- Email: [email protected]
Consideration of obstacle danger level in path planning using A∗ and Fast-Marching optimisation: comparative study
Signal Processing 83 (2003) 2387 – 2396 www.elsevier.com/locate/sigpro
Consideration of obstacle danger level in path planning using A∗ and Fast-Marching optimisation: comparative study P. Melchior∗ , B. Orsoni, O. Lavialle, A. Poty, A. Oustaloup LAP-UMR 5131 CNRS, Universite Bordeaux 1-ENSEIRB, 351 cours de la Liberation, Talence, cedex 33405, France Received 2 December 2002
Abstract Obstacle danger level is taken into consideration in path planning using fractional potential maps. This paper describes the two optimisation methods tested: the A* algorithm and the Fast-Marching technique. The e4ciency of the two approaches is illustrated and compared through a vehicle path planning application in a 5xed obstacle environment. A∗ is a heuristically ordered research algorithm and is complete and admissible. Fast-Marching provides a convex map without local minima and permits real-time evaluation of optimal trajectories. A vehicle path planning application is considered in a 5xed obstacle environment. A speci5c danger level is given to each obstacle. The obtained continuous curve trajectories are compared. ? 2003 Elsevier B.V. All rights reserved. Keywords: Mobile robots; Path planning; Fractional di;erentiation; Weyl fractional integral; A∗ ; Fast marching
1. Introduction For 5xed polygonal obstacles, the connectivity of the robot free space can be captured in a network of nodes and arcs: the roadmap [10,21]. The nodes are the possible states of the robot, and the arcs are the possible transitions between nodes. A variety of methods can be used to obtain roadmaps for path planning, in particular visibility graph [11,16,22,23], retraction approach [4] and freeway method [3]. Finding a trajectory through a roadmap is considered heuristically ordered when the state graph description is not exhaustive but guided by an evaluation function. This function is used to quantify the cost between the current node and the 5nal node. A rough ∗
Corresponding author. Tel.: +33-0540-00-6607; fax +330540-00-6644. E-mail address: [email protected] (P. Melchior). URL: http://www.lap.u-bordeaux.fr
estimation of this cost takes less calculating time than a precise calculation but a compromise should be made. A and A∗ Nilsson’s algorithms [5,16,17] are arti5cial intelligence techniques. The A algorithm supposes all the arc costs are strictly positive and the heuristic function values are positive. A∗ algorithms are special A algorithms, where the heuristic function has the minorance property. As A∗ algorithms are always complete and admissible [5], the number of nodes to be considered can be reduced, and an optimal solution is ensured. The heuristic functions considered in A∗ algorithms depend only on nodes and are independent of time. The form of the trajectory is thus independent of the robot dynamics. However, 5nding an optimal trajectory (with minimal time) requires absence of angular points, contact with obstacles, and speed discontinuities (nonexhaustive list) [2]. Local methods allowing a real-time estimation of the trajectory have been introduced. The better-known
0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1684(03)00191-9
2388
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
local methods in path planning are the :ctitious potential methods [7,8,10,21]. A 5ctitious force 5eld is introduced to take account of the dynamics of the robot and to obtain realistic speeds. The concept of a 5ctitious potential 5eld in path planning is the following [10]: “The potential 5eld concept considers the robot as a charged particle moving under the inGuence of repulsion potentials from obstacles, and attraction potentials of the target”. The attraction (negative) potential Ucible (M) is associated to the target and positive repulsive Coulomb-type potential 5elds, Uk (M), are de5ned for each obstacle. The resulted potential value provides a danger level for each point of the map [8,26]. A force 5eld is also associated to the potential by superposition of gravitational e;ects. The force 5eld F(M) is F(M) = −∇ Ucible (M ) + Uk (M) : (1) k
Thus the force acts in the direction of the strongest decrease of potential. The direction and the norm of the force provide the most appropriate heading for the robot to take. The smoothness of the curve obtained with potential 5eld methods makes practical steering and speed control possible. However, the robot may not 5nd its way through a narrow path. The robot may also be trapped in local minima as the algorithm always indicates the deepest point of potential cups. To avoid the local minima problem, superquadratic potential :elds have been de5ned [21,26]. However, only one obstacle can be considered, and some particular initial and 5nal positions can even require a re-evaluation of the superquadric potential. Moreover, this type of approach does not solve the narrow path problem. These drawbacks limit the practical use of this method. Thus, the existence of local minima often limits the use of the potential 5elds to local planning. Graph description methods, which are global, are limited by their complex calculation. The advantages of the two methods can be combined as in planning by repulsion suggested by the Barraquant and Latombe [1,2], free circles [6] and Krogh [9] methods. However, the word potential is not always used to refer to the same thing as in the Coulomb potential concept. Also, their methods can not take obstacle danger levels into account. Uncertainty about the obstacle position can be used to facilitate navigation [13]. This uncertainty,
corresponding to a hypothetical danger (or potential), can de5ne a 5eld of repulsion used by an A∗ algorithm for example. The danger concept was introduced for military requirements. It permits gradually modifying a trajectory according to the degree of danger of the obstacles or environment. In our approach, the fractional integration order is used as a risk coe4cient for each obstacle. The fractional potential 5eld completes the Coulomb 5eld used in path planning and permits the de5nition of repulsion forces. The potential 5eld norm takes the proximity and the danger level of each obstacle into account. The fractional integration order or risk coef:cient, can precisely quantify the inGuence of identi5ed obstacles over a continuous noninteger range [14,15,18,19,20]. The path planning methods we describe are considered in a 2D environment (operational space) for convenient presentation and illustration. However, the fractional potential methods can easily be applied to multi-dimensional environments (joint space). Section 2 deals with the de5nition of the fractional potential. Then, Sections 3 and 4 describe the two optimisation methods tested: the A∗ algorithm and the Fast-Marching technique. The e4ciency of the two approaches is illustrated and compared through a vehicle path planning application in a 5xed obstacle environment. A speci5c danger level is given to each obstacle.
2. Fractional potential and danger 2.1. De:nition of the fractional potential founded on the Weyl de:nition The fractional potential de5ned by the Riemann fractional integral de5nition [14,15] ensures a continuous passage between the potential created by an isolated charge and the one created by a charge uniformly distributed along an in5nite segment. Using Weyl fractional integral gives a new fractional potential de5nition: 1 q n n Vn (r) = W E(r) = ; (2) W 4 0 r2
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
q 4 0 (n)
∞
r
( − r)n−1 d: r2
In an electrostatic context, energy and potential 5eld of a charge distribution are evaluated from r = +∞ to r. Riemann’s de5nition is not adapted for such an integral reference. Analytical expression of fractional potential founded on Weyl’s de5nition is deduced from (3). Thus ∀n ∈ [0; 2[;
q (2 − n) Vn (r) = : 4 0 r 2−n
Un (r) =
(4)
(5)
Substituting de5nition (4) in Eq. (5), the analytic expression of the Weyl’s normalized fractional potential is deduced ∀n ∈ [0; 2[;
∀r ∈ [rmin ; rmax ];
n−2 r n−2 − rmax : Un (r) = n−2 n−2 rmin − rmax
n=3
0.8
n = 2− segment − n = 1 ponctual charge
0.7 0.6 0.5 0.4 0.3
0.1 0
∀r ∈ [rmin ; rmax ];
Vn (r) − Vn (rmax ) : Vn (rmin ) − Vn (rmax )
n=4
0.9
0.2
The fractional potential is then normalized between 0 and 1. The normalized fractional potential Un (r) must be null over a 5nite distance considered as the maximum distance of inGuence rmax . It must be equal to one at each obstacle to forbid collision. As a singularity appears when r = 0, a 5nite minimum distance rmin is chosen. The normalized fractional potential is therefore a percentage corresponding to the danger level ∀n ∈ [0; 2[;
1
(3)
Normali
Vn (r) =
2389
(6)
Weyl’s fractional potential is thus a compact expression, easy to use. No integration is required for its computation, and the potential shape only depends on the fractional integration order which corresponds to the danger of the obstacle. The charge in the expression of the fractional potential (4) no longer e;ects the normalized fractional potential (6). The curves shape of the normalized fractional potential and thus its inGuence distance, depends only on the fractional integration order n. As Fig. 1 shows it, a continuous variation of order n, allows a continuous and progressive variation of the potential curve. Whatever the distance considered, the more the fractional integration order is important, the more the potential is signi5cant. Thus, the danger zone and the danger itself will be all
0
5
10
rmin = 1
15 Distance
20
25 rmax = L = 30
Fig. 1. InGuence distance of the normalized fractional potential based on Weyl’s de5nition (1 6 n 6 5); coulombian potential created by a ponctual charge (n = 1) and a uniformly distributed charge on a segment (n = 2).
the more signi5cant around the obstacles as the risk coe4cient associated with the obstacles will be important. When n = 1, the fractional potential founded on the Weyl de5nition is V1 (r) =
q 1 : 4 0 r
(7)
This is the Coulombian potential 5eld generated by an isolated charge.
2.2. Fractional map Using (6), the distance map for each obstacle can now be converted into a fractional potential map. The sum of all the maps could de5ne the fractional potential map of the environment, but the obstacle danger level would not be normalized. For each point of the free space, the maximum value of all the potential maps is chosen, thus providing the global fractional map. So, the potential values are normalized. Also, local maximum values of potentials are avoided [6], but this leads to discontinuity of the force 5eld directions. A fractional map example is presented Figs. 2 and 3.
2390
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 2. Fractional 2D map example; the grey range indicates the danger level; the risk coe4cients are arbitrarily 5xed at 1.2 for the square, 1.5 for the chamfered rectangle and 2.5 for the circle; the maximum distance from inGuence is rmax = 20, the minimal distance from evaluation is rmin = L = 1. Fig. 4. Horizontal cross section of the fractional map; threshold of 50% authorized danger level (maximum level).
Fig. 3. Fractional map 3D example—the height is the danger level.
2.3. Fractional road
Fig. 5. Fractional road obtained by thresholding of 50% of the fractional map (white: fractional road; grey: danger zone; black: obstacles).
The operator chooses a threshold of danger level beyond which it estimates that the risk taken by the robot would be too important. A horizontal cross section of the fractional card at the given threshold (Fig. 4), provides some equipotential which separates zones where the danger is too important, from zones of security to be used. The thresholding of potential valleys of the fractional map de5nes a fractional road more or less narrow according to the obstacles dangerousness de5ning these valleys (Fig. 6). Thus, the fractional road reduced the environment in which the robot can move (Fig. 5), but it enables him to avoid collisions. It ensures the respect of a minimal security distance around the obstacles, which depends on their dangerousness degree.
Fig. 6. Between edges A and B of two obstacles, a horizontal cut of the fractional map, corresponding here to a threshold of 50%, de5nes the fractional road which width is represented by the segment C–D: (a) large fractional road allowing the passage in full safety, close to the two obstacles slightly dangerous; (b) narrow fractional road not authorizing the passage near the two dangerous obstacles; (c) fractional road allowing the passage close to the obstacle slightly dangerous and not authorizing the passage near the dangerous obstacle.
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
2391
3. A∗ Algorithm 3.1. Path planning by A∗ algorithm A∗ algorithm is built by de5ning the function of evaluation f(N ) characterizing the path cost. The evaluation function is f(N ) = g(N ) + h(N );
(8)
where g(N ) is the determinist cost, expanded to go from the initial node to the current node, and h(N ) the heuristic function which estimates the minimum-cost path between the current node and the target node. The criterion considered (distance and danger) in the determinist function of the evaluation function of the A* algorithm is thus g(N ) = g(N − 1) + d(N; N − 1)2 + pN2 ; (9) where d(N; N − 1) is the 2D displacement between two successive nodes, pN the potential in the node N and the weighting factor. The heuristic function is the Gying distance without obstacle h(N ) = d(N; Ncible ):
(10)
It veri5es well the minorance property: ∀N ∈ G;
0 6 h(N ) 6 h∗ (N ):
(11)
A∗ algorithms are always complete and admissible: the optimal solution, whenever a solution exists, is thus systematically provided. 3.2. Methodology and examples Obtaining a trajectory by A∗ algorithm requires 4 stages: (1) the environment is 5rst of all modeled by assigning to each obstacle a danger coe4cient; (2) the fractional map, indicating the danger in each point of the free environment for the robot evolution, is then evaluated; (3) the weighting parameter allowing continuity between criteria of distance and danger is then selected; (4) the trajectory is 5nally determined by a development ordered heuristically of the state graph. The trajectories minimizing criteria (9) are determinate for di;erent values of the weighting parameter.
Fig. 7. Trajectory obtained by A∗ algorithm (16 neighbors) for = 0.
The trajectories are obtained by using “Fractional Path Planning Design” module of the CRONE toolbox for Matlab. The results obtained according to the value of the weighting parameter , are given in next paragraphs. 1. Minimum length trajectory ( = 0): The trajectory obtained for a weighting factor = 0, minimizes the sum of the distances between each pixel of the trajectory. The 16 nearest neighbors of the node to be developed are considered (Fig. 7). However, the minimum length trajectory provide by the A∗ algorithm for = 0, is not the discrete approximation. The increase in the number of pixels does not necessarily increase the precision of the solution with respect to the continuous solution. Lastly, the variations with respect to the continuous solution appear irregularly and make the tracking more chaotic. The precision improvement in the trajectory obtained cannot thus be done simply by increasing pixels number, but the number of the acceptable neighbors. This increase being here directly related to the possible orientations of the robot. The increase in the precision is then limited by the size of the smallest obstacle: if the smallest obstacle has a thickness of 1 pixel, only the 4-vicinity and the 8-vicinity can be used; if the smallest obstacle has a thickness of 2 pixels, the 4, 8, 16 and 24-vicinities can be used.
2392
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 8. Trajectory obtained by A∗ algorithm (16 neighbors) for = 8.
2. Trajectory minimizing distance and danger: The trajectory obtained for a weighting factor = 8 minimizing twice distance and danger is represented (Fig. 8). Security is improved by choosing a greater distance from an obstacle, when the resulting increase of trajectory length is minimal. Thus, the trajectory always passes too close to the chamfered rectangle but deviates more quickly from this obstacle. The danger information provided by the fractional map in the criterion leads to two properties: 5rstly, taking into account the danger makes it possible to anticipate the presence of the obstacle (trajectories with continuous curve); secondly, calculations of the danger and displacement length being uncoupled, the discrimination introduced by the potential to move away the trajectories from the obstacles, improves the precision of the result compared to the discretization of the optimal continuous trajectory. The progressive and continuous increase in the weighting parameter , thus authorizes a progressive and continuous passage between the minimal distance trajectory and the minimal danger trajectory. 3. Minimum danger trajectory ( = ∞): Minimum danger trajectory obtained for a weighting factor = 1000, is presented in Fig. 9. The danger of a trajectory will be de5ned as the curvilinear integral of the potential (danger of the en-
Fig. 9. Trajectory obtained by A∗ algorithm (16 neighbors) for = 1000.
vironment) on the trajectory. The danger map towards a target will also be de5ned, providing for each point of the free space, the minimum danger of the available trajectories reaching the target. The minimum danger trajectory is also obtained by gradient descent: successive steps, following the gradient of the danger map towards the target provides the trajectory. The shortest least dangerous trajectories is retained. Curvatures obtained are not always smooth, but in this case they are smooth. Unlike the trajectory of retroaction approach [4], a minimum danger trajectory is not restricted to the minimum potential valley. The greater the trajectory length, the greater the number of potentials to be summed, so the minimum danger trajectory found is both smooth and short.
4. Fast-Marching de nition: Eikonal equation and propagation front Fast-Marching methods and Level Set methods [24,25] are numerical techniques, which solve the Eikonal equation: |∇E(M)|G(M) = 1;
(12)
|∇E(M)| = F(M):
(13)
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
These techniques consider continuous propagation of a wave front with speed G(M) (Eq. (12)). In the equivalent formulation presented in (13), F(M) is the propagation cost evaluated on the propagation front. The “energy” function E(M), solution of (12) or (13), is the curvilinear integral of the front following the faster path. It indicates in any point reached, the minimal cost to reach the target. Particularly, if F(M) is uniform, E(M) is the geodesic distance between the considered point and the target. To adapt the principle to our study, the propagation cost F(M) must be expressed according to the danger. Thus E(M) represents the minimum of the curvilinear integral between the considered point and the target, the mixed criteria between distance and danger. The discretization of the continuous solution (13) obtained with one step is the algorithm basis and ensures the precision and the result consistency. Fast-Marching technique allows fast numerical implementation by using selection or pile algorithm or heapsort. The points (or pixels) of the front to propagate are the pile. The development in tree is complete and balanced [25]. The complexity of the optimisation algorithm is O(M log M ), where M is the total number of points in the free space. We can note that this technique has a wide range of applications such as in image processing [12] and in path planning to give to the robot some distance maps, energy or time to reach the target. 4.1. Mixed criterion (distance-danger) of propagation The Fast-Marching technique is available in the “Fractional Path Planning Design” module of the CRONE toolbox. The potential is particularized considering both displacement and danger Fi; j = 1 + Pi; j ;
(14)
where Pi; j is the fractional potential which correspond a danger and a weighting factor. 4.2. Consistency by modi:ed gradient descent of the convex map obtained by Fast Marching Considering only 3 × 3 mask for the descent gradient, only eight discrete directions are o;ered to the robot. A higher mask allows to increase the number of
2393
allowed orientations. But the choice is not continuous only if the mask is in5nite. The inconsistency origin of the solution obtained is thus identi5ed. A procedure using continuous information given by the convex map obtained by Fast-Marching must be de5ned. The discrete coordinates of the closest neighbors are not su4cient to indicate an orientation, by continuous nature, with the robot. A modi5ed gradient descent is de5ned in three steps: (1) motion following the gradient; (2) memorizing of the fall point; (3) new motion starting from the fall point. The more the number of pixels increases, the more the gradient orientation obtained by Fast-Marching, has di;erent values; when the number of pixels is in5nite, a continuous orientation of the robot can be theoretically obtained. The information given by the gradient is stored by using a modi5ed gradient descent. The continuous coordinates of the fall point are memorized and are used as reference for the following descent of the gradient. 4.3. Results The minimum length trajectory is given by choosing =0; a modi5ed gradient descent is now used to obtain the minimum length. It is represented in Fig. 10. It is a true approximation of a continuous optimal trajectory. The trajectory minimizing twice the motion and the danger for a weighting factor = 5 is represented in Fig. 11. The security is increased while gradually moving away the trajectory from the obstacles according to the minimal length. Thus, the trajectory always passes to the closest to the chamfered rectangle, but deviates more quickly from this obstacle. The use of the fractional map in the criterion allows, as for A∗ algorithm, to anticipate the presence of the obstacle, which leads to smooth trajectories. The progressive and continuous increase of the weighting parameter authorizes a progressive and continuous transition between the minimal distance trajectory and the minimal danger trajectory. The trajectory obtained, for a weighting factor = 1000 is represented in Fig. 12. It does not borrow any more the potential valley, but passes round the obstacle by outside skirting the border of the environment. An angular point seems to appear for the approach by repulsion [4]. This angular
2394
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 10. Minimum length trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 0).
Fig. 12. Optimal trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 1000).
5. Conclusion
Fig. 11. Optimal trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 5).
point results from a strategy in two stages: 5rstly, the target approach remaining in a zone located beyond the maximum distance from the obstacle inGuence; secondly, projection towards the target in the inGuence zone by the shortest path.
Fractional integral ensures a continuous passage between two integer orders of integration, and thus between the coulombian potential shapes generated by a punctual charge and the ones created by a charge uniformly distributed along a straight-line segment. The fractional order is used as a risk coe4cient to characterize the obstacle danger, and controls the potential shape smoothly. So, in path planning design, the fractional potential easily determines the least dangerous of the minimum length trajectories, or the shortest of the least dangerous trajectories. Two approaches are considered in this study: a research ordered heuristically by A∗ algorithm and the establishment of convex maps by the Fast-Marching technique. The A∗ algorithm converges systematically towards the optimal solution whenever a path exists. The weighting factor between length and danger allows smooth passage from maximum security to minimum length trajectories. Danger of obstacles is gradually taken into account to anticipate collision and also to provide smoother trajectories. However, the minimum length trajectory obtained by A∗ algorithm for = 0 is not the discrete approximation of the optimal
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
length continuous trajectory. The danger information given by the fractional map improves the precision of the result with respect to the approximation of continuous trajectory. But the method is inconsistent whatever the value of the weighting factor . The Fast-Marching technique is thus introduced and used as a consistent solution to the problem. The solutions obtained are the true approximations of the continuous and optimal trajectories. Moreover, the convex map can be transferred one for all to the robot and whatever the start point, all the trajectories can be evaluated in real time. The precision and the consistency of the technique come from solution of the Eikonal equation. Thus the level line curvature and the gradient orientation are continuous; the robot can be oriented quasi-continuously. Finally, it can deduced the continuity curvature of trajectories obtained with Fast-Marching’s technique associated to modi5ed gradient descent. A mixed criterion, adapted to the Fast-Marching’s technique, is thus de5ned. The criterion corresponds to the propagation front speed depending twice of the danger and of the displacement length. The criterion allows smooth passage from maximum security to minimum length trajectories. The application domain is not limited to mobile robotics, but can also be used in the automotive domain, to avoid collision. All the results have been obtained through simulations; so we still have to test it on a real robot. As prospects, we have also to extend the method to mobile obstacles and then to unknown obstacles.
References [1] J. Barraquand, J.C. Latombe, Robot motion planning: a distributed representation approach, Internat. J. Robotics Res. 10 (6) (December 1991) 628–649. [2] J. Borenstein, Y. Koren, Real time obstacle avoidance for fast mobile robots, IEEE Trans. System Man Cybernet. 19 (5) (September–October 1989) 1179–1187. [3] R.A. Brooks, Solving the 5nd-path problem by good representation of free space, IEEE Trans. System Man Cybernet. SMC-13 (3) (March–April 1983) 190–197. [4] C. O’Dunlaing, M. Sharir, C.K. Yap, Retraction: a new approach to motion planning, Proceedings of the 15th ACM Symposium on the Theory of Computing, Boston, USA, 1983, pp. 207–220.
2395
[5] H. Farreny, Recherche heuristiquement ordonnUee: gUenUeralisations compatibles avec la complUetude et l’admissibilitUe, Technique et Science Informatiques, RAIRO 16 (7) (September 1997) 25–953. [6] Y.K. Hwang, N. Ahuja, A potential 5eld approach to path planning, IEEE Trans. Robotics Automation 8 (1) (February 1992) 23–32. [7] O. Khatib, Commande dynamique dans l’espace opUerationnel des robots manipulateurs en prUesence d’obstacles, ThUese de Doctorat, ENSAE, Toulouse, 1980. [8] O. Khatib, Real time obstacle avoidance for manipulator and mobile robot, Internat. J. Robotics Res. 5 (1) (Spring 1986) 90–98. [9] B.H. Krogh, C.E. Thorpe, Integrated path planning and dynamic steering control for autonomous vehicles, IEEE International Conference on Robotics and Automation, San Francisco, USA, 1986, pp. 1664 –1669. [10] J.C. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Dordrecht, fourth printing, 1996. [11] J.P. Laumond, Obstacle growing in a non-polygonal world, Inform. Process. Lett. 25 (1) (1987) 41–50. [12] O. Lavialle, F. Angella, P. Baylou, An extension of the minimal path for structure recovery, Proceedings of the IEEE International Conference on Image Processing, Kobe, Japan, October, 1999. [13] N. Lefort-Piat, I. Collin, D. Meizel, Plani5cation des missions de robots mobiles rUefUerencUes par rapport aV l’environnement: tˆache-robot et cartes locales, Rev. Intell. Artif. 11 (4) (1997) 489–516. [14] H. LinarUes, ModUelisation d’environnement par dUerivation non entiVere et commande robuste d’un robot mobile, ThUese de Doctorat de l’UniversitUe Bordeaux 1, Bordeaux, 29 FUevrier 1996. [15] P. Melchior, B. Orsoni, A. Oustaloup, Weyl fractional potential in path planning, Proceedings of the Sixth IEEE ECC’2001, Porto, Portugal, 4 –7 September, 2001. [16] N.J. Nilsson, A mobile automation: an application of arti5cial intelligence techniques, Proceedings of the 1st International Joint Conference on Arti5cial Intelligence, Washington DC, USA, 1969, pp. 509 –520. [17] N.J. Nilsson, Problem-solving Methods in Arti5cial Intelligence, McGraw-Hill, New York, 1971. [18] B. Orsoni, P. Melchior, Plani5cation de trajectoire basUee sur l’utilisation d’un potentiel non-entier, Proceedings of the First IEEE CIFA’2000, Lille, France, 5 –8 Juillet 2000, pp. 819 –826. [19] A. Oustaloup, La dUerivation non entiVere: thUeorie, synthVese et applications, Editions HermVes, Paris, 1995. [20] A. Oustaloup, H. LinarVes, The CRONE path planning, Internat. IMACS J. Math. Comput. Simulation 41 (1996) 209–217. [21] A. Pruski, Robotique mobile, plani5cation de trajectoire, Editions HermVes, Paris, 1996. [22] J.H. Reif, J. Storer, Shortest paths in euclidean space with polyhedral obstacles, Technical Report CS-85-121, Computer Science Department, Brandeis University, 1985. [23] H. Rohnert, Shortest paths in the plane with convex polygonal obstacles, Inform. Process. Lett. 23 (1986) 71–76.
2396
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
[24] J.A. Sethian, A fast marching level set method for monotonically advancing fronts, Nat. Acad. Sci. 93 (4) (1996) 1591–1595. [25] J.A. Sethian, Fast marching methods, SIAM Rev. 41 (2) (1999) 199–235.
[26] R. Volpe, P. Khosla, Manipulator control with superquadric arti5cial potential functions: theory and experiments, IEEE Trans. System. Man Cybernet. 20 (6) (November–December 1990) 1423–1436.
Consideration of obstacle danger level in path planning using A∗ and Fast-Marching optimisation: comparative study P. Melchior∗ , B. Orsoni, O. Lavialle, A. Poty, A. Oustaloup LAP-UMR 5131 CNRS, Universite Bordeaux 1-ENSEIRB, 351 cours de la Liberation, Talence, cedex 33405, France Received 2 December 2002
Abstract Obstacle danger level is taken into consideration in path planning using fractional potential maps. This paper describes the two optimisation methods tested: the A* algorithm and the Fast-Marching technique. The e4ciency of the two approaches is illustrated and compared through a vehicle path planning application in a 5xed obstacle environment. A∗ is a heuristically ordered research algorithm and is complete and admissible. Fast-Marching provides a convex map without local minima and permits real-time evaluation of optimal trajectories. A vehicle path planning application is considered in a 5xed obstacle environment. A speci5c danger level is given to each obstacle. The obtained continuous curve trajectories are compared. ? 2003 Elsevier B.V. All rights reserved. Keywords: Mobile robots; Path planning; Fractional di;erentiation; Weyl fractional integral; A∗ ; Fast marching
1. Introduction For 5xed polygonal obstacles, the connectivity of the robot free space can be captured in a network of nodes and arcs: the roadmap [10,21]. The nodes are the possible states of the robot, and the arcs are the possible transitions between nodes. A variety of methods can be used to obtain roadmaps for path planning, in particular visibility graph [11,16,22,23], retraction approach [4] and freeway method [3]. Finding a trajectory through a roadmap is considered heuristically ordered when the state graph description is not exhaustive but guided by an evaluation function. This function is used to quantify the cost between the current node and the 5nal node. A rough ∗
Corresponding author. Tel.: +33-0540-00-6607; fax +330540-00-6644. E-mail address: [email protected] (P. Melchior). URL: http://www.lap.u-bordeaux.fr
estimation of this cost takes less calculating time than a precise calculation but a compromise should be made. A and A∗ Nilsson’s algorithms [5,16,17] are arti5cial intelligence techniques. The A algorithm supposes all the arc costs are strictly positive and the heuristic function values are positive. A∗ algorithms are special A algorithms, where the heuristic function has the minorance property. As A∗ algorithms are always complete and admissible [5], the number of nodes to be considered can be reduced, and an optimal solution is ensured. The heuristic functions considered in A∗ algorithms depend only on nodes and are independent of time. The form of the trajectory is thus independent of the robot dynamics. However, 5nding an optimal trajectory (with minimal time) requires absence of angular points, contact with obstacles, and speed discontinuities (nonexhaustive list) [2]. Local methods allowing a real-time estimation of the trajectory have been introduced. The better-known
0165-1684/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1684(03)00191-9
2388
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
local methods in path planning are the :ctitious potential methods [7,8,10,21]. A 5ctitious force 5eld is introduced to take account of the dynamics of the robot and to obtain realistic speeds. The concept of a 5ctitious potential 5eld in path planning is the following [10]: “The potential 5eld concept considers the robot as a charged particle moving under the inGuence of repulsion potentials from obstacles, and attraction potentials of the target”. The attraction (negative) potential Ucible (M) is associated to the target and positive repulsive Coulomb-type potential 5elds, Uk (M), are de5ned for each obstacle. The resulted potential value provides a danger level for each point of the map [8,26]. A force 5eld is also associated to the potential by superposition of gravitational e;ects. The force 5eld F(M) is F(M) = −∇ Ucible (M ) + Uk (M) : (1) k
Thus the force acts in the direction of the strongest decrease of potential. The direction and the norm of the force provide the most appropriate heading for the robot to take. The smoothness of the curve obtained with potential 5eld methods makes practical steering and speed control possible. However, the robot may not 5nd its way through a narrow path. The robot may also be trapped in local minima as the algorithm always indicates the deepest point of potential cups. To avoid the local minima problem, superquadratic potential :elds have been de5ned [21,26]. However, only one obstacle can be considered, and some particular initial and 5nal positions can even require a re-evaluation of the superquadric potential. Moreover, this type of approach does not solve the narrow path problem. These drawbacks limit the practical use of this method. Thus, the existence of local minima often limits the use of the potential 5elds to local planning. Graph description methods, which are global, are limited by their complex calculation. The advantages of the two methods can be combined as in planning by repulsion suggested by the Barraquant and Latombe [1,2], free circles [6] and Krogh [9] methods. However, the word potential is not always used to refer to the same thing as in the Coulomb potential concept. Also, their methods can not take obstacle danger levels into account. Uncertainty about the obstacle position can be used to facilitate navigation [13]. This uncertainty,
corresponding to a hypothetical danger (or potential), can de5ne a 5eld of repulsion used by an A∗ algorithm for example. The danger concept was introduced for military requirements. It permits gradually modifying a trajectory according to the degree of danger of the obstacles or environment. In our approach, the fractional integration order is used as a risk coe4cient for each obstacle. The fractional potential 5eld completes the Coulomb 5eld used in path planning and permits the de5nition of repulsion forces. The potential 5eld norm takes the proximity and the danger level of each obstacle into account. The fractional integration order or risk coef:cient, can precisely quantify the inGuence of identi5ed obstacles over a continuous noninteger range [14,15,18,19,20]. The path planning methods we describe are considered in a 2D environment (operational space) for convenient presentation and illustration. However, the fractional potential methods can easily be applied to multi-dimensional environments (joint space). Section 2 deals with the de5nition of the fractional potential. Then, Sections 3 and 4 describe the two optimisation methods tested: the A∗ algorithm and the Fast-Marching technique. The e4ciency of the two approaches is illustrated and compared through a vehicle path planning application in a 5xed obstacle environment. A speci5c danger level is given to each obstacle.
2. Fractional potential and danger 2.1. De:nition of the fractional potential founded on the Weyl de:nition The fractional potential de5ned by the Riemann fractional integral de5nition [14,15] ensures a continuous passage between the potential created by an isolated charge and the one created by a charge uniformly distributed along an in5nite segment. Using Weyl fractional integral gives a new fractional potential de5nition: 1 q n n Vn (r) = W E(r) = ; (2) W 4 0 r2
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
q 4 0 (n)
∞
r
( − r)n−1 d: r2
In an electrostatic context, energy and potential 5eld of a charge distribution are evaluated from r = +∞ to r. Riemann’s de5nition is not adapted for such an integral reference. Analytical expression of fractional potential founded on Weyl’s de5nition is deduced from (3). Thus ∀n ∈ [0; 2[;
q (2 − n) Vn (r) = : 4 0 r 2−n
Un (r) =
(4)
(5)
Substituting de5nition (4) in Eq. (5), the analytic expression of the Weyl’s normalized fractional potential is deduced ∀n ∈ [0; 2[;
∀r ∈ [rmin ; rmax ];
n−2 r n−2 − rmax : Un (r) = n−2 n−2 rmin − rmax
n=3
0.8
n = 2− segment − n = 1 ponctual charge
0.7 0.6 0.5 0.4 0.3
0.1 0
∀r ∈ [rmin ; rmax ];
Vn (r) − Vn (rmax ) : Vn (rmin ) − Vn (rmax )
n=4
0.9
0.2
The fractional potential is then normalized between 0 and 1. The normalized fractional potential Un (r) must be null over a 5nite distance considered as the maximum distance of inGuence rmax . It must be equal to one at each obstacle to forbid collision. As a singularity appears when r = 0, a 5nite minimum distance rmin is chosen. The normalized fractional potential is therefore a percentage corresponding to the danger level ∀n ∈ [0; 2[;
1
(3)
Normali
Vn (r) =
2389
(6)
Weyl’s fractional potential is thus a compact expression, easy to use. No integration is required for its computation, and the potential shape only depends on the fractional integration order which corresponds to the danger of the obstacle. The charge in the expression of the fractional potential (4) no longer e;ects the normalized fractional potential (6). The curves shape of the normalized fractional potential and thus its inGuence distance, depends only on the fractional integration order n. As Fig. 1 shows it, a continuous variation of order n, allows a continuous and progressive variation of the potential curve. Whatever the distance considered, the more the fractional integration order is important, the more the potential is signi5cant. Thus, the danger zone and the danger itself will be all
0
5
10
rmin = 1
15 Distance
20
25 rmax = L = 30
Fig. 1. InGuence distance of the normalized fractional potential based on Weyl’s de5nition (1 6 n 6 5); coulombian potential created by a ponctual charge (n = 1) and a uniformly distributed charge on a segment (n = 2).
the more signi5cant around the obstacles as the risk coe4cient associated with the obstacles will be important. When n = 1, the fractional potential founded on the Weyl de5nition is V1 (r) =
q 1 : 4 0 r
(7)
This is the Coulombian potential 5eld generated by an isolated charge.
2.2. Fractional map Using (6), the distance map for each obstacle can now be converted into a fractional potential map. The sum of all the maps could de5ne the fractional potential map of the environment, but the obstacle danger level would not be normalized. For each point of the free space, the maximum value of all the potential maps is chosen, thus providing the global fractional map. So, the potential values are normalized. Also, local maximum values of potentials are avoided [6], but this leads to discontinuity of the force 5eld directions. A fractional map example is presented Figs. 2 and 3.
2390
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 2. Fractional 2D map example; the grey range indicates the danger level; the risk coe4cients are arbitrarily 5xed at 1.2 for the square, 1.5 for the chamfered rectangle and 2.5 for the circle; the maximum distance from inGuence is rmax = 20, the minimal distance from evaluation is rmin = L = 1. Fig. 4. Horizontal cross section of the fractional map; threshold of 50% authorized danger level (maximum level).
Fig. 3. Fractional map 3D example—the height is the danger level.
2.3. Fractional road
Fig. 5. Fractional road obtained by thresholding of 50% of the fractional map (white: fractional road; grey: danger zone; black: obstacles).
The operator chooses a threshold of danger level beyond which it estimates that the risk taken by the robot would be too important. A horizontal cross section of the fractional card at the given threshold (Fig. 4), provides some equipotential which separates zones where the danger is too important, from zones of security to be used. The thresholding of potential valleys of the fractional map de5nes a fractional road more or less narrow according to the obstacles dangerousness de5ning these valleys (Fig. 6). Thus, the fractional road reduced the environment in which the robot can move (Fig. 5), but it enables him to avoid collisions. It ensures the respect of a minimal security distance around the obstacles, which depends on their dangerousness degree.
Fig. 6. Between edges A and B of two obstacles, a horizontal cut of the fractional map, corresponding here to a threshold of 50%, de5nes the fractional road which width is represented by the segment C–D: (a) large fractional road allowing the passage in full safety, close to the two obstacles slightly dangerous; (b) narrow fractional road not authorizing the passage near the two dangerous obstacles; (c) fractional road allowing the passage close to the obstacle slightly dangerous and not authorizing the passage near the dangerous obstacle.
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
2391
3. A∗ Algorithm 3.1. Path planning by A∗ algorithm A∗ algorithm is built by de5ning the function of evaluation f(N ) characterizing the path cost. The evaluation function is f(N ) = g(N ) + h(N );
(8)
where g(N ) is the determinist cost, expanded to go from the initial node to the current node, and h(N ) the heuristic function which estimates the minimum-cost path between the current node and the target node. The criterion considered (distance and danger) in the determinist function of the evaluation function of the A* algorithm is thus g(N ) = g(N − 1) + d(N; N − 1)2 + pN2 ; (9) where d(N; N − 1) is the 2D displacement between two successive nodes, pN the potential in the node N and the weighting factor. The heuristic function is the Gying distance without obstacle h(N ) = d(N; Ncible ):
(10)
It veri5es well the minorance property: ∀N ∈ G;
0 6 h(N ) 6 h∗ (N ):
(11)
A∗ algorithms are always complete and admissible: the optimal solution, whenever a solution exists, is thus systematically provided. 3.2. Methodology and examples Obtaining a trajectory by A∗ algorithm requires 4 stages: (1) the environment is 5rst of all modeled by assigning to each obstacle a danger coe4cient; (2) the fractional map, indicating the danger in each point of the free environment for the robot evolution, is then evaluated; (3) the weighting parameter allowing continuity between criteria of distance and danger is then selected; (4) the trajectory is 5nally determined by a development ordered heuristically of the state graph. The trajectories minimizing criteria (9) are determinate for di;erent values of the weighting parameter.
Fig. 7. Trajectory obtained by A∗ algorithm (16 neighbors) for = 0.
The trajectories are obtained by using “Fractional Path Planning Design” module of the CRONE toolbox for Matlab. The results obtained according to the value of the weighting parameter , are given in next paragraphs. 1. Minimum length trajectory ( = 0): The trajectory obtained for a weighting factor = 0, minimizes the sum of the distances between each pixel of the trajectory. The 16 nearest neighbors of the node to be developed are considered (Fig. 7). However, the minimum length trajectory provide by the A∗ algorithm for = 0, is not the discrete approximation. The increase in the number of pixels does not necessarily increase the precision of the solution with respect to the continuous solution. Lastly, the variations with respect to the continuous solution appear irregularly and make the tracking more chaotic. The precision improvement in the trajectory obtained cannot thus be done simply by increasing pixels number, but the number of the acceptable neighbors. This increase being here directly related to the possible orientations of the robot. The increase in the precision is then limited by the size of the smallest obstacle: if the smallest obstacle has a thickness of 1 pixel, only the 4-vicinity and the 8-vicinity can be used; if the smallest obstacle has a thickness of 2 pixels, the 4, 8, 16 and 24-vicinities can be used.
2392
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 8. Trajectory obtained by A∗ algorithm (16 neighbors) for = 8.
2. Trajectory minimizing distance and danger: The trajectory obtained for a weighting factor = 8 minimizing twice distance and danger is represented (Fig. 8). Security is improved by choosing a greater distance from an obstacle, when the resulting increase of trajectory length is minimal. Thus, the trajectory always passes too close to the chamfered rectangle but deviates more quickly from this obstacle. The danger information provided by the fractional map in the criterion leads to two properties: 5rstly, taking into account the danger makes it possible to anticipate the presence of the obstacle (trajectories with continuous curve); secondly, calculations of the danger and displacement length being uncoupled, the discrimination introduced by the potential to move away the trajectories from the obstacles, improves the precision of the result compared to the discretization of the optimal continuous trajectory. The progressive and continuous increase in the weighting parameter , thus authorizes a progressive and continuous passage between the minimal distance trajectory and the minimal danger trajectory. 3. Minimum danger trajectory ( = ∞): Minimum danger trajectory obtained for a weighting factor = 1000, is presented in Fig. 9. The danger of a trajectory will be de5ned as the curvilinear integral of the potential (danger of the en-
Fig. 9. Trajectory obtained by A∗ algorithm (16 neighbors) for = 1000.
vironment) on the trajectory. The danger map towards a target will also be de5ned, providing for each point of the free space, the minimum danger of the available trajectories reaching the target. The minimum danger trajectory is also obtained by gradient descent: successive steps, following the gradient of the danger map towards the target provides the trajectory. The shortest least dangerous trajectories is retained. Curvatures obtained are not always smooth, but in this case they are smooth. Unlike the trajectory of retroaction approach [4], a minimum danger trajectory is not restricted to the minimum potential valley. The greater the trajectory length, the greater the number of potentials to be summed, so the minimum danger trajectory found is both smooth and short.
4. Fast-Marching de nition: Eikonal equation and propagation front Fast-Marching methods and Level Set methods [24,25] are numerical techniques, which solve the Eikonal equation: |∇E(M)|G(M) = 1;
(12)
|∇E(M)| = F(M):
(13)
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
These techniques consider continuous propagation of a wave front with speed G(M) (Eq. (12)). In the equivalent formulation presented in (13), F(M) is the propagation cost evaluated on the propagation front. The “energy” function E(M), solution of (12) or (13), is the curvilinear integral of the front following the faster path. It indicates in any point reached, the minimal cost to reach the target. Particularly, if F(M) is uniform, E(M) is the geodesic distance between the considered point and the target. To adapt the principle to our study, the propagation cost F(M) must be expressed according to the danger. Thus E(M) represents the minimum of the curvilinear integral between the considered point and the target, the mixed criteria between distance and danger. The discretization of the continuous solution (13) obtained with one step is the algorithm basis and ensures the precision and the result consistency. Fast-Marching technique allows fast numerical implementation by using selection or pile algorithm or heapsort. The points (or pixels) of the front to propagate are the pile. The development in tree is complete and balanced [25]. The complexity of the optimisation algorithm is O(M log M ), where M is the total number of points in the free space. We can note that this technique has a wide range of applications such as in image processing [12] and in path planning to give to the robot some distance maps, energy or time to reach the target. 4.1. Mixed criterion (distance-danger) of propagation The Fast-Marching technique is available in the “Fractional Path Planning Design” module of the CRONE toolbox. The potential is particularized considering both displacement and danger Fi; j = 1 + Pi; j ;
(14)
where Pi; j is the fractional potential which correspond a danger and a weighting factor. 4.2. Consistency by modi:ed gradient descent of the convex map obtained by Fast Marching Considering only 3 × 3 mask for the descent gradient, only eight discrete directions are o;ered to the robot. A higher mask allows to increase the number of
2393
allowed orientations. But the choice is not continuous only if the mask is in5nite. The inconsistency origin of the solution obtained is thus identi5ed. A procedure using continuous information given by the convex map obtained by Fast-Marching must be de5ned. The discrete coordinates of the closest neighbors are not su4cient to indicate an orientation, by continuous nature, with the robot. A modi5ed gradient descent is de5ned in three steps: (1) motion following the gradient; (2) memorizing of the fall point; (3) new motion starting from the fall point. The more the number of pixels increases, the more the gradient orientation obtained by Fast-Marching, has di;erent values; when the number of pixels is in5nite, a continuous orientation of the robot can be theoretically obtained. The information given by the gradient is stored by using a modi5ed gradient descent. The continuous coordinates of the fall point are memorized and are used as reference for the following descent of the gradient. 4.3. Results The minimum length trajectory is given by choosing =0; a modi5ed gradient descent is now used to obtain the minimum length. It is represented in Fig. 10. It is a true approximation of a continuous optimal trajectory. The trajectory minimizing twice the motion and the danger for a weighting factor = 5 is represented in Fig. 11. The security is increased while gradually moving away the trajectory from the obstacles according to the minimal length. Thus, the trajectory always passes to the closest to the chamfered rectangle, but deviates more quickly from this obstacle. The use of the fractional map in the criterion allows, as for A∗ algorithm, to anticipate the presence of the obstacle, which leads to smooth trajectories. The progressive and continuous increase of the weighting parameter authorizes a progressive and continuous transition between the minimal distance trajectory and the minimal danger trajectory. The trajectory obtained, for a weighting factor = 1000 is represented in Fig. 12. It does not borrow any more the potential valley, but passes round the obstacle by outside skirting the border of the environment. An angular point seems to appear for the approach by repulsion [4]. This angular
2394
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
Fig. 10. Minimum length trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 0).
Fig. 12. Optimal trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 1000).
5. Conclusion
Fig. 11. Optimal trajectory obtained by modi5ed gradient descent on the convex map given by the Fast-Marching technique, for a criterion combining length and danger ( = 5).
point results from a strategy in two stages: 5rstly, the target approach remaining in a zone located beyond the maximum distance from the obstacle inGuence; secondly, projection towards the target in the inGuence zone by the shortest path.
Fractional integral ensures a continuous passage between two integer orders of integration, and thus between the coulombian potential shapes generated by a punctual charge and the ones created by a charge uniformly distributed along a straight-line segment. The fractional order is used as a risk coe4cient to characterize the obstacle danger, and controls the potential shape smoothly. So, in path planning design, the fractional potential easily determines the least dangerous of the minimum length trajectories, or the shortest of the least dangerous trajectories. Two approaches are considered in this study: a research ordered heuristically by A∗ algorithm and the establishment of convex maps by the Fast-Marching technique. The A∗ algorithm converges systematically towards the optimal solution whenever a path exists. The weighting factor between length and danger allows smooth passage from maximum security to minimum length trajectories. Danger of obstacles is gradually taken into account to anticipate collision and also to provide smoother trajectories. However, the minimum length trajectory obtained by A∗ algorithm for = 0 is not the discrete approximation of the optimal
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
length continuous trajectory. The danger information given by the fractional map improves the precision of the result with respect to the approximation of continuous trajectory. But the method is inconsistent whatever the value of the weighting factor . The Fast-Marching technique is thus introduced and used as a consistent solution to the problem. The solutions obtained are the true approximations of the continuous and optimal trajectories. Moreover, the convex map can be transferred one for all to the robot and whatever the start point, all the trajectories can be evaluated in real time. The precision and the consistency of the technique come from solution of the Eikonal equation. Thus the level line curvature and the gradient orientation are continuous; the robot can be oriented quasi-continuously. Finally, it can deduced the continuity curvature of trajectories obtained with Fast-Marching’s technique associated to modi5ed gradient descent. A mixed criterion, adapted to the Fast-Marching’s technique, is thus de5ned. The criterion corresponds to the propagation front speed depending twice of the danger and of the displacement length. The criterion allows smooth passage from maximum security to minimum length trajectories. The application domain is not limited to mobile robotics, but can also be used in the automotive domain, to avoid collision. All the results have been obtained through simulations; so we still have to test it on a real robot. As prospects, we have also to extend the method to mobile obstacles and then to unknown obstacles.
References [1] J. Barraquand, J.C. Latombe, Robot motion planning: a distributed representation approach, Internat. J. Robotics Res. 10 (6) (December 1991) 628–649. [2] J. Borenstein, Y. Koren, Real time obstacle avoidance for fast mobile robots, IEEE Trans. System Man Cybernet. 19 (5) (September–October 1989) 1179–1187. [3] R.A. Brooks, Solving the 5nd-path problem by good representation of free space, IEEE Trans. System Man Cybernet. SMC-13 (3) (March–April 1983) 190–197. [4] C. O’Dunlaing, M. Sharir, C.K. Yap, Retraction: a new approach to motion planning, Proceedings of the 15th ACM Symposium on the Theory of Computing, Boston, USA, 1983, pp. 207–220.
2395
[5] H. Farreny, Recherche heuristiquement ordonnUee: gUenUeralisations compatibles avec la complUetude et l’admissibilitUe, Technique et Science Informatiques, RAIRO 16 (7) (September 1997) 25–953. [6] Y.K. Hwang, N. Ahuja, A potential 5eld approach to path planning, IEEE Trans. Robotics Automation 8 (1) (February 1992) 23–32. [7] O. Khatib, Commande dynamique dans l’espace opUerationnel des robots manipulateurs en prUesence d’obstacles, ThUese de Doctorat, ENSAE, Toulouse, 1980. [8] O. Khatib, Real time obstacle avoidance for manipulator and mobile robot, Internat. J. Robotics Res. 5 (1) (Spring 1986) 90–98. [9] B.H. Krogh, C.E. Thorpe, Integrated path planning and dynamic steering control for autonomous vehicles, IEEE International Conference on Robotics and Automation, San Francisco, USA, 1986, pp. 1664 –1669. [10] J.C. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Dordrecht, fourth printing, 1996. [11] J.P. Laumond, Obstacle growing in a non-polygonal world, Inform. Process. Lett. 25 (1) (1987) 41–50. [12] O. Lavialle, F. Angella, P. Baylou, An extension of the minimal path for structure recovery, Proceedings of the IEEE International Conference on Image Processing, Kobe, Japan, October, 1999. [13] N. Lefort-Piat, I. Collin, D. Meizel, Plani5cation des missions de robots mobiles rUefUerencUes par rapport aV l’environnement: tˆache-robot et cartes locales, Rev. Intell. Artif. 11 (4) (1997) 489–516. [14] H. LinarUes, ModUelisation d’environnement par dUerivation non entiVere et commande robuste d’un robot mobile, ThUese de Doctorat de l’UniversitUe Bordeaux 1, Bordeaux, 29 FUevrier 1996. [15] P. Melchior, B. Orsoni, A. Oustaloup, Weyl fractional potential in path planning, Proceedings of the Sixth IEEE ECC’2001, Porto, Portugal, 4 –7 September, 2001. [16] N.J. Nilsson, A mobile automation: an application of arti5cial intelligence techniques, Proceedings of the 1st International Joint Conference on Arti5cial Intelligence, Washington DC, USA, 1969, pp. 509 –520. [17] N.J. Nilsson, Problem-solving Methods in Arti5cial Intelligence, McGraw-Hill, New York, 1971. [18] B. Orsoni, P. Melchior, Plani5cation de trajectoire basUee sur l’utilisation d’un potentiel non-entier, Proceedings of the First IEEE CIFA’2000, Lille, France, 5 –8 Juillet 2000, pp. 819 –826. [19] A. Oustaloup, La dUerivation non entiVere: thUeorie, synthVese et applications, Editions HermVes, Paris, 1995. [20] A. Oustaloup, H. LinarVes, The CRONE path planning, Internat. IMACS J. Math. Comput. Simulation 41 (1996) 209–217. [21] A. Pruski, Robotique mobile, plani5cation de trajectoire, Editions HermVes, Paris, 1996. [22] J.H. Reif, J. Storer, Shortest paths in euclidean space with polyhedral obstacles, Technical Report CS-85-121, Computer Science Department, Brandeis University, 1985. [23] H. Rohnert, Shortest paths in the plane with convex polygonal obstacles, Inform. Process. Lett. 23 (1986) 71–76.
2396
P. Melchior et al. / Signal Processing 83 (2003) 2387 – 2396
[24] J.A. Sethian, A fast marching level set method for monotonically advancing fronts, Nat. Acad. Sci. 93 (4) (1996) 1591–1595. [25] J.A. Sethian, Fast marching methods, SIAM Rev. 41 (2) (1999) 199–235.
[26] R. Volpe, P. Khosla, Manipulator control with superquadric arti5cial potential functions: theory and experiments, IEEE Trans. System. Man Cybernet. 20 (6) (November–December 1990) 1423–1436.