Motion Planning Using Approximate Cell Decomposition Method

Transcription

1 Motion Planning Using Approximate Cell Decomposition Method Doina Dragulescu, Mirela Toth-Tascau and Lavinia Dragomir Mechanical Department, Faculty of Mechanical Engineering, Bd. Mihai Viteazul No.1, 19 Timisoara (ROMANIA) Abstract. The paper presents two of the approximate cell decomposition methods and illustrates these methods for two different two-dimensional workspaces. The two methods presented paper are: the homogeneous (square-shape) grid method and the non-homogeneous (rectangular-shape) grid method. For the illustration of these methods were used original application-programs. 1. INTRODUCTION A path planning application usually deals with an object to be moved and a workspace cluttered with obstacles. The goal of this application is to find a path for the robot moving from an initial configuration to a final one without colliding with any of the obstacles while optimizing a certain criterion function. The main difficulty in solving the above-mentioned problem is the various shapes of objects (robot and obstacles). There are many methods to robot path planning based on different workspace modeling approaches [1], [2], [3], [4], [5]. Thus, each path planning method deals with a workspace modeling, i.e. e. an approximation of the workspace. In this paper the robot path planning moving in a twodimensional workspace is solved by two planning methods based on the approximate decomposition approaches. The path planning based on the approximate cell decomposition methods has been extended and original software has been developed. For each case, the modeled workspace is searched for a path. The approximate cell decomposition methods are part of the category of the modeling methods using rigid bodies. When the two-dimensional path-planning problems are treated using these methods, the obstacles are considered as solid bodies building rigid configurations, so they occupy a well-specified area of the workspace. Because the shapes of these areas are very complex, and the analysis of the workspace is generally made after an image acquisition and processing of this workspace, it became necessary to simplify their representation in binary form. After the workspace representation form of an image that is, depending on the necessities, more or less processed, the workspace is decomposed in area or volume subsets, named cells. Depending on this decomposition, the path planning will be made using an exact cell decomposition method or an approximate cell decomposition method. As a result of applying these planning-methods there will be obtained an EMPTY cell union, one of these cells is the initial cell C initial and another one the final cell C final. Next, connectivity is constructed and searched; their nodes are cells of the free region and two nodes are connected if and only if the corresponding cells are adjacent, The outcome of the connectivity searching is a sequence of successive cells called channel; this channel connects the initial cell with the final one. The robot is moving along this channel by executing a plan-parallel movement, and the parameters are chosen so that the robot exploits the free space as much as it can. 2. MOTION PLANNING USING APPROXIMATE CELL DECOMPOSITION METHODS All the approximate cell decomposition path-planning methods are based on the same simple principle: the decomposition of the workspace in EMPTY, FULL or MIXED cells: EMPTY cells, obtained by intersection of the free regions between the obstacles and the grid;

2 FULL cells, obtained by intersection of the regions occupied by the obstacles and the grid; MIXED cells that are only partially occupied by the obstacles. The last two categories are always considered as occupied for the robot. So, the obstacles appear with modified boundaries, because there were added to their shape also the MIXED cells belonging to the immediate vicinity of the cells. It is obvious that if the refinement degree of the decomposition is high, than the cells are small and the shape change of the obstacles is not essential. But, usually, the refinement degree is strictly bounded to the dimensions of the rigid robot model that must be entirely contained inside of a cell. The channel representing the path of the robot connects the initial cell with the final cell and, inside of this channel the robot is executing a general movement, if the workspace is three-dimensional, and a plan-parallel movement if the workspace is two-dimensional. In the paper are presented next two approximate cell decomposition methods: homogeneous grid method; non-homogeneous grid method. Figure 1. Expanded obstacles without shape changes in a 2D workspace 3. HOMOGENEOUS GRID METHOD The homogeneous (square-shape) grid method offers good results for a two-dimensional workspace, where the obstacles are represented using polygonal shapes, without any restrictions for the concavities or for the existence of some part with curved side (generalized polygons). In figure 2 there has been illustrated the application of the homogeneous cell decomposition method for the workspace represented figure 1, where there was realized the leading of a mobile minirobot with recognition of the obstacles shapes. Though the decomposition has been made for a very small robot, the shapes of the obstacles have been changed. Around the obstacles it can be seen a layer of MIXED cells. The robot can move only along a channel of adjacent EMPTY cells, where the first cell of the channel is the initial cell C initial and the last one is the final cell C final (figure 2). Figure 2. Decomposition of the workspace shown in figure 1 in equal squares with small dimensions If the robot s dimensions are bigger, the dimensions of the cells will be increased. Two cases can appear: there will not exist a path, or there will be a path constrained to lied in some regions of the free space. It can be observed in figure 3 that if the cells have the edges four times bigger than the ones previous case than exists only EMPTY or MIXED cells and the path can not start from the previous position of the initial cell C initial, because the area of that cell is now part of a MIXED cell. The robot can start only from a new cell, for example C' initial.

3 Figure 3. Decomposition of the workspace presented in figure 1 in equal cell of square shapes with the edges four times bigger It can be observed that if there are used bigger cells than some parts of the free space these can not be used because they are not connected one with the other. These cells can only be used for a local movement of the robot. In reality it is unlikely that such decomposition can satisfy the imposed movements during the accomplishment of the robot s task. 4. THE NON-HOMOGENEOUS GRID METHOD The non-homogeneous (rectangular-shape) grid method assumes all the obstacles of rectangular shapes. The decomposition can be obtained by applying to the workspace a grid with rectangular cells. Their lines are parallel with the edges of the workspace and pass through the vertices of the polygonal obstacles. In this way, every obstacle is covered by a rectangle and there will be EMPTY or FULL cells in the studied workspace. There will be two kinds of s parallel with the workspace edges, every containing EMPTY cells, obtained by intersection with the free regions, and FULL cells, obtained by intersection with the obstacles. Two binary chains can represent every horizontal : the bits corresponding to the first chain reflects the status of the cells (EMPTY or FULL cells), and the second chain reflects the vertical position of the 2D studied workspace. There are six steps of the modeling: Step 1: The representation of every with two binary chains. The order of the bits of the first chain corresponds to the cells order. A bit of the first chain is set to 1 if the corresponding cell is free, and set to if the corresponding cell is occupied. The second chain contains only one bit set to 1, corresponding to the vertical position of the the other bits are set to. Two columns will be obtained reflecting the complete situation of the workspace cells. Step 2: The multiplication of the bit chains obtained in step 1 for every horizontal, so that every chain contains a continuous succession of bits set to 1 and the union of these chains should lead to the chains obtained first step. Step 3: Grouping the chains obtained in step 2 with respect to: generating s; vertical position of the generating s. Step 4: Generation of a new list of chains starting from the previous one, so that: new group i of chain should be generated by combination of every chain of the old group i with every chain of the old group i+1 (i = 1, 2,...); combination of the chains is made using the AND operator for the first chain (left-sided chain), and using the OR operator for the second chain (the rightsided chain); if all the bits of a chain obtained in this way are set to, than that chain will be eliminated from the list, else, the chain will be added to the new list; every time when a new chain is added to the new list, the corresponding old chains will be eliminated from the old list. The step 4 will be repeated until all the possible are obtained. Step 5: Establishing of the free regions represented by the chains remaining list. Step 6: Construction of a non-directed. The nodes correspond to the free regions and the links between the nodes are possible only if the corresponding regions are adjacent or are intersecting. In order to obta path-channel, the initial and the final cell are established for the movement in any free region obtained with the operations of union and intersection of the EMPTY cells according to the described steps of the algorithm. Next, the application of this method will be illustrated, for a workspace representing a section with an horizontal plan of the Robotic Laboratory of the Department of Mechatronics of the Politehnica University of Timisoara. Applying the rectangular grid for this workspace there will be obtained the workspace presented in figure 4.

4 Figure 4. Application of the rectangular non-homogeneous grid for the considered workspace populated with obstacles Steps 2 and 3. Table 2. Steps 2 and 3 number Multiplication of the binary chains from the step 1 Grouping the new chains by the position A B C D Step 1. The first list containing the two columns of chains reflecting the situation of the rectangular cells in the workspace (table 1) will be written. It can be observed that because the horizontal 1 is completely occupied it does not allow the horizontal 11 to take part to the free regions, even if this is completely free. So, the robot cannot pass through the space occupied by the horizontal 1, because this doesn t contain any EMPTY cell that could be used for a link with the 11. Table 1. Step 1 number Binary chain representing the situation of the cells Binary chain representing the vertical position of the The last horizontal does not represent a node, even it is completely free, because it can be seen that next step its intersection with the 11 is empty set. Step 4. Applying repeatedly the steps of the step 4 the following list will be obtained (table 3): Table 3. Step 4. List 4.1 positions 1.2.= = = E = F = = = G =

5 There was eliminated form the list the last two s: since number 1 contains integrally, according to the presented algorithm, it must be eliminated (this represents occupied space); number 11 is blocked by the previous one. Table 4. Step 4. List 4.2 in the positions = 1111 H 111 (1. 2.) (2. 3.) = (2. 3.) (3. 4.) = (3. 4.) (4. 5.) = (4. 5.) (5. 6.) = (5. 6.) (6. 7.) = (6. 7.) (7. 8.) = 1 I 111 (7. 8.) 11 J 111 (8. 9.) Table 5. Step 4. List = (1.2.3.) (2.3.4.) = (2.3.4.) (3.4.5.) in the positions = (3.4.5.) (4.5.6.) = (4.5.6.) 111 K 1111 (5.6.7.) = (5.6.7.) (6.7.8.) = (6.7.8.) (7.8.9.) from the steps = 11 ( ) ( ) = 11 ( ) ( ) = 11 ( ) ( ) = 1 ( ) ( ) = 1 ( ) ( ) 11 Table 7. Step 4. List 4.5 from the steps = 11 ( ) ( ) = 11 ( ) ( ) = ( ) ( ) = 1 ( ) ( ) 1 Table 8. Step 4. List = ( ) ( ) = ( ) ( ) = ( ) ( ) positions L positions M N positions O Table 6. Step 4. List 4.4

6 Table 9. Step 4. List = ( ) ( ) = ( ) ( ) Table 1. Step 4. List = ( ) ( ) positions Intersections positions 1 P After the calculations are finished, the columns of chains will be analyzed for all the possible free region and the nodes of the will be established representing the free regions of the workspace. The establishment of the links between the nodes is made by arranging these nodes in a table, left side will be placed the EMPTY cells of the horizontal s obtained by the operator AND, and on the right side the positions of the cells obtained as vertical union of the s combined with the operator OR. The found order was: The corresponding is represented in figure 5. Table 11. Graph searching The binary chain positions A H M O P B E H M P E F K M N O P F K L M N O P C D 1 11 G 11 1 I J K L N O P A possible path-channel is the succession of nodes A- H-M-O-P and their intersection (figure 6). Figure 5. The corresponding to the decomposition of the workspace represented in figure 4 using the nonhomogeneous grid method

7 a) A d) O b) H e) P b) M f) Free channel Figure 6. A possible path-channel: the succession of the nodes A-H-M-O-P

8 5. CONCLUSIONS Both of the presented methods are easy to implement and allow a good resolution for the two-dimensional workspace. The cells obtained by applying the grid do not in general allow the exact representation of the free space; some free regions cannot be used. There are still important errors geometric model of the workspace, but their influence in robot control is not significant. REFERENCES [1] D. DRAGULESCU, M. TOTH-TASCAU, Planificarea miscarii roborilor industriali, Editura Helicon, Timisoara, 1995 [2] D. DRAGULESCU, M. TOTH-TASCAU, Contributions to cell decomposition method for a two-dimansional workspace, The Second International Conference on Technical Informatics, Timisoara 1996 [3] D. DRAGULESCU, M. TOTH-TASCAU, F. MOLDOVAN, Metoda si algoritm de planificare a traiectoriilor plane, Lucrarile celei de-a II-a Sesiuni de Comunicari Stiintifice a Universitatii Aurel Vlaicu Arad, Sectiunea Inginerie Mecanica, Arad, p.92-96, 1994 [4] J.C.LATOMBE, Robot Motion Planning, Kluwer Academic Publishers, Boston, 1991 [5] M. TOTH-TASCAU, Metode si tehnici de generare a traiectoriilor la roboti industriali, Teza doctorat, Universitatea Politehnica Timisoara, 1996