Covering the path space: a casebase analysis for mobile robot path planning

doi:10.1016/S0950-7051(03)00024-8

Knowledge-Based Systems

Volume 16, Issues 5–6, July 2003, Pages 235-242

https://doi.org/10.1016/S0950-7051(03)00024-8 Get rights and content

Abstract

This paper presents a theoretical analysis of a casebase used for mobile robot path planning in dynamic environments. Unlike other case-based path planning approaches, we use a grid map to represent the environment that permits the robot to operate in unstructured environments. The objective of the mobile robot is to learn to choose paths that are less risky to follow. Our experiments with real robots have shown the efficiency of our concept. In this paper, we replace a heuristic path planning algorithm of the mobile robot with a seed casebase and prove the upper and lower bounds for the cardinality of the casebase. The proofs indicate that it is realistic to seed the casebase with some solutions to a path-finding problem so that no possible solution differs too much from some path in the casebase. This guarantees that the robot would theoretically find all paths from start to goal. The proof of the upper bound of the casebase cardinality shows that the casebase would in a long run grow too large and all possible solutions cannot be stored. In order to keep only the most efficient solutions the casebase has to be revised at run-time or some other measure of path difference has to be considered.

Introduction

The work presented in this paper is a theoretical extension of a research in mobile robotics. We investigate the problem of generating a seed casebase to cover the solution space of a mobile robot. In our earlier work we have implemented a case-based reasoning (CBR) approach to mobile robot path planning and tested it on real robots [1]. In this work we prove the theoretical upper and lower bound of the casebase cardinality to show the feasibility and limitations of our approach.

The rest of this paper is organised as follows. In Section 1.1 we give an insight to the field of robot navigation and explain the relevance of the problem. Section 2 describes our approach and reviews our previous experimental results. In Section 3, we give the basic definitions and state the main robot path space covering problem in Section 4. 5 Lower bound, 6 Upper bound prove (exact) lower and upper bounds, respectively, for the stated covering problem. Section 7 compares the old heuristic approach to the new one presented in the current paper. Section 8 draws some conclusions and discusses the future work.

Our work is motivated by the fact that most of mobile robot applications imply repeated traversal in a changing environment between predefined start and goal points. For example, a mobile robot could be used to transport details and sub-assemblies between a store and production lines. This task implies repeated traversal between the store and the production cells. A mobile robot can also be used for surveillance. This task implies visiting certain checkpoints on a closed territory in a predefined order.

Real environments where these kind of mobile robots have to operate are dynamic by nature. From the point of view of mobile robot navigation, it means that unexpected obstacles can appear and disappear. The nature and density of the obstacles is usually unknown or too difficult to model.

At the same time a mobile robot in a dynamic environment has to fulfill its assignment as fast and safely as possible. This means choosing paths between target points that are most likely unblocked and where the robot does not spend too much time maneuvering between obstacles.

Very few research studies reported so far consider this problem of path selection [2], [3]. Unlike these approaches, we do not assume that the structure of the environment is known a priori. Therefore, our approach is applicable also in cases where very little is known about the environment and where the structure of the environment may change.

Section snippets

System description

Our approach to mobile robot path selection consists of a general world model and of a memory that stores the path traveling experiences for later use. The memory is a casebase that stores the paths traversed in the past.

The world model is a map that permits path planning. Since in a dynamic environment the robot is not able to model all the aspects of its surrounding, the map is always more or less imprecise.

Fig. 1 captures the bottom line of this approach. The global planner receives tasks

Basic definitions

Let [a,b], $a≤b∈ Z$ denote the set {a,a+1,…,b}. We will consider a robot moving on a rectangular grid [0,m]×[0,n] allowing only right and up moves. The robot starts from the point (0,0) (which we consider to be the lower left corner of the grid) and it must reach the point (m,n).

Definition 1

By a path on the grid [0,m]×[0,n] we mean a sequence of grid points $((x_{0},y_{0}),(x_{1},y_{1}),…,(x_{m+n},y_{m+n}))$ such that

(1)
x₀=y₀=0, x_m+n=m, y_m+n=n;
(2)
for each i∈[0,m+n−1], the condition

(x_{i} =x_{i+1} and y_{i} +1=y_{i+1})∨(x_{i} +1=x_{i+1} and y_{i} =y_{i+1})

holds.

The set

Problem statement

We will be looking at the following covering problem in the metric space (P_m,n,d_g).

Problem

For a given integer δ and grid dimensions m and n, find a lower and upper estimate to the cardinality of a subset S⊆P_m,n such that the following conditions hold. $⋃ P∈S B(P,δ)=P_{m,n}$ $∀P′∈S P′∉ ⋃ P∈S\{P′} B(P,δ)$

The lower estimate corresponds to the question about efficient covering. In this setting we ask what is the minimal number of pre-planned paths required in the casebase in order embrace all the possible paths with

Lower bound

Theorem 1

For every $δ∈ N$ and every subset S⊆P_m,n satisfying the properties , , the inequality $|S|≥ π m 2δ+1, n 2δ+1$ holds. Evenmore, there exists such a set S that the properties , are satisfied and equality holds in inequality (4).

Proof

First we consider a special case when $m,n ⋮ 2δ +1$ and a subset T⊆P_m,n defined by the following condition: $T={((x_{0},y_{0}),…,(x_{m+n},y_{m+n}))∈P_{m,n} :∀_{i} ∈[0,m+n] x_{i} ⋮ 2δ+1∨y_{i} ⋮ 2δ+1}.$ We note that for P₁≠P₂∈T the inequality d_g(P₁,P₂)≥2δ+1 holds. Hence, no two different elements of the set T can be

Upper bound

Theorem 2

For every $δ∈ N$ and every subset S⊆P_m,n satisfying the properties , , the inequality $|S|≤ π m δ, n δ, if δ is odd π m δ+1, n δ+1, if δ is even$ holds. Evenmore, there exists such a set S that the properties , and $|S|= π m δ+1, n δ+1$ are satisfied.

Proof

The proof of this theorem is very similar to Proof of Theorem 1. First we consider the case when δ is even and $m,n ⋮ δ+1.$ We define the set T as follows: $T={((x_{0},y_{0}),…,(x_{m+n},y_{m+n}))∈P_{m,n} :∀i∈[0,m+n] x_{i} ⋮ δ+1∨ y_{i} ⋮ δ+1}.$ Assume that we also have a set S corresponding to conditions , .

Similarly

Comparison of approaches

To demonstrate the advantage of the current approach compared to our previous heuristic one, we use parameters from one of our previous real-world experiments. For a grid of 51×67 cells and with the similarity threshold δ=5 the size of the seed casebase would be $π 51 2·5+1, 67 2·5+1 = 4+6 4 =210$ cases.

At the same time, if the heuristic path generation algorithm was used, the new generated paths never covered the solution space although the experiments consisted of more than 500 runs. Another set of our

Conclusions and future work

Our work was motivated by the fact that our previous heuristic technique was not able to generate all possible different solutions to the current path-finding problem. We therefore investigated the possibility to create a seed casebase that covers the whole solution space of the robot.

In this paper, we have proven the lower and upper bound of the solution space. The proof of the lower bound shows that it is realistic to seed the casebase with a solution set that contains a close match to every

References (11)

E. Belogay et al.
Calculating the Hausdorff distance between curves
Info. Process. Lett.
(1997)
M. Kruusmaa, Repeated Path Planning for Mobile Robots in Dynamic Environments, PhD Thesis, Chalmes Univeristy of...
H. Hu et al.
Dynamic global path planning with uncertainty for mobile robots in manufacturing
IEEE Trans. Robot. Automat.
(1997)
K.Z. Haigh et al.
Planning, execution and learning in a robotic agent
AIPS-98
(1998)
C. Vasudevan et al.
Case-based path planning for autonomous underwater vehicles
Proc. 1994 IEEE Int. Symp. Intell. Control
(1994)

There are more references available in the full text version of this article.

Cited by (17)

Biasing Bayesian Optimization Algorithm using Case Based Reasoning
2011, Knowledge-Based Systems
Citation Excerpt :
In this paper, we focus on the combinatorial optimization problems. CBR method has been applied in solving various combinatorial optimization problems up to now; for example, the traveling salesman problem [14–18], the knapsack problem [14,15], and the scheduling problem [19–25]. For these combinatorial optimization problems, the CBR method tries to exploit the knowledge of previously solved problems in solving the new ones.
Studies show that application of the prior knowledge in biasing the Estimation of Distribution Algorithms (EDAs), such as Bayesian Optimization Algorithm (BOA), increases the efficiency of these algorithms significantly. One of the main advantages of the EDAs over other optimization algorithms is that the former provides a trail of probabilistic models of candidate solutions with increasing quality. Some recent studies have applied these probabilistic models, obtained from previously solved problems in biasing the BOA algorithm, to solve the future problems. In this paper, in order to improve the previous works and reduce their disadvantages, a method based on Case Based Reasoning (CBR) is proposed for biasing the BOA algorithm. Herein, after running BOA for solving optimization problems, each problem, the corresponding solution, as well as the last Bayesian network obtained from the BOA algorithm, will be stored as an entry in the case-base. Upon introducing a new problem, similar problems from the case-base are retrieved and the last Bayesian networks of these solved problems are combined according to the degree of their similarity with the new problem; hence, a compound Bayesian network is constructed. The compound Bayesian network is sampled and the initial population for the BOA algorithm is generated. This network will be applied efficiently for biasing future probabilistic models during the runs of BOA for the new problem. The proposed method is tested on three well-known combinatorial benchmark problems. Experimental results show significant improvements in algorithm execution time and quality of solutions, compared to previous methods.
Strategy creation, decomposition and distribution in particle navigation
2007, Information Sciences
Strategy planning is crucial to control a group to achieve a number of tasks in a closed area full of obstacles. In this study, genetic programming has been used to evolve rule-based hierarchical structures to move the particles in a grid region to accomplish navigation tasks. Communications operations such as receiving and sending commands between particles are also provided to develop improved strategies. In order to produce more capable strategies, a task decomposition procedure is proposed. In addition, a conflict module is constructed to handle the challenging situations and conflicts such as blockage of a particle’s pathway to destination by other particles.
Simple agents, smart swarms: a cooperative search algorithm for swarms of autonomous underwater vehicles
2022, International Journal of Systems Science
An A∗based path planning approach for autonomous vehicles
2021, ACM International Conference Proceeding Series
Cloud-based improved Monte Carlo localization algorithm with robust orientation estimation for mobile robots
2019, Engineering Computations (Swansea, Wales)
Robot Path Planning Based on Improved Ant Colony Algorithm
2018, 2018 WRC Symposium on Advanced Robotics and Automation, WRC SARA 2018 - Proceeding

View all citing articles on Scopus

View full text

Covering the path space: a casebase analysis for mobile robot path planning

Abstract

Introduction

Section snippets

System description

Basic definitions

Problem statement

Lower bound

Upper bound

Comparison of approaches

Conclusions and future work

Info. Process. Lett.

Dynamic global path planning with uncertainty for mobile robots in manufacturing

IEEE Trans. Robot. Automat.

Planning, execution and learning in a robotic agent

AIPS-98

Case-based path planning for autonomous underwater vehicles

Proc. 1994 IEEE Int. Symp. Intell. Control