Formations in flow fields

Transcription

1 Formations in flow fields Rick van Meer June 11, Introduction Pathfinding and formations (i.e. an arrangement of agents) are both a part of artificial intelligence and can be used in games. However, it is far from trivial to properly combine these two elements. Several issues arise when formations follow a path. Due to the fact that the agents in a formation are close together it is very likely that their paths overlap each other; the agents continuously have to locally adapt their path. Using a global navigation method, like flow fields, instead of a regular pathfinding method like A* can drastically reduce the time taken to calculate the paths for agents. Global navigation methods do not need a starting point for a path, as it essentially calculates the path towards a goal from every possible starting point. This means that only one flow field is needed to guide all agents towards their shared goal. Another issue remains: the formation will not be able to retain its form along the whole path. Corners and narrow passages may force the formation to adjust its form. Flow fields will not solve this issue because they do not take into account the width of the agent or the formation of agents. The issue needs to be solved locally; the formation itself needs to change in order to follow the path. In this research an attempt has been made to solve this issue. 2 Flow Fields Flow fields are a type of global navigation. A simplified explanation of a flow field would be a field full of vectors that point in the direction one should go in order to get to their goal (Reynolds (1999)). Creating such a flow field can be done in many different ways. A very naive approach would be to start at the goal and iteratively calculate the cost for every neighbor until all cells have a cost assigned. The approach taken in this research is a simplified version of the flow field tiles as explained in Emerson (2013). The creation of a flow field consists of several passes: 1. Tile pass 2. Line of sight pass 3. Integration pass 4. Flow field pass Before explaining these passes it is important to know what a cost field is. 2.1 Cost field The area for which a flow field will be calculated needs to store how traversable it is. A cost field is the data structure used to store said cost. The area is divided into cells (a 100x100 grid for example) and a cost is assigned to each of these cells. The cost specifies how traverable each cell is. Figure 1 is an example of such a cost field. A color gradient from green to red visualizes how costly a cell is and a black cell means that a cell is not traversable; e.g. a wall. The actual values stored for the cost are either 0 (reserved for non-traversable cells) or in the range 1-255, in which a higher number indicates a less traversable cell. A cost of 1 means there is no actual cost in that cell besides having to traverse it. 1

2 Figure 1: The cost field used for the following passes 2.2 Flow field passes In order to generate the flow field only three of the previously mentioned four passes are necessary. The tile pass is purely to increase the performance of the flow field generation. The following is an explanation of the passes in the correct order to generate a flow field Tile Pass Even though flow fields are a global navigation method a big part of the flow field will never be used. When multiple agents move from the top left to the top right of an area it is possible that the bottom half of the flow field is of no use. To take care of this flow field tiles are used, with each flow field tile being a part of the flow field. These tiles can be seen as a big grid overlay on the flow field. Instead of calculating the whole flow field it is decided beforehand which flow field tiles are of use for the situation (taking into account both the start point and the goal). The size of these tiles is important and the preferable size is different per application. Bigger tiles result in a larger portion of the flow field that needs to be generated, but there is a smaller chance that the flow field has to be recalculated. Smaller tiles will be great for performance when generating the flow field, but it takes longer to calculate a path along the tiles and the path might be too small for multiple agents. With a starting point and goal provided A* can be used to calculate a path along the tiles (each tile is a node and the cost is based on the average cost in the tile). This path contains all the tiles that will be used when calculating the flow field. However, it is possible that agents cross other flow field tiles (due to the formation or the flow field itself) or that agents are added that share the same goal and occupy an other flow field tile. An example of this can be seen in figure 2a, which contains the active tiles after the initial pass. Later on, when an agent passes a flow field tile not yet calculated, another flow field tile is activated as can be seen in figure 2b. Unfortunately, whenever a new flow field tile is activated the whole flow field (with all the other active flow field tiles) has to be recalculated. Due to the newly added flow field tile better paths might be created that affect other tiles as well. Not enough time was available to fully finish the tile pass and as a result of this incorrect flow fields can be generated. The tile path calculation does not take into account whether it is possible to travel between tiles, which sometimes causes impossible paths to be generated. A path could have a wall in the middle of it and the flow field will not go beyond that wall: there will be no flow field data available behind that wall. It is possible to solve this issue, as will be explained in section 4. 2

3 (a) Initial flow field tiles used, an external force pushes (b) Flow field tile added upon request and the flow field the agent away from the predicted path has been recalculated Figure 2: Flow field tiles visualized Line of Sight Pass The line of sight pass (LOS pass) is used to find all the cells that are visible from the goal. In this case visible means that a line can be drawn from the cell to the goal without crossing any cells with a cost above 1 (the lowest cost possible). If a cell has line of sight an agent can move towards the goal in a straight line without having to worry about any obstacles. The LOS pass starts a wavefront with the goal as its starting point. A wavefront owes its name to the fact that it moves like the front of a wave. While it moves in every direction it will come across everything that is reachable from its starting point. Figure 3 visualizes how a wavefront propagates through a 5x5 area. (a) (b) (c) (d) (e) (f) Figure 3: A wavefront visualized These wavefronts are used in both the LOS pass and the integration pass, as both passes need to traverse the cost field in a similar fashion. In figure 3 the red cells are the active cells of the wavefront and the actions taken in the LOS and integration pass will take place in these cells. The initial wavefront created by the LOS pass flags every cell as having line of sight. As the wavefront progresses it will set the integrated cost for every cell it passes. The integrated cost is the total cost it would take to travel from that cell to the goal cell (which is trivial as long as the cell has direct line of sight). However, whenever a cell has a cost above 1 it no longer has line of sight and the wavefront will stop (meaning that there is no LOS pass when the goal has a cost above 1). If in addition the cell turns out to be a corner a line segment will be made. This line is the edge between what is visible and what is not visible. The line is created using Bresenham s line algorithm from the corner cell outwards until it reaches the flow 3

4 field border, with the goal as the origin. Along this line every cell gets flagged as wave front blocking. This means that the LOS pass wavefront will not pass this line. As a result of that line the wavefront will never reach any cells that are out of line of sight. The following pseudocode snippet demonstrates how the LOS pass could be implemented: void LOSPass() { if (goalcost > 1) return; while (wavefront.numpoints > 0) { for (cell : wavefront.cells) { if (cell.cost > 1) { if (iscorner(cell)) createwavefrontblockedlines(); // Bresenham s line algorithm cell.stop(); else { cell.flags = einlineofsight; cell.integratedcost = length(cell - goal); wavefront.advancecells(); The resulting LOS field can be seen in figure 4, with white and black respectively being in and out line of sight, green being a LOS corner, and blue being a wave front blocked line. The cube visualizes the goal. Figure 4: The result of the LOS pass on the cost field shown earlier Integration Pass The integration pass integrates the cost for every cell that does not have line of sight. Cells that do have line of sight already have their integrated cost calculated in the LOS pass. The integration pass starts with a list of wavefronts. These wavefronts are created in the LOS pass; every LOS corner is the start of an integration wavefront. The LOS corners are the points at which line of sight stops, everything after this point does not have an integrated cost yet. The wavefronts simultaneously integrate outwards until they hit a border, a wall, or the wave front blocked line. For every cell the integrated cost of the cheapest neighbor and the cost of the cell are added together to get the integrated cost of the cell. With multiple wavefronts active at the same time it is possible that they will collide and integrate cells more than once. To avoid unnecessary integration the wavefront will stop whenever the newly calculated integrated cost is not significantly lower than the old integrated cost. If we would not check for a significant change a wavefront could integrate a previously integrated area without improving the integrated costs 4

5 significantly (a big performance cost for a small gain). The size of significant cost difference depends on how important it is to get the fastest flow field possible. A higher significant cost difference results in less backtracking wavefronts and thus creates the flow field faster. A lower significant cost difference results in a faster flow field, but it takes longer to create that flow field. To implement the integration pass the following pseudocode snippet can be used: void IntegrationPass() { for (wavefront : wavefronts) { if (wavefront.numcells <= 0) { delete wavefront; for (point : wavefront) { float cost = cell.integratedcost; for (neighbourcell : cell) { if (neighbourcell.integratedcost > 0 && neighbourcell.integratedcost + cell.cost < cost - SIGNIFICANT_COST_DIFFERENCE) { cost = neighbourcell.integratedcost + cell.cost; // The wavefront no longer provides a significant better integrated cost if (cell.integratedcost == cost) { celll.stop(); else { cell.integratedcost = cost; wavefront.advancecells(); In the end an integration field like the one in figure 5a is created, with a color gradient visualizing the integrated cost and the cube being the goal. (a) The result of the integration pass on the earlier shown (b) Color gradient from the lowest to the highest integrated cost cost field (please note that flow field tiles are disabled and that this is a section of a bigger terrain) Figure 5: The visualization of an integration field 5

6 2.2.4 Flow Field Pass Of all the passes the flow field pass is the most trivial to implement. This pass generates the actual flow field in which every cell contains a direction vector. The flow field passes iterates over all the cells and finds the cheapest neighboring cell in the Moore neighborhood with a radius of 1. It then generates a direction vector pointing towards that neighbor. This will provide a path from every cell towards the goal when the vectors are followed. The pseudocode would look like this: void FlowPass() { for (cell : flowfield) { if (cell.flags & ehaslineofsight) { cell.direction = enone; else { float cheapestcost = FLT_MAX; Cell cheapestcell = cell; for (neighbourcell : cells) { if (neighbourcell.integratedcost < cheapestcost) { cheapeastcost = neighbourcell.integratedcost; cheapestcell = neighbourcell cell.direction = getdirection(cell, cheapestcell); A resulting flow field can be seen in figure 6, in which the cube is the goal. The arrows in the flow field indicate in which direction an agent should move. When a cell does not contain an arrow the agent is able to move to the goal in a straight line. 6

7 Figure 6: The result of a flow field pass (flow field tiles disabled) 3 Formations A formation as used in this research is an arrangement of agents. Several examples can be seen in figure 7. Moving such formations can be a tricky task (Bjore (2013)). It gets even more difficult if you want the form to be retained. When moving along a path corners and small passages might force the formation to change its form, and that is no easy task. (a) Example of a line formation (b) Example of a square formation (c) Example of a V formation Figure 7: Tactical formations In order to solve the issue of formations not being able to properly follow a path the formation itself needs to change its shape. However, how should this be handled? It would be useful if every formation type would have a properly modified formation for every situation and for all agent amounts; a separate solution for every type of obstacle. In a more realistic and doable scenario the formation would attempt to modify itself on a lower level based on its surroundings. When the upper right part of the formation has to make way for an obstacle it would not be necessary to change the whole formation; the upper right part of the formation can solve it locally (as seen in figure 8). Simply splitting up the formation in a grid-like fashion could potentially solve the issue. However, it would only be able to solve the previously described situation if 7

8 it would be at a small scale. In case the entire right half of the formation has to make way for an obstacle it would no longer take care of the situation properly. It would indeed result in a noticeably better formation, but it can not create more space. Ideally, the right half of the formation would mix with the left half. (a) Formation is moving up and (b) Formation solves the problem hits an obstacle at the upper right locally instead of transforming the part of the formation whole formation Figure 8: Local avoidance in a formation 3.1 Layered Formations Layered formations attempt to solve this by splitting up the formation in different layers, as visualized in figure 9. As can be seen the structure is very similar to a quadtree; every layer has four children. In such a layered formation you end up having multiple grids, every next layer having grid cells twice the size of the previous layer. This method can be seen as a list of modifiable formations, with each next element having a more drastic effect on the formation as a whole. Figure 9: A square formation with 16 agents All of the layers have a position relative to their parent layer in the formation. This structure allows the formation to rearrange layers or agents (the deepest layer contains agents) whenever necessary. As a result of this it is possible to both gradually and partially change the formation. In figure 10 an example is shown in which a square formation gradually changes its form to fit through a narrow passage. In theory this solution works: the formation is able to adjust its form at different scales independent of the rest of the formation. However, in practice another issue arises. The layered formation can change form, but it is unclear when it should do so. It is difficult to detect if a formation should get smaller, and if it should get smaller: how much and on which side? Unfortunately, there was no time available to properly take care of this problem and as a result of this the layered formations have not yet been able to travel across a cost field with walls. 8

9 (a) One agent layer needs more (b) All agent layers need more space space (c) Even more space needed: upper layer changes form the Figure 10: A square formation gradually changes its form 3.2 Formation Movement Even though we have not yet managed to move layered formations through complicated terrains with walls, moving it while using a flow field is still possible. In order to find a correct system to move a layered formation we started with single agents. Moving agents by the use of a flow field is straightforward: the agent queries the flow field for a direction and then uses it as a force to move. However, moving formations requires two different forces: the flow field force and the formation force (to keep an agent at its local position in the formation). The formation force has a strength that is dependent on the distance between the agent and its formation position. The farther the agent is the stronger the force will be. Properly weighing these forces (equal weights gave the best results in our application) creates a formation that both flows and retains its form. Making the step from regular formations to layered formations only changes the way the formation position is calculated. In regular formations the formation position is trivial, in layered formations it is relative to the position of the layer of the agent. These layers themselves move as well, just like the agents do. The layered formations correctly flowed across terrains when these methods were applied. However, there were still some issues. For example, when formations got closer to their goal the flow field force would differ so much from the formation force that it resulted in agents coming to a complete halt. The formation as a whole broke as well: every agent tried to get to the middle point (the goal), as visualized in figure 11. We solved this by gradually increasing the weight of the formation force and decreasing the weight of the flow field force when the formation got closer to its goal. Another problem, which has not been solved yet, is that the formation force could push agents into walls. This problem has not been solved mainly because it is related to layered formation transformations. The formation force weight should be decreased at the same time as the layered formation transformations happen. Due to the fact that we did not have the time to properly implement when these transformations should happen this problem has not been solved either. (a) (b) (c) (d) Figure 11: Flow field force getting too strong in relation to the formation force 9

10 4 Conclusion and Future Work Flow fields have been proven to work before (Emerson (2013)) and layered formations are capable of layered transformations. The difficulty lies in the fact that it is hard to detect when the formation should change. Having layered formations will not automatically make those formations adapt to the environment correctly. They should properly detect when to change, and maybe even more important: when to rewind those changes to get back its original form. Only when the formations are able to do this correctly they will be able to fully use the flow fields to flow towards their goal. Another point of improvement is the tile pass for generating a flow field. In its current state it only works when the area contains no walls. It should identify entrances between tiles and create a proper path based on those entrances. Annotated Hierarchical A* (Harabor and Botea (2008)) could solve this problem. AHA* also takes into account the width of the agents (in our case the width of the formation) which could reduce the amount of transformation needed in the formation. References Bjore, S. (2013). Techniques for formation movement using steering circles. In Game AI Pro. Emerson, E. (2013). Crowd pathfinding and steering using flow field tiles. In Game AI Pro. Harabor, D., & Botea, A. (2008). Hierarchical path planning for multi-size agents in heterogeneous environments. Retrieved from bartak/keps2008/download/paper05.pdf Reynolds, C. W. (1999). Steering behaviors for autonomous characters. Retrieved from 10