to theprevious two cases.


Fig. 6. Y -axis rate distribution Case One.

Fig. 7. Y -axis rate distribution Case Two.

VI. SUMMARY

A two-axis gimbal model was derived to evaluate the per-formance of two methods for stabilizing the LOS pointingvector. With the direct approach, the rate sensors are mountedon the LOS axes and directly measure rate disturbances aboutthe elevation and cross-elevation axes. With the indirect ap-proach, the rate sensors are mounted at the gimbal base, andcompensation requires that these rates be transformed intoequivalent disturbances about the LOS. In this approach, relativerate disturbances such as friction, and imbalance are compen-sated for by tachometer feedback about the torque motor. Theanalysis showed that more error sources were associated withthe indirect approach. The simulation results verified this andindicated that the performance of the indirect approach would

not be as efficient as with the direct approach. In the indirectapproach, several of the noise inputs are coupled to the platformroll disturbance. For both methods, it was shown that a lowerbound on stabilization loop bandwidth was proportional tothe maximum expected roll rate disturbance, coupled in viagimbal geometry. The analysis of the indirect method did notaccount for sampling rate errors or those associated with thegimbal structural rigidity. Essentially, all signals were assumedto be analog. Before choosing a particular method, one musttradeoff the advantage gained with the indirect approach fromthe payload volume reduction versus the penalty paid in LOSdisturbance attenuation performance. Although, indirect LOSstabilization performance may be satisfactory, this approach willnot attain the same performance level as the direct approach,given an equal design effort.


Fig. 8. Y -axis rate distribution Case Three.

APPENDIX A

Equation (13), given in Section III, can be modified by ex-pressing the variables and in terms of the controlledvariables and base disturbances. The following expressions for

and are used to generate this representation:

Using these equations, the first term on the right-hand side of(13) can be expressed as

the second term as

the third term as

and finally, the friction and cable restraint term as

Substituting each of these expressions into (13) results in thefollowing representation for the cross elevation dynamics:

(A1)where

(A2)

(A3)and

(A4)

APPENDIX B

This derivation follows that presented using the GronwellBellman Lemma [7, Th. 16]. The vector norm inequality wasgiven by (35), or

(B1)


Multiplying by and dividing by , one obtains

(B2)Dividing the left side of this expression by the right side andmultiplying by results in

The left hand side can be expressed as the derivative of the nat-ural log of the quantity in the denominator, or

Integrating and raising each side to the power results in

From (B2), one then has

or

ACKNOWLEDGMENT

The authors would like to thank the reviewers whose criticalanalysis and suggestions brought insight into the final editing ofthis paper.

REFERENCES[1] A. K. Rue, Stabilization of precision electro-optical pointing and

tracking systems, IEEE Trans. Aerosp. Electron. Syst., vol. AES-5, pp.805819, Sept. 1969.

[2] , Precision stabilization systems, IEEE Trans. Aerosp. Electron.Syst., vol. AES-10, pp. 3442, Jan. 1974.

[3] B. Friedland and D. Haessig, Jr., On modeling and simulation of fric-tion, ASME J. Dyn. Syst., Meas., Contr., vol. 113, pp. 354362, 1991.

[4] W. J. Bigley and F. Schupan, Wide-band base motion isolation controlfor a mobile platform, in Proc. 1987 American Control Conference,Minneapolis, MN, June 10, 1987.

[5] C. D. Walrath, Adaptive bearing friction compensation based on re-cent knowledge of dynamic friction, Automatica, vol. 20, pp. 717727,1984.

[6] B. Li, D. Hullender, and M. DeRenzo, Nonlinear induced disturbancerejection in inertial stabilization systems, IEEE Trans. Contr. Syst.Technol., vol. 6, pp. 421427, May 1998.

[7] N. Balabanian and T. Bickart, Electrical Network Theory. New York:Wiley, 1969, pp. 727767.

Peter J. Kennedy (M76) received the B.S. degree in mechanical engineeringfrom Lafayette College, Easton, PA, in 1971, the M.S. degree in industrial engi-neering from Lehigh University, Bethlehem, PA, in 1973, and the M.S. degreein electrical engineering from Fairleigh Dickenson University, Teaneck, NJ, in1978.

He has worked in the area of pointing, tracking, and stabilization for the U.S.Army Electronic Warfare Laboratory, Bell Laboratories, SciTec, Inc., NorthropGrumman Corporation, and is presently employed with David H. Pollock Con-sultants, Inc., Westwood, NJ.

Mr. Kennedy is a licensed professional engineer and a member of the Na-tional Society of Professional Engineers, and the Instrumentation, Systems, andAutomation Society (ISA).

Rhonda L. Kennedy (M02) received the B.S. degree in mathematics fromPhillips University, Enid, OK, in 1979 and the M.S. degree in operations re-search from Texas Tech University, Lubbock, TX, in 1981.

She was with Bell Laboratories from 1981 to 1983. Currently she is workingin the area of statistical analysis, modeling and interface design for simulation ofEO/IR sensor systems, and real-time simulations of end-to-end countermeasuresystems. She is presently employed with David H. Pollock Consultants, Inc.,Westwood, NJ.

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