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Tiêu đề Mobile Robots Navigation
Trường học University of Technology
Chuyên ngành Mobile Robotics
Thể loại bài luận
Năm xuất bản 2008
Thành phố Hanoi
Định dạng
Số trang 40
Dung lượng 3,15 MB

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An Artificial Protection Field Approach For Reactive Obstacle Avoidance in Mobile Robots Victor Ayala-Ramirez, Jose A.. This combination results in a safety zone where no other object c

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Method 3 All robots at a time Paths are planned and executed for all robots

simultaneously Whenever collision occurs during plan execution, lower level behavior

causes involved robots to go back for a while, turn left or right for a while and new paths for

those robots are planned This is the fastest method and despite the fact that it is not

collision aware, it is reliable enough

Fig 11 All robots at a time method of repairing the formation

Fig 12 Trails of robots moved to their positions on planned paths (all robots at a time method)

9 Navigation by obstacles with robot formations Formation changing and repairing

The final stage of planning is in checking its soundness by navigating robots in an environment with obstacles We show results of navigating with a team of robots in the initial formation of cross-shape in a crowded environment, see Fig 13 In order to bypass a narrow avenue between an obstacle and the border of the environment, the formation changes to a line, and after bypassing it can use repairing to restore to the initial formation (if it is required), see Figs.14-18 The initial cross-shaped formation is shown in Fig 13 along with obstacles in the environment

Fig 13 Initial formation of robots and the obstacle map

Fig 14 Trails of robots moved to their positions on the cross formation

Trang 3

Method 3 All robots at a time Paths are planned and executed for all robots

simultaneously Whenever collision occurs during plan execution, lower level behavior

causes involved robots to go back for a while, turn left or right for a while and new paths for

those robots are planned This is the fastest method and despite the fact that it is not

collision aware, it is reliable enough

Fig 11 All robots at a time method of repairing the formation

Fig 12 Trails of robots moved to their positions on planned paths (all robots at a time method)

9 Navigation by obstacles with robot formations Formation changing and repairing

The final stage of planning is in checking its soundness by navigating robots in an environment with obstacles We show results of navigating with a team of robots in the initial formation of cross-shape in a crowded environment, see Fig 13 In order to bypass a narrow avenue between an obstacle and the border of the environment, the formation changes to a line, and after bypassing it can use repairing to restore to the initial formation (if it is required), see Figs.14-18 The initial cross-shaped formation is shown in Fig 13 along with obstacles in the environment

Fig 13 Initial formation of robots and the obstacle map

Fig 14 Trails of robots moved to their positions on the cross formation

Trang 4

Reaching the target requires passing by a narrow passage between the border and the

rightmost obstacle To carry out this task, robots in the formation are bound to change the

initial formation They try the line formation, see Figs 14-15

However, making the line formation at the entrance to narrow passage is coupled with some

difficulties: when the strategy all robots at a time is applied, robots at the lower part of the

formation perceive robots at the upper part as obstacles and wait until the latter move into

passage, which blocks whole routine as it is assumed that from the start each robot has a

plan until it does reach goal or until it does collide with another robot

To avoid such blocking of activity the behavior wander was added, see clouded area in Figs

15, 16, which permitted robots to wander until they find that they are able to plan their

paths into the line It can be observed that this wandering consumes some extra time When

the strategy one robot at a time is applied it is important to carefully select the order in which

robots are moved: the robots that have a clear pass to their target positions should go first

Surprisingly, pure behavioral strategy showed good performance in managing with these

difficulties, however, (as we expected) when this strategy is applied, it is time -consuming to

reshape the formation After the line was formed and robots passed through the passage,

see Figs 17-18, the line formation could be restored to the initial cross-shape, if necessary,

with the help of a strategy for repairing formations of section 8.2.1 The results presented in

Figs 14-19 have been witnessing that our approach has proved its usefulness and validity:

in quite complicated obstacle-ridden environments, robots are able to reach the goal The

important feature of this approach is the invariance of the notion of formation with respect

to metric relations among robots: as no metric constraint bounds robots, they are able to

disperse when facing an obstacle with the only requirement being to keep spatial

relationships as set by the betweenness relation imposed upon them

Fig 15 Trails of robots moved to their positions on the line formation

Fig 16 Trails of robots moving in the line formation through the narrow passage

Fig 17.Trails of robots moving in the line formation through and after the passage

Fig 18 Yet another formation change: back to the cross formation

Trang 5

Reaching the target requires passing by a narrow passage between the border and the

rightmost obstacle To carry out this task, robots in the formation are bound to change the

initial formation They try the line formation, see Figs 14-15

However, making the line formation at the entrance to narrow passage is coupled with some

difficulties: when the strategy all robots at a time is applied, robots at the lower part of the

formation perceive robots at the upper part as obstacles and wait until the latter move into

passage, which blocks whole routine as it is assumed that from the start each robot has a

plan until it does reach goal or until it does collide with another robot

To avoid such blocking of activity the behavior wander was added, see clouded area in Figs

15, 16, which permitted robots to wander until they find that they are able to plan their

paths into the line It can be observed that this wandering consumes some extra time When

the strategy one robot at a time is applied it is important to carefully select the order in which

robots are moved: the robots that have a clear pass to their target positions should go first

Surprisingly, pure behavioral strategy showed good performance in managing with these

difficulties, however, (as we expected) when this strategy is applied, it is time -consuming to

reshape the formation After the line was formed and robots passed through the passage,

see Figs 17-18, the line formation could be restored to the initial cross-shape, if necessary,

with the help of a strategy for repairing formations of section 8.2.1 The results presented in

Figs 14-19 have been witnessing that our approach has proved its usefulness and validity:

in quite complicated obstacle-ridden environments, robots are able to reach the goal The

important feature of this approach is the invariance of the notion of formation with respect

to metric relations among robots: as no metric constraint bounds robots, they are able to

disperse when facing an obstacle with the only requirement being to keep spatial

relationships as set by the betweenness relation imposed upon them

Fig 15 Trails of robots moved to their positions on the line formation

Fig 16 Trails of robots moving in the line formation through the narrow passage

Fig 17.Trails of robots moving in the line formation through and after the passage

Fig 18 Yet another formation change: back to the cross formation

Trang 6

Fig 19 Trails of robots in the cross formation in the free workspace after the passage

10 Conclusions and future research

We have proposed a precise formal definition of a formation and we have presented a

Player driver for making formations according to our definition Our definition of a

formation is based on a set of rough mereological predicates which altogether define a

geometry of the space The definition of a formation is independent of a metric on the space

and it is invariant under affine transformations We have examined three methods of

formation restoring, based on a reactive (behavioral) model as well as on decoupled way of

planning We have performed simulations in Player/Stage system of planning paths for

formations with formation change The results show the validity of the approach Further

research will be directed at improving the effectiveness of execution by studying divisions

into sub-formations and merging sub-formations into formations as well as extending the

results to dynamic environments

11 References

Arkin, R.C (1998) Behavior-Based Robotics, MIT Press, ISBN 0-262-01165-4 , Cambridge MA

Balch, T & Arkin, R.C (1998) Behavior-based formation control for multi-robot teams IEEE

Transactions on Robotics and Automation, 14(6), 926-939, ISSN 1042-296X

vanBenthem, J (1983) The Logic of Time, Reidel, ISBN 9-027-71421-5, Dordrecht

Brumitt, B.; Stentz, A.; Hebert, M & CMU UGV Group (2001) Autonomous driving with

concurrent goals and multiple vehicles: Mission planning and architecture

Autonomous Robots, 11, 103-115, ISSN 0929-5593

Uny Cao, Y.; Fukunaga, A.S & Kahng, A.B (1997) Cooperative mobile robotics:

Antecede-nts and directions Autonomous Robots, 4, 7-27, ISSN 0929-5593

Chen, Q & Luh, J Y S (1998) Coordination and control of a group of small mobile robots,

Proceedings of IEEE Intern Conference on Robotics and Automation, pp 2315-2320,

ISBN 0-7803-4300-X, Leuven, May 1998, IEEE Press

Choset, H.; Lynch, K.M.; Hutchinson, S.; Kantor, G.; Burgard, W.; Kavraki, L.E.& Thrun,

S (2005) Principles of Robot Motion Theory, Algorithms, and Implementations, MIT

Press, ISBN 0-262-03327-5, Cambridge MA

Clarke, B L (1981) A calculus of individuals based on connection Notre Dame Journal of

Formal Logic, 22(2), 204-218, ISSN 0029-4527

Das, A.; Fierro, R.; Kumar, V.; Ostrovski, J.P.; Spletzer, J & Taylor, C.J A vision-based

formation control framework IEEE Transactions on Robotics and Automation, 18(5),

813-825, ISSN 1042-296X Gotts, N.M.; Gooday, J.A & Cohn, A.G (1996) A connection based approach to common—

sense topological description and reasoning The Monist , 79(1), 51-75

Khatib, O (1986) Real-time obstacle avoidance for manipulators and mobile robots

International Journal of Robotic Research , 5, 90-98, ISSN 0278-3649

Kramer, J Scheutz, M (2007) Development environments for autonomous mobile robots:

A survey Autonomous Robots, 22, 101-132, ISSN 0929-5593 Krogh, B (1984) A generalized potential field approach to obstacle avoidance control SME-

I Technical paper MS84-484, Society of Manufacturing Engineers, Dearborn MI

Kuipers, B.J & Byun, Y.T (1987) A qualitative approach to robot exploration and map

learning, Proceedings of the IEEE Workshop on Spatial Reasoning and Multi-Sensor

Fusion, pp 390-404, St Charles IL, October 1987, Morgan Kaufmann, San Mateo

CA Ladanyi, H (1997) SQL Unleashed, Sams Publishing, ISBN 0-672-31133-X

De Laguna, T (1922) Point, line, surface as sets of solids J Philosophy, 19, 449-461, ISSN

0022-362-X

Latombe, J (1991) Robot Motion Planning, Kluwer, ISBN 0-792-39206-X, Boston

Ehrich Leonard, N & Fiorelli, E (2001) Virtual leaders, artificial potentials and coordinated

control of groups, Proceedings of the 40th IEEE Conference on Decision and Control,

ISBN 0-780-37061-9, pp 2968-2973, Orlando Fla, December 2001, IEEE Press

Leonard, H & Goodman, N (1940) The calculus of individuals and its uses The Journal of

Symbolic Logic, 5, 45-55, ISSN 0022-4812

Lesniewski, S (1916) O Podstawach Ogolnej Teorii Mnogosci (On Foundations of General Theory

of Sets, in Polish), The Polish Scientific Circle in Moscow, Moscow

Lesniewski, S (1982) On the foundations of mathematics, Topoi 2, 7-52, ISSN 0167-7411 Osmialowski, P (2007) Player and Stage at PJIIT Robotics Laboratory, Journal of Automation,

Mobile Robotics and Intelligent Systems, 2, 21-28, ISSN 1897-8649

Osmialowski, P (2009) On path planning for mobile robots: Introducing the mereological

potential field method in the framework of mereological spatial reasoning, Journal

of Automation, Mobile Robotics and Intelligent Systems, 3(2), 24-33, ISSN 1897-8649

Osmialowski, P & Polkowski, L (2009) Spatial reasoning based on rough mereology: path

planning problem for autonomous mobile robots, Transactions on Rough Sets Lecture

Notes in Computer Science, in print, ISSN 0302-9743, Springer Verlag, Berlin

Player/Stage: Available at http://playerstage.sourceforge.net

Polkowski,L (2001) On connection synthesis via rough mereology, Fundamenta Informaticae,

46, 83-96, ISSN 0169-2968

Trang 7

Fig 19 Trails of robots in the cross formation in the free workspace after the passage

10 Conclusions and future research

We have proposed a precise formal definition of a formation and we have presented a

Player driver for making formations according to our definition Our definition of a

formation is based on a set of rough mereological predicates which altogether define a

geometry of the space The definition of a formation is independent of a metric on the space

and it is invariant under affine transformations We have examined three methods of

formation restoring, based on a reactive (behavioral) model as well as on decoupled way of

planning We have performed simulations in Player/Stage system of planning paths for

formations with formation change The results show the validity of the approach Further

research will be directed at improving the effectiveness of execution by studying divisions

into sub-formations and merging sub-formations into formations as well as extending the

results to dynamic environments

11 References

Arkin, R.C (1998) Behavior-Based Robotics, MIT Press, ISBN 0-262-01165-4 , Cambridge MA

Balch, T & Arkin, R.C (1998) Behavior-based formation control for multi-robot teams IEEE

Transactions on Robotics and Automation, 14(6), 926-939, ISSN 1042-296X

vanBenthem, J (1983) The Logic of Time, Reidel, ISBN 9-027-71421-5, Dordrecht

Brumitt, B.; Stentz, A.; Hebert, M & CMU UGV Group (2001) Autonomous driving with

concurrent goals and multiple vehicles: Mission planning and architecture

Autonomous Robots, 11, 103-115, ISSN 0929-5593

Uny Cao, Y.; Fukunaga, A.S & Kahng, A.B (1997) Cooperative mobile robotics:

Antecede-nts and directions Autonomous Robots, 4, 7-27, ISSN 0929-5593

Chen, Q & Luh, J Y S (1998) Coordination and control of a group of small mobile robots,

Proceedings of IEEE Intern Conference on Robotics and Automation, pp 2315-2320,

ISBN 0-7803-4300-X, Leuven, May 1998, IEEE Press

Choset, H.; Lynch, K.M.; Hutchinson, S.; Kantor, G.; Burgard, W.; Kavraki, L.E.& Thrun,

S (2005) Principles of Robot Motion Theory, Algorithms, and Implementations, MIT

Press, ISBN 0-262-03327-5, Cambridge MA

Clarke, B L (1981) A calculus of individuals based on connection Notre Dame Journal of

Formal Logic, 22(2), 204-218, ISSN 0029-4527

Das, A.; Fierro, R.; Kumar, V.; Ostrovski, J.P.; Spletzer, J & Taylor, C.J A vision-based

formation control framework IEEE Transactions on Robotics and Automation, 18(5),

813-825, ISSN 1042-296X Gotts, N.M.; Gooday, J.A & Cohn, A.G (1996) A connection based approach to common—

sense topological description and reasoning The Monist , 79(1), 51-75

Khatib, O (1986) Real-time obstacle avoidance for manipulators and mobile robots

International Journal of Robotic Research , 5, 90-98, ISSN 0278-3649

Kramer, J Scheutz, M (2007) Development environments for autonomous mobile robots:

A survey Autonomous Robots, 22, 101-132, ISSN 0929-5593 Krogh, B (1984) A generalized potential field approach to obstacle avoidance control SME-

I Technical paper MS84-484, Society of Manufacturing Engineers, Dearborn MI

Kuipers, B.J & Byun, Y.T (1987) A qualitative approach to robot exploration and map

learning, Proceedings of the IEEE Workshop on Spatial Reasoning and Multi-Sensor

Fusion, pp 390-404, St Charles IL, October 1987, Morgan Kaufmann, San Mateo

CA Ladanyi, H (1997) SQL Unleashed, Sams Publishing, ISBN 0-672-31133-X

De Laguna, T (1922) Point, line, surface as sets of solids J Philosophy, 19, 449-461, ISSN

0022-362-X

Latombe, J (1991) Robot Motion Planning, Kluwer, ISBN 0-792-39206-X, Boston

Ehrich Leonard, N & Fiorelli, E (2001) Virtual leaders, artificial potentials and coordinated

control of groups, Proceedings of the 40th IEEE Conference on Decision and Control,

ISBN 0-780-37061-9, pp 2968-2973, Orlando Fla, December 2001, IEEE Press

Leonard, H & Goodman, N (1940) The calculus of individuals and its uses The Journal of

Symbolic Logic, 5, 45-55, ISSN 0022-4812

Lesniewski, S (1916) O Podstawach Ogolnej Teorii Mnogosci (On Foundations of General Theory

of Sets, in Polish), The Polish Scientific Circle in Moscow, Moscow

Lesniewski, S (1982) On the foundations of mathematics, Topoi 2, 7-52, ISSN 0167-7411 Osmialowski, P (2007) Player and Stage at PJIIT Robotics Laboratory, Journal of Automation,

Mobile Robotics and Intelligent Systems, 2, 21-28, ISSN 1897-8649

Osmialowski, P (2009) On path planning for mobile robots: Introducing the mereological

potential field method in the framework of mereological spatial reasoning, Journal

of Automation, Mobile Robotics and Intelligent Systems, 3(2), 24-33, ISSN 1897-8649

Osmialowski, P & Polkowski, L (2009) Spatial reasoning based on rough mereology: path

planning problem for autonomous mobile robots, Transactions on Rough Sets Lecture

Notes in Computer Science, in print, ISSN 0302-9743, Springer Verlag, Berlin

Player/Stage: Available at http://playerstage.sourceforge.net

Polkowski,L (2001) On connection synthesis via rough mereology, Fundamenta Informaticae,

46, 83-96, ISSN 0169-2968

Trang 8

Polkowski, L (2008) A unified approach to granulation of knowledge and granular

computing based on rough mereology: A survey, In: Handbook of Granular

Computing, Pedrycz, W.; Skowron, A & Kreinovich, V., (Eds.), 375-400, John

Wiley and Sons, ISBN 987-0-470-03554-2, Chichester UK

Polkowski, L & Osmialowski, P (2008) Spatial reasoning with applications to mobile

robotics, In: Mobile Robots Motion Planning New Challenges, Xing-Jian Jing, (Ed.),

43-55, I-Tech Education and Publishing KG, , ISBN 978-953-7619-01-5, Vienna

Polkowski, L & Osmialowski, P (2008) A framework for multiagent mobile robotics:

Spatial reasoning based on rough mereology in Player/stage system, Lecture Notes

in Artificial Intelligence, 5306, 142-149, Springer Verlag, ISSN 0302-9743, Berlin

P Ramsey, PostGIS Manual, in: postgis.pdf file downloaded from Refractions Research

home page

J Shao, G ; Xie, J Yu & Wang, L (2005) Leader-following formation control of multiple

mobile robots, In: Proceedings of the 2005 IEEE Intern Symposium on Intelligent

Control, pp 808-813, ISBN 0-780-38936-0, Limassol, June 2005, IEEE Press

sfsexp: Available at http://sexpr.sourceforge.net

Sugihara, K & Suzuki, I (1990) Distributed motion coordination of multiple mobile robots,

In: Proceedings 5th IEEE Intern Symposium on Intelligent Control, pp 138-143, ISBN

9-991-32943-9, Philadelphia PA, Sept 1990, IEEE Press

Švestka, P & Overmars, M.H (1998) Coordinated path planning for multiple robots,

Robotics and Autonomous Systems, 23, 125-152, ISSN 0921-8890

Tarski, A (1929) Les fondements de la géométrie des corps, In: Supplement to Annales de la

Sociéte Polonaise de Mathématique, 29-33, Krakow, Poland

Tarski, A (1959) What is elementary geometry?, In: The Axiomatic Method with Special

Reference to Geometry and Physics, Henkin, L.; Suppes, P & Tarski, A., (Eds)., 16-29,

North-Holland, Amsterdam

Tribelhorn, B & Dodds, Z (2007) Evaluating the Roomba: A low-cost, ubiquitous platform

for robotics research and education, In: 2007 IEEE International Conference on

Robotics and Automation, ICRA 2007, pp 1393-1399, ISBN 1-4244-0601-3, April 2007,

Roma, Italy, IEEE Press

Urdiales, C.; Perez, E.J.; Vasquez-Salceda, J.; Sanchez-Marrµe, M & Sandoval, F (2006) A

purely reactive navigation scheme for dynamic environments using Case-Based

Reasoning, Autonomous Robots , 21, 65-78, ISSN 0929-5593

Whitehead, A N (1979) Process and Reality An Essay in Cosmology 2nd ed., The Free Press,

ISBN 0-029-34570-7, New York NY

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An Artificial Protection Field Approach For

Reactive Obstacle Avoidance in Mobile Robots

Victor Ayala-Ramirez, Jose A Gasca-Martinez, Rigoberto Lopez-Padilla and Raul E Sanchez-Yanez

Universidad de Guanajuato DICIS

Mexico

1 Introduction

Mobile robots using topological navigation require of being able to react to dynamical

obstacles in their environment Reactive obstacle avoidance is also an essential capability

needed by a mobile robot evolving in a cluttered dynamic environment Scenarios like

offices where people and robots share a common workspace are difficult to be modeled by

static maps In such scenarios, robots need to avoid people and obstacles when executing

any other task involving its motion

Another application for reactive obstacle avoidance arises when the robot is building a map

of an unknown environment The robot will need to react to different events during its

exploring task For example, the robot needs to be able to avoid a wall not previously

known or to evade another moving object in its neighborhood

In the past, several researchers have proposed reactive navigation methods Some examples

of these methods are: The artificial potential field (Khatib, 1986) and the elastic band

approach (Quinlan & Khatib, 1993; Lamiraux & Laumond, 2004) proposed originally both

by Khatib; the vector field histogram (Borenstein & Koren, 1990), the dynamic window

approach proposed by (Fox et al., 1997) and the nearness diagram, recently proposed by

(Minguez & Montano, 2004)

Our approach is to build an artificial protection field around the robot and to survey it by

fusing laser range finder and odometry measurements In this work, an approach to reactive

obstacle avoidance for service robots is proposed We use the concept of an artificial

protection field along a robot pre-planned path The artificial protection field is a dynamic

geometrical neighborhood of the robot and a set of situation assessment rules that determine

if the robot needs to evade an object not present in its map when its path was planned This

combination results in a safety zone where no other object can be present when the robot is

executing a motion primitive; a zone where the robot needs to recalculate its path; and some

other zones where the object can perform successfully its navigation task, even if obstacles

are detected near the robot path During the execution of a motion primitive, dynamical

17

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obstacles are detected by using a laser range finder If the obstacles detected in the

neighborhood of the robot path enter the artificial protection field of the robot, reactive

behaviours are launched to recalculate the path online in order to avoid collisions with

them Our method has been tested in an experimental setup both in a simulation platform

and in the real robot in a qualitative manner The robot has demonstrated successful

evolution on these tests for static and dynamic scenarios

The structure of the chapter will be as follows: Firstly, we are going to review recent

approaches in reactive navigation for robots We will also review their usefulness in

topological navigation approaches A second section will describe in detail what are the

specific features of the proposed artificial protection field approach and a critical

comparison of its characteristics against some other algorithms for obstacle evasion in the

recent literature Test protocols used to validate our approach will be inspected in detail in a

subsequent section We will show results in a custom-developed robotic simulation platform

for our robot We will also show experimental tests that have been implemented on a

Pioneer P3-AT mobile robot named XidooBot under several scenarios Our main findings

will then be discussed and we finish our proposal with a section giving our conclusions and

perspectives of future work

2 Reactive Navigation Methods

Robot navigation using reactive methods has been studied extensively Here we present

main approaches and recent methods proposed in literature

One of the first approaches used for reactive navigation is the potential field (PF) approach

(Khatib, 1986) proposed the generation of an artificial potential field that repels the robot

from the obstacles and that attracts it toward its goal Main problem of this method is the

emergence of isopotential regions because of the potential selection for the environment

elements; that traps the robot in local minima regions, impeding it to attain its goal This

drawback limits the applicability of the method in complex environments Recently (Antich

& Ortiz, 2005) have proposed to define a behaviour based navigation function that

combined with the PF approach can partially overcome main drawbacks of the PF approach

PF methods are global planning methods so they can be used also for motion planning

A variation of the PF methods is known as the elastic band (EB) approach EB methods

propose to deform an a-priori computed path when obstacles not considered at planning

time are detected during the execution of a give trajectory Some examples of this approach

are the works by (Quinlan & Khatib, 1993) for manipulator robots and (Lamiraux &

Laumond, 2004; Lamiraux et al., 2004) for mobile robots As said before, EB methods require

a pre-planned path, so they can not be used for exploration tasks

The vector field histogram (VFH) is a reactive navigation method proposed by (Borenstein

& Koren, 1991) The main idea is to represent the free space surrounding the current

position of a robot using an occupancy grid A polar histogram is created and the robot

selects the direction with the maximal cell count of free space as a preferential orientation

for its motion This method is essentially a local navigation method to avoid obstacles

(Fox et al., 1997) have originally proposed the dynamic window (DW) approach to avoid collision with obstacles in a reactive way In this method, the dynamic restrictions of differential and synchro-drive steered robots are taken into account to generate arc motion primitives that avoid intruding elements The optimization of the motion primitives find the optimal values for the translational and rotational velocities with respect to the current target heading, robot velocity and clearance

A more recent approach is presented by (Minguez & Montano, 2004; Minguez, 2005; Vikenmark & Minguez, 2006) as the nearness diagram (ND) approach This method models objects and free space in the proximity of the robot The robot recognizes its situation with respect to the task to be done The robot takes then consequent actions (motion laws) according to the assessment Recently, (Li et al., 2006) have proposed the hybridization of this technique with the DW approach Main contribution of this improvement is to increase the speed of the mobile robot even in troublesome scenarios

Some other methods have also been proposed recently for reactive navigation Some of them use fuzzy logic to control the reactive motion of the robot Some examples of this approach are the works by (Mester, 2008) and (Larson et al., 2005) Some other consider also the identification of the behaviours associated to dynamic obstacles by using Bayesian approaches, as for example, (Lopez-Martinez & Fraichard, 2008) and (Laugier et al., 2008) However, they are not very related with our approach even if they are alternatives for reactive robot motion

3 The Artificial Protection Field Approach

Our approach is based in the concurrent execution of two tasks: the execution of a navigation command and the obstacle detection task The navigation commands implement the planned path in a static and known environment configuration We define the robot

pose by using the (x,y,) coordinates The motion primitives link two poses by using advance and rotate primitives Nevertheless, each time an obstacle is detected the motion command is stopped and a re-planning process is spawned In the following, we give the details of the above procedure implemented in a mobile robot platform

3.1 Obstacle Detection

In any reactive navigation method, the robot needs to acquire information about its surroundings in order to detect obstacles In our robot, obstacles are detected by using the laser range finder (LRF) sensor The LRF measures acquired by the mobile robot are classified to determine a safety condition for it In particular, they are classified according to its closeness to a protection zone around the robot We call such safety zone an artificial protection field (APF)

3.2 Artificial Protection Field

The APF is defined in terms of three restrictions:

Firstly, we consider a region where the robot can freely execute the motion commands

without re-planning its path, namely the minimal obstacle free space E The shape of the

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obstacles are detected by using a laser range finder If the obstacles detected in the

neighborhood of the robot path enter the artificial protection field of the robot, reactive

behaviours are launched to recalculate the path online in order to avoid collisions with

them Our method has been tested in an experimental setup both in a simulation platform

and in the real robot in a qualitative manner The robot has demonstrated successful

evolution on these tests for static and dynamic scenarios

The structure of the chapter will be as follows: Firstly, we are going to review recent

approaches in reactive navigation for robots We will also review their usefulness in

topological navigation approaches A second section will describe in detail what are the

specific features of the proposed artificial protection field approach and a critical

comparison of its characteristics against some other algorithms for obstacle evasion in the

recent literature Test protocols used to validate our approach will be inspected in detail in a

subsequent section We will show results in a custom-developed robotic simulation platform

for our robot We will also show experimental tests that have been implemented on a

Pioneer P3-AT mobile robot named XidooBot under several scenarios Our main findings

will then be discussed and we finish our proposal with a section giving our conclusions and

perspectives of future work

2 Reactive Navigation Methods

Robot navigation using reactive methods has been studied extensively Here we present

main approaches and recent methods proposed in literature

One of the first approaches used for reactive navigation is the potential field (PF) approach

(Khatib, 1986) proposed the generation of an artificial potential field that repels the robot

from the obstacles and that attracts it toward its goal Main problem of this method is the

emergence of isopotential regions because of the potential selection for the environment

elements; that traps the robot in local minima regions, impeding it to attain its goal This

drawback limits the applicability of the method in complex environments Recently (Antich

& Ortiz, 2005) have proposed to define a behaviour based navigation function that

combined with the PF approach can partially overcome main drawbacks of the PF approach

PF methods are global planning methods so they can be used also for motion planning

A variation of the PF methods is known as the elastic band (EB) approach EB methods

propose to deform an a-priori computed path when obstacles not considered at planning

time are detected during the execution of a give trajectory Some examples of this approach

are the works by (Quinlan & Khatib, 1993) for manipulator robots and (Lamiraux &

Laumond, 2004; Lamiraux et al., 2004) for mobile robots As said before, EB methods require

a pre-planned path, so they can not be used for exploration tasks

The vector field histogram (VFH) is a reactive navigation method proposed by (Borenstein

& Koren, 1991) The main idea is to represent the free space surrounding the current

position of a robot using an occupancy grid A polar histogram is created and the robot

selects the direction with the maximal cell count of free space as a preferential orientation

for its motion This method is essentially a local navigation method to avoid obstacles

(Fox et al., 1997) have originally proposed the dynamic window (DW) approach to avoid collision with obstacles in a reactive way In this method, the dynamic restrictions of differential and synchro-drive steered robots are taken into account to generate arc motion primitives that avoid intruding elements The optimization of the motion primitives find the optimal values for the translational and rotational velocities with respect to the current target heading, robot velocity and clearance

A more recent approach is presented by (Minguez & Montano, 2004; Minguez, 2005; Vikenmark & Minguez, 2006) as the nearness diagram (ND) approach This method models objects and free space in the proximity of the robot The robot recognizes its situation with respect to the task to be done The robot takes then consequent actions (motion laws) according to the assessment Recently, (Li et al., 2006) have proposed the hybridization of this technique with the DW approach Main contribution of this improvement is to increase the speed of the mobile robot even in troublesome scenarios

Some other methods have also been proposed recently for reactive navigation Some of them use fuzzy logic to control the reactive motion of the robot Some examples of this approach are the works by (Mester, 2008) and (Larson et al., 2005) Some other consider also the identification of the behaviours associated to dynamic obstacles by using Bayesian approaches, as for example, (Lopez-Martinez & Fraichard, 2008) and (Laugier et al., 2008) However, they are not very related with our approach even if they are alternatives for reactive robot motion

3 The Artificial Protection Field Approach

Our approach is based in the concurrent execution of two tasks: the execution of a navigation command and the obstacle detection task The navigation commands implement the planned path in a static and known environment configuration We define the robot

pose by using the (x,y,) coordinates The motion primitives link two poses by using advance and rotate primitives Nevertheless, each time an obstacle is detected the motion command is stopped and a re-planning process is spawned In the following, we give the details of the above procedure implemented in a mobile robot platform

3.1 Obstacle Detection

In any reactive navigation method, the robot needs to acquire information about its surroundings in order to detect obstacles In our robot, obstacles are detected by using the laser range finder (LRF) sensor The LRF measures acquired by the mobile robot are classified to determine a safety condition for it In particular, they are classified according to its closeness to a protection zone around the robot We call such safety zone an artificial protection field (APF)

3.2 Artificial Protection Field

The APF is defined in terms of three restrictions:

Firstly, we consider a region where the robot can freely execute the motion commands

without re-planning its path, namely the minimal obstacle free space E The shape of the

Trang 12

boundary of this region can be arbitrarily defined by using a polar defined function r()

The value of this function r is taken as the maximal distance being free of obstacles at an

orientation  with respect to a robot-centered coordinate system That is,

The second restriction considers the distance of the current position of the robot with respect

to the goal of the motion primitive, named d g That is, we do not want to react to an obstacle

in E which is farther than the goal The zone satisfying this restriction is represented by G

in Figure 1, and it is shown in yellow Given that we take a Euclidean distance metric, the

shape of E is a half circle centred on the robot reference point, that is

Finally, the critical space C (see Figure 1) is a safety zone to avoid collisions of the robot

with obstacles If we take a critical distance d c , region C is defined as follows:

Reactive behaviours of the robot are launched after recognizing a danger context To

recognize which behaviour has to be taken by the robot, we use the LRF measures as input

information As usual, the sensor provides us with a range distance to the next obstacle at a

pre-defined set of orientations We process this set of measurements to infer the free zone L

in front of the current position of the mobile robot We assess then the situation that the robot is facing Our goal is to classify the situational context of the robot as a class of the following list of exhaustive states: Obstacle free situation (labelled as O f), a low risk

situation (labelled as O l ), a medium risk situation ( O m ) and a high risk situation ( O h)

Obstacle Free Situation O f

This is the ideal situation for the robot because it has not detected an obstacle in E That situation lets the robot to continue with the execution of the current motion primitive Formally this is represented by:

Low Risk Situation O l

When an object invades the minimal free space of the robot, we consider the event as a low risk situation if the obstacle is farther than the goal position, i.e.,

LE G

Medium Risk Situation O m

The robot is facing a medium risk situation if the following three conditions are fulfilled:

 There is an object invading the minimal free space

 The target position for the motion primitive is closer than the current position of the robot

The obstacle does not enter into the critical space C

Formally, that implies,

High Risk Situation O h

A high risk situation is the event where the robot has detected an object invading ist critical space In a formal way,

Figure 2 shows how obstacles are detected and used to assess the robot situation

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boundary of this region can be arbitrarily defined by using a polar defined function r()

The value of this function r is taken as the maximal distance being free of obstacles at an

orientation  with respect to a robot-centered coordinate system That is,

The second restriction considers the distance of the current position of the robot with respect

to the goal of the motion primitive, named d g That is, we do not want to react to an obstacle

in E which is farther than the goal The zone satisfying this restriction is represented by G

in Figure 1, and it is shown in yellow Given that we take a Euclidean distance metric, the

shape of E is a half circle centred on the robot reference point, that is

Finally, the critical space C (see Figure 1) is a safety zone to avoid collisions of the robot

with obstacles If we take a critical distance d c , region C is defined as follows:

Reactive behaviours of the robot are launched after recognizing a danger context To

recognize which behaviour has to be taken by the robot, we use the LRF measures as input

information As usual, the sensor provides us with a range distance to the next obstacle at a

pre-defined set of orientations We process this set of measurements to infer the free zone L

in front of the current position of the mobile robot We assess then the situation that the robot is facing Our goal is to classify the situational context of the robot as a class of the following list of exhaustive states: Obstacle free situation (labelled as O f ), a low risk

situation (labelled as O l ), a medium risk situation ( O m ) and a high risk situation ( O h)

Obstacle Free Situation O f

This is the ideal situation for the robot because it has not detected an obstacle in E That situation lets the robot to continue with the execution of the current motion primitive Formally this is represented by:

Low Risk Situation O l

When an object invades the minimal free space of the robot, we consider the event as a low risk situation if the obstacle is farther than the goal position, i.e.,

LE G

Medium Risk Situation O m

The robot is facing a medium risk situation if the following three conditions are fulfilled:

 There is an object invading the minimal free space

 The target position for the motion primitive is closer than the current position of the robot

The obstacle does not enter into the critical space C

Formally, that implies,

High Risk Situation O h

A high risk situation is the event where the robot has detected an object invading ist critical space In a formal way,

Figure 2 shows how obstacles are detected and used to assess the robot situation

Trang 14

Fig 2 Situation assessment in a service robot using the APF

3.4 Reactive Behaviour

During the execution of a motion command, the robot is polling its sensors in order to assess

the navigation situation When a no risk (O f ) or low risk ( O l) situation is detected, the

motion primitive continues its execution A medium risk situation O m causes the robot to

execute the following actions: 1) Stop motion execution, 2) Re-plan its path to get an

obstacle free trajectory, and 3) Execute the modified path If a high risk situation O h is

recognized, the service robot interrupts inmediately ist motion to avoid collision with the

intruding object An activity diagram of the actions taken as a reaction to the situation

assessment is shown in Figure 3

3.5 Path Re-Planning

If the robot recognizes a medium risk situation, it stops execution of the current motion

primitive It also launches a path re-planning process that uses LRF measurements as input

information The purpose of this process is to find an intermediate goal

s g* in the free space perceived by the robot and that minimizes some metric related to the path

To do so, we analyse some points in the free-space L (see Section 3.3) detected by the LRF

The feasible intermediate target goal S g consists of all the points in a given set of radial

distances from the current position of the robots that belong also to L Another characteristic

that all points in S g have is that an inspection windos W centered on them presents no

collision with any obstacle The inspection window extendes from an angle   to an

angle  , where  is the angular coordinate of the point in S g being tested with respect

to the robot-centered coordinate system, and  is the width of the inspection window

All target goals in S g are then attainable We carry out an optimization process over all these points and we choose the optimal one according to the objective function being optimized

Activity Diagram0

act Activity Diagram0 act

Perception System Robot Control

Motion Control

Robot Setup

Load Path File

Execute Motion Primitive

Acquire LRF Data Assess

Robot Situation

Select Next Primitive

[Obstacle Free or Low Risk]

/ More primitives

Stop Motion Primitive

Insert Modified Path Segment

Abort Motion Plan

[High Risk]

Motion End [No]

Fig 3 Activity diagram for the reactive behaviour of the service robot

Trang 15

Fig 2 Situation assessment in a service robot using the APF

3.4 Reactive Behaviour

During the execution of a motion command, the robot is polling its sensors in order to assess

the navigation situation When a no risk (O f ) or low risk ( O l) situation is detected, the

motion primitive continues its execution A medium risk situation O m causes the robot to

execute the following actions: 1) Stop motion execution, 2) Re-plan its path to get an

obstacle free trajectory, and 3) Execute the modified path If a high risk situation O h is

recognized, the service robot interrupts inmediately ist motion to avoid collision with the

intruding object An activity diagram of the actions taken as a reaction to the situation

assessment is shown in Figure 3

3.5 Path Re-Planning

If the robot recognizes a medium risk situation, it stops execution of the current motion

primitive It also launches a path re-planning process that uses LRF measurements as input

information The purpose of this process is to find an intermediate goal

s g* in the free space perceived by the robot and that minimizes some metric related to the path

To do so, we analyse some points in the free-space L (see Section 3.3) detected by the LRF

The feasible intermediate target goal S g consists of all the points in a given set of radial

distances from the current position of the robots that belong also to L Another characteristic

that all points in S g have is that an inspection windos W centered on them presents no

collision with any obstacle The inspection window extendes from an angle   to an

angle  , where  is the angular coordinate of the point in S g being tested with respect

to the robot-centered coordinate system, and  is the width of the inspection window

All target goals in S g are then attainable We carry out an optimization process over all these points and we choose the optimal one according to the objective function being optimized

Activity Diagram0

act Activity Diagram0 act

Perception System Robot Control

Motion Control

Robot Setup

Load Path File

Execute Motion Primitive

Acquire LRF Data Assess

Robot Situation

Select Next Primitive

[Obstacle Free or Low Risk]

/ More primitives

Stop Motion Primitive

Insert Modified Path Segment

Abort Motion Plan

[High Risk]

Motion End [No]

Fig 3 Activity diagram for the reactive behaviour of the service robot

Trang 16

Let us consider, for example that our objective is to minimize closeness to the target position

s t in the interrupted motion command Then, for all s k in S g:

where for example,  could be an Euclidean norm

4 Test and Results

We have run tests firstly in a custom-developed simulation platform The goal of these

experiments was to define the best parameters for the implementation on a real P3-AT

robotics platform named XidooBot We have then executed motion scripts that tested

exhaustively all the contexts that could arise in the reactive navigation task

The first set of tests has been performed in a custom-developed simulation environment A

scenario with multiple objects is shown in Figure 4 Here, we can observe the re-planning of

a new trajectory once the robot finds an obstacle in its APF After that, an obstacle-free

trajectory is executed until it reaches its goal The half-elliptical safety zone at the end of the

trajectory is specifically indicated in the Figure

Fig 4 Simulation of a script composed of a single straight-line segment

In Figure 5, the simulation of a more complex trajectory is given Here, point A is an

intermediate goal and the final one is point B We observe a reactive behaviour zone in the

trajectory between the start point and point A From A to B, its necessary beginning with a continuous re-planning of the trajectory until the goal is attained executing a straight trajectory

Fig 5 Execution of a script composed of multiple straight-line segments

A second phase of tests was run in the real platform for robotics experiments We have also setup scenario configurations that cover all the cases foreseen in the APF approach The reactive navigation system has been tested qualitatively The test procedure includes the execution of a set of paths in an environment where obstacles were added dynamically during the motion commands execution Figure 6 shows a successful motion command execution because no obstacle was detected along the execution of the motion primitives Even if an obstacle is near the target position of the robot, path is not re-planned because the object does not interfere the robot motion

Fig 6 A reactive navigation scenario (case I) for a Pioneer P3-AT

Trang 17

Let us consider, for example that our objective is to minimize closeness to the target position

s t in the interrupted motion command Then, for all s k in S g:

where for example,  could be an Euclidean norm

4 Test and Results

We have run tests firstly in a custom-developed simulation platform The goal of these

experiments was to define the best parameters for the implementation on a real P3-AT

robotics platform named XidooBot We have then executed motion scripts that tested

exhaustively all the contexts that could arise in the reactive navigation task

The first set of tests has been performed in a custom-developed simulation environment A

scenario with multiple objects is shown in Figure 4 Here, we can observe the re-planning of

a new trajectory once the robot finds an obstacle in its APF After that, an obstacle-free

trajectory is executed until it reaches its goal The half-elliptical safety zone at the end of the

trajectory is specifically indicated in the Figure

Fig 4 Simulation of a script composed of a single straight-line segment

In Figure 5, the simulation of a more complex trajectory is given Here, point A is an

intermediate goal and the final one is point B We observe a reactive behaviour zone in the

trajectory between the start point and point A From A to B, its necessary beginning with a continuous re-planning of the trajectory until the goal is attained executing a straight trajectory

Fig 5 Execution of a script composed of multiple straight-line segments

A second phase of tests was run in the real platform for robotics experiments We have also setup scenario configurations that cover all the cases foreseen in the APF approach The reactive navigation system has been tested qualitatively The test procedure includes the execution of a set of paths in an environment where obstacles were added dynamically during the motion commands execution Figure 6 shows a successful motion command execution because no obstacle was detected along the execution of the motion primitives Even if an obstacle is near the target position of the robot, path is not re-planned because the object does not interfere the robot motion

Fig 6 A reactive navigation scenario (case I) for a Pioneer P3-AT

Trang 18

A second example is shown in Figure 7 When the motion execution is launched, the robot

detects a medium risk situation, so it re-plans its trajectory We can see in Figure 7 what are

the sub-goals chosen by the robot in order to reach the originally planned target position

A more complex environment is shown in Figure 8 We have several obstacles near the

planned path (shown in blue) In the figure, we can see the executed path (shown in red)

after reacting to the presence of obstacles

A closed path execution is shown in Figure 9 Our reactive strategy is applied when during

the execution of a path segment an obstacle appears We show the planned path in blue and

the modified executed path in red As we can see, the robot passes by the control points of

the motion primitive sequences without problems

Fig 7 A reactive navigation scenario (case II) for a Pioneer P3-AT

Fig 8 A reactive navigation scenario (case III) for a Pioneer P3-AT

Fig 9 A reactive navigation scenario (case IV) for a Pioneer P3-AT

5 Conclusion and Perspectives

In this work, we have presented a reactive obstacle avoidance approach based on a activity paradigm Our system has demonstrated to be robust in qualitative tests developed

situated-on a dynamically changing envirsituated-onment We have shown tests in a custom-developed simulation platform and in the real robot Our approach is based in the risk assessment made by the robot by using an artificial protection field to avoid moving obstacles

In the near future, we will evaluate quantitative performance of the method by analyzing pose errors after the re-planning step both in simulation and in the real robotic platform XidooBot Fusion of several sensors information will be done in order to improve the robustness of this approach This method will be integrated with a topological navigation system

6 References

Antich, J & Ortiz A (2005), Extending the potential fields approach to avoid trapping

situations, Proc Int Conf on Intelligent Robots and Systems, 2005 (IROS 2005), pp 1386–1391, August, 2005, IEEE Press

Borenstein, J & Koren, Y (1990), Real-time obstacle avoidance for fast mobile robots in

cluttered environments, Proc of the 1990 IEEE Int Conf on Robotics and Automation (ICRA1990), pp 572–577, Cincinnati, Ohio, May, 1990, IEEE Press Fox, D.; Burgard, W & Thrun, S (1997), The dynamic window approach to collision

avoidance, IEEE Robotics & Automation Magazine, Vol 4, No.1 (March, 1997), pp 23–

33

Khatib, O (1986), Real-time obstacle avoidance for manipulators and mobile robots,

International Journal of Robotics Research, Vol 5, No 1 (1986), pp 90–98

Trang 19

A second example is shown in Figure 7 When the motion execution is launched, the robot

detects a medium risk situation, so it re-plans its trajectory We can see in Figure 7 what are

the sub-goals chosen by the robot in order to reach the originally planned target position

A more complex environment is shown in Figure 8 We have several obstacles near the

planned path (shown in blue) In the figure, we can see the executed path (shown in red)

after reacting to the presence of obstacles

A closed path execution is shown in Figure 9 Our reactive strategy is applied when during

the execution of a path segment an obstacle appears We show the planned path in blue and

the modified executed path in red As we can see, the robot passes by the control points of

the motion primitive sequences without problems

Fig 7 A reactive navigation scenario (case II) for a Pioneer P3-AT

Fig 8 A reactive navigation scenario (case III) for a Pioneer P3-AT

Fig 9 A reactive navigation scenario (case IV) for a Pioneer P3-AT

5 Conclusion and Perspectives

In this work, we have presented a reactive obstacle avoidance approach based on a activity paradigm Our system has demonstrated to be robust in qualitative tests developed

situated-on a dynamically changing envirsituated-onment We have shown tests in a custom-developed simulation platform and in the real robot Our approach is based in the risk assessment made by the robot by using an artificial protection field to avoid moving obstacles

In the near future, we will evaluate quantitative performance of the method by analyzing pose errors after the re-planning step both in simulation and in the real robotic platform XidooBot Fusion of several sensors information will be done in order to improve the robustness of this approach This method will be integrated with a topological navigation system

6 References

Antich, J & Ortiz A (2005), Extending the potential fields approach to avoid trapping

situations, Proc Int Conf on Intelligent Robots and Systems, 2005 (IROS 2005), pp 1386–1391, August, 2005, IEEE Press

Borenstein, J & Koren, Y (1990), Real-time obstacle avoidance for fast mobile robots in

cluttered environments, Proc of the 1990 IEEE Int Conf on Robotics and Automation (ICRA1990), pp 572–577, Cincinnati, Ohio, May, 1990, IEEE Press Fox, D.; Burgard, W & Thrun, S (1997), The dynamic window approach to collision

avoidance, IEEE Robotics & Automation Magazine, Vol 4, No.1 (March, 1997), pp 23–

33

Khatib, O (1986), Real-time obstacle avoidance for manipulators and mobile robots,

International Journal of Robotics Research, Vol 5, No 1 (1986), pp 90–98

Trang 20

Koren, Y & Borenstein, J (1991), Potential field methods and their inherent limitations for

mobile robot navigation, Proc IEEE Int Conf Robotics and Automation, April,

1991, pp.1398-1404

Lamiraux, F.; Bonnafous, D & Lefebvre, O (2004), Reactive path deformation for

nonholonomic mobile robots, IEEE Trans on Robotics and Automation, Vol 20, (Dec

2004), pp 967–977

Lamiraux, F & Laumond, J.-P (2004), Reactive obstacle avoidance and trajectory

optimization for non-holonomic systems: two problems, one solution, Proc World Automation Congress 2004, vol 15, pp 473–478, June, 2004

Larsson, J.; Broxvall, M & Saffiotti, A (2008), Laser Based Intersection Detection for Reactive

Navigation in an Underground Mine, 4th IEEE Int Conf on Intelligent Systems,

2008 (IS '08), Vol 1, pp 2-45 -2-50, Nice, France, September 2008

Laugier, C.; Vasquez-Govea, D.A.; Yguel, M.; Fraichard, T & Aycard, O (2008), Geometric

and Bayesian models for safe navigation in dynamic environments, Intelligent

Service Robotics, Vol 1, No 1 (2008), pp 51-72

Li, G.; Wu, G & Wei, W (2006), ND-DWA: A Reactive Method for Collision Avoidance in

Troublesome Scenarios, Proc of the 6th World Congress on Intelligent Control and Automation, 2006 (WCICA 2006), Volume 2, pp 9307–9311, 2006

Martinez-Gomez, L & Fraichard, T (2008), An efficient and generic 2D Inevitable Collision

State Checker, Proc Int Conf on Intelligent Robots and Systems, IROS 2008, Sept

2008, pp 234–241

Mester, G (2009), Obstacle-slope avoidance and velocity control of wheeled mobile robots

using fuzzy reasoning, Proc Int Conf on Intelligent Engineering Systems (INES 2009), April, 2009, pp 245–249

Minguez, J & Montano, L (2004), Nearness Diagram (ND) Navigation: Collision Avoidance

in Troublesome Scenarios, IEEE Trans On Robotics and Automation, Vol 20, No 1

(Feb 2004), pp 45-59

Minguez, J (2005), Integration of planning and reactive obstacle avoidance in autonomous

sensor-based navigation, Proc Int Conf on Intelligent Robots and Systems, (IROS 2005), pp 2486–2492, August 2005

Quinlan, S & Khatib, O (1993), Elastic bands: Connecting path planning and control Proc

of IEEE Int Conf on Robotics and Automation, Vol 2, pp 802–807, May 1993 Vikenmark, D & Minguez, J (2006), Reactive obstacle avoidance for mobile robots that

operate in confined 3D workspaces, Proc IEEE Mediterranean Electrotechnical Conference, MELECON 2006, pp 1246–1251, May 2006

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