purdue university school of aeronautics and astronautics

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PURDUE UNIVERSITY ~ ~~~ SCHOOL OF AERONAUTICS AND ASTRONAUTICS I

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(hASA-CE-160k56) BIALXSIS CE bLB CCNPIGUBA?ICNS &SING GCB PGB €IIFC!I l lODELIIIG l a t e r i a S t a t u s €&Fort (Purdue Uaiv,) 108 p

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Unclas 45360

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West Lufuyetfe, Indiana 47907

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ILIEMO: Infomial Subject: Analysis of NLR Configurations Using OCM for Pilot Modeling From: M.H. Dcijeske To: D.K. Schmidt

INTRODUCTION

The purpose of this memo is to present a summary of the results obtained from an analytic handling qualities analysis of the rate-commandattitude-hold aircraft configurations originally presented in an NLR report by Mooij[’]. Pilodvehicle performance is evaluated using an optimal control technique for pilot modeling. Numerical and graphical results for a closed-loop frequency-domain analysis are presented and discussed and comparisons with experimental results are made. Finally, the results from this study are compared with those from another study that dealt with similar configurationsi2].

NLR CONFIGURATIONS

The aircraft configurations studied by Mooij were variations of a medium-weight, twin- engine jet transport (a modified Foker F28W6000) with a rate-commandattitude-hold control system for both pitch and roll. Two aircraft and a total of 12 configurations were studied. The theta-to-stick and gamma-to-stick transfer functions for the 12 configurations are presented in Appendix I. As shown in Table 1, the 12 configurations constitute three groups: E,F and G. Further details concerning the aircraft configurations and control system implementation are contained in Chapter 7 of [ 13.

Pa ra-,e t e r

v a r i e d r a t i o n n

Conf i r u -

I

1 1

s h o r t - E - 1

p e r i o d E- 2

f r e q u e n c y E - 3 4.71

E - 4 E-5:’

n unie r a t o r E-5:.

4 .71 t ime F- 3 c o n s t a n t F- 2

F- 1

R a n oe uv r e G - 1

enhancement G- 2

1 1

3.41 G- 3 G-4

I --1 E q u i v a l e n t t r a n s f e r

f u n c t i o n pa ramete rs Table 1 NLR Configuration Sets

- 2 -

CRITICAL TASK MODELING


Two separate tasks were modeled in this analysis: (1) The precise control of attitude (i.e., attitude tracking) and (2) The precise control of flight path (i.e., flight-path tracking). The modeling of the attitude and flight-path tracking tasks are discussed in detail in [3] and [4] respectively. The multi-loop nature of the flight-path tracking task is reflected by the block diagram in Fig.1. The describing functions Pu,,Pu and P, in Fig.1 are estimated via the OCM. Using block diagram manipulation, one may reduce Fig.1 to Fig.2 and obtain the equivalent single-loop pilot describing function, Peq.

Figure 1 : Flight-Path Tracking Task

Figure 2 : Equivalent Flight-Path Tracking Task

- 3 -

Similarly, the nature of the precision attitude tracking task is reflected by the block diagram in Fig.3. In this case, Pee and Po may be combined in order to find a single-loop equivalent pilot function as shown in Fig.4. The equations for the above manipulations are presented in Appendix II. I

Figure 3 : Attitude Tracking Task

Figure 4 : Equivalent Attitude Tracking Task

PILOT MODELING USING AN OCM

The pilot-related parameters used in the OCM for the flight-path and the attitude tracking tasks are presented in Tables 2 and 4 respectively. The values selected for z,the pilot’s observation delay, and TN, the pilot’s neuromuscular lag time constant, were chosen to represent the human operator in his most aggressive mode. This is done in order to expose handling qualities digs in the various configurations by modeling the pilot’s most aggressive control techniquesi4]. The rationale behind the selection of the other parameters is discussed in [3] and [4]. Sample input files for the local implementation of the OCM code (PIREP) are presented in Appendix m.

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RESULTS

Presented in Appendix V are the graphical results of the frequency domain analysis of the flight-path and attitude tracking tasks. Included are representations of the pilot functions Pye,Pee,Py and Po as well as the equivalent single-loop pilot function P,. Open-loop and closed-loop Bode plots for all 12 configurations are also presented. Quantities such as crossover frequency, therefore, can be determined directly from these plots.

A number of key quantities (ref. fig.2 and 4) for both the flight-path and attitude tracking tasks are summarized in tables 3 and 5 respectively. The following definitions are used:

5NS

TOTAL COST

crossover frequency - the frequency at which the magnitude of the open- loop pilot/vehicle system equals 0 dE3

phase angle of the equivalent pilot function P, at the crossover frequency

phase angle of PW at lower frequencies

bandwidth - the frequency at which the closed-loop pilothehicle response has phase = -90 deg

pilot phase compensation for the attitude tracking task adjusted to be comparable to a quantity discussed by Neal-Smith 5NS = $Peg I Bw + 57.3 z Bw + tan-'(TN Bw)

maximum magnitude of the closed-loop pilotjvehicle response for the attitude tracking task

predicted rms gamma error

predicted rms theta error

predicted rms stick rate

value of OCM objective function

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Parameter

I I I I I I I

I I'

Value

Config.

~~

Observation Vector

Objective Function Weights

Observation Thresholds

E- 1 E-2 E-3 E-4 E-5

F-3 F-2 F- 1

G- 1 e - 2 G-3 G-4

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7; = [ye i e 9 y 9 i 9 e 9 61 qi = 1 9 0 0 0 9 0 01 rF, = 0 , gi, to yield TN

TYe , Tu, To = 0.05 deg Tie, Ti , Tb = 0.18 deglsec

- 5 -

Observation Noise Ratio

T;lble 2 : Pilot Model Parameters

-20 dB All Observed Variables

Command Signal (n=white noise)

0Jn = .25/(s2 + .5s + .25) yc/ec = S / ( s + .5)

Fractional Attention

fi = 0.3333 All Observed Variables

Observation Delay z=O.2 sec

Neuromuscular Lag zNz0.2 sec

Motor Noise Variance

Control Input I F, (Stick Force in lbs.)

Table 3 : Performance Measures for Flight-Path Tracking Task

8.00/ - 5.17/5.67 4.00/ - 3.00/ - 2.33/3.67

3.33/ - 4.331 - 6.3 3f7.67

5.67/6.00 4.171 - 4.00/5.3 3 3.001 -

0,

(radsec)

1.800 1.819 1.858 1.892 1.922

1.967 1.97 1 2.077

1.935 2.342 2.348 2.344

57.19 55.63 53.31 46.66 40.09

, 36.06 36.12 23.69

38.09 -53.24 -59.7 1 -6 1.07

61.20 58.43 55.16 54.07 53.72

50.07 50.02 29.86

52.88 48.62 45.53 43.20

.4843

.4743

.4562

.4414

.4287

.4099

.408 1

.3740

.3897

.3410

.3455

.3502

Total Cost

21.77 19.07 15.18 13.29 12.17

9.43 5.7 1 3.37

15.17 7.85 5.02 3.84

.3544

.3399

.3140

.2933

.2764

.2528

.2506 2094

.2384

.17 19

.1707

.1740

-7 T V ? 4 ’

Pxanie ter

Observation Vector

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Value

7; = [e, 9 e , , 0 9 e1

Table 4 : Pilot Model Parameters

Objective Function Weights

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S i = [ 1 , O , O , O ] rF, = 0 , gk, to yield 2,

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Observation Thresholds Tee, TO = 0.05 deg Tie , Ti, = 0.18 deg/sec

E-5

F-3 I

Observation Noise Ratio

G- 1 G-2

-20 dB All Observed Variables

Fractional Attention

f; = 0.5 All Observed Variables

Observation DeIay

Neuromuscular Lag

z=O.2 sec

~ N z 0 . 2 sec

0% OF, Total Cost

Motor Noise Variance I -25 dB

(deg) 1.186 1.179 1.163 1.146 1.132

1.107 1.076 1.07 1

1.143 1.141 1.139 1.137

Control Input F, (Stick Force in lbs.)

(n=white noise)

(lbs/sec)

15.83 13.85 11.08 9.68 8.92

7 .OO 4.7 1 2.42

8.93 8.86 8.80 8.73

Table 5 : Performance Measures for Attitude Tracking Task

IF E-2 1.737 1.747 1.774 1.804 1.829

1.872 1.928 1.939

1.810 1.814 1.817 1.821

26.21 69.04 4.155 24.86 68.22 4.188 21.13 66.05 4.265 16.29 62.99 4.320 11.54 59.82 4.354

6.03 55.98 4.39 1 0.64 52.79 4.480

-4.30 48.90 4.465

12.63 60.40 4.335 12.18 59.99 4.335 11.57 59.54 4.335 11.09 59.15 4.335

1.832 1.806 1.751 1.697 1.654

1.578 1.488 1.478

1.687 1.681 1.674 1.668

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DISCUSSION

Using the data presented, one should be able to address the following questions:

1. Are there correlations between model-based performance measures and experimental pilot ratings, and if so, are these correlations comparable to the results presented in [3] and [4].

2. Do the results of the attitude-tracking task offer any additional insight beyond that gained via the flight-path tracking task results.

3. How do the results for the NLR configurations compare with the results for the TIFS configurations as presented in [2].

When analyzing the model-based performance memcs for the flight-path tracking task, a number of trends were expected. Anderson and Schmidti4] have shown previously that in the flared landing task, a strong correlation exists between pilot rating and achievable bandwidth (or crossover frequency), required pilot phase compensation, rms gamma error, and total cost. Comparing these quantities for the E configuration set in Figs.5-8, we note that there is strong correlation between experimental and model-based performance memcs within this subset of the configurations.

Pilot workload (and thereby pilot rating) has been shown to correlate with the pilot phase compensation required. Pilot phase compensation here refers to the phase angle of the equivalent single-loop pilot function, P,,, at crossover. In Fig.6 we see that there is a consistent trend for configurations E-1 to E-5. Configuration E-5 has the best pilot rating (CH 2.33) and the lowest pilot phase required (40 deg). Likewise, configuration E-1 has the worst pilot rating (CH 8.0) and the highest pilot phase required (57 deg).

A model-based measure of pilothehicle performance is the rms gamma-error computed by the OCM. Fig.7 shows the correlation between pilot rating and rms gamma-error for the E configuration set. In addition, total cost is a model-based measure that has correlated with pilot rating in previous studies[51. Fig.8 shows the correlation between total cost and pilot rating for the E configuration set.

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8 -

n I 0 - 6 - D C - c e e 4 -

n 0 - -

2 -

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Crossover Frequency vs. Pilot Rat ing (E conf~gurotions)

0 E-1

0

0

0 0 E-5

0

1.7 1.8 1.9 2.0

crossover frequency (rad/sec)

Figure 5

Fiiot Phase Comp. vs. Pilot Rat ing (E configurations)

10

i E-5 0 2 1

E-1 0

0

0 0

o ! I #

30 40 50 60

pilot phose comp. (deg)

Figure 6

70

R M S Gamma Error vs. Pilot Rating

0 -

n I V - 6 - 0) C - c e c 4 -

n 0 - -

2 -

( E configurations)

1

n I V - 6 - 0, K - c e c 4 -

n 0 - -

2 -

E-5 0 0

0

E- 1

0

0

o ! 1 I I I I

0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.40 0.49 0.50

rms gamma error (dag )

Figure 7

Total Cost vs. Pilot Rating ( E configurotlons)

10

8 €-1 0

0 E-5 G

0

0

0 1 0.20 0.22 0.24 0.26 0.28 0.3Q 0.32 0.34 0.36 0.38 0.40

total cost from OCM

Figure 8

- 10-

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While the model-based measures for the E configuration set correlated well with experimental pilot rating, such is not the case across all of the configuration sets. It would be exceptional if we could find a single model-based performance metric that always correlated with pilot rating in a predictable manner. However in this case, because of the lack of consistent correlations over all configurations studied, we need to take a closer look at the configuration dynamics in the attempt to explain the subjective pilot ratings.

Bode Analysis

In the attempt to explain the experimental pilot ratings for configuration sets F and G, the vehicle-alone dynamics (namely y/F, and €IF,) for configuration sets E,F and G are now considered. Vehicle transfer functions are derived and presented in Appendix I. Bode plots for y/F, and 8/F, are presented in Figs.9-14.

For good closed-loop stability margins in a tracking system, the desired shape of the open-loop frequency response in the crossover region is well known (i.e. constant slope of -20 dB/decade and constant -90 deg phase). Assume that the critical task for a pilot in the approach and flared landing task is the precise control of flight-path angle (or sink rate). If this is indeed the case, we would expect to find the configurations that have a y/F, frequency response more like K/s in the region of crossover to also be rated better by the pilots. Note that crossover frequency for all configurations studied is below 2 rad/sec. From Figs.l0,12 and 14 we find that the y/F, frequency responses that appear more Kls- like in the critical frequency range (0.5-2.0 radsec) invariably correspond to the better rated configurations. More specifically, those configurations with (y/F,)-phase tending to be more uniformly equal to -90 deg in the critical frequency range also tend to be rated better.

Comparison of NLR and TIFS Analyses

A pilothehicle analysis of pitch-rate command aircraft configurations in an approach and landing task was also conducted by Wendelf2]. In Wendel’s analysis, 26 configurations, originally studied using Calspan’s Total Inflight Simulator (TIFS) were evaluated using the OCM for pilot modeling. Of those 26 configurations, six had dynamics free of any added dynamics due to pre-filters,etc. A quick comparison of results from the two analyses is made here in an attempt to find some sort of consistency between studies.

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Config.

1- 1 1-2 1-3

2- 1 2-2

7- 1

- 11 -

Pilot Rating

6.00 6.83 5.33

6.50 3.75

2.75

The 5 pitch-rate comnmnd configurations from Wendel's TIFS study having dynamics comparable to the NLR configurations discussed are presented in Table 6. Configuration 7-1 is also included in this comparison for it represents an aircraft with conventional dynamics. The 8/F, and y/F, transfer functions for the 6 configurations presented in Table 6 are presented in Appendix N. Model-based metrics for the flight-path tracking task are summarized in Table 7. Configuration 7-1, which has the best pilot rating out of the 6, has a very small value for required pilot phase compensation (PqIo, = 7 deg). Configuration 7-1 also has the largest crossover frequency (0, = 1.90 radsec). Presented in Fig.16 are the frequency responses for y/F, for configurations 1-1,2-1 and 7-1. Note that the shape of the Bode plots for configurations 1-1 and 2-1 in the region of crossover are very similar, as are their respective pilot ratings (CH=6). Configuration 7-1, looking much more like K/s in the region of crossover (0.5-2.0 radsec), e m s a much better pilot rating (CH=2.75)!

Table 6 : TIFS Configuration Sets

0, (radsec)

1.65 1.85 1.80

1.65 1.65

1.90

Config.

1-1 2.79 0.8 0.38 1-2 2.76 0.8 0.72 1-3 2.73 0.8 1 .oo

2- 1 1.78 0.6 0.38 2-2 1.75 0.6 0.72

7- 1 2.84 0.8 0.72

SPeq ' 0, Bw (deg) (radsec)

39.2 1.86 24.0 1.96 16.2 1.96

60.0 1.79 43.8 1.82

7.0 2.06

0.93 - -

- 0.16 0.01

Table 7 : Performance Measures for Flight-Path Tracking Task for Comparable TIFS Configurations

- 12-

-270. - -360.

NLR Vehicle Frequency Responses 60. I U

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40.

20.

0.

-20.

-40.

-60.

0.

- - -

I I I - thrta/rtlok : E-1 - - - thota/rtick : E-3 . . . . . . . thrto/rtick : E--3

1

e, 0, -180. C 0

-

-270. 0) v)

0 -360. I 0. .01 0.1 1 .o 10. 100.

frequency - rad/sec

Figure 9

m

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Q, 73 3

C P

+J .-

; P 0) 73

I e, tn C 0

0) v) 0 L 0.

-

NLR Vehicle Frequency Responses

40.

20.

0.

-20.

-40.

-60.

- 1 3 -

m

I U

a, U 3

C 0

+J .-

E P a, U

I a, P C U

a, u) U I

-

n

m

I 7J

a, U 3

c 0 13

.cI .-

E P a, 73

i a, P f U

a, u) U I

-

n

NLR Vehic l e Frequency Responses 60.

20. 40‘ h.... -40. -=O- 1 y,.. ............... . ‘1. . ’.

-60. I I I I -theto/dlok : F--3 - - - thota/otlok : F-2 ....... thota/stlck : F-1

0. ............

-90.

-180.

-270.

-360.

.o 1 0.1 1 .o 1 0 . 100.

frequency - rad/sec

Figure 11 .

N L R Vehicle Frequency Responses

40.

-gamma/rtlck : F-3 - - - gamma/dlck : F-2 ....... gamma/rtick : F-1 0. 1

-270.

-360. .01 0.1 1 .o 10. 100.

frequency - rad/sec

Figure 12

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NLR Vehicle Frequency Responses 60 I

40.

20.

0.

-20.

-40.

-60.

0.

- theto/mtlck : ' 0-1 - - - thrto/atick : C-2 , . , . . . . . theta/8tick : 0-3 , -90.

-180.

-270.

-360.

60,

40.

20.

0.

-20.

-40.

-60.

0.

-90.

- 1 80.

-270.

-360.

.01 0 . 1 1 .o 10. 100.

frequency - rad/sec

Figure 13

N L R Vehicle Frequency Responses

I I 1 -1 - gornma/mtick : G-1 - - - gammo/mtlck : C-2 . . . . . . , gornmo/stick : C-3

1

.Ol 0.1 1 .o 10. 100.

frequency - rad/sec

Figure 14

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I I 1 I I I I. I I I I I I 1

m

I

73

0) U 3

c P 0

& .-

E P al U

I al P C 0

a, v) 0 L

-

a

TlFs Vehicle Frequency Responses

- theta/.tlok : 1-1 - - - thota/mtlck : 2-1 ....... thota/atlck : 7-1

90. , 1 ........ .............

0. 4 ....... I I

-270. I 1 I .01 0.1 1 .o 10. 100.

frequency - rad/sec

Figure 15

TlFs Vehicle Frequency Responses 20. ,

0.

-20.

-40.

-60.

-80. -I - gammo/mtlck : 1-1 - - - gamma/mtick : 2-1 ....... gamma/atlck : 7-1

........... - ........ .......

-90. - -180. - \

-270. S I .01 0.1 1 .o 7 0; 100.

frequency - rad/sec

Figure 16

- 16-

Attitude Tracking Analysis

Up to this point, very little has been said about the analysis of the attitude tracking task. Certainly for an aircraft to have acceptable performance in an outer-loop such as flight- path (refer to Fig.l), it must first have good inner-loop, attitude dynamics. This is the reason for conducting an attitude tracking analysis - to insure that the configuration under study has good attitude dynamics before beginning the flight-path analysis. Another reason for conducting an OCM analysis of the attitude tracking task might be that attitude tracking has been argued by some to be the critical task.

Yet, as stated earlier, it is hypothesized that the critical task for approach and landing is the precise control of flight-path angle. If this hypotheses is true, one should expect to find correlation between experimental pilot ratings and model-based memcs from the flight-path tracking analysis. Having shown through this study that there is a strong correlation between the y/F, frequency response in the region of crossover and experimental pilot rating offers evidence that precise control of flight-path is indeed the critical task. For this reason, it is more sensible to draw conclusions based upon the results from the flight-path analysis and not the attitude analysis.

CONCLUSION

This informal memo was written in order to summarize the results of an analytic han- &ng qualities analysis of the rate-command, attitude-hold aircraft configurations originally presented in an NLR report by Mooij. While the discussion contained is rather lim- ited in scope, there were a few key ideas that were brought forward:

1. Model-based performance metrics, such as required pilot phase compensation, appear to correlate well with experimentally obtained pilot ratings within a configuration class, but not necessarily over all classes. This was true for both the TIFS database as well as for the NLR study.

2. Vehicle configurations that exhibit Ws behavior in the flight-path response, especially in the frequency range 0.5-2.0 rad/sec, elicit better pilot ratings in the landing task.

3. For the approach and landing task, the OCM analysis of the attitude tracking task, while a necessary step in the complete analysis procedure, does not seem to offer as much insight as the OCM analysis of of the flight-path tracking task. This offers evidence for the hypotheses that the critical tusk for a pilot in the approach and landing task is the precise control of flight-path.

Mooij, H., " Criteria for tow-Speed Longitudinal Handling Qualities," National Luchten Rumtevamlaboratorium (NLR), Netherlands, December 1984.

[31

[41

r51

Wendel, T., " Closed-Loop Pilot1 Vehicle Analysis of Approach and Land- ing of Pitch Rate Command Aircraft," M.S. Thesis, Purdue University, December 1985.

Bacon, B., Schmidt, D., '' An Optimal Coutrol Approach to PilotlVehicle Analysis and the Neal-Smith Criteria," Journal of Guidance and Control, Vo1.6, Sept.-Oct. 1983, pp. 339-347.

Anderson, M., Schmidt, D., " Closed-Loop PilotlVehicle Analysis of the Approach and Landing Task," Journal of Guidance and Control, Vol.10, Mar.-Apr. 1987, pp. 187-193.

Hess, R., " Prediction of Pilot Opinion Rating Using an Optimal Pilot Model," Human Factors, Vol.19, Oct. 1977, pp. 459-475.

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1.000 )5** 5 ( -0. 5780E-01) + J( 11.39 )Saw 5 ( -1. 100 1 + J( 20. 12 IS** 4 ( -10.00 + J( 12. 17 IS** 3 ( -0. 3682 ) + J( 3.555 IS*-* ( -9. 3682 1 + J( 0. l&06 )S+* 1 ( 0. 1 + JC

0. 1 G a m r n a / s t i c k E-1

0. 9667 IS** 4 ( -0. 1 6 6 O E - 0 1 1 + J( 1.394 IS** 3 ( -0. 3920 1 + J( 7. 564 IS * * 2 ( -0. 5168 1 + 3( c-. 929 IS** 1 ( -0. 5168 1 + J( 0. 4652E-01)

0. 0. 0.

0. 3600 -0. 3600

0.

0. 0.

2. 670 -2.670

( 1.000 )S** 6 ( -0. 5780E-01) + J( 0. I ( 11.39 IS** 5 ( -1.100 ) + J( 0. 1 ( 20.12 1399 4 ( -10.00 1 + J( 0. 1 ( 12. 17 IS** 3 ( -0. 3682 1 + J( 0.3600 1 ( 3.555 I S * * 2 ( -0. 3582 1 + J( -0. 3600 ( 0. 1686 IS*+ 1 ( 0. 1 + J( 0. 1 ( 0. 1 C o n f i g u r a t i a n E-2

i, ----------------------------------------------------------- Thota/stick E-2

( 11.00 )S-+* 3 ( -0. 8310E-*S1) + J( 0. f ( 13. 77 1s+* 2 ( -0. 3620 ) + J( 0. 1 ( 4. bo4 IS*+ 1 ( -0.7oco 1 + J( 0. 1 ( o. 2958 )

3 ( 0. 1 + J( 5 ( -0. 6450E-01) + J( 4 ( - I . cao 1 + J( 3 ( -10.90 1 + J( c 7 ( -0. 4226 + J( 1 ( -0. 4224 1 + J(

0. 0. 0. 0.

0. 4203 -0.4z03

0. 0.

2.570 -2. 670

( 1. GOO ) S S S i? ( 0. 1 + J( 0. 1 ( 11.99 IS** 5 ( -0. 645GE-01) + J( 0. 1 ( 21.23 1 3 U - F 4 ( -1. 030 1 + J( 0. 1 ( 14. 37 )Si+++ 3 ( -10.00 1 + J( 0. )

( 4. ,577 I s l e * 2 ( -G. 3220 1 + J( G. 42.03 )

( 0.2475 )S+U 1 ( -3. 9226 1 + J( -0. 3TO3 1 ( 9 ) ____________--______---------------------------------------

9 r e f ' e r t a l a s t p a g e o f A p p e n d i x I far derivation

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1.

14. 20 19. 97 7. 747

0. 5140

1.000 12. 19 23.31 19. 94 7. 719

0. 4631 0.

1. 441 2. 402 11. 62 6. 768

0. 1092

1.000 12. 19 23. Et1 19.94 7.719 0. 4631

0.

Thetalstick E-3 IS** 3 ( -0.8310E-01) + J( 0. IS** 2 ( -0. 6170 1 f J( 0. IS** 1 ( -0. 7060 1 + J( 0. 1

6 ( -0. 7240E-01) + J( 5 ( -0. SEI70 1 + J(

3 ( -0. 5651 + J( 2 ( -0. 5651 ) + J(

4 ( -10.00 1 + J(

1 ( 0. 1 + J (

Gammalstick E-3 4 ( -0. 166OE-01) + J( 3 ( -0.6170 1 + J( 2 ( -0. 5168 1 + J( 1 ( -0. 5168 1 + J(

0. 0 . 0.

0. 5733 -0.5733

0.

0. 0.

2. 670 -2.670

IS** 6 ( -0. 7240E-01) + J( 0. IS** 5 ( -0. 9870 + J( 0. IS** 4 ( -10.00 ) + J( 0. IS** 3 ( -0. 5651 1 + J( 0. 5733 )SO* 2 ( -0. 5651 + J( -0. 5733 IS** 1 ( 0. 1 + J( 0. 1

17. 00 25. 91 10. 36

0. 7331

1.000 12.39 26. 38 25. 50 10.82

0. 6827 0.

1. 719 3. 059 14. 07 3. 577 0. 1551

1.000 12.39 26. 38 25. A 0 10.92

0.6827 0.

Theta/stick E-4 IS** 3 ( -0.9310E-01) + J( IS** 2 ( -0. 7060 ) + J( )S** 1 ( -0. 7350 ) + J ( )

IS** b ( 0. 1 + J ( IS** 5 ( -0. 7560E-01) + J ( IS** 4 ( -0. 9030 + J( IS** 3 ( -10.00 + J( )Sa* 2 ( -0. 7070 1 + J( )S** 1 ( -0. 7070 ) + J( 1

)S** 4 ( -0. 166OE-011 + J ( IS** 3 ( -0. 7350 + J( IS** 2 ( -0. 5168 + J ( IS** 1 ( -0. 5168 1 + J ( )

G a r n m a l s t i c k E-4

)S** 6 ( 0. 1 + J ( I S + * 5 ( -0.7550E-01) + J ( IS** 4 ( -0. 9030 1 + J( IS** 3 ( -10.00 + J( IS** 2 ( -0. 7070 1 + J ( 1s** 1 ( -0. 7070 ) + J( )

0. 0. 0.

0. 0. 0. 0.

0. 7072 -0. 7072

0. 0.

2. 670 -2. 670

0. 0. 0. 0.

0. 7072 -0. 7072

I I

I I I I I I 1. I I I I I I I

Configuration E-5 -----------------------------^-------------------------------

Thetalstic k E-5 ( 19. 20 )SO* 3 ( -0.8310E-011 + J ( 0. 1 ( 31.85 )SO* 2 ( -0. 7060 1 + J ( 0. ( 14. 31 IS** 1 ( -0. 8700 + J ( 0. 1 ( 0.9800 1

1.000 1 2 . 5 9 28. 97 31. 83 14. 37

0.9352 0.

1.951 3.746 16. 2 5 12. 82

0.2084

Is** 6 ( -0. 7750E-01) + J(

)SO* 5 ( -0. 5380 ) + J( IS%++ 4 ( -10.00 1 + J ( )Si?* 3 ( -0. 8398 ) + J ( I S * * 2 ( -0. 8383 1 + J( IS** 1 ( 0. 1 + J( 1

IS+* 4 ( -0. 1660E-011 + J ( IS** 3 ( -0. 8700 1 + J ( IS** 2 ( -0. 5168 1 + J( IS** 1 ( -0. 5168 1 + J (

Gammalstic k E-5

0. 0. 0.

0. 8581 -0. 3581

0.

0. 0.

2. 670 -2. 670

( 1.000 IS** 6 ( -0.7750E-01) + J ( 0. 1 ( 12.59 IS** 5 ( -0. 8380 1 + J( 0. 1 ( 28.97 IS** 4 ( -10.00 1 + J( 0. 1 ( 31.83 IS * * 3 ( -0. 8388 1 + J( 0.8581 1 ( 14. 37 )SO* 2 ( -0. 8388 I + J( -0.a5ai 1 ( 0.9352 IS** 1 ( 0. + J ( 0. 1 ( 0. 1 Configuration F-3

Theta/stick F-3 ( 26.90 IS** 3 ( -0.8310E-01) + J ( 0. ( 37.73 IS** 2 ( -0.6210 1 + J( 0. ) ( 14.76 IS** 1 ( -0. 7060 1 + J( 0. 1 ( 0.9801 1

1.000 12. 59 28.97 31. 33 14. 37

0. 9352 0.

2. 730 4. 5 6 2 22.02 12.90

0. 2081

)S** b ( 0. ) + J ( IS** 5 ( -0.7750E-01) + J ( )S** 4 ( -0.83~10 1 + J( )Sa* 3 ( -10.00 1 + J( IS** 2 ( -0. 8388 + J( IS** 1 ( -0. 8388 1 + J ( 1

IS*+ 4 ( -0. 1660E-01) + J( IS** 3 ( -0. 6210 1 + J( IS** 2 ( -0. 5168 1 + J( )S** 1 ( -0. 5168 + J( 1

Gamma/stick F-3

0. 0. 0. 0.

0. 8581 -0. 8581

0. 0.

2. 370 -2. 670

( 1.000 I S * + 6 ( 0. 1 + J ( 0. 1 ( 12.59 IS** 5 ( -0.7750E-011 + J ( 0. 1 ( 28.97 )S** 4 ( -0. 8380 1 + J( 0. 1 ( 31.83 )S** 3 ( -10.00 I + J( 0. 1 ( 14.37 ) S a * 2 ( -0. 8398 + J( 0.3531 1 ( 0.9352 IS+* 1 ( -0. 8388 1 + J ( -0.5591 1 ( 0. 1

Configuration F-2


T h e t a l s t i c k F-2 ( 44. a0 )S** 3 ( -0.8310E-01) + J( 0. 1 ( 52.02 I S * * 2 ( -0. 3720 1 + J( 0. 1 ( 15. 78 IS** 1 ( -0. 7C60 1 + J( 0. 1 ( 0.9777 1

1.000 12.59

31. 33 14.37

0. 9352

28. 97

0.

4.547 6 . 467 35.48 13. 10

0. 2077

b ( -0. 7750E-011 + J( 0. 5 ( -0. 8380 ) + J( 0. 4 ( -10.c10 1 + JC 0.

2 ( -0. €3583 1 + J( -0. 3531 1 ( 0. 1 + J( 0.

3 ( -0. a383 + J( o. 8581

Gamma/ s t i c k F-2 4 ( -0. 1660E-011 + J( 0. 3 ( -0. 3720 1 + J( 0. 2 ( -0. 5168 + J( 2.670 1 ( -0. 5168 1 + J( -2.670

1.000 12. 59 28.97 31. 83 14.37

0 . 9352 0.

9. 130 1 1 . 29 69. 47 13. 71

0. 2085

6 ( -0. 7750E-01) + J( 5 -0. 8580 1 + J(

3 ( -0. 8339 1 + J(

1 ( 0. ) + J(

4 ( -10.00 1 + J(

2 ( -0. 8388 1 + J(

Gamma/stick F-1 4 ( -0. 16eOE-01) + J( 3 ( -0. 1860 1 + J( c 7 ( -0. 5168 + J( 1 ( -0. 5168 1 + J(

0. 0. 0.

0. 85al

0. -0.8581

0. 0.

2. 670 -2. 670

I I


I

I.

1.

18. 50 26. 36 9 . 923 0. 6986

1.000 12. 31 25 . 53 26.22 9. 835 0. 6527

0.

1. 880 3. 031 11.43 a. 5 5 9

-0. 126lE-

1.000 12. 31 25. 5 9 26.22

0. 6527 9. a35

0.

Thetalstick G-1 )S** 3 ( -0. 9100E-011 + J( 0. )S** 2 ( -0. 4940 ) + J( 0. IS** 1 ( -0. 8100 1 + J( 0. )

Is** 6 ( -0. 8350E-01) + J ( 0. IS** 5 ( -0. 5520 1 + J ( 0. ) S * % 4 ( - 1 0 . G O 1 + J( 0. IS** 3 ( -0. 8354 1 + J( 0.3475 IS** 2 ( -0. 8354 1 + J( -0. 8475 IS** 1 ( 0. 1 + J( 0.

Gamma/stick G-1 )S** 4 ( 0. 1470E-02) + J( 0. IS** 3 ( -0. 8400 1 + J ( 0 . IS*+ 2 ( -0. 3868 1 + J( 2.295 IS** 1 ( -0. 3868 1 + J( -2. 293

a 1 1

I S * * 6 ( -0. 8350E-01) + J( 0. IS** 5 ( -0.5520 1 + J( 0. IS** 4 ( -10.00 1 + J( 0. IS+* 3 ( -0. 8354 1 + J( 0 .8475 )S** 2 ( -0. 8354 I + J( -0.8475 IS+* 1 ( 0. 1 + J( 0. 1

I 1 I I I I I. I I I I I I I

Configuration 6-2 ---------------------------------~------------------------- Theta/stick 6-2

( 18. 80 )S+* 4 ( -0. 9110E-01) + J( 0. ( 30.06 IS * * 3 ( -0. 1980 1 + J( 0. )

( 15.21 1599 2 ( -0. 5250 1 + J( 0. ( 2. 684 IS** 1 ( -0. 7350 1 + J( 0. ( 0. 1398 1

( 1.000 ( 12. 51 ( 29.05 ( 31.33 ( 15. 08 ( 2. 620 ( 0. 1305 ( 0.

IS** 7 ( -0.8350E-01) + J( 0. )

)S** h ( -0.2000 1 + J( 0. 1 )Sa+ 5 ( -0.5520 1 + J( 0. )

IS** 4 ( -10.00 1 + J( 0. 1 )S** 3 ( -0.5354 1 + J( 0 .8475 1 IS++* 2 ( -0.8354 1 + J( -0 .8475 1 )S** 1 ( 0. 1 + J( 0. 1

Gamma/stick 6-2 ( 4. 332 IS** 5 ( 0. 1470E-02) + J( 0. 1 ( 7.681 IS** 4 ( -0. 2190 1 + J ( 0. 1 ( 15. 82 )S** 3 ( -0. 6 5 8 0 + J( 0. I ( 10. 88 IS** 2 ( -0. 4488 1 + J( 1. 589 1 ( 1. 684 )S** 1 ( -0. 4488 + J( -1.589 1 ( -0. 2499E-02)

( 1.000 IS** 7 ( -0.8350E-01) + J( 0. 1 ( 12. 51 IS** 6 ( -0. 2000 1 + J( 0. 1 ( 28.05 IS+* 5 ( -0. 5520 + J( 0. ( 31.33 IS** 4 ( -10.00 1 + 3( 0. ( 1s. 08 IS** 3 ( -0. 8354 I + J ( 0.8475 I ( 2. 620 IS** 2 ( -0. 8354 + J( -0 .8475 )

( 0. 1305 >sa* 1 ( 0. 1 + J ( 0. I ( 0. 1 -----------------------------------------------------------

I

I I


I.

I.

19. 10 30. 01 15. 22 2 . 6 9 9 0. 1421

1.000 12. 5 1 28. 05 31. 33 15 .08 2. 620 0. 1305

0.

7. 098 12. 48 20.26 11 .00 1. 666

-0. 2473E-

1. GOO 12. 51 28.05 31 .33 15. 08 2 .620

0 . 13C5 0.

Thetalstick 6-3 IS** 4 ( -0. 9310E-01) + J( 0. )S** 3 ( -0. 1960 + J ( 0. IS** 2 ( -0. 5860 1 + J ( 0. IS** 1 ( -0. 6960 + J( 0.

IS++* 7 ( -0.835OE-01) + J ( 0. IS** b ( -0.2000 + L'( 0. )SO* 5 ( -0. 5520 1 + J( 0. IS** 4 ( -10.00 + J( 0. IS** 3 ( -0.9354 1 + J( 0.8475 IS** 2 ( -0. 8354 1 + J( -0.3475 IS** 1 ( 0. 1 + J( 0. I

Gammalstick 6-3 )S*W 5 ( 0. 1470E-02) + J ( 0. IS** 4 ( -0. 2590 + J ( 0. IS** 3 ( -0.4600 ' + J ( 0. )S*u 2 ( -0. 5203 1 + J ( 1.311 IS** 1 ( -0. 5203 + J ( -1.311

02 1

IS** 7 ( -0. 8350E-01) + J( 0. IS** 6 ( -0.c7000 + J( 0. IS** 5 ( -0. 5 5 2 0 1 + J ( 0. IS** 4 ( -10.00 1 + J( 0. IS** 3 ( -0. 8554 1 + J( 0.8475 IS** 2 ( -0. a354 ) + J( -0. 8475 IS** 1 ( 0. 1 + J( 0. I

e

Configuration G-4 ........................................................... T h e t a / s t i c k 6-4

( 19. 40 )S** 4 ( -0.9140E-01) + J ( 0. 1 ( 29.98 )Sa* 3 ( -0. 1940 1 + J( 0. ( 15. 19 IS** 2 ( -0. 6300 + J ( 0. 9420E-01) ( 2. 680 IS*+ 1 ( -0. 6300 + J ( -0. 942OE-01) ( 0. 1396 1


( 1.000 ( 12. 51 ( 2’3.05 ( 31. 33 ( 15. 08 ( 2. 620 ( 0. 1305 ( 0.

)Sa* 7 ( -0. 8350E-01) + J ( 0. )SE* b < -o.,?oco 1 + J ( 0. 1 IS** 5 ( -G. 5520 1 + J( 0. )S** 4 ( -10.00 1 + J( 0. 1 I S * * 3 ( -0.8354 1 + J ( 0.8475 1 IS** 2 C -0. 8354 + J( -0.8475 1 )S*U 1 ( 0. ) + J( 0. 1 1

Gamma/stic k 6-4 ( 9.578 IS*+ 5 ( 0. 147OE-02) + J( 0. 1 ( 16. 89 IS** 4 ( -0. 3072 1 + J( 0.9310E-01) ( 24.43 IS** 3 ( -0.3072 + J ( -0. 9310E-01) ( 11.35 )SU+ 2 ( -0.5755 1 + J ( 1. P88 1 ( 1. 703 IS** 1 ( -0.5755 1 + J ( .-1. 188 1 ( -0. 2529E-02)

I 13. I I9 I I ‘. I I I I I

I

L

1.

I

I I 1 I 1 I I. I I I I I I 1

I I I

I L b

L

I PP

1 - I

Pea

I. I I I I 1 I I. I I I I I I I

I 1.

J

APPEbIOI , ( I I I - OCM I n p u t F i l e For C d n f i q E - 1 231nR1.3 Tracking T a s k I

I I 1 I I I

t=. 21 i d e n t u = 2 ,

. v u ( 1 )=-25. I

I. I I I I I I I

OCM I n p u t F i l e f o r C o n f i g . E-1 T h e t a Tracking T a s k 8

1 I. I I I I

I I

( 1. 000 IS** 5 c 0. ) + de 0. I ( 4. 866 IS** 4 -0. 3320~-01) + J( 0. 1 ( 10.19 IS * * 3 ( -0.434b + J( 0. 1 ( 4. 203 IS** 2 ( -2. 174 1 + Ji 1.757 1 I 0.2925 )S** 1 ( -2. 174 1 + J( -1.757 1 ( 0. 1

( 1 . O c ) O I S * * 5 ( 0. 1 + J ( 0. 1 ( 5.452 I S * * 4 ( -0. 3560e-011 + J ( 0. 1 ( 12-67 IS+* 3 { -1. 190 1 + J( 0. 1 ( 3.302 IS** 2 ( -2. 113 1 + J ( 1.72h 1 ( 0. 3154 IS** 1 ( -2. 1 1 3 1 + J ( -1.726 1 ( 0. 1

Gammalstic k 1-3 4 0. 5230e-0115** 4 ( 0.2230~-02) + J ( 0. 1 t 0.2022 IS** 3 ( -2.000 1 + J( 0. 1 ( 1.075 IS** 2 ( -0.9343 1 + J ( 3.5’92 1 ( 1.756 IS** 1 -0. 9343 1 + Ji -3. 432 1 ( -0. 392le-021

I 1. coo )S** 5 0. 1 + 3 1 0. 1 4 5.452 IS** 4 ( -0. 356Oe-011 + J ( 0. 1 ( 12.67 IS** 3 ( -1. 190 1 + J ( 0. 1 ( 3.302 IS** 2 ( -2. 113 1 + J ( 1.725 1 ( 0. 3154 IS** 1 ( -2. 113 1 + J ( -1.726 1 ( 0. 1

( 1.060 I S * * 5 0. 1 + J ( 0. 1 ( 2.702 )SU* 4 ( -0. 7390e-01) + J ( 0. 1 ( 4.415 )S** 3 ( -0.4997 1 + JI 0. 1 ( 1. 990 1399 2 ( -1.063 1 + J ( 1.423 1 ( 0. 1156 IS** 1 -1. 064 1 + J ( -1.423 1 ( 0. 1

Gamma/stick 2-1 ( 0. 2435e-Ol)S*+ 4 . ( 0. 1450~-02) + J ( 0. 1 ( 0. 5770e-O11S** 3 ( -2.000 1 + J( 0. 1 ( 0.1932 IS** 2 -0. 1355 1 + J( 2. 5 9 9 1 ( 0.3301 IS * * 1 ( -0. 1955 1 + J( -2. 5 9 8 1 ( -0. 479le-031

( 1.009 IS++* 5 ( 0. I + J ( 0. 1 ( 3.005 IS** 4 ( -0. 4830e-011 + J ( 0. 1 ( 5. 036 IS** 3 ( -9. 921Q 1 + J( 0. 1 ( 3. oa5 IS** 2 ( -1.033 1 + J ( 1.425 1 ( 0. 1375 IS** 1 ( -1.013 1 + J ( -1.425 1 I 0. 1

Gammalstic k 2-2 { 0.2435e-01 IS** 3 ( -2. 000 1 + J( 0. 1 ( 0. 7639e-01)S++ 2 ( -0. 5686 1 + J[ 2.623 1 ( 0.2308 IS** 3 ( -0. 5686 + J( -2. 623 1 < 0.3503 )

( 1.000 IS** 4 ( -0.4930e-01) + J( 0. 1 ( 3.005 )SO* 3 ( -0. 9310 1 f J( 0. 1 t 5. 08& IS** 2 ( -1.033 I + J ( 3.425 1 ( 3.085 IS** 1 -1. 033 1 + J ( -1 .425 1 ( 0.1375 1

( 3 . 000 IS** 4 I -9. 1610e-01) + 3( 0. lb33 1 4. 630 IS** 3 ( -0. 363Oe-031 + 3( -a. 1633 1 3.253 )S** 2 t -2.324 1 + J ( 3.633 I

( 0.3355 IS** 1 ( -2.324 1 + J ( -1.&33 1 ( 0. 2177 1

Gamma/stick 7-1 < 0.4159e-91)S** 3 0. 4000e-021 + J ( 0. 1 ( 0. 4673e-0115** 2 ( -0. 5644 1 + J( 3.935 1 I 0.6580 IS** 1 < -0. 5544 1 + J ( -3. 933 1 ( -0. 2533e-02)

APPENCIX V

N L R Cc.nf igur-at ion E-1 : G o m m o Trocking

so. -

40. -

20. -

0. -

-20. {’

‘54 -- b

...- .-.. e..

. . - - . . .

+ a += .t ++ =*

+ + + + +++ + + a

LEGEND: - pilot reeponae to gamma error + pilot response to gamma

pilot reapanse to theta

270. -

IBQ. A ’

SO. -

0. -

-go. -

- 1 80.

*t + II * II

+ Ir *** ;t ;&e I

i

*

# - a*\

. . . . . * . . -

1 I 1

frequency - rod /sec

N LR

8 6.

40.

20-

0.

-2..

-40.

Gonf igur -u t i c ln E--2 : G a m m a Tracking

\ .-.e- .... ...

. .

LEGEND: . pilot reeponae to gommo error + pilot reaponse to gamma Y pilot r e s p o n s e to theto

270.

1ao.

90.

0.

-90.

+ tg j t

i i

I i mf

I

0 I -180.

U

I .1 1 . 10. 1 0 0 .

frequency - rad/sec

t d L R Conf igurat ion €--3 : Gornrna Tracklng

_. \ .---.... .... . ...

-20, + + ++ + +++

-40- -3, - 1 1 , 10, 1 0 0 ,

frequency - r a d / s e c

LEGEND: - pilot reeponae to gamma error

+ pilot reaponse t o gomma r pilot ra~sponsa t o theta

270,

1 ao,

90.

0.

-!2 0,

+ = + I+ I+ m +

. . . = +

m +

- 1 €30. I I I 1

.1 1 . 7 0. 1 0 0 .

I frequency - rod/sec

I m

270. -

lao-

90. -

0. -

-QO. -

- 1 80.

m

-!t*& + m + I+ .I+

.I + . .

. . . . a + . . - 1(+

m+ II+ rct

*G I + I( +

I I 1

.- C

I F .-

t

IdLH C o n i i y u r o t i o n E - 4 : G a m m a Trocking

40.

Bo- 1 + -20. + ++ + + + + i + .I++

-40 , I I I - 1 1 . IO, 1 0 0 ,

f requency - rad/sec

LEGEND: - pilot reeponae to gomma error + pilclt rsaponse t o gamma i pilot response t o theta

NLH

-

-

-

-

BO.

40.

20.

0,

-20..

-40-

. . . . . . .

7.-, ++ + + + + +++p

Configuraticbn E - 5 : G o t n r n a Tracking

I + Y ++ =I

I I I

. . . . . ... ...... \

-

-

-

. . - II+ - . . . . e . . a +

I+ m+

w+ =k

II + = I I 1

LEGEND: - pilot reeponae t o gamma e r r o r + pi lot reaponse t o gamma at pilot reaponse t o theta

270.

1 BO,

90,

0.

-90,

- 1 80.

m +

+ I t + a + I+

I 1 . w

m: r

rg

t4LR C , o n f i g u r a t i o n F--3 : Gan- ma Track.in(j

f . .

--.-

a + si f +

-40- 1 I 1 - 1 1. I O ,

f r e q u e n c y - rad/sec

LEGEND: . pilot reeponae to gamma error

+ pilot reaponae to gamma i pilot reaponae t o theta

270.

180,

9 0 .

0.

-90-

- 1 80,

1 m + i+

. . i + r + -

* . 1 M +

1 0 0 -

+ M +

i m+ r, Y

.1 1 . 1 0 , 1 0 0 . 1

frequency - rad/sec

I I I. I I I I I I

I, Q,

I" I

Ig C

I " a, m

BO.

40.

20.

0.

-20.

-40 .

. . . . .

- 1 1 - I O. 1 0 0 ,

f r e q u e ncy - r a d / s e c

LEGEND: . pilot reeponae ta g a m m a error

+ pilot r e s p o n s e to g a m m a pilot r e a p a n s m t o theta

270.

1 so,

9 0.

+ m +

+ *+ w

-9 0,

. . E +

t - M . .

+

- 1 80. I I 1 I

.1 1 . IO. 1 0 0 .

f r e q u e n c y - r a d / s e c

14 L F-'

BO.

40.

0.

-20.

-44- I I I 1 - 1 1 . 1 0 - 1 0 0 .

frequency - rad/sec

LEGEND: . pilot reeponae t o gamma e r r o r

+ pilot response t o gamma *! pilot response t o theto

270.

1 am

90.

0.

-90.

-I ao.

+ % ++

11 1 1 +

+ + . . a +

11 11 + : - 3 s + +

M +

e .

I

*,++ M + M +

1( I( la L-F****. . ...

I I 1 ,1 1 . IO. 100.

frequency - rad/sec

I '

C

L

I CP I$ I

4 0 .

20- . . . .

-20. O- 1 + m +?e + + ++

+ ++ + + + + + +*

- 1 1 . 1 a- 100-

frequency - rad/s6c

LEGEND: . pilot reeponae to gamma e r r o r

+ pilct reapense t o gamma )* pilot reapclnse t o t h e t a

270.

1 am

9 0.

0.

-9Q-

-1 80. . 1 1 . IO. 1 0 0 .

frequency - rod/sec

I I I.

1. I I r g I ' I F 0

I: 0

1% I.

Ql

-20. 7 .

1: -40,

I I I I I I

++ +++cH***u*sl + + + + +++ - *til* = 3s

i *t* 3s I I I

I .4 l -K Configuration

90. -

0. -

-go, -

- 1 80.

. .

++ It +

+ i +It

+T mi 1( i

*++ r + a

* . . . . . - * .

1

c-7; - 2' :

.. ._.......-I---

G a m m o Trackirig

- 1 1 , IO, 1 0 0 ,

f requency - r a d / s e c

LEGEND: - pilot reeponae t o gamma e r r o r + pilot response to gamma Y pilot response t o theta

I I

I I

I I I.

I. I I

I; I; I n , m

I.

0

0

1:

1

t4LK

-60

4 0 .

20.

0.

-20.

-40 ,

+++ ++++*

m j+++ + I #(% M i m* >

... Bt + Y

s* 4

I I I - 1 1 . IO. 1 0 0 .

frequency - rad/sec

LEGEND: . pilot reeponae to qornma error

+ pilot reaponse to gamma *i pilot reaponse t o thsta

270.

180,

90.

0.

-9 0,

- 1 80.

*(

+ ** 1 ++

I I 1 . I 1 . 10. 1 0 0 .

frequency - rad/sec

I I I. I I I I I I

TI

0

- B O .

40.

20-

Q.

-20.

- 4 0 -

1

. .

. 0 . ..... ...... -.---

+++ + + +++ j+++ + ,It lE= z222ia s- i

IR +* I *a .c

I I I - 1 1 . 1 CI- 1 0 0 -


LEGEND: . pilot reeponae to gamma error

+ pilot reHponse to gamma *I pilot reaponse to theta

270.

1 €lo-

9 o.

0.

-90,

- 1 80.

I + *E

++

+ I . . .

I I I I

I I I I .1 1 . IO. 100.

frequency - rad/sec

Id LR

80.

C o n f i g u r a t i o n E- 1 : G a m m a Trocklng

40.

. . .

\ --c '..__ ... . ...

0.

-20.

-40-

LEGEND: equivalent m i n g l e loop pilot function

180.

9 0,

0.

-9 0.

-1 80.

-270.

. . . . . - - - -

I I 1 . 1 1. 10. 1 0 0 .


I I I. b W

ti .- 6

I‘

E I I I. I I I

- I f”

i I

0

60. 1

40. -

20- -

0. -

-20. -

-40.

NLR Configuration E--2 : G a m m a T r - m c k i n g

I I 1

. . . .


LEGEND: . equivalent m i n g l e loop pilot function

780.

Q Q ,

0.

-90.

- .

II io-’aQ- -270. m .1 1 . 10. 1 0 0 ,

frequency - rad/sec I

I I I. I m 73

I; .-

I!

I ' 0

E I I

I I I

Iv

1.

B

B d 19

I

I

N L R

(30.

Conf igurat icn E--5 : Gamma Trocklng

40.

20-

Q.

-20.

cs --.- .-... .. . .-.

-40.

- 1 1 , 10,


LEGEND: - equivalent m i n g l e l o o p pilot function

180.

0 0 ,

0.

-90.

- 1 UO,

-270.

. . - . .

1 1 1 .1 1 . 10. 1 0 0 .

frequency - rad/sec

I I I

td LR

40.

20.

0.

-2..

Conf igurat ion E--4 : G a m m a Tracking

-40 , I - 1

.. \ --.- . ...... . ..

I I I 1 , 1 0 - 1 0 0 ,

frequency - rad/sec

LEGEND: . equivalent s i n g l e loop p i l o t function

180.

8 0,

0.

-90.

-1 80-

-270.

. . . .

I .1

I I I 1 . 10. 100.


m 1 Iy

N L R Configuration E - 5 : Garnma Tracking

60.

40.

20-

0.

-20.

-40.

- 1 1 . 1 0 , 1 0 0 .

frequency - rad/sec

LEGEND: equivalent m i n g l e loop pilot function

180,

90,

a.

-90,

-1 ao-

-270. I .1

. - - .

. I

I 1.

I 10.

I 1 0 0 .

I I I'

I I I I

I< l a ,

I

P)

rd ~ f i

60,

40.

20-

0.

-20,

-40-

Configuration F-3 : G a m m a Tracking

. . - ...--._ ... . .

1 , 1 0 - 1 0 0 -


LEGEND: equtvolent m i n g l e loop pilot funct ion

i ao.

0 0,

Q.

-90,

- 1 80,

-270.

- . .

I I

frequency - rad/s:ec - 1 1 . 7 0. 1 0 0 .

I 1 .

frequency - rad/s:ec

..

. .

1 - 1

I I 7 0. 1 0 0 .

I I I.

t.4 L R

SO.

4 0 .

20.

0.

-2..

-40.

C c , n f i g u r - a t i m r l F-2 : G a m m a Track lnq

1 , IO.

frequency - rad/sec

LEGEND: . equivalent m i n g l e l o o p pilot function

180.

9 0.

0.

-9 0.

- 1 so.

-270.

. . . .

. 1 1 . IO. 1 0 0 .

freauencv - r o d / s e c

I d L R

BO.

40.

20-

0.

-20.

-40-

C c , n i i g u r r o t i c . r - t F- 1 : Gamma Track ing

.....-,-- e . .

.. . .

. .

- 1 1 , 1 0 , 1 0 0 ,

f re q u e n cy - rad/sec

LEGEND: . e q ~ ~ i v a l e n t einqle loop pilot function

180.

90,

0.

-90.

- 1 so.

-270.

. .

. .

I I 1 .1 1 . IO. 1 0 - 3 .


N L H Configuration G - I : G a m m a Traching

I. BO.

I 4 0 .

I I

Q! W 3 e C

20..

0.

-20. I -t

I. I

-40-

- 1 1 . IO, 1 0 0 ,


LEGEND: . equivalent single Iaop pilot function

180.

9 0,

0.

-9 0.

- 1 €50.

-270. .1 1 . 10. 1 0 0 .

I I I. I I I I 1 I

I I

r 4 L H Cor-1figurotic.n e---2 : G o r n r n a TracLing

B O -

40.

20.

0.

-20.

-40-

. .. ...... -.--- * .

. . . . . * .

- 1 1 , 7 0, 1 0 0 -

frequency - rad/sec

LEG END: - equivalent eingle loop pilot function

180.

9 0 .

0.

-9 0.

-1 ao.

-270.

. . . . .

. *

I I 1 .1 1. 10. 1 0 0 .

f requency - rod/sec

-. . .. ..

.1 1. 10. 1 0 0 .


I ' Q)

L

NLR Configuration G-3 : G o m r n o Tracking

SO.

40.

20.

0.

-20.

-40 , I -1

. . . - * ...........*----

I 1 ,

I IO,


LEGEND: - equivalent m i n g l e loop pilot function

I

I" I

1.1 CP

I " 0

180,

8 0,

* . .

0.

-go.

-1 ao,

-270.

...

I 1 0 0 -

I .1

I 1 .

I I 10. 1 0 0 .


NLR Corlf igurotiorl G-4- : G a m m a Tracking

40. - m '0

20. - I a> W 3

t P

4J 0. - .-

-2.. -

-40 .

1 I 1.

I I 1

. - . ......_....e*-*-

- 1 1 , 7 0, 1 0 0 ,

frequency - rad/sec

LEGEND: 1. - equivalent m i n g l e loop pitot function

U

i ao.

90,

0.

-90.

-1 so,

-270.

-.. .. . . . . . .-.

.1 1 . 10. 1 0 0 .

frequency - rod/sec

I I

1% I' I S .-

I F I I 1. I I I

E

I.

a)

t

B I

f i Ii I

40. -

N L H Configuration E- 1 : G a m m a Tr-ocking

. . . .

eo.

40.

+ I +

-40- I I L

t - 1 1 , 10, 1 0 0 ,


LEGEND: . gamma to gamma command - open + g o m m a to gomrna command - closed

180,

5) 0,

0.

-9 0.

-1 so.

-270. 1 . 10. 100.

freauencv - rad /sec

u

-

-

-

1

-

t-4 LR

ad.

. . .

+ ++

I I &

1

40.

20-

0.

-20.

-40,

Configurotion E--2 : Gamma Trocking

. .

++*+ + + + + + + . + ---\

-?

+

LEGEND: . g a m m o

+ gamma

180.

90,

0.

-90.

to gamma commond - o p e n to gamma command - C los6d

+ +++ ++

+ + + + +

0 -270.

*1 1 . 10. 1 0 0 .

frequency - rad/sec

I I

k

I s

I I I. I I

I.

I ' V

I f

U

C

a, F I

f Iu

I&

CP

E

I

~.ILR

8 0 .

40.

20-

0.

-20.

-40-

C m n f i g u r a x t i o n E--3 : G a m m a Tracking

++*+ + +++ + + . . +

.-- \

+

I +

1 I


+ 1

- 1 1 , 10, 1 0 0 .

LEGEND: gamma

+ gamma

180.

80.

0.

-Q0.

-1 a a

-270.

+ ++

to g a m m a command - o p e n to g a m m a command - closed

+ +++ + + + +

+ + +

* . +% - . . .'.j+ . . t %.

I I 1 . 1 1 . 10. 1 0 0 .

frequency - rad/sec

I 1 I. I I I

m

I T)

Q> W 3

C 0 0

3 .-

E

CP

N L R

80.

40.

20-

0.

-20.

-40 ,

C u n f i g u r a t i o n E - 4 : G a m m a Trocking

++*+ + + + + + + * . +

- - e \

+ +

1 , 1 0 - 1 0 0 .

frequency - rad/sec

LEGEND: - g a m m a to gamma command - open

+ g a m m a to gamma command - closed

180.

9 0,

0.

-9 0,

-1 SO.

-2770. .1 1 . IO. 1 0 0 .

frequency - r o d / s e c

N L R

450-

40.

20-

0.

-20.

-40.

Configuration E - 5 : Garnmo Trocklng

* +? ++

- - * \

+ + + + + + - '.

+ +

+

LEGEND: - gamma

+ gamma

180.

go-

0.

-90.

- 1 so,

-270.

to gamma command - o p m n to gamma cornmond - closed

+ +++ + + + +

+ + +

I .1

I 1 .

I I IO. 1 0 0 .

frequency - rod/sec

I I I. I m T)

I '

NLR Configuration F-3 : G a m m a Trocking

cia- 1

. . . . . 20.

40- i . - *

,?++++

*."\

+ + + + + + + + + -

-20. j t +

+ -40- I +

I I I - 1 1 , IO. 1 0 0 -


LEGEND: - gamma to gamma commond - o p e n + gamma to gamma cornmond - c losed

I

4 80.

8 0-

0.

-90.

+ +++ ++ + +

+ + +

.1 1 . 7 0, 1 0 0 .

frequency - rad/sec

I I I. I I I

I I

f..J L F:

BO.

40.

20-

0.

-2..

-40 -

I

m

Cc\niigurotic\rl F-2 : G a m m a Tracking

1 , I O .

frequency - rad/sec

LEGEND: g a m m a to g a m m a command - opgn

+ g a m m a to g a m m a command - closed

1 BO.

90 ,

0.

-90.

- 1 ao,

-270.

+ +++ + + + +

+ + +

.1 1. IO. 100.

frequency - rad/sec

I I I. I I I I I I

I

I s

r.1 L F:

SO.

40.

20-

0.

-200.

- 4 0 -

Configuration F- 1 : Garnmo Track ing

+ +

+ I

- 1 1

1 . I

1 0 -

frequency - rad/sec

LEGEND: gamma

+ g a m m a

180.

90,

0.

-9 0.

- 1 eo.

-270.

I 1 0 0 .

to gamma command - open to gamma command - closed

1 + ++ + +++ + + + + + + + + *+*

.1 1 . 7 0. 100.

frequency - rad/sec

1 I I. I I I

CP

I:,

r d L F i

BO.

40.

20.

-20..

-40 .

Configuration G - I : G a r n n ~ o Tracking

+ +++ ++*+

+ + *-1, 1 , , * + + , - 1 1 - I a- 1 0 0 ,


LEGEND: gamma to gommo command - open

+ gamma to gamma command - c l o s s d

180.

9 0,

0.

-90.

- 1 SO.

-270.

+ ’ + + + + + + + + + + + ~

I + +++

++ + . . . . . * a * . * + +

. 1 1 . IO. 100,

f reauencv - rad / s e c

+ +

-40- I I

I I 1 - 1 1 , 10, 1 0 0 ,

frequency - rQd/SeC

LEGEND: gamma to gamma command - open

+ gamma to gammo command - closed I

180.

I. -270, .1 1 . 10. 100.

frequency - rod/sec

I I I.

60. -

40 . -

20- -

0. 1.

-20. -

-40 .

I I I I I I

. . . .

+ + + + +++ ++++\ **. *.. ..

+ +

+ I I

-L

1

NLR Configuration G--3 : G o m m o Trocking

- 1 1 , 1 0 , 1 0 0 ,

frequency - rad/sec

U -

I I

I

b 1:

LEGEND: - gamma to gamma command - o p s n + gammo to gommo commond - c l o s e d

180.

9 0,

0.

-9 0.

-1 so,

-270. 1 . 7 0. 1 0 0 .

frequency - rad/sec

m

I I ”

& .- I & u I E

- 80.

40.

20.

0.

-20,

-40-

- 1 1 - IO, 1 0 0 ,


LEGEND: gamma to gomma command - o p e n

+ gamma to gamma commond - closed I

T3

I ’

0

180,

9 0.

0.

-90.

-1 ao.

-270. . 1 1 . 10. 1 0 0 .


I I

I I I I I I

U 1 SQ-

9 0.

E.

-9 G.

- 1 eo. 1. 10. 100.

+ T + +' ++

1

I I I 1

270.

i ao.

90.

0.

-90-

- 1 80. . 1 1 . IO. 1 0 0 .


I ++ -4 Cf. 1 I I i

++ + + + + +

+ +

+

0.

--so.

- 1 B O . .1 1. 1 0 . 100.

frequerlcy - rod/ - -eC

I I I B I I

I

I JI f . !

4 C). -

0 . -

-20. -

+ +

r 7 .- L / CIA

1 8 0 .

90.

G.

-GIG-

- 1 eo.

++ + +

+ +

+ +

+ t . . . . . . . . . . .

+ + + + + + + + +

++ +,

.1 1 . IO. 1 0 0 . - f req u e rl cy r a d /' 5 e c

I 1 I.

. . . . . .

+ + + + + + +

+ + + ++ A + +

+ +

LEGEND: . pitat reeponae t a theta error

+ pitat rsapclnse t o the ta

270.

1 ao.

9 0.

0.

-9 0,

- 1 80.

+ +

+ +

+ +

+ + + + + + + + + + + + +

. 1 1 . 10. 100.


I 8

e 0.

3 1:) .

2 0 .

0.

-20.

- 4 0. I

- 1

. . . . . . . .

+ ++ + + + +

+ + ++- +

+ + I 1 .

frequency - r u ci j’s e c:

270.

1 ao,

9 0.

0.

-9 0-

- 1 80.

+ ++ +

+ +

+ +

.. . ... + . . + ’ + + +

++ ++

+% % + + + +

1 1 0 0 .

. 1 1 . IO. 100.

f requency - rad/E.:ec

r , 7 .-. L .( 1,'.

i eo,

9 0.

0.

-90-

-1 80. . 1 1. 1 0 . 100.


1 I I 1 I I

4 r:, . -

-20. -

-413- -

_ . . . .

270.

1 a m

9 0.

0.

-9 0-

- 1 80. .1 1 . IO. 100.

f r e q u e n c y - r a d / - 4ec

. . . . .

r + ++ + + + *

' ++++A + +

++

LEGENP: . pilclt t - e m p a n s e tc, t h e t a error

+ pilclt r m a p o n 3 e t o theta

270,

1 eo,

9Q.

0.

-90-

- 1 ao.

++ + + +

+ +

+ + . .. . . . . .

+ + + + +

+ ++ + + +

+ +

+ + . .. . . . . .

+ + + + + + + + + + + + + +

I I I

+ + + + + + + + 4

. I 1 . 3 0. 100.

frequency - rad/sec

U

t . 1 3 -

4 I.-J . --

2c. -

0. -

4 -20. -

-40.

I ’ . -. _ _ -

. . . . . .

. .

+ ++ + + + +

+ + + ++ A i +

++ I 1 I

. -- r L 18‘ t

CL

I

. I 7 .

270.

1 eo.

9 0.

0.

-9 0.

- 1 eo.

+ +

+ + +

+ . + . . . . . ..

- .

+ + + + +

I

I I I 1 . l 1. IO. 100.

frequency - rod/- dec

1 . 1 9 . -

4 7,. -'

>' L

0. -

4 -2u. -

-40-

-1

1 E O -

9r3.

-I ++ + + + +

+ + + ++ A + +

++ I 1 - 7 0. 1 0 0 -

0.

-% 0-

-1 eo.

0 . -

-%0- -

-1 eo.

++ 4-

+ + +

+ . . . . . ._ . .

+ + + + + + + + + + + + + +

I I 1

+ +

+ +

I + + + + + + + + + + I

+ I I I 1

1 I

L

1

L a i

. . . . . .

+ ++ + + + +

. ++++A + +

++

270.

1 eo.

9 0.

0.

--9 0,

- 1 80.

+ ++ + +

+ +

+ +

+ . . . - . _ _ . . .

. . + + + + + + + + + + + + + +

I I I . l 1 . IO. 1 0 0 .

f r equency - rod/sec

1 I I.

i

1 I I 1

I I t I B I

I, Qr

20.

0.

- 4- c',

. . '.

. . . . . . .

r I L _ - - . : 1 t b c ~ " t . 0 7 t-ac I - a i r i g

-

-

1 I I.

I I 1

1 1 I u I I

. . . .

B i n g l e l o o p pilot function

- e .

180.

9 0,

0.

-9 0.

-I am

-270.

75 . .

I

I I a?

T! 7

. . . _

I I I 1 I I I. I I I I I l I

I J 1 f

CCj.

0.

-40.

- 1 1 - IO. -ic*a- f r e q u e n c y - r a d j s e c

LEGEND: equivalent m i n g l e loop pilot f u n c t i o n

I . 1

I I I 1 . IO. 100.

f re y u e n c ~ , , ~ - rad/sec

I I I. I I I I I I

eo.

4 0.

20-

0.

7 - - i I-' .

- 4 0. I - 1

I 1 -

I IO.

I 1 0 0 .


LEGEND: equivalent e i n g l e I a o p pilot function

180.

9 0.

Q.

-9 0.

- 1 so-

-270.

I

. . ._ . .

1 I 1 . I 1 . 7 0. 1 0 0 .


I I I. I I I I I I

B Y I

u I t-l I m

m 0

. . .

LEGEND: . equivalent single Iaop pilet function

18rj.

9 Q.

0.

-9 0.

-1 80,

-270.

. ..... . .

I I 1 . 1 1 . IO. 100.

f r e a u e n c v - rod /sec

I I

I I I I 1 I I. I I I

I.

1 - 1

I 1 -

180.

90.

0.

-9..

- I am

-270.

. . . ...

.1 1 . IO. 1 0 0 .

frequency - rad/sec

I I

I I I I I, IC I.

I.

I I I I 1 I I p s

I

80-

4 o.

20.

0.

-20.

-40-

. . .

. . . .

- 1 1 - 1 0 - 1 0 0 .


LEG END: equivalent e i n g l a loop pilot function

180.

9 0,

0.

-9 0.

- 1 so-

-270. . 1 1 . 1 0 . 100.


I I

I I I I I I I. 1 1 I I I I I

1.

I. 270. c

c c71 .

I I I 1

4 0.

Q.

-20.

-40-

..-- .. -.. \

. . . .

LEGEND: equi;iaIent s i n g l e l cop pilot func:ticln

1 1 B O . I B O . -

90, -

. . . . . 0. - . -

-9G. -

- 1 8 0 - -

c 270. I I 1 . 1 1 . IO. 100.

f r e q u e rl c y - t-ad/sec

I - ( ‘ ( l ’ - I

1 8 0 . -

90. - CP

I 1 2

I Q. - I, -

1: -30. -

I ! L la

--lea, -

I. -270.

2 iJ .

. ...* . .

I I I

. .

LEG EN C’: equivalent Binale l o o p pilot function

frequency - i-ad/sec 1

1 1 1 1 I

m -E

a! -D 3

E 4 - a .-

E

I I. 1

1 I 1 I

I

[LI 6'1 0 i LL

. , . .

180,

9 0.

0.

-3 0.

- 1 so-

-270.

.1 1 . IO. 100.

frequency - r-od/sec

f 4 1 f 1: I-

4 0 . - '

I -20.

LEG E td D: equivalent eingle loop pilot function

f r e q u e n c y - rod / - 3 C C

1 I I.

I 1 I

E O -

G.

-20.

-4 El-

m& m 6 m a r m -

++ + + ++

LEGEND: open . theta tc. t h m t a c ~ , m m a r l d -

+ the ta t r _ . thBta ~ = . z ~ m m a r r l d - c: I CZI E. e d

180.

9 0,

0.

-90.

-1 80,

-270.

+ ++

.. ,/

+ + + + + + +

+ + + + +

* . + . +

+ + +'. + * + * +. y.

1 I I 100.

I .1 1. IO.

frequer-Icy - r a d ,/'s;.ec

-&v. -

4 0 .

2 G .

0.

-20.

. .

- 4 0 . I I I +

1

LEGENC>: . theta to theta cctrnrnarld - open

+ theta tc, theta cmmmand - c l o s m d

1 8 0 ,

9 0,

0.

-go.

- 1 ao.

-270.

7

* - ? + + + + + + + + +

+ + + + +

- . + + + +'. +. + ' + ' ++'.

I I I

+- .

I + ' + ' ++'.

I I I .1 1 . I a. 100.

frequency - rod /-. 4cc c - x

'I

U

L

3 r:.* .

1 - i c : .

0.

. .

LEG E t 4 D:

180.

9 0.

rJ.

-90.

-1 ao.

--- 2 7 0.

theta c.=,mmand - op-n t h e t a cclmmartd - c l c l s m d

+ + + +

+ + +

++'

. I 1. IO. 100.

f requency - r o d / s e c

I Bo

I 1 a?’ w

I I I 1 1 I I

LEG EN D: . t he t a tc theto cc~mmand - open

+ t he t a tc t h e t o ~ c ~ r n m o n d - cIaBed

’ 8 Q * 1

1 -90.

+ +. I +

+ +

* . + +

-l j , .-”: . , , +‘ + ’

-270. - 1 1 . IO. 100,


I

l e ( . . -

90- -

a. I.

-90. -

-1ao. -

-270.

-40 - I + I I 1

? + + + +++ +

+ + I

+ + + +

- . + + .. + * +. +* +' . + * + ' + ' y.

I I i

- 1 1 . 1 0 - 7 00-


LE G E P4 D : Io . theta tc. theta c o m m a n d - open + theta tc the ta c m m r n o n d - c1oE-d

I

U

I '

I 1 I.

I I I u 1 I I i

e 0-

40 .

0.

-20.

-40 .

-1

. . .

1 +

LEG EN D: . theta t o theta cc~rnrnand - open + theta tc:. thbta caminand - clozed

180.

9 0,

0.

-9 0.

- 1 so,

- e, L70.

+ + + + + + + + + +

+ +

+' +' . +. + ' + *. +,.

I

I 1 I 1 .1 1 . IO. 1 0 0 .

frequency - rad/- dec

2 13 I I . -0

0.

I I - 1 1 - IO.

f r - e q u e n cy - raci/sec

LEG END: . theta tc- theta cclmrnand - open

+ theta tc: thf l ta cctmrnand - closad

1 'do.

. . .

+ ++ + + ++ + +++

0.

1

1 . IO. 100. frequency - rod/sec

I I

I I I I I I I.

i.

1ao. -

90, -

0. 1,

-90. -

-iao. -

-270.

I I I I 1 I I

+ ++ + +++ + + +

+ + + + + +

* * . + ..+

4 +. +. +' + *. + . + . + . I I 1

6 0 .

40.

212)-

0.

-20.

-40-

- 1 1. 7 0, 1 0 0 ,

frequency - ruc i / sec

LEGEND: open . theta to theta command -

+ theta tc thQta c~lrnrnarld - closed

'I I I

I I I I I I I. I I I I 1 I I

I

I.

I.

- 4 0 . - I -b

I I

LEG END: . theta tc. theta cgrnmand - open

+ theto t c th-to command - c l o s m d

1 8 0 .

9 0.

0.

-9 0.

+ ++

~ * -

+ +++ + + + + +

+ +

. . +. + ' .+

+ /. +'

-180. - +'. + ' + ' ++'.

-2770. I -e'

I 1 . 1 1 . 7 0. 1 0 0 .


I I I I I I

i

I "

4 0.

20.

0.

-2..

-40 - + I

- 1 I 1.

I I I O - 1 0 0 -

f t'F g u e 1-1 cy - rad/, CFr,

LEG END: . t he ta to thBta ccmmartd - open

+ the ta t c

1 EO.

9 0,

0.

-90.

-1 230.

-270.

theta carnrnand - c l o m e d

+ + + + + + +

+ + + + +

* . + ' . +

Y +. +'. + -

I I 1 .1 1 . IO. 1 0 0 .


I I

I I I I I I I. I I I I 1 I I

I

I.

I.

4 0.

0.

---eQ_

. .

++ + + + +

LEGEND:

1 ao.

9 0-

C.

-9 0.

-1 am

2 7 Q .

theta command - open theta cctmrnand - c:lclBed

+ + + + + + + + +

+ +

. . + + ' . +

-t +. +'. +. +' + .. 2.

I 1 1 .1 1 . IO. 100.

frequency - rod/sec

-4-1- 1 + 1 1 I

- 1 1 - I O , 100-

frequency - rad/',* -=ec

LEG EN E a : . t h e t o t c th- ta cammarid - open

+ theta t n th-ta carnrnand - closed

I 1 180,

I I I I

0.

-9 0.

- 1 80.

-270.

i t + + + + + + + +

-I * +.

I".? ++ + + + + + + +

+ + +

* . + + * . + - * +.

+'. + * +- + .. \..

I I I

+'. + * + ' + ..

I \.. I I I

.1 1 . 10. 100.

frequency - rad/s.ec

purdue university school of aeronautics and astronautics

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