Time Domain Pilot Model Identification Using ...

AIAA Modeling and Simulation Technologies Conference and Exhibit 18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7109

Time Domain Pilot Model Identification Using Maximum Likelihood Estimation T. Berger, * P.M.T. Zaal, † M. Mulder ‡ and M.M. van Paassen‡ Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands

Currently, multi-channel pilot models parameter estimation is done using a two-step frequency domain technique—identifying a non-parametric frequency response and fitting a parametric model to it. A time domain identification method would only require one step— directly fitting a parametric pilot model to the time domain data. Time domain identification has additional advantages in that the forcing functions used do not have specific limitations and multi-channel identification can be accomplished with only one forcing function. This paper displays the results of single- and multi-channel pilot model identifications done using the MATLAB MMLE3 toolbox. Identification was performed first on Simulink simulations and then on actual data collected using the SIMONA Research Simulator at the Delft University of Technology.

I. Introduction Traditionally, multi-channel pilot models can be identified using several frequency domain techniques. Such methods however, require two steps. First a non-parametric identification of the frequency response functions is made. Following that, a parametric model is fit to the non-parametric response using either a Fourier Coefficient method1-5 or ARX6. A time domain identification method only requires one step—directly fitting a parametric pilot model to the time domain data. Such an identification method may have several advantages over frequency domain methods. The forcing functions used do not need to be sums of sine waves, as the Fourier Coefficient method requires. For example, the transient response to a step input may be used to perform the identification in the time domain. Furthermore, it may be possible to identify a multi-channel pilot model with just a target signal or just a disturbance signal, but not both, as frequency domain methods require. This study is concerned with testing a method for performing multi-channel pilot model identification in the time domain. The method looked at was the Maximum Likelihood Estimation method. This method was originally developed to identify aerodynamic coefficients of aircraft, but can be applied to a pilot model. The pilot model that is used is the Van der Vaart model, which uses central visual and motion perception paths with constant gains and time delays, along with neuromuscular dynamics, to represent the pilot control behavior. A brief explanation of pilot model identification is given in Section II, followed by a discussion about the MLE method in Section III. The method was initially used to identify a pilot model simulation done in MATLAB Simulink. Information about the simulation setup is presented in Section IV. The identification results are presented in Section V including the method’s sensitivity to initial parameter guesses, as well as robustness to remnant noise gain. In addition to the simulations, experimental data collected using the SIMONA Research Simulator (SRS) were also analyzed. Here the method was used to make both single- and multi-channel identifications. A discussion of the results is presented in Section IV.

II. Pilot Model Identification The pilot model which is analyzed in this paper, in the closed-loop, along with the aircraft dynamics being controlled, is shown in Figure 1. The Pilot model analyzed has two separate parts—the error response, shown by the *

Stanford University, Department of Aeronautics and Astronautics (currently with the University Affiliated Research Center, NASA Ames Research Center). † Ph.D. Candidate, Control and Simulation Division, Faculty of Aerospace Engineering. ‡ Associate Professor, Control and Simulation Division, Faculty of Aerospace Engineering. 1 American Institute of Aeronautics and Astronautics Copyright © 2008 by Delft University of Technology. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

HPe block, and the rate response, shown by the HPx block. The aircraft dynamics being controlled, the AC block, is a double integrator, representing a roll response. The pilot task, used in simulations and experiments, consists of following a target signal, ft, while correcting a disturbance signal, fd. Both signals nominally consist of a finite number of harmonics at discrete frequencies. n

Pilot

e

ft

fd

u

HPe

AC

x

―

x

HPx

Figure 1: Closed-loop system used for simulation. The specifics of the pilot model used, which is a derivative of that proposed by Hosman7 and Van der Vaart8, are shown in Figure 2. The model is composed from a combination of a linear part and a remnant noise signal, to account for nonlinearities. The pilot model contains two channels. The first includes the pilot’s response to an error signal, between the desired aircraft attitude, commanded by the target signal, and the actual aircraft attitude. The dynamics of this channel are described by proportional and lead gains, Kvis and tlead, respectively, as well as a time delay tdvis. The second channel is the pilot’s response to vestibular motion perception of the attitude of the aircraft. It first contains a double differentiator since the inputs to the semi-circular canals are accelerations. That is followed by the dynamics of the semi-circular canals, Hvest(ω), a proportional gain, Kvest, and a time delay, tdvest. Finally, the pilot’s neuromuscular dynamics, Hnm(ω), are accounted for, in addition to the remnant noise signal, n. n

Kvis(1+j ωtlead)

e

e− jωtdvis

Hnm(ω)

u

―

x

(jω)2

Hvest(ω)

Kvest

e − jωtdvest

Figure 2: Multi-channel pilot model used for identification. The semi-circular canal dynamics, which were fixed in this identification method, are given by:

H vest (ω ) =

1 + jωtv1 1 + jωtv 2

with, tv1=0.1 second and tv2=6.0 seconds. While the neuromuscular dynamics are given by:

2 American Institute of Aeronautics and Astronautics

(1)

H nm (ω ) =

f nm 2 f nm 2 + 2d nm f nm jω + ( jω )

2

(2)

where, fnm is the neuromuscular frequency and dnm is the neuromuscular damping.

III. Maximum Likelihood Estimation The method used to identify pilot models in this study was the MMLE3 Identification Toolbox for State-Space System Identification using MATLAB. This method allows for identification of continuous linear time-invariant state-space systems using the Maximum Likelihood method.9 In order to define a state-space model for the pilot model used, a 5th-order Pade approximation was used for all time delays. In this case, the Output Error method was used. For the case of a stable, single output system with no state noise, but with white Gaussian measurement noise, the output error method gives the Maximum Likelihood estimates.9 A. Optimization Algorithm in MLE The MMLE3 Toolbox uses an algorithm which minimizes the cost function:

J=

[

1 N ~T ∑ z k (θ )W~z k (θ ) 2 k =1

]

(3)

where,

θ

is the vector of parameter estimates

~z is the difference between the measured observation vector and predicted observation vector, as a function of the

current parameter estimate vector W is a weighting function equal to the inverse of the sample innovation covariance, RR, which is given by:

RR =

1 N

N

∑ ~z ~z k

T k

(4)

k =1

Calculations of the gradient and second gradient of the cost function are performed to obtain optimum estimations. The gradient of the cost function and the Fisher information matrix are calculated in the following manner. Since RR is a diagonal matrix, the weighting function W can be written as the square of a matrix, W1, (ie

W = W1T W1 ). Thus, the expression for the cost function can be rewritten as: J=

[

1 N ~T z k (θ )W1T W~ z k (θ ) ∑ 2 k =1

]

(5)

z → zˆ , the gradient of the cost function at the ith iteration becomes: Using the transformation W1~ ∇θ J ( i ) =

∂J ∂θ

θ =θˆ ( i ) T

⎡ ∂zˆ (θ ) ⎤ =⎢ zˆ (θ ) ⎣ ∂θ ⎥⎦ θ =θˆ ( i )

3 American Institute of Aeronautics and Astronautics

(6)

The second gradient approximation of the cost function becomes: T

2

∇θ J

(i )

2 ⎡ ∂zˆ (θ ) ⎤ ∂zˆ (θ ) ∂ zˆ (θ ) =⎢ + zˆ (θ ) ∂θ 2 ⎣ ∂θ ⎥⎦ ∂θ θ =θˆ( i ) T

⎡ ∂zˆ (θ ) ⎤ ∂zˆ(θ ) ≈⎢ ⎣ ∂θ ⎥⎦ ∂θ θ =θˆ( i )

(7)

The optimization algorithm implemented in the MMLE3 Toolbox updates the parameter estimation vector

θˆ (i +1) using the equation: θˆ (i +1) = θˆ ( i ) + ξ ( i )

(8)

ξ ( i ) = −(∇θ2 J + λI ) ∇θ J

(9)

where, −1

The constant λ is selected to help further reduce the magnitude of the cost function, since in the regions far from the minimum, although the algorithm points in the proper direction, it may not have sufficient magnitude. Within the algorithm, the value of λ is regulated such that it becomes smaller as the cost function gets closer and closer to its minimum.10 Once parameter estimates are produced, an evaluation of their accuracy is also performed, using Cramer-Rao Lower Bounds (CRLB). According to the Cramer-Rao inequality, the covariance matrix of parameters estimated using an unbiased estimator is greater than or equal to the inverse of the Fisher Information Matrix. Since the Maximum Likelihood Estimator is asymptotically unbiased, the parameter error covariance matrix approaches its limit, and is well approximated by9:

E[(θˆ − θtrue )(θˆ − θtrue )T ] ≈ H −1

(10)

where, H is the Hessian matrix. The standard deviation of a parameter can therefore be approximated by its CramerRao Lower Bounds9:

σ i ≈ CRLBi ≈ [ H −1 ]1/ii 2

(11)

B. MLE Pilot Model Parameter Estimation Toolbox In order to run the identification, the user must have the appropriate input and output data, whether from a simulation or from an experiment, the type of model to be identified, and the rate at which the data was sampled, which is used by the MLE routine to calculate a discrete time state space representation of the model. The model selected to be used for identification determines which parameters will be identified. The available models are a single channel, central visual only, pilot model or a multi channel, central visual and vestibular, model. Once the model is selected and the input and output data are loaded, the identification is ready to be run. The initial guesses for the parameters may also be passed to the identification toolbox, however, a default set is used if none are supplied. The default values are those estimated by Van der Vaart in a similar experiment8, and are given in Table 1.

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Table 1: Default values for parameter initial guesses. Kvis [-] tlead [s] tdvis [s] Kvest [-] tdvest [s] fnm [rad/s] dnm [-]

Single Channel Model 0.32 1.68 0.39 ― ― 12 0.3

Multi Channel Model 0.17 2.93 0.32 1.59 0.29 12 0.3

IV. Simulation A. Method Parameter estimation was first done on data obtained by running simulations in MATLAB Simulink. The pilot model used for the multi-channel simulations contains a central visual path, a vestibular path, and neuromuscular dynamics, as described in Section II. The multi-channel pilot model also contains a remnant noise signal. This signal was generated by passing white noise through a second-order shaping filter. The gain on the noise was nominally set to 0.4. The simulations run contained both a target signal and a disturbance signal. The nominal signals used were composed of a sum of several sine wave forms at different prime frequencies. Furthermore, the disturbance signal was pre-filtered with the inverse aircraft dynamics. This was done to make sure that enough high frequency content made it through the double integrator dynamics and into the pilot model, in order to be able to properly identify the neuromuscular dynamics. The frequencies used and their amplitudes are presented in Table 2, as well as the prime value of each frequency. The phases were all set to zero. The simulations were run with a sampling frequency of 50 Hz, and so a sampling time of 0.02 second was used to convert the continuous time state space representation of the pilot model into a discrete time representation. Table 2: Amplitudes and Frequencies of the disturbance and target signal harmonics. Disturbance Signal Amplitude [deg] Frequency [Hz] 0.0031 0.061 0.0071 0.0977 0.0115 0.1343 0.0193 0.2075 0.0284 0.3418 0.0363 0.5615 0.0407 0.7202 0.0488 1.001 0.059 1.2939 0.0755 1.6724 0.1033 2.1729 0.1308 2.5757

Prime [-] 5 8 11 17 28 46 59 82 106 137 178 211

Amplitude [deg] 0.0876 0.0764 0.0613 0.043 0.025 0.012 0.0081 0.0052 0.0038 0.0029 0.0024 0.0021

Target Signal Frequency [Hz] 0.0732 0.1099 0.1587 0.2319 0.354 0.5737 0.7446 1.0132 1.3062 1.6968 2.1851 2.6001

Prime [-] 6 9 13 19 29 47 61 83 107 139 179 213

B. Results 1. Parameter Estimation For the preliminary test of the MLE algorithm, a simulated data was analyzed. Ten simulations were run and analyzed. The results are presented in Figure 3, including error bars indicating the Cramer-Rao lower bounds calculated by the MLE toolbox. In addition, for each parameter the bias of the average of the ten runs is shown as well as the variance of the results of the ten runs. The actual values of each parameter used in the

5 American Institute of Aeronautics and Astronautics

Bias = 0.018425, Variance = 0.0056497

Bias = 0.046479, Variance = 0.00027184 tdves t [s]

Kvis [-]

1 0

1

2

3

0.2

4

5 6 7 8 9 10 Run Number Bias = -0.13578, Variance = 2.1994

tlead [s]

10 0 -10

1

2

3

5 6 7 8 9 10 Run Number Bias = -0.0083988, Variance = 9.8952e-005 dnm [-]

0.3

1

5 6 7 8 9 10 Run Number Bias = -0.4024, Variance = 0.5997 2

3

4

15 10 5

4

0.35

tdvis [s]

0.3

fnm [rad/s]

-1

0.4

5 6 7 8 9 10 Run Number Bias = -0.027552, Variance = 0.00017209 1

2

3

4

1

2

3

4

0.3 0.28 0.26

Kvest [-]

0.25

1

2

3

4

5 6 7 8 9 10 Run Number Bias = -0.18399, Variance = 0.084205

4

9 10

ML E MMLE

Initial Guess Actual Values

2 0

5 6 7 8 Run Number

1

2

3

4

5 6 7 8 Run Number

9 10

Figure 3: Parameter estimation for multi-channel simulations.

Target [deg]

0.5

0

0.5

0

-0.5 1 0.5 0 -0.5

Aircraft Respose [deg]

Pilot Respose [deg] (Output)

yddot Signal [deg/s 2] (Vestibular Input)

Error Signal [deg] (Central Visual Input)

Disturbance [deg]

-0.5

5

0

-5 2 1 0 -1 1

0

-1

0

10

20

30

40

50

60

70

t [s]

Figure 4: Time history of one multi-channel simulation.

6 American Institute of Aeronautics and Astronautics

80

90

tdvest [s]

0.2

1

2

3

5 6 7 8 9 10 11 Run Number Bias = -0.20923, Variance = 0

5

1

2

3

5 6 7 8 9 10 11 Run Number Bias = -0.0028258, Variance = 3.3896e-033

0.4 0.3

Kvest [-]

0.2

1

2

3

4

5 6 7 8 9 10 11 Run Number Bias = -0.37742, Variance = 5.4234e-032

4

1

2

3

4

5 6 7 8 9 10 11 Run Number Bias = -0.57474, Variance = 1.3884e-029

1

2

3

4

1

2

3

4

15 10 5 6 7 8 9 10 11 Run Number Bias = -0.04159, Variance = 3.3896e-033

1 0.5 0

5 6 7 8 Run Number

9 10 11

ML MMLE E

Initial Guess Actual Values

2 0

0.5

5

4

dnm [-]

tdvis [s]

0

1

0

4

tlead

[s]

0

Bias = 0.048224, Variance = 3.3896e-033

fnm [rad/s]

Kvis [-]

Bias = 0.014353, Variance = 8.4741e-034 0.4

1

2

3

4

5 6 7 8 Run Number

9 10 11

Figure 5: Sensitivity of parameter estimation to initial conditions. simulation and the initial guess are also shown on the plot. For these analyses, the initial guess was set to the parameter values used in the simulation. Overall, the parameter estimates of the multi-channel simulations show higher biases than those for the single loop simulations. For the multi-channel case, the MLE method is best at estimating the visual channel lead gain and time delay, with biases of 2.6% and 4.6% respectively, and the neuromuscular dynamics, with biases of 3.4% for the neuromuscular frequency and 9.2% for the neuromuscular damping. Biases for all parameters are given in Table 3. Figure 4 shows a sample time history of a multi-channel simulation. The top two sets of axes display the target and disturbance signals. The next three sets of axes show the error signal and the second derivative of the attitude signal which are the two inputs to the pilot, and the pilot’s response. These were the time histories fed to the MLE identification toolbox. Finally, the attitude of the piloted aircraft is shown. 2. Sensitivity to Initial Conditions A sensitivity analysis was done by taking one set of input/output data and running the identification on it while varying the initial guesses. The first run was done with the initial parameter guesses set to the same values used in the pilot model to generate the data. Subsequently, ten identification runs were made while varying each of the initial guesses a random amount within ±40% of the value used. The results are given in Figure 5. The variance in the results obtained is effectively 0, and it is clear that the MLE algorithm has low sensitivity to initial conditions. 3. Sensitivity to Remnant Noise Several runs were made with varying gains on the remnant noise signal, each run consisting of 10 simulations. The noise gain was varied between 0.1 and 0.8, corresponding to a variation in the ratio of the power of the remnant noise signal to that of the piloted control signal of between 3% and 32%. At each run, the 10 simulations were analyzed and the bias of the mean of the results as well as the variance of the results were recorded. The biases are plotted in Figure 6 and the variance in Figure 7.

7 American Institute of Aeronautics and Astronautics

400

tdvest Bias (% of value)

Kvis Bias (% of value)

200 100 0

200 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%) fnm Bias (% of value)

tlead Bias (% of value)

200

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

100 0

20 10 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

Kvest Bias (% of value)

10 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

dnm Bias (% of value)

tdvis Bias (% of value)

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

20

40 20 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

40 20 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

0.2 0

tlead Variance

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

200 100 0

0.02 0.01 0

Kvest Variance

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

4

5

2 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

dnm Variance

tdvis Variance

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

tdvest Variance

0.4

fnm Variance

Kvis Variance

Figure 6: Parameter estimation bias versus remnant noise gain.

0.05

0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

0.4 0.2 0

0 10 20 30 40 Remnant Noise to Control Signal Power Ratio (%)

Figure 7: Parameter estimation variance versus remnant noise gain.

8 American Institute of Aeronautics and Astronautics

tdvest [s]

1.5

0.2

0

Nom

Dist

fnm

tlead [s]

4

2

0

Nom

Dist

Nom

Dist

Targ D(step) T(step)

Nom

Dist

Targ D(step) T(step)

Nom

Dist

Targ D(step) T(step)

13 12 11

1 dnm [-]

[s] tdvis

0.5

10

Targ D(step) T(step)

3 2 1 0

1

0

Targ D(step) T(step)

[rad/s]

Kvis [-]

0.4

Nom

Dist

Targ D(step) T(step)

0.5

0

Kvest

[-]

4 MLE MMLE

2

0

Actual Values

Nom

Dist

Targ D(step) T(step)

Figure 8: Parameter estimation for multi-channel simulations with different forcing functions. Figure 6 shows a significant affect of the remnant noise gain on the bias of the estimation of the vestibular gain. As expected, the other parameters also show an increase in bias and variance with increase in remnant noise gain. The decrease in the bias and variance of some parameters at high remnant noise gain is because parameter estimation failed on some of the simulations with high remnant noise gain, and so the results are of a more limited set of estimations that succeeded. 4. Sensitivity to Forcing Functions In order to test the MLE method’s ability to identify multi-channel parameters with just one forcing function, parameter identification was done using just a target signal and later just a disturbance signal. Furthermore, the signals used were varied between the nominal sum of sine waves used in the analyses above and a step input. The step input used was a positive step for half the duration of the simulation followed by a negative step for the remainder of the simulation. Step inputs were not used for any manned experiments, but for those cases they could be put in at random intervals. In the cases of disturbance only signals, the gain on the disturbance signal was raised from its nominal value of 1 to 4. For those cases, the gain on the remnant noise signal was also raised by a factor of four. Figure 8 shows the parameter estimation results using different forcing functions. From left to right, the results show parameter estimates using the nominal disturbance and target functions, then disturbance only, followed by target only, disturbance only step input, and finally target only step input. The values on the plot are the mean of 100 simulations for each case, and the error bars represent the 95% confidence interval for each case. Table 3 summarizes the biases in the parameter estimates for the different forcing functions used. For the target signal only case, the parameter estimates for the central visual loop were slightly worse than in the two forcing function nominal case. However, for the disturbance signal only case, there was a significant increase in the bias of all parameters except the visual channel gain and neuromuscular dynamics. The neuromuscular dynamics and vestibular channel gain and time delay were identified best in the cases using a target only step input.

9 American Institute of Aeronautics and Astronautics

Table 3: Percent biases in the mean of parameter estimation of 10 simulations for different forcing functions. Nominal Kvis Bias [%] tlead Bias [%] tdvis Bias [%] Kvest Bias [%] tdvest Bias [%] fnm Bias [%] dnm Bias [%]

12.37 -13.32 -3.63 -10.50 15.88 -3.76 -9.67

Disturbance Only -6.28 -27.20 406.72 41.26 144.95 -3.59 -17.14

Target Only 8.97 -40.16 -5.81 -10.89 38.50 -11.90 -35.51

Disturbance Only – Step 0.97 -1.29 15.12 -3.82 19.36 1.70 -33.60

Target Only – Step 12.61 -7.68 0.38 -2.58 7.81 -0.81 -9.29

V. Experiments A. Method Two sets of experimental data were analyzed using the Maximum Likelihood Estimation method. The first set of data used was from a disturbance rejection experiment run using five test subjects. Five test subjects were each exposed to three different test conditions. Each condition had a central visual indication, but a varying degree of motion, ranging from no motion, to reduced motion, to full motion. The data were analyzed using a single loop lumped pilot model. The experiment was carried out at the Delft University of Technology using the SIMONA Research Simulator. The second set of data that was analyzed was gathered in an experiment revisiting the Hosman and Van der Vaart tracking experiment.11 The experiment dealt with analyzing compensatory target following and disturbance rejection tasks, and as with the first set of data, was carried out at the Delft University of Technology using the SIMONA Research Simulator, and five different test subjects. This time however, two separate types of forcing functions were constructed, so the results are grouped into 10 categories. The two types of forcing functions differ in the distribution of their sinusoid amplitudes. The first uses the Hosman and Vander Vaart spectrum11 while the second uses the McRuer 6-4 spectrum12. In addition, each experimental run was done with either a target forcing function, for a target following task, or a pre-filtered disturbance forcing function, for a disturbance rejection task. Again, three conditions were looked at here—central visual only, central visual and motion, and central and peripheral visual and motion. B. Results 1. Single Loop The results for a lumped model identification are shown in Figure 9. From left to right of each group of parameter estimates, the condition of the experiment were changed from central visual and no motion, to central visual and reduced motion, to central visual and full motion. As expected, the results show that going from no motion to full motion, the lumped model, which only uses a central visual channel, shows an increase in the central visual proportional gain, a decrease in the central visual lead gain, a decrease in the time delay, and an increase in both the neuromuscular frequency and damping. For the third test subject, the MLE toolbox was not able to identify a model for the full motion case. 2. Multi-channel The data from the second experiment described above was used to do multi-channel parameter estimation. In this case, all runs were done with just one forcing function at a time. Figure 10 shows the results for the disturbance rejection task, and Figure 11 for the target following task. The missing points on the figures are cases where the MLE toolbox could not identify the parameters. From left to right of each group of parameter estimates, the conditions of the experiment were changed from central visual and motion only to central and peripheral vision and motion. As in the single loop case, the results obtained in the multi-channel parameter estimation match the expected behavior. The addition of peripheral visual cues have a larger effect in target following tasks than in disturbance rejection tasks11. Furthermore, there is a general increase in the motion perception path associated with additional cueing to the test subject.

10 American Institute of Aeronautics and Astronautics

20

fnm [rad/s]

Kvis [-]

1.5 1

15 10

0.5

5

0

1

2

3 4 Test Subject

0

5

2

3 4 Test Subject

5

1

2

3 4 Test Subject

5

[-]

1.5

1.5

1

dnm

tlead [s]

2

1

1

0.5

0.5

1

2

3 4 Test Subject

0

5

[s]

0.4

tdvis

0.5

0.3

MLE MMLE

Initial Guess

0.2 0.1

1

2

3 4 Test Subject

5

Figure 9: Single loop parameter estimation from experimental data using five test subjects. From left to right of each group of parameter estimates, the condition of the experiment were changed from central visual and no motion, to central visual and reduced motion, to central visual and full motion.

0

[s]

2

3

4

5 6 7 8 Test Subject

0

1

2

3

4

5 6 7 8 Test Subject

0

9 10

5

-5

tdvis t2

1

0.2

fnm [rad/s]

tlead t1

[s]

-5

0.4

2

2

3

4

5 6 7 8 Test Subject

9 10

1

2

3

4

5 6 7 8 Test Subject

9 10

1

2

3

4

5 6 7 8 Test Subject

9 10

20

2

1 0

1

40

0

9 10

dnm [-]

Kvis K1

[-]

tdvest [s] t4

5

1

2

3

4

5 6 7 8 Test Subject

9 10

0 -2

Kvest [-] K4

10 MLE MMLE

5 Initial Guess 0

1

2

3

4

5 6 7 8 Test Subject

9 10

Figure 10: Multi-channel parameter estimation from experimental data using ten test subjects with a disturbance rejection task. From left to right of each group of parameter estimates, the condition of the experiment were changed from central visual and motion, to central and peripheral visual and motion.

11 American Institute of Aeronautics and Astronautics

0

[s] tlead t1

2

3

4

5 6 7 8 Test Subject

2

3

4

5 6 7 8 Test Subject

0

9 10

2

1

0.2

1

2

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5 6 7 8 Test Subject

9 10

1

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5 6 7 8 Test Subject

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1

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9 10

10

2

0.5 0

1

20

0

9 10

dnm [-]

[s]

1

4

0

tdvis t2

tdvest [s] t4

1

0.4

fnm [rad/s]

Kvis K1

[-]

2

1

2

3

4

5 6 7 8 Test Subject

9 10

1 0

Kvest [-] K4

10 MLE MLE MMLE

5 Initial Guess 0

1

2

3

4

5 6 7 8 Test Subject

9 10

Figure 11: Multi-channel parameter estimation from experimental data using ten test subjects with a target following task. From left to right of each group of parameter estimates, the condition of the experiment were changed from central visual and motion, to central and peripheral visual and motion.

VI. Conclusion The results presented in this paper demonstrate the ability of the Maximum Likelihood Estimation method to identify pilot model parameters. The identification is all done in the time domain, which has several advantages over frequency domain techniques currently employed for pilot model identifications, including the freedom to choose forcing functions and the ability to identify a multi-channel pilot model with just one forcing function. A pilot model specific toolbox was developed around the MMLE3 MATLAB toolbox. It was then tested using multi-channel computer simulations. The sensitivity of the method to initial conditions, as well as its robustness to parameters such as pilot model remnant noise gain and forcing function type were tested. The method was then tested on experiment data gathered by two PhD candidates at the Delft University of Technology using the SIMONA Research Simulator. Here the method performed well identifying single-loop models. For the multi-channel case, the method gave good results for most cases, however for other cases no results were obtained. This was mainly due to the method trying to assign a negative value to the time delays. Future work on improving this identification technique will include the ability to place limits on the parameter values, in addition to supplying the method initial parameter guesses. The design of forcing functions will also be studied further, to see its effect on the parameter estimates. Finally, more experimental data should be analyzed, even from previous experiments, and the results compared to other pilot model identification methods.

References 1

Van Paassen, M. M., Biophysics in Aircraft Control. A model of the neuromuscular system of the pilot’s arm, Ph.D. thesis, Faculty of Aerospace Engineering, Delft University of Technology, 1994. 2 Mulder, M., Cybernetics of tunnel-in-the-sky displays, Ph.D. thesis, Faculty of Aerospace Engineering, Delft University of Technology, 1999. 3 Kaljouw, W. J., Mulder, M., and Van Paassen, M. M., “Multi-loop Identification of Pilot’s Use of Central and Peripheral Visual Cues,” Proceedings of the AIAA Modeling and Simulation Technologies Conference and Exhibit, Providence (RI), No. AIAA-2004-5443, Aug. 16–19 2004.

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4 Mulder, M., Kaljouw, W. J., and Van Paassen, M. M., “Parametrized Multi-Loop Model of Pilot’s Use of Central and Peripheral Visual Motion Cues,” AIAA Modeling and Simulation Technologies Conference and Exhibit, San Francisco (CA), No. AIAA-2005-5894, Aug. 15–18 2005. 5 Löhner, C., Mulder, M., and Van Paassen, M. M., “Multi-Loop Identification of Pilot Central Visual and Vestibular Motion Perception Processes,” Proceedings of the AIAA Modeling and Simulation Technologies Conference, San Francisco, (CA), No. AIAA-2005-6503, Aug. 15-18, 2005. 6 Nieuwenhuizen, F. M., Zaal, P. M. T., Mulder, M., Van Paassen, M. M., and Mulder, J. A., “Modeling Human MultiChannel Motion Perception and Control Using Linear Time-Invariant Models,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 999–1013. 7 Hosman, R. J. A. W., Pilot’s perception and control of aircraft motions, Ph.D. thesis, Faculty of Aerospace Engineering, Delft University of Technology, 1996. 8 Van der Vaart, J. C., Modeling of perception and action in compensatory manual control tasks, Ph.D. thesis, Faculty of Aerospace Engineering, Delft University of Technology, 1992. 9 Milne, G. W., “MMLE3 Identification Toolbox for State-Space Identification using MATLAB,” Control Models, 2000 10 Juliana, S., Cessna Citation II Aircraft Aerodynamic Model Parameter Identification, Master Thesis, Delft University of Technology, 2001. 11 Pool, D. M., Mulder, M., Van Paassen, M. M., and Van der Vaart, J. C., “Effects of Peripheral Visual and Physical Motion Cues in Roll-Axis Tracking Tasks,” Journal of Guidance, Control, and Dynamics, accepted for publication 2008. 12 McRuer, D. T., Graham, D., Krendel, E., and Reisener, W., “Human Pilot Dynamics in Compensatory Systems. Theory, Models and Experiments With Controlled Elements and Forcing Function Variations,” Tech. Rep. AFFDL-TR-65-16, Systems Technology Inc. & the Franklin Institute, 1965.

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