A MODIFIED TARGET MANEUVER ESTIMATION TECHNIQUE USING PSEUDO-ACCELERATION INFORMATION

IckHoWhangt TaeKyungSungt JangGyuLee4 Dept. of Control and Instrumentation Eng.

& The Automation and Systems Research Institute

Seoul National University, Seoul, Korea

ABSTRACT

This paper presents a tracking filter with maneuver compenoa- tion technique using pseudo-acceleration measurements or pseudo- acceleration residuals, instead of filter residuals which is com- monly used in the conventional techniques to detect and compen- sate maneuvers. The pseudo-acceleration measurement which is more sensitive to maneuvers is defined. The pseudo-acceleration residual is formed using the pseudo-acceleration measurement. The proposed filter is developed then based on the pseudo accel- eration residuals. In developing the proposed filter a new target model which has a stepwise change of nominal acceleration level is employed. It is shown that the proposed filter detects maneu- vers more sensitively than existing methods. Consequently an improvement of tracking performance can be achieved with the new filter.

1. INTRODUCXION

A tracking filter is to estimate the position, velocity, and ac- celeration of a target in real time. For its recursive structure and capabilities of sequential data processing, a Kalman filter is widely used as a tracking filter, which is called Kalman tracking filter. To implement this Kalman tracking filter, a model which assumes target dynamics as a constant velocity motion had been tradition- ally used. But the filter with this model can not efficiently cope with a maneuvering target environment.

In 1970, Singer[l] proposed a model which considers a maneu- ver as a Markov first order process with zero-mean and correlation time 7 . A tracking filter with Singer's model shows good perfor- mance for low maneuvers but its performance is degraded in case of high maneuvers. Berg(Z] improved Singer's model by adding a mean jerk term on the assumption of the coordinate turn. Song et. a1.[3] interpreted the acceleration of a maneuvering target as a random perturbation around the coordinate turn. They proposed an acceleration model by adding a mean target acceleration term into Berg's model. But their performances are still unsatisfactory for a highly maneuvering target. In this paper, a new maneuver model is proposed by considering the acceleration as a pertur- bation around a nominal acceleration called a maneuver level. In the model, each maneuver level is modeled as a step function, and a maneuver is regarded as a transition of a maneuver level. To use the model, a maneuver level and its changing time should be obtained. Since the Singer's model can be interpreted as pertur- bations around zero maneuver level, the proposed model includes it. Moreover, this model can describe the acceleration of a high level maneuver as well as a low level maneuver, because it can be done simply by changing maneuver levels.

In 1973, McAlay and Denlinger[7] propoeed an adaptive tracker which employed maneuver detection and correction technique. This filter detects a maneuver using the fact that the Kalman tracking filter residuals are increased if a maneuver occurs. And it compensates the maneuver by increasing error covarianceg. Since this method does not compensate filter estimates directly, i t may

t Doctoral Student $ Associate Professor, member AIAA

Copyright O 1992 American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

require a long time to compensate maneuvers. Thereafter, many adaptive tracking filters such as a variable dimension filter by Bar-Shalom[8], a multiple model technique by Moose[Qll], and an input estimation technique by Chan et. a1.[5,6] were suggested.

The input estimation technique regards a maneuver as an in- duced accelerations from command input, and estimates the mag- nitude of the input using least square method. But this technique shows poor performance in case of a low maneuvering target b cause of its tendency to over-compensation. To complement this problem, Bogler[4] proposed an adaptive filter with a revised in- put estimation technique which estimates the starting time of a maneuver as well as its magnitude. However, Bogler's technique needs too long a window to estimate a low level maneuver. It requires a complex filter structure and a lot of computations as well. So, it is suitable way to adapt a tracking filter to maneuvers that $nger's model is used for a low level maneuver and a maneu- ver detection technique is used for a high level maneuver. In this paper, a new tracking filter scheme which requires a short window length is proposed by combining the Kalman tracking filter with process noises and a new maneuver estimation method.

The existing maneuver detection methods commonly exploit the property that the Salman tracking filter residuals are un- biased in case of no maneuver, but they are increased when a maneuver occurs. A Kalman tracking filter which uses a model with process noises like Singer's regards it as a process noise. And it tends to change the estimates to reduce the residuals. This ef- fect, called Q-effect, suppresses the increase of residuals caused by the maneuver. In case of low maneuvering targets, the Q-effect enables to maintain the track without maneuver detection. It can also track a slowly varying maneuver input. But, in this case, the estimates have some biases. Moreover, Q-effect cannot track a high maneuver but makes some insensitiveness to maneuver de- tection, which degrades the performance.

In this paper, in order to reduce the Q-effect on maneuver detection, pseudo-acceleration residuals are defined from the esti- mates a t the time of a maneuver. Since they are independent of the estimates after a maneuver occurs, they are more sensitive to maneuvers than the Kalman tracking filter residuals. In this pa- per, a new maneuver detection and estimation method is derived by minimizing the sum of normalized pseudo-acceleration residu- als. Combining this method and the Kalman tracking filter using the proposed model makes a new tracking filter. In the filter, the magnitude and starting time of a maneuver level are estimated a t every time stage for maneuver detection. Since the filter is derived with pseudbacceleration residuals, it is more sensitive to the maneuver than any other tracking filter employing detection techniques with Kalman tracking filter residuals. Also a simplified method using the pseudcxw.celeration measurements is proposed, which drastically reduces computation time while maintaining a satisfactory performance.

2. M O D I F I E D M A N E U V E R E S T I M A T I O N T E C H N I Q U E U S I N G PSEUDO-ACCELERATION

INFORMATION

2.1 Target Models

A target dynamics model has to be simple for real-time im- plementation, while representing the target motion effectively. In 1970, Singer[l] proposed a maneuver model as a first order Markov process with zer*mean and correlation time r as follows,

where a(t) is an acceleration and w(t) is a zero-mean white noise with variance ~ { w ~ ( r ) ) = qb(t- r). Since the target accelerations can be interpreted as perturbations around zero, Singer's model cannot cope with high maneuvering target. In order to overcome this problem, in this paper, a maneuver level is introduced. A maneuver level is defined as a nominal point of acceleration. Then the target acceleration is modeled as a perturbation around the maneuver level. If a target changes its maneuver level from bl to bz at time t,, the acceleration a(t) can be represented by Eq. (2).

a(t) = a ( t ) + b~ + (ba - b ~ ) l ( t - t.) (2 )

In the equation, a( t ) behaves along Eq. (1) and l(t) is a unit step function. From Eqs. (1) and (2), the dynamics of the acceleration a(t) can be obtained.

1 i ( t ) = - - a ( t ) + w( t ) + (bp - bl)6(t - t,)

1 1 = - - a ( t ) + ;{bl + (ba - bl) l ( t - t , ) )

+(ba - bl)6(t - t.) + w(t) (3)

of position, velocity, and acceleration, can be written as follows.

For the sake of the implementation in a digital computer, dis- cretization of Eq. (4 is needed. Let the sampling time be T, and assume that t, = n T' Define zk be a state vector at time t = kT. By the similar method to [I] and [3], the discrete form of Eq. (4) can be obtained as follows,

Fzk + Gbl + wk ; k < n F[zr + B(ba - b l ) ] + Gba + wk ; k = n Fzk + Gba + wk ; k > n

(5)

where

and wk = [wi w; w;IT is a zero-mean white noise vector with the same variance as in [I].

In this model, a maneuver is represented as a transition of maneuver levels. So, to construct a filter using this model, a maneuver level change must be checked at every time stage, and a new maneuver level and its starting time should be estimated. Since a maneuver level is modeled as a step function, this model is suitable to the input estimation techniques.

2.2 Pseudeacce le ra t ion Measurementa a n d Pseudeacce le ra t ion Residuals

When a target flights in constant velocity, its states do not change abruptly. Its velocity k e e p a constant, and its position increases in proportion to the velocity. A maneuver input, how- ever, makes the severe changes of states, which enlarges Kalman tracking filter residuals. A Kalman tracking filter which uses the target model with process noises like Singer's considers it as a process noise. It reduces the increase of residuals by changing the state estimates. So, the estimates after a maneuver occurs are influenced by the maneuver. This effect will be called Q-effect. Q-effect suppresses the increase of residuals while a maneuver en- larges the residuals. In case of low maneuvers, Q-effect prevents the residuals from increasing, so the tracking is possible. But, in case of large maneuvers, the tracking performance is degraded because Q-effect cannot suppress the increase of residuals suffi- ciently. Hence, a maneuver detection and correction method is needed to track a highly maneuvering target.

The conventional maneuver detection techniques are baaed on the residuals. Because Q-effect decrease0 residuals which are influ- enced by a maneuver, the conventional maneuver detection meth- ods are insensitive to a maneuver. To reduce the Q-effect. in this section, a pseudo-acceleration residual is defined using the esti- mates at the time of a maneuver. Since it is independent of the estimates after a maneuver occurs, it should be more sensitive to a maneuver than Kalman tracking filter residuals.

Now, examine the effect of the maneuver on residuals in one dimensional case. Assume that the state zk is made up of po- sition, velocity, and acceleration, denoted by rk, vk, ak, and a measurement zk is given by Eq. (6),

where H = [I 0 01 and nk is a zero mean white Gaussian random noise with variance R. Let a posterior estimate cf z i be kk and its error covariance matrix be Pk. b u r n i n g that the current maneuver level is b, the Kalman filter for the system of Eqs. (5) and (6) is given by

where the Kalman gain Kk is given by P ~ H ~ [ H P ~ H ~ + R]-', and a prior error covariance matrix P; is equal to FPk-'FT +Q.

Since the acceleration is modeled as perturbation around the current maneuver level, b is approximately same as dk. If T is small enough, the residual 6k+1 in Eq. (8) can be approximated by Eq. (lo),

where 4, Oh, dk are the estimates of rk, vk, ak respectively. Define a peeudo-acceleration meaeurement as follows.

Subtracting irk from both sides of Eq. ( l l ) , Eq. (12) can be obtained.

T 2 T a + i k + ~ - a,+,) = z k + ~ - 11, +TO, + - a k ) 2 (12)

The comparison between Eqs. (10) and (12) shows that Eq. (12) does not contain ik. Since the estimates are influenced by Q- effect, the pseudo-acceleration measurements are more sensitive to maneuvers than Kalman tracking filter residuals.

Now, examine the properties of pseudo-acceleration measure ments. Assume that the current maneuver level b is correct. From Eqs. (5), (6), and (12), Eq. (13) can be obtained.

where wi is a position process noise. Since the current maneuver

level is correct, all estimates of the Kalman tracking filter are unbiased, and nk and w i are both zero mean white Gaussian. Therefore, - 8k is Gaussian with zero mean. Its variance is given by Eq. (14),

where Kk(2) and Kk(3) are second and third element of the Kalman gain vector KL. But if the target changes its maneuver - - - level, i ik - iik provides a biased value since vk - ek and a t - bk have biases.

The estimates a t the time of a maneuver are more useful to maneuver detection since they are not influenced by Qeffect. A pseudo-acceleration residual ia defined using the estimates a t the time of a maneuver. In this way, the pseudo-acceleration residuals are more sensitive to the maneuver than the original residuals. Let b and n be the new maneuver level and i b starting time respectively, and k be the current time. The pseud*acceleration residual 6i-1 is defined as follows,

i f k < n b (new maneuver level) i f k > n

The variance of 6:- , denoted by Stw1, is given by Eq. (16) where Sk-l is obtained Bom Eq. (14) and that ia independent of k after maneuvering if R is time-invariant.

2.3 Maneuver Est imation Using Pseud*acceleration

2.3.1 Overview

When residuals are used in maneuver detection, the perfor- mance is degraded as mentioned in the previous section. In order to overcome this problem, a new tracking filter is derived using the target model in section 2.1 and the pseudo-acceleration reaid- uals. In the proposed target model an acceleration is represented as a perturbation around the maneuver level. To implement the filter using this target model, it is necessary to estimate an ac- curate maneuver level and its changing time. In 1987, Bogler(41 proposed a method to estimate input magnitude and its starting time using Magill bank of N filters. It is obtained by optimizing the performance index which is made up of Kalman tracking filter residuals. But it is somewhat insensitive to maneuvers because of Q-effect. In this section, a performance index is defined as the sum of normalized pseudo-acceleration residuals, and a new ma- neuver level estimation scheme is derived by minimizing it. Also, in order to reduce the computation time, a simplified method is proposed using pseudtxmderation measurements.

2.3.2 A N e w M e t h o d Of Maneuver Est imation

Assumption that a new maneuver level b is started at time n, and current time is k. Define the performance index J (n) as follows.

In the e uation, 6Ll and Si"_ are given by Eqs. (15) and (16). Since J(Q is a function of b andn, it must be minimzed over both b and n. Differentiating Eq. (17) wit! respect to b and setting it be zero, the optimal maneuver level b can be obtained as follaws.

By inserting Eq. (18) into Eq. (17), J(n) becomes only a function of n.

Let I be a window length. Rearranging the Eq. (19), Eq. (2O)can be obtained if Rk is time invariant,

where

The optimal fi can be obtained by finding i which minimizes J ( i ) in Eq. (20)overi= k-1,k-(I-1) , . . - ,k-1. Sinceonly AJ(n) can be changed in the window in Eq. (20), it is sufficient to check AJ(i) over i = k - I, k - (I - I), . . . , k ; 1 for estimating n.

The estimate of the maneuver level b in Eq. (18) is computed using informations from maneuver starting time n to current time k. If k - n is too small, an inaccurate maneuver level may be obtained because of the lack of data. Thus, it is necessary to restrict the smallest value of k-n, named least entimation interval. Note that the filter becomes sensitive to the measurement noise if the least estimation interval is too short, while the maneuver detection is delayed if it is too long.

Now, the pmpensation equation of the filter state will be derived using b and fi which have been obtained from Eqs. (18) and (20). Assumed is that the maneuver level is changed from bo t o b a t time n. Also assumed is that the values of b and n are exactly known. The Kalman filter for the system in Eq. (5) becomes as follows.

[ I - K ~ + I H ] ( F J I + G ~ o ) + K ~ + I x ~ + I i f k < n

&*+I = [ I - K ~ + I H ] ( F [ & + B(b - bo)] + Gb) + K ~ + I x ~ + I i f k = n

( [I - K;+IH](F& + Gb) + K ~ + I z ~ + I if k > n

Propagating lZi from i = m to m + j with Eq. (21), Eq. (22) can be obtained where m < n and j = 1,2,3,..-.

where

M =

Let the maneuver detection time(or current time be k. Because the maneuver is not detected from n to k, & shoul 1 be propagated without ~!(b - bo) in that interval. Hence, the filter states are to be compensated in the following way.

In the equation, 2; is a compensated state and ii, 8 are the estimates of n and b respectively.

To compensate the error covariance Pk, the covariance of b - 8 should be computed. Under the assumption that the estimate of n is correct, Eq. (24) is obtained by inserting Eq. (15) into Eq. (18).

where

k-n

V D ~ = [ T a E ( j - f 1'1'

When i , j > n, the covariance of pseudo-acceleration residuals are given as follows.

where

In Eqs. (24), (2fi) and (27), it is assumed that ii is equal to n. But In pract~ce, n is somewhat different from n. The state error covariance should be compensated as follows.

2.3.3 Simplified Method of Maneuver Estimation

From Eqs. (24) and (25), the error c o v a r i ~ c e of 8 is given by Eq. (26). Note that the error covariance of b - bo is same as that of 8.

k-n k-n-1 1 1

v N 2 = 2 R [ x ( j - S)' - x ( j + i ) ( j - 5 ) ] ,= 1 j= 1

Pr = Pk + T: V [Ti ]= + errors due to d (28)

In this equation, is the compensated error covariance matrix. Since the errors due to A are small compared with the first two terms when ii is close to n, the last term in Eq. (28) may be neglected.

Since a maneuver is defined as a transitioqof a maneuver level, a maneuver detection logic checks whe*er Ib - bo( is sufficiently small. Using the error covariance of b given by Eq. (26 , the following inequality has to be tested a t every time stage to etect a maneuver.

d

In the equation, T h means a specified threshold. In summary, from Eqs. (18) and (20), the estimates of the

maneuver level and its starting time are computed. Using the estimate of maneuver level, maneuver detection logic is tested according to Eq. (29). If it satisfies, the state and error covariance are compensated by Eqs. (23) and (28). After the compensation, the filter is propagated using the new maneuver level b.

In the case that R is time invariant, V is given by Eq. (27) because SLl is independent of i for i > n.

In the above section, a new maneuver estimation scheme was proposed. In order to implement this method, however, a large amount of computations is needed. Especially, the estimation of maneuver starting time n requires a great deal of computa- tions, because AJ(m) in Eq. (20) should be computed for all m = k-1,k-(1-I), . .- ,k-1. In thisection,anewmethodfor estimating n is introduced, which can quite reduce the number of computations.

Recall the properties of i ik- i ik discussed in section 2.2. ilk -irk has no bias when the current maneuver level is correct and should have some biases when a maneuver occurs. Using this property, another method for estimating maneuver starting time can be derived. Define y ( m ) , f l m ) and D(m) as follows.

If the maneuver starts a t time n, it is expected that D(n) has maximum value over D(k - I), .. - , D(k - 1). Hence, m which maximizes D(m) over m = k - I, - - a , k - 1 can be regarded as 8. In comparison with Eq. (20), D(n) in Eq. (32) requires drastically less computations. In combination with Eqs. (la), (29), (23),

(28), this new estimation method for n can be another efficient estimator.

Monte Carlo simulation results of the proposed maneuver es- timation technique, its simplified form, and Bogler's input esti- mation method[4] are compared for one dimensional case. For the simplicity of notation, the filter derived in section 2.3.2 and its

Table 1: Maneuver Detection Time & Estimated Starting Time

I

detection time 11 53.370 1 0.808 1 52.740 1 1.354 1 54.604 1 0.907 startinn time 11 49.408 1 1.136 1 48.780 1 1.578 1 50.490 1 0.866

1st detection in 50-601secl

1 # of no detection 11 0 0 4

Filter I 1 Filter I1 1 Bogler mean I s.d. I mean I s.d. I mean I s.d.

simplified form in section 2.3.3 will be nemed as Filter I and Fil- ter I1 respectively. The measurement model is given by Eq. (6). The proposed target model is employed both for the proposed filters and the Bogler's filter. Other simulation conditions are as follows.

sampling interval T : 1 [see] maneuver correlation time r : 10 process noise variance q : 2 x ! x 5' variance of measurement noise R : 100[m2] maneuver detection threshold : 3u level (about 99%) window length 1 : 5 least estimation interval : 3 the number of Monte Carlo runs : 100

For performance analysis, three cases of maneuver scenarios, which include cases of sudden and slow changes of maneuver levels, are considered as shown in Fig. 1. In order to examine the detection performance, mean and standard deviation of the first detection time and estimated maneuver starting time ii for the case 1 are tabulated in Table 1. In the table, mean and standard deviation are obtained from the detections in 50 - 60[sec] and detections after 60[sec] are considered as no detection. Since the least ek timation interval is 3, the detection time should be 53[sec], and ii be 50[sec]. In comparison with Bogler's method, Filter I de- tects the maneuver faster and has a smaller standard deviation of detection time. It is because pseudeacceleration residuals are more sensitive to the maneuver than the Kalman tracking filter residuals. Though Filter I1 has the fastest detection time, the estimate of maneuver starting time is less accurate than Filter I. And it has larger standard deviation both in detection time and 8. Since Filter I1 makes inaccurate ii and many detections, it has large RMS state estimation errors.

Fig. 2 shows the mean velocity and acceleration errors over 100 Monte Carlo runs for Case 1. In the figure, both of Fil- ter I and Filter I1 have little biases regardless of maneuver levels, while Bogler's filter has a little biases. Bogler's filter computes the maneuver level which minimize the square sum of normalized residuals. Since residuals are influenced by Q-effect, the esti- mates obtained from the residuab have some inaccuracy, which produces biases. On the other hand, because the proposed filters are less influenced by Q-effect, they can estimate the maneuver level more precisely and have little biaees. Though they makes more detections than Bogler's because of their tendency to sen- sitiveness, they show smaller mean errors. In particular, Filter I has low levels in both mean and RMS errors.

Fig. 3 and Fig. 4 show the mean velocity and acceleration er- rors for the maneuver scenarios of Case 2 and Case 3, respectively. During the interval in which the maneuver inputs are gradually increasing or decreasing, Kalman tracking filter tracks the maneu- ver input by Q-effect. Because Q-effect suppresses the increase of residuals, Bogler's filter hardly detects the maneuver and it does not change the maneuver level adaptively. So, it has some biases. In the new model, a maneuver level is modeled as a step function which has different properties from the maneuver input in that interval. So, the proposed filters also have some biases although they detect the maneuver.

In case 3, the input gradually changes the maneuver level from 15[m/sec2] to 0[m/sec2]. Bogler's filter cannot detect the change

of maneuver level. So it k e e p some biases in spite of the end of the maneuver level change. The proposed filters, however, can adapt its maneuver level properly because they can detect the change of maneuver level more aensitively. Hence, they k e e p little biaeee no matter how the maneuver level changes gradually.

In this paper, a tracking filter with a modified input estima- tion technique is proposed. Pseudo-acceleration meosuremente and pseudo-acceleration residuals, which are more sensitive to a target maneuver than the Kalman tracking filter residuals, are introduced. With the pseudo-acceleration residuala and a new target model representing transitions of nominal accelerations, a new method for tracking a maneuvering target is derived. Using the pseudo-acceleration measurements, a simplified method is also proposed that can reduce the number of computations drastically.

Simulation results show that these new filters are more sensi- tive to maneuver than a filter using Kalman tracking filter resid- uals, and have little biases regardless of maneuver level changes. In comparison with the pro@ filter, its simplified form pro- duce less precise estimation of the maneuver starting time and consequently have larger error deviation, while computations are greatly reduced.

5. REFERENCES

[I] R.A. Singer, 'Estimating Optimal Tracking Filter Perfor- mance for Manned Maneuvering Targets," IEEE Trans. Aerosp. Electron. Syst., Vol. AES-6, Jul., 1970

(21 R.F. Berg, "Estimation and Prediction for Maneuvering Target Trajectories," ZEEE Iltons. Automat. Contr., Vol. AG28, Mar., 1983

[3] T.L. Song, J.Y. A h , C. Park, 'Suboptimal Filter Design with Pseudomeasurements for Target Tracking," ZEEE Bans. Aerosp. Electron. Syst., Vol. AES-24, Jan., 1988

141 P.L. Bogler, "'Ikacking a Maneuvering Target Using Input Estimation," IEEE Bans. Aerosp. Electron. Syst., Vol. AES-23, May, 1987

[5] Y.T. Chan, A.G.C. Hu, J.B. Plant, "A Kalman Filter Based Tracking Scheme with Input Estimation," ZEEE Bans. Aerosp. Electron. Syst., Vol. AES-15, Mar., 1979

[6] Y .T. Chan, J.B. Plant, J.R.T. Bottornley, 'A Kalman Tracker with a Simple Input Estimator," ZEEE Bans. Aerosp. Elec- tron. Syst., Vol. AES-18, Mar., 1982

[7] R.J. McAulay, E. Denlinger, 'A Decision-directed Adaptive Tracker," IEEE Bans. Aerosp. Electron. Syst., Vol. AES- 9, Mar., 1973

[8] Y. Bar-Shalom, K. Birmiwal, 'Variable Dimension Filter for Maneuvering Target Tracking," ZEEE IZtonr. Aerosp. Electron. Syet., Vol. AES-18, Sep., 1982

[9] R.L. Moose, 'An Adaptive State Estiamteion Solution to the Maneuvering Target Problem," ZEEE Iltons. Automat. Contr., Vol. AC20, Jun., 1975

[lo] N.H. Gholson, R.L. Moose, 'Maneuvering Target Tracking Using Adaptive State Estimation," ZEEE Bans. Aerosp. Electron. Syst., Vol. AES-13, May, 1977

[ l l ] R.L. Moose, H.F. Vanlandingham, D.H. McCabe, "Mod- eling amd Estimation for Tracking Maneuvering Targets," IEEE Bans. Aerosp. Electron. Syst., Vol. AES-15, May, 1979

VEL. ERRORS [CASE 21 8

6

MANEUVER INPUT TYPES

......... BOGLER // - FILTER I

[CASE 2 ]

-14 -

......... BOGLER - FILTER I -- FILTER I1

.....

Figure 1: Maneuver Input Scenarios

VEL. ERRORS [CASE11 2 0 ,

I , , , , , , , , , ,

Figure 3: Mean Velocity and Acceleration Errors of Case 2

-10 -

10

R s V) O u 2 -5

........... u BOGLER -10 ---- FILTER I

FILTER I1

VEL. ERRORS [CASE 31

- 1 8 , 1 1 T 1 1 r I I 1 1 0 20 40 80 I W 120 140 1 0 180 200

ACC. ERRORS [CASE 21

............ BOGLER

-----. ----- FILTER I

FILTER II FILTER II

0 20 40 m m loo 120 140 180 180 po

TIME[SEC]

ACC. ERRORS [CASE11 16 14

12 h ., ;< ,

-3 .... .,.. >., : i . . I...

- 5 , . . . . . . . . . . . . . . . . . . . .

0 20 40 m 80 1W 120 140 1 0 180 200 10

8

< 0

Ow 4

I : i$ .......... u -3 BOGLER B

-- - FILTER I

-10 FILTER II -12 -1 4 -16

0 20 40 m 80 loo 120 140 1m 180 200

TIME[SEC]

ACC. ERRORS [CASE 31

........ BOGLER ----- FILTER I

FILTER II

Figure 2: Mean Velocity and Acceleration Errors of Case 1

Figure 4: Mean Velocity and Acceleration Errors of Case 3