Original paper
Soft Computing 6 (2002) 441 ± 448 Ó Springer-Verlag 2002 DOI 10.1007/s00500-001-0159-1
Application of generalised neural network for aircraft landing control system D. K. Chaturvedi, R. Chauhan, P. K. Kalra
Abstract It is observed that landing performance is the most typical phase of an aircraft performance. During landing operation the stability and controllability are the major considerations. To achieve a safe landing, an aircraft has to be controlled in such a way that its wheels touch the ground comfortably and gently within the paved surface of the runway. The conventional control theory found very successful in solving well de®ned problems, which are described precisely with de®nite and clearly mentioned boundaries. In real life systems the boundaries can't be de®ned clearly and conventional controller does not give satisfactory results. Whenever, an aircraft deviates from its glide path (gliding angle) during landing operation, it will affect the landing ®eld, landing area as well as touch down point on the runway. To control correct gliding angle (glide path) of an aircraft while landing, various traditional controllers like PID controller or state space controller as well as maneuvering of pilots are used, but due to the presence of non-linearities of actuators and pilots these controllers do not give satisfactory results. Since arti®cial neural network can be used as an intelligent control technique and are able to control the correct gliding angle i.e. correct gliding path of an aircraft while landing through learning which can easily accommodate the aforesaid non-linearities. The existing neural network has various drawbacks such as large training time, large number of neurons and hidden layers required to deal with complex problems. To overcome these drawbacks and develop a non-linear controller for aircraft landing system a generalized neural network has been developed.
T thrust in pounds W total weight in pounds Cl lift coef®cient Cd drag coef®cient v indicated aircraft velocity, ft/sec r air density in slugs per cubic ft s wing area in square ft q dynamic pressure in per sq. ft Mcg pitching moment in pound ft Cmcg pitching moment coef®cient c wing mean aerodynamic chord length in ft u vertical velocity, ft/sec A aspect ratio of aircraft Cdf parasite drag coef®cient e aircraft ef®ciency factor
1 Introduction In order to achieve a safe landing, an aircraft has to be so controlled in such a way that its wheels touch the ground comfortably within the paved surface of the runway. While designing automatic ¯ight control system, it is observed that landing performance is the most typical phase of aircraft performance. During landing operation the stability and controllability are major considerations so that aircraft should touch the ground gently. The basic limitation of conventional control theory is the need to know the precise mathematical model of the system to be controlled. This information is seldom available. The effect of disturbances and un-modeled dynamics on the performance of the system must also Keywords Generalized Neural Network, Aircraft, Landing be taken into account for real-life systems. Practically, one rarely has precise knowledge of the Control System, Backpropagation system model. Furthermore, the system model may vary with time, e.g., the dynamic equations of an aircraft near List of symbols sea-level are very different from those of the same aircraft L lift in pounds at high altitudes. In such cases, to maintain the controller D drag in pounds performance, it is necessary to use an adaptive controller, so that it could adopt the variations in system parameters and operating conditions. D. K. Chaturvedi (&), R. Chauhan Dayalbagh Educational Institute, Dayalbagh, Agra (UP), India 282005 e-mail:
[email protected] Fax: 91-562-281226
P. K. Kalra Department of Electrical Engineering, Indian Institute of Technology, Kanpur (UP), India 208016 e-mail:
[email protected]
2 Aircraft landing system The aircraft performance characteristics such as maximum speed, rate of climb, time to climb, range and take-off are all predicted from estimates of the variation of the lift, drag and thrust forces as functions of angle of attack, altitude and throttle setting respectively. The forces acting on aircraft during ¯ying are lift (L), drag (D), thrust (T)
441
442
and weight (W) of the aircraft. The ¯ight path of the aircraft can be controlled within the limitations of its aerodynamic characteristics and structural strength, through control over the equilibrium angle of attack (a), angle of side slip (b), angle of bank (/) and the output of the power plant. The APPROACH AND LANDING exercise deals with landing of the aircraft from the turn on to downwind position to the completion of the landing run. A good landing follows a steady approach; hence it is important that the circuit be standardized so that the ®nal approach is such as to facilitate a good landing. The ®nal landing approach begins when the ¯ight path is aligned with the runway in preparation for straight ahead descent and landing and ends when aircraft contacts the landing surface. Thus ®nal approach may be considered to have ®ve distinct phases listed below and also shown in Fig. 1 [1, 7, 14, 15]. i. ii. iii. iv. v.
Level approach Approach descent Round out Float Touchdown
2. Engine assisted landing approach, in which the descent should be started slightly earlier than for the glide approach. The approach path is adjusted with the throttle, speed being maintained by the elevators. This approach is recommended for all modern aircraft due to following advantages: i.
Permits a ¯atter approach as the glide path of some aircraft is uncomfortably steep. ii. The approach path can be adjusted as required. iii. The stalling speed is low; the approach can be carried at a speed, thus reducing the landing run. iv. Use of engine improves elevator and rudder control of propeller-driven aircraft. v. Landing is safer, quicker and more accurate.
3 Drawbacks of existing landing control system i. The ILS is a pilot interpreted system, pilot is directly depending on the signals transmitted by ILS, he has to blindly follow path shown. Even slight mistake can lead to disastrous landing. ii. The ILS serviceability checks and calibrated are to be carried out by the ¯ying aircraft only and ground technicians can not ensure its serviceability merely on ground. iii. GCA system requires a high-resolution radar and high talented ground controller to guide pilot [6, 7, 10, 13]. iv. In GCA system radar picks up the aircraft position and display it on the screen and ground controller in turn guide the pilot, but little parallax error in this may take away from the actual center line of the runway [2, 3, 17, 18]. v. Atmospheric conditions are drastically changing and conventional controller is unable to cope up these conditions. vi. The inherent non-linearities in system components and cognition level of human being have led researchers to consider neural network techniques to build a non-linear ANN controller a with high ef®ciency of performance.
The ®nal approach involves a descent from the circuit height (1000 ft) to the landing check height. The pilot should plan the approach so that the pilot gets suf®cient time to judge the descent and position himself to carry out the straight ®nal approach from at least 400 ft. This gives suf®cient time to retrim the aircraft and concentrate on the ®nal approach. The ®nal turn-in should be adjusted so as to line up with the landing path (glide path) and s-turn should be avoided on the ®nal approach. The adjustment of the rate of descent will depend on the pilot's judgement and type of approach. There are two types of landing approaches [19, 20, 22, 23]: 1. Glide landing approach in which the pilot must judge the point from where to start the approach, taking the prevailing wind into consideration. The steepness of the descent can be adjusted by intelligent use of ¯aps. Full ¯aps being lowered only when positive of making the intended touchdown point safely. On no account should he The existing simple neural networks have numerous allow his speed to drop below that recommended, in an de®ciencies as stated below: effort to stretch his glide. This approach is done under the in¯uence of the force of gravity and without the use of the i. The number of neurons required in hidden layers is engine. large for complex function approximation. ii. The number of hidden layers required for complicated functions may be greater than three. Though it has been reported that a network with only three layers can approximate any functional relation [8, 9, 11], it is found that the training time required is very large, which can be computationally very expensive. iii. The fault tolerant capabilities of the existing neural networks are very limited. iv. Existing neural networks require a large number of unknowns to be determined for complex function approximation. This increases the requirement of the minimum number of input±output pairs. v. The training time required is dependent to a large extent on the type of input±output mapping chosen Fig. 1. Final approach extends from line upto touch down (like DX Y, X Y, etc.) [5].
vi. The training time is also dependent on the normalization ranges used for the input and output data [16]. In the present work an automatic controller for aircraft landing system has been developed using Neural Networks to increase the accuracy and ef®ciency of the control system at the time of landing [21]. The aim of this controller is to restore the glide path, if it deviating from the Fig. 2. Generalized neuron model desired path in the shortest time. The following ideas were formulated to overcome the 3.2 de®ciencies of the existing neural networks: i.
In Fuzzy Logic Systems, the compensatory aggregation operators perform better than the individual min and max operators. This is because an appropriate weighted combination of these two is used in the compensatory operator [4]. In a similar way the Sum and Product neurons could be combined to develop a generalized neuron, which may be expected to perform better than the individual neurons. ii. Development of a generalized neuron model, which exhibits characteristics of all types of existing neurons. Such neuron models will be general enough to accommodate properties of simple as well as higher order neurons. iii. The neuron model should be ¯exible enough to accommodate variations in mappings and hence drastically reduce the total number of neurons in the neural network. iv. The generalized neuron model should reduce the requirement of the total number of hidden layers. This would result in a neural network model, which is computationally ef®cient. v. The smaller number of neurons needed to model any function in the generalized neural network should reduce the number of free parameters associated with the neurons, thus reducing the amount of data required for training.
Standard BKP learning algorithm of generalized neuron model The following step are involved in the training of Generalized Neural Network Step 1 The output of R-part of generalized neuron is OR f1
RwRi xi xoR Step 2 The output of p-part of generalized neuron is
Op f2
pWpi xi xop as
Step 3 The output of Generalized Neuron can be written
Oi WR OR
1
WR Op
Step 4 After calculating the output of Generalized Neuron in the forward pass of feed forward back propagation neural networks, it is compared with the desired output to ®nd the error and then it is minimized to train the GNN model. Hence in this step the output of the GNN with a single ¯exible generalized neuron model is to be compared with the desired output to get error for ith set of input.
Error Ei
Yi
Oi
Then it is necessary to calculate the sum-squared error for convergence while training.
EP 0:5RE2i
The above ideas have been incorporated in the generA multiplication factor of 0.5 has been taken for simplialized neuron structure developed. This neuron model has fying the calculations. been used to develop a Generalized Neural Network Step 5 Reversed pass for modifying the connection (GNN), which is used in the development of a landing strength. control system. (a) Weight associated with the R and R-parts of the Generalized Neuron
3.1 Development of generalized neuron model In the generalized neuron model both summations as well as product have been taken as the aggregation functions and the output of these aggregation functions have been passed through the Sigmoidal and Gaussian functions respectively. The ®nal output of the neuron is a function of the two outputs OR , Op with the weights WR and
1 WR ) as shown in Fig. 2. Hence the output of the neuron becomes Oi WR OR
1
WR Op
WR
k WR
k where
1 DWR
DWR gdR
OR and
Op aWR
k
dR R
Yi
1
Oi
(b) Weight associated with the inputs of the R-parts of the Generalized Neuron
WRi
k WRi
k where
1 DWRi
The above mentioned neuron model is known as sumDWRi gdRj Xi aWRi
k mation type compensatory neuron model, since the output and of the Sigmoidal and Gaussian functions have been dRj Rdk WR OR
1 OR summed up.
1
443
444
An error signal is de®ned at the output of the dynamic (c) Weight associated with the input of the p-parts of the plant as the difference between the state vector of the plant Generalized Neuron and the desired state vector, back-propagation of the error Wpi
k Wpi
k 1 DWpi vector through plant state space equations is effected where by means of multiplication of the error vector with the DWpi gdpj Xi aWpi
k 1 Jaccobian matrix of the plant (a matrix containing the and derivatives of the state variables of the plant) [25]. The resulting signal is the then back propagated through dpj Rdk
1 WR Op
2
pWi Xi Xop neural network controller and the adaptive weights of the controller are adjusted according to the back-propagation 4 algorithm. Backpropagation through time algorithm Figure 3a essentially composed of a layered arrangeThe feed forward neural network studied in the previous ment of controller and plant equations blocks. The consection was a static structure in the sense that no time troller is a neural network. If the block containing the dependency existed between inputs and desired outputs. plant equations P, were replaced by neural network copy The situation is different when the network to be of P (it can be shown that any continuous nonlinear adapted is embedded in a dynamic structure. In our ap- function can be approximated to an arbitrary degree of plication, for example, a neural network is used to control precision by a two layer neural network [14]), the unrava dynamic plant. It has been shown above that the discrete eled system of Fig. 3a would become a giant layered neural state vector x
n bVLD of the system controlled by network with inputs x
0 and Cl, output x
N and desired robust controller eventually converged to a steady state output d
N. The back propagation algorithm could then value equal to xsteadystate , but this convergence was slow. be applied to train such a network. By doing so, the error The neural network controller that replaces the robust gradient de®ned at the output of the network are backcontroller should make the system converge to the same propagated through the neural network copy of Plant P steady state vector while limiting the duration and mag- and controller C blocks, from x
N back to x
0; hence the nitude of the transient. Such an operation cannot be per- name back-propagation through time. This approach was formed instantaneously. In addition, the value of the ®rst introduced by Nguyen and Widrow and was sucdesired control signal (i.e. the desired output of the neural cessfully applied to number of applications in the area of network) is not known a priori, only the desired steady nonlinear neural control [26]. state of the system is known. The simple static back In this paper we adopt a slightly different approach by propagation algorithm is not directly applicable; it must be which we avoid the introduction and training of a neural generalized to the present dynamic structure. The result of network copy of the plant equations. The basic idea is that this generalization is referred to as the back propagation instead of building a neural network copy or emulator of through time algorithm. plant P back-propagated error gradients through it, it is
Fig. 3. a Neural network controller and plant model for aircraft control unfolded in time. b Block diagram of landing control system using GNN
possible to directly back-propagate the error gradient through the plant equations. A feed-forward Generalized neuron model is shown in Fig. 2. The input xi are multiply by the adaptive weights wRi and Wpi , the output is obtained by passing the sum (R) and product (p) aggregation functions of weighted input through Sigmoidal ( f1 ) and Gaussain ( f2 ) functions respectively with weights WR and
1 WR .
assumptions have been taken for the development of the generalized neural network controller i. ii. a.
Oi WR f1
RwRi xi xoR
1 WR f2
pWpi xi xop
b.
Initially weights are chosen small random values, the weights adjusted after each presentation of new input patterns. The adaptation rule is given by:
iii.
DWpi
T
g d
e e=dW
a
iv.
The gliding angle should be 5±7 and angle of attack should be 2±4 . The landing performance can be divided into two main phases Transition from threshold to touch down including rounds out and ¯oat. Braked ground run. Here only transition from threshold to touch down is considered. The aircraft should start gliding after approaching the screen height with constant gliding angle. Air density is constant at all altitude, since speed taken is indicated air speed.
The block diagram of generalized neural network controller and aircraft model structure is shown in Fig. 3b. The controller should make the aircraft to follow the correct glide path. The aircraft variables and GNN controller parameters are given in the Table 1. The gliding angle is sensed at every time instant at every time instant and if there is any deviation in the gliding angle will be DWpi g di xi corrected by the GNN controller of aircraft. The inputs where di 2ek dxh =dxi and output of GNN controller are given in Table 2 and Kth term of the error gradient vector obtained by backreceived from the aircraft model block. The aircraft propagating through the neural network is proportional to dynamic model can be represented by the following the derivative of the kth output of the network with respect equations (as derived in the Appendix) to the ith input. The matrix containing the derivative of the outputs of P with respect to the input is called Jocobian b tan 1
Cd/Cl
M1 p matrix. v
2 W cos b=
Cl r s
M2 Building a neural network emulator of the plant and 2
M3 L 0:5 Cl v r s back propagating error gradients through it is nothing other than approximating the true Jacobian matrix of the 2
M4 D 0:5 Cd v r s plant using a neural network technique. Whenever the equations of the plant are known beforehand, they can be used to compute analytically or numerically, the elements of the Jocobian matrix. Error gradient at the input of the Table 1. Values of variables and parameters plant is then obtained by multiplying the output error gradient by the Jacobian matrix. This approach avoids the Variable Value introduction and training of a neural network emulator, which brings a substantial saving in the development time. Lift coef®cient 0.96 0.0875 In addition, the true derivatives being more precise than Desired L/D ratio Aspect ratio 9 the one obtained approximately with a neural network 3000 lb emulator, the controller training is faster and more pre- Weight 0.02 cise. One disadvantage of this method, however is that a Cdf Ef®ciency 0.8 neural network controller included in such a structure Wing area 100 ft2 cannot track changes in the plant (such as parameter Air density 0.00248 variation due to wearing of certain parts, etc.), if such Generalized neural network controller parameters considerations are an issue, the neural network emulator Learning rate 0.0001 method is a better choice. Again, the Jacobian approach is Lambda 0.10 only applicable if an analytical description of the plant is Momentum 0.3 known beforehand. Otherwise, a plant neural network emulator can be used to identify the plant to back propagate the error gradients. Table 2. GNN controller variables
where g is the learning rate, W is the weight associated with the signal, e is the error vector i.e. the difference between the actual and desired output. It can easily be shown [26] for a detailed derivation that Eq. (a) is equivalent to
5 Development of landing control system using GNN The generalized neural network developed here is used to control the aircraft during landing using back propagation through time learning algorithm [25, 26]. The following
Input variables
Output variable
Lift force (L) Drag force (D) Velocity (v) Gliding angle (b)
Control action for lift coef®cient (Cl)
445
The vector M consisting on [bvLD] and fed to the GNN controller after a unit delay (i.e. at
t 1 time instant). The generalized neural network controller output u can be determined from the past value of vector M. The GNN controller modi®es the lift coef®cient depending on the control action u and the old value of Cl, and drag coef®cient by the following equation
Cl Cl u Cd Cdf Cl2 =3:141 A e : 446
Which in turn affect the lift and drag forces at every time instant (Dt). The ratio of D/L which also equals to Cd/Cl is compared with desired ratio of Cd/Cl (i.e. 0.0875) and if there is any error then the controller will take a corrective action by changing the weights of the controller during learning. The one way to implement GNN controller is to build a neural network emulator of the aircraft model and back propagation error gradient learning algorithm can be used. This is nothing but approximating the true Jacobean matrix of the plant using neural network. In this work, a slightly different approach has been adopted by which the introduction and training of a huge neural network for copying the aircraft model can be avoided using aircraft equations. The basic idea is that instead of building a neural network copy of aircraft equations using back propagation error gradients learning algorithm, which brings a substantial saving in development time. In addition, the true derivatives bring more precise than the one obtained by approximately with a neural network emulator of aircraft model. The controller training is faster and more precise.
6 Simulation results The above developed GNN controller for aircraft landing control system has been simulated with initial lift coef®cient = 0.96 and other variables given in Table 1. The GNN controller uses the four input variables (i.e. lift force, drag force, velocity and gliding angle) for calculating the control action u, which is ®nally modifying the lift coef®cient Cl in turn drag coef®cient Cd. Then the ratio of Cd to Cl is compared with the desired ratio and the discrepancy is used for changing the weights of GNN controller as mentioned above. This process will be repeated till the discrepancy becomes zero. When the discrepancy is zero means the aircraft is following the correct glide path. The simulation experiments have been performed on the computer and the simulation results obtained from the aircraft model with GNN controller are given in Table 3. Also the system is simulated with a PID controller for the same inputs and the results obtained is shown in Table 4. Figures 4±7 shows the variations of lift coef®cient with respect to drag coef®cient, Cl2 and Cd, time pro®les of gliding angle and velocity respectively. 7 Conclusion The generalized neural network control has been developed for aircraft landing control system, which controls
the lift coef®cient Cl and drag coef®cient Cd, which controls the lift and drag forces and ultimately correct the angle of attack by changing elevator angle as per the velocity and deviation in the gliding angle of aircraft. The following conclusions have been drawn: Table 3. Simulation results of GNN Cl
Cd
b
v
Time
4.289745 4.569534 3.807501 3.052516 2.557884 2.259584 2.074728 1.956167 1.878235 1.826201 1.791118 1.767323 1.751122 1.740066 1.732510 1.727340 1.723801 1.721377 1.719716 1.718578 1.717799 1.717264 1.76898 1.76646 1.76474
0.833179 0.925332 0.660654 0.431755 0.309124 0.245621 0.210215 0.189097 0.175892 0.167374 0.161766 0.158024 0.155505 0.153663 0.152640 0.151850 0.151310 0.150941 0.150688 0.150515 0.150397 0.150316 0.150260 0.150222 0.150196
0.189972 0.201850 0.172148 0.140769 0.122011 0.108369 0.101035 0.096404 0.093399 0.091412 0.090082 0.089185 0.088576 0.088128 0.087879 0.087686 0.087554 0.087463 0.087401 0.087359 0.087330 0.087310 0.087296 0.087287 0.087280
74.813024 72.327003 79.033864 88.494391 96.089215 103.118216 107.672751 110.925259 113.227821 114.846160 115.970648 116.762429 117.306505 117.712575 117.940931 118.118959 118.241297 118.325304 118.382960 118.422516 118.449647 118.468254 118.481013 118.489761 118.495759
0.1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0
Table 4. Simulation results of PID controller Cl
Cd
b
v
Time
4.289745 5.467310 4.791444 4.175710 3.552528 3.057884 2.859584 2.294728 2.156166 1.907823 1.842630 1.790208 1.777303 1.762122 1.751155 1.735676 1.730091 1.728831 1.722367 1.721971 1.719978 1.718796 1.718572 1.716898 1.716646 1.716474
0.833179 0.961323 0.953321 0.865065 0.531345 0.409124 0.345621 0.280215 0.2181097 0.1958911 0.187374 0.171766 0.168024 0.165505 0.163663 0.162640 0.162350 0.162110 0.161989 0.161985 0.161965 0.161917 0.161816 0.161760 0.160211 0.160186
0.189972 0.210890 0.201850 0.182148 0.151669 0.132069 0.118249 0.103535 0.098704 0.095699 0.093412 0.092182 0.091850 0.090136 0.089128 0.088679 0.087586 0.087554 0.087550 0.087491 0.087399 0.087350 0.087320 0.087300 0.087298 0.087280
74.813024 73.717020 70.702300 76.145864 85.494391 94.089215 100.282160 104.812751 109.1925259 112.2278201 114.946160 116.970648 117.262429 117.516405 117.671250 117.931561 118.121233 118.223197 118.332304 118.378960 118.432456 118.450767 118.461254 118.479803 118.488895 118.48796
0.1 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0
447 Fig. 4. Coef. of list vs coef. of drag
Fig. 7. Time pro®le of velocity
i.
Fig. 5. Sq. of coef. of lift vs coef. of drag
The GNN controller is adaptive in nature and consisting of learning capabilities make the controller superior than conventional controllers. ii. The GNN controller can incorporate the non-linearities involved in the system and cognition level of human beings. iii. The GNN controller is relatively ef®cient and accurate than conventional controllers. iv. The GNN technique can be used for design and parameter calculation of aircraft after the successful application in landing control system. v. The PID controller is giving more oscillations in results and if the suddenly atmospheric conditions changes it could not cope up with that effectiveness as GNN can do.
Appendix ± Aircraft dynamic model while landing There are four forces lift (L), drag (D), thrust (T) and weight (w) are acting through center of gravity of an aircraft along different axis under different level of ¯ight as shown in Fig. A1. Summation of forces along X- and Zaxis and total moments acting about the Y-axis, yields the equations of static equilibrium for the aircraft in straight symmetric ¯ight [6]. RFx 0; Fig. 6. Time pro®le of gliding angle
RFz 0;
RMcg 0 :
Considering the above cases following equations can be derived
Fig. A1. Forces acting on an airplane during landing
T cos a
D W sin b 0
T sin a
L W cos b 0
1
2
RMcg 0
3
RMcg Cmcg q s c
4
From the above equations following conclusions have been drawn. i.
448
In Eq. (4) the pitching moment coef®cient (Cmcg) is a function of Cl. The equilibrium can be stabilized if the components of aircraft are proportioned to allow Cmcg = 0. Mcg = 0 at some useful lift coef®cient (Cl). This useful lift coef®cient can be calculated from the required lift, which is used for equilibrium. ii. Since thrust T is a function of aircraft speed and throttle control, the rate of climb
R=C or rate of descent
R=S is regulated through throttle control. Hence the rate of climb or rate of descent is a function of thrust. iii. Velocity of aircraft
V for a given wing loading
W=s and altitudes are purely a function of lift coef®cient and the lift coef®cient is a function of angle of attack (a). iv. Along shallow path of curve aircraft velocity
v is a function of the angle of attack or lift coef®cient. In this paper only gliding approach (Engines are cut off) landing performances (i.e. round out, ¯oat and touch down) are considered. The aircraft will glide under the in¯uence of the force of gravity and without the use of engine. Changing elevator angle with the help of actuator operation will control the velocity of aircraft. By changing the elevator angle the angle of attack will be changed and hence, the gliding angle is controlled. Since the angle of attack is relatively small angle, therefore cos a 1 and sin a 0. The Eqs. (1±4) can be rewritten as
D W sin b
or
sin b D=W
5
L W cos b
or
cos b L=W
6
The lift forces, drag and drag coef®cient can be given as
L 0:5 Cl v2 r s
7
D 0:5 Cd v2 r s
8
2
Cd Cdf Cl =3:141 A e
9
Divide Eqs. (5)±(6) and (7)±(8)
tan b D=L Cd/Cl or
b tan 1
D=L tan 1
Cd/Cl
10
From Eq. (6) and (7)
W cos b 0:5 Cl v2 r s p v
2 W cos b=
Cl r s
11
References
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