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view/download model file: formation_perfect.nlogo
This is an extension of “formation_newton.nlogo,” for the book entitled “Physicomimetics: Physics-Based Swarm Intelligence.”
Multiple particles use F = ma and a “split Newtonian” force law to self-organize into a perfect triangular or square lattice.
This is the first simulation for Chapter 4 of the book, which pushes the envelope of physicomimetics. This simulation creates perfect lattices, by making minor modifications to the existing framework.
First, we assume that each particle has an attribute assigned to it at initialization. This attribute is composed of two integers: m and n. This attribute serves to indicate where a particle should be with respect to another particle.
Second, we require that all particles share a global coordinate system (which was never assumed in the prior simulations). This can be accomplished easily - each particle is assumed to have a digital compass that tells the particle which way is north.
Finally, we deliberately break Newton’s third law to a small degree, by modifying the force law. We do this to improve performance. The reason this is important is because we want to stress that physicomimetics “mimics” physics, as opposed to merely copying physics. We are task-driven. If alterations to standard physics improves performance, that is quite acceptable.
Click SETUP AGENTS to initialize the particles, and click MOVE AGENTS to have them move.
The CLEAR button will clear the graphics, which becomes handy when particles have their pens down (more on this below).
The NUMBER_OF_PARTICLES slider allows you to control the number of particles created at initialization. Changing this slider while the simulation is running will have no effect. However, you can increase the number of particles while the simulation is running by using the ONE IS BORN button. However, the ONE IS BORN button does not change the number of initial particles when the simulation is restarted by clicking SETUP AGENTS.
The ONE IS BORN button creates a new particle.
There is also a TOGGLE FORMATION button that allows you to choose between square and triangular lattices.
All other sliders will affect the simulation when it is running.
Particles are initialized in a random cluster in the middle of the graphics pane, and self-organize into a perfect square or triangular lattice.
The GRAVITATIONAL_CONSTANT controls the G parameter in the split Newtonian force law. The POWER controls the value of “p” in the generalized law. The DESIRED_SEPARATION is the desired distance between neighboring particles.
The GOOD GRAVITATIONAL CONSTANT monitor uses the theory established in Chapter 3 to provide guidance on a good value of G, when you change the DESIRED_SEPARATION.
Again, FRICTION is enabled. This allows the system to stabilize.
Note that when creating triangular lattices, angular momentum is not conserved. But note that linear momentum is still conserved. The reason for this is explained in Chapter 4 of the book.
The red dot in the simulation shows the center of mass of the system. If the Conservation of Linear Momentum holds in both the x- and y-dimensions, the red dot will not move. This simulation includes a monitor for the Angular Momentum.
This model allows you to add particles. Adding particles allows you to see how scalable the system is, and illustrates just how nicely new particles are incorporated into the lattice.
See how the FRICTION changes behavior. Click SETUP AGENTS, lower the friction, and click MOVE AGENTS. What happens?
Similarly, change the GRAVITATIONAL_CONSTANT or the POWER.
Change the DESIRED_SEPARATION while the system is running. Try changing it slowly and then change it quickly. What happens?
Add particles and watch their trajectories.
See how the G PHASE TRANSITION value is affected when you change the GRAVITATIONAL_CONSTANT, the POWER and the FORCE_MAXIMUM.
Toggle between triangular and square formations. Note how robust the system is - the formations are torn apart and then self-repair.
Find a more elegant way to compute the (m, n) attribute.
Consider changing the model so that the (m, n) attribute is swapped between particles, as opposed to having particles move so much.
Extend this model to create perfect honeycomb formations.
Try the Lennard-Jones force law instead.
Note, in order to change any NetLogo simulation, you must have the source code (i.e., “formation_perfect.nlogo”) downloaded to your computer, as well as NetLogo itself. You can not change the code when you are running the simulation with your browser.
This simulation allows the user to create new particles.
To create a particle, the “hatch” command is used - this clones an existing particle and moves it away from the original. Also, the new particle has the “pen down,” which means that you will see the path that the particle takes. If the graphics pane becomes too busy, click on the CLEAR button.
This code has two procedures (“attract” and “repulse”) where values of arguments are passed to the procedures.
This is an extension of “formation_newton.nlogo” to create perfect lattices.
Chapter 17 (by Sanza Kazadi) provides alternative mechanisms for creating perfect lattices. His chapter includes videos of these techniques.
To see the first papers that outlined the work presented in this simulation:
Gordon, D. F., Spears, W. M., Sokolsky, O., and Lee, I. (1999) Distributed spatial control, global monitoring and steering of mobile physical agents. In Proceedings of IEEE International Conference on Information, Intelligence, and Systems.
Spears, W. M., and Gordon, D. F. (1999) Using Artificial Physics to control agents. In Proceedings of IEEE International Conference on Information, Intelligence, and Systems.
The latter paper is the first to show the perfect formations.
If you mention this model in an academic publication, we ask that you include these citations for the model itself and for the NetLogo software:
- Spears, W. M. and Spears, D. F. (eds.) Physicomimetics: Physics-Based Swarm Intelligence, Springer-Verlag, (2011).
- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
Copyright 2011 William M. Spears. All rights reserved.
Permission to use, modify or redistribute this model is hereby granted, provided that both of the following requirements are followed:
a) this copyright notice is included, and
b) this model will not be redistributed for profit without permission from William M. Spears. Contact William M. Spears for appropriate licenses for redistribution for profit.
; William M. Spears September 2011
; Perfect Triangular and Square Lattices Using the Split Newtonian Force Law
; For research and educational use only
breed [particles particle] ; Introduce the "particle" breed
globals [total_lmx total_lmy total_angular_mom G p FR D
center_of_mass_x center_of_mass_y FMAX DeltaT
square? square_button?]
turtles-own [hood deltax deltay r F Fx Fy v vx vy dvx dvy mass view
lmx lmy theta lever_arm_x lever_arm_y lever_arm_r angular_mom
squarem squaren hexm hexn pm pn km kn cx cy]
to setup
clear-all ; Clear everything
; Create and initialize particles
create-particles Number_of_Particles [setup-particles]
set square_button? true ; Start with a square lattice
update-info
; Computes center of mass and displays location
set center_of_mass_x (sum [xcor * mass] of particles) / (sum [mass] of particles)
set center_of_mass_y (sum [ycor * mass] of particles) / (sum [mass] of particles)
ask patch (round center_of_mass_x) (round center_of_mass_y)
[ask patches in-radius 4 [set pcolor red]]
reset-ticks
end
to run-and-monitor
if (count turtles < 1) [user-message "Please click HALT and then SETUP AGENTS first" stop]
update-info
ask particles [ap-particles]
ask turtles [move]
; Computes center of mass and displays location
set center_of_mass_x (sum [xcor * mass] of particles) / (sum [mass] of particles)
set center_of_mass_y (sum [ycor * mass] of particles) / (sum [mass] of particles)
ask patch (round center_of_mass_x) (round center_of_mass_y)
[ask patches in-radius 4 [set pcolor red]]
set total_lmx sum [lmx] of particles ; Total linear momentum, x-component
set total_lmy sum [lmy] of particles ; Total linear momentum, y-component
set total_angular_mom sum [angular_mom] of particles ; Total angular momentum of objects
tick
do-plots
end
to setup-particles ; Set up the particles
setxy (random-normal 0 20) (random-normal 0 20) ; Start in a cluster
set heading random 360 ; Everyone has a random heading
set vx 0 set vy 0 set mass 1 ; Start with no motion and mass = 1
set shape "circle" set size 5
ifelse ((who mod 2) = 0)
[set color white]
[set color yellow] ; Different colors for square formations
set theta 0
init-m-n
end
; Terribly ugly code to set up the (m,n) attributes. It would
; be wonderful if someone could make this more elegant - hint hint.
to init-m-n
; For square lattices
let ring (1 + (floor (sqrt (who / 4))))
let index (who - ((ring - 1) * (ring - 1) * 4))
ifelse (index < (2 * ring - 1)) [set squarem (1 + index - ring)] [
ifelse (index <= (4 * ring - 2)) [set squarem ring] [
ifelse (index < (6 * ring - 3)) [set squarem (5 * ring - index - 2)] [set squarem (1 - ring)]]]
ifelse (index < ( 2 * ring)) [set squaren (ring - 1)] [
ifelse (index < (4 * ring - 2)) [set squaren (3 * ring - 2 - index)] [
ifelse (index < (6 * ring - 2)) [set squaren (0 - ring)] [set squaren (index - 7 * ring + 3)]]]
; For triangular lattices, good up to 126 particles
ifelse (who = 0) [set ring 1] [
ifelse (who < 7) [set ring 2] [
ifelse (who < 19) [set ring 3] [
ifelse (who < 37) [set ring 4] [
ifelse (who < 61) [set ring 5] [
ifelse (who < 91) [set ring 6] [set ring 7]]]]]]
ifelse (who = 0)
[set index 0]
[set index (who - ((3 * ring * ring) - (9 * ring) + 7))]
ifelse (index < ring) [set hexm (2 * index - (ring - 1))] [
ifelse (((3 * ring - 3) <= index) and (index <= (4 * ring - 4))) [set hexm ((8 * ring) - 8 - (2 * index) - (ring - 1))] [
ifelse ((ring <= index) and (index < (2 * ring - 2))) [set hexm index] [
ifelse (index = (2 * ring - 2)) [set hexm (2 * ring - 2)] [
ifelse (((2 * ring - 2) < index) and (index < (3 * ring - 3))) [set hexm ((4 * ring) - index - 4)] [
ifelse (((4 * ring - 4) < index) and (index < (5 * ring - 5))) [set hexm (3 * ring - index - 3)] [
ifelse ((5 * ring - 5) < index) [set hexm (index - (7 * ring) + 7)] [
if (index = (5 * ring - 5)) [set hexm (2 - 2 * ring)]]]]]]]]
ifelse (index < ring) [set hexn ring - 1] [
ifelse (((3 * ring - 3) <= index) and (index <= (4 * ring - 4))) [set hexn 1 - ring] [
ifelse ((ring <= index) and (index < (3 * ring - 3))) [set hexn (2 * ring - index - 2)] [
if (index > (4 * ring - 4)) [set hexn (index + 5 - 5 * ring)]]]]
end
; Attractive force, x- and y-components
to attract [wx wy]
set Fx (Fx + wx * F * (deltax / r))
set Fy (Fy + wy * F * (deltay / r))
end
; Repulsive force, x- and y-components
to repulse [wx wy]
set Fx (Fx - wx * F * (deltax / r))
set Fy (Fy - wy * F * (deltay / r))
end
to ap-particles ; Run artificial physics
set Fx 0 set Fy 0 ; Initialize force components to zero
set vx (1 - FR) * vx ; Slow down according to friction
set vy (1 - FR) * vy
set hood [who] of other particles ; Get the IDs of everyone else
foreach hood [
set deltax (([xcor] of particle ?) - xcor)
set deltay (([ycor] of particle ?) - ycor)
set r sqrt (deltax * deltax + deltay * deltay)
set view 1.5 ; For triangular lattice
if (square?) [ ; For square lattice
ifelse ((who mod 2) = (? mod 2))
[set view 1.3 set r (r / (sqrt 2))] ; See Chapter 3 for details
[set view 1.7]
]
ifelse (square?)
[set pm squarem set pn squaren set cx 0.5 set cy 0.5
set km ([squarem] of particle ?) set kn ([squaren] of particle ?)]
[set pm hexm set pn hexn set cx 0.25 set cy 0.433
set km ([hexm] of particle ?) set kn ([hexn] of particle ?)]
if (r < view * D) [ ; The generalized split Newtonian law
set F (G * mass * ([mass] of turtle ?) / (r ^ p))
if (F > FMAX) [set F FMAX] ; Bounds check on force magnitude
ifelse (r > D) [attract 1 1] [
ifelse (((deltax < 0) and (pm < km) and (deltay < 0) and (pn < kn)) or
((deltax > 0) and (pm > km) and (deltay > 0) and (pn > kn))) [attract 2 2] [
ifelse (((deltax < 0) and (pm < km)) or ((deltax > 0) and (pm > km))) [attract 2 2] [
ifelse (((deltay < 0) and (pn < kn)) or ((deltay > 0) and (pn > kn))) [attract 2 2] [
ifelse (((pn = kn) and (abs(deltay) > cy * D)) or ((pm = km) and (abs(deltax) > cx * D))) [attract 2 2] [
ifelse ((not square?) and (pn != kn) and (abs(deltay) < cy * D)) [repulse 0 3] [
ifelse ((not square?) and (pm != km) and (abs(deltax) < cx * D)) [repulse 3 0] [repulse 1 1]]]]]]]]
]
set dvx DeltaT * (Fx / mass)
set dvy DeltaT * (Fy / mass)
set vx (vx + dvx) ; The x-component of velocity
set vy (vy + dvy) ; The y-component of velocity
set v sqrt (vx * vx + vy * vy)
set deltax DeltaT * vx
set deltay DeltaT * vy
if ((deltax != 0) or (deltay != 0))
[set heading (atan deltax deltay)]
end
to move ; Move the turtle
fd (sqrt (deltax * deltax + deltay * deltay))
set lmx (mass * vx) ; Linear momentum of the turtle
set lmy (mass * vy)
set lever_arm_x (xcor - center_of_mass_x)
set lever_arm_y (ycor - center_of_mass_y)
set lever_arm_r sqrt (lever_arm_x * lever_arm_x + lever_arm_y * lever_arm_y)
if (((vx != 0) or (vy != 0)) and ((lever_arm_x != 0) or (lever_arm_y != 0)))
[set theta (atan (mass * vy) (mass * vx)) - (atan lever_arm_y lever_arm_x)]
set angular_mom (lever_arm_r * mass * v * (sin theta)) ; Angular momentum of the turtle
end
to update-info ; Update information from the sliders
set G Gravitational_Constant
set p Power
set FMAX Force_Maximum
set FR Friction
set DeltaT Time_Step
set D Desired_Separation
set square? square_button?
end
to do-plots
set-current-plot "Linear and Angular Momenta" ; Select the Momenta plot
set-current-plot-pen "Lmx" ; Select the Lmx pen
plot total_lmx ; Plot the linear momentum, x-component
set-current-plot-pen "Lmy"
plot total_lmy ; Plot the linear momentum, y-component
set-current-plot-pen "Angular"
plot total_angular_mom ; Plot the angular momentum
end
; Create a new particle
to one-is-born
if (count (particles with [mass = 1]) > 0) [
ask one-of particles with [mass = 1]
[hatch 1 [set deltax 0 set deltay 0 pd init-m-n ; Clone an existing particle
ifelse ((who mod 2) = 0)
[set color white]
[set color yellow] ; Different colors for square formations
setxy xcor + (random-normal 0 (D / 2))
ycor + (random-normal 0 (D / 2))]]
]
end
; Toggle between triangular and square formations
to toggle-formation
if (square_button? != 0) [set square_button? not square_button?]
end