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TRANSCRIPT
Flight Dynamics Identification of a Helicopter in Hovering Based on Flight Data
Ren Qianqian School of Control Science and Engineering
Faculty of Electronic Information and Electrical Engineering
Dalian University of Technology Dalian , China
Feng Yi School of Control Science and Engineering
Faculty of Electronic Information and Electrical Engineering
Dalian University of Technology Dalian , China
Abstract—A method to identify the flight dynamics model of a helicopter in hovering is put forward in this paper. First, a mechanism model of the helicopter is given and a great lot of helicopter flight tests are performed based on this model. Then, flight data obtained from the tests are used in the proposed stepwise regression identification approach to ascertain the parameters of flight dynamics model and a state-space model of the helicopter in hovering is built. Finally, flight experiments are carried out and the results validate the correctness and accuracy of the model.
Keywords- Helicopter; System Identification; Regression Method; State-space Model
I. INTRODUCTION Flight control is the core problem in autonomous flight of
unmanned helicopter. The success of a flight control system directly depends on the accuracy of the helicopter model. Helicopter is a typical multi-input and multi-output system. Also, it has complicated aerodynamic characteristics [1], including wind effects, rotor flapping, aerodynamic interaction between different parts of helicopter, and so on. In addition, helicopter is a non-linear, non-minimum phase system with high degree of coupling between channels. So, it is very difficult to build accurate mathematical model of a helicopter. Though theoretical model is relatively easy to obtain, but it can not reflect the dynamic behaviors precisely. Using flight data to identify flight dynamics model becomes a hot topic in this field.
Nowadays, researches on aircraft state estimation and parameters identification have developed to a quite high level in many countries.Time domain and frequency domain methods are two generally approaches. U.S.military and NASA developed together a frequency domain identification algorithm CIFER[2], including structure identification and parameters identification. CIFER can be applied to identify many types of helicopters. It works well on modeling for CH-47[3], S-76[4], UH-60MU[5], Fire Scout[6] and so on. Transfer functions and state-space equation models for systems can be obtained using this method and the results are satisfied. In the field of time domain identification, KennetbW Iliff[7] proposed a Maximum Likelihood Estimation algorithm in the early 70s of last century and it was used to identify the dynamic model of aircrafts.
Afterwards, many researchers applied the algorithm to identify helicopter models and got three-dimensional dynamic model of helicopters with vertical and horizontal separation. Because of this separation , the identification results can not reflect the coupling between different channels . Time domain and frequency domain identification techniques have both advantages and disadvantages. In this paper, time domain method is used because it is easier to obtain time domain data in the flight tests. The coupling between channels is also taken into consideration in the process of identification.
II. MECHANISM HELICOPTER MODEL The identification object in this paper is a single rotor,
conventional layout light helicopter whose take-off weight is 300 kilograms. Compared with a miniature helicopter, it can meet higher load requirements. The forces and moments acted on the helicopter are calculated using blade element theory[8]
and a nonlinearized model with six degree-of-freedom (6-DOF) can be described with the following quations:
2 2
( ) ( )
( ) ( )
( ) ( )
( ) sin
( ) sin cos
( ) cos cos
sin sec cos sec
cos sin
xx yy zz xz
yy zz xx xz
zz xx yy xz
a
a
a
I p I I qr I r pq L
I p I I rp I r p M
I p I I pq I p qr N
Xu rv qw gMYv pw ru gMZw qu pv gM
q r
q r
p q
θ
φ θ
φ θ
ψ φ θ φ θ
θ φ φ
φ
• •
•
• •
•
•
•
•
•
•
= − + + +
= − + − +
= − + + +
= − + −
= − + +
= − + +
= +
= +
= + sin tan cos tanrφ θ φ θ+
(1)
Based on small perturbation theory[9], parameters of the helicopter near equilibrium can be expressed as :
xxx e Δ+= (2)
Then the force and moment acted on the helicopter can be written in the following approximate form:
2010 International Conference on Artificial Intelligence and Computational Intelligence
978-0-7695-4225-6/10 $26.00 © 2010 IEEE
DOI 10.1109/AICI.2010.346
514
2010 International Conference on Artificial Intelligence and Computational Intelligence
978-0-7695-4225-6/10 $26.00 © 2010 IEEE
DOI 10.1109/AICI.2010.346
514
e
col lat lon pedcol lat lon ped
X X X X X X XX X u v w pu v w p
X X X X X Xq rq r
φ θ ψφ θ ψ
δ δ δ δδ δ δ δ
∂ ∂ ∂ ∂ ∂ ∂ ∂= + Δ + Δ + Δ + Δ + Δ + Δ + Δ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂+ Δ + Δ + Δ + Δ + Δ + Δ∂ ∂ ∂ ∂ ∂ ∂
Using small perturbation theory, non-linear helicopter model is trimmed and the linearized equations of motion for the full 6-DOF can be written as:
CxyBuAxx
=+= (3)
][ rqpwvux ψθφ=][ ψθφrqpwvuy eee=
Where wu ,,ν are translational velocities in three orthogonal directions of the fuselage fixed axis system.
ψθφ ,, are Euler angles defining the orientation of the body axes relative to the earth. rqp ,, are angular velocities. Here, ][ pedlonlatcolu δδδδ= , where colδ is the installation angle of rotor’s root which equals to the sum of main rotor collective pitch angle and linear pre-torsion angle. In this paper, linear pre-torsion angle is 8°; latδ is
lateral cyclic pitch; lonδ is longitudinal cyclic pitch and pedδ
is tail rotor collective pitch angle. According to the trim calculations in hovering, the followings can be obtained:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
0.4800-0.00970.56500000.0287-0.12600.0006-0.00310.5002-0.42670000.0148-0.00150.03830.34362.0295-2.2085-0000.1782-0.2775-0.01091.00000.0005000.0000-0.00000000.0005-1.00000000.0000-0000.0006-0.0000-1.000000.00000.0000-000
0.00000.0015-0.0293-00.00550.0046-0.4518-0.0366-0.0500-0.03910.2255-0.2449-00.00009.80000.0198-0.0350-0.00120.0000-0.23530.2255-09.8000-00.0002-0.0011-0.0200-
A
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
0.4700-0.00000.00000.89060.0078-0.3911-0.00000.01390.41670.00011.85770.8127-
00000000000000.00020.0000-2.0418-
0.04590.00000.20640.0903-0.0000-0.20640.0000-0.0011-
B
III. SYSTEM IDENTIFICATION METHODS Discretizing the state-space model obtained from the
previous section, discretized dynamics models are defined with the following equations:
( 1) ( ) ( )( 1) ( )
d d
d
x k A x k B u ky k C x k
+ = ++ =
(3)
Supposing,
11 1 19
1 9
91 9 99
... .... ... . ... .
... .... ... . ... .
... ...
j
d i ij i
j
a a a
A a a a
a a a
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
11 1 14
1 4
91 9 94
... .... ... . ... .
... .... ... . ... .
... ...
j
d i ij i
j
b b b
B b b b
b b b
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
dA ， dB are the matrixes need to be identified in equation (3). pilot command vector ( )u k and state variables ( )x k at any given time k , ( )1:k n∈ can be obtained using flight tests data.
The main purpose of this section is to obtain accurate dA and dB from the flight data using iterative regression algorithm step by step.
From equation (3), the ith state varaible at step time k + 1 can be defined as:
9 4
1 1( 1) ( ) ( )i ij j ij j
j jx k a x k b u k
= =
+ = +∑ ∑ (4)
( )x k , ( )u k are all given, so the input vector can be
defined as ( ) [ ( ) ( )]Tio k x k u k= . The parameter vector at the kth step time is formed by the parameters to be estimated of elements of matrices A and B.
[ ]1 2 3 4 5 6 7 8 9 1 2 3 4( )i i i i i i i i i i i i i iAB k a a a a a a a a a b b b b=
Assuming ( )ie k is the error in the iterative process, then equation(4) can be written as ( ) ( ) ( ) ( )i i ix k ab k io k e k= + . The error ( )ie k should be minimized so that dA and dB can reflect the systems dynamics accurately. Supposing n times of iterations are needed.
(2) (1) (1) (1)......
( ) ( 1) ( 1) ( 1)
i i i
i i i
x ab io e
x n ab n io n e n
= +⎧⎪⎨⎪ = − − + −⎩
(5)
These equations (5) are summarized with the following formulation:
*desX AB IO e= +
where,
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[ ][ ]
[ (2) ... ( )]
(1) ... ( 1)
(1) ... ( 1)
Tdes des des
T
T
X x x n
IO io io n
e e e n
⎧ =⎪⎪ = −⎨⎪
= −⎪⎩
(6)
The following cost function J is defined to minimize the error:
2
1
( ) ( )n
T
kJ AB e We We k
=
= =∑ (7)
W is selected to be diagonal. If W equals to eye, it means that the error coefficients have the same weight, so the values of weight factor can be adjusted according to actual needs. In the process of iteration, newer points are given greater weight than the older points, so the weight for the kth iteration are chosen to be ( ) n k
ijW k γ −= ( γ <1). Therefore, the weight gets smaller as n-k increasing and approaches to zero at last while it gets larger as n-k decreasing. In this way the above conditions can be satisfied. The smaller γ is, the faster the algorithm can track as well as bigger estimation differences for each iteration. Supposing the parameters to be identified are time-invariant, then the cost function equation can be developed as:
( ) ( * ) ( * )2 * * *T T
T Tdes des
T Tdes des des
J AB e We X AB IO W X AB IOX WX X WAB IO IO AB WAB IO
= = − −
= − + (8)
To minimize the cost function, its derivative is defined as equal to zero.
( ) 2 2 0T T Tdes
J AB X WIO IO AB WIOAB
∂ = − + =∂
(9)
T T TdesX WIO IO AB WIO= (10)
Transforming equation (10): T TdesIO WX IO WIOAB=
The matrix to be identified can be obtained:
1( )T TdesAB IO WIO IO WX
∧−= (11)
The matrix ( 1)IO N + can be written as:
[ ]( 1) (1) (2) ... ( 1) TIO N io io io N+ = + (12)
The term TIO WIO can be developed by introducing a recurrence relationship [10] .
1
1
1
( 1) ( 1) ( ) ( ) ( )
( ) ( ) ( 1) ( 1)
( ) ( ) ( 1) ( 1)
NT T
kN
N k T T
kT T
IO N WIO N io k w k io k
io k io k io N io N
IO N WIO N io N io N
γγ
γ
+
=
−
=
+ + =
= + + +
= + + +
∑
∑ (13)
Defining 1( ( ) ( ))TIO k WIO k − as ( )P k , then
1
1
1 1
( 1) ( ( 1) ( 1))( ( ) ( ) ( 1) ( 1) )( ( ) ( 1) ( 1) )
T
T T
T
P N IO N WIO NIO N WIO N io N io NP N io N io N
γγ
−
−
− −
+ = + += + + += + + +
(14)
Using the following analytical formula [7]
1 1 1 1 1 1( ) ( )A BCD A A B C DA B DA− − − − − −+ = − +
If 1( ) , ( 1), 1, ( 1)TA P N B io N C D io Nγ −= = + = = + , the following equation can be obtained:
1
( ) ( )( 1) ( 1)*
( ) ( )1 ( 1) ( 1) ( 1)T T
P N P NP N io N
P N P Nio N io N io N
γ γ
γ γ
−
+ = − +
⎡ ⎤+ + + +⎢ ⎥⎣ ⎦
(15)
For the same reason,
( 1) ( 1)( ) ( ) ( 1) ( 1)
Tdes
Tdes des
IO N WX NIO N WX N io N x Nγ
+ + =+ + +
(16)
Substituting Eq(15)and Eq(16)into(11), it can be obtained:
1
( )( 1) ( ) ( ) ( 1) ( 1) * (
( ) ( ) ( )( 1) 1 ( 1) ( 1) ( 1) )
Tdes des
T T
P NAB N IO N WX N io N x N
P N P N P Nio N io N io N io N
γγ
γ γ γ
∧
−
⎡ ⎤+ = + + +⎣ ⎦
⎡ ⎤− + + + + +⎢ ⎥⎣ ⎦
(17)
In Eq(11), estimated parameter matrix is defined as:
( ) ( ) ( ) ( )TdesAB N P N IO N WX N
∧=
It is substituted into Eq(17):
1
( )( 1) ( ) ( 1) ( 1)
( ) ( )( 1) 1 ( 1) ( 1) *
( 1) ( ) ( 1) ( 1) ( 1)
des
T
T Tdes
P NAB N AB N io N x N
P N P Nio N io N io N
io N AB N io N io N x N
γ
γ γ
∧ ∧
−
∧
+ = + + +
⎡ ⎤− + + + +⎢ ⎥⎣ ⎦
⎡ ⎤+ + + + +⎢ ⎥⎣ ⎦
(18)
Defining 1
( ) ( )( 1) ( 1) 1 ( 1) ( 1)TP N P NK N io N io N io Nγ γ
−⎡ ⎤+ = + + + +⎢ ⎥⎣ ⎦
(19)
Then,
( )( 1) ( ) ( 1) ( 1)
( 1) ( 1) ( ) ( 1) ( 1)
des
Tdes
P NAB N AB N io N x N
K N io N AB N io N x N
γ∧ ∧
∧
+ = + + + −
⎡ ⎤+ + + + +⎢ ⎥⎣ ⎦
(20)
This formula can be simplified:
( 1) ( ) ( 1)( ( 1) ( 1) ( ))TdesAB N AB N K N x N io N AB N
∧ ∧ ∧+ = + + + − + (21)
Substituting Eq(19) into Eq(15), the expression of P becomes:
1( 1) ( ( 1) ( 1) ) ( )TP N I K N io N P Nγ
+ = − + + (22)
This is the whole process of identifying the matrix AB. The flow chart of this identification process is as follows:
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Figure 1. Flow chart of identification process
For each step, the parameters matrices A and B are obtained. Each pair of matrices are tested for the entire signals and the pairs which make the minimum error are selected. The error can be defined by equation (23). Then the pair with minimum I is chosen as our identification result.
92
1 1
( ( ) ( )* ( ))n
desi i i
i k
I X k AB k IO k= =
= −∑∑ (23)
IV. FLIGHT TESTS AND MODEL IDENTIFICATION At the beginning of the tests, manual operations are carried
out to make the helicopter take off and keep it in hovering. Then it is transferred to automatic control model. Basic four feedback loops PID control method is used. PID parameters are adjusted to stabilize the helicopter in hovering. Control inputs are main rotor collective pitch, lateral cyclic pitch, longitudinal cyclic pitch and tail rotor collective pitch. The control inputs are recorded for each step. Machine body is equipped with a strap-down AHRS to obtain real-time recording of the helicopter’s state variables in hovering. It includes a three-axis gyro, a three-axis accelerometer, a three-axis magnetic heading of dollars and a GPS. Thus the data desX and IO needed for identification are achieved from the AHRS. Precision of tests data directly affect system identification accuracy. Therefore, it is required that data obtained in the experiments close to the true values to the maximum extent to improve the identification accuracy. A lot of noises and interferences affect the helicopter in the flight tests, so the tests data need to be processed to filter out noises and disturbances before identification. Firstly, a low pass filter is used to filter high frequency noises; then the trend of data items and the median values are wiped out; finally, the tests data are smoothed using Polynomial Centre Interpolation fitting algorithm. Afterwards, introducing these processed tests data into the algorithm operations given in section II, the matrices A and B are both identified. The coupling between yaw channel and other channels is relatively small, so yaw angle can be removed from these states to simplify the calculation in the process of identification. The matrices A and B obtained from the process of identification are:
0.0196 0.0014 0.0204 0.0005 0.0376 0.0508 0.1743 0.10950.0009 0.036 0.0144 0.0049 0.0187 0.0113 0.1077 0.03460.0499 0.0373 0.4339 0.0123 0.0405 0.028 0.0916 0.0319
0.0498 0.1581 5.7448 3.9744 10.6731 7.0587A
− − − −− − − −
− − − − − −− − − − −
=3.0826 10.184
0.0413 0.0655 1.639 3.0683 1.9605 4.0207 22.3698 5.9220.0443 0.396 2.1116 3.2331 6.5585 14.9893 57.191 34.7412
0.0148 0.0077 0.1801 0.8901 3.3501 0.5935 6.3854 0.63240.0514 0.0013 0.082 0.4776 9.8524 0.294
−− − −− − −
− − −8 2.7781 2.4798
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
−⎢ ⎥⎣ ⎦
0.0957 0.2463 0.0764 0.25820.1007 0.04 0.1333 0.08490.7414 0.4388 0.1604 0.076669.7839 19.5575 2.7752 36.0032
29.6938 16.2502 0.4227 18.3259122.857 15.0804 47.9762 96.50432.0714 7.0285 6.9233 4.440121.8654
B
− − −− −− −−
=− −
−− − −− 24.763 0.2157 3.8452
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
−⎢ ⎥⎣ ⎦
V. IDENTIFICATION RESULTS VERIFICATION AND ANALYSIS The identification results can be validated in time domain.
Using a group of data of control input as input signal of the
identified model, a group of state variables can be obtained through calculation. The block diagram of this process is as follows:
, ,φ θ Ψ
Figure 2. Block diagram of state equation simulation
Calculated state variables and measured values from flight tests can be compared in these figures (3), (4), (5) and (6).
Calculate matries 1( 1) ( ) ( 1) ( 1) ( ) ( 1)TK k P k io k io k P k io kγ
−⎡ ⎤+ = + + + +⎣ ⎦
Calculate matries ( 1) ( ) ( 1)( ( 1) ( 1) ( ))TdesAB k AB k K k x k io k AB k
∧ ∧ ∧+ = + + + − +
Initialize matries P andAB，P is diagonal .AB get from mechanical calculation
Calculate matries 1( 1) ( ( 1) ( 1) ) ( )TP k I K k io k P kγ
+ = − + +,k=k+1
N
initialize 0 1γ< ≤
( ) ?J AB ξ<
Y
NY
K>=n ?
end
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Figure 3. Pitch angle
Figure 4. Roll angle
Figure 5. Pitch angle rate
Figure 6. roll angle rate
These figures tell that errors between calculated state variables and measured values are relatively small in the low frequency band while large in the high frequency domain. This is determined by many factors. Firstly, small perturbation theory used in the model building ignores a lot of disturbances.
Secondly, low pass filter with relatively high cut-off frequency is selected to fully reflect the control response. However, body dynamics model of the helicopter is actually a low frequency model. This brings identification error in the high frequency domain. Thirdly, due to vibration, wind and other ignored influences, there exist a lot of high-frequency noise during the attitude measurement process. These all affect the accuracy of the identification.
VI. CONCLUSION Firstly, a state-space model of a helicopter in hovering is
built using mechanism modeling method. Thus the structure of the identification object and initial parameters are determined. Then targeting at minimum gradient of error function, the parameters are adjusted gradually through stepwise iteration with flight data and finally identification results are obtained. Flight experiments indicate the calculated output of the model and the measured output have high degree of agreement and verify that the identified model can accurately reveal dynamic characteristics of the helicopter in hovering. However, there are still some problems need to be solved. For example, data miss during flight data collection, outside wind influence, sensor measure errors and so on. Therefore, more flight tests are to be performed to further prove the identification results.
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