roadmap based path planning - a report

7/29/2019 Roadmap Based Path Planning - A Report

1/18

Roadmap Based

Path Planning

Anirudh Singh


2/18

1

ABSTRACT

Path planning is an integral part of game development. In fact,

there are a number of virtual world models which depend on

planning for the motion of entities. A lot of algorithms are

available for this purpose like scripting, grid-based search, local

reactive methods but most are difficult to implement in real-time

and require a lot of post-processing to obtain clear roadmaps. This

motivates us to look for innovative ideas to develop roadmaps as

well as answer path queries for single / multiple entities in real-

time. Two of the methods discussed in this report create their

roadmaps based on the Voronoi diagram to generate smooth and

collision-free paths. I also discuss the behavior of a group of

entities moving together. I also discuss another method that

addresses the issue of movement in 3-D and is based on a

modification of the A* algorithm but it requires to two additional

post-processing steps.


3/18

2

Contents1. Introduction

3

2. Path Planning: Single Entity3

2.1.Roadmap Generation 4

2.1.1.Creating samples4

2.1.2.Improving the roadmap4

2.2.Answering queries

5

3. Path Planning: Multiple Entities6

3.1.Backbone path and corridor

6

3.2.Movement in corridor 7

3.3.Coherence of group

7

4. Camera Motion8

5. Results 96. Conclusion

9


4/18

3

7. References9


5/18

4

1. Introduction:

Path planning in a virtual environment is a vital part for the

movement of virtual entities like characters, vehicles or a camera.

The entities not only have to traverse through the ground plane

(2-D) but in many cases having open terrain simulations, have to

plan their motion in 3-D. Also in some cases, the motion of a

group of entities in a virtual world has to be planned, for example

the movement of soldiers in a game. Some methods typically used

for planning the above are scripting, grid-based search, local

reactive methods and flocking.

Scripting involves explicitly defining the paths that must be

followed by the entities. This is a very time-consuming process and

the user is easily able to identify the repletion in the paths

followed by the entities. It also is very complicated when a numberof entities are present.

For real-time applications such as games, it is very important to

decide how the virtual world is to be spatially segregated for

planning effectively. Grid based methods are fast and useful as

long as the environment is small because they work by dividing

the whole world into a grid of cells.

Reactive methods are based on the idea of adapting a pre-

computed path to the obstacles near the path which were not


6/18

5

considered previously. This method is fairly simple but often leads

to deadlock situations where the entity cannot decide where to

move, e.g. it can get stuck in a corner of a room.

Thus, for any path planning algorithm must meet the following

requirements:

o The path should have the lowest costo It should be fast robust and accurateo Should be generated automaticallyo Should be applicable to different world maps

2. Path Planning: Single Entity

The following sections discuss the method used for generation of

the roadmap for the entities to traverse. The method which isproposed is a slight variation of the Probabilistic Roadmap (PRM)

method that is based on the Vornoi diagram.


7/18

6

2.1. Roadmap Generation

The Voronoi diagram defines a set of points in the world space

that are equidistant from any two closest obstacles. The roadmap

that is to be developed has to satisfy the following criteria:

o It should have a certain minimum amount of clearance fromthe obstacles

o The paths should be smooth and continuouso The paths should be created fast and be short

2.1.1. Creating Samples

Sampling involves the creation of vertices and edges on the

Voronoi diagram to get a graph which will finally be used for

planning. The method involves iteratively picking up a sample c

from the free space and retracting to the Voronoi diagram by

means of the following steps. The closest point cc to c on any

obstacle near it is found (Fig 1 a) and the point from copposite cc

is marked as c. This point is moved away from c till the closest

obstacle to itchanges (Fig 1 b). Now a binary search between c

and c leads to the vertex cv (Fig 1 c) which almost lies on the

Voronoi diagram (with an error of)


8/18

7

(a) (b) (c)

Figure 1 Creating Samples

The above steps are followed till the required number of vertices

are located. The vertices are then connected using to form the

edges wherever they do not collide with the obstacles.

2.1.2. Improving the Roadmap

The edges obtained from the previous step do not lie on the

Voronoi diagram but often get too close to the obstacles (Fig 2 a).

Thus these edges have to be retracted back to the Voronoi

diagram till they have a pre-specified amount of clearance form

the obstacles. This is done by first finding the mid-point of the

edge close the obstacle and retracting this point to the Voronoi

diagram using the method described above. The edge is then split

into two different segments of equal parts (Fig 2 b). This is done

repeatedly until the desired clearance is achieved or till the length

of the retracted edges shorter than a pre-defined limit (Fig 2 c).


9/18

8

(a) (b) (c)

Figure 2 - Retracting Edges

The roadmap so obtained is still made up of straight lines and

sharp turns. Thus parts of these line segments are replaced by arcs

of a circle to obtain a smooth curve called a circular blend. The

degree of a vertex is given by the number of incoming line

segments at that vertex. For any vertex with a degree of two or

more, the mid-points of every pair of incoming line segments are

found and are connected using arcs of circles (Fig 3 a, b). In some

cases, adding a circular blend decreases the clearance at that

edge. In such cases, the blend is replaced by another circular blend

of smaller radius till the desired clearance is obtained (Fig 3 c).

(a) (b) (c)

Figure 2 Circular Blends


10/18

9

2.2. Answering the queries

Once the above steps are completed, the roadmap is ready and it

can be searched in real-time to plan a path in the environment. An

example of a roadmap is shown in figure 4 a. The initial and final

positions are connected with the closest vertices via circular blends

and then the roadmap is searched using Dijkstras algorithm for

the shortest path between these vertices. The final path looks

somewhat like shown in figure 4 b.


11/18

10

(a) (b)Figure 4 Roadmap and Final Path

3. Path Planning: Group of Entities

It is often required in virtual environments to plan the motion of a

group of entities for e.g. movement of a group of computer

controlled soldiers in a game. Most current techniques to solve

this problem involve the use of planning for each entity separately

and keeping them together using flocking. But this is often difficult

in complicated environments and the difficulty increases with


12/18

11

obstacles. While planning the paths for a group of entities in the

virtual world, the following things should be kept in mind:

o The entities must avoid collisions with obstacles along thepath and amongst themselves

o They should always stay together as a groupThe entities are modeled as discs moving in a plane. Path planning

is done by first finding a backbone path for a single entity and

then defining a corridor around that path. All entities must stay

inside this corridor.

3.1. Backbone Path and corridor

A backbone path is a path in the world where the clearance at

every point on the path is at least equal to the radius of the

enclosing circle of the largest entity. It is preferred to have as

much clearance along the path as possible so as to allow for

coherent movement of the entities. A corridor is defined along the

path as the radius of the largest circle centered at every point

along the path, which does not intersect with any obstacles. The

movement of the entities is restricted to the corridor.

If generated using a non-optimal roadmap, the backbone path

could be very close to the obstacles at some points (Fig 5 a). This

would result in the corridor being very narrow at some points

along the path. But, when an optimal roadmap is used, like the


13/18

12

one described in section 3, then sufficient clearance is obtained

along the path (Fig 5 b).

(a) (b)

Figure 5 Backbone path and Corridor

3.2. Movement in the Corridor

Artificial force-fields technique is used to monitor the movement

of the entities in the group. Every entity is assigned an attraction

point on the backbone path which is the maximum advance point

p(Fig 5b) such that the entity is still inside the circle centered at p

with radius equal to the clearance at that point. This keeps the

entities in the corridor and there is repulsion among the entities to

avoid collisions.

3.3. Coherence of Group


14/18

13

The groups dispersion should be upper-bounded to keep it

coherent so as not to allow the entities to stray off the backbone

path. The lateral dispersion is already bounded by the corridor

width. The longitudinal dispersion, though, is not bounded. Both

the dispersions can be manipulated to control the behavior of the

group along the path. A high longitudinal dispersion results in a

stretched out group (Fig 6 a). Whereas low longitudinal and high

lateral dispersion will result in the group being huddled together

(Fig 6 b).


15/18

14

(a) (b)

Figure 6 Group Coherence

4. Camera Motion

A camera is essential to every virtual environment as it is the

portal through which the user can see the world. Thus there can

be a single or multiple cameras in a virtual world and they are to

be allowed to move and rotate to change the viewing angle and

position. In many situations, a camera moving automatically in the

environment is of great use (e.g. an architectural walk for users

through housing projects to give a feel of the site). The roadmap

construction discussed above is very suitable for this application as

the paths generated are smooth and continuous. There are two

more parameters to be considered. First, the speed of the camera

should adapt to the path for a smooth motion of the camera. Also,

there should be a maximum limit on the acceleration and

deceleration of the camera to avoid abrupt changes in speed. The


16/18

15

idea gets clear on looking at figure 7 where (a) shows the path to

be followed by the camera, (b) shows the corresponding maximum

speed limits along the path and figure 7 (c) shows the actual

speed of the camera considering the bounds of acceleration and

deceleration. Thus it figures that the speed of the camera will be

slower around the curves and will increase along straight paths.

(a) (b) (c)

Figure 7 Camera path, speed and acceleration

The second parameter to be kept in mind is the viewing angle of

the camera. The camera should always point in the direction where

it is going to be after some time (say a second) in order give a

better idea of where it heading. The position along the path where

the camera would point to depends on the speed of the camera.

4. Results


17/18

16

Computation of the roadmap shown in figure 4 (a) took around 1

second on a P 4, 2.4 GHz machine. That is nominal considering the

fact that the roadmap can be generated once when the virtual

environment is being created / modeled and saved for future use.

Once the roadmap is obtained, answering the path queries hardly

take any time. It was observed that for a roadmap consisting of

1000 vertices and 3000 edges, the path was calculated in less than

10 ms. Even for a group of 50 to 100 entities, the path was

generated without much processor usage (< 1 percent). Moreover

the behavior of the entities could easily be varied by changing thedispersion parameters. Even the application for planning a camera

motion in a virtual environment generates smooth camera motion

along the path to the destination selected by the user. The user is

also allowed to move freely in the virtual environment.

5. Conclusion

In this report, I have looked at two path planning methods one

which talks about generating real-time smooth paths for a single

entity and the other for a group of entities. This approach is very

useful in large environments where grid based search fails. The

method described is a 2-D method, though it can be expanded to

allow the motion of the entities in 3-D.


18/18

17

The roadmap can be customized as per the requirements of the

programmer, allowing him to add or delete paths, attach weights

to certain paths to make them more preferable. It is also very easy

to add obstacles to the environment as the roadmap will simply

modify accordingly.

7. References

1 Automatic Construction OF High Quality Roadmaps for PathPlanning, D. Nieuwenhuisen, A. Kamphuis, M. Mooijekind, M.

H.Overmars, Dec 2004

2 Generic Path Planning For Real Time Applications,Christopher Niederberger, Dejan Radovic, Markus Gross

3 A Recursive Approach to roadmap based path planning,David W. Dougall, James K. Archibald

4

Roadmap Path Planning, Brian C. Williams, Oct 20035 Algorithms For Sensor-based Robotics: Roadmap Methods, G.

D. Hager, Nov 2006

6 Automated Planning: Theory and Practice Planning inRobotics, Dana Nau, 2005

7 Route Planning & Physically-based Modeling Using GPUAccelerated Computation, Ming C. Lin, May 2004

8 Robot Motion Planning, Dr. John Xiao

roadmap based path planning - a report

Documents