Path Planning and Navigation of Mobile Robots

in Unknown Environments

Torvald Ersson and Xiaoming Hu

Optimization and Systems Theory / Centre for Autonomous Systems

Royal Institute of Technology, SE 100 44 Stockholm Sweden

(ersson@math.kth.se, hu@math.kth.se)

Abstract

For a robot trying to reach places in at least partially

unknown environments there is often a need to replan

paths online based on information extracted from the

surroundings. In this paper is is assumed that the sens-

ing range of the robot is short compared to the length

of the paths it plans and the environment is modeled as

a graph consisting of nodes and arcs. The replanning

problem is solved using the network simplex method .

The applicability of the planner is demonstrated by inte-

grating it with a navigation control strategy. Simulation

results show that the approach works well.

1 Introduction

Path planning and navigation for mobile rob ots, in par-

ticular the case where the environment is known, is a

well studied problem, see, for example, the bo ok by

Latombe [4] and the references therein. In practice,

however, one problem is that often no complete knowl-

edge about the environment is available. Having a de-

tailed map with all the obstacles marked seems to be

unrealistic for most situations. In many outdo or appli-

cations the robots can determine their co ordinates by

using, for example, GPS. However the knowledge about

the surroundings may often be very limited. Under such

conditions there is to o much uncertainty for a very de-

tailed plan to make sense. For preplanning purposes a

coarser choice is probably go od enough. Additionally it

is important to be able to replan the path online based

on new information obtained by sensors while navigat-

ing.

A natural way of updating plans is to ?rst select a path

based on the present knowledge, then move along that

path for a short time while collecting new information.

Based on the new ?ndings the path is then replanned.

This methodology is often used in the literature for path

planning in unknown areas. One of the original moti-

vations for studying this problem was the terrain acqui-

sition problem, where a robot is required to produce a

complete map of an unknown terrain. In many publica-

tions graph methods are used for solving the task. For

example, [6] considers disjoint convex p olygonal obsta-

cles and presents proofs on convergence. [5] does not use

any constraint on the shape of the obstacles and tries to

minimize the length of the path during the operation.

In [7] results for more general graphs are presented.

In this paper an online path planning algorithm, based

on the so-called network simplex method, is proposed.

Compared with the aforementioned graph metho ds, the

information stored about the environment is strictly in-

tended for path planning and less details about the ob-

stacles are needed. Although the algorithm of this pa-

per to some extent is similar to the

D

algorithm [8, 9],

the two strategies di?er in a number of ways. Firstly,

here a di?erent discretization scheme is used. Instead of

modeling the environment as a set of squares this paper

uses a graph representation. Secondly, while producing

a path from the start to the goal

D

has to solve the

shortest path problem for every possible starting p oint.

The algorithm of this paper does not require these ex-

tra solutions. Here the shortest path problem is thought

of as a minimum cost ?ow problem and solved by the

network simplex metho d. Furthermore, in this paper

the feasibility of the algorithm is justi?ed by integrat-

ing it with a generic navigation control strategy, since

the topic here is

online

path planning.

It is worth to mention that as a contrast a completely

di?erent online path planning approach is presented in

[11]. There a path is computed using steepest gradient

descent on an up dated harmonic function.

It should be emphasized that in this paper the range of

the sensors is assumed to be limited. A requirement for

the metho d to work is that the sensing range is longer

than the length of the longest arcs of the graph. I.e. the

lower the sensing range the closer the no des need to be

placed. In [10] the sensing constraint is relaxed and an

extension of [8, 9] with special box patterns for sparse

environments is presented.

This paper is organized as follows. First the path plan-

ning approach, using network simplex, is presented. A

more general description of the planning approach is

followed by an example where parameter values have

been chosen. Then a navigation controller is described

and integrated with the path planning. Finally simula-

tion results illustrate the performance of the combined

strategy.

2 Path Planning

Three basic assumptions used in this approach are:

•

The robot has a short sensing range compared to the

size of the region of interest.

•

It senses radially from its position. I.e. obstacles can

block the sensing in some directions.

•

It knows its coordinates and orientation (for example,

via GPS).

2.1 Discretization

For our modeling it is necessary that the local region be-

ing considered is discretized. Discretizing the problem

obviously excludes lots of solutions. However in many

cases no exact path following is needed and a coarse plan

is satisfactory. Another concern might be that in some

cases the planning can result in a non-smooth path. For-

tunately this matter can be dealt with using a control

algorithm that smoothens the motion.

In this paper the terrain is described by nodes connected

by arcs. Each arc has a speci?ed cost for moving along

it. Another straight forward approach, used by [8], is to

split the environment into squares/rectangles of equal

size. Then one assigns a cost for a transition between

two neighboring squares.

A similar description of the environment is to ?rst place

nodes in a grid pattern, then introduce arcs between

neighboring nodes and assign costs to these arcs. (Natu-

rally the positions of the nodes need to be known.) Two

obvious representations are shown in ?gure 1 a) and b).

(An arc drawn between two nodes in the pictures should

be interpreted as two arcs, one in each direction.) The

two patterns in ?gure 1 a) and b) are, as can be seen in

?gure 2, almost equivalent to the approach using squares

in a regular pattern. The node-arc representation can

however easily be extended to more advanced patterns

such as the one in ?gure 1 c). This structure allows

more planning options and makes shorter and smoother

a)

b)

c)

Figure 1:

a) ”4-star” b) ”8-star” c) “16-star”.

paths possible. Note that some of the arcs would corre-

spond to moving between non neighboring squares. For

Figure 2:

”4-star” and ”8-star” mapped onto a square pat-

tern.

a node arc approach to be interesting one, as previously

mentioned, needs to have a sensing range that is longer

than the length of the longest arcs. Apart from that

restriction any kind of no de arc network can be used.

The cost of an arc is naturally dependent on the dis-

tance between the no des, but other aspects need to be

taken into consideration as well. It is easy to include

knowledge about the environment as e.g. lower costs of

arcs in “friendly” terrain and very high costs for arcs af-

fected by obstacles such as rocks, steep descents, creeks,

fences, walls etc.. Of course this very high cost is ideally

in?nity, but in a world of computers in?nity is replaced

by a value much greater than the costs of the arcs that

can be used.

This scheme for discretizing the surroundings makes

it really easy to use the well known network simplex

method for the path planning.

2.2 Minimum Cost Flow

For the sake of completeness this linear optimization

problem is described.

Consider a network consisting of

n

nodes plus arcs con-

necting them. The unit cost for using the arc going

from no de

i

to node

j

is

c

ij

. The ?ow going that way

is denoted

x

ij

. At node

i

there can be a ?ow supply or

demand denoted

b

i

(supply:

b

i

>

0,demand:

b

i

<

0).

The problem is balanced if

n

i

=1

b

i

= 0. For every no de

in?ow minus out?ow equals demand/supply. The prob-

lem of minimizing the total ?ow cost of this network can

be stated as the following linear optimization problem:

minimize

n

i

=1

n

j

=1

c

ij

x

ij

(1)

s.t.

n

i

=1

x

ij

?

n

i

=1

x

ki

=

b

i

, i

= 1

..n

(2)

x

ki

?

0

(3)

Only node-pairs (

ij

) corresp onding to arcs need to be

considered.

Network simplex

[2] is a well known e?cient method for

solving such a problem. It works with so called basic

solutions of the problem. Such a solution corresponds

to a spanning tree. (A graph of arcs that connects all

nodes of the network and do es not include cycles.) The

three steps of the metho d can be summarized as follows:

Step1.

Start with a given basic feasible solution. Such

a solution can be created by constructing a spanning

tree.

Step 2.

Compute node prices

?

i

for each node

i

. This

is done by solving

?

i

?

?

j

=

c

ij

(4)

for each

i, j

corresponding to a basic arc. For the above

equation to have a unique solution one no de price has

to be assigned a value, e.g.

?

n

= 0. The reduced costs

r

ij

for non basic arcs are computed using

r

ij

=

c

ij

?

(

?

i

?

?

j

)

.

(5)

If all

r

ij

are nonnegative the solution is optimal. Oth-

erwise go to step 3.

Step 3.

A new spanning tree is created by adding an

arc to form a cycle and then eliminating another arc

from the cycle. Select a non basic arc with negative

reduced cost

r

ij

to enter the basis. Addition of this arc

to the spanning tree will pro duce a cycle. Intro duce a

?ow of value

?

around this cycle. As

?

is increased some

old basic ?ows will decrease.

?

is chosen as the smallest

value that makes the ?ow in a basic arc zero. That

zeroed arc now goes out of the basis. The new solution

obtained in this step is then plugged into step 2.

The problem described above is more general than a

shortest path problem. To model a path starting at

node

k

and ending at node

l

a supply of one is placed at

k

,

b

k

= 1, and a demand of one at

l

,

b

l

=

?

1. All other

b

i

are equal to zero. in this case a basic solution includes

the actual path ?ow represented by arcs carrying a ?ow

of one, as well as arcs with zero ?ow. The latter ones

are needed for creating a spanning tree.

Assume that network simplex has produced an optimal

solution. What happens if a few costs

c

ij

are changed ?

The solution is still both basic and feasible so step 2 is

entered. If

c

ij

is changed for a basic arc, there might be

a need to recompute all node prices before checking a

large number of reduced costs of the network. If no cost

of any basic arc is changed only the

r

ij

corresponding

to the updated non basic arcs need to be checked.

Network simplex is not the fastest possible choice for

solving a shortest path problem from scratch. However

when having an optimal solution and changing a few

costs it is quite e?cient.

2.3 Path Planning Algorithm

The general idea is that a path is planned based on what

is known right now. When the robot gets new informa-

tion it considers choosing another path. Gradual learn-

ing about the surroundings results in better and better

plans. Information about the environment is translated

to arc costs

c

ij

. If the terrain is completely unknown

the arcs can initially be assigned costs as if they were

not a?ected by obstacles. Otherwise information from

e.g. maps can be incorporated as suitable arc costs.

Step 1. Planning

Based on the present knowledge a path from the current

node to the goal node is planned using network simplex.

If the cost of this path is so high it is clear that an

arc with ’in?nite’ cost is used the algorithm must be

terminated. This means no feasible path exists within

the network. If the cost is reasonable carry on to step

2.

Step 2. Moving and sensing

The platform moves along the ?rst arc that is part of

the solution while collecting information about the en-

vironment. At the end of the arc a new node is reached.

If that node is the goal node the task is completed, oth-

erwise step 3 follows.

Step 3. Reposing the problem

Unless the arc has been used before and nothing has

changed the system now knows more about the sur-

roundings and has updated a number of arc costs

c

ij

.

This new information can be used for replanning. In or-

der for the new planning problem to be correctly posed

the supply of one is moved from the previous no de to

the current one. The previous base solution is slightly

modi?ed to be a starting guess to the new problem. The

only adjustment needed is setting the ?ow of the latest

arc used to zero. In case no new information has been

added this is an optimal solution. The algorithm now

returns to step 1.

2.4 Some Practical Issues of Online Path Plan-

ning

Sometimes there are several paths with the same total

cost, as in the case of a ?at area almost free of obsta-

cles. In these cases the proposed metho d will cho ose one

of them arbitrarily. For some vehicles the choice does

not matter that much, but for others it can make a big

di?erence. Many mobile platforms, especially those in

outdoor applications, are nonholonomic. One implica-

tion of this is that the range of steering is limited. Even

if that range is su?cient a more winding path with un-

necessary many turns is often unwanted.

Therefore ways of avoiding such choices are of interest.

To prevent the path planner from cho osing jerky paths

one should lower the cost of going straight by a small

quantity

?

. This means that if a solution starting with a

turn has the same cost as one starting by going straight

before anything is adjusted, the

?

will make the metho d

pick the arc going straight. (?gure 3) Once the robot

has left the node the arc going straight will regain the

cost it had before

?

was subtracted.

?

should be sig-

ni?cantly smaller than the cost of an arc. It should

be mentioned that in general an introduction of bigger

temporary cost reductions can cause deadlo ck problems.

In order to show the applicability of the approach one

b)

a)

Figure 3:

a) If all the arcs have equal cost this winding so-

lution is an optimal one. b) If there in every it-

eration is a slightly lower cost for going straight,

this is the solution chosen when the ?rst action

is a step to the right.

has to answer the following question: what costs can be

estimated during motion? If the sensing range is only

marginally longer than the length of the longest arcs it

implies that:

•

The robot can estimate the costs of the arcs that in-

tersect the path.

•

It can estimate the costs of the arcs starting and end-

ing at the nodes it visits.

In case of a ”4-star” pattern there are no intersecting

arcs. For the ”8-star” pattern the only arc intersec-

tion occuring is between diagonal arcs. If a ”16-star”

pattern is considered the situation is much more com-

plicated. When this node-arc mo del is used there are

three di?erent arc types. In ?gure 4, it is shown how

other arcs intersect the three in the interior of the grid.

(Close to the borders the situation is slightly revised.)

These cutting arcs will be visible when moving along

the indicated types.

From a logical point of view often even more arcs can

be measured. Consider a visible arc that is not blocked

by obstacles. Assume that one of its neighb or arcs is

parallel to it and lies within the range of the robot sen-

sors. Then they both lie on the same line and within

the sensing range. In that case it is possible to estimate

the cost of that more distant arc as well.

b)

c)

a)

Figure 4:

Arcs of a) type 1, b) type 2 and c) type 3 are

shown as thick arrows. Intersecting arcs are

drawn as thin arrows.

One could of course choose a denser grid. In such a case

the sensing range may widely exceed the length of the

longest arcs of the network. That means that more arcs

are visible from the no des and arcs used by the mobile

platform.

2.5 Some Drawbacks

Discretizing a continuous problem is often a goo d way of

making it easier, but doing so often intro duces unwanted

side e?ects. There are some obvious drawbacks with

this graph approach. Firstly the solutions obtained are

optimal for the discrete network problem, but not for

the original problem. In ?gure 5 a) this is demonstrated.

Imagine that the terrain is ?at and free of obstacles. The

real optimum is shown as a dashed line and the network

optimum as white arrows.

Secondly in some cases an unfortunate choice of network

might cause the discretized problem to be infeasible,

even though the original problem has a solution (?gure

5 b)). Lo cal online modi?cation of the network is one

possible way of dealing with such situations.

goal

start

obstacles

b)

a)

Figure 5:

a) The optimal solution for the discretized prob-

lem is clearly di?erent than the optimal one for

the original problem. b) The original problem is

feasible, but the discretized is not.

2.6 Maze Example

If the environment is fairly uncomplicated a robot with-

out memory or planning skills could still have a sat-

isfactory performance. To show the p otential of the

proposed planner a tough example with lots of possi-

ble deadlocks was designed. The maze-like environment

chosen would cause many less sophisticated approaches

to fail. As seen in ?gure 6 the planner did not have a

problem ?nding a way through in a rational manner.

When the example was tested on a computer with a

360 MHz UltraSPARC II processor the average plan-

ning task at a node to ok about 70 ms to ?nish. A grid

consisting of 50x50 no des connected with “16-stars” was

used. The distance from a no de to its closest neighbors

was 1 length unit and the length of the longest arcs was

2.23. The sensing range used was 2.3 length units and

?

was set to 0.1. The area was initially assumed to be

obstacle free. For obstacle free arcs the costs were equal

to the length of the arcs. Arcs blocked by walls were

assigned a cost 10000 times higher than their length.

3 Integration of Online Path Planning

and Control

Good and robust path tracking control strategies are

needed in many applications and this has b een a well

studied topic. Naturally, how good a path following con-

trol is depends a lot on how well the path is planned.

Many path planning strategies pro duce a smo oth path

where the curvature, for example, is minimized. The

path planning algorithm of this paper, which is designed

for navigation in an unknown environment, is quite dif-

ferent. Since a path pro duced by the algorithm is not

even smooth, it is sometimes di?cult to follow it exactly.

However, when the mobile robot navigates in an at least

partially unknown environment, where online path plan-

ning is almost unavoidable, following the planned path

exactly is not the top priority. The important thing is

0 5 10 15 20 25 30 35 40 45 50

?50

?45

?40

?35

?30

?25

?20

?15

?10

?5

0

Figure 6:

Solution of the maze example. The dots and

arrows show how the robot moves. Walls are

drawn as thick lines and the thin lines are tem-

porary plans on how to reach the goal in the top

right corner. The starting point is situated in

the bottom left corner.

that the robot follows the path reasonably well in a very

robust way. The robustness requirement is easily under-

standable since small hinders and external disturbances

should always be taken into consideration, especially in

outdoor applications.

In this section, by integrating the path planning algo-

rithm with the so called virtual vehicle control strategy

[3] , it is shown that the algorithm serves well for the

purpose of navigating in an unknown environment. Of

course many other control approaches could be inter-

esting as alternatives. However here the aim is just to

show that the planner can be used in practice.

3.1 The Virtual Vehicle Approach

For the sake of simplicity, only paths in a 2D-plane are

considered. Suppose that a planned reference path is

parameterized as:

x

d

=

p

(

s

)

, y

d

=

q

(

s

)

,

0

?

s

?

s

f

(6)

where subindex

d

stands for desired. Consider a vehicle

and a reference as shown in ?gure 7. Now assume that

p

(

s

)

2

+

q

(

s

)

2

= 0 and let

?

x

=

x

d

?

x,

?

y

=

y

d

?

y, v

= ?

x

2

+ ?

y

2

?

?

=

?

d

?

?, ?

= (?

x

)

2

+ (?

y

)

2

(7)

where

v

is of course the vehicle speed and (

x, y

) a ref-

erence point on the mobile platform, for example, the

mid point of the vehicle’s front axle for a car-like robot.

?

denotes the orientation of the platform.

In the virtual vehicle approach the parameter

s

is

treated as a function of the time

s

(

t

). Its dynamics

is governed by a di?erential equation that contains the

tracking error

?

. Thus

s

(

t

) is called the virtual vehicle

(that the actual one should track). Suppose that one

(p(s),q(s))

?

(x,y)

?

y

?

x

d

?

?

Figure 7:

Virtual vehicle geometry.

can control the rotational and translational velocity of

the platform. (They can be viewed as higher-level con-

trols. For platforms that do not have direct control over

the velocities, one needs to design the actuator control

so that these velocity controls are realized.) Then the

following generic control algorithm is given by [3]:

?

?

?

?

?

?

s

=

ce

?

??

v

0

v

p

(

s

)

2

+

q

(

s

)

2

?

=

k

?

?

+ ?

?

d

v

=

??cos

(?

?

)

(8)

where

?

= ?

?

is the rotational velo city and

?

,

c

,

?

and

k

are some positive constants. As can be seen in the

translational velo city control law, reverse drive is used

when the magnitude of the deviation from the desired

direction is too big.

In [3] it is shown that this control strategy is stable

and quite robust for the two di?erent platforms dis-

cussed there. Furthermore, the tracking error can be

tuned arbitrarily small asymptotically but remains pos-

itive (note if

?

= 0 then

?

d

is ill-de?ned).

Since the position and orientation of our platform are

assumed to be known the above closed-loop control ap-

proach can easily b e applied here.

3.2 Path Planning Online Combined with the

Virtual Vehicle Approach

It is obvious that in order to use the virtual vehicle

approach, the path has to be smo oth (

C

1

). Therefore

there is a need to somewhat mo dify the path produced

by the planning algorithm.

In fact, if relaxing the problem slightly one can say that

a node is reached if the robot gets within a certain dis-

tance from it. This will de?ne a “goal disc” centered

at the no de point. Between such discs trying to track a

straight line along an arc seems reasonable. The radius

of the goal area is limited by the fact that one needs

to be close enough to evaluate outgoing arcs from the

node. Otherwise the planning would be crippled by a

lack of necessary information.

From an initial position (usually at the border of a goal

disc) the vehicle will try to track a straight line leading

to the next node area using the virtual vehicle approach

presented above. When the robot gets there and plan-

ning has taken place the next node along the path is

determined and the same procedure takes place again.

This will go on until the ?nal goal is reached. (?gure 8)

If the starting node has position ?

r

1

= [

x

1

, y

1

]

T

and the

next node is lo cated at ?

r

2

= [

x

2

, y

2

]

T

the line between

them can be described by ?

r

= ?

r

1

+ (?

r

2

?

?

r

1

)

s

s

f

, yielding

p

(

s

) =

x

1

+ (

x

2

?

x

1

)

s

s

f

q

(

s

) =

y

1

+ (

y

2

?

y

1

)

s

s

f

(9)

and

p

(

s

) =

x

2

?

x

1

s

f

, q

(

s

) =

y

2

?

y

1

s

f

.

(10)

Clearly

p

(

s

)

2

+

q

(

s

)

2

= 0 unless

r

1

=

r

2

. The reference

could have

s

starting at zero or at a relatively low value.

In simulation it turns out that the vehicle chases the ref-

erence point like a dog would chase a rabbit, rather than

being in front or beside it. Since the robot chases the

reference p oint, the reference line might extend beyond

the next node point (

s > s

f

).

ideal planned path

actual motion with

relaxed conditions

Limit for sufficient info about

outgoing arcs from the node

Circle where a new goal

point is chosen

Figure 8:

Relaxation of the control problem.

If the reference is smo oth and long enough the method

is guaranteed to work, but the switching of reference

paths at the nodes could cause problems.

3.3 Example of Online Path Planning Combined

with Control

The combined planning and control package was tested

on the previous maze example.

In the simulation, a car-like robot model [1] is used:

?

?

?

?

x

=

v cos

(

?

+

?

)

?

y

=

v sin

(

?

+

?

)

?

?

=

v

l

sin

(

?

)

(11)

where (

x, y

) is the mid point of the vehicle’s front axle,

?

is the steering angle and the other components have

the same de?nitions as ab ove. This mo del works for

both forward and backward motion as well as switching

continuously between them.

With this mo del we ?rst need to ?nd the corresp ond-

ing steering control from the generic rotational velocity

control:

?

=

sin

?

1

(

l

v

(

k

?

?

+ ?

?

d

))

.

(12)

The simulation result is displayed in ?gure 9, where the

steering angle is assumed to be saturated at +

/

?

? /

5.

Figure a) is too small to show any details but as seen in

b) the vehicle moves quite nicely along the planned path.

It should be noted that the robot was relatively small

(turning radius ab out 0.2 length units) and drove slowly,

but the outcome of the test is undoubtedly positive.

0 10 2 0 3 0 4 0 5 0

? 50

? 45

? 40

? 35

? 30

? 25

? 20

? 15

? 10

?5

0

3 0 30 . 5 31 3 1. 5 32 3 2. 5 33 3 3. 5

? 21

? 20 .

5

? 20

? 19 .

5

? 19

? 18 .

5

? 18

b)

a)

Figure 9:

a) The actual path through the maze. b) Zoomed

detail of the maze example. An

?

represents a

node chosen by the planner.

4 Conclusions

The paper presents an online path planner for unknown

or partially unknown environments and shows that it

can deal with really di?cult problems, such as mazes.

Furthermore, the applicability to navigation is demon-

strated by combining the planner with path following

control. Simulation results for the integrated online

path planning and path following control module sug-

gests that the method has a good chance of being fea-

sible for real applications.

Acknowledgments

Special thanks to Dr. Stefan Feltenmark for his help

with the code and his advice about network methods.

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Robust Control

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