Proceedings of the 34th Chinese Control Conference July 28-30, 2015, Hangzhou, China
Quantum Genetic Algorithm Based Homing Trajectory Planning of Parafoil System TAO Jin, SUN Qinglin, ZHU Erlin, Chen Zengqiang College of Computer and Control Engineering, Nankai University, Tianjin 300071 E-mail:
[email protected] Abstract: Homing Trajectory planning is a core task of autonomous homing of parafoil system. This paper analyzes and establishes a mathematical model, and then proposes a homing trajectory planning method of parafoil system based on quantum genetic algorithm. By using the quantum bit with superposition state to encode the control law and introducing the quantum rotation gate to realize evolution of chromosomes, the proposed method guides and realizes homing trajectory optimization design. Simulation results under different initial states indicate that this method is effective for homing trajectory optimization of parafoil system. Compared with the chaos particle swarm optimization algorithm, the homing precision is even better. Key Words: Quantum genetic algorithm; Parafoil system; Homing trajectory planning
1
Introduction
Parafoil system is made up of a traditional airfoil umbrella, loads and GNC (Guidance Navigation & Control) equipment. It is a kind of precise airdrop landing system with good pneumatic performance, excellent gliding ability and easy handling property. In view of many advantages, it has been widely used in military, aerospace and civil fields, such as weapons or supplies aerial delivery, spacecraft recycling, aerial photography and entertainments, etc. For some years, with the introduction of GPS navigation technology, and the development of measurement technology and control science, autonomous homing of parafoil system becomes practicable. Homing trajectory design and optimization is overwhelmingly crucial to realize autonomous homing, and the pros and cons of homing trajectory to a great extent affect the result of homing. At present, a variety of homing control methods have been recommended, which are blind-angled control method, multi-stage control method and optimal control method, the last one is now widely concerned by academics. The so-called optimal control method can be described as searching for a particular trajectory from the initial point to the target that meets specific performance under certain constraints. Currently, exploration in this field is relatively less. Pearson [1] took the lead in carrying out some researches in 1972ˈhe solved the optimal control problem which could reach the target without any constraints by differential dynamic programming method. Some domestic scholars have been undertaking related research since 2000, Xiong [2] adopted conjugate gradient method to solve the problem that is transformed into two-point boundary value problem by pontryagin minimum principle. Zhang [3], Gao [4] used gauss pseudo-spectral method to transform the problem into parameters optimization with a series of algebraic constraints, and got a satisfactory solution. Liu [5], Jiao [6] respectively adopted the improved particle swarm
* This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61273138.
algorithm in optimizing homing trajectory of parafoil system. This paper starts from the description of the problem, and analyzes the homing trajectory optimal design problem, then establishes the corresponding mathematical model, further proposes a homing trajectory planning method of parafoil system based on quantum genetic algorithm. This method can effectively improve the efficiency of genetic operators and convergence abilities by means of the quantum state of superposition, entanglement and interference based on the theory of quantum mechanics and quantum computing. Simulation results show the efficiency of this method.
2
Quantum Genetic Algorithm
Quantum Genetic Algorithm (QGA) [7, 8, 9] is the combination of classic genetic algorithm (GA) and quantum computing. The properties of quantum parallel and entanglement are used in QGA, in addition, the multiple state quantum bit encoding method and quantum rotation door are adopted at the same time to effectively improve the search capability. 2.1
Quantum Bit
Quantum bit is a two-state quantum system, which acts as the smallest information storage unit of physical media. The difference between quantum bit and classical bit is that it can be in two different superposition of quantum state at once, which can be symbolized as: M D 0 E1 (1) Where (D , E ) represent two complex numbers, which specify the probability amplitude of corresponding states and meet with D 2 E 2 1 . 0 or 1 represent the state of spin down or spin up respectively. Explicitly, the quantum bit is in the state of 0 and 1 simultaneously. For QGA, the storage and expression of genes are by means of quantum bit, which is in the state of 0 or 1 , or their arbitrary superposition. That is to say, genes’ expressions are no longer certain information, but all possible information. Any operations of genes would also
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apply to all possible information. The diversity index of QGA is better than GA. With D 2 or E 2 approaching to the state of 0 or 1 , chromosomes will converge to a single state. 2.2
Quantum Measurement
The purpose of quantum measurement is to obtain a set of definite solution through individual measurements of the population. The solution can be presented as: P(t ) p1t , p2t ,, pnt (2)
^
`
Where n represents the number of generator unit, t represents the evolution generation, p tj represents the binary coding of the generation volume of the jth generator unit, its length is m . Each bit is in the state of 0 or 1 , which is decided by D it
2.3
2
2
or Eit .
Quantum Update
The quantum gate acts as the actuator of evolution that is chose according to specific problems. The quantum rotation gate is more suitable according to the calculation characteristic of QGA. The quantum rotation gate operates as follows: ªcos(T i ) sin(Ti ) º U (Ti ) « (3) » ¬sin(Ti ) cos(T i ) ¼ The update process is described as follows: ªD i' º ªD i º ªcos(T i ) sin(T i ) º ªD i º « ' » U (T i ) « » « »« » ¬ E i ¼ ¬sin(Ti ) cos(T i ) ¼ ¬ E i ¼ ¬ Ei ¼ Where >D i
(4)
Ei @ represent the probability amplitude of T
the ith bit of chromosomes before updating, while T ª¬D i' Ei' º¼ represent the probability amplitude after updating.
3.2
The flight control of parafoil system is realized by manipulations of left or right control ropes on its trailing edge. Xiong [2] established a six degree of freedom movement equation of parafoil system. Simulations of the six degree of freedom model show that the increase of unilateral lower deviator leads to the decrease of gliding ratio and roll angle. Consequently, the unilateral lower deviator is usually limited within a certain range for it will damage holistic stability of parafoil system when reducing to a certain degree. Accordingly, the following assumptions are established to simplify the model: z The horizontal and vertical velocities remain unchanged under ignoring effects caused by the variation of atmospheric density. z The horizontal wind field is fixed and known. z The control response of parafoil system is without any delay. Based on the above assumptions and wind coordinate system, the original point is chose where flared landing is began to implement. The motion equation of parafoil system can be simplified as: x vs cos(\ ) ° ° y vs sin(\ ) (5) ® °\ u ° z v z ¯ ( Where x, y, z ) mean the location information of parafoil system, vs means its horizontal velocity, vz means its vertical velocity, \ means its heading angle, \ means its yaw rate, u denotes the control quantity with the scope of [umax , umax ] , and there exists a one-to-one relationship between u and the unilateral lower deviator.
Ti represents the quantum gate rotation angleˈits size and
4
symbol are determined by the adjustment strategy which is designed beforehand.
4.1
3
Mathematical Model
The dynamic model of parafoil system is highly complex and nonlinear, and always has strong coupling. Therefore, the particle model is typically used in place of the complex high degree of freedom model in homing trajectory planning. This way can greatly simplify calculations in process. 3.1
Wind coordinate System
The wind coordinate system is often adopted in homing trajectory design of parafoil system. The direction of each coordinate axis in the wind coordinate system is consistent with the geodetic coordinate system. Its original point moves with the airflow. The original point of wind coordinate system is coincide with the original point of geodetic coordinate system at the time when flaredlanding being implemented. Thus, the degree and direction of wind can be transformed into the shift position of the original point.
Establishment of Particle Model
Analysis of homing trajectory planning Problem description
The problem of homing trajectory planning of parafoil system can be described as the pursuit of control law which can make the specific performance optimal under certain dynamic constraints. The requirements and conditions of homing trajectory optimization can be summarized as follows: z The landing site is close to the target point. z The landing direction is against the wind. This is the necessary condition of flaredlanding through which can reduce the speed of landing and avoid the damage of recycled material in the process of landing. z The feasible manipulation and low energy consumption is required. 4.2
Constraints and objective functions
Based on the particle model and assumptions mentioned above, constraints and objective functions are formulized as follows: z Initial conditions t0 is known, x(t0 ) x0 , y (t0 ) y0 , z (t0 ) z0 , \ (t0 ) \ 0 . z Terminal constraints
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z 0 / v z , x (t f )
tf
0 , y (t f )
0 , z (t f )
0.
\ (t f ) \ wind r (2 n 1)S , \ wind represents the velocity of horizontal wind. This constraint is to ensure the landing direction is against the wind. z Control constraint u d umax , umax is the allowed maximum control quantity, and also matches the minimum turning radius. z Objective functions According to requirements and conditions mentioned above, three objective functions are designated as follows: ° J1 ° ° ®J2 ° ° J3 ° ¯
t
min( ³ f u 2 dt ) 0
min( x(t f )2 y(t f )2 )
(6)
min(cos(\ (t f ) 1))
Where J1 denotes the minimum of energy consumption,
J 2 denotes the minimum of landing error, J 3 denotes the demand condition of flaredlanding. For convenience to solve the problem, the multi-objective optimization is transformed into a single objective optimization by the method of weighed factor. The objective function can be transformed into: J f1 u J1 f 2 u J 2 f3 u J 3 (7) Where f1 , f 2 and f 3 are weighed factors which are nonnegative. The weighed factors should be valued according to requirements of the actual project.
5
QGA based homing trajectory planning method
5.1
The essence of homing trajectory optimization of parafoil system is a kind of optimal control problem. The optimal solution cannot be directly abstained via QGA for the search space is a functional space. In order to facilitate encoding and decoding, the first thing to do is to transform the optimal control problem into a parameter optimization issue. Shared parametric methods mainly include direct discrete method, multiple parameter interpolation method and function approximation method. In order to simplify encoding and improve the expression ability of control law, non-uniform b-spline is used for fit the control law in this paper. Consequently, the chromosome of genetic space is formed by control vertices of b-spline basis function. Through this kind of function approximation method, less dimension control parameters is used for acquire all forms of complex control curves, the encoding and decoding of QGA and subsequent optimization computation complexity are greatly simplified. Non-uniform b-spline curve is defined as: n
¦d N i
i,k
( s)
Assuming that the initial population is qi , (i 0,1, , N ) , where N is the size of the population, and also the number of control laws, qk denotes the control law u . Each u can be considered as a chromosome. For u can be expressed by non-uniform b-spline curve, control vertices di are corresponding to each gene locus on the chromosome. To encode u by quantum bits, and its chromosome is defined as: qkt
(8)
i 0
Where di , (i 0,1, , m) represent control vertices, N i , k ( s ) , (i 0,1, , n) represent the kth of b-spline basis
t t ªD1,1 D1,2 D1,t n « t t t «¬ E1,1 E1,2 E1, n
t t D 2,1 D 2,2 t t E 2,1 E 2,1
(10)
D 2,t n D mt ,1 D mt ,2 D mt , n º » E 2,t n E mt ,1 E mt ,2 E mt , n »¼
Where t represents the tth generation of qi , j represents the jth individual chromosome, m represents the number of the control vertices, n represents the number of quantum bit that each gene contains. 5.2
Control law encoding and population initialization
u ( s)
function which is decided by si , (i 0,1, , n k 1) , that can be deduced by De-Boor-Cox recursive formula as: 1 si d s si 1 ° N i ,0 ( s ) ® others ° ¯0 (9) ® s si si k 1 si ° N (s) N i , k 1 ( s ) N i 1, k 1 ( s ) ° i,k si 1 si si k 1 si 1 ¯ The support interval of kth b-spline basis function is [ si , si k 1 ] , k 1 Node intervals are included. Namely, the b-spline basis function is related to as much as k 1 nodes, and has nothing to do with the other nodes. More information can be found in literature [10].
Adjustment strategy of quantum rotation gate
Here, a general adjusting strategy is designed beforehand. The adjustment strategy is shown in table 1. Table 1: Adjustment Strategy of Quantum Rotation Gate f ( x) !
'T i
s (D i , E i )
xi
besti
0
0
False
0
0
0
0
0
0
0
True
0
0
0
0
0
0
1
False
0.01ʌ
+1
-1
0
f1
0
1
True
0.01ʌ
-1
+1
f1
0
1
0
False
0.01ʌ
-1
+1
f1
0
1
0
True
0.01ʌ
+1
-1
0
f1
1
1
False
0
0
0
0
0
1
1
True
0
0
0
0
0
f (best )
Di Ei ! 0
Di Ei 0
Di
0
Ei
0
Where xi is the ith bit of the current chromosome, besti is the ith bit of the best chromosome, f ( x ) is the object function, s (D i , E i ) is the direction of rotation angle, 'T i is the rotation angle, its value is determined by the strategy listed in table 1. The adjustment process is to compare the fitness value f ( x ) of the current measured value of q tj (t ) with the current optimal individual fitness
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value f (best ) , if f ( x) ! f (best ) , than adjust the corresponding quantum bit of q tj (t ) to make (D i , Ei )
Initial state
A
B
C
revolute to the direction of xi , otherwise, adjust the
x0 m
-1000
-3000
6000
y0 m
0
3000
4000
\ 0 rad
ʌ
ʌ/3
-2ʌ/3
Table 2: Initial State Parameter Settings
corresponding quantum bit of q tj (t ) to make (D i , Ei ) revolute to the direction of besti . 5.3
Process of QGA
The process of QGA is described as follows: z Initial the population Q(t0 ) , so as to randomly generate n chromosomes encoded by quantum bits. z Measure every individual of population Q(t0 ) , to get the definite solution P(t0 ) . z Value the fitness of P(t0 ) . z Reserve the best individual of population and its fitness. z Determine whether the process can end, if meet the end condition then quit, otherwise continue to calculate. z Measure every individual of population Q(t ) , so as to get the definite solution P (t ) . z Value the fitness of P (t ) . z Use the quantum rotation gate U (t) to update, to get a new population Q(t 1) . z Reserve the best individual of population and the best fitness. z Add 1 to the number of iteration t , jump to step 5DŽ
6 6.1
6.2
Simulation results and analysis
The simulation results are shown in Fig. 1, Fig. 2 and Fig. 3, which respectively represents the homing trajectory of parafoil system in the initial state A, B and C. Fig.1 show the simulation results of the homing trajectory of parafoil system and the corresponding optimal control curve of the initial state A, namely when the initial position is closer to the target. It can be seen from Fig.1(a) and Fig.1 (b) clearly that the trajectory show a trend that the parafoil system flies far away from the target at the early time, and it has significant indirect turn stage and no obvious glide stage. The purpose of consecutive turns is consuming extra height to move to the target fast. The more obvious it turns, the larger is the corresponding control quantity. From Fig.1(c), it can be see that the control curve is similar to the low-frequency cosine curve among the whole process of homing.
Simulation and analysis Simulation conditions and parameter settings
Based on the above method and idea, homing trajectory planning of parafoil system via QGA is simulated by Matlab. The Euler method is used for calculate the trajectory by iterated integral along the time orientation. QGA is adopted as the optimization tool. Basic parafoil system motion parameter settings are as follows: the horizontal velocity vs 15m / s , the vertical velocity vz 5m / s , the initial height h 2000m , the maximum control quantity umax vs Rmin 0.12 . Non-uniform b-spline is set as follows: the size of control vertices m 5 ˈthe power of b-spline k 2 . Parameter settings of QGA are as follows: the population size N 40 , the largest number of iterations tmax 200 , the number of quantum bit of each gene n 20 , using the quantum rotation gate mentioned above to update the population, weighed factor are set as: f1 2000 , f 2 1 , f3 10000 . In order to demonstrate configurations and control characteristics of the homing trajectory of parafoil system comprehensively and vividly, three types of initial motion states are simulated hereinafter. The initial state parameter settings are shown in Table 2.
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(a) Horizontal trajectory
(b) 3D trajectory
(c) Control curve Fig. 1 Homing trajectory of the initial state A
(c) Control curve Fig. 2 Homing trajectory of the initial state B
Fig.2 show the simulation results of the homing trajectory of parafoil system and the corresponding optimal control curve of the initial state B, namely when the initial position is far away from the target and still can be reachable. It can be seen from Fig.2(a) and Fig.2(b) that the glide stage increases obviously , and the corresponding control quantity is very small during gliding, this can make the parafoil system approaching toward the target fast and energy saving. Turning at the start enable the parafoil system adjust heading angle so as to fly toward to the target. Turning at the end aims to adjust direction against the wind for meeting the condition of flaredlanding. While turning, the corresponding control quantity becomes larger (as in Fig.2(c)).
Fig.3 show the simulation results of the homing trajectory of parafoil system and the corresponding optimal control curve of the initial state C, namely when the initial position is too far away from the target and unreachable. It can be seen from Fig.3(a) and Fig.3(b) that the parafoil system has landed before reached the target. Among the whole process of homing, the first step is to turn sharp, then glide a long period of time so as to approach the target, afterward, adjust its direction to against the wind before landing.
(a) Horizontal trajectory
(a) Horizontal trajectory
(b) 3D trajectory
(b) 3D trajectory
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robustness and extensive adaptability. The optimized trajectory designed by QGA allows the parafoil system to implement homing in a nearly optimal way. Compared with results optimized by chaos particle swarm optimization algorithm, the landing precision is better and the direction deviation is smaller. The follow-up work maybe introduce quantum crossover, mutation and disaster operations into QGA, and adjust the rotation angle of quantum gate according to the dynamic evolutionary process, and it will make the method more applicable in homing trajectory planning of parafoil system.
References
(c) Control curve Fig. 3 Homing trajectory of the initial state C
[1]
Simulation results of initial state A and B which target is reachable are listed in Table 3. Simulation results of the same initial position in literature [6] (A and B are respectively corresponding to pos7 and pos2) are compared. It can be seen from Table 3 that the parafoil system can accurately get to the target and turn its direction upwind before flared landing following the trajectory designed by QGA. Compared with simulation results optimized by chaos particle swarm optimization algorithm, landing precision and direction deviation of QGA is more accurately, the total energy consumption is almost at the same level. Table 3: Simulation Results State
'xt f / m
'yt f / m
\ t / rad
usum
A
0.11
-0.49
3.12
1.2
f
Pos7
-0.3
1.3
3.00
1.3
B
0.96
-0.15
3.10
1.2
Pos2
2.6
-1.3
2.94
1.2
7
Conclusion
In this paper, homing trajectory planning of parafoil system based on QGA is discussed, and the basic principle of QGA is given. Chromosomes are encoded by quantum bits with the superposition state, and updated by the quantum rotation gate. The proposed method guides and realizes homing trajectory optimization design of parafoil system. Simulation results show that this method is a kind of effective way for homing trajectory planning with strong
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