How to Draw Visibility Graph for Robot Path Planning
For path planning algorithms based on visibility graph, amalgam a visibility graph is very time-consuming. To reduce the computing time of visibility graph construction, this newspaper proposes a novel global path planning algorithm, bidirectional SVGA (simultaneous visibility graph structure and path optimization by ). This algorithm does not construct a visibility graph before the path optimization. However it constructs a visibility graph and searches for an optimal path at the aforementioned fourth dimension. At each step, a node with the lowest interpretation price is selected to be expanded. Co-ordinate to the status of this node, dissimilar through lines are drawn. If this line is free-collision, it is added to the visibility graph. If not, some vertices of obstacles which are passed through by this line are added to the OPEN listing for expansion. In the SVGA process, only a few visible edges which are in relation to the optimal path are drawn and the nigh visible edges are ignored. For taking reward of multicore processors, this algorithm performs SVGA in parallel from both directions. By SVGA and parallel operation, this algorithm reduces the computing time and space. Simulation experiment results in different environments prove that the proposed algorithm improves the time and infinite efficiency of path planning.
1. Introduction
Intelligent mobile robots accept been widely used not simply in military activities but also in ceremonious life. Navigation is the fundamental problem of the mobile robot technology. The quandary of navigation could be summarized: "Where am I? Where do I go? How do I become there? [i]" The offset two issues are well-nigh localization and mapping, and the last problem is nearly path planning.
Path planning is to determine a collision-free path between the get-go and target position by a performance criterion such equally the distance, time, and energy consumption [2]. According to whether an surround is known or non, there are two categories of path planning algorithms, namely, local and global path planning. A mobile robot plans a feasible path in an unknown or dynamic environment by the data which is gotten from sensors. This is known as local or online path planning. Global path planning is to search for an optimal path based on consummate data about stationary obstacles and is also known as offline path planning.
Classical local path planning approaches include artificial potential field, vector field histogram, dynamic windows, and bug. Withal these algorithms suffer from some drawbacks, such as trapping in local minima and unsmooth planned path. To overcome such limitations, there have emerged some proposals combining evolutionary or heuristic algorithms with classical path planning approaches, such equally parallel evolutionary artificial potential field (PEAPF) [3] and EgressBug [4]. Global path planning consists of two steps: environmental modeling and path optimization. Configuration space (C-Infinite) is a cardinal approach to environmental modeling trouble. Prison cell decomposition and roadmap are well known environmental modeling approaches based on C-Space. The simplest prison cell decomposition is grid with a stock-still resolution. The master difficulty of this approach is how to determine the size of prison cell. Some improved approaches are proposed to solve this difficulty, such as fan-shaped grid map [five] and quad-tree grid map [six]. Voronoi diagram and visibility graph are two main roadmap approaches. Voronoi diagram is proposed by Dunlaing and Yap. This approach constructs a roadmap by using points equidistant from two or more obstacles [7]. Artificial intelligent path planning methods apply modern artificial intelligent technology to mobile robot path planning [8]. Artificial neural nets, fuzzy logic, genetic algorithm, and particle swarm optimization are widely practical technologies in the path planning enquiry. These artificial intelligent technologies overcome many drawbacks of classic approaches, and they reinforce the robot intelligence. Only these technologies are difficult to implement path planning independently, and they need to be combined with classic approaches [9]. Speedily exploring random tree (RRT) is a sampling-based path planning algorithm and is being implemented heavily in recent years [10].
Visibility graph modeling is to construct a compact, undirected graph that registers visibility amidst vertices of obstacles [eleven]. For its simplicity, visualization, and completeness, visibility graph is still useful in many applications. Withal, constructing a visibility graph is very time-consuming. Some algorithms aiming efforts at reducing computing time are proposed by scholars. Tran et al. proposed a parallel-oriented visibility graph structure algorithm. This algorithm divides the environment into some parts and constructs the visibility graph for each part in parallel [12]. Zhang et al. apply a simplified visibility graph suitable for path planning algorithm to model surround [thirteen]. By ignoring redundant obstacles which do not touch on the issue of path planning, information technology improves the efficiency of path planning. Some techniques are used to better the time complexity such as reducing the amount of visibility edges, simplifying obstacles to rectangles, and combining the tiny obstacles [14, 15]. Some intelligence algorithms such as quantized algorithm [16] and ant colony [17] are used in path planning based on visibility graph to amend the computational efficiency.
The above improved algorithms all firstly construct a visibility graph and so search for an optimal path based on this visibility graph. Constructing a visibility graph is very time-consuming and the optimization efficiency is drastically decreased with the increasing of border number. To improve the computational efficiency, this paper proposes an improved global path planning algorithm, bidirectional SVGA (simultaneous visibility graph construction and path optimization by ). Ii approaches to increase the performance of global path planning are introduced to the proposed algorithm: (i) Information technology does not construct the visibility graph before the search process, just it constructs the visibility graph and searches for the path at the aforementioned time. Whether a directly line between ii vertices is collision-gratis or non is the main process of the visibility graph construction. This process is called visibility judgment in short. In order to improve the efficiency of global path planning, the chief purpose of the proposed algorithm is to reduce the executions of visibility judgment. 2 strategies are used in this arroyo. One is to make use of the relationship between the start position, target position, and all vertices. Many vertices unrelated to the path planning result are ignored and visibility judgment between them does not need exist executed. The other is to brand apply of the heuristics search. If a line between 2 vertices is more related to the optimal path, the visibility judgment of this line is executed earlier. Information technology causes that the visibility judgment less related to the optimal path may not be executed. (2) SVGA is performed in parallel from both directions. Past taking advantage of multicore processors, it improves the efficiency of global path planning.
The rest of this paper is organized as follows. Section 2, at outset, introduces the central concepts of global path planning based on visibility graph. The global path planning algorithm based on bidirectional SVGA is presented in Department 3. The performance analysis is presented in Department four. The experimental results are provided in Department 5. Finally, Section half-dozen concludes this paper.
2. Global Path Planning
Global path planning can be described equally a search trouble in a iv-atom formulation : (i) : search space. (ii) : start position. (iii) : target position. (iv) : obstacle fix.
If a path series is a solution, , , and . Global path planning aims to search for an optimal solution in all solutions.
2.one. Visibility Graph
Visibility graph is constructed by joining the lines which connect the first position, target position, and vertices of obstacles. These lines do not intersect obstacles. In other words, these lines are visible. The vertices and lines form a visibility graph . Figure 1 shows a visibility graph where grayness polygons correspond obstacles, is the start position, and is the target position. After a complete visibility graph is synthetic, the shortest path is then identified by some search algorithms.
2.2. Algorithm
Dijkstra is a goal-directed search algorithm. Based on Dijkstra, adds a potential office to the priority key of each node in the queue [18, xix]. The potential function is an estimation of the path length through the vertex . represents the estimated shortest path length between the starting time and target position through the vertex . is the real path length from the start position to the vertex , and is heuristics function which estimates the shortest path length from the vertex to the target position.
iii. Bidirectional SVGA
Classic global path planning algorithms based on visibility graph search for a path after the complete visibility graph roadmap is constructed. The time complexity of visibility graph construction is O(N iii), where is the number of vertices. To reduce the visibility graph construction time, this newspaper proposes the bidirectional SVGA algorithm which does non construct a complete visibility graph. This algorithm simultaneously constructs the visibility graph and searches for the optimal path by . In the optimization procedure only related edges are added to the visibility graph, and the added edges equally bear upon the optimization process. From both directions, SVGA is performed in parallel, for it can take advantage of multicore processors [20]. One direction is from the start position to the target position, and it is called forwards SVGA. The other is from the target position to the start position, and information technology is called backward SVGA.
For convenience, unidirectional SVGA is firstly introduced. Information technology performs SVGA from one direction. Figure 2 shows this procedure.
At each pace of the optimization procedure, one through line is drawn. The through line is a directly line which tin can pass through obstacles to connect two vertices. In other words, the line would be the shortest path between the ii vertices supposing the robot can laissez passer through obstacles. According to this line, related vertices are added to the OPEN list for expansion.
Firstly, a through line from the showtime position to the target position G is drawn. Information technology can be seen that if the robot wants to go to G, it must laissez passer past the vertex A 1 or A 2. A 1 and A 2 are added to the Open listing. The node A 1 with the lowest interpretation cost is expanded, and a through line from South to A 1 is drawn. Considering this line does not laissez passer through whatsoever obstacle, it is added to the visibility graph. So a through line from A one to G is drawn, and B 1 and B two are added to the Open up list. Next A 2 is selected to be expanded. The line from S to A 2 is added to the visibility graph, for the line is free-collision. A through line from A 2 to One thousand is fatigued. C 1 and C 2 are added to the OPEN listing. B 1 is selected for its lowest estimation cost. A through line from A 1 to B 1 is drawn. Because this line is free-collision, it is added to the visibility graph and B one is connected to be expanded. A through line from B 1 to Thousand is drawn, and it is likewise added to the visibility graph for there is no collision. Because the optimization arrives at G, this algorithm is finished and the optimal path is output.
Bidirectional SVGA is similar to unidirectional SVGA, and the difference is that bidirectional SVGA is performed in parallel from both directions. For parallel performance, at that place are two Open lists, OPENF and OPENB. They are used in forward and backward SVGA, respectively. where represents the th node. Each node is related to a vertex, and index represents the index of this vertex in the vertex set V. prev represents the preview node. condition represents whether there is a visible edge between this node and preview node or non. Unlike from the archetype algorithm, in this algorithm is not the real path length but an estimated path length from the commencement position to the current node. represents and it is calculated as where represents the straight-line distance from the preview node to the current node. hn represents the heuristics function . Information technology is the straight-line distance from the current node to the target position. Nodes of OPENF are not deleted in this algorithm for the same node might exist visited repeatedly. priority represents the access priority and visit count. A dead circle in the optimization process tin can exist avoided by the priority holding. SVGA is based on the graph search and each node is identified by alphabetize and prev. It means that when alphabetize and prev are both equal, the nodes are the aforementioned. A node is represented as in a simplified form. The structure of OPENB is the aforementioned as OPENF.
Two Airtight lists, CLOSEDF and CLOSEDB, are used in this algorithm. Each node is defined as where index is the index in V. is the existent path length between the first position and the current node.
The proposed algorithm consists of two steps: initialization and optimization. In the initialization process, two Open lists and two CLOSED lists are initialized. The first position, target position, and vertices of all obstacles are added to the vertex set 5.
Simply edges between adjacent vertices of some obstacles are added to the border ready Due east.
The target position K is added to OPENF, and the kickoff position S is added to OPENB. where D SG is the Euclidean distance betwixt S and Thousand, priority = 0 represents that the node has not been expanded, and status = 0 represents that the visibility between the 2 nodes is unknown.
Two CLOSED lists are initialized as The node is the initial node if prev equals 0.
But frontward SVGA is specified. Astern SVGA is the aforementioned as forward SVGA. At each step of the optimization process, the node with the lowest interpretation cost is selected to be expanded. The estimation part is defined as where MAX represents the max possible distance from S to Chiliad. After this expansion, the priority value of this node minuses 1.
Different through lines are drawn according to the status value of this node. A through line from the preview node to this node is fatigued provided that status equals 0. A function is used to determine whether a line betwixt the vertices and is visible or not. is the set of boundaries of obstacles. If the line intersects any obstacle, the function returns imitation. If this line is visible, it is added to E, and this node is added to CLOSEDF, and status is assigned to 1. where cNode represents the current node. If this node exists in CLOSEDB, information technology means that frontward and backward SVGA encounter the current node. The optimization is finished and the optimal path is output. If this line passes through an obstruction, no more than two vertices of this obstacle are added to OPENF. If a vertex of this obstacle is uttermost from the line in whatever direction of two directions and is not in OPENF, information technology is added to OPENF. where is the index of this vertex in Five, is the directly-line distance from preview node to the vertex, and D PG is the straight-line distance from the vertex to G. The pseudo-lawmaking of adding vertices of the crossed obstacles to the OPEN listing are shown beneath. Process (01) For (o in crossedObs) (02) = maxDistanceFromTop(o, line); (03) If () (04); (05) End If (06) If (moreFarLineSouth2M exists) (07); (08) End If (09) = maxDistanceFromBotton(o, line); (x) If (exist(, node.prev) == false) (xi) addVertex(, node.prev); (12) Terminate If (thirteen) If (moreFarLineSiiChiliad(, node) == true exists(, Due south) == false) (xiv)addVertex(, S); (xv) End If (sixteen) End For Finish Procedure
If the status value of this node equals 1, a through line from the current node to 1000 is drawn. If this line has been in E or is complimentary-standoff, the optimization is finished and the optimal path is output. If the line passes through an obstacle, no more than two vertices are added to OPENF. where is the altitude from the current node to the new node.
The flowchart of frontward SVGA is shown as Figure 3. Parallel performance causes unlike stop criterions. If the following 4 end criterions are satisfied, bidirectional SVGA is over and the optimal path or no path is output. (i) Forward SVGA arrives at . (ii) Backward SVGA arrives at . (iii) Forward and backward SVGA meet the aforementioned node. (4) The priority value of the node with the everyman estimation price is less than 0.
4. Functioning Analysis
iv.1. Completeness
If SVGA satisfies the following conditions in an environment with limited vertices, it is complete. (1) SVGA is guaranteed to observe a solution when at that place is a viable path betwixt the start and target positions. (2) SVGA is guaranteed to exist finished in finite steps when at that place is no path.
SVGA tin plan a path by through lines when in that location is a feasible path between the start and target positions. At each footstep, a robot passes through the vertices far away from the through line to avoid the obstacles. These vertices are added to the OPEN list for expansion. If there are obstacles between these vertices and the offset position or the target position, SVGA continues drawing the through lines and the vertices of these obstacles are besides added to the Open list. By the finite through lines, a mobile robot can avoid the obstacles to arrive at the target position.
When all possible paths are explored, the priority of every node is less than null. It indicates there is no path between the start and target positions and SVGA is over.
Considering bidirectional SVGA executes unidirectional SVGA at most twice, it is complete when unidirectional SVGA is complete.
iv.2. Optimality
Firstly, the path planned by SVGA is proved to be optimal, so the path planned by bidirectional SVGA is proved to exist optimal.
2 conditions are required for optimality. One is that the incomplete visibility graph constructed past SVGA includes all edges on the optimal path. The other is that SVGA can find the optimal path from this incomplete visibility graph.
The starting time step is to testify that the incomplete visibility grapy includes all edges on the optimal path. If the through line does non cross whatsoever obstacle, it is an optimal path. This line is added to the visibility graph as a visible edge and incomplete visibility graph includes the border of this optimal path. If this line is invisible, there are some obstacles between S and 1000. There are only two means which let a robot brim an obstacle through the optimal path. The first is that the robot goes through a vertex of this obstruction which is the farthest from the line in any management of ii directions. The second is that the robot goes through a position which is far away from this obstacle. It tin can exist seen from Figure 2 that if the robot skirts the obstacle O 1, it must laissez passer through the vertex A 1 or A 2 or a position which is farther from the through line than A one or A two.
(a) The robot skirts an obstacle by the vertex P which is the farthest from the trough line in whatsoever direction of two directions. The vertex P is added to the Open up list when SVGA draws the through line . Considering P is on the optimal path, it must be selected for expansion. The through line is drawn. If this line is visible, information technology is added to the visibility graph. If this line is invisible, the robot should skirt the crossed obstacles to arrive at P. The robot must go through the vertices of these obstacles which are farthest from the through line in any management of two directions, or far abroad from these obstacles. By this recursive inference based on the two ways, information technology tin can exist concluded that the vertices on the optimal path from S to P are added to the Open up list, and the edges on the optimal path are added to the visibility graph.
(b) The robot skirts an obstruction by a position which is far away from this obstacle. Information technology is based on the post-obit 2 cases.
The first case is that there is a vertex Q of the other obstacle which is farther away from the through line than the vertex P of this obstruction. The robot goes from S to Q or goes from Q to G past skirting this obstacle. When the through line is drawn, the vertices P and Q are both added to the OPEN listing. The through lines from Southward to Q and from Q to G are drawn. If these lines are visible, they are added to the visibility graph. If these lines are invisible, the robot skirts these obstacles by the two ways.
The other example is that at that place is an obstacle between S and P or between P and K, and the vertex Q of this obstacle is further away from the through line than P. If this obstruction is betwixt Due south and P, the through lines and are drawn. The edges between these vertices are added to the visibility graph by through lines. If this obstacle is between P and G, the nodes and are both added to the OPEN list for expansion. If there are some obstacles between these vertices, that the showtime condition is satisfied can be proven by the recursive inference based on the ii ways.
No matter in any case, the visibility graph synthetic by SVGA includes all edges on the optimal path by this recursive inference based on the two ways.
The next step is to evidence that SVGA can discover the optimal path based on this visibility graph. The estimation office of SVGA is nondecreasing. is supposed as the successor of node x. where is Euclidean altitude between and x. Past the general triangle inequality, it can be concluded that Considering the estimation function is nondecreasing, SVGA determines a node to exist expanded according to a nondecreasing sequence. The first selected target node for expansion is the optimal solution [21]. The estimation price of all succor nodes is not less than the outset selected target node.
It is assumed that forward and astern SVGA meet the same node when the expanded node of 1 directional SVGA is in the CLOSED listing of the other directional SVGA. Bidirectional SVGA is over when forward and backward SVGA meet the same node. This node may exist the get-go position, target position, or the vertex of an obstruction. The path from the outset position to this vertex found by forward SVGA and the path from target position to this vertex found by backward SVGA are both optimal, so the constitute path is optimal.
4.three. Complication
iv.3.1. Fourth dimension Complexity
SVGA uses algorithm to determine a node to be expanded and through lines to determine vertices to be added to the OPEN list. The time complexity of algorithm is O(Northward 2). SVGA adds new nodes by visibility judgment of a through line. The fourth dimension complexity of the part is O(Due north), so the fourth dimension complexity of SVGA is O(Due north 3), the same grade as complete visibility graph construction. But a structure of the complete visibility graph involves all vertices. This construction needs to call the part times. SVGA ignores the vertices which are independent of the optimal path, and the involved vertices are much fewer than the vertices of consummate visibility graph structure. In Figure 1, there are 28 vertices. The complete visibility graph construction needs to call the office 756 times. By ignoring the most vertices, SVGA only phone call the part seven times. Information technology can exist seen that the calculating time of SVGA is far lower than the complete visibility graph construction and bidirectional SVGA is faster than SVGA for parallel functioning.
iv.iii.2. Infinite Complexity
The huge retentiveness requirements of global path planning algorithms based on visibility graph are the visible edges. The infinite complexities of complete visible graph and SVGA are both S(N 2), but retentiveness requirements of SVGA are fewer than the complete visibility graph. The complete visibility graph keeps all visible edges in memory, but SVGA ignores the most visible edges and keeps a few visible edges related to the optimal path. As shown in Figures one and 2, consummate visibility graph includes 107 visible edges and SVGA but includes 33 visible edges. The number of visible edges in bidirectional SVGA is no more than merely twice the number of SVGA.
5. Experimental Results and Give-and-take
The simulation experiments are carried out to validate the effectiveness of SVGA and to compare SVGA with some global path planning algorithms based on visibility graph. The beginning algorithm searches for an optimal path by afterwards a consummate visibility graph is synthetic. It is chosen complete VG + in curt. The second algorithm uses a simplified visibility graph to implement environmental modeling and to search for an optimal path [thirteen]. It is called simplified VG + in short. In this algorithm, redundant obstacles which do non touch the path planning outcome are removed past because positions of obstacles, the outset and target points. The third algorithm uses parallel-oriented visibility graph to implement environmental modeling and to plan a path [12]. This algorithm combines the modified visibility graph and parallel computation to improve calculating time. It is called POVG + in short. The fourth is unidirectional SVGA and the concluding is bidirectional SVGA. For a off-white comparison between these algorithms, they are tested by using the aforementioned second environment with obstacles.
Firstly, the environment shown in Figure 4(a) is used to exist tested for the comparing between five algorithms. There are 6 obstacles and 29 vertices including Southward and Thousand. The position of S is at (43, 101), and the position of 1000 is at (476, 425). All algorithms except POVG + can detect the optimal path, and the length of this path is 574.61.
(a) Surroundings
(b) Complete VG +
(c) Simplified VG +
(d) POVG +
(e) Unidirectional SVGA
(f) Bidirectional SVGA
The complete VG + algorithm constructs a consummate visibility graph shown as Figure 4(b). The construction of this visibility graph takes 0.66 s and 106 visible edges are added to the visibility graph. The function is chosen 812 times. This algorithm uses algorithm to search for an optimal path based on the visibility graph. The optimization process takes 0.05 s. At the cease of the optimization, the size of the OPEN list is 67. It tin can be seen that the main phase of this algorithm is to construct the visibility graph, and information technology is very fourth dimension-consuming. Simplified VG + algorithm constructs a simplified visibility graph shown as Figure iv(c). By ignoring the redundant obstacles for the optimal path, the number of visible edges is reduced to 38. This algorithm takes 0.39 due south, including the visibility graph construction time 0.36 s and the optimization fourth dimension 0.03 due south. In the visibility graph construction process, the office is called 240 times. The size of the Open up list is xiii at the end of the optimization. POVG + takes 0.31 south to find the optimal path, and it is shown in Figure 4(d). This algorithm divides the environment into the three regions. In each region, this algorithm constructs the visibility graph and finds the path, respectively. At last, 3 subpaths are combined to a complete path. POVG + adds 70 edges to the visibility graph and calls the office around 90 times for each region.
By ignoring the vertices contained of the optimal path, SVGA improves the path planning efficiency. Past half dozen through lines and 4 edges, unidirectional SVGA finds the optimal path. It merely calls the part ten times. It takes 0.06 s, and the size of the Open up list is six. This process is shown in Figure four(e). Because parallel performance causes dissimilar execution sequences, the time and space consumption of each performance for bidirectional SVGA are a petty different from 1 another even though in the same surroundings. Figure iv(f) shows the event of one performance. In this performance, the running fourth dimension is 0.05 s, and the sum size of OPENF and OPENB is 15. Comparison with the other iii algorithms, whether unidirectional or bidirectional SVGA improves the efficiency of global path planning based on visibility graph enormously. For computational time, there are 90 percent improvement comparing with the commencement algorithm, eighty-4 pct improvement comparison with the 2d algorithm, and fourscore pct improvement comparing with the tertiary algorithm. Comparing unidirectional SVGA, bidirectional SVGA takes less computational time and more computational space.
Next, another comparing is carried out in a complex environs. This surroundings is shown in Figure five(a). There are fifteen obstacles and 165 vertices in the surround. Due south is at (62, 12) and Thousand is at (375, 486).
(a) Circuitous Environment
(b) Complete VG +
(c) Simplified VG +
(d) POVG +
(e) Unidirectional SVGA
(f) Bidirectional SVGA
As shown in Figure 5(b), complete VG + draws 790 visible edges. The total running time is 24.79 due south, including the visibility graph structure time 23.59 southward and the optimization time i.twenty s. The time of visibility graph construction increases speedily for the role is called 27060 times with the growth of the vertex number. In fact, many visibility judgments are contained of the optimal path searching. A large number of visible edges crusade the more than fourth dimension and space consumption on the path optimization. At the finish of the optimization, the size of the OPEN listing is 325. Simplified VG + takes 21.32 due south and constructs 663 visible edges. The running time of the simplified visibility graph structure is 20.17 southward, and the running fourth dimension of the path optimization is 1.fifteen s. At the end of the optimization, the size of the OPEN listing is 310. From Figure 5(c), it can be seen that there is a piffling improvement comparing with complete VG + for most obstacles are related to the optimal path. POVG + takes 7.84 south to discover a feasible path. It divides the environment to four regions and is shown in Figure five(d). This algorithm adds 323 edges to the visibility graph and calls the part 7918 times for all regions. Unidirectional SVGA takes one.34 s to find the optimal path, and the size of the Open list is 129. Effigy v(eastward) shows the optimization process. By the through lines and the heuristics search, the optimal path can exist quickly found by SVGA. This algorithm simply called the function 65 times. Bidirectional SVGA takes 0.97 s to detect the optimal path and information technology shown in Figure 5(f). At the cease of the optimization, the sum size of OPENF and OPENB is 173. Whether in time or space consumption, unidirectional and bidirectional SVGA are far improve than the other three algorithms.
To farther validate the efficiency of the proposed algorithm, v algorithms are compared in a 150 × 150 area. The max length of edges is thirty, and the vertex number of each obstacle is no more than 10. Five categories of environments are used in this comparison. The numbers of obstacles in different categories are non the same, and they are half-dozen, 9, 12, 15, and 18, respectively. Each category of environment is randomly generated 100 times. The results are shown in Table 1.
| | ||||||
| Algorithms | Number of obstacles | Boilerplate number of vertices | Average number of edges in Due east | Average running time/southward | Best running time/southward | Worst running time/s |
| | ||||||
| Complete VG + | 6 | 41 | 171 | 0.82 | 0.67 | 1.38 |
| Simplified VG + | 76 | 0.35 | 0.002 | 1.19 | ||
| POVG + | 131 | 0.32 | 0.nineteen | 0.45 | ||
| Unidirectional SVGA | 36 | 0.05 | 0.002 | 0.19 | ||
| Bidirectional SVGA | 40 | 0.04 | 0.002 | 0.17 | ||
| | ||||||
| Complete VG + | nine | sixty | 288 | ii.84 | i.38 | 5.58 |
| Simplified VG + | 183 | one.87 | 0.002 | v.18 | ||
| POVG + | 174 | 0.87 | 0.64 | 1.15 | ||
| Unidirectional SVGA | 59 | 0.12 | 0.002 | 0.34 | ||
| Bidirectional SVGA | 64 | 0.09 | 0.002 | 0.31 | ||
| | ||||||
| Complete VG + | 12 | 79 | 405 | half dozen.75 | 4.87 | 7.32 |
| Simplified VG + | 308 | five.06 | 0.27 | 6.50 | ||
| POVG + | 203 | ane.85 | 1.44 | 3.12 | ||
| Unidirectional SVGA | 84 | 0.18 | 0.01 | 0.85 | ||
| Bidirectional SVGA | 92 | 0.14 | 0.01 | 0.76 | ||
| | ||||||
| Complete VG + | fifteen | 95 | 535 | 11.96 | seven.38 | 16.42 |
| Simplified VG + | 410 | 9.xviii | 6.24 | 17.45 | ||
| POVG + | 235 | 2.73 | 1.54 | iv.74 | ||
| Unidirectional SVGA | 117 | 0.40 | 0.17 | 1.63 | ||
| Bidirectional SVGA | 131 | 0.32 | 0.14 | ane.47 | ||
| | ||||||
| Complete VG + | 18 | 113 | 601 | 17.28 | fourteen.82 | 31.18 |
| Simplified VG + | 516 | xv.02 | viii.73 | 32.29 | ||
| POVG + | 269 | 4.12 | two.89 | 7.26 | ||
| Unidirectional SVGA | 140 | one.03 | 0.21 | iv.07 | ||
| Bidirectional SVGA | 167 | 0.81 | 0.18 | four.39 | ||
| | ||||||
It can be seen whether unidirectional or bidirectional SVGA is better than the other three algorithms in any environs. Even though in the environs with 18 obstacles, the proposed algorithm tin can detect an optimal path in one second for most test cases. However the boilerplate running time of complete VG + is 17.28 southward and 31.xviii s in the worst status.
Unidirectional SVGA, bidirectional SVGA, and simplified VG + all firstly draw a direct line between the start and target position. If this line is collision-free, the search processes of the iii algorithms are over. If there are very few obstacles or no obstacles which affect the robot going to the target position, unidirectional SVGA, bidirectional SVGA, and simplified VG + all can detect a path quickly. Based on these environments, the running times of three algorithms are all around 0.002. It tin can be seen from Table one that the best running fourth dimension of the 3 algorithms are all 0.002 which is accurate to the 3rd-decimal place. But, in virtually conditions, the running time of bidirectional SVGA is less than unidirectional SVGA and simplified VG + .
The space complexity of the SVGA algorithms is also ameliorate than the other three algorithms. It tin exist ended that the SVGA algorithms tin can improve the time and space complexity of the path planning based on visibility graph roadmap. The efficiency deviation between unidirectional SVGA and bidirectional SVGA is that unidirectional SVGA takes more than computational time and less computational space, and, on the contrary, bidirectional SVGA takes less computational fourth dimension and more computational space.
half-dozen. Conclusions
This newspaper presents a global path planning algorithm based on bidirectional SVGA. This algorithm constructs the visibility graph and searches for an optimal path simultaneously. Information technology takes advantage of the vertex positions and heuristics search, and most visibility judgments between two vertices are ignored. By reducing executions of visibility judgment, this algorithm improves the efficiency of path planning on fourth dimension and space. Much of the recent improvement in computer speed is due to multicore processors. This algorithm takes advantage of multicore processors and adapts the path planning to parallel processing. From both directions, information technology executes SVGA in parallel.
To validate the efficiency of the proposed algorithm, different simulation environments are tested. These environments include a simple environment, a circuitous environment, and 5 different categories of environment in a 150 × 150 surface area. These simulation experiments all validate the effectiveness of the SVGA algorithm.
The proposed algorithm is to better the global path planning based on visibility graph. It is assumed that the environs is known and static, and the vertex positions of all obstacles should be known in accelerate. In many applications, environments are dynamic or unknown. The further research is to apply SVGA in the dynamic environments.
Competing Interests
The authors declare that they take no competing interests.
Acknowledgments
This work is financially supported by the National Natural Science Foundation of China (no. 61101197). Taizhi Lv would similar to thank Jiangsu Overseas Research & Training Program for Academy Prominent Young & Centre-Aged Teachers and Presidents for financial support. Taizhi Lv likewise would like to give thanks the Qianfan project of Jiangsu Maritime Establish for financial support.
Copyright
Copyright © 2017 Taizhi Lv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted utilise, distribution, and reproduction in whatever medium, provided the original piece of work is properly cited.
Source: https://www.hindawi.com/journals/jr/2017/8796531/
0 Response to "How to Draw Visibility Graph for Robot Path Planning"
Post a Comment