2017 IEEE 3rd International Conference on Control Science and Systems Engineering
Path Planning for Carrier Aircraft Based on Geometry and Dijkstra's Algorithm
Jing Zhang, Jia Yu, Xiangju Qu
Yu Wu
School of Aeronautic Science and Engineering Beihang University Beijing, China
[email protected] e-mail:
[email protected],
[email protected]
College of Aerospace Engineering Chongqing University Chongqing, China e-mail:
[email protected]
Abstract—Safety and efficiency of carrier flight deck are two important factors that have great effects on the comprehensive performance of carrier. This paper studies on path planning algorithms for carrier aircraft on the deck in consideration of safety and efficiency requirements. Firstly, a kinematic model of the carrier aircraft, obstacle avoidance models and boundary constraints are proposed. Subsequently, search zone for path planning is designed based on geometry. Then, the optimal path is computed according to Dijkstra's Algorithm and the details are described by steps. Finally, an example is simulation to verify the rationality and the effectiveness of the path planning algorithms. Keywords-carrier aircraft; path planning; obstacle avoidance; geometry; Dijkstra's algorithm
I.
INTRODUCTION
As a main weapon of naval aviation, carrier aircraft is an important force to maintain superiority in the ocean battlefield. Its mobility is a significant technical indicator to measure the capabilities of carrier in combat and comprehensive support, and it is also a key and difficult factor in aircraft design. Because capacity of the flight deck often becomes a bottleneck of the mission in optimizing the schedule of carrier aircraft, it is critical to make carrier aircraft move from the initial position to the target position quickly. Meanwhile, safety is as important as the operational efficiency for the flight deck. As an example, a Nimitz-class aircraft carrier, has a limited deck space which is much smaller than a general airport, and its flight deck is 332.9 meters long and 76.8 meters wide. In addition, there are a number of carrier aircraft and an island on the deck, so path planning for carrier aircraft on the limited deck space safely is a very important and difficult issue. Nowadays, artificial scheduling is applied to path planning on most of the decks, which is mainly based on the work experience of scheduling staffs. In view of defects of artificial scheduling, automatic path planning for carrier aircraft on the limited deck has become a hot research topic. Currently, there are some literatures about path planning algorithms for aircraft, and these algorithms can be divided into three-dimensional path planning [1], [2] and twodimensional path planning [3]-[6]. In detail of the twodimensional path planning algorithms, chaotic approaches are used to plan path for uninhabited combat air vehicle [3],
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[4]. A dynamic weight heuristic function was designed based on an improved A* algorithm [5]. And variants of the quantized visibility graph for efficient path planning is proposed based on plane geometry [6]. These algorithms compute costs of the candidate paths according to some way points, and the optimal path must be smoothed after these way points are confirmed or the carrier aircraft will fail to track the optimal path. In this paper, we introduce a path planning algorithm for carrier aircraft on deck based on geometry and Dijkstra's Algorithm [7]-[9], and it is produced in three steps: (i) describes carrier aircraft and obstacles by mathematical models, (ii) finds all way points and computes the costs of every pairs of way points, (iii) finds the optimal path according to Dijkstra's Algorithm. As a result, the optimal path do not need to be smoothed and carrier aircraft can track it directly. II.
CONCEPTUAL MODEL
A reasonable path planning for the carrier aircraft is necessary in order to improve operation efficiency of the flight deck. Key factors of path planning are mission requirements and constraints. The mission requirements are safety and efficiency of the flight deck, and the constraints contain maneuver performance of the traction system, the environment constraints and mission requirements. This paper extracts the key factors of path planning for carrier aircraft and builds a conceptual model as presented in Fig. 1.
Figure 1. Conceptual model.
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III.
MATHEMATICAL MODEL
According to the conceptual model in Fig. 1, kinematic model of the carrier aircraft, obstacle avoidance models and boundary conditions will be built at first. A. Kinematic Model of Carrier Aircraft As a single object, carrier aircraft has three degrees of freedom when it moving on the deck. Therefore, its position on the deck and attitude of its body axes can be used to express its movement. In order to obtain the kinematic model, this paper describes a simplified geometric model of the carrier aircraft, as shown in Fig. 2. In Fig. 2, the point G is the geometric center of the aircraft, and its coordinate is (x, y) on the deck coordinates. M, which is used to describe attitude of the aircraft, is the angle between the body axes and x-axis. And E is the deflection angle of the nose wheel while v is the speed of the drive wheels. Point I is instantaneous center of the carrier aircraft. Therefore, a differential equation of the traction system motion can be expressed as follows.
x vG cos J ° ® y vG cos J °M v tan E AB ¯
J =M +arctan tan E BG AB
vG = v cos J M
B. Obstacle Avoidance Models It is easier for obstacle avoidance modeling if the carrier aircraft and the obstacles are replaced by their feature shapes. Single aircraft, multiple aircraft and the island are all obstacles for the path planning on the deck. Single aircraft can be simplified as a smallest circle which containing itself and this circle named feature circle [10]. Multiple aircraft can be simplified as a smallest rectangular which is named a feature rectangular and containing themselves [11]. Although boundary of the island is an irregular graphics, it can be simplified as a rectangular too. In order to avoid a collision between two aircraft, they must satisfy the next equation.
O1O2 ! R1 +R 2
where O1 and O2 are centers of these two feature circles, while R1 and R2 are feature radii [12]. As illustrated in Fig. 3, (6) will be satisfied if the point O1 is outside the threaten zone, and there is no collision between these two aircraft. Similarly, a threaten zone of the multiple carrier aircraft is presented in Fig. 4. The point O1 should be outside the threaten zone in order to avoid a collision. Furthermore, we can obtain the obstacle avoidance model of the island refer to Fig. 4. C. Boundary Conditions The Initial position and attitude of aircraft are known, and the target position and attitude are also required, so that the boundary conditions can be described as (7) and (8).
X t t
> x0 , y0 ,M0 @
T
0
According to maneuver performance of the aircraft, E has its limit as described as (4).
E d E max
Furthermore, minimum turning radius of the traction system can be computed according to (5).
R min
AB tan E max
2
BG 2
Figure 3. Obstacle avoidance model of single carrier aircraft.
Figure 2. Geometric model of carrier aircraft.
Figure 4. Obstacle avoidance model of multiple carrier aircraft.
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X t t
> xT , yT ,MT @
T
T
B. Path Planning in Consideration of Obstacles As illustrated in Fig. 6, circle O is the threaten zone of an obstacle and it is overlaps with the edge of original search zone. Therefore, the search zone should be expanded, and the Convex Hull which contains these five circles will be the new search zone. Of course, the new search zone also should be expanded if its edge is overlaps with another threaten zone, until the final search zone is obtained. Subsequently, we will find the optimal path in the final search zone. The final search zone is demonstrated in Fig. 7. As depicted in Fig. 7, there are three obstacles in the final search zone. One of these obstacles represent a threaten zone of multiple carrier aircraft, and we divided it into four circles in order to find the optimal path easily. Moreover, all these circles are numbered as Oi (i=1, 2, … , N) and N=10 in Fig. 7. From the previous we can see that all candidate paths are consists of arcs and tangent segments of these circles. Therefore, we should find all tangent segments in the search zone and we will describe these tangent segments using vectors as represented as (10).
In (8), t=T is the time when the aircraft reaches the target position. IV.
PATH PLANNING ALGORITHM
The objective function of path planning is proposed to improve operation efficiency of the flight deck [13]. The velocity of the aircraft fluctuates within a narrow range, so we can regard it as uniform motion. The objective function is shown as follow.
J
min
1 v
T arg et
³
dl
Initial
where J is the objective function of path planning which is established to minimize the total elapsed time, and it equivalent to minimize the total length of the path. A. Path Planning without Considering Obstacles In Fig. 5, A and B* are the initial position and the target position respectively, while arrows at A and B* represent the initial attitude and the target attitude respectively. Obviously, the optimal planning path is the straight line between A and B* without considering the maneuver performance, environmental constraints or boundary conditions. In order to make the final attitude fulfil the requirement, we take the point B as a way point and the line BB* will be the last part of the candidate paths. Furthermore, attitude of BB* is the same as the target attitude, while its length is equal to R1. As a result, the point B is took as the new target position for path planning. There are two circles on either side of point B and their radius are equal to R1. Point A is the same as point B. The planning path cannot overlap with these circles in consideration of the maneuverability constraints. Therefore, all candidate paths consists of two pieces of arc and one tangent segment according to geometry theorem. For example, the path A-C-D-B is a candidate path as shown in Fig. 5. Furthermore, we get a Convex Hull which contains these four circles, and it is easy to know that all candidate paths must be in the Convex Hull. The Convex Hull is named the search zone.
Lj
ª¬Oi1 , Ornti1j , Posni1j ,Oi 2 , Orntij2 , Posnij2 , l j º¼
1 d i1 i 2 d N , Ornti1j , Orntij2
r1
Figure 6. Search zone expanded due to obstacles.
Figure 7. The final search zone.
Figure 5. Path planning without considering obstacles.
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Step 5. Let i=i+1. If i
