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    172 J. GUIDANCE AND CONTROL VOL. 3, NO. 2 ARTICLE NO. 78-1307R Optimizing the Gains of the Baro-Inertial Vertical Channel Massachusetts Institute of Technology, Cambridge, Mass. William S. Widnall* and Prasun K. Sinhat Intermetrics, Inc., Cambridge, Mass. The selection of the three gains in the baro-inertial vertical channel has been formulated as a stochastic op- timal control problem, where the objective is to minimize the mean-square error of the indicated vertical velocity. The optimal set of gains is surprisingly different from a conventional set of gains, and it provides a significant performance improvement. Sensitivity of the results to the statistical assumptions is explored. Ap- proximate analytical formulas are presented giving the optimal gains and pole locations as a function of the assumed statistics of the sources of error. A time domain simulation also exhibits the performance improvement. Introduction THE first aircraft inertial navigation systems were primarily two-channel systems that provided horizontal navigation data.J-3 Inertial navigators instrumenting three channels were introduced for missile navigation and guidance. In addition, the value of inertially derived vertical velocity was recognized in aircraft applications involving flight path angle determination and precision weapon delivery. It is well known that the altitude channel of a pure inertial navigation mechanization, in which gravity magnitude is computed as a function of the indicated altitude, is unstable.4-6 Near the surface of the Earth, the time constant of this exponential instability is about 10 min. Hence, for typical cruise navigation durations, the vertical channel of a terrestrial inertial navigator must be stabilized by some external altitude reference. The most commonly used external altitude reference in aircraft is a barometric altimeter. The optimal time-varying combination of the inertial and barometric data may be obtained using a Kalman filter.7 However, in applications not demanding the minimum navigation errors or in which the computer capacity is severely limited, a simple mechanization is commonly used in which the difference between the in- dicated and barometric altitude is fed back through constant gains or simple transfer functions to stabilize the altitude navigation variables.8 A typical set of constant-gain baro- inertial mechanization equations, which is analyzed in detail in this paper, is vz=fz-g(h,L) + Coriolis terms-fc2 (/*-/*,,) -da ba=k3(h-hb) (Ib) (Ic) Received June 26, 1978; presented as Paper 78-1307 at the AIAA Guidance and Control Conference, Palo Alto, Calif., Aug. 7-9, 1978; revision received July 3, 1979. Copyright © 1979 by W.S. Widnall and P.K. Sinha. Published by the American Institute of Aeronautics and Astronautics with permission. Reprints of this article may be ordered from AIAA Special Publications, 1290 Avenue of the Americas, New York, N.Y. 10019. Order by Article No. at top of page. Member price $2.00 each, nonmember, $3.00 each. Remittance must accompany order. Index category: Guidance and Control. *Associate Professor, Dept. of Aeronautics and Astronautics. Associate Fellow AIAA. tSenior Engineer, Navigation and Analysis Dept. where h is^the indicated altitude, vz is the indicated vertical velocity, 6cr is the computed vertical acceleration error, fz is the measured vertical specific force, g is the magnitude of gravity computed as a function of indicated altitude and latitude, and klt k2, k3 are the loop gains. This third-order vertical channel mechanization is superior to a second-order mechanization, which omits the £a equation, because it has zero vertical velocity error due to any bias vertical ac- celeration error such as accelerometer bias or gravity com- putation error. The Litton CAINS (Carrier Aircraft Inertial Navigation System) implements such a third-order baro-inertial vertical channel. In the CAINS, the three gains have been chosen so that the characteristic equation of the errors has a triple pole at the complex frequency s= - 1/r, where T is the desired time constant. For such a triple pole placement, it can be shown one chooses the gains to be A:, =3/7 k3=l/r3 k2=3/T2+2g/R (2) where R is the geocentric radius. In the CAINS, the time constant has been chosen to be T = 100 s. We have no literature explaining the designer's choice of the triple pole and its location. Perhaps the triple pole configuration was arbitrarily selected to reduce the gain-setting problem from parameters (kl9 k2> k3) to one parameter (T). Speculating further, perhaps the time constant of 100 s was an order-of- magnitude choice, selected so as to be faster than the 571 s time constant of the pure-inertial vertical-channel instability yet slower than the typical barometric error fluctuations associated with short-term aircraft maneuvers. This choice would be expected to both stabilize the vertical channel and provide some smoothing of the barometric altimeter errors. Regardless of the reasoning, the CAINS has peformed well in its intended applications. We shall refer to the CAINS set of three gains, given by Eq. (2) with r = 100 s, as the baseline set. Some applications have more demanding vertical velocity requirements than the CAINS requirements. In such ap- plications it may be necessary to optimize the vertical channel gains to reduce the vertical velocity errors. One such ap- plication was the use of the Magnavox X-set GPS navigator in the demonstration of pinpoint bombing on a target whose absolute coordinates were known. The X-set GPS navigator includes a barometric-inertial navigation subsystem and a GPS X-set receiver whose outputs are combined by a Kalman filter. An error analysis by Ausman9 predicted that the two largest contributions to bomb miss distance would be due to Downloaded by UNIVERSITY OF STRATHCLYDE on February 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/3.55966
    MARCH-APRIL 1980 OPTIMUM GAINS FOR A BARO-INERTIAL VERTICAL CHANNEL 173 the altitude navigation error and the vertical velocity navigation error at bomb release. Flying at 450 knots, at 5000- ft altitude above ground level, with a level release, and using low-drag bombs, Ausman indicated that the anticipated altitude and vertical velocity navigation errors of 19 m and downrange bomb miss distance. The absolute altitude error of the integrated navigation system is caused by the bias-like 0.75 m/s would contribute 24 m and 17 m, respectively, to wal Fig. 1 Baro-inertial vertical channel error model. wbl errors in the GPS measurements. The choice of baro-inertial gains has little or no effect on the absolute altitude accuracy. However, the absolute vertical velocity errors are a noticeable function of both the GPS measurement errors and the baro- inertial errors. The integrated system vertical velocity errors can be reduced if first the baro-inertial vertical velocity errors are minimized. There is no reason to assume that the triple pole gain set provides the best performance. Other pole placements might provide superior performance. To obtain new insights into the effect of the gains, we have formulated the vertical channel gain setting problem as a parameter optimization problem for the control of a stationary stochastic process. This paper presents the optimization problem putational results, analysis of the results, and a time-domain simulation of the performance with the recommended gains. formulation, com- Before proceeding, we comment that any constant gain set (including our optimized set) will be less optimal than ap- propriate time-varying gains that take into account the nonstationary nature of the inertial and barometric altimeter errors. The optimal time-varying combination of the inertial and barometric data may be obtained using a Kalman filter. Of course it is also possible to select effective time-varying gains with non-Kalman approaches. One example of the use of time-varying gains is provided by Whalley.10 Whalley points out the large error in barometric altitude when a supersonic aircraft passes through Mach 1 due to shock waves moving past the static pressure port. Whalley suggests eliminating this source of error by switching out the air data automatically from say Mach 0.95 to Mach 1.1. This could be mechanized by programming klt k2, k3 to be zero in this Mach interval. Another example of the use of time-varying gains is provided by Ausman and associates.11'12 They note that in -subsonic flight the largest source of barometric altimeter error is often the scale factor error due to the atmosphere not having the standard-day temperature-vs-altitude profile. In climbs and dives the scale factor error induces significant vertical velocity error into a constant gain baro-inertial vertical channel. Ausman and associates designed and im- plemented vertical channel mechanizations that reduce the gains during climbs and dives, while observing the baro-bias shift due to scale factor error. The estimated baro-bias shift is automatically subtracted from the baro-inertial error feed- back so that loop transients due to the scale factor error are minimized. Also before proceeding, we comment that additional ex- ternal data may be useful in reducing the effect of non- standard-day temperature. Blanchard13 proposes using in- flight measured temperature data, in place of the standard- day lapse-rate assumption, to relate more accurately pressure changes to altitude changes. Formulation of the Gain-Optimization Problem The error model for the vertical channel is shown in Fig. 1. The positive feedback with gain 2g/R is the destabilizing effect of normal gravity being calculated at the closed-loop altitude, which is in error by dh. The error state da is a random walk modeling any bias or slowly varying error in the vertical acceleration due to accelerometer bias, gravity anomaly, or error in the Coriolis terms. The white noise wa2 into the in- tegration provides the random walk. The white noise wal models short correlation time acceleration error, such as the effect during a short maneuver of vertical accelerometer scale factor error and input axis misalignment. The error state db is a random walk modeling any bias or slowly varying error in the altitude indicated by the barometric altimeter. Physical error sources include zero setting error, static pressure measurement error, variation in the height of a constant pressure surface, and scale factor error due to nonstandard day temperature. The white noise wb2 into the integrator provides the random walk. The white noise ww models short correlation time altimeter error, such as due to changes in the angle of attack or sideslip angle during a maneuver, or due to altimeter quantization or other noise. It is difficult to suggest appropriate values for the spectral densities of the four independent white noises in this stochastic model. Nevertheless, it must be done for the analysis to proceed. Table 1 shows the nominal values of the noise spectral densities that have been selected. These somewhat arbitrary numerical values have been arrived at by the following considerations . For the short correlation time acceleration error, a typical amplitude could be 200 fig. This error could be caused by a vertical accelerometer input axis misalignment of 200 ^ rad (41 arc-sec) together with a horizontal maneuver acceleration of one g. The typical duration of a horizontal maneuver is assumed to be of the order of 60 s. For a repeated series of random aircraft maneuvers, the autocorrelation function of the acceleration error would have area approximately / = (200xlO-6xlOms-2)2x(60s) = -3 (3) The area of the autocorrelation function equals the low- frequency value of the spectral density. For the white noise, whose autocorrelation function is a Dirac delta function with area Qah the spectral density Qal applies at all frequencies. Of interest is the response of the vertical channel at frequencies lower than the higher frequencies of the short-correlation acceleration error. So the low-frequency density of Eq. (3) is used for the spectral density of the white noise. Table 1 Nominal values of noise spectral densities White noise for Short correlation time acceleration error Acceleration error random walk Short correlation time altimeter error Altimeter error random walk Noise density symbol 0.1 Qa2 Qb, Qu Noise density value 2 . 4 x l O -4m2s 1.0xlO-9m2s -3 -5 100m2 s 100m2 s'1 Downloaded by UNIVERSITY OF STRATHCLYDE on February 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/3.55966
    174 W.S. WIDNALL AND P.K. SINHA J. GUIDANCE AND CONTROL The acceleration error random walk models the slowly varying error due to accelerometer bias shifts, changes in the gravity anomaly, and changes in the error in the Coriolis terms. If over a period of 1000 s, the accelerometer bias were expected to shift 100 fig, the appropriate noise spectral density for the random walk is Qa2 = (lOOx 10~6 x 10ms -2)2/(1000s) = l.Ox 10~9 m2s ~5 (4) For the short correlation time altimeter error, it is assumed that repeated random fluctuations of the order of 10 m may be present in the baro-indicated altitude, and that these errors persist for correlation times of the order of 1 s. To match the low-frequency spectral density of this error, the white noise error model should have density ew=(10m)2x(ls)=100m2s (5) The altimeter error random walk models the slowly varying error due to: changes in the static pressure measurement error (due to speed changes), variations in the height of a constant pressure surface (the weather pattern of highs and lows), and scale factor error (related to nonstandard-day temperature and nonzero aircraft climb or descent rate). For an at- mospheric scale factor error of 3% and an aircraft climb or descent rate of 33 m/s (6500 ft/min), the error rate of the baro-indicated altitude is 1.0 m/s. If the climb or descent continues for 100 s, the change in altimeter error will be 100 m. Assume that the aircraft trajectory is characterized by a random sequence of such climbs and descents. The ap- propriate noise spectral density for the random walk is then where near the surface of the Earth c=2g/R = 3.07 x 10~6 s ~2. For the steady-state solution to exist and be equal to Eq. (9), the set of loop gains must yield a stable system. Therefore, the permissible values of the gains are in the regions defined by kj>0, k2-c>0, k3>0, k1(k2-c)-k3>0 (10) The explicit computation of the mean-squared vertical velocity error is used in a computer program that seeks a set of gain values that minimizes the mean-squared error. The pattern search algorithm of Hooke and Jeeves15 has been utilized. The algorithm does not require explicit gradient information. The natural frequencies (poles) of the closed-loop portion of the baro-inertial vertical channel are the three roots of the characteristic equation s3 + kjs2 + (k2 -c)s+k3 = 0 (11) the locations of the three poles plt p2, p3. In such a case, the roots of the cubic Eq. (11) are computed according to the known formulas for those roots. When the time constant of a pole is mentioned, it is defined to be the inverse of the real part of the complex frequency of the pole. Optimization Results To provide a baseline design and performance against which to compare the optimized performance, the mean- squared velocity error is evaluated for the set of gains, Eq. (2), which place a triple pole at T = 100 s For a candidate set of gains, it is often of interest to inspect ew=(100m)2/(100s)=100m2s-1 (6) A:7=3.0xlO~2s k2 = 3.0307 x 10 -4s The mean-squared error of the indicated vertical velocity has been selected as an appropriate performance index. Note that with the random walk error models for acceleration error and for altimeter error, only the vertical velocity error has a stationary and finite mean-square value. All other error states have mean-square values that grow unbounded with time. Referring to Fig 1, 60 tracks 5a+ (2g/R)5h as this sum wanders off, and dh tracks db as it wanders off. The mean-squared vertical velocity error may be computed as an explicit function of the input noise spectral densities and of the loop gains. One first calculates four transfer functions Hf(s) independent white noise input. The power spectrum of the vertical velocity error is then relating the vertical velocity error response to each The mean-squared velocity error, with the nominal (Table 1) values of noise spectral densities, is found to be (6v)2 =0.818 m2s ~2 = (0.904 m/s)2 (12) (13) Using the Hooke and Jeeves pattern search procedure, the gains that minimize the mean-square velocity error, with the nominal noise densities , are found to be 7 = 1.003s-1 A:2=4.17xlO-3s-2 (14) The corresponding value of the mean-square velocity error is (7) where Q, is the spectral density of the rth white noise. The mean-square value of the vertical velocity error is the integral of the power spectrum wr2=E^-. (8) The four integrals are evaluated using an appropriate table of integrals.14 The result is 2[kI(k2-c)-k}\ 2[k1(k2-c)-k3] + (k3+ck1)2k1]Qb2 2k3[k1(k2-c)-k3] (dv)2 =0.418 m2 s ~2 = (0.647 m/s)2 (15) This is a significant performance improvement relative to the baseline case. The rms velocity error is 30% lower. The three poles associated with the gain set, Eq. (14), are located at Pj = -0.998s-1 p ,p = -2.082xl0^3 ±/2.34x 10~4 S"1 They have time constants of T7 = 1.002s 480.3s (16) (17) The optimized gains and resulting pole placements (and time constants) are radically different from the baseline triple pole set. One time constant is a factor of 100 faster; the other two time constants are a factor of 5 slower. (9) Downloaded by UNIVERSITY OF STRATHCLYDE on February 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/3.55966 2 3
    MARCH-APRIL 1980 OPTIMUM GAINS FOR A BARO-INERTIAL VERTICAL CHANNEL 175 Table 2 Contributions to mean-square velocity error of nominal noise densities Mean-square velocity error , (m/s) 2 Triple pole, r=100s 0.018 0.00175 0.00018 0.798 Optimized set 0.0291 0.0275 0.00087 0.361 Noise density Qa, c£ Total 0.818 = (0.904)2 0.418 = (0.647)2 where The individual contributions of the various white noises to the mean-square velocity error are shown in Table 2, for the nominal values of the white noise spectral densities. The data of this table show that the mean-square velocity error is dominated by the altimeter error random walk (Qb2), while the contribution of short correlation time altimeter error (Cw) is least. To obtain further insight into the nature of the optimal solution and to exhibit the sensitivity of the optimal solution to obtain to the noise density assumptions, the optimal solution has been computed for various values of the noise spectral densities. Table 3 shows the results for four cases in which one of the noise densities is increased while holding the other three densities at their nominal (Table 1) values. With the optimized gains being so different from the baseline gains, it is interesting to ask: for what set of input noise densities are the "triple pole" gains optimal? From Table 2, one notes that in the "triple pole" case, the con- tributions of the altimeter error noises to the mean-square velocity error would be more nearly equal if Qbl were 10 times larger and Qb2 were 100 times smaller. The optimization program has been rerun with these altered values for noise density. The results are presented as the last case in Table 3. These results demonstrate that the baseline triple pole set is close to being an optimal set if the random walk component of the altimeter error is significantly smaller and if the short correlation time altimeter error is somewhat larger than the nominal assumed values. Analysis of Results In all cases presented in the previous section (except the greatly reduced Qb2 case) the dynamics of the optimal third- order vertical channel are that of a fast first-order loop nested inside a slower second-order system. From Table 3 it is clear that the fast pole frequency is simply related to the first gain Pi~-kt (18) With such a fast real pole, the characteristic equation of the third-order vertical channel can be factored as K j (19) (20) This can be shown by multiplying the two factors in Eq. (19) The correct characteristic equation of the third-order vertical channel, Eq. (1 1), in terms of k'2 is Comparing Eqs. (21) and (22), it is clear that sufficient conditions for the factorization of Eq. (19) to be true are (22) k'2
    176 W.S. WIDNALL AND P.K. SINHA J. GUIDANCE AND CONTROL and the assumed noise densities, two of which have equal values. A dimensionally correct expression that also gives the right numerical value is discovered to be (26) Similarly, using Eq. (27) in Eq. (28) and applying Eq. (30) one obtains k2/kt (31) The preceding formulas for the gain ratios are in excellent agreement with the numerical results in Table 3, for all cases that have the nested fast loop. An approximate formula for the location of the slower This formula is in excellent agreement with the numerical results in Table 3 (except for the greatly reduced Qb2 case). The formula appears valid under the same conditions that give rise to the nested fast loop. A remarkable conclusion is that the first gain is optimized based only on the relative poles as a function of the noise densities may be derived by assuming k2 « k2 and by using Eqs. (30) and (31) in Eq. (25). strengths of the two noise densities in the assumed altimeter error model. One may derive additional useful formulas for the other gains as a function of the noise densities as follows. The explicit formula for the mean-square velocity error / as a function of the noise densities was given in Eq. (9). Assume that the nested fast loop conditions Eqs. (23) and (24) apply. Also assume that k2>c. Note that the numerical results in- dicated that the contribution of the term proportional to Qbl was negligible. Assume that the gradient of this term with respect to k2 and k3 is also negligible. Delete this term from the analysis. An approximate formula for the cost (mean square velocity error) is then a2/Qb2] * (32) This formula also is in excellent agreement with the numerical results. An interesting observation supported by the computer results is that when the nested fast loop is optimal, the op- timal second and third gain ratios (k2/kj and k3/ k j) as well as the optimal second and third pole locations are not a function of the assumed density of the short-correlation-time altimeter error. The computer results showed a factor of 5 increase in Qbl producing a shift in the optimal time constant of less than 0.1%. When the strength of the altimeter error random walk Qb2 is sufficiently large relative to the strength of the other sources of error, that is when Qai/Qb2<2c Qa2/Qb2
    MARCH-APRIL 1980 OPTIMUM GAINS FOR A BARO-INERTIAL VERTICAL CHANNEL 177 35 30 10 5 \ 200 400 600 800 1000 1200 TIME (SECS) Fig. 2 Trajectory altitude history. n————i————i————i— 300 1800 1000 800 TIMECSEZCS) 1 0 800 600 800 TIMECSECS) 100®. 1800 Fig. 4 Performance with baseline gains. 1000 1200 200 400 800 600 TIME (SECS) Fig. 3 Trajectory heading history. P2,p3 « -Vc= - 1.75 x 10 r2,TJ«//Vc=571s (36) (37) (38) These limiting results are a function only of the destabilizing gravity gradient c. Note that the nominal case, the increased Qbl case, and especially the increased Qb2 case have computed results (Table 3) approaching this limiting case. An important conclusion is that even if the measured specific force is perfect (zero Qal and Qa2), the feedback gain ratios k2lkl and k$lkl must be maintained at certain nonzero values to stabilize the vertical channel and to minimize the effect of the gravity computation error. These required values correspond to an upper limit on the optimal double pole time constant of 571 s. When the strength of the acceleration short correlation error Qa] is important in the sense that al /Qb2 - Qal /Qb2 >2-Jc2 + Qa2 then the optimal gain ratio k 2lk l simplifies to and Eq. (32) yields two real poles at (39) (40) (41) (42) Fig. 5 Performance with optimized gains. 800 1000 ' 1£00 These formulas approximate computed results obtained in the increased Qal case (Table 3). When the strength of the acceleration error random walk Qa2 is important in the sense that al /Qb2 <2^2 + Qa2/Qb2 then the optimal gain ratios are (44) (45) Downloaded by UNIVERSITY OF STRATHCLYDE on February 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/3.55966
    178 W.S. WIDNALL AND P.K. SINHA J. GUIDANCE AND CONTROL and the associated pole locations are (Qa2/Qb2) * (46) These formulas approximate the computed residuals obtained in the increased Qa2 case (Table 3). Simulated Performance The vertical channel gains obtained by the optimization procedure have been evaluated using a time-domain three- channel simulation that includes detailed models for the sources of acceleration and altitude errors and that exhibits the dependence of these sources of error on the aircraft trajectory. Error sources included in the simulation are listed in Table 4. The aircraft trajectory simulated represents a F4 tactical mission profile. Figure 2 shows the altitude vs time. Figure 3 shows the heading vs time. This is a high dynamic trajectory. During the first descent the aircraft executes a "figure-eight" maneuver. During the remainder of the flight while climbing and diving. In some of these maneuvers, maximum bank angles of 70 deg are used, with associated load factors of 3g. One pull-up has a load factor of 5g. The vertical channel performance results with the baseline gains, Eq. (12), and with the optimized gains, Eq. (14), are exhibited in Figs. 4 and 5. The optimized gains do provide a lower level of vertical velocity error. A noticeable con- sequence of the fast (r =1 s) loop around the indicated altitude is the increased noise content in the indicated altitude due to the baro-error fluctuations. This may be a disad- vantage of the optimized gain set in some applications. the aircraft is performing rapid zigzag evasive maneuvers Conclusions The fundamental assumptions underlying the results of this analysis are that the most important sources of error in the third-order vertical channel may be adequately modeled by random walks and white noises as presented earlier. If these assumptions are correct, then the following conclusions are obtained. The most significant source of error in the vertical channel is the fluctuation in the altimeter bias (such as due to altimeter scale factor error and nonzero vertical velocity). The second most significant source of error is the short-correlation time acceleration error (such as due to specific force measurement error during a maneuver). Altimeter noise at the assumed level has negligible effect. The optimal choice of the gain set is radically different from the baseline gain set, which provides a triple pole at of-magnitude larger than that in the baseline set. The optimal value of kl is a function only of the relative strengths of the altimeter bias fluctuation and the altimeter noise. For the assumed relative strengths in the nominal case, the optimal gain value recommended is kl = 1.0 s ~ *. The optimal values for k2 and k3 are sensitive to the assumed numerical values for the noise densities. However, the optimal gain ratios /k and k are relatively //isensitive to the assumed k /k noises. The gain ratios recommended by the nominal optimal solution are k2/kt =4.2x 10~3 s ~l and k3/kj =4.4x10~6 s ~2. This choice of gains will place a fast pole at r = 1 s and a double pole at r =480 s. This time constant of the optimal double poles is slower than the time constant of the baseline triple poles. T = 100 s. The optimal gains include a kl that is two orders- The optimal gains produce a significant performance improvement compared with the baseline case. The rms vertical velocity error is reduced 30%. The recommended value for kt perhaps should be accepted with some degree of skepticism. However, the recommended values for the gain ratios k2/k1 and k3/kj can be adopted with some confidence, because of their low sensitivity to the assumed noise values. The low sensitivity is a result of the gain ratios approaching a fundamental limit imposed by the destabilizing feedback of the gravity computation error. If the short-correlation time acceleration error important than assumed in the nominal case, the optimal gain ratio k2lkl increases and the optimal double pole splits into two real poles. On the other hand, if the acceleration error bias (such as due to accelerometer bias and gravity anomaly) is shifting more than assumed in the nominal case, both optimal gain ratios are increased and the optimal double pole splits into a complex conjugate pole pair. A detailed baro-inertial error simulation has exhibited the reduced vertical velocity errors that can be obtained with the optimized gains. It provides confidence that the fundamental assumptions of the stochastic analysis are sound. is more Because of the very long settling time associated with the recommended optimized gains, one should also implement a faster set of gains for use in the ground alignment mode. References Draper, C.S., Wrigley, W., and Hovorka, J., Inertial Guidance, Pergamon Press, New York, 1960. 2McClure, C.L., Theory of Inertial Guidance, Prentice-Hall, Englewood Cliffs, N. J., 1960. 3Markey, W.R., The Mechanics of Inertial Position and Heading Indication, John Wiley & Sons, New York, 1961. Pitman, G.R., Jr. (ed), Inertial Guidance, John Wiley & Sons, 5O'Donnel, C.F., Inertial Navigation, Analysis and Design, 6Broxmeyer, C., Inertial Navigation Systems, McGraw-Hill, New 7Farrell, J.L., Integrated Aircraft Navigation, Academic Press, New York, 1962, p. 41. McGraw-Hill, New York, 1964, pp. 35-36. York,1964, p.148. New York, 1976, pp. 21-22. Wiley & Sons, New York, 1969, pp. 317-319. 8Kayton, M. and Fried, W.R., Avionics Navigation Systems, John 9Ausman, J.S., "GPS Bombing Demonstration Error Analysis," Litton Guidance & Control Systems Division presentation to General Dynamics Electronics Division, May 1976. 10Whalley, R., "Inertial Techniques in Height Measurement," Journal of the Institute of Navigation, Vol. 19, No. 1, 1966, pp. 61- 67. 11 Ausman, J.S. and Kouba, J., "Airborne Ran^e Instrumentation System (ARIS)," Vol. II (Mechanization Equations), Litton Guidance and Control Systems Document No. 402029, Woodland Hills, Calif., Nov. 1972, pp. 25-28. 12Ausman, J.S., et al., "Close Air Support System (CLASS) F-4D Flight Test," Vol. I (Summary), Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio, TR-73-363, Sept. 1973, pp. 96-102. 13Blanchard, R.L., "A New Algorithm for Computing Inertial Altitude and Vertical Velocity," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-7, Nov. 1971, pp. 1143-1146. 14Newton, G.C., Gould, L.A., and Kaiser, J.N., Analytic Design of Feedback Controls, John Wiley & Sons, New York, 1961. 15Hooke, R. and Jeeves, T.A., "Direct Search Solution of Numerical and Statistical Problems," Journal of the Association for Computing Machinery, Vol. 8, April 1961, pp. 212-229. 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