A dynamical mobility model Beyond the roboticle metaphor roboticle - - PowerPoint PPT Presentation

A dynamical mobility model

Beyond the roboticle metaphor roboticle metaphor : individual and collective autonomous robots moving around. How to monitor and control them. Suggestions for a Robotics Course Robotics Course. . Antonio D'Angelo

• Dept. of Math. And Computer Science

University of Udine Email: antonio.dangelo@dimi.uniud.it Web: www.dimi.uniud.it/~dangelo

Implementing Heat Control

• To this aim we need to define two different

scalar quantities

 the temperature distribution, which acts as

stigmergic information (cfr. pheromone levels within ant systems)

 the diffusivity, which explicitly considers the

roboticle distribution around the temperature source (cfr. the walking ants from the home position to the target position and reversely)

Navigation around Obstacles

• Let us consider a robot which tries to reach a

given target position by navigating inside an environment disseminated of obstacles.

• A path to follow is generated by considering the

quality of obstacles; namely,

• the robot is required to qualify its moving around
• bstacles by extracting information through a

proper interaction,

• causing the expected trajectory to be really

covered.

Source of Heat

• If we model the autonomous robot within the

framework of roboticles, each obstacle is a source of heat affecting the environment, which results in a perturbation of roboticle current moving.

• How it can really happen?

 roboticle senses obstacle;  it understands its quality by  generating an appropriate temperature field which, in turn,  affects the robot movement.

Intelligent Obstacle Avoidance

• During the navigation across the free space around the

disseminated obstacles in the environment,

• each of them can be understood as a source of information

which help the robot to reach the designated target position.

• Let us consider its trajectory covering while it passes near

an obstacle.

• We can refer its movement to the obstacle by introducing a

frame of reference centered on it frame of reference centered on it with both cartesian cartesian and polar polar coordinates to be used with the transformation fomulas

x=r cos y=r sin

Thermal Field

• From a preceding slide the thermal fields stems

from a temperature gradient grad T which gives raise to the heat flux H, as a consequence of the diffusivity ρ.

• The diffusivity

diffusivity should represents how the environment responses to the applied thermal

• gradient. Let us assume such a quantity to be

characterized by the follow formula ≡  H⋅ r=H 1 xH 2 y

Temperature Decay

• The first main consequence of the preceding definition is that the

decay rate of the temperature obeys the inverse distance law, namely,

• We can easily convice ourselves of this formula by considering the

following chain of equalities namely, which implies the term inside parentheses to be identically null.

dT dr =−1 r

=  H⋅ r=− r⋅grad T =−r dT dr

1r dT dr =0

Obstacle Anisotropy

• Any obstacle is an hint
• bstacle is an hint to drive robots towards

their targets. It can happen because

 obstacles are generalized forms of stigmergy and  the temperature gradient is the implementation in

the roboticle framework

• To this aim it is very useful to associate to each
• bstacle the polar pattern which assigns a

different quality to the obstacle with the respect to the direction under which the obstacle is sensed.

Anisotropy Measurement

• The quantity which determines this obstacle aptitude can be easily
• btained from the chain of equalities appearing below

where the following identities have been used

• Rearranging

where Ψ is the angle which identifies the cartesian component of the heat flux.

dT d  =dT dx dx d  dT dy dy d  =−y dT dx x dT dy = 1 [ y− dT dx −x− dT dy ]= H 1 y−H 2 x H 1 xH 2 y

H 1=− dT dx H 2=− dT dy

− dT d  = H 2 x−H 1 y H 1xH 2 y = tan −tan  1tan  tan =tan −

Temperature

• Putting all together, it yields to

and, because dT must be an exact differential form, the angle Ψ only depends on the robot direction φ from the obstacle point of view.

• In the special case where the heat flux is directed from the obstacle to

the robot (Ψ = φ), the temperature field is given by namely, the obstacle is isotrope

• bstacle is isotrope.
• In this case robot behavior doesn't depend on the way it is

approaching the obstacle.

−dT =dr r tan −d 

r=ae

−T

Building Stigmergy

• The temperature distribution around an obstacle is built by

combining different contributions through the following relations

• Each component defines the heat flux which originates from

the object having a fixed angle apart the direction towards the robot.

• A parametric example with one component

=1 2...n tan i= y mi x

Monopolar Source

• Each Ψi appearing in the preceding formula is termed

the i-th driving direction of the heat flux

• The most simple temperature distribution comes from a

single driving direction

• The most general heat flux satisfying the given

constraint takes the form

• And, then, the diffusivity

tan = y m1 x

H 1=m1 xQ x , y H 2= yQ x , y

=m1 x

2 y 2Q x , y

Monopolar Temperature

• The exploitation of the temperature distribution requires the integration of

the direction dependent part of the temperature increment dT where we have introduced the auxialiar variable s = tanφ, for convenience.

• Putting all together it yields to
• and, then, the final formula

2tan −d =2 s m1 −s 1 s

2

m1 ds 1s

2=

1−m1ds

2

m1s

21s 2

= ds

2

m1s

2− ds 2

1s

2

−2dT=2dr r  ds

2

m1s

2− ds 2

1s

2=2dr

r dln m1s

2

1s

2

a

2e −2T=r 2m1sin 2cos 2=x 2m1 y 2

Thermal Field around the Object

• The figure plots the thermal

field around the object with T = 1, and the parameter m taking values 1 1 (black), 0.1 0.1 and 0.9 0.9 (blue), 0.3 0.3 and 0.7 0.7 (red) and 0.5 0.5 (green).

• The plotted values should

be compared with the heat flux directions appearing in a previous slide.

Dispersion Factor

• By combining the relations expressing the temperature

distribution and the diffusivity in this specific case, it yields to where is said the dispersion factor dispersion factor around the obstacle centered in the origin of the frame of reference.

• This quantity is referred to the (temporal) asymptotic

distribution of roboticles around the obstacle due to the assigned heat flux.

=W e

−2T

W =a

2Q x , y

Simple Dispersion

• Let us consider a dispersion factor having the quadratic

form where a a is a quantity which takes track of the unit length measure and p p and q q are general parameters which define the quality of W.

• By so doing the cartesian components of the heat flux are
• and then, by partial derivation, through the well-known

formula

W =a

2 px 2qy 2

H 1=m1 px

3m1qxy 2

H 2=pxy

2qy 3

F=3m11 px

2m13qy 2

M =2m1q− p xy

Isotropical Behavior

• If we choose the parameters p

p and q q taking its values such that

• the roboticle behavior takes the isotropical form

where τ and ω0 have the usual meaning with the aim to weigh the relative strength of the dissipative component

• f the motion with the respect to the conservative one.

3m11 p=m13q= 1 2 2m1q− p=0 F= x

2y 2

2 M =0 xy

Dissipative/Conservative Tradeoff

• The dissipative/conservative tradeoff is given by the constant

m = ω0τ yielding to

• and the relationship is depicted by the following figure

m= 3m1

2−1

m133m11

Isotropical Motion

• With the given source of heat, originating from the obstacle, the

dynamical law of the roboticle takes the form

• yielding to the following trajectory equation
• namely,

u=− x −0 x=−m1  x v=− y 0 y= m−1 

EdS=vdx−udy=m1  ydxm−1  xdy= xy  d [m−1ln x a m1ln  y a ]

E= xy  e

S= xy

a

2  m

 y x 

Isotropical Trajectories

• Really speaking the isotropy property must be referred

to the dissipation function which doesn't depend on direction whereas the internal momentum does.

• The figure below depicts this particular situation

Dipolar Source

• A temperature distribution generated by two

driving directions takes the form and, then, the heat flux

• so that the diffusivity becomes

tan = y m1 x  y m2 x 1− y m1 x y m2 x = m1m2xy m1m2 x

2−y 2

H 1=[m1m2 x

2−y 2]Q x , y

H 2=[m1m2 xy]Q x , y

=m1m2 x

2m1m2−1 y 2 xQ x , y

Dipolar Stigmergy

• When the heat flux originates from an object with two driving

directions, the stigmergy looks like the figure below

• where parameters m

m1

1 and m

m2

2 are chosen so that the

temperature distribution be an hint for the roboticle motion.

Dipolar Temperature

• In this case, the temperature distribution around the
• bstacle is obtained by integrating the formula

which yields to where

−dT =m1 m2 x

2− y 2dxm1m2 xy dy

m1m2 x

2m1m2−1 y 2

a

2 x 2n−1e −2T=[ p x 2 y 2

n−1 ]

n

n= m1m2 m1m2−1 p=m1m2

Dipolar Dispersion

• Also in this case we have the general formula
• but now the dispersion factor takes the form

=W e

−2T

W =a

2 x 2n−1Q x , y

Tripolar Source

• The temperature distribution generated by three driving

directions takes the general form

• whereas the heat flux is

where

• and the diffusion

tan =m1m2m1m3m2m3 x

2 y− y 3

m1m2m3 x

3−m1m2m3 xy 2

H 1=b3 x

3−b1 xy 2Qx , y

H 2=b2 x

2 y−b0 y 3Q x , y

=[ b3 x

4b2−b1x 2 y 2−b0 y 4]Q x , y

b1=m1m2m3b0 b2=m1m2m1m3m2m3b0 b3=m1m2 m3b0

Unstructured Dispersion

• When we take the auxiliar function Q(x,y) to take a constant

value, the diffusivity also depends on temperature distribution. If we assume Q(x,y) = 1, it yields to

• and, then, the dissipative function and the internal momentum
• The dynamical law is

H 1=b3 x

3−b1 xy 2

H 2=b2 x

2 y−b0 y 3

F=b23b3x

2−b13b0 y 2

M =−2b1b2 xy

u=−23b3−b1x v=23b0−b2 y

Tetrapolar Source

• A much more entangled situation stems from an heat

source originating from four or more driving directions, such as

• where

with b0 an arbitrary given constant.

tan = b3 x

3 y−b1 xy 3

b4 x

4−b2 x 2 y 2b0 y 4

b1=m1m2m3m4b0 b2=[m1m2m3m4m2m3m4m3m4]b0 b3=[m1m2m3m4m3 m4m1m2]b0 b4=m1m2m3m4b0

Trajectory Generation

• During the motion the roboticle moves between
• bstacles usually with the aim to reach some

given target position.

• Accordingly with the strategic plan the roboticle

tries to use obstacles as a cues (suggestions) by a properly selected interpretation.

• The interpretation becomes active by assigning

to the obstacle a suitable thermal field.

• The real motion comes from the derived

dynamical law.

Conclusions

• In the preceding discussion we have

 considered a general method to generate a thermal

field for each obstacle encountered during the roboticle motion,

 provided an effective interpretation which motivates

the presence of the obstacle,

 argued some properties about the relationship

between the diffusivity ρ ρ on one side and the temperature T T and the dispersion W W on the other,

 dealt with some meaningful example by analyzing the

relevant parameters involved in the model.