This paper presents a control theoretic approach to modeling human response errors (HRE) in the flight simulation domain. The human pilot is modeled as a supervisor of a highly automated system. The synthesis uses the theory of optimal control pilot modeling for integrating the pilot's observation error and the error due to the simulation model (experimental error). Methods for solving the HRE problem are suggested. Experimental verification of the models will be tested in a flight quality handling simulation

Ntuen, Celestine A.

English

NASA Technical Reports Server

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MODELING HUMAN RESPONSE ERRORS IN .c_
_ .YN_-IETIC FLIGHT SIMULATOR DOMAIN _t \o
C_,e_ineA.Ntue. _ 9 3 " 3[9 4 6_4
Human-Machine Systems Engineering Lab.
Department of Industrial Engineering
North Carolina A&T State University
Greensboro, NC 27411
This project is being supported by:
NASA Center of Research Excellence
(NASA-CORE)
School of Engineering
North Carolina A&T State University
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Research Memorandum # HMSEL-CORE-0191
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August 1992
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Modeling Human Response Errors In
Synthetic Flight Simulator Domain
Celestine A. Ntuen
HMSEL - NASA CORE
/
ABSTRACT
This paper presents a control theoretic approach to modeling human response errors
(I-IRE) in the flight simulation domain. The human pilot is modeled as a supervisor of a highly
automated system. The synthesis uses the theory of optimal control pilot modeling for
integrating the pilot's observation error and the error due to the simulation model (experimental
error). Methods for solving the HRE problem are suggested. Experimental verification of the
models will be tested in a flight quality handling simulation.
I. INTRODUCTION
The use of flight simulator in pilot training is as old as flying itself. However, it was
not until the late part of 1940s that the human pilot was considered as a part of the simulation
model (ref. 3). In this respect, the human pilot is considered to be a complex servo-mechanicai
system whose position in the simulation loop represents that of a sub-optimal controller (ref. 1,
4, 5, 16).
Control theoretic models have been shown to be very robust mathematical tools for
modeling servo systems (ref. 5, 7, 10, 17). Whether the human is modeled as an observer (ref.
5, 6, 7), a controller (ref. 6, 8, 14, 21), a supervisor (ref. 9, 13), or a problem solver in fault
diagnosis domain (ref. 12, 13, 17), the most important goal is to predict the human performance
and behavior in a human-machine interaction system (ref. 12, 15, 20).
The application of control model in the human pilot training simulation have been
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promising and accepted as the conventional approach to modeling the pilot handling quality
fidelity (ref. 6, 11). There is one important drawback in the current control models for flight
simulators. That is, the performance of the model is based solely on the knowledge of the plant
response before control is applied. In this sense, the human response error is a simplistic
assumption of a Gaussian wide noise with zero mean and variance which depends on the plant
dynamics.
In this paper consideration is given to human response error (HRE) models which are
additive components of both the model representation error and the experiment error
respectively. The HRE models are conceptualized with generality in mind thereby allowing the
simulationist the flexibility to experiment on a variety of flight handling quality (FHQ) tests.
Methods for solving the HRE problem are suggested.
Symbols
aJ_(i=o,i;je {m, o})
m
B_
C_
D,
d
E(.)
e
%
eo
%
F
HRE
i
J'o
J
K
L.(.)
M
m
Nm(" )
characteristic gain function of the term in second-order error dynamic
equation
control vector for simulated model
control vector for reference model
coefficient matrix for the reference system output
coefficient of error matrix for the reference system output
cardinality index
expected value operator
additive human response error term
simulation model error
observed model error
expected latent error of expected input and conditional control
cost functional of least-square equation for eo
Human Response Error
index (i --, o, 1)
HRE cost functional
index operator, j c {m, o}
scaling factor
a function describing unknown dynamic input response
expected value of v 2
index for "model"
known part of system dynamics for the simulation model
J
iL_
z ;
[]
i i
i - "
n
o
P
U.
U,
V
R
r
X_
X,
Y,
z.o
_(_)
_(_)
_(_)
¢_(j_)
II,
cardinality index
index of observation in reference model
covariance matrix of error estimate
control vector for simulation model reference control vector
control vector for reference model
error term
real number in Euclidian space
reference model index
simulation model state vector
reference model state vector
output vector from reference model
response disparity distribution function
simulation error term described by neurodynamic function of the operator
auto correlation function of x(t)
cross correlation function of x and y
power-spectral density of (t)
cross-spectral density of x(t) and y(t)
THE HRE MODELING PROBLEM
A. The optimal control model for HRE problem (OCM/HRE)
The OCM/HRE system of interest are derived by the following dynamic equations,
/(M = N_(x,t) + L_(x,u,t) + _ (t) ......... (i)
.=2
where X c R n is plant state vector whose components may represent aircraft dynamics such as
velocity, flight path angle, and altitude; U e R* is a control vector whose components may
represent flap deflection, pitch roll angle, and elevator deflection; Nm(x,t) is a known part of the
system dynamics in the model before control is applied (i.e., the initial system state);
L,,(x, u, t) represent (the unknown response plant dynamics when the control vector u is applied,
and _7(t) is e R ° is an unknown disturbance vector or the neuromotor noise of he human pilot.
The time variable t represents time. Equation (1) represents the OCM/HRE model.
B. The classical OCM
Following the classical optimal control model (OCM); see, e.g; Ref 5. Let us define a
linear quadratic time invariant reference model which generates a desired trajectory (see ref. 2)
= A r X r + B r U r ........ (2)
and the measurement
Yr = Cr Xr + Dr Ur .............................. (3)
u
m
! _==:
_U
l
!b
_ Jj
_L
is observed in the reference model r, where X, is the reference plant state vector, U_ is the
referenc6 pilot control vector, Y, is the observed system output from the reference model (i.e;
output vector utilized by the pilot in performing the control task). A_ eR '_ is a constant stable
system matrix, B_ e R _'d is a constant control vector and C_ E R p'_ and D, e R_Xd; Y_ e R v. Note
that C_ is a known matrix;
Let v = Dr Ur................................ (4)
with E(v) = o ................................ (5)
E(vv r) = M .................................. (6)
where M is a known p x p positive matrix.
C. The HRE Model
We are interested in modeling the response errors in the system. Starting from the
reference model; let us suppose that we had an estimate of the state before the simulation
(measurements)'are made; which we will call X,, where
[(x,.-i) 7] -- ,.r........................... (7)
where J is a known n x n positive matrix. Observing equation (3) shows that Y_ is a weighted-
least-square of the estimate vector X. The usual criterion (ref. 2, 7) is a minimization of a
quadratic form
Um
r
= =
z=
= i/2[(x,-2,)_.J-'(x,-2r)+(Yr-C,X,)TM-'(Z,-CrX,)].................(S)
To determine Xr, consider the differentialof equation (8):
c c: ............. I
In order that dJ = 0 for arbitrary dxa', the coefficient of dx z, in equation (9) must vanish:
(J-'+ c_,M-_Cr)X_= J-_2, + c_,_-_Y,)= (J-_+ c[ M-_c,)Xr+ c_ M-_(Y,-c,2r)
=
_,= _, + P cr,M-'(y,-crx,)....................(1o)
Where p-1 = j-l + Crr M-l Cr ................ (ii)
U
m
P is the covariance matrix of the error in the estimate Xr, that we have
U p = _ [(L-x,)(_-x,)7]........................(11b)
Theorem I: The observation error estimate e o = Xr-X r (see ref.2).
w
r_
w
L
L
b
Proof"
By adding and subtracting X_ in the eo term we have
e ° = Xr-X" + _- . ........................... (12a)
= 2,-x, + Pc_ M-' [m,ur-c,(2,-x,_] ............... (12bl
Since X,-X r and D r U, are independent, it follows equation (12b) that
E(e o ero) = (I-KCr) J(I-KCr) r + KMK r............ (13)
and I is a unit matrix.
m_
where K PC r M -I (14)
Premultiplying equation (11) by P and postmultiplying by J, we have
w J = P + PCrr M-* C r J ........... , ............... (15)
iw or P + (I - KCr) J ............................ (16)
w
By using equation (16) in equation (13):
i
UB
E(eoero) = P - P CrrKr+KMK r = P - PCrr M -I C r P+PCrr M -I C r P = P .... (17)
Thus, we have establisheda model for observation error, e• in equation (12b and their
computing propertiesin equations (13)-(17).
We are now interested in establishing the existence of model error era. To do this, we
can introduce the command vector U(t) into equation (1). By rewriting equation (1) with the
BmUm(t) component we have:
/(m = Nm(x,t) + Lm(x,u,t)-B_U.(t) + _(t) + B_U_(t) ..... (18)
 fro = Nm(x,t) + Zm(x,u,t) ÷ B=U_(t) + _(t) ..... (19)
where the term Z,.(x,u,t) is defined by
Zm(x,u,t ) = L.(x,u,t) - B=U=(t) ...... _ ....... (20)
Bm E R _'a is a known constant matrix of rank d selected from the experimental model. Next, we
define a model error e= to be the difference between the plant state vector and the reference
vector,
L
E
,,elm,
=:w
z
r .
U
e m = X_ - X r.................................. (21)
Therefore the total human response error (I-IRE) comprises of the model error and reference
error vectors respectively. That is
IIR____EE= e m + e o................................ (22)
w
i =-4
i R=:_
w
U
E
J
=
D. Properties of HRE
There various properties of HRE that need to be investigated experimentally.
Case 1: If the model state vector X_ is absent, then % = 0 thus,
HRE = eo which is the classical method of state estimation.
the properties discussed under section C above.
Thus HRE has all
Case 2: if the reference state vector X_ is absent, then X_ describes the synthetic
simulation model whose validity is by experimental observation only. In this
case HRE = %. However, there is an error or experimental bias introduced by the
difference between unknown (latent) response L,,(x,u,t) and the input control
BmUm(t) as defined by Z_(x,u,t) in equation (20). Let _m define this error such
that
am : e[(zm(x,u,t)]...........................(23)
Then, HRE = em + _m ......................... (24)
Case 3: If e,, + eo = 0, then, we say that the simulation model described by X._ has a high
fidelity. This is never attained in reality.
!
Case 4: The order of the system.
From equation (21): % = X_ - X,, and the time rate of change of the error %, is
_m = X,,, - Xr ................................. (25)
X_(to), Xm(to) given. Similarly;
= em + _o ................................ (26)
is the time rate of change of HRE. Clearly, HRE can be modeled as a second-order system with
= =
L the minimization criterion defined by
Jo = 1/2 Ilell2................................ (27)
w
b
z
w
U
iml
w
U
l m
4U
w
where e = HRE.
Case 5: HRE is a second-order error dynamic system. This property follows directly from case
4 above. Since eo and em are independent, we can define the error dynamic equations
by:
m
_ = a_ em + ao em : 0 ........................ (28)
0 0
i_. + ai eo = ao eo = 0 .......................... (29)
Where a_ (i = o, 1; j E(m,o}) is the characteristic gain vector associated with each
system of equation.
Case 6: I__(x,u,t) can be determined experimentally as follows: using the second-order gradient
method, we guess a control parameter u(t=0) and determine X(t=0) from N.,(x(t=0),
u(t=0) =0, and then Lm(X(0), u(0)). We can then determine the first and second
derivatives of I__(x,u,t) with respect to u. Thus, we can approximate the (L_ vs u)-
curve by a quadratic curve:
[OL.] [OL'](U-Ua) 2 .... (30): Lm(Xo,Uo,O) ÷ (U Uo)÷
Case 7: Time frequency property of I-IRE
Previous human response models in the aircraft simulation domain have been described
by Taylor (ref. 18, 19) in terms of time frequency and power spectrum density functions. In
a particular case in which em = eo, the autocorrelation function describing HRE is found by
172
1 _z2fo(t)
= -- fo(t + r) dt ................ (31)
T
where fo(t) is fitted distribution describing the observation error, eo. In this case the power
LL_
L
D
spectrum is
1
_,_ (o_) = _T
T/2
__2_(T)eJ"'.'dt .................... (32)
If eo ;_ e.. during the period of observation T;
172
_mo(7) = _ fo(t)
T _ 2
fm(t + T)dt ............... (33)
defines the crosscorrelation function of fo(t) and fro(t); and
b-Jm:a
_:._A
L
w
where
m
q_mo(J _) = Fro(n)
T_
F(n)F°(n) T
Fo(n ) ........................ (34)
Exp(-j n _m)_o(r)dt ........... (35)
• w
J
r_
CONCLUDING REMARKS
The discussion in this paper is geared towards modeling human response errors in a
synthetic simulation domain in which flight handling qualities are the main tasks. The following
conceptual contributions are prevalent to this paper.
1. We model the HRE as a component of two types of errors: the model error constructed
around the simulation domain; and the reference error which is the theoretical state space
model commonly used. In addition, we introduce the concept of experimental latent
error which is the disparity between the theoretical input vector and the human input
response at a given state space.
|u
w
E
w
E
D
D
v
.
3.
We discuss the various properties of HRE and their implications.
We formulate a cost minimization model of a simulation environment interms of the
HRE function.
4. We demonstrate how the HRE model can be used in both the time and frequency
domains.
It should be noted here that the discussions in this paper needs further theoretical proofs
as well as actual experimentation to warrant their applications in flight handling quality
characterization.
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Modeling human response errors in synthetic flight simulator domain

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