Abstract:
A method is presented for path planning after changes in task space. In one embodiment, the method is applied to planning a path for a robot arm. The method identifies areas in the configuration space which are affected by the changes in task space. Cost waves can then be repropagated in these affected areas to allow for planning in N dimensions and using space variant metrics. The method is also adapted to use in the presence of phantom obstacles.

Description:
This is a continuation of U.S. application Ser. No. 07/166,599, filed Mar. 9, 1998 now U.S. Pat. No. 4,949,277. 
    
    
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the fascimile reproduction by anyone of the patent document of the patent disclosure, as it appears in the Patent and trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to path planning and in particular to repropagating cost waves in a configuration space after some aspect of that space has been changed. 
     The invention also relates to path planning in a configuration space in which the location of obstacles is not known. 
     2. Prior Art 
     The field of path planning is one with many applications. The most common application is to controlling robots, for instance robot arms such as are used in the space shuttle. Other applications include electronic maps, traffic control, emergency vehicle control, and emergency exit systems. 
     The path planning problem, as applied to robots, typically involves getting a robot from a start point to a goal point while avoiding obstacles. Automating multi-dimensional path planning for robots is one of the great historical problems of robotics. 
     The present invention is an improvement on the invention disclosed U.S. patent application Ser. No. 123,502, now abandoned which is incorporated herein by reference as background material. That application disclosed, amongst other things, propagating cost waves through a configuration space by budding, using a space-variant metric. 
     After budding, some aspect of the configuration space may change, for instance, if an obstacle is removed or a goal added. In such a case, it may be inefficient to bud the entire configuration space again because only a small part of the configuration space may be affected. 
     Another problem which arises after a change in configuration space is that the precise location of the changes, particularly in obstacle location, may not be known. An approach to this problem is set forth in V. Lumelsky, “Algorithmic and Complexity Issues of Robot Motion in an Uncertain Environment”, Journal of Complexity 3, 146-182 (1987); and V. Lumelsky, “Dynamic Path Planning for a Planar Articulated Robot Arm Moving Amidst Unknown Obstacles”, Automatica, Vol. 23., No. 5, pp. 551-570 (1987). This approach suffers from certain shortcomings. For instance, the method disclosed is only able to deal with two dimensions. The method, also does not have a memory for previously encountered obstacles. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the invention to deal with changes in a configuration space. 
     It is a second object to deal with changes in configuration space in the context of a space-variant metric. 
     It is a further object of the invention to deal with changes in a configuration space using budding. 
     It is still a further object of the invention to deal with changes in condition in multiple dimensions. 
     It is another object of the invention to deal with unsensed or partially sensed changes in conditions in the context of a space-variant metric. 
     It is yet another object of the invention to deal with unsensed or partially sensed changes in conditions using budding. 
     It is yet a further object of the invention to deal with unsensed or partially sensed changes in conditions with improved abilities to incorporate new data into the memory of the path planes. 
     These and other objects of the invention are achieved herein using a method referred to herein as differential budding. 
     The method involves identifying a region in configuration space which is affected by a change of conditions in task space. Precisely selected states referred to as a perimeter and which define this region are then placed on a “sifting heap” from which they are budded. 
     In the case of an added obstacle, the perimeter includes the front edge of the obstacle as well as surrounding a group of states which are connected via direction arrows to the back edge of the added obstacle. 
     In the case of removed obstacles, the perimeter states are neighbors of the removed obstacle states. 
     In the case of added goals, the perimeter states are the added goals themselves. 
     In the case of partially sensed changes in conditions, the changes in conditions are detected while following a precalculated path. Most changes in conditions can then be dealt with as an added obstacle, a removed obstacle, an added goal, or a removed goal. In one embodiment of the invention, the object first encounters a portion of the change in conditions iteratively follows a newly calculated path until the change in conditions is taken into account. 
     Further objects and advantages will become apparent in what follows. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     The invention will now be described using several examples with reference to the following figures: 
     FIG. 1 is a summary flow chart of the differential budding method. 
     FIGS. 2 and 3 are flow charts giving more details of box  110  of FIG.  1 . 
     FIGS. 4 and 5 are flow charts giving more details of box  120  of FIG.  1 . 
     FIGS. 6 and 7 are flow charts giving more details of box  130  of FIG.  1 . 
     FIG. 8 a  shows a task space with a two link robot following a minimum joint motion path. 
     FIGS. 8 b,    9 , and  10  show the effects of the method of FIGS. 1-7 on the configuration space corresponding to the task space of FIG. 8 a.    
     FIG. 11 a  shows the task space of FIG. 8 a  with the obstacle removed. 
     FIG. 11 b  shows a configuration space which has been altered by removing an obstacle and to which the method of FIGS. 1-7 has been applied. 
     FIG. 12 shows the configuration space of FIG. 11 b  after budding and with a new path. 
     FIG. 13 a  shows the task space of FIG. 11 a  with newly added goals. 
     FIG. 13 b  shows a configuration space at an intermediate status resulting from the newly added goals of FIG. 13 a.    
     FIG. 14 shows the configuration space of FIG. 13 b  at a second intermediate status. 
     FIG. 15 shows the configuration space of FIG. 14 after budding and with a new path. 
     FIG. 16 shows the configuration space of FIG. 15 with a goal state removed. 
     FIG. 17 shows the configuration space of FIG. 16 at an intermediate status resulting from the application of the method of FIGS. 2-7. 
     FIG. 18 shows the configuration space of FIG. 17 after budding and with a new path. 
     FIG. 19 a  shows the task space resulting in FIG. 19 b.    
     FIG. 19 b  shows the configuration space of FIG. 18 with a newly added goal state and a new obstacle. 
     FIG. 20 shows a configuration space of FIG. 19 b  at an intermediate status resulting from the application of the method of FIGS. 2-7. 
     FIG. 21 shows the configuration space of FIG. 20 after budding and with a new path. 
     FIG. 22 shows the configuration space of FIG. 21 with the obstacle and the goal both moved slightly. 
     FIG. 23 shows the configuration space of FIG. 22 at an intermediate status resulting from the application of the method of FIGS. 2-7. 
     FIG. 24 shows the configuration space of FIG. 23 after budding and with a new path. 
     FIG. 25 a  shows a task space with a goal and a phantom obstacle. 
     FIG. 25 b  shows a configuration space corresponding to the task space of FIG. 25 a  before the phantom obstacle is sensed. 
     FIG. 26 shows the configuration space of FIG. 25 b  with the phantom obstacle discretized. 
     FIG. 27 shows the configuration space of FIG. 26 in an intermediate status after the robot first encounters the phantom obstacle and after the application of the method of FIGS. 2-7. 
     FIG. 28 shows the configuration space of FIG. 27 after budding. 
     FIG. 29 shows the configuration space of FIG. 28 in an intermediate status after the robot encounters the phantom obstacle for the second time and after the application of the method of FIGS. 2-7. 
     FIG. 30 shows the configuration space of FIG. 29 after budding. 
     FIG. 31 shows the configuration space of FIG. 30 in an intermediate status after the robot encounters the phantom obstacle for the third time and after the application of the method of FIGS. 2-7. 
     FIG. 32 shows the configuration space of FIG. 31 after budding. 
     FIG. 33 shows the configuration space of FIG. 32 with the newly calculated path. 
     FIG. 34 a  shows a task space with one goal and a two-link robot following a path which minimizes the distance of travel of the end-effector. 
     FIG. 34 b  shows the configuration space corresponding to FIG. 34 a.    
     FIG. 35 a  shows the task space of FIG. 34 a  with a newly added obstacle. 
     FIG. 35 b  shows the configuration space of FIG. 34 b  with an outline of the newly added obstacle. 
     FIG. 36 shows the configuration space of FIG. 35 at an intermediate status after adding the obstacle and after applying the method of FIGS. 2-7. 
     FIG. 37 shows the configuration space of FIG. 36 after budding. 
     FIG. 38 shows the configuration space of FIG. 37 with a new position for the obstacle indicated in outline. 
     FIG. 39 shows the configuration space of FIG. 38 at an intermediate status after moving the obstacle and after applying the method of FIGS. 2-7. 
     FIG. 40 a  shows the task space corresponding to FIG. 40 b.    
     FIG. 40 b  shows the configuration space of FIG. 39 after budding and with a new path. 
     FIG. 41 a  shows a task space with a two-link robot and a number of goals. 
     FIG. 41 b  shows a configuration space corresponding to the task space of FIG. 41 a  with cost waves propagated using a metric which minimizes joint motion. 
     FIG. 42 a  shows the task space of FIG. 41 a  with the robot following a newly calculated path. 
     FIG. 42 b  shows the configuration space of FIG. 41 b  after application of the method of FIGS. 1-7. 
     FIG. 43 shows the configuration space of FIG. 42 with a goal state removed. 
     FIG. 44 shows the configuration space of FIG. 43 at an intermediate status after removing the goal and after applying the method of FIGS. 2-7. 
     FIG. 45 shows the configuration space of FIG. 44 after budding. 
     FIG. 46 shows a flow chart for dealing with phantom changes in conditions. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     1. Overview of the Method 
     FIG. 1 gives a general overview of steps used in “differential budding”, a method for regenerating a configuration space with changes in obstacles and goals. The resulting configuration space provides information necessary to generate a series of set points to be followed for an optimal path. 
     In box  100 , a configuration space filled with direction arrows and costs_to_goal is assumed. Information about changed goal and obstacle states is also assumed to be provided. These states are already transformed from task space to configuration space. It should be noted that the method will work if the configuration space is merely initialized, with UNCOSTED in each cost-to-goal, and no direction arrows. 
     In box  110 , information about newly added obstacle states and newly removed goal states is used to initialize the corresponding states in configuration space. 
     In box  120 , all states ‘influenced’ by the added obstacle and removed goal states are initialized to have UNCOSTED values and no direction arrows. ‘Influenced’ neighbors are those that are on paths that point to or through these added obstacle or removed goal states. The ‘perimeter’ of states around this region is obtained and added to the heap for later ‘budding’. 
     In box  130 , information about newly removed obstacle states and newly added goal states is used to initialize the corresponding states in configuration space. The heap is set up so that ‘budding’ can take place. 
     In box  140 , the states on the heap accumulated as a result of boxes  110 ,  120 , and  130  are ‘budded’ resulting in a stable (valid) configuration space. Given a starting state, an optimal path of set points can be read out by following the direction arrows in configuration space to the goal state. 
     Two special structures are used for efficiency: a sifting heap, and a sifting array. The ‘sifting heap’ is a regular heap except that it keeps at most one copy of a tuple (state) in the heap even though many requests to add extra copies may be made. Whenever the word “heap” is used herein, a “sifting heap” is intended. The ‘sifting array’ (used to store unique states of the perimeter) is similarly a regular array except that it keeps at most one copy of a tuple (state). Some of the arrays referred to herein are sifting and some are not. In both cases, sifting is achieved by maintaining flags in each state of the configuration space that report if a state is in the heap or in the perimeter array. Then all that is needed when adding or removing states from the heap or perimeter is to modify these flags correctly. 
     A. Check added obstacles and removed goals. 
     The method in box  110  is detailed by the flowchart of FIG.  2  and FIG.  3 . The method of box  120  is detailed by the flowchart in FIG.  4  and FIG.  5 . The method of box  130  is detailed by the flowchart of FIG.  6  and FIG.  7 . 
     The ‘check added obstacles and removed goals’ method of box  110  includes three main parts. 
     The first part is shown in boxes  200 ,  210 , and  220 . Box  200  tests whether the end of the “added obstacle” list has been reached. If the result of the test in box  200  is “Yes”, control passes to box  230 . If the result of the test in box  200  is “No”, control passes to box  210 . Box  210  gets the next obstacle state ‘S’ from the added obstacle list. Box  220  then clears the direction arrows field and sets the cost_to_goal field to INF in the state ‘S’. After box  220 , control is returned to box  200 . 
     Boxes  200 ,  210 , and  220  thus constitute a loop which initializes each ‘added obstacle’ state (‘S’) in configuration space to have no direction arrows and an INF (INFINITE) cost_to_goal. 
     The second part of the “check added obstacles and removed goals” method of box  110  is shown in boxes  230 ,  240  and  250  of FIG. 2, and boxes  300 ,  310 ,  320 ,  330 ,  340 , and  350  of FIG.  3 . It should also be noted that line  201  of FIG. 2 connects to line  301  of FIG. 3; line  202  of FIG. 2 connects to line  302  of FIG. 3; and line  203  of FIG. 2 connects to line  303  of FIG.  3 . 
     In this second part of box  110 , the ‘front edge’ of the obstacle and the ‘back edge’ of the added obstacle are obtained. The ‘front edge’ contains those neighbor states that are on the ‘goal side’ of the added obstacle. These states are neighbors of the obstacle region, have cost values other than INFINITY or UNCOSTED, and are not pointing toward any of the obstacle states. The ‘back edge’ contains those neighbor states that are not on the front edge. These states have direction arrows that point into the added obstacle. 
     Box  240  signifies a test for the end of the “added obstacle” list. If the end has been reached, control passes via line  201 / 301  to box  360 . If the end has not been reached, control passes to box  250 , where an added obstacle ‘A’ is retrieved. 
     Then at box  300 , it is determined whether all of the neighbor states ‘N’ of the obstacle state ‘A’ have been considered. If all of the neighbor states ‘N’ have been considered, control returns via line  303 / 203  to box  240 . If a neighbor state ‘N’ has not been considered, control passes to box  310 , where that state ‘N’ is retrieved. 
     Box  320  indicates a test of whether the state ‘N’ contains a direction arrow that points to ‘A’. If state ‘N’ does not contain such a direction arrow, control passes to box  340 . If state ‘N’ does contain such a direction arrow, the state ‘N’ is added to the heap at box  330 . If the state ‘N’ is added to the heap at box  330 , ‘N’ is considered to be part of the ‘back edge’ of the added obstacle. After box  330 , control returns to box  300 . 
     In box  340 , ‘N’ is tested for whether its cost_to_goal field contains INF. If ‘N’ does have a cost_to_goal of INF, control is returned to box  300 . If ‘N’ has a cost_to_goal other than INF, it is added to the ‘perimeter list’ at box  350 , and is considered part of the ‘front edge’ of the obstacle. 
     The third part of the “check added obstacles and removed goals” method of box  110  is shown in boxes  360 ,  370  and  380  of FIG.  3 . These three boxes form a loop. In box  360 , it is determined whether the end of the “removed goals” list has been reached. If the end has been reached, the “check added obstacles and removed goals” method is complete. If the end has not been reached, the next goal state ‘G’ is retrieved at box  370 . This goal state ‘G’ is then added to the heap at box  380  and control is returned to box  360 . Thus, in this loop, each of the removed goal states in the ‘removed goal list’ is added to the heap. 
     B. Clear influence. 
     The ‘clear influence’ method of box  120  includes two main parts. The first part determines the ‘perimeter of influence’ due to the newly added obstacle and removed goal states, and the second part takes the perimeter and puts it into the heap. 
     The first part of the “clear influence” method of box  120  is shown in boxes  400 ,  410   420 ,  430 ,  440 ,  450 ,  460  and  470  of FIG. 4, and boxes  500 ,  510  and  520  of FIG.  5 . It should be noted that line  401  of FIG. 4 connects to line  501  of FIG. 5, line  402  of FIG. 4 connects to line  502  of FIG. 5, line  403  of FIG. 4 connects to line  503  of FIG. 5, and line  404  of FIG. 4 connects to line  504  of FIG.  5 . 
     At the beginning of the ‘clear influence’ method, the heap consists of neighbor states at the back edge of any added obstacles and any removed goal states. At box  400 , the heap is tested to see if it is empty. If the heap is empty control passes via line  401 / 501  to box  530 . 
     If the heap is not empty, then, at box  410 , a minimum cost state ‘S’ is retrieved from the heap and and a variable, uncostflag, corresponding to ‘S’ is set to “NO”. At box  420 , it is then determined whether all neighbors, ‘N’, of ‘S’ have been considered. If all the neighbors have been considered, control passes to box  460 . 
     If all of the neighbors have not been considered, the variable uncostflag is tested at box  430 . If uncostflag has a value “No” then, at box  440 , ‘S’ is added to the perimeter, if ‘S’ has not already been added to the perimeter. If uncostflag has a value “Yes”, then, at box  450 , the cost_to_goal field of ‘S’ is set to UNCOSTED; the direction_arrows field of ‘S’ is cleared; and all neighbor from the ‘Bag’ array are emptied into the heap which sifts them, i.e. it does not store any duplicate states. 
     At box  460  a next neighbor ‘N’ is retrieved. At box  470  the cost_to_goal field of ‘N’ is tested to see if it is either INF or UNCOSTED. If it is neither, control passes via line  402 / 502  to box  520 . If it is either INF or uncosted, control passes via line  403 / 503  to box  500 . 
     At box  500 , ‘S’ is checked to see if it has a direction arrow pointing to ‘N’. If not, control passes to box  520 . If so, the uncostflag associated with ‘S’ is set to “YES”. 
     In box  520  the neighbor ‘N’ is temporarily stored in the Bag array. Control then returns via line  504 / 404  to box  420 . 
     Thus, during the first part of the “clear influence” method of box  120 , each state ‘S’ in the heap, that points to any neighbor ‘N’ that has a cost_to_goal that is either INFINITE or UNCOSTED is identified. Such a state ‘S’ must therefore have been ‘connected’ via direction arrows to the back of a new obstacle state or to a removed goal state. Any neighbor that is not INFINITE or UNCOSTED is a candidate for expanding the search and is temporarily stored in the ‘Bag’. The ‘Bag’ is a standard array containing states. After all neighbors of ‘S’ are examined, if ‘S’ is connected, then all neighbors of ‘S’ that are in the ‘Bag’ are added (“emptied”) to the heap, which sifts them, and ‘S’ is reinitialized to have cost_to_goal of UNCOSTED and no direction arrows. If ‘S’ is not connected, ‘No’ branch from box  430 , then ‘S’ must be a member of the ‘perimeter’ and the neighbors that are in the ‘Bag’ are NOT added to the heap. 
     The second part of the “clear influence” method of box  120  is shown in boxes  530 ,  540 , and  550  of FIG.  5 . At box  530 , the perimeter list, stored in a sifting array, is checked to see if it is empty. If so, the clear influence method ends. If not, a perimeter state ‘P’ is retrieved  540  and added  550  to the heap, if not already in the heap. After box  550 , control is returned to box  530 . This second part of box  120  thus reads the states in the “perimeter list” and stores them in the heap. 
     Thus the perimeter list can be said to surround an influenced or “affected region” of the configuration space, and this terminology has been used herein. The term “influenced” is used herein particularly to refer to the “clear influence” step  120  and is distinguished from the term affected, which is used more broadly as explained below. However, this terminology is not meant to imply that the surrounded region is contains the only states which may be changed as a result of differential budding. States on or outside the perimeter may have their cost_to_goal or direction_arrows values changed as a result of budding the heap in box  140  of FIG.  1 . 
     C. Check removed obstacles and added goals. 
     The ‘check removed obstacles and added goals’ method of box  130  has three main parts. 
     The first part is shown in boxes  600 ,  610  and  620 . At box  600  it is determined whether the end of the ‘removed obstacle’ list has been reached. If the end has been reached, control passes to box  630 . If the end has not been reached, a next obstacle state, ‘S’, is retrieved from the list at box  610 . In this state, ‘S’, the direction_arrows field is cleared and the cost_to_goal field is set to UNCOSTED. This first part is, then, a loop which initializes each ‘removed obstacle’ state in configuration space to have no direction arrows and an UNCOSTED cost_to_goal. These are appropriate values, since these now unoccupied (removed) obstacle states have no presumed costs or arrows. These removed obstacle states can be said to be the region affected by the obstacle removal, and this terminology is used herein. However, it should be noted that states outside this “affected region” may still have their cost_to_goal and direction_arrows values changed as a result of budding in box  140 . The states surrounding the removed obstacle can be said to be the perimeter, but they require a less sophisticated treatment than the perimeter of the removed goals and added obstacles. 
     The second part of box  130  is shown in boxes  630 ,  640  and  650  of FIG.  6  and boxes  700 ,  710 ,  720 , and  730  of FIG.  7 . It should also be noted that line  601  of FIG. 6 connects to line  701  of FIG. 7, line  602  of FIG. 6 connects to line  702  of FIG. 7, and line  603  of FIG. 6 connects to line  703  of FIG.  7 . 
     This second part of box  130  starts at the top of the removed obstacle list, at  630 . A test is performed at  640 , as in box  600 , to determine whether the end of the “removed obstacle” list has been reached. If the end has been reached, control passes via line  601 / 701  to box  740 . If the end has not been reached, a next obstacle state ‘R’ is retrieved at  650 . Then, in box  700 , it is determined whether all neighbors of ‘R’ have been considered. If so, control is passed via lie  703 / 603  to box  640 . If not, a next neighbor state ‘N’ of ‘R’ is retrieved at  710 . If ‘N’ has a cost_to_goal field which has a value of UNCOSTED or INF, then control is returned to box  700 . If ‘N’ has a cost_to_goal field which is not UNCOSTED and not INF, then ‘N’ is added to the heap at  730 . 
     Thus, in this second part of box  130 , all neighbors of a removed obstacle state that have a cost_to_goal other than ‘INFINITY’ or ‘UNCOSTED’ are added to the heap, which sifts them. These states together form an edge that is adjacent to the removed obstacle. These neighboring states will help (via the ‘budding’ mechanism of box  140 ) to fill in the vacated area where the ‘removed obstacle’ states reside. 
     The third part of box  130  is shown in boxes  740 ,  750 ,  760 , and  770  of FIG.  7 . At box  740 , a test is performed to determine if the end of the “added goals” list has been reached. If the end has been reached, the ‘check removed obstacles and added goals’ method terminates. If the the end has not been reached, a next goal state ‘G’ is retrieved at  750 . At  760 , for the goal ‘G’, the direction_arrows field is cleared and the cost_to_goal field is set to zero. Then, at  770 , the goal ‘G’ is added to the heap and control is returned to box  740 . 
     Thus, this third part of box  130  is a loop which takes each newly added goal state from the “added goals” list, clears the direction arrows, sets the cost_to_goal to zero, and adds each goal state to the heap. Thus the added goal states are both the “affected region” and the perimeter as that terminology is used herein. However, it should be noted that states outside this “affected region” may still have their cost_to_goal and direction_arrows values changed as a result of budding in box  140 . 
     In summary, then, the “affected region” as that term is used herein means states that have been cleared as part of the clear influence step  120 , and states corresponding to removed obstacles and added goal states which are “influenced” are the ones cleared during the clear influence step  120 . 
     Appendix A contains source code performing the method of FIGS. 1-7. In addition to code for executing the method, Appendix A has code for producing a graphic simulation. The source code is in the ‘C’ language. In printing this source code, the printer has changed the character sequence “\n” in the “printf” statements to “ 0 ” (zero). Therefore, in order to use this code, the programmer will have to change these zeroes back to “\n”. 
     This concludes the differential budding aspect of this algorithm. At this point, the states that reside in the heap are ‘budded’ box  140  according to the ‘budding method’ found in U.S. patent application Ser. No. 123,502 now abandoned. This produces the final updated configuration space from which an optimal path can be produced. 
     2. Simplified Examples 
     The method of FIGS. 1-7 will now be illustrated with reference to some simplified examples. In these examples, a coarse configuration space is used, so that values of the direction_arrows and cost_to_goal fields will be clearly visible. In addition, the various steps used in the method are more clearly visible on the coarse configuration space. 
     A. Adding an obstacle. 
     FIG. 8 a  illustrates a task space with a two-link robot  801  following a minimum joint motion path to a goal  802 . An obstacle  803  has been added to the task space, after the minimum joint motion path was calculated. 
     FIG. 8 b  shows a coarse configuration space corresponding to the task space of FIG. 8 a.  In the coarse configuration space, states appear as squares in the table. The locations of the states are identified by shoulder and elbow angle. States have cost to goal values marked as numbers in the squares. Direction arrows are shown leading from one state to another. For instance, the state corresponding to a shoulder angle of 72° and an elbow angle of 36° is indicated as [ 72 , 36 ]. State [ 72 , 36 ] has a cost to goal of 0.26 and two direction arrows pointing respectively to states [ 36 , 36 ] and [ 36 , 0 ]. It should be noted that each illustrated direction arrow is associated with the state where it starts. Each direction arrow points to the state where its arrowhead appears. The goal is indicated with a target symbol at state [ 288 , 288 ]. The path to the goal is indicated by blackened states, e.g. at  805 . 
     In the configuration space of FIG. 8 b,  the newly added obstacle first appears as a line polygon  804 , which blocks the path  805 . The line polygon  804  has a finer resolution than the configuration space, and will be discretized in later figures. 
     In the configuration space of FIG. 9, the obstacle has been transformed into states [ 180 , 180 ], [ 180 , 216 ], [ 216 , 180 ], and [ 216 , 216 ]. FIG. 9 shows the configuration space after the ‘clear influence’ phase of box  120 . The heap now contains all configuration states that are ‘boxed’ such as [ 72 , 72 ], [ 72 , 180 ] etc. These coordinates are read [shoulder angle, elbow angle]. The states that have actually been cleared during the ‘clear influence’ stage  120  show a ‘U’ (meaning UNCOSTED) as their state cost_to_goal. All of these states were traceable (via direction arrows in FIG. 8) to the back of the added obstacle. 
     In FIG. 10, a stable configuration space has been calculated and is shown after ‘budding’ the states that were in the heap after FIG.  9 . 
     B. Removing an obstacle. 
     In the task space of FIG. 11 a,  the obstacle  803 , previously added to the task space, has been removed again. 
     FIG. 11 b  illustrates the effect of removing the obstacle  803  on the configuration space. Boxes  600 ,  610 , and  620  yield the UNCOSTED values that are at states [ 180 , 180 ], [ 180 , 216 ], [ 216 , 180 ], and [ 216 , 216 ]. Boxes  630 ,  640 , and  650  of FIG. 6 along with  700 ,  710 ,  720 , and  730  of FIG. 7 result in the ‘boxed states’ that are on the perimeter of the influenced or affected region and therefore reside on the heap as of FIG.  11 . The cost_to_goal fields of states [ 180 , 180 ], [ 180 , 216 ], [ 216 , 180 ] and [ 216 , 216 ] are set to UNCOSTED, and the ‘boxed states’, e.g. [ 144 , 144 ], [ 144 , 180 ], [ 144 , 216 ], etc. shown are in the heap. 
     The result of budding the ‘boxed states’ from FIG. 11 is the stable configuration space shown in FIG.  12 . 
     C. Adding goals. 
     FIG. 13 a  shows the task space with newly added goals  1301  and  1302 . 
     In FIG. 13 b,  the newly added goals are shown at goal states: [ 108 , 36 ], [ 36 , 108 ], [ 252 , 72 ], [ 72 , 252 ] and [ 216 , 216 ]. The goal states are cost-initialized to zero and their direction arrows are cleared. The pre-existing goal is shown at state [ 288 , 288 ]. 
     In FIG. 14, these new goals are shown in the heap. The ‘boxed states’ that are also goal states show boxes through the goal (target) symbol. 
     In FIG.  15 . the final (stable) configuration space is shown after budding the heap, with accommodations for the new goals. A new path is also shown at  1501 . 
     D. Removing goals. 
     FIG. 16 shows the removal of goal  1301 , which corresponds to state [ 216 , 216 ]. 
     In FIG. 17, the area influenced by the removed goal of FIG. 16, formerly at [ 216 , 216 ], is shown by the area of ‘UNCOSTED’ states, e.g. [ 144 , 144 ], [ 180 , 144 ], [ 216 , 144 ], etc. The perimeter of these UNCOSTED states, along with the front edge of the obstacle, are shown as ‘boxed states’. The ‘boxed states’ are in the heap and will be budded next. The states that are actually cleared during the ‘clear influence’ step  120 , show a ‘U’ (meaning UNCOSTED) as their state cost_to_goal. Prior to the ‘clear influence’ step of box  120 , all of these states were traceable (via direction arrows as shown in FIG. 8 b ) to the back of the removed goal, formerly at [ 216 , 216 ]. 
     In FIG. 18, the result of Budding the configuration space starting with the ‘boxed states’ that were in the heap is shown. A new path appears in FIG.  18 . The new path leads through states [ 144 , 144 ], [ 108 , 108 ], and [ 108 , 72 ] to the goal state at [ 108 , 32 ]. 
     E. Adding goal and obstacle. 
     In FIGS. 19 a  and b, a new goal located at state [ 252 , 252 ] and a new obstacle in the [ 72 , 72 ] vicinity are added at the same time. 
     In FIG. 20, the states of the perimeter of the influence of the discretized obstacle, and the new goal state, are stored on the heap after ‘clearing the influence’  120  of the new obstacle, and are shown as ‘boxed states’. 
     FIG. 21 shows the resulting configuration space and the new path  2101  that the robot would follow. 
     F. Moved obstacle and moved goal. 
     In FIG. 22, the obstacle has moved slightly, from  2201  to  2202 , and a goal is also moved from [ 252 , 252 ] to [ 288 , 288 ]. In this situation, all four main procedures are performed simultaneously: adding and removing goal states, and adding and removing obstacle states. By moving the obstacle, some new obstacle states may be added, in this case [ 72 , 36 ], [ 36 , 72 ]; and some obstacle states may be removed, in this case [ 108 , 72 ] and [ 72 , 108 ]. State [ 72 , 72 ] is unaffected. The goal at [ 252 , 252 ] can be ‘moved’ by adding a goal at the new location, [ 288 , 288 ] in this case, and removing the goal at the old location, [ 252 , 252 ] in this case. 
     FIG. 23 shows the configuration space of FIG. 22 as it appears after the operations described in FIGS. 2-7 but before budding. 
     In FIG. 23, the cost_to_goal of states [ 108 , 72 ] and [ 72 , 108 ] which were formerly part of the obstacle are set to UNCOSTED. The goal at [ 180 , 180 ] is included on the heap because it is at the perimeter of the influence of the removed goal state [ 252 , 252 ]. The goal state at [ 288 , 288 ] is in the heap because it is going to be budded. In addition to added goals, other states on the heap are shown in ‘boxed states’ at the perimeters of affected areas. The cost_to_goal fields of states in the affected areas have been set to uncosted. 
     FIG. 24 shows the stable configuration space which results from budding the states which are on the heap as of the time of FIG.  23 . 
     3. Treatment of Unknown or Unforeseen New Information 
     In all of the previous examples, perfect knowledge of the environment, that is, of all of the obstacles in the environment, was assumed. The invention can also handle the situation where some new information is discovered while the robot is already moving on a prescribed path. This arises when a robot does not have a vision sensor, or has a faulty one, but does have a proximity or tactile sensor of some kind. This can also arise where stored information becomes out of date. In many cases a robot may be able to learn about an environment simply by moving around, acquiring information about new obstacles or goals or removed obstacles or goals as it encounters them, and incorporating this knowledge into its configuration space. The robot can also adapt to a changing environment even if the vision sensors are not perfect. This next example shows how the method of FIGS. 1-7 can solve this problem. It should be noted that although the following description relates to added obstacles, it is equally applicable to removed obstacles, and removed and added goals. 
     A. Coarse configuration space. 
     In FIG. 25 b,  there is a goal state at [ 180 , 180 ] and the starting state is [ 36 , 36 ]. An unsensed (also called a phantom) obstacle is located in the area of [ 108 , 72 ], [ 72 , 108 ], [ 72 ,  72 ], [ 108 , 36 ] and [ 36 , 108 ]. According to the sensed, but factually incomplete information, the robot should be able to move from state [ 36 , 36 ] to [ 180 , 180 ] without obstruction. As the robot actually performs the move, the tactile or proximity sensors will report that an obstacle has been encountered at location [ 72 , 72 ]. FIG. 25 a  shows a task space corresponding to FIG. 25 b.    
     In FIG. 26, the ‘phantom’ obstacle is shown discretized and in half tone. Note that the direction arrows point through the phantom, because the planner is not aware that this obstacle actually exists. 
     In FIG. 27, the robot takes its first step along the prescribed path, but at its next step senses an obstacle state at [ 72 , 72 ]. The newly discovered obstacle state is treated the same as a newly added obstacle. The perimeter of configuration space that this new obstacle state affects is shown in FIG. 27 as ‘boxed states’ and is added to the heap. States affected by the newly discovered obstacle at [ 72 , 72 ] have their cost_to_goal fields set to UNCOSTED. Thus FIG. 27 shows a configuration space in which the portion of the phantom obstacle corresponding to state [ 72 , 72 ] has been sensed and in which the method of FIGS. 1-7 has been performed, inasfar as it relates to newly added obstacles. 
     In FIG. 28, the stable configuration space corresponding to FIG. 27 is shown. FIG. 28 is generated by budding the states which are on the heap at the time of FIG.  27 . FIG. 28 thus takes into account the newly discovered information about the phantom obstacle. 
     In FIG. 29, the robot has followed the newly calculated path, shown in FIG.  28 . The robot thus moved from the position of FIG. 28 (state [ 36 , 36 ]) to state [ 36 , 72 ]. The robot then tried to move to state [ 72 , 108 ]. At [ 72 , 108 ], however, the robot sensed another obstacle, which is therefore indicated in black meaning that it has been recognized as an obstacle. 
     At this point, it should be noted that many techniques can be used for following the path with sensing capabilities enabled. The method shown is simple. If an obstacle is sensed, then the immediate neighbor states are also checked for obstacle existence. Therefore, in this case, the path follower has also sensed the obstacle state [ 36 , 108 ]. State [ 36 , 108 ] is therefore also indicated in black, meaning that it has been recognized as an obstacle. 
     In practice, if the motion controller of a robot has a proximity and/or tactile sensors and could pinpoint the point at which contact with an obstacle occurred, then the entire set of obstacle states in the configuration space corresponding to that point could be directly determined. If many obstacle states can be obtained en masse, only one differential budding “session” will be necessary to compensate for one obstacle point in task space. In the above example, only a few states at a time are discovered requiring several differential budding sessions. It can be seen, then, that, although the path following mechanism does not actually impact the method as it applies to phantom obstacles, it does affect the number of times the method is applied. It should be noted that if the object is a car, the path following mechanism is a driver, who can make fairly sophisticated observation about changes in the environment which can be reported back. 
     The edge of the ‘cleared influence’ area is once again added to the heap and is shown by the ‘boxed states’. 
     In FIG. 30, the stable configuration space shown is the result of ‘budding’ the states on the heap as of the time of FIG.  29 . In the configuration space of FIG. 30, the portions of the phantom obstacle at [ 108 , 36 ] and [ 108 , 72 ] remain unsensed. Therefore direction arrows continue to point at those portions. 
     In FIG. 31, the robot has followed the newly planned path to state [ 72 , 86 ], because the sensed information did not show an obstacle at this state. It then attempted to move to [ 108 , 72 ] which is a phantom obstacle state. According to the present path following mechanism, [ 108 , 36 ] was therefore also sensed as a phantom obstacle state. At this point, the entire ‘phantom obstacle’ has been sensed. Therefore the entire phantom obstacle, consisting of states [ 36 , 108 ], [ 72 , 72 ], [ 72 , 108 ], [ 108 , 36 ], and [ 108 ,  72 ], is now indicated in black. ‘Boxed states’ are on the heap, indicating the perimeter of the area affected by the newly discovered portions of the phantom obstacle. The ‘boxed states’ do not entirely surround the obstacle, because only part of the obstacle is newly discovered. States whose direction arrows previously pointed to the undiscovered part of the obstacle have had their cost_to_goal fields set to UNCOSTED. 
     Although the present example resulted the sensing of the entire ‘phantom obstacle’, it is often necessary to only sense part of the obstacle before an optimal path is found that does not lead into the obstacle. 
     FIG. 32 shows the stabilized configuration space of FIG. 31 after the states on the heap have been budded. 
     In FIG. 33, the path has been marked on the configuration space of FIG.  32 . The robot can now follow the prescribed path from [ 72 , 36 ] to [ 108 , 0 ], [ 144 , 324 ], [ 180 , 288 ], [ 180 , 252 ], [ 180 , 216 ], and finally to the goal [ 180 , 180 ] without further difficulties. 
     This technique works equally well with other metrics, in higher resolutions and dimensions, and with different numbers of neighbors. 
     B. Fine configuration space. 
     A finer resolution ( 64  by  64 ) example with  16  neighbors, i.e., sixteen possible direction arrows for each state, using ‘distance minimization of the end-effector’ criterion as the variable metric follows. 
     FIG. 34 a  shows a task space in which a two-link robot follows a path determined by a criterion which seeks straight line motion of the end effector. 
     FIG. 34 b  shows the configuration space which corresponds to FIG. 34 a.  The states in this configuration space are only indicated by their direction arrows, because the resolution is too fine to indicate cost to goal values or to allow squares to be marked around the states. The configuration space of FIG. 34 b  shows the space variant metric which results from the criterion of seeking straight line motion of the end effector. 
     FIG. 35 a  shows the task space of FIG. 34 a,  with a newly added obstacle  3501 . The newly added obstacle  3501  blocks the previously planned path. 
     FIG. 35 b  shows the configuration space of FIG. 34 b,  with the newly introduced obstacle  3502  shown in a non-discretized form. 
     FIG. 36 shows the configuration space of FIG. 35 b,  but with the newly added obstacle discretized. Those states whose direction arrows previously pointed to the obstacle  3601  have now had their cost_to_goal fields set to UNCOSTED. UNCOSTED is indicated in FIG. 36 by the states which are now white. These states now appear to be white because, in addition to being UNCOSTED, they have no direction arrows. The newly added obstacle  3601  has affected two regions. One affected region  3602  is adjacent to the portion of the obstacle  3601  which would strike the forearm of the robot. A second affected region  3603  is adjacent to the portion of the obstacle  3601  which would strike the protruding elbow end of the forearm link. The perimeter includes those points adjacent to the white (cleared and UNCOSTED) area  3602 ,  3603 , as well as the region in front of the obstacle. The perimeter region is not ‘boxed’ because the graphical congestion in the fine resolution example prohibits it. Nevertheless the perimeter states are on the heap. 
     FIG. 37 shows the completed configuration space, corresponding to FIG.  36 . In order to get from FIG. 36 to FIG. 37, the perimeter states of FIG. 36, i.e. those on the heap, were budded. The new path that is prescribed, taking the new obstacle into account, is shown at  3701 . 
     FIG. 38 shows the configuration space of FIG. 37 with the obstacle in a position which appears moved away from the goal. The discretized area  3801  is where the obstacle moved from, and the polygonal shape  3802  is the non-discretized region describing approximately where the obstacle has moved to. 
     FIG. 39 shows the configuration space of FIG. 38 after the method of FIGS. 1-7 has been applied. As in FIG. 36, the area affected by the movement of the obstacle is now white. The perimeter of the white area is on the heap, although the resolution of the figure is too fine to show the perimeter area as boxed. It can be seen that not very much of the configuration space has to be recalculated in this case. 
     FIG. 40 b  shows the configuration space of FIG. 39 after the perimeter states have been budded. The resulting path is shown at  4001 . 
     FIG. 40 a  shows the task space corresponding to the configuration space of FIG. 40 b.    
     FIG. 41 a  shows a task space in which there are no obstacles, but a number of goals, e.g.  4101 , have been added. 
     FIG. 41 b  shows a configuration space in which cost waves have been propagated from a single goal, using the metric induced by the criterion of moving the end-effector in a straight line. As in FIG. 41 a,  no obstacles are appear. Goal states corresponding to the newly added goals of FIG. 41 a  have been superimposed on the configuration space of FIG. 41 b  resulting in nine newly added goal states. e.g.  4102 . 
     In applying the method of FIGS. 1-7 to the space of FIG. 41 b,  the newly added goals must be added to the heap. However no ‘clear influence’ step needs to be applied. It is only necessary to bud the new heap. 
     FIG. 42 b  shows the result of budding the goal states which were added to the heap at the time of FIG. 41 b.  In addition, a path is shown from a given starting state to the appropriate goal. 
     FIG. 42 a  shows the task space corresponding to the configuration space of FIG. 42 b.    
     FIG. 43 shows the configuration space of FIG. 42 b  where the goal state  4301 , which is roughly in the center, is removed, and the other nine goal states are unchanged. 
     FIG. 44 shows the result of ‘clearing the influence’  120  of the removed goal state. As before, the area from which influence has been cleared is indicated in white, signifying that the direction arrows have been cleared and the cost_to_goal fields have been set to UNCOSTED. Again, the states on the perimeter of the white area are on the heap. 
     FIG. 45 shows the stable configuration space resulting from budding the perimeter states which were on the heap at the time of FIG.  44 . 
     FIG. 46 is a flow chart which summarizes the above-described approach to dealing with unsensed or partially sensed changes in conditions. 
     In box  4601  the method determines a next state in a precalculated path, by following direction_arrow values from a current states. 
     In box  4602  the method sends a setpoint to a motion controller for the robot, based on the next state. 
     At box  4603  a test is performed to determine whether a sensor associated with the motion to the next state has detected new information. If no new information is detected, control returns to box  4601 . 
     If new information is detected there are two options. If the new information is not an absolute obstacle to movement, control may optionally return via the dotted line  4604  to box  4601 . The dotted line  4604  represents a situation in which movement is chosen over calculation of a new optimum path. The dotted line  4604  is chosen, in other words, when movement is more important than strict optimality. 
     While the dotted line  4604  is taken, a processor could be incorporating new information into a configuration space while motion continues. Such parallel processing could be useful when the new information does not immediately affect the motion on the precalculated path. For instance, if the method is applied to control of emergency vehicles, an emergency vehicle might report a traffic blockage in a lane of opposing traffic. Such a report need not affect the progress of the emergency vehicle which reports the blockage. In this case of the emergency vehicle, the vision of the driver is the proximity sensor. 
     If the dotted line  4604  is not taken, and new information is sensed, control passes to box  4605 . In box  4605 , the new information is incorporated into configuration space. At  4606 , differential budding occurs, for instance using boxes  110 - 140  of FIG.  1 . 
     After  4606 , control returns to box  4601 . 
     Appendix A also shows code for dealing with phantom changes in conditions.