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 obstacles 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 with both cartesian cartesian and frame of reference centered on it polar coordinates to be used with the transformation fomulas polar 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, dT dr =− 1 r ● We can easily convice ourselves of this formula by considering the following chain of equalities r ⋅ grad T =− r dT =  H ⋅ r =− dr namely,  1  r dT dr = 0 which implies the term inside parentheses to be identically null.

Obstacle Anisotropy ● Any obstacle is an hint obstacle 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 obstacle 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 obtained from the chain of equalities appearing below dy ]= H 1 y − H 2 x d  = dT dT d   dT dx d  =− y dT dy dx  x dT dy = 1 [ y − dT dx − x − dT dx dy H 1 x  H 2 y where the following identities have been used H 1 =− dT H 2 =− dT dx dy ● Rearranging d  = H 2 x − H 1 y − dT H 1 x  H 2 y = tan − tan  1  tan  tan = tan − where Ψ is the angle which identifies the cartesian component of the heat flux .

Temperature ● Putting all together, it yields to − dT = dr r  tan − d  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 − T r = ae namely, the obstacle is isotrope obstacle is isotrope. ● In this case robot behavior doesn't depend on the way it is approaching the obstacle.

Building Stigmergy ● The temperature distribution around an obstacle is built by combining different contributions through the following relations tan  i = y = 1  2  ...  n m i x ● 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

Monopolar Source ● Each Ψ i 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 y tan = m 1 x ● The most general heat flux satisfying the given constraint takes the form H 1 = m 1 xQ  x , y  H 2 = yQ  x , y  ● And, then, the diffusivity 2  y 2  Q  x , y  = m 1 x

Monopolar Temperature ● The exploitation of the temperature distribution requires the integration of the direction dependent part of the temperature increment dT s − s 2 m 1  1 − m 1  ds 2 2 ds ds 2 − ds 2tan − d = 2 2 = = 2 2  1  s 2  2 1  s  m 1  s m 1  s 1  s 1  s m 1 where we have introduced the auxialiar variable s = tan φ , for convenience. ● Putting all together it yields to 2 r  dln m 1  s 2 2 − 2dT = 2dr ds 2 − ds 2 = 2dr r  2 m 1  s 1  s 1  s ● and, then, the final formula − 2T = r 2 e 2  m 1 sin 2  cos 2 = x 2  m 1 y 2 a

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 − 2T = W e where 2 Q  x , y  W = a 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.

Simple Dispersion ● Let us consider a dispersion factor having the quadratic form 2  px 2  qy 2  W = a 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 3  m 1 qxy 2 2  qy 3 H 1 = m 1 px H 2 = pxy ● and then, by partial derivation, through the well-known formula 2  m 1  3  qy 2 F = 3 m 1  1  px M = 2  m 1 q − p  xy

Isotropical Behavior ● If we choose the parameters p p and q q taking its values such that  3 m 1  1  p = m 1  3  q = 1 2  m 1 q − p = 0 2  ● the roboticle behavior takes the isotropical form 2  y 2 F = x M = 0 xy 2  where τ and ω 0 have the usual meaning with the aim to weigh the relative strength of the dissipative component of the motion with the respect to the conservative one .

Dissipative/Conservative Tradeoff ● The dissipative/conservative tradeoff is given by the constant m = ω 0 τ yielding to 2 − 1  3  m 1 m =  m 1  3  3 m 1  1  ● and the relationship is depicted by the following figure

Isotropical Motion ● With the given source of heat, originating from the obstacle, the dynamical law of the roboticle takes the form u =− x − 0 x =− m  1 v =− y  0 y = m − 1 x   ● yielding to the following trajectory equation EdS = vdx − udy = m  1 ydx  m − 1 xdy = xy  d [ m − 1  ln  x a  m  1  ln  y a ]   ● namely, m E = xy S = xy  y e 2  x   a

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 y y m 1 x  m 2 x =  m 1  m 2  xy tan = y y 2 − y 2 m 1 m 2 x 1 − m 1 x m 2 x and, then, the heat flux 2 − y 2 ] Q  x , y  H 1 =[ m 1 m 2 x H 2 =[ m 1  m 2  xy ] Q  x , y  ● so that the diffusivity becomes 2  m 1  m 2 − 1  y 2  xQ  x , y  = m 1 m 2 x

Dipolar Stigmergy ● When the heat flux originates from an object with two driving directions, the stigmergy looks like the figure below ● where parameters m m 1 1 and m m 2 2 are chosen so that the temperature distribution be an hint for the roboticle motion .