Attitude Control of a Bias Momentum Satellite Using Moment of Inertia

Transcription

1 I. INTRODUCTION Attitude Control of a Bias Momentum Satellite Using Moment of Inertia HYOCHOONG BANG Korea Advanced Institute of Science and Technology HYUNG DON CHOI Korea Aerospace Research Institute Analysis on the attitude controller based upon moment of inertia distribution for a bias momentum satellite is discussed. Spacecraft moment of inertia distribution is represented in the form of product of inertia terms in the system inertia matrix. The product of inertia between orthogonal body axes of the satellite is used to build a controller which controls the nutational motion caused by the angular momentum of the wheel. The attitude controller in the pitch axis controlling the pitch motion as well as nutational dynamics in the roll/yaw planes is analyzed in detail. Analytic expressions using linearized equations are derived providing further insight into the dynamic coupling effect among orthogonal body axes. Manuscript received January 19, 2001; revised August 17 and October 22, 2001; released for publication September 28, IEEE Log No. T-AES/38/1/ Refereeing of this contribution was handled by T. E. Busch. Author s addresses: H. Bang, Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Kusong-Dong, Yusong-Gu, Daejon, , Korea, (hcbang@fdcl.kaist.ac.kr); H. D. Choi, Guidance and Control Department, Korea Aerospace Research Institute, P.O. Box 113, Yusung-Gu, Daejon, , Korea, (hdchoi@kari.re.kr) /02/$17.00 c 2002 IEEE Bias momentum spacecraft is a popular stabilizing technique applied to a number of spacecraft missions, in particular geostationary communication satellites [1 5]. The primary reason is simplicity of system while maintaining required pointing accuracy. A single momentum wheel is usually aligned along the orbit normal direction while the wheel itself retains a certain angular momentum level [3]. The wheel speed is usually adjusted about the nominal value to control the pitch error. The angular momentum of the wheel results in gyroscopic stiffness effect about the orthogonal body axes (or roll/yaw). The gyroscopic stiffness contributes to maintaining pointing accuracy with respect to external disturbance inputs. Bias momentum of the spacecraft in the pitch axis causes roll/yaw nutational motion by initial body rates [3]. The nutational motion is a periodic motion whose frequency is dependent upon the magnitude of angular momentum of the wheel and moment of inertia of spacecraft body. There are a number of techniques available for the nutational mode control. One efficient approach is to utilize moment of inertia distribution [6 8]. The off-diagonal terms in the inertia matrix cause dynamic coupling between pitch and roll/yaw axes. For instance, the product of inertia in the pitch and yaw planes of the spacecraft body is a primary source of coupling between the pitch and roll/yaw dynamics. This property can be made use of indirectly controlling the nutational mode through a pitch control loop. In other words, the momentum wheel is engaged in controlling pitch attitude error as well as the nutational mode. The control law could be an active type taking measurements of pitch as well as roll angles provided by an Earth sensor. In this work, we discuss controller design for a pitch bias momentum spacecraft model for nutational mode control using the product of inertia property. The dynamic coupling effect has been drawn through explicit expressions between angular responses and wheel input torque. The angular responses include not only pitch but roll/yaw degrees of freedom arising from the coupling effect. A PD (proportional plus derivative) feedback control approach is investigated by the root-locus plot with a varying feedback gain as well as magnitude of the product of inertia term. Even if the controller design itself is not new as a typical PD type, the principle of controller design is viewed from different perspectives. The performance of the controllers design and analysis is verified mostly by simulation study. II. BASIC EQUATIONS The schematic configuration of an orbiting pitch bias momentum satellite is presented in Fig. 1. The momentum wheel is aligned along the orbit normal IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 1 JANUARY

2 where (T x,t y,t z ) are body components of external torque input such as gravitational, environmental, and other control torques. The linearized kinematic relationships between Euler attitude angles and body angular rates can be approximated as [1 3]! x ' _ Ã! 0 Á,! y ' _ Á +! 0 Ã,! z ' _ µ +! 0 (7) Fig. 1. Orbiting pitch bias momentum satellite. direction. It is assumed that the wheel itself is attached to a pivot device being used to rotate the momentum vector about the roll (y) axisat and angular rate _. This generates yaw (x) control torque, which also could be used in the roll/yaw nutation control. As mentioned already, the spacecraft possesses product of inertia in the yaw/pitch (x z) plane of the body. The spacecraft momentum of inertia is therefore given by 2 3 I x 0 I xz 6 I = 4 0 I y 0 I xz 0 I z 7 5: (1) This product of inertia term ( I xz ) produces dynamic coupling between roll/yaw and pitch axes. For analysis of the coupling effect, we choose to set up dynamic equations of motion and derive some explicit expressions. The angular momentum vector of the above system including spacecraft and the momentum wheel attached to a pivot device is written as H =(I x! x I xz! z + hsin )b 1 + I y! y b 2 +(I z! z I xz! x + hcos )b 3 (2) where! =(! x,! y,! z ) T represent a vector of body angular velocity, (b 1,b 2,b 3 ) are unit vectors along each body axis, and h and are angular momentum of the wheel and pivot angle, respectively. Application of the Euler s equation to the angular momentum vector as dh dt +! H = T (3) leads to the following sets of nonlinear governing differential equations of motion I x _! x I xz _! z +(I z I y )! y! z I xz! x! y + hcos! y + h _ cos + _ hsin = T x (4) I y _! y +(I x I z )! x! z + I xz (! 2 x!2 z ) h(! x cos! z sin )=T y (5) I z _! z I xz _! x +(I y I x )! x! y + I xz! y! z hsin! y h _ sin + _ hcos = T z (6) where Ã,µ,Á represent yaw, pitch, and roll attitude angles defined about the orbiting reference frame [2, 3]. The linearized kinematic relationships are substituted into (4) (6) yielding equations of motion linearized with respect to small attitude angles as I x à Ixz µ + p1 à + p 2 _ Á + h _ cos + _ hsin = Tx (8) I y Á + p3 Á + p 4 _à 2I xz! _ µ + h! z sin = T y (9) I z µ Ixz Á +2Ixz! 0 _ Á + Ixz! 2 0 à hsin! y h _ sin + _ hcos = T z (10) where the parameters (p i (i = 1,2,3,4)) are defined as p 1 =( I y + I z )! 2 0 +! 0 hcos p 2 =( I x I y + I z )! 0 + hcos p 3 =(I z I x )! 0 2 +! 0 hcos p 4 =(I x + I y I z )! 0 hcos : In the process of linearizing (9), a secular term ( I xz! 0 2 ) has been dropped. This is because the magnitude is so small and it is not directly related to the coupling effect among orthogonal axes. Dropping the secular term simplifies the equations without loosing physical significance of the problem. The product of inertia term explicitly shows up in the linearized governing equations of motion. As it can be shown in (8) (10), the attitude dynamics are coupled about three axes through I xz term. Now we need to derive some input/output relationships. The output variables consist of angular responses while the input is the wheel torque command. The control input applied to the pitch axis possesses control authority about the pitch axis as well as roll/yaw axes through the product of inertia term. To have better understanding on the dynamic coupling, we derive explicit relationships in the form of transfer functions by using the linearized governing equations of motion for the input and resultant output responses. In order to derive transfer functions, the pivot angle ( ) is set to zero tentatively. Even if =0 is a special case of the pivot angle, it does not significantly affect the interpretation of the physical nature of the system. This is because the goal here is to derive transfer functions between angular responses and wheel control input with a particular value of the pivot angle. As a consequence, (8) (10) can be 244 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 1 JANUARY 2002

3 simplified into I x à Ixz µ + p1 à + p 2 _ Á = Tx (11) I y Á + p3 Á + p 4 _à 2I xz! 0 _ µ = Ty (12) I z µ Ixz à +2Ixz! 0 _ Á + Ixz! 2 0 à = _ h + T z : (13) The external control inputs are set T x = T y = T z =0 so that only the momentum wheel is taken as input. Next the Laplace transform is taken over the above equations which leads to (I xz s 2 + p 1 )ª(s) I xz s 2 (s)+p 2 s (s) = 0 (14) p 4 sª(s)+(i y s 2 + p 3 ) (s) 2I xz! 0 s (s) = 0 (15) I xz ( s 2 +! 2 0 )ª(s)+2I xz! 0 s (s)+i z s2 (s) = _ H w (s) (16) where the capital letters denote Laplace transform variables of the attitude angles as ª(s)=L[Ã(t)], (s)=l[á(t)], (s)=l[µ(t)]: Equations (14) and (15) are rearranged as Ix s 2 ½ ¾ ½ + p 1 p 2 s ª(s) Ixs s 2 ¾ = (s): p 4 s I y s 2 + p 3 (s) 2I xz! 0 s (17) In other words, ½ ¾ ª(s) = 1 ½ ¾ A(s) (s) (18) (s) B(s) where each polynomial is written as =I x I y s 4 +(p 1 I y + p 3 I x p 2 p 4 )s 2 + p 1 p 3 A(s)=I y I xz s 4 +(p 3 I xz 2p 2 I xz! 0 )s 2 (19) B(s)=( p 4 I xz +2I xz I x! 0 )s 3 +2p 1 I xz! 0 s: Equation (18) is also rewritten as ª(s)= A(s) (s), B(s) (s)= (s): (20) The above set of equations are substituted into (16) to yield (s) H _ w (s) = (21) (s) where (s) is now a polynomial represented as (s)=i z s 2 +I xz ( s 2 +! 0 2 )A(s)+2I xz! 0 sb(s): (22) Equation (21) thus shows the input/output relationship between the wheel angular momentum change or input torque and pitch angle response. Note that the numerator polynomial contains extra terms multiplied by I xz.ifi xz is equal to zero, then the pitch transfer TABLE I Roots of Polynomials in Transfer Functions Polynomials Zeros of Polynomials (s) 0,0, 2: i, 7: i 2: i, 7: i A(s) 0,0, 1: B(s) 0, 1: i function reduces to a simple single axis rigid body equation (1=I z s 2 ) as expected. In addition, the transfer functions for roll and yaw axes are derived by making use of (20) as follows ª(s) H _ w (s) = A(s) (s) (23) and (s) H _ w (s) = B(s) (s) : (24) Note that if there is no product of inertia term (I xz ) then the polynomials A(s),B(s) trivially reduce to zero, thus the transfer functions become undefined. Thus the product of inertia term directly causes the dynamic coupling effect about the body axes orthogonal to each other. It should be noted that the transfer functions in (21) (24) consist of a common denominator polynomial ( (s)) and different numerator polynomials. The numerator polynomials for pitch and yaw axes are fourth order, while for roll axis the numerator is a third-order polynomial. Table I presents the zeros of numerator and denominator polynomials of the transfer functions derived in the above. In this table, the material properties of the spacecraft are taken as I x = 17,581, I y = 16,533, I z = 3,529, and I xz = 482 (in-lbf-sec 2 ) while the nominal angular momentum of the wheel is h = 475(in-lbf-sec) for a typical geostationary bias momentum satellite. Thus the orbital rate (! 0 )is7: rad/s while the altitude of the orbit is equal to 37,786 km from the Earth s surface. As it can be shown, the zeros of the denominator polynomial consist of orbital ( 7: i), nutational ( 2: i), and pure rigid mode frequencies. The zeros of B(s) are located at the origin and imaginary axis as a characteristic of minimum-phase systems. This can be partly explained by (11) (13) as the wheel torque ( h) _ is directly connected to the roll angle through I xz term. Meanwhile the yaw numerator polynomial (A(s)) is a nonminimum-phase system with an unstable zero. Fig. 2 shows Bode diagrams for the magnitude responses of each transfer function. The Bode diagrams reflect the results of Table II. For pitch axis response, the orbital rate mode is cancelled between the numerator and denominator. The pole and zeros are alternating in the roll axis response. BANG & CHOI: ATTITUDE CONTROL OF A BIAS MOMENTUM SATELLITE USING MOMENT OF INERTIA 245

4 Fig. 3. PD controller block diagram. Fig. 3 shows a block diagram for the proposed controller. The error signal is a linear combination of the pitch (µ) error and weighted roll (Á) error as e(t)=µ(t)+ká(t) (25) where e(t) is error signal and k is a constant weighting factor on the roll. The pitch and roll angles are usually measured by an Earth sensor which is a primary motivation of the error signal defined in the above equation. Then the time rate of angular momentum change of the wheel from Fig. 3 is written as _ H w (s)= sh w (s)= G c (s)e(s) (26) where H w (s) is Laplace transform of the wheel torque, G c (s) is the PD compensator, and E(s) is the Laplace transform variable of the error signal prescribed as E(s)= (s)+k (s), G c (s)=k P + K D s (27) III. Fig. 2. Bode diagram for each channel (magnitude plot only. CONTROL LAWS AND TREND ANALYSIS The dynamic coupling examined in the previous section can be extended into a controller design. The key principle of the controller design is to employ the pitch momentum wheel for both pitch error and roll/yaw nutational mode control. For this goal the control command should comprise both pitch and roll(or yaw) attitude signals. A simple but practical PD controller is analyzed. The PD control approach is not new but has been tried for a wide classes of spacecraft attitude control. The PD control here is taken as an example for the analysis of active control using product of inertia. A. PD Control Law In the PD type pitch axis controller, the error signal combined with a PD compensator is fed back to the wheel in the form of wheel torque command. where K P,K D are typical design parameters of the PD control. By making use of (23) we can rewrite the error signal as E(s)= 1+k B(s) (s) (28) therefore H _ w (s)= (K P + K D s) 1+k B(s) (s) = C p (s) (s) (29) where C p (s) plays a role of a new compensator for simultaneous control of roll/yaw nutation and pitch error. Now it appears that (21) and (29) constitute pitch dynamics and associated controller independent of other attitude variables. For this configuration of system we can derive a closed-loop system characteristic equation from Fig. 3 in the form 1 (s) C p (s)=0 In other words, =1+ (s) (K P + K D s) 1+k B(s) =0: (30) (s)+(k P + K D s)[+kb(s)] = 0: (31) 246 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 1 JANUARY 2002

5 Fig. 4. Root-locus of closed-loop equation with varying k. The above equation represents a characteristic equation for the closed-loop system by application of the PD controller. The stability of the overall system is determined by the roots of the characteristic equation. The closed-loop characteristic equation determines the system stability as well as dynamic responses under the control. The characteristic equation is a sixth-order polynomial, for which the order is attributed to the coupling effect among different body axes. As it can be shown, the closed-loop roots are determined by feedback gains K P, K D,andk. B. Root-Locus Analysis In the previous discussion, it was shown that the feedback gain (k) on the roll angle determines the closed-loop system dynamics. Now we are interested in the trace of roots of the characteristic equation of the closed-loop system with varying k. This would set up a practical baseline for selecting the roll control gain. Fig. 4 shows the root-locus of (31) or (34) for which the gain k is varied continuously. The feedback gains of the pitch axis are selected as K P =20(s/rad) and K D = 150 (s/rad), respectively. The control gains (K P,K D ) are decided from desired closed-loop pitch dynamics which can be obtained by setting k =0. In other words, pure pitch control is designed to determine the desired feedback gains K P and K D. From Fig. 4 it can be shown that as the gain k increases from a negative value to zero, the real part of the nutational mode approaches zero closer. This behavior illustrates the effectiveness of the non-zero gain k on the nutational mode control. The orbital mode close to the origin is shown to be almost invariant with respect to k. The rigid body mode is also affected a noticeable amount by k. According to the root-locus plot, the sign of k should be negative for the given configuration of the controller. The magnitude of k should not set to be an arbitrarily large number, instead it should be determined by a compromise between the closed-loop rigid and nutational modes. The dynamic coupling is primarily created by the product of inertia in x z plane of the spacecraft body. The size of I xz term therefore may have a dominant effect on the overall dynamics. In order to examine the effect of the size of the product of inertia on the closed-loop stability, we introduce a parameter such that I xz = Ixz 0 where is a magnification factor with respect to the nominal value Ixz 0.Fig.5showsthe Fig. 5. Root-locus of closed-loop equation with varying. closed-loop root-locus with respect to variable. As increases it can be shown that the real part of the nutational mode moves further left in the complex plane. This again quantitatively demonstrates the effectiveness of the product of inertia term on the nutational mode control. The rigid body mode on the other hand becomes less stable with increasing. Thus the original pitch damping efficiency is decreased in this case by increasing the product of inertia effect. The product of inertia therefore needs to be selected based upon the relative effectiveness of the rigid and nutational modes stabilization in a manner similar to the case of varying k. Based upon the PD control law coupled with the roll signal measurement, simulation has been performed. Fig. 6 shows pitch and roll/yaw responses, respectively, where the roll angle feedback gain (k) is set to zero. The initial nutational motion is not eliminated with k = 0. On the other hand, Fig. 7 presents simulation results with roll feedback control for which the feedback gain k is set to 1. In both cases the wheel pivot angle has been set to zero while the nominal angular momentum of the wheel is taken as 475 (in-lbf-sec) for about 220 s of nutational period. In the simulation results, we can see that the nutational mode is effectively stabilized. The pitch response is also shown to be mixed with the nutational mode under the application of the roll feedback signal. It is noteworthy that there exist off-set in roll/yaw attitude responses. This is because direct roll/yaw attitude control is not feasible by using the pitch axis wheel. Only nutational mode is stabilized as explained in the following section. C. Pointing about Roll Axis The PD control law in (25) and (26) is implemented using the error signal which consists of both pitch and roll angles. The error signal can accommodate a reference value for off-set pointing purpose. For instance, for a constant reference pitch angle, µ ref, the error signal can be modified into e(t)=µ(t) µ ref + ká(t): (32) BANG & CHOI: ATTITUDE CONTROL OF A BIAS MOMENTUM SATELLITE USING MOMENT OF INERTIA 247

6 Fig. 6. Simulation results with pure pitch control. Fig. 7. Simulation results with simultaneous pitch and roll control. Application of the PD control command with appropriate control parameters will lead to stable responses, especially a constant off-set pitch angle. The roll off-set or pointing is not feasible, however, since the control actuator (wheel) itself is aligned along the pitch axis instead of roll axis. For more explanation, the error signal with a reference roll angle (Á ref ) can be written as e(t)=µ(t)+k(á(t) Á ref ): (33) The above error signal, however, is regarded as being equivalent to e(t)=[µ(t) ká ref ]+ká(t) (34) and the roll error becomes pitch error scaled by the roll weighting factor. Thus direct roll pointing by the pitch wheel is not allowed even in the presence of a product of inertia term. Also, the transfer functions between the error signal and pitch/roll variables are presented in reference to the block diagram in Fig. 3 as and (s) E(s) = (K P + K D s)b(s) (s)+(k P + K D s)(+kb(s)) : (36) It is worthwhile to note that the difference in the expression of the two transfer functions lies in the numerator polynomials (B(s),). The numerator polynomial in the roll transfer function contains a zero at the origin while in the pitch axis has zeros only at the imaginary axis. The zeros at the origin in the numerator polynomial are cancelled by that of denominator which results in loss of controllability. Hence tracking of the constant off-set command in the roll axis is not possible. Obviously, off-set command in pitch is possible if pitch signal in (25) is replaced by pitch error which is the difference between actual pitch and desired pitch off-set. The yaw off-set pointing is not feasible since yaw pointing is essentially equivalent to roll pointing in terms of pointing capability. D. Nutation Control Network (s) E(s) = (K P + K D s) (s)+(k P + K D s)(+kb(s)) (35) The concept of nutation control network is introduced in [3]. The structure of the nutation control 248 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 1 JANUARY 2002

7 result of Fig. 8 illustrates the effectiveness of the nutation control network in controlling nutational mode. The system stability is not however always guaranteed under the nutation control network. It turns out that over a certain critical value of K nut the system stability is destroyed. Thus preliminary root-locus type analysis may be essential in designing the nutation control network for a stable system. Fig. 8. Root-locus analysis for varying nutation control network gain. network is given by the following form K G nut (s)= nut s (1 + s 1 )(1 + s 2 ) : (37) The nutation control network is used to connect the roll error to the pitch control loop. The differentiator in the numerator is intended to control the constant roll error while the denominator poles pass the nutational mode only. The time constants 1 and 2 are selected in such a way that the filter itself is a band-pass filter for the nutational mode [3]. The control signal consists of pitch error with a PD controller and roll error with the nutation network as follows _ H w (s)= (K P + K D s) (s)+g nut (s) (s): (38) Now by making use of (20) between (s) and (s), the above equation can be rewritten as H _ w (s)= (K P + K D s)+g nut (s) B(s) (s): By making use of the input/output equation for the pitch axis in (21) and the new control law of (41), the closed-loop system characteristic equation is constructed as 1 (s) (39) (K P + K D s)+g nut (s) B(s) =0: (40) The above equation can be rewritten, by substituting the original expression of G nut (s), as (s)(1 + 1 s)(1 + 2 s) +[ K nut B(s)(K P + K D s)(1 + 1 s)(1 + 2 s)] = 0: (41) Hence the nutation control filter gain K nut affects the closed-loop system stability together with 1 and 2. The trace of the closed-loop system roots with varying K nut in (41) is presented in Fig. 8. The time constants ( 1, 2 ) of the network are set close to the nutational frequency of the system. As K nut increases, the nutational mode poles are more damped. Not much change is observed over the rigid mode. The IV. CONCLUDING REMARKS Analysis on the attitude control for a pitch bias momentum spacecraft with a product of inertia term produced some useful physical insight into the problem. Analytical expressions based upon linearized governing equations of motion for input/output relationships have been derived. A closed-loop characteristic equation was derived and used for stability analysis by the root-locus approach. Direct roll pointing capability by the pitch control loop alone turned out to be infeasible. The proposed scheme was verified also through simulation work. The analytical results gained through this study may contribute to enhancing the effectiveness and performance of attitude controller of future pitch bias momentum spacecraft. REFERENCES [1] Junkins, J. L., and Turner, J. D. (1986) Optimal Spacecraft Rotational Maneuvers. Amsterdam: Elsevier, 1986, [2] Wie, B. (1998) Space Vehicle Dynamics and Control. New York: AIAA, Education Series, [3] Sidi, M. J. (1997) Spacecraft Dynamics and Control. New York: Cambridge, 1997, [4] Hubert, C. (1981) Spacecraft attitude acquisition from an arbitrary spinning or tumbling state. Journal of Guidance, Control, and Dynamics, 4, 2 (1981), [5] Ross, I. M. (1996) Formulation of stability conditions for systems containing driven rotors. Journal of Guidance, Control, and Dynamics, 19, 2 (1996), [6] Philips, K. (1973) Active nutation damping utilizing spacecraft mass properties. IEEE Transactions on Aerospace and Electronic Systems, 9, 5 (1973), [7] Devey, W. J., Field, C. F., and Flook, L. (1977) An active nutation control system for spin stabilized satellites. Automatica, 13 (1977), [8] Fox, S. (1986) Attitude control subsystem performance of the RCA Series 3000 satellite. In Proceedings of AIAA 11th Communications Satellite System Conference, San Diego, CA, Mar , 1986, paper CP. BANG & CHOI: ATTITUDE CONTROL OF A BIAS MOMENTUM SATELLITE USING MOMENT OF INERTIA 249

8 Hyochoong Bang was born in Korea in He received his B.S. and M.S. degrees in aeronautical engineering from Seoul National University, Seoul, Korea, in 1985 and 1987, respectively. He received his Ph.D. from Texas A&M University, College Station, TX, in From 1992 to 1994, he worked as a research Assistant Professor at the U.S. Naval Postgraduate School conducting spacecraft attitude control research. From 1995 to 1999 he worked at the Korea Aerospace Research Institute in the area of geosynchronous spacecraft attitude control and mission design. During 1999 and 2000, he was an Assistant Professor at Chungnam National University. Sine 2001 he has been an Assistant Professor at Korea Advanced Institute of Science and Technology. His current research interests include spacecraft attitude control, real-time control, and ground-based experimental mechanics for various aerospace applications. Hyung Don Choi received his B.S. degree in mechanical engineering from Seoul National University, Seoul, Korea, in He studied at Korea Advanced Institute of Science and Technology and received M.S. and Ph. D. degrees in mechanical engineering in 1983 and 1988, respectively. He joined Korea Aerospace Research Institute in He was a principal investigator for the guidance and control system of the KSR-II sounding rocket and is currently the head of the Guidance and Control Department. His research interest include guidance and control system design of rocket system with special emphasis on development of actuators and thrusters applicable to sounding rockets. 250 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 1 JANUARY 2002