A LUMPED PARAMETER MODEL OF TURBOPROP AIRCRAFT ...

AIAA 2001-4374

A LUMPED PARAMETER MODEL OF TURBOPROP AIRCRAFT OPERATING ON GRAVEL RUNWAYS Shane D. Pinder*, Trever G. Crowe†, Peter N. Nikiforuk‡ University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9

ABSTRACT

TAKEOFF PERFORMANCE MONITORING

The purpose of an aircraft Takeoff Performance Monitoring System (TOPMS) is to provide to the pilot information pertaining to the level of safety with which a takeoff is proceeding. The authors have developed a theoretical dynamic model to investigate the feasibility of using an observer system during the roll and takeoff phases of aircraft operation to provide to the pilot the information that is needed to manoeuver safely. This model was validated using a prototype device installed in a 19-passenger commercial turboprop aircraft.

The "critical engine failure recognition speed" (V1) is defined as the speed above which takeoff could safely continue if the most critical engine failed.2 In the event that the aircraft has reached a point such that the remaining runway is equal to the required safe stopping distance and has not yet reached a speed of V1, the takeoff must be aborted. It is standard practice for pilots to reduce takeoff thrust to a level that would allow acceleration to V1 followed by deceleration to rest such that the entire length of the runway would be used. This practice prolongs the service life of the aircraft engines.

Unlike previous work in this field, this investigation focussed on various factors that are unique to the far-northern environment. Further, the Global Positioning System(GPS) was proposed as the sole source of kinematic information. This provided the possibility that a TOPMS could be devised that would require no additional ground-based installation. The results of a practical investigation1 that was conducted to validate the theoretical model and signal processing technique appears in AIAA 2001-4170. The investigation showed that it was possible to predict the displacement of an aircraft to within 15 [m], the length of the test aircraft, in sufficient time to aid the pilot in decision-making.

* †

‡

Doctoral Student, Department of Mechanical Engineering Associate Professor, Department of Agricultural and Bioresource Engineering Dean Emeritus, College of Engineering

Copyright © 2001 by Shane D. Pinder. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

One of the most common aviation accident events continues to be runway overrun during takeoff or landing.3 In the case of takeoff runway overrun, the problem is often associated with engine power loss. This problem is further aggravated in inclement weather where runway surfaces are contaminated by water or ice. Pilots of multi-engine aircraft must evaluate a complex set of variables in situations involving varying winds, limited control of ground traction, and necessary application of reverse thrust. Some parameters of importance in a TOPMS include wheel bearing viscous friction, aircraft drag, runway slope, engine thrust, aircraft velocity, position relative to the end of the runway, and frictional coefficient between the aircraft tires and the runway. Previous work in the design of takeoff performance monitors4,5 accounted for such variables individually, which led to a large uncertainty in the predicted displacement. The most significant uncertainty pertained to the measurement of runway friction coefficient. In far-northern regions, however, the majority of airports consist of gravel runways. On gravel runways, the principal means of reducing speed is through the

1 American Institute of Aeronautics and Astronautics

application of reverse thrust. While braking is available to aircraft operating in these circumstances, it is used sparingly and only when absolutely necessary. As a result, a measurement of runway frictional coefficient may not be needed. This serves to improve the likelihood that a monitor specifically suited to gravel runways can be successfully developed. The relative importance of the remaining parameters remains to be determined. It is expected that some parameters may be negligible and the influence of others may be combined.

D = D 3 v a2 + 2 D 3 v a v + D 3 v 2 ,

(3)

D = D1 + D 2 v + D 3v 2 ,

(4)

or,

where:

D n are constant parameters for a given aircraft geometry,

v a is the component of wind in the direction The objectives of this theoretical examination were to develop a dynamic model of a turboprop aircraft in contact with the ground, to conduct an uncertainty analysis of the predictive aspect of the model, and to devise a signal processing technique that would permit real-time determination of model parameters.

of the runway, and;

v is the speed of the aircraft relative to the ground. Similarly, thrust,

T = T 0 + T 3 v /2a ,

LUMPED PARAMETER MODEL Assuming that all necessary quantities can be measured, a takeoff performance monitor would require a method to project how the speed, position, and acceleration of an aircraft might change in the future based on measurements taken in the past. This necessitates the availability of a mathematical model describing how these parameters vary with respect to one another.

where:

T 0 is a parameter representing the throttle setting, and;

T 3 is a parameter that accounts for increased thrust at higher engine inlet air pressures. As in the derivation for drag,

T = T0 + T 3 ( v a + v ) 2 .

(6)

T = T 0 + T 3 v a2 + 2 T 3 v a v + T 3 v 2 ,

(7)

T = T1 + T2 v + T3 v 2 ,

(8)

The force of drag on an aircraft,

D = D 3 v /2a , where:

(1)

(5)

Expanding,

D 3 is a constant parameter for a given aircraft geometry, and; or,

v / a is the speed of the aircraft relative to the air. Applying the convention that a headwind is positive and aircraft speed is positive forward,

D = D 3 (v a +v ) 2 . Expanding,

where:

T n are constant parameters for a given throttle setting.

(2)

Simple relationships exist for viscous friction,

F = F2 v ,

2 American Institute of Aeronautics and Astronautics

(9)

and for the component of weight in the direction of motion,

W = W1 , where:

ds =

v dv . P1 + P2 v + P3 v 2

(16)

(10)

F 2 and W 1 are constants provided that the

If the instantaneous position and speed are known, the displacement at a reference speed can be determined through integration. The displacement,

runway slope is constant. v2

s 2 − s1 = ∫

Grouping similar parameters and applying Newton’s Second Law,

a = P1 + P2 v + P3 v 2 , where:

(11)

v1

where:

Pn are parameters representing the net force

v dv , P1 + P2 v + P3 v 2

s 1 is the instantaneous position; s 2 is the predicted position at the reference

per unit mass acting on the aircraft, and;

speed;

a is the acceleration of the aircraft in the direction of motion.

v 1 is the instantaneous speed, and; v 2 is the reference speed.

To use this model for the prediction of later displacement requires an equation describing the displacement as a function of speed. From fundamental kinematics,

ds , and; v = dt a =

dv , dt

The solution to this integral,

ds dv . = v a

(13) where:

is found. Incorporating the model,

v dv , a

c=

P2 + P22 − 4 P1 P3

d =

2 P3

,

(18)

v1

, and;

P2 − P22 − 4 P1 P3 2 P3

v2

,

(19)

(20)

(14)

Rearranging, an equation describing the differential displacement,

ds =

c ln v + c − d ln v + d s 2 − s1 = P3 ( c − d )

(12)

where s and t represent position and time, respectively. This can be solved to describe the differential time,

dt =

(17)

can be used to conduct an uncertainty analysis for the measured quantities in the model. The constants described by equations (19) and (20) are used simply to make the manipulation of equation (18) more manageable.

(15) UNCERTAINTY ANALYSIS An assessment of the sensitivity of the theoretical model to uncertainties in the measured quantities required an uncertainty analysis. In the prototype

3 American Institute of Aeronautics and Astronautics

∂ P1 =1 , ∂a

device and filter, Parameter 1, P1 , is calculated as a function of the measured acceleration, the filtered values of P2 and P3 , and the measured speed. Thus,

P1 = a − P2 v − P3 v 2 ,

(21)

and the partial derivatives of P1 ,

∂ P1 = −v , ∂ P2 ∂ P1 = −v 2 , and ∂ P3

(22)

(23)

 v2 +c c ( v1 − v 2 ) +  ln ∂ ∆s ∂ c  v 1 + c ( v 1 + c )( v 2 + c ) = ∂a ∂ P1  P3 ( c − d )  

(24)

describe how Parameter 1 varies with the measured quantities. While speed is also a measured quantity, the uncertainty of the speed measurement is not included in this analysis. In the most conservative case, speed uncertainty is negligible with respect to the uncertainty of the other measured quantities.6 This is due primarily to the fact that speed is a quantity that is measured directly with a GPS receiver, while the other measurements considered are derived quantities. The partial derivatives of displacement with respect to the measured quantities,

   − ∂d  ∂P 1  

 v2 +d d (v1 −v 2 ) +  ln ( v 1 + d )( v 2 + d )  v1 + d  P3 ( c − d )  

     

 v2 +c v +d  − d ln 2  c ln  v1 + c v1 +d   ∂ c ∂d   + −   P 3(c −d ) 2  ∂ P1 ∂ P1      

 v2 +c c(v1 −v 2 )  +  ln  ∂ ∆s ∂ c  v 1 + c ( v 1 + c )( v 2 + c )  ∂ d = −  ∂P ∂ P2 ∂ P2  P3 ( c − d ) 2      v +c v +d  − d ln 2  c ln 2  v1 + c v1 +d   ∂c ∂d   + −   P 3(c −d ) 2  ∂ P2 ∂ P2      

 v2 +d d (v1 −v 2 ) +  ln ( v 1 + d )( v 2 + d )  v1 + d  P3 ( c − d )  

4 American Institute of Aeronautics and Astronautics

     

and;

(25)

(26)

 v2 +c  v2 +d c(v1 −v 2 )  d (v1 −v 2 ) + +  ln   ln ∂ ∆s ∂ c  v 1 + c ( v 1 + c )( v 2 + c )  ∂ d  v 1 + d ( v 1 + d )( v 2 + d ) = −  ∂ P3  ∂ P3 ∂ P3  P3 ( c − d ) P3 ( c − d )        v2 +c v +d   v2 +c v +d  − d ln 2 − d ln 2  c ln   c ln  v1 + c v1 +d   v1 + c v1 + d   ∂c ∂d   + − +     P 3(c −d ) 2 P32 ( c − d )  ∂ P3 ∂ P3          

     

(27)

together with partial derivatives,

∂c = − P22 − 4 P1 P3 ∂ P1

(

∂c = ∂ P2

∂c = ∂ P3

(

1 + P22 − 4 P1 P3

−

1 2

,

(28)

  ∂ P   P2 − 2 P3  1    ∂ P2    , 2 P3

)

−

1 2

(

− P2 − P22 − 4 P1 P3 − 2 P3 P22 − 4 P1 P3

)

−

1 2

 ∂P   P1 + P3 1  ∂ P3  

2 P32

∂d = P22 − 4 P1 P3 ∂ P1

(

∂d = ∂ P2

∂d = ∂ P3

)

(

1 − P − 4 P1 P3 2 2

)

−

1 2

(30)

(31)

  ∂ P   P2 − 2 P3  1    ∂ P2    , and; 2 P3

)

−

1 2

(

)

−

1 2

 ∂P   P1 + P3 1  ∂ P3  

2 P32

An experimental investigation that was performed to

,

,

− P2 + P22 − 4 P1 P3 + 2 P3 P22 − 4 P1 P3

form the basis of the uncertainty analysis.

(29)

(32)

,

(33)

validate the theoretical model is described in Pinder et al.1 A prototype takeoff performance monitor was installed in a 19-passenger British Aerospace 3112

5 American Institute of Aeronautics and Astronautics

operated by an airline servicing far-northern Canadian airports. During a typical takeoff, the pilot typically made changes in control settings until the aircraft reached a speed of 30 [m/s]. After reaching this speed, the filter typically required two seconds to converge to model parameters. At an instantaneous speed of 35 [m/s], and based on conservative parameter values,

P1 = 3 .0 [m/s2],

= −6 .3 × 10 3 [s·m], and;

P3 = −0 .00014 [m-1], partial derivative values,

∂c = −2 .2 × 10 1 [s], ∂ P1

(34)

∂d = 2 .2 × 10 1 [s], ∂ P1

(35)

∂c = −1.2 × 10 3 [m], ∂ P2

(36)

∂d = −5 .9 × 10 3 [m], ∂ P2

(37)

∂c = −1.6 × 10 5 [m2/s], and; ∂ P3

(38)

∂d = −1.2 × 10 6 [m2/s], ∂ P3

(39)

are obtained. These quantities result in partial derivatives of displacement with respect to measured quantities,

∂ ( s 2 − s1 ) ∂c ∂d = 6 .3 − 0 .21 ∂a ∂ P1 ∂ P1 (40)

(41)

∂ ( s 2 − s1 ) ∂c ∂d = 6 .3 − 0 .21 − 170 ∂ P3 ∂ P3 ∂ P3 = −1.2 × 10 6 [m2].

P2 = −0 .020 [s-1], and

= −1.4 × 10 2 [s2],

∂ ( s 2 − s1 ) ∂c ∂d = 6 .3 − 0 .21 ∂ P2 ∂ P2 ∂ P2

(42)

From a previous study of the accuracy of a GPSderived measurement of acceleration,7 it is estimated that the filtered measurement of acceleration is accurate to within 0.05 [m/s2]. Parameter 3 is a characteristic of the aircraft engines and the aerodynamic drag coefficient of the aircraft. Therefore, Parameter 3 changes very slowly throughout the service life of the aircraft engines. It is estimated that the uncertainty in this measurement is far less than 1.0%. This estimate was based purely on the theoretical foundation of Parameter 3 as a characteristic of the aircraft. In a functional takeoff performance monitor, the device would project the displacement that would occur between the instantaneous speed and a decision speed, V1. This displacement would be added to the projected displacement that would occur when decelerating from V1 to rest. The total displacement would be compared to the instantaneous measurement of remaining runway length, and the difference would be displayed to the pilot as a margin of safety. There are several factors that could affect the actual margin of safety, most notably the reaction time of the pilot. Assuming that pilot would compensate for reaction time by selecting a comfortable margin of safety, the required accuracy of the margin measurement must be selected. For larger aircraft, a larger margin would be selected. It is therefore appropriate to establish required accuracy based on the length of the aircraft. In the most conservative case, a takeoff rejection is initiated at V1 and the pilot has selected a margin of safety that would be completely consumed by reaction time. In this instance, the remaining runway would be completely consumed during the deceleration phase. It is therefore desirable that the runway remaining when the aircraft has come to rest is no less than one aircraft

6 American Institute of Aeronautics and Astronautics

length. The authors have therefore selected the length of the aircraft, measured from nose to tail, as the required accuracy in the measurement of projected displacement. The aircraft used in this experimental investigation measured 15 metres from nose to tail.

Each state may also be assigned some random variability to uncouple neighbouring states. For instance, it would not be uncommon to describe the dynamics of an aircraft during its takeoff roll based on its position, s , speed, v , acceleration, a , and jerk, j,

If the uncertainty in the projected displacement,

∂ ( s 2 − s1 ) ∆s = ∆a ∂a , (43) ∂ ( s 2 − s1 ) ∂ ( s 2 − s1 ) + ∆P2 + ∆P3 ∂ P2 ∂ P3

s  1 v     = 0 a  0     j k 0 where:

is to be kept less than 15 [m],

1.4 × 10 2 (5.0 × 10 −2 ) + 6 .3 × 10 3 ( ∆P2 )

+1.2 × 10 6 ( 1.4 × 10 −6 ) < 15 [m],

(44)

then ∆P2 < 0 .0010 [s-1]. This corresponds to a 5% allowable uncertainty in Parameter 2. In reality, the acceleration uncertainty is likely far less than that estimated, due to the signal processing technique that will be described shortly. The contribution of the uncertainty in Parameter 3 is about 10% of the total error. Regardless, a large uncertainty in Parameter 2 is permissible. This result is particularly useful. While Parameter 2 depends on such variable factors as wind speed and direction, a large uncertainty is acceptable. This is due to the manner in which the filter calculates Parameter 1 as a function of Parameters 2 and 3 and measured acceleration.

SIGNAL PROCESSING Customarily, the states in a Kalman Filter are time derivatives of one another. This stems from the rigidity of the continuous Kalman Filter, which requires that all states be related to one another through differentiation in a homogeneous domain. The discrete Kalman Filter is not limited in this way. Based on the dynamics pertaining to the particular application, the designer typically chooses a high derivative to identify as a random process. The lower states are then dependent on the random variable.

dt

0

1

dt

0

1

0

0

0  s  0  0     0 v  +   , (45) 0  d t a      0  j  k −1 q j 

d t is the difference in time between the measurements subscripted k − 1 and k , and;

q j is a zero-mean random variable. Such a filter functions best when jerk most closely resembles a zero-mean random process, though this is usually an approximation of reality. For small time steps, it may be considered a reasonable approximation. In the model,

a = P1 + P2 v + P3 v 2 ,

(46)

jerk can be found through differentiation,

j=

da = P2 a + 2 P3 va , dt

(47)

or,

(

)

j = P1 P2 + P22 + 2 P1 P3 v

+3 P2 P3 v 2 + 2 P32 v 3

,

(48)

and is clearly not a zero-mean process. The higher derivatives are also non-zero. On the other hand, velocity derivatives of the model,

7 American Institute of Aeronautics and Astronautics

da = P2 + 2 P3 v , dv

(49)

d 2a = 2 P3 , and; dv 2

(50)

d 3a =0 , dv 3

GPS-derived data to instantaneously and continuously determine three parameters that can be used to adequately predict the distance required to reach any particular speed. These three parameters take into account the majority of factors affecting the motion of the aircraft. The use of a lumped parameter model in conjunction with a carefully designed Kalman Filter has made it possible to design a prototype takeoff performance monitor that is less susceptible to the uncertainties present in the factors affecting an aircraft during takeoff. While it remains to be determined whether this level of accuracy will be attainable on other similar aircraft, it appears likely that a takeoff performance monitor designed for turboprop aircraft on gravel runways will be capable of predicting

(51)

provide an alternative method of observer construction. The third velocity derivative of acceleration is a zeromean process. Without approximation, this can be considered a random process. Based on this knowledge, a novel observer,

 s   v     a  =  da   dv   d 2 a dv 2  k

1 0  0  0  0

dt

0

0

1

dt

0

0

1

dv

0

0

1

0

0

0

was constructed. This was a model for a Kalman Filter that was capable of an optimal estimation despite reference to a non-linear model. Note that the matrix relating state variables from one step to the next, known as the state transition matrix, contained both differential time and differential speed. As a result, the state transition matrix varied from step to step. As a result of this signal processing technique, the predictions of the displacement of the test aircraft were accurate to within 15 [m] within a few seconds of the pilot finalizing control settings. This result held for 176 takeoffs recorded under varying weather conditions.

0  s  0   v  0  a  +   d v   da dv  2 1   d a dv 2  k −1

0 0   0,   q 4   q 5 

(52)

displacement with a level of uncertainty approximately equal to the length of the aircraft.

ACKNOWLEDGEMENTS The management and maintenance staff of Transwest Air generously provided crucial advice as well as space onboard a Jetstream 31 aircraft for the installation of the prototype device. Neil Larsen and Jonathan Tonn deserve special thanks. Daniel Aspel of TR Labs, Saskatoon, provided collaborative technical support. Their efforts are sincerely appreciated.

REFERENCES CONCLUSIONS 1. A theoretical dynamic model of an aircraft in contact with the ground has been devised. While it is widely understood that many factors influence the motion of an aircraft in contact with the ground, the proposed model suggests that the sufficient information exists in

Pinder, S.D., Crowe, T.G., and Nikiforuk, P.N., “A Practical Investigation of a Takeoff Performance Monitor for Turboprop Aircraft,” Proceedings of the AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August 6-9,

8 American Institute of Aeronautics and Astronautics

2001. 2.

Wagenmakers, J., Aircraft Performance Engineering, Prentice Hall, 1991.

3.

Transportation Safety Board of Canada, “TSB Statistical Summary, Aviation Occurrences, 1996," Minister of Public Works and Government Services Canada, 1997.

4.

Srivatsan, R., “Design of a Takeoff Performance Monitoring System,” Ph.D. Thesis, University of Kansas, 1985.

5.

Middleton, D.B., Srivatsan, R., and Person, L.H., Jr., “Flight Test of Takeoff Performance Monitoring System,” NASA TP-3403, 1994.

6.

Pinder, S.D., “Aircraft Takeoff Performance Monitoring in Far-Northern Regions,” Ph.D. Thesis, University of Saskatchewan, 2001.

7.

Pinder, S.D., Crowe, T.G., and Nikiforuk, P.N., “Application of the Global Positioning System in Determination of Vehicular Acceleration,” Proceedings of the AIAA International Communication Satellite Systems Conference, Oakland, California, April 10-14, 2000, pp. 831-837.

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