measuring symmetry of moving stick creatures...imunus katehe. the first three creatures come from...
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Poznan University of Technology
Institute of Computing Science
Piotrowo 2, 60-965 Poznan, Poland
Measuring symmetryof moving stick creatures
Wojciech Jaskowski, Maciej Komosinski
Technical Report RA-020/06 December 2006
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Measuring symmetryof moving stick creatures
Wojciech Jaśkowski and Maciej Komosinski
Technical Report RA-020/06
Abstract
This work summarizes various approaches to measuring symmetry of static and movingcreatures. It outlines some problems with estimating symmetry of moving creatures: deter-mination of the temporal direction of movement, aggregation of changing lifespan symmetryto a single value, and indeterminism of computer simulations.
1 Introduction
The ubiquitous symmetry around us is still one of the mysteries of this world. Symmetry is presentin physical and mathematical laws and theories, and those considered most beautiful are usuallythe most symmetrical. Of course, this kind of beauty has a subjective nature.Mathematically speaking, symmetry is an intrinsic property of a mathematical object which
causes it to remain invariant under certain classes of transformations (such as rotation, reflection,inversion, or more abstract operations).Symmetry has long dominated in architecture and it is an unifying concept for all cultures of
the world. Some famous examples include the Pantheon, Gothic churches and the Sydney OperaHouse.There is no agreement among scientists why symmetry in biology is such a popular evolutionary
concept, but this phenomenon must be certainly related to the properties of the physical world.According to one of the hypotheses [7], a bilaterally symmetrical body facilitates visual perception,as it is easier for the brain to recognize while in different orientations and positions. Anotherpopular hypothesis suggests that symmetry evolved to help with mate selection. It was shownthat females of some species prefer males with the most symmetrical sexual ornaments [8, 9].For humans, there are proved positive correlations between facial symmetry and health [16], andbetween facial symmetry and perception of beauty [11].It is a common intuition that bilateral symmetry resulted from the direction of movement of
living creatures. This view was supported by some biological studies suggesting that there is apositive correlation between locomotive efficiency and morphological symmetry [1, 4, 12]. Symme-try of living creatures is only approximate and it is disturbed by such morphological elements asheart that is positioned asymmetrically. It is interesting to note that of all the reports of largescale asymmetry [10, 2], none of the investigated asymmetric features directly affected locomotion.On the other hand, in the world of flowers, symmetry (usually radial) is common and it is certainlynot related to locomotion.Symmetry is a very old evolutionary concept. The oldest known bilaterally symmetrical organ-
ism is Vernanimalcula that lived about 600 million years ago. Nowadays, the vast majority of themulticellular organisms exhibit either bilateral or radial symmetry. Radial symmetry is a featureof some marine species like sea anemone, jellyfish, sea stars, etc. Most animals are bilaterallysymmetrical (e.g. mammals). Notable exceptions among animals are the sponges.
1.1 Motivations
There is no objective measure of symmetry. The only thing that can be assessed objectively iswhether an object is entirely symmetrical or not. The natural language is not sufficiently preciseto express intermediate values of symmetry. We say that something is nearly symmetrical, butwe are not able to say that something is symmetrical to a certain degree, and we are not able
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to specify this degree numerically in the same manner as, for instance, angles can be described.This lack of expressions in natural languages describing partial symmetry is reasonable because,as stated above, many objects in the real world are symmetrical. However, symmetry is not sucha popular concept in artificial worlds and in order to study the phenomenon of symmetry and itsimplications, there is a need for defining a numerical, fully automated and objective measure ofsymmetry for creatures living in artificial environments.The natural, “binary” notion of symmetry is insufficient for this application. The advantage
of numerical measure of symmetry is not only in that it allows determining the extent to what anobject is symmetrical, but also in that it allows to state if one object is more symmetrical thananother.In everyday speech, we tend to use the term symmetry not only for static objects, but also for
entities that move, change their shape, position and direction. However, we usually think aboutstatic symmetry, i.e. symmetry of a motionless entity. In a real (and artificial) world environment,a creature usually changes its morphology during lifetime — for example, it uses its muscles tobend some parts of its body. Therefore, it is possible that a statically symmetrical creature wouldbecome asymmetrical while interacting with the environment and vice versa. The value of staticsymmetry highly depends on the exact moment in which a creature is captured. This situationis also true in the real world, and despite the fact that moving animals are most of the time notsymmetrical, we tend to say that they are – because of their “basic” shape we have in mind.In previous work, we addressed the need to measure symmetry of (artificial) 3D stick creatures
or constructs and proposed a numerical measure of symmetry [5]. The method operated on the“embryo”, i.e. a phenotype grown from genotype, but not yet interacting with the environment.It was a static symmetry. In this report we will propose several methods of measuring dynamicsymmetry, i.e. a symmetry of moving creatures, operating on the phenotype in motion. In thecontext of artificial life research, this work adds another automatic tool that helps a human examineand evaluate virtual creatures. The numerical measure of symmetry follows the numerical similaritymeasure of pairs of creatures/constructs [6]. Such measures are especially useful when a researcherfaces the need to analyze, systematize or classify large populations or sets of individuals, and theseare often encountered in artificial life experiments dealing with evolution and creation. Similarityand symmetry estimates act as simple decision support tools, as they aid a researcher in analysisand interpretation of experimental results. These tools are also helpful for nonprofessionals, asthey help reduce complexity and size of experimental data and make it more comprehensible.Other important motivations for this study come from the real world. We would like to mea-
sure the degree of symmetry in motion because this is a property of a method of locomotion.Dynamic symmetry is a property that captures a feature of the motion pattern and its mechanicalaspect. Other features include discovering whether (or: to what degree) the movement is periodicor chaotic, how dynamic it is, how effective it is, etc. As we consider a realistic and quite gen-eral creature model, results of this research can have important implications in understanding ofevolution on Earth and methods of locomotion both in living animals and designed robots.
1.2 Related work
A continuous symmetry measure for chemical molecules has been developed in [13, 14]. The sameresearch group defined a symmetry measure for raster images as a quantifier of the minimum’effort’ required to transform a given shape into a symmetric shape [15], but these approaches arenot suitable for the model of creatures considered in this report.The (static) numerical measure of symmetry for artificial creatures was previously considered
by Josh Bongard in [3], where the correlation between symmetry and locomotive efficiency forcreatures in 3-dimensional space was studied. However, that measure of symmetry was definedonly for simple tree-like creatures consisting of spherical units of identical sizes and masses. Thoseunits could be connected to each other by links of uniform length with no mass. Links betweenunits were constrained to only six cardinal directions. Creatures were facing a fixed direction. Inthe first step of the algorithm, the plane of symmetry was chosen as a vertical plane that intersectsthe unit whose horizontal position is closest to the average horizontal positions of all the units. Inthe second step, the symmetry sym(c) for creature c was computed using the following equation[3]:
sym(c) =4pl
(2n− 1)− p− l(1)
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where n was the total number of units creature c consisted of; 2n − 1 was the total numberof units and links creature c consisted of; p was the number of pairs of units lying outside of theplane of symmetry and symmetrical about that plane; and l was the number of pairs of linksnot contained in the plane of symmetry and symmetrical about that plane. Let us notice thataccording to the Eq. (1), if a creature is perfectly symmetrical, then sym(c) = 1.0.The measure described above is sufficient for the simple creature model assumed in [3]. Never-
theless, it is not general enough to be successfully applied to more realistic creature models. Thisis why we developed much more general measure of symmetry applicable to 3D stick creatures [5].The most important features of this measure are described in Sec. 1.3. So far, no research hasbeen performed on measuring symmetry of creatures in motion.
1.3 Static measure of symmetry
1.3.1 Creature model
The measure of symmetry introduced here evaluates 3-dimensional creatures consisting of con-nected, variable length sticks (rods). Creatures can also be equipped with other elements such asreceptors, neurons, effectors etc., but only the skeleton (i.e. sticks) is taken into consideration formeasure calculation (see Fig. 1). The model resembles stick insects from the order of Phasma-todea, and it can also be used for other creatures with compatible internal structure (e.g. boneskeleton).
Figure 1: An example of a 3D stick creature. Structures containing cycles (closed loops) are alsoallowed.
1.3.2 Definition of the symmetry measure
Let us denote the symmetry value of a creature c as sym(c). We introduce five conditions thatsym(c) is expected to fulfill:
• The Symmetry Condition. If c is perfectly bilaterally symmetrical, then sym(c) = 1.0.
• The Asymmetry Condition. If c is completely asymmetrical, then sym(c) = 0.0.
• The Common Sense Condition. If c1 is more symmetrical than c2, then sym(c1) > sym(c2).
• The Proportional Difference Condition. The difference between sym(c1) and sym(c2) shouldmatch the difference in anatomical symmetry between c1 and c2.
• The Scalability Condition. The proposed measure should be robust against scaling: forcreature c2 that is a scaled version of c1 (body enlarged or diminished), we expect sym(c2) =sym(c1).
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It is important to note that although The Symmetry Condition and The Scalability Condition canbe verified in an objective way, The Common Sense Condition and The Proportional DifferenceCondition are of a subjective nature. Also, it is hard to imagine a “completely asymmetrical”creature (The Asymmetry Condition).Let us denote symmetry of a creature c about plane p as sym(c, p). We say that “a creature
is symmetrical” if it is symmetrical about any plane, therefore we are looking for a plane of thehighest symmetry:
sym(c) = maxp
(sym(c, p)) (2)
For a detailed description of the algorithm that determines sym(c, p) refer to [5].
2 Measuring symmetry
The problem of determining a symmetry of moving creatures is hard, because there is no mathemat-ical definition of symmetry for objects that change their shape. In this section we propose severalmethods to determine the measure of symmetry of moving creatures, and discuss the advantagesand disadvantages of each of them.In the following points we will consider different approaches to the measure of symmetry in
movement starting with the most trivial ones. All symmetry measures will be accompanied withexamples of their application to the four Framsticks creatures:
1. Basic Quadruped,
2. Bulldog,
3. Rototiller,
4. Imunus Katehe.
The first three creatures come from the Framsticks database and were designed by hand, whereasImunus Katehe was evolved for speed (this is why is shape is more chaotic). All the four creaturesare shown in Figure 2. The reader is encouraged to see how these creatures move in the Framsticksenvironment.
2.1 Static symmetry
2.1.1 Grown phenotypes
Static symmetry is computed basing on the genotype of a creature. Thus it is independent of theinfluence of the environment and cannot be used as an estimate of a symmetry in motion (seeSection 1.1). However, static evaluation is a good starting point.
The values and planes of symmetries for the four examined creatures are shown in Figure 3. Aswe see, two hand-designed creatures (Basic Quadruped and Bulldog) are completely symmetrical.Rototiller is almost symmetrical (0.85) because of a few asymmetrical bulges. Imunus Katehe isnearly symmetrical (0.95) despite the fact that its shape looks very irregular. Its high value ofsymmetry is caused by the fact that its body is nearly flat. All flat (2D) objects are symmetricalin terms of a bilateral symmetry in 3D.
2.1.2 Phenotypes situated in environment
A phenotype encompasses physical appearance and constitution. It is determined to a large extentby genotype, but it is also influenced by environmental factors. In the case of Framsticks system,a creature placed in the environment is influenced by simulated physical forces. In effect, its initialshape can change. Because of the resilience forces, its shape can even oscillate for some time untilall the forces counterbalance and the creature finally stabilizes. By situated symmetry we will calla static symmetry for a such a steady creature. Of course, the shape of a situated creature can
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(a) Basic Quadruped (b) Bulldog
(c) Rototiller (d) Imunus Katehe
Figure 2: The four considered creatures.
(a) Basic Quadruped (1.000) (b) Bulldog (1.000)
(c) Rototiller (0.850) (d) Imunus Katehe (0.956)
Figure 3: Symmetry planes of the four considered creatures. Symmetry values are given in brackets.
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Table 1: Static symmetries for grown and situated phenotypes.genotype grown situated average situated std.dev.
Basic Quadruped 1.000 0.978 0.064Bulldog 1.000 0.999 0.000Rototiller 0.850 0.843 0.022
Imunus Katehe 0.956 0.991 0.000
depend on the way it is placed in the environment (position, force used to place it, orientation,elevation, etc.), but also on some random fluctuations, if present in the environment. This iswhy in order to determine the final symmetry value, the process of stabilizing and computing thesymmetry is repeated 10 times.In Table 1, static symmetries for grown phenotypes and situated phenotypes are shown.
There is no clear relation between the values of grown and situated symmetry values. For BasicQuadruped, Bulldog and Rototiller, situated symmetry is a little smaller than grown symmetry,but for the “flat” Imunus Katehe, it is the opposite.The results shown in Table 1 suggest that the symmetries of creatures situated in the environ-
ment are similar to those computed for “embryos”. This is because the creatures we consider havestiff body structure and the world is flat, so bodies do not change much when they are situated intheir stable orientation. This would change if bodies were less rigid, gravity was higher, the worldwas steep, etc. Since each creature is evaluated in a world (whether it was created or evolved), thesituated symmetry is generally a more suitable estimate of symmetry than the grown one.The situated symmetry only considers body shape in a single moment of life. In the next
section, we are going to propose dynamic symmetry measures that take into account informationabout body shape from some period of life.
2.2 Dynamic symmetry
2.2.1 Paths of moving creatures
Before we propose dynamic measures of symmetry, we will present moving patterns of creatures,because some intuition about the movement will be useful for further deliberations.In the Framsticks environment, movement of creatures results from using bones (sticks), mus-
cles, senses and the brain. For example, creatures can smell out food and change their direction ofmovement accordingly. We, however, only consider creatures that move without any purpose, i.e.patterns of their movement are only based on the senses of touching ground and 3D orientation oftheir body parts. Three of the considered creatures were designed to demonstrate various ways ofmoving. Imunus Katehe, on the other hand, was evolved for speed.
In Figure 4, exemplary 3D paths from 5000 steps of simulation were shown. Each point ingraphs is the center of gravity of a creature in some moment of time. The vertical (z axis)coordinate was enlarged for some paths for better readability. The dotted line on the bottomplane is the projection of the 3D path. Note that scales of the graphs are different. It is interestingto note the characteristic oscillations on the vertical z axis. The most regular ones can be observedfor Rototiller. The movement of Imunus Katehe seems to be the most chaotic. Only for BasicQuadruped is the horizontal path (2D projection) approximately a straight line. Bulldog andImunus Katehe move in an unpredictable way.This has to do with randomness of the world and characteristics of computer simulations. In an
ideal, flat world, a creature built exactly in the same way each time would behave in a precisely thesame way. We do not want to simulate such an ideal world because this is unrealistic1, therefore
1Virtual worlds are often considered “ideal” because they are simulated on computers, but this is not the wholetruth. Various sources of computation errors exist in computer simulations, and even the smallest errors canpropagate, grow and result in completely different behaviors. Some creatures are robust against such errors andwill damp them, but some – on the contrary – will amplify them leading to the “butterfly effect”. The Imunus
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we add random noise to creatures (initial state for each neuron is randomized, and creature bodiesare disturbed randomly).In a simulated world, some creatures (Basic Quadruped, Rototiller) move in the same way
regardless of the initial conditions, while other creatures (Imunus Katehe) move in a chaotic way.This effect is shown in Figure 5 where paths for 10 different Framsticks simulations are shown onthe same graph for every creature. Bulldog is somewhere between those two categories: its pathshave certain similarities (e.g., Bulldog always turns right), but they are different.
2.2.2 Dynamic 3df symmetry
We define dynamic 3df 2 symmetry as average symmetry for a creature that moves. Graphs in Fig.7 show how the symmetries of creatures change over time. Those graphs correspond to movementpaths shown in Fig. 6. The oscillations of symmetry value are quite regular for all the creatures,but again, the most regular ones are for Rototiller. The value for time step 0 is equal to thesituated symmetry introduced in Subsection 2.1.2. Notice that for Basic Quadruped and Bulldog,it starts from nearly 1.0, but quickly deteriorates. This effect is clearly visible for Bulldog.
Katehe is an example of a creature which is very sensitive to any change in its simulation. Its genotype is XL(FFr(LX[*] [N, 0:8.806,5:-0.774,si:2] [N, 0:4.179,6:-1.93] [|, 1:0.77,r:1,p:0.25] [Sin, t:0,f0:0.0628319]),(XXCw(qX[@, 5:3.037] AXCsCsr(fX [Sin, 4:6.133] [|, 1:2.907, p:0.25] [Sin, t:0]), CQX[|, 1:6.255][Sin, t:0,f0:0.0628319]),SX)). Even if the simulation is totally deterministic, this creature may behave in acompletely different way depending on the initial position in the world. A little difference in initial coordinates, asthey are numbers of limited precision that are subject to arithmetic operations, produces differing outcomes whichare used step by step for subsequent computation (see Figure 5).This situation is yet another motivation to add random noise to computer simulations, evaluate performancemultiple times and calculate averages. Note that if the initial state of the neural network is randomized, then thebehavior of evolved creatures may depend on the random non-zero initial values, and require them to perform asexpected (i.e. creatures evolved with random initialization of their neural networks may not work when they areinitially zeroed).23df comes from 3 degrees of freedom and concerns allowed positions of the symmetry plane.
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Figure 5: 10 paths for four considered creatures. Note that the scales are different in each graph.
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Figure 6: Movement paths of four examined creatures for a single simulation run (5000 steps).Starting points were marked with blue triangles.
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Figure 7: The values of symmetry in time for the simulation run considered in Fig. 6.
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Table 2: Dynamic 3df symmetries (dynamic 3df), their standard deviations (std.dev.) and maximaland minimal values.
creature dynamic 3df std.dev. min maxBasic Quadruped 0.825 0.070 0.663 0.999Bulldog 0.684 0.075 0.500 0.999Rototiller 0.873 0.037 0.757 0.983
Imunus Katehe 0.849 0.071 0.668 0.998
The dynamic 3df symmetry values are presented in Table 2. According to this symmetrydefinition, the most symmetrical creature is Imunus Katehe. This is against intuition, since it isthe most irregular creature among the four. But again, the high value comes in this case fromthe fact that Imunus Katehe is flat and moves low over the surface, thus its plane of symmetry ishorizontal. Interestingly, Bulldog, the leader both according to the grown and situated symmetry,is the least symmetrical according to the dynamic 3df measure. It also obtained the lowest value ofsymmetry while running (0.500). Due to its method of movement, Rototiller has the most stablesymmetry (see standard deviation in Table 2).
2.2.3 Dynamic 2df symmetry
The problem with the static symmetry measures (grown and situated) and dynamic 3df symmetryis seen best for the Imunus Katehe creature, which is flat. Speaking about symmetry of real worldcreatures, we usually mean a vertical bilateral symmetry. This is related to the vertical asymmetryof the environment we live in, which manifests itself in the force of gravitation which is directeddownwards. Taking into account vertical planes only was impossible for the grown symmetry,because it is computed for a genotype that does not exist in an environment. But now we considercreatures moving in the artificial world, where the force of gravitation exists, just like in the realworld. Therefore our considerations can be restricted to vertical planes as potential symmetryplanes. For a creature c, vertical symmetry (2df symmetry) is defined as
sym(c) = maxp
(sym(c, p)), (3)
where p is a vertical plane and sym(c, p) is a symmetry of creature c about the plane p (seeSection 1.3.2). sym(c, p) is defined in [5].
In Figures 8–10, results of a standard 5000 steps simulation are shown. Figure 8 presentsthe values of dynamic 2df symmetry over time. When compared with 7, it can be observed thatthe symmetry stays on the similar level for Basic Quadruped and Rototiller, while it is slightlydeteriorated for Bulldog and significantly deteriorated for Imunus Katehe.Very interesting observations can be made based on Fig. 9, which shows the direction (as an
angle) of the symmetry plane for consecutive time steps. The direction of Basic Quadruped is nearlyconstant and the direction of its symmetry plane is also approximately constant. For Rototiller thedirection of plane of symmetry changes linearly and this corresponds to the characteristic, circularmovement of this creature. Notice that the deviations from a certain trend are usually aroundπ/2. This means that for some moment t, a symmetry plane is found which is perpendicular tothe plane t− 1.Figure 10 shows planes of symmetry plotted on the paths of creatures. It can be observed that
for Basic Quadruped and Bulldog, the planes are usually directed in accordance to the directionof creatures movement (although exceptions exist). On the other hand, the best symmetry planes
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Figure 8: The values of vertical (2df) symmetry in function of time for four considered creatures.
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Figure 9: Plane directions for vertical (2df) symmetry in function of time for the simulation runcorresponding to paths in Fig. 6. Note that the graphs are vertically wrapped and the ranges are[−π/2, π/2].
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Figure 10: The creature 2D paths (red) with vertical planes shown (green). For clarity, every 10thplane was shown. Compare those graphs to the corresponding graphs shown in Fig. 9.
Table 3: Dynamic 2df symmetries (dynamic 2df), their standard deviations, and maximal andminimal values.
creature dynamic 2df std.dev. min maxBasic Quadruped 0.829 0.055 0.638 0.999Bulldog 0.488 0.092 0.345 1.000Rototiller 0.807 0.057 0.660 0.943
Imunus Katehe 0.501 0.083 0.313 0.780
for Rototiller were usually perpendicular to the direction of its movement. No clear relation canbe found for Imunus Katehe.Table 3 presents the values of dynamic 2df symmetry for four examined creatures. This time, the
value of symmetry of Imunus Katehe is more realistic, since its flat shape does not help anymore asthe plane of symmetry is always vertical. Imunus Katehe has the lowest maximal value of symmetry,because while moving, it never becomes symmetrical or even nearly symmetrical. Notice that boththe standard deviation and the difference between min and max values are highest for Bulldog.Although it seems symmetrical, Bulldog changes its body shape significantly during its movement.
2.2.4 Determining direction of creature movement
The symmetry plane of vast majority of high-level animals coincide most of the time with theirdirection of movement. It is interesting to know what is the symmetry of a creature along the lineit moves. In order to do this, we have to determine the direction of movement for every step ofsimulation. Since the creatures move in different (usually not straight) ways, it is quite a challenge(recall the paths in Fig. 6).We devised various algorithms to approach this problem, some of them being inspired by
existing methods approximating a set of points with a function of some form. The simplest wayto determine the direction for simulation step t is to compute the difference between creature
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Figure 11: The original paths (red) and the ones smoothed using a low pass filter (blue) .
positions for step t and t − 1. This is natural, since difference is a discrete version of differential.This approach is, however, prone to all the fluctuations and oscillations of creature movement.After comparing various methods for determining direction of movement, we have chosen thefollowing approach: in a preliminary step, we smooth the movement path using a low pass filter.In another words, for every point pt in step t, we compute the average for points that belong tothe time window of width w centered in time t. The parameter w controls how much the pathis smoothed. It is hard to find an objective way of determining w, however it certainly has to begreater than the oscillation periods that are seen in figures showing creature paths. We assumedw = 200.Figure 11 shows original (red) and smoothed (blue) paths for the four creatures. Notice that
all the small fluctuations and oscillations disappeared, while the shape of each path is preserved.In Fig. 12, vectors tangent to the smoothed paths are presented.
2.2.5 Dynamic 1df symmetry
Having the directions of movement for every point of creature path determined, we can nowcompute the dynamic 1df symmetry defined as the symmetry for a vertical plane coinciding withthe movement direction. Symmetries defined in this way are shown in Fig. 14 in function oftime. Imunus Katehe is most affected by this symmetry definition. Based on the graphs and thesymmetry values (see Table 4), one can conclude that its symmetry is very weakly correlated withits direction of movement. The symmetry of Basic Quadruped is still very high.
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Figure 12: Smoothed 2D paths (blue) with determined directions of movement for every step(green). For better clarity directions are shown for every 20th step. Compare the figure with Fig.10.
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Figure 13: Movement directions based on the smoothed paths over time for the simulation runcorresponding to paths in Fig. 11. Note that the graphs are vertically wrapped and the ranges are[−π/2, π/2].
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Figure 14: The values of vertical (1df) symmetry over time for the four considered creatures.
Table 4: Dynamic 1df symmetries (dynamic 1df), their standard deviations and maximal andminimal values.
creature dynamic 1df std.dev. min maxBasic Quadruped 0.769 0.073 0.549 0.945Bulldog 0.371 0.092 0.098 0.613Rototiller 0.528 0.139 0.123 0.909
Imunus Katehe 0.232 0.091 0.039 0.694
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Figure 15: The values of vertical (1df) symmetry in function of time.
2.2.6 Soft dynamic 1df symmetry
Restricting the orientation of vertical symmetry plane to the precise direction of movement seemstoo radical. Therefore, this constraint is relaxed in the soft dynamic 1df symmetry, so the p inequation 3 is only approximately vertical. We allow the plane p to have 20 degrees of freedomvertically and horizontally around the determined direction of movement.In Figure 15, changes of the soft dynamic 1df symmetry are illustrated. Averaged values are
shown in Table 5. We believe that this measure is better than the dynamic 1df symmetry, becauseit is prone to small oscillations in the direction and pattern of movement. The four creaturesordered by the decreasing value of soft dynamic 1df symmetry are:
1. Basic Quadruped
2. Rototiller
3. Bulldog
4. Imunus Katehe
This order follows a subjective feeling based on the observation of movement of those creatures byhumans.
3 Summary and future work
In this report, we described a few approaches to determining symmetry of moving creatures. Aproblem of determining the direction of movement was described and solutions were presented. Sixapproaches to estimating symmetry were reported: two static (grown, situated) and four dynamic(3df, 2df, 1df, and soft 1df).
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Table 5: Soft dynamic 1df symmetries (soft 1df), their standard deviations and maximal andminimal values.
creature soft 1df std.dev. min maxBasic Quadruped 0.777 0.063 0.588 0.950Bulldog 0.475 0.062 0.162 0.768Rototiller 0.688 0.109 0.154 0.932
Imunus Katehe 0.327 0.119 0.090 0.737
The last approach (soft 1df) seems most suitable to be used in the forthcoming experimentson symmetry of creatures in motion, however, it can be further improved after larger scale experi-ments are performed. Having the measure of motion symmetry, interesting research can be carriedout. This includes testing what is the relation of dynamic symmetry and speed of movement orcapability to cope with difficult (hilly) environment. With the body of a creature fixed, one willbe able to evaluate various neural controllers in terms of the symmetry of movement patterns theygenerate. The symmetry of creatures that have been evolved can also be examined, and the dy-namic symmetry estimate can be added to the fitness formula to promote or discourage constructsthat move in a symmetrical way.A related research will concern other ways of evaluating the way creatures move: to what degree
the movement is periodic or chaotic, how dynamic it is, how effective it is, etc.In all experiments reported here, the MechaStick simulator was employed. It is a simple sim-
ulation engine that does not support rigid body dynamics but is fast and produces realistic en-vironments and physical interactions. Better correspondence to the real world conditions will beprovided by employing the ODE (Open Dynamics Engine) in all simulations. This engine, albeitslower, will support better collision detection among parts of creature body, which is currentlydisregarded.
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