A MULTI-PURPOSE OFF-LINE PATH PLANNER BASED ON AN A SEARCH ALGORITHM

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Proceedings of

The 1996 ASME Design Engineering Technical Conferences and

Computers in Engineering Conference August 18-22, 1996, Irvine, California

96-DETC/MECH-1134

A MULTI-PURPOSE OFF-LINE PATH PLANNER BASED ON AN A* SEARCH ALGORITHM

Arturo L. Rankin and Carl D. Crane III

Center for Intelligent Machines and Robotics

300 MEB, University of Florida Gainesville, Florida, 32611, U.S.A.

ph: (352) 392-0814 fax: (352) 392-1071 email: alr@https://www.wendangku.net/doc/7816912839.html,

ABSTRACT

Efficient navigation of an autonomous mobile robot through a majority of which search for shortest euclidean-distance paths for well-defined environment requires the ability of the robot to plan robots free of nonholonomic constraints. For a broad overview of paths. An efficient and reliable planar off-line path planner has been path planning for robots see (Latombe, 1990). Deo and Pang developed that is based on the A* search method. Using this (1984) classify 79 shortest-path algorithms according to problem method, two types of planning are accomplished. The first uses a type, input type, and the technique used to solve the problem. The map of all known obstacles to determine the shortest-distance path most popular shortest-path algorithms used for off-line path from a start to goal configuration. The second determines the planning are the classic graph searching methods, such as, Dijkstra's shortest path along a network of predefined roads. For the most algorithm (Dijkstra, 1959) and the A* search method (Hart et al.,complicated environment of obstacles and roads, a near-optimal 1968, Nilsson, 1971). These methods usually involve the searching piecewise-linear path is found within a few seconds. The planner of a visibility graph (Jarvis, 1983, Lozano-Perez and Wesley, 1979)can generate paths for robots capable of rotation about a point as or a tangent graph (Lui and Arimoto, 1991).

well as car-like robots that have a minimum turning radius. For car-Some algorithms (Brooks, 1983, Donald, 1984) search a set of like robots, the planner can generate forward and reverse paths.curves that are, in general, equidistant from obstacles. Others divide This software is currently implemented on a computer controlled the work space into grid cells (Grevera, 1988, Jarvis, 1985,Kawasaki Mule 500 all-terrain vehicle and a computer controlled Verbeek et al., 1986) and search for the shortest cell-to-cell path.John Deere 690 excavator.

Nonholonomic path planning methods (Barraquand and Latombe,INTRODUCTION

The purpose of a path planner is to obtain a sequence of This paper describes the implementation of an A* search recommended moves that will direct a robot from a specified initial algorithm used for two types of path planning for autonomous configuration to a specified goal configuration in a manner, so as, to mobile robots. The first uses a map of all known obstacles to avoid contact with all known obstacles. The sequence of moves determine the shortest collision-free path from a start configuration recommended by a path planner defines a path guaranteed to exist to goal configuration. The second determines the shortest path in free-space.

along a network of predefined roads, where a road is a preferred If the environment is largely static and well defined, and the travel path. The planner can generate paths for nonholonomic only constraint on the vehicle is that it does not collide with (steered-wheeled) vehicles and vehicles capable of rotation about a obstacles, the motion planning problem can be divided into two point (differential-drive and omnidirectional). For nonholonomic phases: finding a collision-free path and executing the collision-free robots, the planner can generate forward and reverse paths.

path. The initial phase is often called off-line path planning.

The literature is replete of off-line path planning algorithms, the 1989, Pin and Vasseur, 1990) search for paths for car-like robot vehicles. A few methods (Krogh and Thorpe, 1986, Warren, 1989)use artificial potential fields for off-line path planning.

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The major contributions of this paper are the technique used to graph of nodes. This is accomplished by generating a configuration generate paths for nonholonomic vehicles, the ability to use the space (Lozano-Perez and Wesley, 1979). Configuration space is the obstacle-avoiding planner for a wide range of mobile robots space in which the robot is represented as a point. The obstacles are (steered-wheeled, differential-drive, or omnidirectional), the mapped into this space by "growing" or expanding their boundaries,application of an A* search to accomplish two types of path and the work-space boundary is mapped into this space by dilating planning, and the laying of the foundation for the merging of the two its sides. The robot vehicle is symbolically shrunk down to a path planners into one multi-purpose path planner.

reference point by expanding all the obstacles and dilating the work-The output of the A* search is a set of via-points that define the space boundary by a scaling width, r , that accounts for the physical shortest path when they are connected by line segments. The extent of the robot vehicle. The task is then to find a safe path for a planner can be divided into four parts: defining a map of the point rather than a polygon.

environment, the creation of configuration space, the creation of The ability of the robot to get around the side of a known pseudo-obstacles, and the search algorithm.

obstacle depends to a large extent on the scaling width. For a DEFINING THE ENVIRONMENT

The site of operation is assumed to contain some regions where would allow corner points to barely touch the vehicle upon planning the robot vehicle is prohibited from operating. Obstacles are a path. The addition of a constant provides a safety factor.

represented as polygons for which the position of each vertex is The selection of r for a steered-wheeled vehicle is somewhat known. An obstacle is any region of space that we require the more complex. The scaling width is chosen based on the worst-case vehicle to avoid. Notice that obstacles can be subterranean or sub-corner clearance (cc). The path found by the search algorithm is planar, such as, craters, ditches, and valleys, or above surface, such then actually a lane of the width two times the scaling width. By as, debris, other vehicles, buildings, trees, telephone poles, and including a safety margin in the scaling width, a certain amount of equipment.

tracking and positioning error in path following is explicitly allowed A map of the environment, stored in a data file, contains known for.

obstacle data, work-space boundary points, the vehicle's dimensions The inner angle at each convex vertex of the polygonal and minimum turning radius, and points that define the center of obstacles is required to be 90E or more. Under the assumption that each road. Circular obstacles are represented by N-sided polygons.the turns from one lane to another can be approximated by a circular The boundary polygon and the obstacle polygons can be either arc, the corner clearance can be calculated as:

convex or concave. As a preprocessing step, this data is read by the cc = omroc - (omroc - r )/sin 1 - vwidth/2.0,

planner, and a map of the environment is stored in memory.

where omroc is the operational minimum radius of curvature, vwidth CREATION OF CONFIGURATION SPACE

An A* search requires the reduction of the environment to a

straddles one side of an obstacle and makes a 90E turn onto a lane

s differential-drive vehicle, a rule of thumb is to choose r to be s greater than one-half the diameter of the smallest circle that encompasses the robot vehicle. A one-half diameter expansion s s is the vehicle width, and 1 is half of the inner angle at the vertex in question (Figure 1).

The worst-case corner clearance occurs when the vehicle

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straddling another side of the same obstacle. Table 1 illustrates the worst-case corner clearance for several different values of the scaling width, r.s

Table 1. Estimating Corner Clearance

vehicle scale OMROC lane clearance width (ft)width (ft)(ft)width (ft)(ft)4.0 6.012.512.0 1.314.07.012.514.0 2.724.0

7.7

12.5

15.4

2.93

The first row of Table 1 shows that if r is chosen to be 1.5s times the vehicle width, there will exist a lane having at least a width of 12 feet from the start position to the goal position, centered on the path that is produced by the search algorithm, and guaranteed to be in free space. Figure 2 illustrates that the width of the lane about the path found by the planner will be, at minimum, two times the scaling width.

An additional restriction is made when planning paths for steered-wheeled vehicles. The length of each polygon side must be greater than or equal to "minlen", given by the equation:

minlen = (omroc - r ) [csc " + csc " ],

s 12where the angles " and " are one-half the inner angles for the 12vertices on the ends of the side in question. This protects against planning paths that contain u-turn type turns that do not respect the vehicle's minimum radius of curvature constraint.

CREATION OF PSEUDO-OBSTACLES

Greater care is necessary for determining an optimal path for a steered-wheeled vehicle than a differential-drive vehicle. A piecewise-linear path would suffice as a feasible path for a differential-drive vehicle. A steered-wheeled vehicle, however,requires lanes wide enough to ensure that turning onto other lanes can be accomplished without invading forbidden space.

The same search algorithm is used to determine the optimal path for the different types of robot vehicles. There is one difference, however, in the way the environment is set up for a steered-wheeled robot than for a differential-drive robot. When a search is going to be made for a steered-wheeled robot, pseudo-obstacles are defined around the start and goal positions.

Pseudo-obstacles are fictitious obstacles introduced in a region to ensure that kinematic constraints are not violated in that region by the search algorithm in determining an optimal path. As an example, consider two points, A and B, that are separated by 50 feet and lie in an obstacle-free environment. The goal position B is due east of the start position A. The vehicle is positioned at A, but is

oriented due north. The path planner is asked to find the shortest path for the vehicle from A to B. A straight-line path would suffice for a differential-drive vehicle, as the vehicle would need only to rotate about A until oriented eastward and then execute the path.For a steered-wheeled robot, however, the width of the lane would have to be at least two times the vehicle's minimum radius of curvature for the straight-line path to be feasible. Pseudo-obstacles eliminate the need for such an exorbitant scaling width.

Figure 3 illustrates a pseudo-obstacle placed at the start position A. The start pseudo-obstacle is oriented so that the line segments AD coincides with the start orientation of the vehicle.Line segment AD is checked to see if it lies in free space. If so, then D, the pseudo-start position, becomes the point at which the search algorithm begins its search. The rectangular region bounded by EFGH is treated as forbidden space. The pseudo-start point is placed an infinitesimal distance outside the rectangular region EFGH.

By using pseudo-obstacles at the start position and the goal position, the path found is guaranteed not to contain any turns less than 90E , where a path angle is the inner angle between two path segments. The line segment EH is four times the vehicle's minimum turning radius, and the line segment HG is two times the vehicle's minimum turning radius. ADHG and ADEF are both feasible paths.

The goal pseudo-obstacle acts as a one-car garage in that it ensures that the vehicle arrives at the goal position at the desired orientation. Further, the pseudo-obstacles ensure that the initial and final turns in the planned path respect the minimum turning radius associated with the vehicle. In planning a forward path, the start pseudo-obstacle orientation coincides with the vehicle's initial heading and, the goal pseudo-obstacle orientation is rotated 180E from the vehicle's goal heading. A reverse path for a steered-wheeled robot can be planned by simply rotating these fictitious obstacles 180E .

THE A* SEARCH ALGORITHM

The A* search algorithm described in this paper is used to generate shortest-distance collision-free paths in environments containing polygonal obstacles and shortest road routes along a network of predefined roads or preferred paths. The same algorithm is used for both types of off-line path planning. The factor that distinguishes what type of path is planned by the A* search is the predefined graph of nodes that is used during a search.

This section of the paper provides a general description of A* search algorithms and discusses an implementation for collision-free path planning and shortest road route planning.

General Description of an A* Search

The search algorithm is required to produce the shortest-distance path (SDP) from the start configuration to the goal configuration. The problem of determining the SDP can be reduced to finding the shortest path through an undirected graph. The most common graph used is a visibility (or adjacency) graph. A visibility graph can be viewed as a two dimensional symmetric matrix A, where the element A[i,j] contains the cost of traveling from node i to node j. This cost is usually a distance but it can include other factors, such as a terrain weight.

Off-line path planning problems have traditionally been approached using either the "mathematical approach" or the "heuristic approach". The mathematical approach is most concerned with determining an exact solution. Typically, these algorithms will examine each node of a graph in an ordered fashion. The heuristic approach uses known information about the problem's domain to develop algorithms that are computationally efficient. Typically, a search that uses a heuristic approach does not require the examination of every node in the graph. An exact solution, however, is not guaranteed by a heuristic search.

An A* search algorithm is a conglomerate of the two approaches described above. Information from the problem's domain is strategically incorporated into a formal mathematical approach to reduce the computation time, while yet guaranteeing to find the minimum cost path, if a path exists. Thus, although an A* search uses a heuristic, it is said to be algorithmic.

An A* search algorithm is optimal in the sense that it requires the examination of the smallest number of nodes in determining the minimum cost path. The output of an A* search is a set of via-points which represents a subset of the nodes in the searched graph. The SDP is obtained by connecting the via-points with line segments.

The algorithm A* is considered a family of algorithms. The choice of the heuristic h determines the family to which the algorithm belongs. At any given intermediate point during a search, several incomplete paths may exist. The incomplete path chosen to expand upon is the one with the lowest current estimate of the total

path length, where the total path length is the sum of the distance already traveled (backward cost) and an estimate h of the distance remaining (forward cost) to the goal.

The heuristic h can range from zero to the straight-line distance (as used in this implementation) for a SDP problem. The more accurate the estimate of remaining distance, the faster the algorithm will find the shortest-distance path. As long as the heuristic does not over-estimate remaining distance, the optimal path is guaranteed to be found.

Implementation of the A* Search

All nodes in the graph are numbered sequentially, starting with the start position. All obstacles, including pseudo-obstacles, are numbered in a clockwise fashion. If the path is required to remain within a specified work-space boundary, the boundary vertices are numbered as well. The last node N to be numbered is the goal node.

The task is to find the SDP from node 0 to node N.

Two linked-lists are maintained during the search: an open list (O) and a closed list (C). The elements placed in each list contain information, such as, the previous node i, the current node j, the backward cost (BC) from node 0 to node j, and an estimate of the total cost (TC) from node 0 to node N. This estimate is the sum of the backward cost and the heuristic estimate of the forward cost from node j to node N. Node i indicates how node j was reached.

The closed list is an ordered list, arranged on the basis of the total cost in each element in the list. The closed list is sorted from smallest to largest total cost. For each path segment investigated during the search, an element is placed in the closed list. Elements are sequentially removed from the top of the closed list and placed at the top of the open list until either an element is found in which j=N, i.e., the current node is the goal node, or the closed list is empty. If the goal node can be reached, the SDP can be constructed from the final open list.

A path segment is investigated during a search by "expanding"

on a node. To expand on a node i means to find all nodes j visible to node i that have not previously been expanded on, and place an element in the sorted closed list that specifies an estimate of total distance to the goal through each visible node j. Node j is visible from node i if the line segment connecting the nodes does not intersect the bounding line segments of polygons that define exclusion zones.

Obstacle-Avoiding Path Planner

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the environment.

Consider the example in Figure 4. The process begins by expanding on node 0, the start node. Nodes 1, 2, and 4 are visible from node 0. Elements for each of these nodes are placed in the closed list except for node 1 (Figure 5), since node 1 does not represent a tangent point. The first element in the closed list indicates that in moving from node 0 to node 2, the distance thus far traveled is 4.47. It is estimated that 8.59 is the total distance that needs to be traveled along this route to reach the goal node. The results of expansion #1 suggests that it will be shorter to take a route through node 2 than through node 4.

Next, the top element in the closed list is placed at the top of the open list. The search is to continue from node 2. The visible nodes from node 2 are 0, 1, 3, and 5. Node 0 is not considered since it has already been expanded on.

In expansion #2, elements for nodes 1, 3, and 5 are added to the closed list. The top element of the closed list is again transferred to the top of the open list. Since node j of this element is the goal node,the search is ended. The shortest path specified by the open list is 0-2-5.

The most expensive part of the search procedure is determining the visible nodes from a specific node. If the robot is differential drive or omnidirectional and the environment is not likely to change,then visibility can be established prior to searching for shortest paths by generating a visibility graph (VGRAPH) as a preprocessing step.Otherwise, visibility is determined when needed.

A visibility graph indicates where straight-line travel between nodes is possible. Figure 6 shows the visibility graph for the environment in Figure 4. A VGRAPH is a two dimensional symmetric matrix stored as a lower-triangular, nondiagonal,dynamically allocated two dimensional array, within which a 1indicates truth or visibility and a 0 indicates that visibility between the two nodes in question is blocked. Once the graph is established,a search can approve a candidate move by simply reading the appropriate element of the visibility matrix.

When determining visibility as needed, the line segment from the current node to each candidate node must be checked for intersection with the sides of all obstacles and the work-space boundary until an intersection is found, or until all sides have been considered. If j is visible from i , it is trivial to determine the euclidean distance between the two nodes. Otherwise, the distance between the nodes is an extremely large number that symbolizes infinity. While it is traditional for a visibility graph to store the

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distances between nodes, here integers are used to indicate visibility search. If the specified start or goal position is not in free to conserve computing memory.

space, an appropriate error message is returned.

The best way to increase the efficiency of an A* search where visibility is determined as needed is to reduce the number of nodes that have to be checked for visibility during a search. Several steps have been taken to improve the efficiency of this type of A* search:

a.If the user has specified that the path to be planned must remain inside the user-defined work-space boundary, then all convex boundary vertices are tagged as a preprocessing step to indicate they are not candidate move-to nodes during a search.Node h in Figure 7 is a convex boundary vertex. The shortest path will never require you to go into one of these corners along the way.

b.All concave obstacle vertices are tagged as a preprocessing step to indicate they are not candidate move-to nodes during a search. Node f in Figure 7 is a concave obstacle vertex. The shortest path will never require you to go into a "nook" along the way. This step allows obstacles to contain convex and concave vertices.

c.An array containing all the nodes keeps track of which nodes have been expanded on. It would be wasteful to expand on the same node more than once.

d.All nonpseudo-obstacle and boundary nodes in the graph are tagged as a preprocessing step to indicate whether they are in free space or in forbidden spac

e. Pseudo-obstacles are expanded and their nodes are tagged as the first step during each search. Then, all other nodes are checked to see if any lie within the pseudo-obstacles. This allows obstacles to intersect.All nodes tagged as in forbidden space are not used within a

e.The algorithm may or may not consider backward moves (see Figure 7). In environments that are not obstacle dense,backward moves may be ignored. When the angle 1 between the vector from the current node a to the goal node g and the vector from the current node to a candidate node b exceeds some critical angle, the candidate node is not considered.When backward moves are ignored, however, the path found is not guaranteed to be the SDP.

f.If a node on an expanded obstacle is not a tangent node with respect to the current node, it is not considered. In Figure 7, nodes d and e are tangent nodes with respect to node a , but node c is not.

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Figure 10. Computer controlled Kawasaki Mule all-terrain vehicle.

Figure 11. Computer controlled John Deere excavator.

g.During a search, the visibility of a candidate point is road number of -1, which serves as a flag that this element and all checked against an obstacle only if the vector from the connections to the node represented by this element should be candidate node to the current node is bounded by the vectors removed immediately after a search, as this node is valid for only a from the candidate node to the tangent nodes on the obstacle.subset of off-road points, i.e., those points that lie on the line defined In Figure 8, vector a is not bounded by the tangent vectors to the obstacle A (vectors b and c ). It is not necessary, therefore,to check to see if each line segment of obstacle A intersects the line segment from the current node to the candidate node.

Roads Path Planner

For the roads planner, a partial visibility or adjacency list is generated as a preprocessing step. In Figure 9, two roads are defined by a set of two center points each, (1,2) and (3,4). Each node in the graph is represented by an element in the horizontal (major) linked-list. An element for each intersection point is added to the end of the adjacency list just prior to a search. Intersection elements are arbitrarily given a road number of 0. In Figure 9, the two roads intersect at the point labeled node 5.

The minor elements under a major element indicates which nodes can be reached from the node represented by the major element. For example, the adjacency list shows that nodes 2 and 5can be reached from node 1. It is said that a connection exists between nodes 1 and 2 and nodes 1 and 5.

The specified start and goal positions are not required to lie on the network of predefined roads. Simply making a connection to the nearest road node from an off-road point, however, will not suffice for finding the SDP since the total path length is dependant on the on-road path length as well as the lengths of as much as two off-road connections. Connections are made from an off-road point to the closest point on each road and temporarily added to the adjacency list.

When the closest point on a road is not a point that is already in the adjacency list, a new temporary element is added to the end of

the adjacency list to represent this point. This element is given a by the off-road connection but within the outer work-space boundary.

A search is conducted from node s to node g, the start and goal points as shown in Figure 9. The length of off-road connections are weighted by a factor of two. The net effect is that it costs twice as much to travel off-road than it does to travel on-road. The algorithm works to force a path towards on-road travel. All off-road connections are checked to determine if they are collision-free.

Constructing the Optimal Path

At every intermediate stage during a search, several candidate paths may exist. Expansion occurs at the node at the end of the path that is estimated to yield the SDP. Once a search has been successful, the final open list will contain not only the SDP but the other uncompleted candidate paths. The SDP must be extracted from this list containing extraneous information.

The SDP from node 0 to node N can be described by the ordered set of nodes p[i], for i=1...n, where p[1]=0, and p[n]=N.Recall that the nodes 0 and N represent the start and goal positions respectively. The entire set p[i] can be constructed from the final open list when a search has been successful. There is one basic rule that must be followed in extracting the optimal path from the final open list, however.

Given a node y known to exist along the optimal path,the node x from which y is reached can be determined by moving down the open list until the level closest to the bottom that contains y as the current node entry

start point

goal point

start goal point

point

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Figure 12. Example shortest-distance collision-free path.

Figure 13. Example path along a network of roads.

is reached. The node in the corresponding previous node entry is x.

The node at the top of the final open list yields the last two not required.

nodes in the optimal path, i.e. ...p[n-1] and p[n]. The rest of the optimal path can be extracted by continuously working backwards from the node p[i] at the left of the partially specified path p. The steps are as follows:

1. Let y=the node to the left of the partially specified path.

2. The node x from which y is reached can be found by each new vertex, and determining which vertices can be used in a applying the given rule. Insert this node at the left of the search. The nodes that can not be used in a search are nodes that lie partially specified path.

outside the work-space boundary, nodes that lie inside one or more 3. Repeat steps 1 and 2 until the inserted node x=0.

convex nodes on the dilated work-space boundary.

RESULTS AND CONCLUSIONS

An off-line path planner based on the A* search method was weighting all off-road distances by a scaling factor. The scaling developed on a Silicon Graphics workstation using the C factor times the euclidean distance of an off-road path segment programming language and is currently implemented on a computer yields the cost of traveling on the off-road path segment.

controlled Kawasaki Mule all-terrain vehicle (Figure 10) and a The obstacle-avoiding planner takes less than 7 seconds to plan computer controlled John Deere excavator (Figure 11). Two types a path in the same environment. The reason the obstacle-avoiding of path planning can be accomplished using this algorithm: the planner takes longer to plan a path than the roads planner is because planning of shortest routes along a network of predefined roads the obstacle-avoiding planner determines visibility as needed.(Figure 12) and the planning of shortest-distance collision-free paths Determining visibility as needed eliminates the need for N memory in an environment where the locations of obstacles are known locations, where N is the number of obstacle and boundary nodes in (Figure 13). On the vehicles, the path planner runs on a VME the graph.

computer, operating under the VxWorks operating system.

The obstacle-avoiding path planner returns a status that The use of pseudo-obstacles at the start and goal position indicates the SDP was found or the reason why a path was not enables the planning of forward and reverse paths for steered-found. The error-related status messages are:

wheeled vehicles which contain a minimum turning radius. Pseudo- 1. The start position is not in free space.obstacles ensure that the initial and final turns in a path are feasible,

2. The goal position is not in free space.and that the last leg of a path coincides with the goal heading. For differential-drive or omnidirectional vehicles, pseudo-obstacles are For an environment that contains 90 center-line road points and 170 obstacle and work-space boundary points, the entire preprocessing stage takes less than 5 seconds when running on the on-board computer. The preprocessing stage includes determining the intersection points for the set of roads and establishing an adjacency list for road points, expanding all obstacle polygons and dilating the boundary polygon by a scaling width and numbering expanded obstacles, concave nodes on expanded obstacles, and Planning road paths is virtually instantaneous. Road paths can be planned from and to points that do not lie on roads. Start and goal points are forced to be connected with a nearby road by 2

3. A path could not be found.Dijkstra, E. W., 1959, "A Note on Two Problems in Connexion with

4. The start heading is blocked by an obstacle or the boundary.

5. The goal heading can not be achieved.

Errors concerning the start and goal heading apply to only steered-wheeled vehicles. A start or goal position is reported to be in forbidden space if it lies outside the work-space boundary or inside an expanded obstacle. A path can not be found when an obstacle, or a set of intersecting obstacles, divide the work space into two regions, one containing the start position and the other containing the goal position. An option exists to allow the planning of paths out of obstacles but never into one.

The planner has proven to be robust and efficient in environments where obstacles are not clustered or exorbitant. The execution time for a search increases exponentially with each additional node to the graph. In obstacle dense environments, overestimating the size of obstacles can lead to significantly longer paths than required or no acceptable path at all. Overestimating the size of obstacles by adding a highly subjective safety factor to the scaling width (as well as modeling circular obstacles as polygons) can close off passage-ways that may have been acceptable. FUTURE WORK

In the future, the roads path planner and the obstacle-avoiding path planner will be merged into one planner. This will eliminate the scenario where the planned path has segments that run nearly parallel to a nearby road. Different shaped start and goal pseudo-obstacles will be experimented with in an effort to allow paths planned for car-like robots to contain a combination of forward and reverse maneuvers.

ACKNOWLEDGEMENTS

The authors wishes to thank Wright Laboratories, Tyndall Air Force Base, Florida, for funding this work.

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赏析艺术手法(解析版)

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