- Email: [email protected]
A Theoretical Approach to Potential Fields within the Configuration Space Using Green's Functions
Copyright @ IFAC Mobile Robot Technology, Jejudo Island, Korea, 2001
A THEORETICAL APPROACH TO POTENTIAL FIELDS WITHIN THE CONFIGURATION SPACE USING GREEN'S FUNCTIONS B. Curto, V. Moreno, F. J. Blanco'
• Computer Science Department, University of Salamanca ,Spain. E-mail: [email protected]
Abstract: In this paper a new procedure to obtain directly the repulsive potential surface at the Configuration Space (C-space) due to the presence of obstacles within the robot environment is presented. This method is derived from a theoretical approach that has been succesfully applied to evaluation of obstacle representation within the C-space for mobile or articulated robots. It will be shown that it it is possible to find a new theoretical interpretation that unifies this obstacle representation (C-obstacle) with the repulsive potential fields that can be considered at the C-space. Main advantage of the method comes from the fact that the computation of the C-obstacles is not needed neither the subsequently potential evaluation over the computed C-space, i.e the resulting potential field is obtained in a unique computation step. In this way, the potential evaluation can be done without any aditional computational cost respect to the C-obstacle evaluation. As the results are based on a general formalism, it can be applied to any kind of robot (links shape, kinematics , ...); hence, it constitutes an important result that may be used with path-planning techniques that employ potential fields and within any other robotics-related studies. Copyright @20011FAC Keywords: Potential fields , Green's functions, Configuration Space, Path planning
1. INTRODUCTION
force field , and these forces are constituted by two main components: an attractive one that leads the robot to goal configuration and a repulsive one that pushes away from the obstacles. If these forces are applied to the robot actuators, it moves towards its goal configuration without colliding with any obstaCles, so it is not necesary to compute any path in the configuration space (C-space).
Potential field techniques have been used in order to prm'ide qualitative and quantitative information about robotic environments. Hence , they are related to those approaches that prosecute a global goal, an intelligent robotic system , and more precisely with path planning and reactive control capabilities. These techiques , which are broadly used at other scientific fields , were introduced in robotics research by (Khatib, 1986) which , initially, proposed its implementation with the aim of avoiding obstacles in real time. The main idea behind this approach is that the robot is moving within a
Also, potential fields information can be used to increase the low-level control quality. Instead of a simple control system following a previously planned trajectory, obstacle information can be taken into account within a control strategy in terms of a generated repulsive potential, as in (Blanco et al. , 1998). By using these strategies, obstacle avoidance can be improved and it must
1 This work has be done with the support of the Junta de Castilla y Len SA02-00F.
13
From this representation the potential field at the C-space is obtained but the local minima in existence are not avoided, so it is necessary to include powerful tools to escape from them.
be included in a direct way in the low-level control loop. The force that supports the robot can be specified by F(q) = -'VV(q) where V(q) is a non-negative escalar function defined at the robot C-space. A possible interpretation is that the function V (q) determines the potential energy for the robot (which in C-space is considered as a point) within a configuration; this fact determines the technique denomination.
So, as it can be seen, the examined works make reference to path planning techniques that use potential fields where the main aim is avoiding the local minima. But, they does not take into account the high computational cost of the potential surface evaluation at the at the C-space. The approach proposed by (Blanco et al. , 1998) is rather faster because the potential field at the C-space is computed directly from the C-obstacles representation. Simultaneously, the roadmap at the C-space is obtained, so the search problem is simplified. At the path planning stage, search algorithms are applied in order to guide the search graph , and potential fields are used at their heuristic definition to avoid narrow regions of the Cspace. In (Mantegh et al ., 1997) harmonic functions are used to guarantee that there exist no local minima, so that this goal is reachable.
One advantage , that is directly deduced from the specification of the force as the potential gradient, becomes from the fact the system behaviour can be predicted without an exact knowledge of the robot movements sequence. More precisely, if Cspace is restricted and the potential function is time invariant , it can be shown (Khatib , 1986) that the robot will move to a local potential minimun. An important property is the additive character for these functions . So, several potential sources can be considered independently in order to provide for the robot 's different capabilities. If the sum of several potentials is used , the robot 's behavior will be a combination of the desired capabilities.
But at all these works , the potential approach leads to a very high computational cost , due to every tries need several stages. At (Barraquand and Latombe, 1991) (Kavraki, 1994), first the potential evaluation at workspace is performed and next its representation at the C-space is obtained. And at (Blanco et al., 1998) the C-space evaluation is followed by a potential surface computation. Also, there is not a solid mathematical base that leads us to obtain the potential surface and its interpretation.
Nevertheless , if some potentials with opposing aims are added , the robot can fall into a local minimum. These ones prevent the potential field method from directly solving the problem of collision-free robotic movement. Since the method lets us guide the robot towards a desired configuration with minimal computational cost as compared with a free space representation, there is great interest on its adaptation to the path-planning research field. The main difficulty appears when the robot follows a direction opposite t.o gradient and it arrives to a local minimum that is not the goal configuration. This problem can be solved if special potential functions are used, called navigation junctions, that do not present local minima and do have a minimum potential at goal configuration. Analytical navigation functions have been proposed (Koditscheck , 1987), although they are used only in very unique environments. In some papers, it is possible to find numeric navigation functions (Barraquand and Latombe, 1991) defined over discrete representations of the configuration space. When a discrete configuration space is used as a basis , in order to avoid the local minimum on potential surface, random movements have been applied , see (Barraquand and Latombe, 1991 ) (Kavraki, 1994). In other works , other path planning techniques are used together with potential fields , as in (Barraquand and Latombe , 1991) , where a potential field without local minima at the workspace is computed by using a roadmap.
In order to solve these limitations , at this work we are going to present a mathematical formalism that can be used to obtain a potential representation at C-space directly from the workspace representation and the robot . Its based on another previous formalism that has been applied successfully to compute representation of C-obstacles for various kinds of robots , both mobile and articulated, (Curto and Moreno , 1997). Its advantages can be traced back to the highly optimized procedure that is obtained. In fact, it has been used with a mobile robot in a changing environment (Blanco et al., 1998) , which shows its powerful capabilities. However, as we shall show , we propose a new interpretation (using the Green's functions ) for our previous formalism from which we can find a procedure to obtain potential surfaces at the Cspace in a quite direct performing an only step. So in this way, the evaluation of the C-obstacles and the potential surfaces that they generate can be seen as two applications of only formalism with two definitions of the potential functions. In the following the article is organized as follows: in section II the needed theoretical bases that will
14
the robot A at a given configuration q collides with the obstacles, it is necessary to evaluate CB(q) (1) and, furthermore, the value of A(q,x) . In order to propose an example we can consider a simple robot, a rigid object with circular shape that moves freely in R2. So, we have that a configuration q is parameterized by q = (x r , Yr) where Xr and Yr are the robot center coordinates if the cartesian coordinates functions (x , y) are used at the robot workspace. In this case, equation can be written as
be used in the following are presented. At section Ill, we will explain the interpretation that is the main purpose of this article. We make use of a well known theory, that is the Green's function. In order to show its applicability we consider a mobile robot as case of study in section IV, where several surfaces are obtained that can be seen as the main result of this research work. Finally, in section V we will present the main conclusions.
2. THEORETICAL BASIS
I if (x , y) E A(x y ) A( Xr,yr,x , y ) = { Oif(x )dA r,r , Y l" (xr,Yr )
In this section , we are going to present the formalism that has been applied i~ order to obtain the representation of obstacles in C-space . As we shall see , the formalism is based on the definition of a nev,,' function , CB , that allows us to evaluate the C-obstacles. Next , we will show an application of the formalism, where it can observed how the computation can be highly optimized. Afterwards, the basis of theory of Green functions will be briefly presented. With these basics we will able to propose an interpretation of the exposed formalism in order to obtain a procedure to compute the potential fields within the C-space.
CB(xr , Yr)
=
J
A'(O 0 x' ') , , ,Y
~
CB(q )
{ 1 if x E B 0 if x rt B
J
J
A (q. x )B(x)dx
Vq E C
0 si (x' , y' )
E A(o,o) rl l"
A
(4)
(0 ,0)
A (o,o )(x r - x , yr - y)B(x , y)dxdy
(5 )
So , the CR function can be evaluated considering the function that describes the robot at a single configuracion (0 , 0). This is a key element to consider in order to reduce the computational load . Moreover, the evaluation can be done if the convolution theorem is taken into account ; if this theorem is applied to (5) then
R be the function defined by
=
= { 1 si (x', y')
being (x', y' ) = (x - x r , Y - Yr) and A (o.o) is the subset of points that represent the robot at configuration q = (0,0). Now if we consider the function A(o ,o)(x , y) = A'(O , O, -x , -y) clearly, CB(x r , Yr) can be calculated with (3) as
where A(q ) is the set of points of W that represents the robot at configuration q , and B is the subset of vF constituted by the obstacles. Let CB: C
A(xr , Yr , x , y)B(x , y)dxdy (3)
As it has been stated in (Curto and Moreno , 1997) , the evaluation can be strongly simplified in such a way that a new function A' defined over C' x VV ' into R is used and it is independent of some configuration parameters. In particular it can be determined that A(xr , Yr,x , y ) = A'(O,O , X - xr , y - Yr) where
In the following , W will designate the set of points at the workspace and C will be the set of configurations for the robot. Let A : C x 11" ~ {O , I} and B : HT ~ {O , I} be the function s defined by
=
2
So the expression for the C-obstacles (1) would be
2.1 CB definition
I if x E A(q ) A (q, x ) = { 0 if x rt A(q ) B(x)
()
(1)
\Ve define CB f as the subset of C that verifies CBr = {q E C j CB(q ) > O}. it is possible to prove the following theorem Theorem 1. Let CB = {q E C jA,(q ) n B i 0} be the definition of the C-obstacles region proposed by (LozanoPerez , 1983). Then it follows that CB = CBr .
In this particular case, the C-obstacles evaluation can be achieved by performing the pointwise multiplication of the Fourier transform of the function that represents the robot at the configuration (0, 0) and the Fourier transform of the function that represents the obstacles in the robot workspace. In order to evaluate the previous expression in a computer the A and B functions have to be discretized , obtaining two bitmaps as a result.
From the previous result, it is straightforward to see that q E Cjree +-t CB(q ) = 0, where CIree is the subset of C that corresponds with those configurations in which the robot does not intersect any obstacle. So , in order to know whether 15
It is significant, in this case, that the function represents the potential generated by a point charge placed at r'. By considering the previous expression (8) it becomes
2.2 Green's functions
The previous definition (1) can be seen as an integral expression that through Green's functions can be related to a differential equation. These kinds of relationships are used typically when a physical formalism , such as the potential theory, is developed.
q;(r)
=
J
G(x, x')f(x')dx'
= <5(x -
x')
=
r
where Q represents the charge that. produces the generated potential field
(6)
3. INTERPRETATION OF CB DEFINITIOI\' In this section , we will present a new approach to understand the proposed mathematical formalism. If the well known theory of Green 's functions is used then a new interpretation for the proposed formalism can be found. In this way, the function A (q, x) takes a special meaning that will lead to a definition of a procedure to obtain artificial potential surfaces at the C-space in a fairly optimized way.
(7)
J
G(x , x')f(x')dx'
It can be observed that the problem to find Green 's function of an operator L is equivalent to invert L. In this way, equations as Ly = f can be solved as y
= L- )f = K f =
(12)
Q
The expression can be also written as LK = I where I has the same property as the Dimc delta function <5 and K is the integral operator which is defined by Kf
p(r') d3 r' r - r' I
cjJ(r) -+ -
where C(x, x') is Green 's function related to the differential operator L. If this one is applied to G(x, :r') it can be shown that
LG(x, x')
JI
If a serial Taylor's expansion is performed and if I r I is large enough then
Let the equation Ly = f where L is some linear differential operator and f is a given function. In terms of Green's functions the solution for this equation will be
y(x)
=
The Green's function theory can be applied to get an interpretation, from a different point of view , for the proposed expression (1) used to compute the obstacle representation at the C-space.
J
G (x, x')f(x')dx'
CB(q ) =
J
.4(q , x)B(x)dx
If it can be found a function G that satisfies (7) and, if the solutions c(x ) of the homogeneous equation Lq; = 0 are known , then the general solution of Lq; = F(x) can be written as
By comparing this expression with (8) , i.e. the solution for the differential equation, there must exist a differential operator L in such a way that
J
L[CB(q)] = B(x)
q;(x)
= dx ) +
G(x, x')F(x')dx'
where the related Green's function G would be A(q , x) , <:nd it must comply with
A quite interesting application of these functions is found in Electromagnetic theory, which considers attractive / repulsive potentials, A differential equation (Poisson equation ) allows us to find the spacial electromagnetic potential dJ(r ) due to a charge distribution p( r )
v 2 d>(r ) = -4r.p (r )
L[A (q,x)]
By using th e Fourier transform method, it is posible to solve the equation
= _ 2.. I 47i r
1,
- r
(10 )
I
(14 )
We want to relate an integral expression (1) with a differential equation in order to obtain an interpretation from a physical point of view for the C-obstacles. As it is well known , differential equations are broadly used in different research areas in order to model the system 's behavior, i.e. , a body that is moving under a force/potential field.
and the corresponding Green's function will be G (1'. r')
= <5(q , x)
The goal is not to find explicitly the differential operator L that satisfies (14), since the operator will depend on the function A( q, x) that, as it has been exposed, defines the robot at a configuration for each workspace point.
(9)
\,2G (r, r' ) = <5(r - r' )
(13)
(8 )
(11 )
16
Once we have established the relationship between Green 's functions and C-obstacles, it is possible to find the analogy with some physical magnitudes. From the equation (13), we can see the similarity with Poisson equation (\7 2 q\(r) = -471'p(r)) where the electric potential q\ is defined for a distribution of charges p.
considered in such a way that the function A(q, x) will be given by
( , Yr,x,y ) = { Vif(x,Y)EA(xY)(16) AXr 0 if(x )dA r , r ,Y '" (Xr,Yr) where A(xr,Yr) is the subset of the workspace points that represents the robot at the configuration (xr,Yr) and V is a constant. If 11 = 1 the proposed situation is the same (2) as the case in which the goal is the obtention of the obstacles representation at the C-space for this circular mobile robot .
So, in a general way, it can be interpreted that CB corresponds with a distribution of artificial potential and that the obstacle B (x) is the cause that originates this distribution. Also, if the expression (14) is considered , the robot A(q , x ) will be the artificial potential related with a point potential source, i.e, a point obstacle.
If we consider the obstacles of the workspace that appear in Figure 1, then we can obtain the
For each differential equation a concrete differential op erator L, which is related with a Green's function , appears and determines the solution of this differential equation as in the exposed example of Poisson equation (11).
representation of the C-obstacles that appears at this same figure . To be precise , the robot radius is 10cm and the workspace dimensions is (lmx1m ) where we can find two obstacles.
Nevertheless, there exist many situations where the consideration of Green 's function instead the differential operator is more useful , even it is not necessary to know it explicitly. So , if Green's function A( q, x ) is defined it is possible to obtain the potential distribution related with the obstacles for a desired behavior. This approach will be used in the next sections in order to obtain the artificial potential surfaces.
•-• . .
-~~ c --------------~
4. CASE OF STUDY: A MOBILE PLATFORM
Fig. 1. Considered workspace and resulting Cobstacles at the C-space
In order to validate the proposed interpretation , it will be applied to a particular robot: the circular mobile platform considered previously. This is a simple example that can be extended easily to any kind of platform. In this case the expression for the C-obstacles was
Figure 1 can be understood in such a way that the obstacles projection can be seen as an artificial potential distribution if we accept the following hypothesis. We will consider that a point obstacle produces a barrier potential, so potential effects appear at the limits of C-obstacles region and they do within the rest of C-space. So the utility for this potent;al will be in collision detection.
If Green' s functions theory is taken into account , there must exist a differential operator L such that
L[CB (x r , Yr , x , V)]
= B (x , y )
where the associated Green's function A(x r , Yr , x, y ) must satisfy
L [A (x" Yr , x , y) ] = 8(xr - x , Yr - y )
Another potential definition that could be very useful can be done if an electrostatic-like potential is considered, so that it decreases as the distance to obstacles increases . In this way Green 's function will be as follows •4 (XT>
In this case, CB (x r, Yr, x , y ) is the artificial potential generated by a distribution of sources B (x , y) that corresponds to the obstacles. The concrete shape of this distribution depends on the choice of A(x,·,Yr , x.y ), that will be the potential due to a point source placed at (xr, Yr ).
Yro x, Y ) -
{oo11/ I
r
if (x , y) E A (xr,Yr ) (1- ) 'f ( )d A ( 1 x , Y '" (Xr ,Yr )
where F is a constant and r is the minimum distance from a point to the robot. The same algorithm for C-obstacle computation that has been proposed is valid for this potential evaluation. Instead of the binary matrix that represents the robot we will use a bidimensional matrix A that contains a discretized representation of
I'
5. CONCLUSIONS
this distri bu tion (17). This expression will define a new Green's function and its representation can be found in Figure 2.
Main results that are obtained in this work agrees with those ones that appears at several research works that are related with the potential field approach. Nevertheless, the main advantage of the proposed method comes from the fact that the procedure to compute the potential surface is based on the same formalism that leads to C-obstacle evaluation. So, it is not necessary to perform any additional computation over the workspace and the C-space. By using the Green's theory it is only necessary to evaluate the potential related to an puntual obstacle at the workspace. In this way, a theoretical formalism that makes use of some well established mathematical tools can be applied to ,)L>tain the potential that can be useful at several path planning approaches. Moreover , the computational cost can be highly optimized, so the procedure can be used when temporal restrictions have to be considered.
lO
8
A
4
2
o 50 50
y
-50
-50
x
6. REFERENCES
Fig. 2. Green's function considered for a repulsive potential As it has been explained, it can be seen as the potential generated by an obstacle placed at the workspace origin. By applying the algorithm for the same workspace we obtain directly the potential surface. The results appears at Figure 3. It can be seen that the obstacles generate a repulsive potential that decreases as distance increases, as is the case for an electrostatic potential. C-.pau potDttimfid d
"
Fig. 3. Repulsive potential surface at C-space It is a sound result since it is not necessary to perform any special computation in order to obtain potential surface at C-space.
18
Barraquand , J. and J. C. Latombe (1991 ). Robot motion planning: a distribuited representation approach. Int. 1. of Robotics Research 10(6),628-649. Blanco , F . J ., V. Moreno and B . Curto (1998). Path planning method for mobile robots in changing environments . In: Proc. of Intelligent Components for Vehicles - ICV'98. pp. 425-430. Curto, B. and V. Moreno (1997). Mathematical formalism for the fast evaluaton of the configuration space . In: Proc. of the 1997 IEEE Int . Symp. on Comp . Intelligence in Robotics and A utomation. pp . 194-199. Kavraki, L. E . (1994). Random networks in configuration space for fast path planning. Ph. d. dissertation. Stanford University. Khatib, O. (1986 ). Real-time obstacles avoidance for manipulators and mobile robots. International Journal of Robotics Research 5(1 ),9098. Koditscheck , D. E . (19a7). Exact robot navigation by means of potential functions:some topological considerations. In: Proceedings of the IEEE International Conference on Robotics and Automation. pp. 1-6. LozanoPerez , T . (1983). Spatial planning: A configuration space approach. IEEE Transactions on Computers 32 , 108-120. Mantegh, I., M. Jenkin and A. Goldenberg (1997) . Solving the find-path problem: a complete and less complex approach using the bie methodology. In: Proceedings of the Symposium an Computers Intelligent IEEE CIRA .
A THEORETICAL APPROACH TO POTENTIAL FIELDS WITHIN THE CONFIGURATION SPACE USING GREEN'S FUNCTIONS B. Curto, V. Moreno, F. J. Blanco'
• Computer Science Department, University of Salamanca ,Spain. E-mail: [email protected]
Abstract: In this paper a new procedure to obtain directly the repulsive potential surface at the Configuration Space (C-space) due to the presence of obstacles within the robot environment is presented. This method is derived from a theoretical approach that has been succesfully applied to evaluation of obstacle representation within the C-space for mobile or articulated robots. It will be shown that it it is possible to find a new theoretical interpretation that unifies this obstacle representation (C-obstacle) with the repulsive potential fields that can be considered at the C-space. Main advantage of the method comes from the fact that the computation of the C-obstacles is not needed neither the subsequently potential evaluation over the computed C-space, i.e the resulting potential field is obtained in a unique computation step. In this way, the potential evaluation can be done without any aditional computational cost respect to the C-obstacle evaluation. As the results are based on a general formalism, it can be applied to any kind of robot (links shape, kinematics , ...); hence, it constitutes an important result that may be used with path-planning techniques that employ potential fields and within any other robotics-related studies. Copyright @20011FAC Keywords: Potential fields , Green's functions, Configuration Space, Path planning
1. INTRODUCTION
force field , and these forces are constituted by two main components: an attractive one that leads the robot to goal configuration and a repulsive one that pushes away from the obstacles. If these forces are applied to the robot actuators, it moves towards its goal configuration without colliding with any obstaCles, so it is not necesary to compute any path in the configuration space (C-space).
Potential field techniques have been used in order to prm'ide qualitative and quantitative information about robotic environments. Hence , they are related to those approaches that prosecute a global goal, an intelligent robotic system , and more precisely with path planning and reactive control capabilities. These techiques , which are broadly used at other scientific fields , were introduced in robotics research by (Khatib, 1986) which , initially, proposed its implementation with the aim of avoiding obstacles in real time. The main idea behind this approach is that the robot is moving within a
Also, potential fields information can be used to increase the low-level control quality. Instead of a simple control system following a previously planned trajectory, obstacle information can be taken into account within a control strategy in terms of a generated repulsive potential, as in (Blanco et al. , 1998). By using these strategies, obstacle avoidance can be improved and it must
1 This work has be done with the support of the Junta de Castilla y Len SA02-00F.
13
From this representation the potential field at the C-space is obtained but the local minima in existence are not avoided, so it is necessary to include powerful tools to escape from them.
be included in a direct way in the low-level control loop. The force that supports the robot can be specified by F(q) = -'VV(q) where V(q) is a non-negative escalar function defined at the robot C-space. A possible interpretation is that the function V (q) determines the potential energy for the robot (which in C-space is considered as a point) within a configuration; this fact determines the technique denomination.
So, as it can be seen, the examined works make reference to path planning techniques that use potential fields where the main aim is avoiding the local minima. But, they does not take into account the high computational cost of the potential surface evaluation at the at the C-space. The approach proposed by (Blanco et al. , 1998) is rather faster because the potential field at the C-space is computed directly from the C-obstacles representation. Simultaneously, the roadmap at the C-space is obtained, so the search problem is simplified. At the path planning stage, search algorithms are applied in order to guide the search graph , and potential fields are used at their heuristic definition to avoid narrow regions of the Cspace. In (Mantegh et al ., 1997) harmonic functions are used to guarantee that there exist no local minima, so that this goal is reachable.
One advantage , that is directly deduced from the specification of the force as the potential gradient, becomes from the fact the system behaviour can be predicted without an exact knowledge of the robot movements sequence. More precisely, if Cspace is restricted and the potential function is time invariant , it can be shown (Khatib , 1986) that the robot will move to a local potential minimun. An important property is the additive character for these functions . So, several potential sources can be considered independently in order to provide for the robot 's different capabilities. If the sum of several potentials is used , the robot 's behavior will be a combination of the desired capabilities.
But at all these works , the potential approach leads to a very high computational cost , due to every tries need several stages. At (Barraquand and Latombe, 1991) (Kavraki, 1994), first the potential evaluation at workspace is performed and next its representation at the C-space is obtained. And at (Blanco et al., 1998) the C-space evaluation is followed by a potential surface computation. Also, there is not a solid mathematical base that leads us to obtain the potential surface and its interpretation.
Nevertheless , if some potentials with opposing aims are added , the robot can fall into a local minimum. These ones prevent the potential field method from directly solving the problem of collision-free robotic movement. Since the method lets us guide the robot towards a desired configuration with minimal computational cost as compared with a free space representation, there is great interest on its adaptation to the path-planning research field. The main difficulty appears when the robot follows a direction opposite t.o gradient and it arrives to a local minimum that is not the goal configuration. This problem can be solved if special potential functions are used, called navigation junctions, that do not present local minima and do have a minimum potential at goal configuration. Analytical navigation functions have been proposed (Koditscheck , 1987), although they are used only in very unique environments. In some papers, it is possible to find numeric navigation functions (Barraquand and Latombe, 1991) defined over discrete representations of the configuration space. When a discrete configuration space is used as a basis , in order to avoid the local minimum on potential surface, random movements have been applied , see (Barraquand and Latombe, 1991 ) (Kavraki, 1994). In other works , other path planning techniques are used together with potential fields , as in (Barraquand and Latombe , 1991) , where a potential field without local minima at the workspace is computed by using a roadmap.
In order to solve these limitations , at this work we are going to present a mathematical formalism that can be used to obtain a potential representation at C-space directly from the workspace representation and the robot . Its based on another previous formalism that has been applied successfully to compute representation of C-obstacles for various kinds of robots , both mobile and articulated, (Curto and Moreno , 1997). Its advantages can be traced back to the highly optimized procedure that is obtained. In fact, it has been used with a mobile robot in a changing environment (Blanco et al., 1998) , which shows its powerful capabilities. However, as we shall show , we propose a new interpretation (using the Green's functions ) for our previous formalism from which we can find a procedure to obtain potential surfaces at the Cspace in a quite direct performing an only step. So in this way, the evaluation of the C-obstacles and the potential surfaces that they generate can be seen as two applications of only formalism with two definitions of the potential functions. In the following the article is organized as follows: in section II the needed theoretical bases that will
14
the robot A at a given configuration q collides with the obstacles, it is necessary to evaluate CB(q) (1) and, furthermore, the value of A(q,x) . In order to propose an example we can consider a simple robot, a rigid object with circular shape that moves freely in R2. So, we have that a configuration q is parameterized by q = (x r , Yr) where Xr and Yr are the robot center coordinates if the cartesian coordinates functions (x , y) are used at the robot workspace. In this case, equation can be written as
be used in the following are presented. At section Ill, we will explain the interpretation that is the main purpose of this article. We make use of a well known theory, that is the Green's function. In order to show its applicability we consider a mobile robot as case of study in section IV, where several surfaces are obtained that can be seen as the main result of this research work. Finally, in section V we will present the main conclusions.
2. THEORETICAL BASIS
I if (x , y) E A(x y ) A( Xr,yr,x , y ) = { Oif(x )dA r,r , Y l" (xr,Yr )
In this section , we are going to present the formalism that has been applied i~ order to obtain the representation of obstacles in C-space . As we shall see , the formalism is based on the definition of a nev,,' function , CB , that allows us to evaluate the C-obstacles. Next , we will show an application of the formalism, where it can observed how the computation can be highly optimized. Afterwards, the basis of theory of Green functions will be briefly presented. With these basics we will able to propose an interpretation of the exposed formalism in order to obtain a procedure to compute the potential fields within the C-space.
CB(xr , Yr)
=
J
A'(O 0 x' ') , , ,Y
~
CB(q )
{ 1 if x E B 0 if x rt B
J
J
A (q. x )B(x)dx
Vq E C
0 si (x' , y' )
E A(o,o) rl l"
A
(4)
(0 ,0)
A (o,o )(x r - x , yr - y)B(x , y)dxdy
(5 )
So , the CR function can be evaluated considering the function that describes the robot at a single configuracion (0 , 0). This is a key element to consider in order to reduce the computational load . Moreover, the evaluation can be done if the convolution theorem is taken into account ; if this theorem is applied to (5) then
R be the function defined by
=
= { 1 si (x', y')
being (x', y' ) = (x - x r , Y - Yr) and A (o.o) is the subset of points that represent the robot at configuration q = (0,0). Now if we consider the function A(o ,o)(x , y) = A'(O , O, -x , -y) clearly, CB(x r , Yr) can be calculated with (3) as
where A(q ) is the set of points of W that represents the robot at configuration q , and B is the subset of vF constituted by the obstacles. Let CB: C
A(xr , Yr , x , y)B(x , y)dxdy (3)
As it has been stated in (Curto and Moreno , 1997) , the evaluation can be strongly simplified in such a way that a new function A' defined over C' x VV ' into R is used and it is independent of some configuration parameters. In particular it can be determined that A(xr , Yr,x , y ) = A'(O,O , X - xr , y - Yr) where
In the following , W will designate the set of points at the workspace and C will be the set of configurations for the robot. Let A : C x 11" ~ {O , I} and B : HT ~ {O , I} be the function s defined by
=
2
So the expression for the C-obstacles (1) would be
2.1 CB definition
I if x E A(q ) A (q, x ) = { 0 if x rt A(q ) B(x)
()
(1)
\Ve define CB f as the subset of C that verifies CBr = {q E C j CB(q ) > O}. it is possible to prove the following theorem Theorem 1. Let CB = {q E C jA,(q ) n B i 0} be the definition of the C-obstacles region proposed by (LozanoPerez , 1983). Then it follows that CB = CBr .
In this particular case, the C-obstacles evaluation can be achieved by performing the pointwise multiplication of the Fourier transform of the function that represents the robot at the configuration (0, 0) and the Fourier transform of the function that represents the obstacles in the robot workspace. In order to evaluate the previous expression in a computer the A and B functions have to be discretized , obtaining two bitmaps as a result.
From the previous result, it is straightforward to see that q E Cjree +-t CB(q ) = 0, where CIree is the subset of C that corresponds with those configurations in which the robot does not intersect any obstacle. So , in order to know whether 15
It is significant, in this case, that the function represents the potential generated by a point charge placed at r'. By considering the previous expression (8) it becomes
2.2 Green's functions
The previous definition (1) can be seen as an integral expression that through Green's functions can be related to a differential equation. These kinds of relationships are used typically when a physical formalism , such as the potential theory, is developed.
q;(r)
=
J
G(x, x')f(x')dx'
= <5(x -
x')
=
r
where Q represents the charge that. produces the generated potential field
(6)
3. INTERPRETATION OF CB DEFINITIOI\' In this section , we will present a new approach to understand the proposed mathematical formalism. If the well known theory of Green 's functions is used then a new interpretation for the proposed formalism can be found. In this way, the function A (q, x) takes a special meaning that will lead to a definition of a procedure to obtain artificial potential surfaces at the C-space in a fairly optimized way.
(7)
J
G(x , x')f(x')dx'
It can be observed that the problem to find Green 's function of an operator L is equivalent to invert L. In this way, equations as Ly = f can be solved as y
= L- )f = K f =
(12)
Q
The expression can be also written as LK = I where I has the same property as the Dimc delta function <5 and K is the integral operator which is defined by Kf
p(r') d3 r' r - r' I
cjJ(r) -+ -
where C(x, x') is Green 's function related to the differential operator L. If this one is applied to G(x, :r') it can be shown that
LG(x, x')
JI
If a serial Taylor's expansion is performed and if I r I is large enough then
Let the equation Ly = f where L is some linear differential operator and f is a given function. In terms of Green's functions the solution for this equation will be
y(x)
=
The Green's function theory can be applied to get an interpretation, from a different point of view , for the proposed expression (1) used to compute the obstacle representation at the C-space.
J
G (x, x')f(x')dx'
CB(q ) =
J
.4(q , x)B(x)dx
If it can be found a function G that satisfies (7) and, if the solutions c(x ) of the homogeneous equation Lq; = 0 are known , then the general solution of Lq; = F(x) can be written as
By comparing this expression with (8) , i.e. the solution for the differential equation, there must exist a differential operator L in such a way that
J
L[CB(q)] = B(x)
q;(x)
= dx ) +
G(x, x')F(x')dx'
where the related Green's function G would be A(q , x) , <:nd it must comply with
A quite interesting application of these functions is found in Electromagnetic theory, which considers attractive / repulsive potentials, A differential equation (Poisson equation ) allows us to find the spacial electromagnetic potential dJ(r ) due to a charge distribution p( r )
v 2 d>(r ) = -4r.p (r )
L[A (q,x)]
By using th e Fourier transform method, it is posible to solve the equation
= _ 2.. I 47i r
1,
- r
(10 )
I
(14 )
We want to relate an integral expression (1) with a differential equation in order to obtain an interpretation from a physical point of view for the C-obstacles. As it is well known , differential equations are broadly used in different research areas in order to model the system 's behavior, i.e. , a body that is moving under a force/potential field.
and the corresponding Green's function will be G (1'. r')
= <5(q , x)
The goal is not to find explicitly the differential operator L that satisfies (14), since the operator will depend on the function A( q, x) that, as it has been exposed, defines the robot at a configuration for each workspace point.
(9)
\,2G (r, r' ) = <5(r - r' )
(13)
(8 )
(11 )
16
Once we have established the relationship between Green 's functions and C-obstacles, it is possible to find the analogy with some physical magnitudes. From the equation (13), we can see the similarity with Poisson equation (\7 2 q\(r) = -471'p(r)) where the electric potential q\ is defined for a distribution of charges p.
considered in such a way that the function A(q, x) will be given by
( , Yr,x,y ) = { Vif(x,Y)EA(xY)(16) AXr 0 if(x )dA r , r ,Y '" (Xr,Yr) where A(xr,Yr) is the subset of the workspace points that represents the robot at the configuration (xr,Yr) and V is a constant. If 11 = 1 the proposed situation is the same (2) as the case in which the goal is the obtention of the obstacles representation at the C-space for this circular mobile robot .
So, in a general way, it can be interpreted that CB corresponds with a distribution of artificial potential and that the obstacle B (x) is the cause that originates this distribution. Also, if the expression (14) is considered , the robot A(q , x ) will be the artificial potential related with a point potential source, i.e, a point obstacle.
If we consider the obstacles of the workspace that appear in Figure 1, then we can obtain the
For each differential equation a concrete differential op erator L, which is related with a Green's function , appears and determines the solution of this differential equation as in the exposed example of Poisson equation (11).
representation of the C-obstacles that appears at this same figure . To be precise , the robot radius is 10cm and the workspace dimensions is (lmx1m ) where we can find two obstacles.
Nevertheless, there exist many situations where the consideration of Green 's function instead the differential operator is more useful , even it is not necessary to know it explicitly. So , if Green's function A( q, x ) is defined it is possible to obtain the potential distribution related with the obstacles for a desired behavior. This approach will be used in the next sections in order to obtain the artificial potential surfaces.
•-• . .
-~~ c --------------~
4. CASE OF STUDY: A MOBILE PLATFORM
Fig. 1. Considered workspace and resulting Cobstacles at the C-space
In order to validate the proposed interpretation , it will be applied to a particular robot: the circular mobile platform considered previously. This is a simple example that can be extended easily to any kind of platform. In this case the expression for the C-obstacles was
Figure 1 can be understood in such a way that the obstacles projection can be seen as an artificial potential distribution if we accept the following hypothesis. We will consider that a point obstacle produces a barrier potential, so potential effects appear at the limits of C-obstacles region and they do within the rest of C-space. So the utility for this potent;al will be in collision detection.
If Green' s functions theory is taken into account , there must exist a differential operator L such that
L[CB (x r , Yr , x , V)]
= B (x , y )
where the associated Green's function A(x r , Yr , x, y ) must satisfy
L [A (x" Yr , x , y) ] = 8(xr - x , Yr - y )
Another potential definition that could be very useful can be done if an electrostatic-like potential is considered, so that it decreases as the distance to obstacles increases . In this way Green 's function will be as follows •4 (XT>
In this case, CB (x r, Yr, x , y ) is the artificial potential generated by a distribution of sources B (x , y) that corresponds to the obstacles. The concrete shape of this distribution depends on the choice of A(x,·,Yr , x.y ), that will be the potential due to a point source placed at (xr, Yr ).
Yro x, Y ) -
{oo11/ I
r
if (x , y) E A (xr,Yr ) (1- ) 'f ( )d A ( 1 x , Y '" (Xr ,Yr )
where F is a constant and r is the minimum distance from a point to the robot. The same algorithm for C-obstacle computation that has been proposed is valid for this potential evaluation. Instead of the binary matrix that represents the robot we will use a bidimensional matrix A that contains a discretized representation of
I'
5. CONCLUSIONS
this distri bu tion (17). This expression will define a new Green's function and its representation can be found in Figure 2.
Main results that are obtained in this work agrees with those ones that appears at several research works that are related with the potential field approach. Nevertheless, the main advantage of the proposed method comes from the fact that the procedure to compute the potential surface is based on the same formalism that leads to C-obstacle evaluation. So, it is not necessary to perform any additional computation over the workspace and the C-space. By using the Green's theory it is only necessary to evaluate the potential related to an puntual obstacle at the workspace. In this way, a theoretical formalism that makes use of some well established mathematical tools can be applied to ,)L>tain the potential that can be useful at several path planning approaches. Moreover , the computational cost can be highly optimized, so the procedure can be used when temporal restrictions have to be considered.
lO
8
A
4
2
o 50 50
y
-50
-50
x
6. REFERENCES
Fig. 2. Green's function considered for a repulsive potential As it has been explained, it can be seen as the potential generated by an obstacle placed at the workspace origin. By applying the algorithm for the same workspace we obtain directly the potential surface. The results appears at Figure 3. It can be seen that the obstacles generate a repulsive potential that decreases as distance increases, as is the case for an electrostatic potential. C-.pau potDttimfid d
"
Fig. 3. Repulsive potential surface at C-space It is a sound result since it is not necessary to perform any special computation in order to obtain potential surface at C-space.
18
Barraquand , J. and J. C. Latombe (1991 ). Robot motion planning: a distribuited representation approach. Int. 1. of Robotics Research 10(6),628-649. Blanco , F . J ., V. Moreno and B . Curto (1998). Path planning method for mobile robots in changing environments . In: Proc. of Intelligent Components for Vehicles - ICV'98. pp. 425-430. Curto, B. and V. Moreno (1997). Mathematical formalism for the fast evaluaton of the configuration space . In: Proc. of the 1997 IEEE Int . Symp. on Comp . Intelligence in Robotics and A utomation. pp . 194-199. Kavraki, L. E . (1994). Random networks in configuration space for fast path planning. Ph. d. dissertation. Stanford University. Khatib, O. (1986 ). Real-time obstacles avoidance for manipulators and mobile robots. International Journal of Robotics Research 5(1 ),9098. Koditscheck , D. E . (19a7). Exact robot navigation by means of potential functions:some topological considerations. In: Proceedings of the IEEE International Conference on Robotics and Automation. pp. 1-6. LozanoPerez , T . (1983). Spatial planning: A configuration space approach. IEEE Transactions on Computers 32 , 108-120. Mantegh, I., M. Jenkin and A. Goldenberg (1997) . Solving the find-path problem: a complete and less complex approach using the bie methodology. In: Proceedings of the Symposium an Computers Intelligent IEEE CIRA .