- Email: [email protected]
The CRONE path planning
MATHEMATICS AND COMPUTERS IN SIMULATION
ELSEVIER
Mathematics and Computers in Simulation 41 (1996) 209-2 17
The CRONE path planning A. Oustaloup CRONE-LAP/ENSERB,
*, H. Linares
Universite’ de Bordeaux I, 351, Cows de la Lib&ration, 33405 Talence Cedex, France
Abstract
This article presents a trajectory planning strategy with a time law which, using a generalized potential, makes it possible to take into account the risk presented by each obstacle. Section 2 defines the generalized potential at a given point created by an obstacle, the risk coefficient of which is given. This potential is then used in Section 3 to obtain a trajectory for a mobile robot in an environment cluttered with known obstacles. Section 4 presents the use of this technique in an example.
1. Introduction 1.1. The various planning methods Trajectory planning can call upon various methods. The graphic methods, such as those using the graph of the geodesic lines or the Voronoi’ graph, are fast, but, when used as such the result obtained is all too often sub-optimal or impossible to implement. These methods are therefore generally used to find an initial trajectory that is subsequently improved by another algorithm. Other methods use a direct determination of an optimal trajectory, according to a predefined criterion, in a modelling of the space of the possible configurations of the robot. Mention must be made, among these methods, of those presented in [5,8,9,16]. A proper use of the two previous techniques consists of creating an artificial potential representing a repulsion with respect to the obstacles, and, eventually, an attraction for the target position. There are a large number of strategies using this potential such as [ 1,3,6,7,14], each with their own specific features. 1.2. Introduction of the notion of danger The origin of the notion of danger for a trajectory lies in military requirements. Route plans through the enemy lines are proposed in [2] and [lo], with the introduction of a risk factor into the search criterion. * Corresponding author. E-mail: [email protected]. 0378-4754/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0378-4754(95)00071-2
210
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and Computers in Simulation 41 (1996) 209-217
This notion of danger, which makes it possible to modify a trajectory, is therefore of interest in trajectory planning for conventional mobile robots. A trajectory planning strategy with a time law was thus developed which, by using a generalized potential, makes it possible to take account of the risk presented by each obstacle. This is the CRONE path planning, covered in this article. 1.3. Plan of the article Section 2 presents the generalized potential on which this planning is based. The generalized potential created by an obstacle at a given point is defined as the derivative of the order n’ (real n’ E [-2, - 11) in relation to the distance from the obstacle, of the Coulomb electrical field created by a (unitary) punctual electric charge situated on the surface of the obstacle at the point of orthogonal projection of the potential calculation point. A risk coefficient is associated with each obstacle in the environment, which is simply the order of derivation used for the calculation of the generalized potential created by that obstacle. The generalized potential of the complete environment created by all the obstacles is therefore the result of the algebraic sum of the generalized potentials of each obstacle. Section 3 presents two methods making it possible to obtain a trajectory within this generalized potential. The first method uses equipotential curves which define, in the free space, routes the width of which depends on the risk coefficients associated with the obstacles. In this route, a trajectory can therefore be determined which will observe the level of the equipotential, i.e. the risk that is accepted. The second method uses the electromechanical analogy enabling the assimilation of this potential to a speed. In this virtual speed potential, the robot may therefore select a trajectory which will minimize a simple travel time criterion. Section 4 is devoted to the study of an example illustrating the value of this trajectory planning technique.
2. Generalization
of the potential
2.1. The Coulomb potential In the literature, the potential most frequently used to create a repulsion with respect to obstacles is a Coulomb potential, i.e. a potential which decreases in l/r, where r is the distance from the obstacle. This potential is the result of the Coulomb field created by an isolated charge in a vacuum. An isolated charge in fact creates an electrical field of the following type: EC
--’ u 47FEOr2 ’
where q is the charge, EOis the vacuum permittivity, r the distance to the charge and u the radial unit vector. Given the circular symmetry, these vectors are projected on the radius from the centre of the charge. The potential can thus be written as follows: r
V=-
s
Edxz-4
00
4lT&O cc
This notion is then extended to the complete obstacles, considering each one as being isolated, and by taking as distance (r) this to the closest obstacle.
A. Oustaloup, H. Linar&/Mathematics
and Computers in Simulation 41 (1996) 209-217
211
2.2. The problem of differentiation In certain cases, it can be extremely valuable to differentiate the obstacles according to their nature. Certain obstacles can in fact be more “dangerous” than others. The notion of danger must be taken here in the widest sense of the term: it may correspond to a given criterion that the robotician wishes to apply to differentiate the obstacles. An attempt will therefore be made to construct a trajectory passing close to the danger-free obstacles but farther away from the high-risk obstacles. The notion of generalized potential stems from this idea.
2.3. From the Coulomb potential to the generalized potential The essential idea of the generalized potential is to modify the curvature of the potential resulting from an obstacle according to the associated risk. As with the Coulomb potential, this potential results from the electrical field. Nevertheless, the structure of the repulsive charge representing the obstacle will take into account the risk represented by that obstacle, leading to a modification of the appearance of the potential of the obstacle according to the risk it presents. First of all, let the repulsive charge be considered as no longer being unique and at the level of the edge of the obstacle, but forming a charge “thickness”, uniformly distributed from the surface of the obstacle to infinity. The calculation of the potential thus becomes
It can be seen that an additional integration is created when moving from a single charge to a uniformly distributed charge. To preserve continuity, the natural deduction is to consider those potentials corresponding to the integrations the order of which is between 1 and 2. The techniques developed in the CRONE team (Robust Control of a Non-Integer Order) of the University of Bordeaux I make it possible to calculate these integrations of a non-integer order. The problem therefore is to study the derivatives of the order - 1 to -2 of the function l/r2. The definition of the non-integer derivative [ 1 l] makes it possible to establish the following equation: n
,-n-1
E(r) = -___ r(-n)
u(r)*E(r).
The numeric calculation of the generalized potential, performed on a microcomputer, is based on the algorithm described in Appendix 1 of [l 11, which results from the generalization of the definition of the right-hand derivative. It is therefore now possible to calculate the potentials for the orders of derivatives between - 1 and -2. After normalization, the potentials thus obtained have the appearance illustrated in Fig. 1.
212
A. Oustaloup, H. Linar>s/Mathematics and Computers in Simulation 41 (19%) 209-217
Fig, 1. Generalized potential according to the distance from the obstacles for different risk factor values (from -2 to - 1).
2.4. The generalized potential of an environment The generalized potential of an environment is simply defined as the algebraic sum of the generalized potentials created by the various obstacles present in that environment. The distance from the object, the potential of which is calculated, must however be known at each point. This information is provided by a small simple algorithm taken from discrete geometry [4]. It is then possible, at each of these points, to calculate the generalized potential due to the obstacle as described above. The robot may go outside of the image of the known environment, as, not having any information, it will interpret the area as being obstacle-free. This must be avoided, because if the robot moves outside of the environment for which it has the necessary obstacle data, it is no longer possible to control the risks of collision. To solve this problem, a fictitious obstacle is added to the existing obstacles, in the form of a surrounding wall which follows the periphery of the image of the known environment. The robot will thus find itself in a potential “paddock” that it will no longer leave. An order of derivation of - 1 will be attributed to the surrounding wall so as not to confine the trajectories too much within the centre of the environment image.
3. A trajectory in the potential 3.1. The route It is shown in Section 2 that the generalized potential of an environment models the danger inherent to that environment according to the risk represented by each of the constituent obstacles. Thus, the equipotential curves define those areas presenting a risk below the threshold defining each equipotential. These areas are called routes. If a risk value acceptable for the robot is chosen, the potential will give the route where the robot can move while observing that risk.
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and Computers in Simulation 41 (1996) 209-217
213
In that route, an optimal trajectory will be researched in relation to a criterion that takes account of the distance covered and of the travel time. Two types of algorithm have been tested. The first is a fast algorithm which determines an initial trajectory for a later local optimization. It is based on the algorithm presented in [6]. The principle is a tracing in an arborescence of the free space which progresses into the generalized potential. The second method is based on a researching A* algorithm. This algorithm is slower, but has the advantage of making it possible to obtain a trajectory taking account of the turning circle of the mobile base. The strategy consists of progressing within the route, pixel by pixel, trying to take the shortest possible path. The turning circle constraint is simply introduced when a new pixel is observed, by checking that it is accessible to the robot. The greater the risk limitation constraint the thinner the route. There is, therefore, the danger of the robot not being able to reach its objective if the initial or final position is outside of the route, or if two obstacles too close together are positioned across the route. In such cases, the risk constraint must therefore be reduced, or the route abandoned.
3.2. Trajectory
with a time law
The potential was calculated in consideration of an electrical field and therefore of an electrostatic force. The basic principle of dynamics provides that a force is neither more nor less than the image of an acceleration (for a given mass). Moreover, according to the principle of kinematics, the integration of acceleration in relation to time gives us a speed. Thanks to electromechanical analogy, the integration of the electrical field in relation to space can be interpreted as the integration of acceleration in relation to time. Force(r) = qE(r)
+
Force(f) = my(t)
4 /dr V,,,(r)
= -
s
4 /df Edr
+
Vspeed(t) =
s
y dt
Such a vision of the potential makes it possible to consider that, at each point, it gives the opposite of the maximum speed that can be accepted by the robot, namely, for a given position in the plan, the speed of the robot is limited according to the proximity of the obstacles to that point and to the risk they present. The point being to go as fast as possible, the robot should, at each point, travel at the maximum speed authorized by the generalized potential. However, at the robot starting and target points, its speed is nil, which means that it has to accelerate to reach the speed to which the potential around it corresponds. A cone in the space of the potential thus corresponds to this acceleration. The slope of the cones depends on the acceleration and deceleration capacities of the robot. With the addition of these cones, the generalized potential is therefore the perfect representation of the travel possibilities of the robot to go from the starting point to the target point. Within this speed space, a trajectory can therefore be found, obtained by a somewhat specific trajectory determination algorithm, developed from a classic type A* determination. Once these calculations have been completed, the robot is in a two-dimensional space where the speed at each point is a datum. With safety being ensured by the use of the generalized potential, the cost criterion to
214
A. Oustaloup, H. Linarks/Mathematics
Fig. 2 (left panel). Environment.
and Computers in Simulation 41 (1996) 209-217
Figs. 3 (middle panel) and 4 (right panel) Trajectory by Coulomb potential.
be used is simply the travel time, namely the sum of the [distance/speed] the calculation points.
ratios for all the intervals between
4. Results from an example 4.1. Presentation of the environment The working environment is given in the form of a standard BitMap format image, which visualizes the various obstacles, the approach of which has a greater or lesser degree of risk (see Fig. 2). To compare the routes obtained by using a Coulomb potential (i.e. with an order of derivation equal to - 1) and a generalized potential, the planning tests are performed within the same space. Risk coefficients are attributed to the various obstacles to be avoided. 4.2. Calculation by the Coulomb potential With the Coulomb potential, all the objects have the same weight, i.e. the same risk coefficient is attributed to all the obstacles. The path obtained is therefore unique. The trajectory thus calculated to go from point A to point B is visualized in Fig. 3 for the case of a holomic robot, and in Fig. 4 for the case where the turning circle is 4 pixels. 4.3. Calculation by the generalized potential The generalized potential of the environment illustrated in Fig. 5 is the result of the sum of the potentials that correspond to each of the obstacles. A cut through this space on a horizontal plane provides a route within the meaning of the definition in Section 3. The distance away from the surrounding wall is relatively small even for a low imposed risk, since the risk coefficient (order of derivation) associated with the wall is - 1. On the other hand, for obstacle 4 in Fig. 2, the route moves away considerably when the accepted risk value is lowered, because this obstacle has a risk coefficient of -2. The trajectory obtained by the first planning algorithm, for an accepted risk of up to 50%, can be seen in Fig. 6. Fig. 7 illustrates the trajectory obtained by the A* algorithm.
A. Oustaloup, H. Linart?s/Mathematics
and Computers in Simulation 41 (1996) 209-217
215
Fig. 5. 3D display of the generalized potential.
Figs. 6 (left panel) and 7 (middle panel). Trajectory by route in the generalized potential. Fig. 8 (right panel)Trajectory in the speed space.
From these figures, it can be seen that the path has shifted from that obtained with the Coulomb potential (Figs. 3 and 4). This shift corresponds to an avoidance of obstacle 4, which has a high risk coefficient (-2). For the trajectory with a time law, the path providing the best compromise between travel time and robot safety must be determined. The generalized potential is the same as in the previous case, but with the addition of acceleration and deceleration cones. The trajectory calculated by the A* algorithm for the same risk coefficient values is illustrated in Fig. 8.
5. Conclusion This article deals with a new application of the non-integer derivative in robotics [ 121. Indeed, after the edge detection [ 131, the non-integer derivative is here applied to a trajectory planning technique. This technique uses the generalized potential which is the non-integer integration of the electrical field created by a coulombian charge. It allows to obtain the optimal safe trajectory for a robot moving in a known obstacle-
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and Computers in Simulation 41 (1996) 209-217
strewn environment, by taking into account the various risk coefficients of all the obstacles present in this environment. The results obtained show the efficiency of this strategy. Other works, such as non-integer motion generation or fractal landscape generation, are in process of development.
6. Glossary [15] Danger Risk Safety
State or situation with a potential of damage to people, the robot or the environment. Measurement of the level of the danger combining a measurement of the occurrence of an undesirable event with a measurement of its consequences Absence of circumstances likely to cause either human death or the degradation or loss of equipment or property.
A situation where there is no danger to fear, in particular following mastery of the associated risks (elimination, reduction or control). Environment The entire known area in which the robot moves, with its obstacles. Space Specific modelling of the environment. Free path Trajectory ensuring the absence of collisions with the obstacles. Route Area of the space in which the risk is below a given limit. Surrounding wall An obstacle added artificially to the environment to prevent the robot from leaving the known environment. References [l] J. Barraquand, B. Langlois and J.-C. Latombe, Robot motion planning with many degrees of freedom and dynamic contraints, Proc. Int. Symp. Robotics Research (Tokyo), August 1989, 74-83. [2] P.-L. Bergier, Planification de missions etcontre-mobilid: deux utilisations d’algorithmes de larobotique mobile, Congr& EC2 Avignon 1991 (Les syst. experts et leurs appli.), 101-108. [3] J. Borenstein and Y. Koren, Real-time obstacle avoidance for fast mobile robots, IEEE Trans. Systems Man Cybemet. 19(5) (1989) 1179-1187. [4] J.-M. Chassery and A. Montanvert, GComCtrie disc&e en analyse d’images, Trait6 des nouvelles technologies Hermes, 1991. [5] Th. Fraichard, C. Laugier and G. LiCvin, Robot motion planning: the case of non-holonomic mobiles in a dynamic world, IEEE Int. Workshop on Intel]. Robots and Syst. IROS 1990. [6] Y.K. Hwang and N. Ahuja, A potential field approach to path planning, IEEE Trans. Robot. Autom. 8( 1) (1992) 23-32. [7] B.H. Krogh and C.E. Thorpe, Integrated path planning and dynamic steering control for autonomous vehicles, IEEE Int. Conf. on Robot. and Autom., San Francisco (1986) 1664-1668. [8] J.P. Laumond, L.A.A.S.-C.N.R.S., Finding collision-free smooth trajectories for a non-holomic mobile robot, Proc. Int. Joint Conf. on Artificial Intelligence, August 1987. [9] J.P. Launiond, T. Simeon, R. Chatila and G. Giralt, L.A.A.S.X.N.R.S., Trajectory planning and motion control for mobile robots, IUTAM/IFAC Symp. Dynamics and Controlled Mechanical Systems, Ziirich (June 1988). [lo] T.R. MacMillan, C.E. Gerber, J.M. Sackett and PD. Holden, Knowledge based route planning, IEEE, NAECON 1990 Dayton Convention Center, May 1990, 1001-1007. [ 1 l] A. Oustaloup, La commande CRONE, Trait6 des nouvelles technologies Hermes, 1991. [ 121 A. Oustaloup, La derivation non entibre, Traid des nouvelles technologies Herr&s, 1995.
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and Computers in Simulation 41 (1996) 209-217
211
[ 131 A. Oustaloup, B. Mathieu and I? Melchior, Edge extraction based on non-integer derivation, ECCTD-91; IEEE, Copenhague, Danemark, 3-6 Septembre 1991 [ 141 E. Rimon and D.E. Koditschek, Exact robot navigation using artificial potential functions, IEEE Trans. Robot. Autom. 8(5) (1992) 501-518. [ 151 Universite Bordeaux I/IUT “A” & CEA/INSTN Grenoble, Assises intemationales des formations universitaires et avancees dans le domaine des sciences et techniques du danger, Actes INFORISK-Salon du management des risques Bordeaux Lac, Jan 1993. [16] D. Zhu and J.-C. Latombe, Robotics Laboratoy, Stanford University, Constraint reformulation in a hierarchical path planner, IEEE Int. Conf. on Robot. and Autom. Vol. 3, May 1990, 1918-1923.
ELSEVIER
Mathematics and Computers in Simulation 41 (1996) 209-2 17
The CRONE path planning A. Oustaloup CRONE-LAP/ENSERB,
*, H. Linares
Universite’ de Bordeaux I, 351, Cows de la Lib&ration, 33405 Talence Cedex, France
Abstract
This article presents a trajectory planning strategy with a time law which, using a generalized potential, makes it possible to take into account the risk presented by each obstacle. Section 2 defines the generalized potential at a given point created by an obstacle, the risk coefficient of which is given. This potential is then used in Section 3 to obtain a trajectory for a mobile robot in an environment cluttered with known obstacles. Section 4 presents the use of this technique in an example.
1. Introduction 1.1. The various planning methods Trajectory planning can call upon various methods. The graphic methods, such as those using the graph of the geodesic lines or the Voronoi’ graph, are fast, but, when used as such the result obtained is all too often sub-optimal or impossible to implement. These methods are therefore generally used to find an initial trajectory that is subsequently improved by another algorithm. Other methods use a direct determination of an optimal trajectory, according to a predefined criterion, in a modelling of the space of the possible configurations of the robot. Mention must be made, among these methods, of those presented in [5,8,9,16]. A proper use of the two previous techniques consists of creating an artificial potential representing a repulsion with respect to the obstacles, and, eventually, an attraction for the target position. There are a large number of strategies using this potential such as [ 1,3,6,7,14], each with their own specific features. 1.2. Introduction of the notion of danger The origin of the notion of danger for a trajectory lies in military requirements. Route plans through the enemy lines are proposed in [2] and [lo], with the introduction of a risk factor into the search criterion. * Corresponding author. E-mail: [email protected]. 0378-4754/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0378-4754(95)00071-2
210
A. Oustaloup, H. Linar&/Mathematics
and Computers in Simulation 41 (1996) 209-217
This notion of danger, which makes it possible to modify a trajectory, is therefore of interest in trajectory planning for conventional mobile robots. A trajectory planning strategy with a time law was thus developed which, by using a generalized potential, makes it possible to take account of the risk presented by each obstacle. This is the CRONE path planning, covered in this article. 1.3. Plan of the article Section 2 presents the generalized potential on which this planning is based. The generalized potential created by an obstacle at a given point is defined as the derivative of the order n’ (real n’ E [-2, - 11) in relation to the distance from the obstacle, of the Coulomb electrical field created by a (unitary) punctual electric charge situated on the surface of the obstacle at the point of orthogonal projection of the potential calculation point. A risk coefficient is associated with each obstacle in the environment, which is simply the order of derivation used for the calculation of the generalized potential created by that obstacle. The generalized potential of the complete environment created by all the obstacles is therefore the result of the algebraic sum of the generalized potentials of each obstacle. Section 3 presents two methods making it possible to obtain a trajectory within this generalized potential. The first method uses equipotential curves which define, in the free space, routes the width of which depends on the risk coefficients associated with the obstacles. In this route, a trajectory can therefore be determined which will observe the level of the equipotential, i.e. the risk that is accepted. The second method uses the electromechanical analogy enabling the assimilation of this potential to a speed. In this virtual speed potential, the robot may therefore select a trajectory which will minimize a simple travel time criterion. Section 4 is devoted to the study of an example illustrating the value of this trajectory planning technique.
2. Generalization
of the potential
2.1. The Coulomb potential In the literature, the potential most frequently used to create a repulsion with respect to obstacles is a Coulomb potential, i.e. a potential which decreases in l/r, where r is the distance from the obstacle. This potential is the result of the Coulomb field created by an isolated charge in a vacuum. An isolated charge in fact creates an electrical field of the following type: EC
--’ u 47FEOr2 ’
where q is the charge, EOis the vacuum permittivity, r the distance to the charge and u the radial unit vector. Given the circular symmetry, these vectors are projected on the radius from the centre of the charge. The potential can thus be written as follows: r
V=-
s
Edxz-4
00
4lT&O cc
This notion is then extended to the complete obstacles, considering each one as being isolated, and by taking as distance (r) this to the closest obstacle.
A. Oustaloup, H. Linar&/Mathematics
and Computers in Simulation 41 (1996) 209-217
211
2.2. The problem of differentiation In certain cases, it can be extremely valuable to differentiate the obstacles according to their nature. Certain obstacles can in fact be more “dangerous” than others. The notion of danger must be taken here in the widest sense of the term: it may correspond to a given criterion that the robotician wishes to apply to differentiate the obstacles. An attempt will therefore be made to construct a trajectory passing close to the danger-free obstacles but farther away from the high-risk obstacles. The notion of generalized potential stems from this idea.
2.3. From the Coulomb potential to the generalized potential The essential idea of the generalized potential is to modify the curvature of the potential resulting from an obstacle according to the associated risk. As with the Coulomb potential, this potential results from the electrical field. Nevertheless, the structure of the repulsive charge representing the obstacle will take into account the risk represented by that obstacle, leading to a modification of the appearance of the potential of the obstacle according to the risk it presents. First of all, let the repulsive charge be considered as no longer being unique and at the level of the edge of the obstacle, but forming a charge “thickness”, uniformly distributed from the surface of the obstacle to infinity. The calculation of the potential thus becomes
It can be seen that an additional integration is created when moving from a single charge to a uniformly distributed charge. To preserve continuity, the natural deduction is to consider those potentials corresponding to the integrations the order of which is between 1 and 2. The techniques developed in the CRONE team (Robust Control of a Non-Integer Order) of the University of Bordeaux I make it possible to calculate these integrations of a non-integer order. The problem therefore is to study the derivatives of the order - 1 to -2 of the function l/r2. The definition of the non-integer derivative [ 1 l] makes it possible to establish the following equation: n
,-n-1
E(r) = -___ r(-n)
u(r)*E(r).
The numeric calculation of the generalized potential, performed on a microcomputer, is based on the algorithm described in Appendix 1 of [l 11, which results from the generalization of the definition of the right-hand derivative. It is therefore now possible to calculate the potentials for the orders of derivatives between - 1 and -2. After normalization, the potentials thus obtained have the appearance illustrated in Fig. 1.
212
A. Oustaloup, H. Linar>s/Mathematics and Computers in Simulation 41 (19%) 209-217
Fig, 1. Generalized potential according to the distance from the obstacles for different risk factor values (from -2 to - 1).
2.4. The generalized potential of an environment The generalized potential of an environment is simply defined as the algebraic sum of the generalized potentials created by the various obstacles present in that environment. The distance from the object, the potential of which is calculated, must however be known at each point. This information is provided by a small simple algorithm taken from discrete geometry [4]. It is then possible, at each of these points, to calculate the generalized potential due to the obstacle as described above. The robot may go outside of the image of the known environment, as, not having any information, it will interpret the area as being obstacle-free. This must be avoided, because if the robot moves outside of the environment for which it has the necessary obstacle data, it is no longer possible to control the risks of collision. To solve this problem, a fictitious obstacle is added to the existing obstacles, in the form of a surrounding wall which follows the periphery of the image of the known environment. The robot will thus find itself in a potential “paddock” that it will no longer leave. An order of derivation of - 1 will be attributed to the surrounding wall so as not to confine the trajectories too much within the centre of the environment image.
3. A trajectory in the potential 3.1. The route It is shown in Section 2 that the generalized potential of an environment models the danger inherent to that environment according to the risk represented by each of the constituent obstacles. Thus, the equipotential curves define those areas presenting a risk below the threshold defining each equipotential. These areas are called routes. If a risk value acceptable for the robot is chosen, the potential will give the route where the robot can move while observing that risk.
A. Oustaloup, H. Linar&/Mathematics
and Computers in Simulation 41 (1996) 209-217
213
In that route, an optimal trajectory will be researched in relation to a criterion that takes account of the distance covered and of the travel time. Two types of algorithm have been tested. The first is a fast algorithm which determines an initial trajectory for a later local optimization. It is based on the algorithm presented in [6]. The principle is a tracing in an arborescence of the free space which progresses into the generalized potential. The second method is based on a researching A* algorithm. This algorithm is slower, but has the advantage of making it possible to obtain a trajectory taking account of the turning circle of the mobile base. The strategy consists of progressing within the route, pixel by pixel, trying to take the shortest possible path. The turning circle constraint is simply introduced when a new pixel is observed, by checking that it is accessible to the robot. The greater the risk limitation constraint the thinner the route. There is, therefore, the danger of the robot not being able to reach its objective if the initial or final position is outside of the route, or if two obstacles too close together are positioned across the route. In such cases, the risk constraint must therefore be reduced, or the route abandoned.
3.2. Trajectory
with a time law
The potential was calculated in consideration of an electrical field and therefore of an electrostatic force. The basic principle of dynamics provides that a force is neither more nor less than the image of an acceleration (for a given mass). Moreover, according to the principle of kinematics, the integration of acceleration in relation to time gives us a speed. Thanks to electromechanical analogy, the integration of the electrical field in relation to space can be interpreted as the integration of acceleration in relation to time. Force(r) = qE(r)
+
Force(f) = my(t)
4 /dr V,,,(r)
= -
s
4 /df Edr
+
Vspeed(t) =
s
y dt
Such a vision of the potential makes it possible to consider that, at each point, it gives the opposite of the maximum speed that can be accepted by the robot, namely, for a given position in the plan, the speed of the robot is limited according to the proximity of the obstacles to that point and to the risk they present. The point being to go as fast as possible, the robot should, at each point, travel at the maximum speed authorized by the generalized potential. However, at the robot starting and target points, its speed is nil, which means that it has to accelerate to reach the speed to which the potential around it corresponds. A cone in the space of the potential thus corresponds to this acceleration. The slope of the cones depends on the acceleration and deceleration capacities of the robot. With the addition of these cones, the generalized potential is therefore the perfect representation of the travel possibilities of the robot to go from the starting point to the target point. Within this speed space, a trajectory can therefore be found, obtained by a somewhat specific trajectory determination algorithm, developed from a classic type A* determination. Once these calculations have been completed, the robot is in a two-dimensional space where the speed at each point is a datum. With safety being ensured by the use of the generalized potential, the cost criterion to
214
A. Oustaloup, H. Linarks/Mathematics
Fig. 2 (left panel). Environment.
and Computers in Simulation 41 (1996) 209-217
Figs. 3 (middle panel) and 4 (right panel) Trajectory by Coulomb potential.
be used is simply the travel time, namely the sum of the [distance/speed] the calculation points.
ratios for all the intervals between
4. Results from an example 4.1. Presentation of the environment The working environment is given in the form of a standard BitMap format image, which visualizes the various obstacles, the approach of which has a greater or lesser degree of risk (see Fig. 2). To compare the routes obtained by using a Coulomb potential (i.e. with an order of derivation equal to - 1) and a generalized potential, the planning tests are performed within the same space. Risk coefficients are attributed to the various obstacles to be avoided. 4.2. Calculation by the Coulomb potential With the Coulomb potential, all the objects have the same weight, i.e. the same risk coefficient is attributed to all the obstacles. The path obtained is therefore unique. The trajectory thus calculated to go from point A to point B is visualized in Fig. 3 for the case of a holomic robot, and in Fig. 4 for the case where the turning circle is 4 pixels. 4.3. Calculation by the generalized potential The generalized potential of the environment illustrated in Fig. 5 is the result of the sum of the potentials that correspond to each of the obstacles. A cut through this space on a horizontal plane provides a route within the meaning of the definition in Section 3. The distance away from the surrounding wall is relatively small even for a low imposed risk, since the risk coefficient (order of derivation) associated with the wall is - 1. On the other hand, for obstacle 4 in Fig. 2, the route moves away considerably when the accepted risk value is lowered, because this obstacle has a risk coefficient of -2. The trajectory obtained by the first planning algorithm, for an accepted risk of up to 50%, can be seen in Fig. 6. Fig. 7 illustrates the trajectory obtained by the A* algorithm.
A. Oustaloup, H. Linart?s/Mathematics
and Computers in Simulation 41 (1996) 209-217
215
Fig. 5. 3D display of the generalized potential.
Figs. 6 (left panel) and 7 (middle panel). Trajectory by route in the generalized potential. Fig. 8 (right panel)Trajectory in the speed space.
From these figures, it can be seen that the path has shifted from that obtained with the Coulomb potential (Figs. 3 and 4). This shift corresponds to an avoidance of obstacle 4, which has a high risk coefficient (-2). For the trajectory with a time law, the path providing the best compromise between travel time and robot safety must be determined. The generalized potential is the same as in the previous case, but with the addition of acceleration and deceleration cones. The trajectory calculated by the A* algorithm for the same risk coefficient values is illustrated in Fig. 8.
5. Conclusion This article deals with a new application of the non-integer derivative in robotics [ 121. Indeed, after the edge detection [ 131, the non-integer derivative is here applied to a trajectory planning technique. This technique uses the generalized potential which is the non-integer integration of the electrical field created by a coulombian charge. It allows to obtain the optimal safe trajectory for a robot moving in a known obstacle-
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strewn environment, by taking into account the various risk coefficients of all the obstacles present in this environment. The results obtained show the efficiency of this strategy. Other works, such as non-integer motion generation or fractal landscape generation, are in process of development.
6. Glossary [15] Danger Risk Safety
State or situation with a potential of damage to people, the robot or the environment. Measurement of the level of the danger combining a measurement of the occurrence of an undesirable event with a measurement of its consequences Absence of circumstances likely to cause either human death or the degradation or loss of equipment or property.
A situation where there is no danger to fear, in particular following mastery of the associated risks (elimination, reduction or control). Environment The entire known area in which the robot moves, with its obstacles. Space Specific modelling of the environment. Free path Trajectory ensuring the absence of collisions with the obstacles. Route Area of the space in which the risk is below a given limit. Surrounding wall An obstacle added artificially to the environment to prevent the robot from leaving the known environment. References [l] J. Barraquand, B. Langlois and J.-C. Latombe, Robot motion planning with many degrees of freedom and dynamic contraints, Proc. Int. Symp. Robotics Research (Tokyo), August 1989, 74-83. [2] P.-L. Bergier, Planification de missions etcontre-mobilid: deux utilisations d’algorithmes de larobotique mobile, Congr& EC2 Avignon 1991 (Les syst. experts et leurs appli.), 101-108. [3] J. Borenstein and Y. Koren, Real-time obstacle avoidance for fast mobile robots, IEEE Trans. Systems Man Cybemet. 19(5) (1989) 1179-1187. [4] J.-M. Chassery and A. Montanvert, GComCtrie disc&e en analyse d’images, Trait6 des nouvelles technologies Hermes, 1991. [5] Th. Fraichard, C. Laugier and G. LiCvin, Robot motion planning: the case of non-holonomic mobiles in a dynamic world, IEEE Int. Workshop on Intel]. Robots and Syst. IROS 1990. [6] Y.K. Hwang and N. Ahuja, A potential field approach to path planning, IEEE Trans. Robot. Autom. 8( 1) (1992) 23-32. [7] B.H. Krogh and C.E. Thorpe, Integrated path planning and dynamic steering control for autonomous vehicles, IEEE Int. Conf. on Robot. and Autom., San Francisco (1986) 1664-1668. [8] J.P. Laumond, L.A.A.S.-C.N.R.S., Finding collision-free smooth trajectories for a non-holomic mobile robot, Proc. Int. Joint Conf. on Artificial Intelligence, August 1987. [9] J.P. Launiond, T. Simeon, R. Chatila and G. Giralt, L.A.A.S.X.N.R.S., Trajectory planning and motion control for mobile robots, IUTAM/IFAC Symp. Dynamics and Controlled Mechanical Systems, Ziirich (June 1988). [lo] T.R. MacMillan, C.E. Gerber, J.M. Sackett and PD. Holden, Knowledge based route planning, IEEE, NAECON 1990 Dayton Convention Center, May 1990, 1001-1007. [ 1 l] A. Oustaloup, La commande CRONE, Trait6 des nouvelles technologies Hermes, 1991. [ 121 A. Oustaloup, La derivation non entibre, Traid des nouvelles technologies Herr&s, 1995.
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[ 131 A. Oustaloup, B. Mathieu and I? Melchior, Edge extraction based on non-integer derivation, ECCTD-91; IEEE, Copenhague, Danemark, 3-6 Septembre 1991 [ 141 E. Rimon and D.E. Koditschek, Exact robot navigation using artificial potential functions, IEEE Trans. Robot. Autom. 8(5) (1992) 501-518. [ 151 Universite Bordeaux I/IUT “A” & CEA/INSTN Grenoble, Assises intemationales des formations universitaires et avancees dans le domaine des sciences et techniques du danger, Actes INFORISK-Salon du management des risques Bordeaux Lac, Jan 1993. [16] D. Zhu and J.-C. Latombe, Robotics Laboratoy, Stanford University, Constraint reformulation in a hierarchical path planner, IEEE Int. Conf. on Robot. and Autom. Vol. 3, May 1990, 1918-1923.