Framsticks

The picture on the right shows sample forces calculated in the Framsticks physical simulator, Mechastick.

The body is made from parts (points) and joints (sticks, rods).

In each simulation step:

#define NEURO_MAX 10.0
input=getWeightedInputSum();
velocity=force*(input-state)+inertia*velocity;
state+=velocity;
if (state>NEURO_MAX) state=NEURO_MAX;
else if (state<-NEURO_MAX) state=-NEURO_MAX;
tmp=state * sigmo;

if (tmp<-30.0) output = -1;          //safer than exp(-tmp)
          else output = (2.0/(1.0+exp(-tmp))-1.0); // -1..1

if (tmp<-30.0) output = 0;           //safer than exp(-tmp)
          else output = (1.0/(1.0+exp(-tmp)));     //  0..1

• state, velocity – internal variables
• sigmo, force, inertia – neuron parameters (properties)
• input, tmp – temporary variables

  D (differentiate)	Calculate the difference between the current and the previous input values.
  Thr (threshold)	Outputs lo if the input signal is below the t threshold. Outputs hi otherwise. t, lo and hi are parameters.
  * (constant output)	Outputs the value of 1.
  Rnd (random noise)	Outputs random values (uniform distribution) in the range of –1..+1
  Sin (sinus generator)	The output sinusoid has frequency f and initial phase t. The output frequency depends on the neuron input value. Parameters: f0 (base frequency). The output frequency f = f0 + current_neuron_input_value. t (time) – initial phase.
  Fuzzy (fuzzy control system)	See this paper for details; see this movie for demonstration.

The N neuron in detail

Sigmoid neurons (with short name N) use a simple weighted sum of input signals. Excitation influences neuron state, which has some inertia. Stronger signals can change the state faster than weak signals. Output is flattened to a [-1,+1] range using basic sigmoidal function. See examples below: (input/state/output)

Simple excitation. The state goes up when the input is positive and falls down when theinput reaches zero. Note that in this example the neuron's state can fall below zero due to its inertia.
Short but strong impulse gives similar results to a weak and long one.
In this example a strong signal causes saturation of the neuron (its state goes very high). Later signal changes do not influence the output.

N parameters

• Force, noted as 'fo', value range: 0..1, default 0.04
  
• Inertia, noted as 'in', value range: 0..1, default 0.8
  
• Sigmoid, noted as 'si', any real number accepted, default 2.0
  

Force=0.1, inertia=0 Slow state change. Note the instant reaction after the input signal pulse when inertia is disabled.
Force=0.1, inertia=0.8 With inertia enabled, the neuron's state rises above the input pulse amplitude, and then drops below zero. The final state is achieved after several oscillations.
Force=1, inertia=0 Maximum force coefficient results in an instant input to output propagation.

The third, sigmoid coefficient changes the output function. Detailed formulas which describe the work of the N neuron are as follows:

velocity_t = velocity_t–1
inertia + force
(input_t – state_t–1)

state_t = state_t–1 + velocity_t

outputt =

2 1 + e (-velocityt sigmoid)

– 1

where
input: weighted sum of neuron inputs,
velocity: analogous to physical velocity,
state: internal state (analogous to physical location),
output: output signal.
Subscripts represent the time moment.

The following pictures show sample usage of the sigmoid parameter.

Sigmoid=2.0 Default.
Sigmoid=10.0 High values nearly produce a threshold function.
Sigmoid=0.5 Low values give a nearly linear output function.






Special neurons: muscles and receptors

Basic muscles (actuators) and receptors (sensors) are illustrated below.

A muscle neuron ('|' or '@' in genotype) can change the relative orientation of the controlled stick (relative to the previous stick). This simple 2-stick creature with a bending muscle in the middle is described by the genotype "XX[|...]". The joint stays straight when the signal is equal to 0. Positive and negative values bend the constructon in the opposite directions. [animation]

	The "G" receptor (gyroscope, equilibrium sense) gives information about the stick's orientation relative to the gravity force. A gyroscope mounted on a horizontally aligned stick sends 0 to its outputs. Vertical position is perceived as -1 or +1 depending on which end is higher. [animation]
	The "T" receptor (touch) can be imagined as a whisker attached to the stick. Its relaxed state is -1 (nothing detected in the whiskers' range). The signal value grows as the stick gets closer to any material object. Reaches 0 when the stick touches the ground. Higher values mean that the stick is pushed into the ground. [animation]
	The "S" receptor (smell) excitation depends on the sum of the neighbor energy sources (energy balls and other creatures). Output range is from 0 (nothing detected) to 1 (maximum). Rich or closer energy sources smell 'stronger' than small or distant ones. [animation]

Formulas for basic effectors and receptors

Neuron type	How simulated
| (muscle) [animation]	Changes the relative orientation of the joint's parts by rotating the second part around the first part's Z axis. Parameters: p (power) – muscle strength. Affects the movement velocity and the maximum force the muscle can exert. r (range) – Affects movement range, maximal value 1.0 means full range (-180°..+180°). The following rule applies: Angle = Range * Signal * 180° Sample creature (f1 genotype XX[|]) //0 p: p:1, m=2 p:2 j:0, 1, dx=1 j:1, 2, dx=1 n:j=1, d=| Neutral position Singal = 0 Angle = 0° Muscle active Signal = 0.2 Angle = 36° Explanation: The joint is attached to parts #1 and #2. Part #2 was rotated around the Z axis of Part #1, so its position and orientation was changed. The above pictures were created using the "standard-xyz" OpenGL visualization style, which can be used to investigate part orientations in creatures.
@ (muscle)	Changes the relative orientation of the joint's parts by rotating the second part around the first part's X axis. Parameters: p (power) – muscle strength. Affects the movement velocity and the maximum force the muscle can exert. This musle always uses the full movement range (-180°..+180°). Angle = Signal * 180° Sample creature (f1 genotype XX[@]) //0 p: p:1, m=2 p:2 j:0, 1, dx=1 j:1, 2, dx=1 n:j=1, d=@ Neutral position Singal = 0 Angle = 0° Muscle active Signal = 0.25 Angle = 45° Explanation: The joint is attached to parts #1 and #2. Part #2 was rotated around the X axis of Part #1. Part #2 was not moved, because it lies on the X axis (in creatures built from f1 genotypes, all parts lie on the X axis). The above pictures were created using the "standard-xyz" OpenGL visualization style, which can be used to investigate part orientations in creatures.
G (gyroscope) [animation]	state = (part1.z – part2.z) / stick_length
T (touch) [animation]	if touches: state = distance of part and ground (which is equivalent to positive depth) if does not touch: check along x orientation of the part state = –1.0 if there are no objects closer than 1.0 distance state = ... (intermediate negative values) state = 0.0 if a touched object is just at the T part
S (smell) [animation]	state = (sum_all_energy_sources[ energy / distance2 ])/100 (if distance<1, then use 1)
Water (water detector / water pressure indicator) [animation]	above surface: state = 0.0 below surface: state = depth
Energy (energy level) [animation]	state = 1.0 for initial energy level (a newborn organism) state = ... (intermediate values) state = 0.0 for no energy (creature dies)

All neuron types: summary

N (Neuron) Standard neuron	supports any number of inputs provides output value does not require location in body Properties: Inertia (in) float 0..1 Force (fo) float 0..999 Sigmoid (si) float -99999..99999 State (s) float -1..1
G (Gyroscope) Equilibrium sensor. 0=the stick is horizontal +1/-1=the stick is vertical	does not use inputs provides output value should be located on a Joint
T (Touch) Touch sensor. -1=no contact 0=just touching >0=pressing, value depends on the force applied	does not use inputs provides output value should be located on a Part
S (Smell) Smell sensor. Aggregated "smell of energy" experienced from all energy objects (creatures and food pieces). Close objects have bigger influence than the distant ones: for each energy source, its partial feeling is proportional to its energy/(distance^2)	does not use inputs provides output value should be located on a Part
* (Constant) Constant value	does not use inputs provides output value does not require location in body
| (Bend muscle)	uses single input does not provide output value should be located on a Joint Properties: power (p) float 0.01..1 rot.range (r) float 0..1
@ (Rotation muscle)	uses single input does not provide output value should be located on a Joint Properties: power (p) float 0.01..1
D (Differentiate) Calculate the difference between the current and previous input value. Multiple inputs are aggregated with respect to their weights	supports any number of inputs provides output value does not require location in body
Water (Water detector) Output signal: 0=on or above water surface 1=under water (deeper than 1) 0..1=in the transient area just below water surface	does not use inputs provides output value should be located on a Part
Energy (Energy level) The current energy level divided by the initial energy level. Usually falls from initial 1.0 down to 0.0 and then the creature dies. It can rise above 1.0 if enough food is ingested	does not use inputs provides output value does not require location in body
Ch (Channelize) Combines all input signals into a single multichannel output; Note: ChSel and ChMux are the only neurons which support multiple channels. Other neurons discard everything except the first channel.	supports any number of inputs provides output value does not require location in body
ChMux (Channel multiplexer) Outputs the selected channel from the second (multichannel) input. The first input is used as the selector value (-1=select first channel, .., 1=last channel)	uses 2 inputs provides output value does not require location in body
ChSel (Channel selector) Outputs a single channel (selected by the "ch" parameter) from multichannel input	uses single input provides output value does not require location in body Properties: channel (ch) integer
Rnd (Random noise) Generates random noise (subsequent random values in the range of -1..+1)	does not use inputs provides output value does not require location in body
Sin (Sinus generator) Output frequency = f0+input	uses single input provides output value does not require location in body Properties: base frequency (f0) float -1..1 time (t) float 0..6.28319
Delay (Delay)	uses single input provides output value does not require location in body Properties: delay time (t) integer 1..1000
Nn (Noisy neuron) Propagates weighted inputs onto the output, but occassionally generates a random value	supports any number of inputs provides output value does not require location in body Properties: Error rate (e) float 0..0.1
Sf (Smell food) Detects only food, not other creatures (in experiments with food in group #1)	does not use inputs provides output value should be located on a Part
Thr (Threshold) if (input>=t) then output=hi else output=lo	uses single input provides output value does not require location in body Properties: threshold (t) float low output value (lo) float high output value (hi) float

Experimental neurons

Do not use unless you know what you are doing. Some of these neurons can be unstable, many are used for special purposes.

Nu (Unipolar neuron [EXPERIMENTAL!]) Works like standard neuron (N) but the output value is scaled to 0...+1 instead of -1...+1. Having 0 as one of the saturation states should help in "gate circuits", where input signal is passed through or blocked depending on the other singal.	supports any number of inputs provides output value does not require location in body Properties: Inertia (in) float 0..1 Force (fo) float 0..999 Sigmoid (si) float -99999..99999 State (s) float -1..1
Fuzzy (Fuzzy system [EXPERIMENTAL!]) Refer to publications to learn about this neuron	supports any number of inputs provides output value does not require location in body Properties: number of fuzzy sets (ns) integer number of rules (nr) integer fuzzy sets (fs) string fuzzy rules (fr) string
VEye (Vector Eye [EXPERIMENTAL!])	uses single input provides output value should be located on a Part Properties: target.x (tx) float target.y (ty) float target.z (tz) float target shape (ts) string perspective (p) float 0.1..10 scale (s) float 0.1..100 show hidden lines (h) integer 0..1 output lines count (each line needs four channels) (o) integer 0..99 debug (d) integer 0..1
VMotor (Visual-Motor Cortex [EXPERIMENTAL!]) Must be connected to the VEye and properly set up.	supports any number of inputs provides output value does not require location in body Properties: number of basic features (noIF) integer number of degrees of freedom (noDim) integer parameters (params) string
Sti (Sticky [EXPERIMENTAL!])	uses single input does not provide output value should be located on a Part
LMu (Length muscle [EXPERIMENTAL!])	uses single input does not provide output value should be located on a Joint Properties: power (p) float 0.01..1