text stringlengths 128 2.05k |
|---|
In general, our approach is more organic in contrast to the typical symbolic approach (direct encoding of pitch, roll, yaw angles, use of pattern generators using Gaussian functions etc.). The biological inspiration is not practiced as an end in itself but rather introduces more robustness in computations and it allows... |
One focus of our current research track is to design fitness landscapes by using appropriate controller designs. We investigate possibilities of smoothing the fitness landscape by a sophisticated interaction between the controller design and the mutation operator. We test whether it is useful to maximize the causality ... |
The investigated scenario is a modular-robotics variant of gait learning in simulation. Initially, we connect five modules in a simple chain formation as the body formation itself is not yet in our focus. The task is to move as far as possible by utilizing the hinge in each module only (no wheels). |
Artificial Homeostatic Hormone Systems In AHHS, sensors trigger hormone secretions, which increase hormone concentrations in the robot. These hormones diffuse, integrate, decay, interact and finally, affect actuators. We have analyzed AHHS controllers in single robots before Schmickl et al., 2010 Schmickl and Crailshei... |
2.1 AHHS1 We call the AHHS, initially presented in Schmickl et al., 2010 Schmickl and Crailsheim, 2009 , AHHS1. An AHHS consists of a set of hormones and a set of rules. On the one hand, it defines production/decay rates and diffusion coefficients for each hormone. On the other hand, it defines by rules the production ... |
Schmickl et al., 2010 2.2 AHHS2 Based on AHHS1 we designed an improved variant called AHHS2. The guiding principle of this improved controller design was to gain higher evolvability by creating smoother fitness landscapes. There were three main changes. |
First, we introduced an additional rule type that implements nonlinear hormone-to-hormone interactions in the general form of [MATH] , where [MATH] is the considered hormone concentration and |
[MATH] is the hormone concentration of the influencing hormone that triggers the considered rule. The idea is to increase the intrinsic dynamics (basically transient behavior before equilibria are reached) of the hormone network even without significant sensor input. |
Second, a rule is not just triggered by exceeding or falling below a threshold but is linearly weighted within a trigger window (i.e., a tent function with a maximum of 1, defined by a center and a width, see eq. below). |
Third, the mutation of rule types in the form of discrete switches seemed to be too radical. This was overcome by introducing a concept of weights for rule types. Now, each rule can operate as any rule type at the same time. Each rule has a weight for each of the five rule types summing up to one (see Fig. ). The influ... |
below. A mutation will now only change two rule weights by reducing one by [MATH] and adding [MATH] to the other weight. In a well adapted controller we would expect that the weights of a rule are mainly concentrated on one or at most two rule types. Other weight distributions should be transitional only because specia... |
The mathematical closed-form of this concept using the example of a linear hormone rule type is [EQUATION] where [MATH] is the hormone amount that is to be added to the considered hormone at time [MATH] [MATH] is the weight of the linear hormone rule (see Fig. ), [MATH] is the index of the input hormone and [MATH] is i... |
is the dependent dose, [MATH] the fixed dose. [MATH] is called trigger function and defined by [EQUATION] for trigger window center [MATH] and trigger window width [MATH] . For a more detailed introduction of AHHS2 and for a comparison of the AHHS approach to the standard ANN approach, see Hamann et al., 2010 |
Note that the rule parameters (fixed dose, input hormone, trigger window etc.) are correlated via the rule types. For example, the input hormone is used for both the linear and the nonlinear hormone rule. If we would allow independent parameters for each rule type the genome (encoding of the controller) size would be i... |
Investigated scenarios Our main focus is on the field of modular robotics and our main concern is whether we are able to evolve fast locomotion in the gait learning task. Still, we tested the AHHS approach also in an inverted pendulum task as well, due to its lower computational complexity. |
3.1 Inverted pendulum In addition to the gait learning task, we tested the AHHS approach in a task that is easier to handle: balancing the inverted pendulum (see Fig. ). The computational demand of the gait learning task is very high due to the sophisticated simulation of physics. We satisfy the need for a simulation o... |
3.2 Gait learning in multi-modular robotics Gait learning in legged robotics is a commonly studied task in evolutionary robotics as reported by |
Nelson et al., 2009 . However, here we investigate gait learning in multi-modular robotics. Each module consists of one hinge and we connect five modules. These five hinges are controlled decentrally although the modules have a low-level communication channel by means of diffusing hormones. |
In contrast to the standard tasks of gait learning and collision avoidance, the challenge of gait learning in multi-modular robotics is more complex. The resulting gait is emergent due to the decentral and cooperative control of the actuators. In addition, there are several conceptionally different solutions, that is, ... |
In each module the same controller is executed. Therefore, the gait learning task includes several sub-tasks. The organism has to break the symmetry (head and tail), synchronize through collective cooperation, and start moving into a common direction. This synchronization aspect is similar to the gait learning task for... |
All of this work is based on simulations as the actual hardware is not yet available (see Fig. for a current prototype of Symbrion and Replicator SYMBRION, 2010 REPLICATOR, 2010 ). We use the simulation environment Symbricator3D by Winkler and Wörn, 2009 that was developed for these projects. We use the current prototy... |
We have tested the AHHS controllers with two variants of the simulation framework. In the first version, the forces in the joints, that connect the modules, were damped and small displacements of the modules at the joints were allowed (i.e., simulation reacts moderately to big forces). It turned out that caterpillar-li... |
We start the scenario with five robot modules which are simply connected in a chain. Initially this robotic organism is placed in the center of the arena. In order to increase the complexity of the gait learning task, the central area is surrounded by a low wall forming a square (its height is about half the height of ... |
The fitness is defined by the covered distance of the organism. It is an aggregate fitness function Nelson et al., 2009 that evaluates the organism’s performance as a whole. Although the organisms might achieve advancements early in the evolutionary run, there is a bootstrapping problem. For example, the downward proxi... |
Results and discussion 4.1 Inverted pendulum The evolutionary runs of the inverted pendulum were performed with a population of 200 randomly initialized controllers. The AHHS was set to 15 hormones. For AHHS1 60 rules were used and 15 for AHHS2. The runs were stopped after 200 generations. Linear proportional selection... |
For this task we configured AHHS with a left and a right compartment. The left compartment incorporates the left actuator [MATH] , the left proximity sensor, the sensors giving the angles of the pendulum when it is in the left half etc. and for the right compartment respectively. |
The comparison of the best controllers of each run is shown in Fig. 5(a) . In this scenario, AHHS2 performs significantly better than AHHS1 although in terms of evolution speed there is no significant difference (see Fig. 5(b) ). The AHHS2 design is the better choice in this task. The cause of the advantage of AHHS2 ov... |
One of the best evolved AHHS2 controllers showing interesting behavior is analyzed in the following . While it is not possible to keep the pendulum in the upper equilibrium for longer time due to noise, the controller still tries to maximize the time the pendulum is close to the upper equilibrium mostly by small displa... |
See Fig. for the sensor, hormone, and actuator dynamics. This sample run begins with an initial ( [MATH] move of the crab from the center to the outer left due to transient dynamics of [MATH] in the left compartment (see Fig. 7(a) ). This motion implements the up-swinging of the pendulum and is followed by ten small di... |
4.2 Gait learning The evolutionary runs of the gait learning task were performed with a population of 20 randomly initialized controllers. The configuration of the AHHS was set to 5 hormones. The number of rules was varied between 20 and 300. The runs were stopped after 200 generations. Linear proportional selection wa... |
In the first version of the simulation (damped joints), the evolved behaviors reach high fitness values for all investigated settings of the AHHS (see Fig. ). Directly approaching the wall yields a fitness of about 0.7, getting one half of the modules over the wall yields a fitness of 0.8, and a fitness of above 1 is r... |
Using the second version of the simulation (fixed joints), we have tested smaller differences in the number of rules between AHHS1 and AHHS2. The results show that the more realistic simulation of the joints complicates the evolution of fast locomotion. However, the favoring of caterpillar-like locomotion is reduced si... |
The comparison of the best evolved behaviors is shown in Fig. 10(a) and the speed of evolution is shown in Fig. 10(b) . 55% of the AHHS2-runs with 50 rules and 38% of the AHHS1-runs with 80 rules reach a best fitness that is within 80% of the theoretical maximum fitness of about 1.7. Significant results are only reache... |
One important aspect in the differences between the two controller types seems to be the different triggering of rules in AHHS1 and AHHS2. The behaviors of AHHS1 clearly show more fast-paced movements. With damped joints this seems to be a disadvantage as smooth movements are less likely. Using the fixed joints this so... |
The evolved structures are complex and the underlying processes are often counter-intuitive. The in-depth analysis of individual behaviors is alleviated by considering the number of steps a rule has been active (triggered). Typically, about one third of the rules trigger never or very seldom. |
4.3 Post-evaluation and analysis We have investigated the behavior of one of the best evolved AHHS2 controllers in the second version of the simulator. It shows a dynamic caterpillar-like motion . It is noticeable that the rules show characteristics of specialization and optimization. For example, often the (floating) ... |
[MATH] , and [MATH] . The angle of the hinge is mainly controlled by hormones [MATH] and [MATH] (see Fig. 11(a) . High values of [MATH] turn the hinge towards [MATH] while any value of |
[MATH] turns the hinge towards [MATH] . As a reinforcing effect there is a hormone rule that decreases [MATH] , if [MATH] [MATH] shows the influence by diffusion of hormones through the organism (see Fig. 11(b) . A decreasing concentration in the back module is consequently followed by a decrease in the second last, mi... |
Finally, we investigated the influence of mutations. The leading design paradigm of AHHS2 was to improve the causality of the mutation operator (small changes in genome result in small changes in the behavior). This was done exemplarily by taking an evolved controller from each type. For both we produced 35 controllers... |
Conclusion and Outlook We have reported the application of our hormone control approach to the domain of evolutionary modular robotics. The automatic synthesis of controllers, that facilitate locomotion of organisms built from five robot modules, has been effective in a majority of the evolutionary runs. Almost all evo... |
Whether the redesigned controller AHHS2 is generally superior to the original AHHS1 design is still an open question. However, in case of the inverted pendulum it performs significantly better. In the gait learning scenario AHHS2 shows a higher diversity and behaviors with smoother movements resulting in more reliable ... |
There are many open issues and this research track is rather at its beginning. Our future research will include the following. The different possibilities of initializations need to be investigated extensively. For example, the controllers could be initialized with specialized sensor, hormone, and actuator rules (i.e.,... |
# Source: arxiv 1011.4675 # Title: Nonlinear threshold Boolean automata networks and phase transitions # Sections: all # Downloaded: 2026-03-02T08:58:30.763930+00:00 |
Nonlinear threshold Boolean automata networks and phase transitions Abstract In this report, we present a formal approach that addresses the problem of emergence of phase transitions in stochastic and attractive nonlinear threshold Boolean automata networks. Nonlinear networks considered are informally defined on the b... |
[MATH] Université Joseph Fourier de Grenoble, TIMC-IMAG, AGIM, Faculté de médecine, 38700 La Tronche, France [MATH] Université d’Évry – Val d’Essonne, IBISC, 91000 Évry, France |
[MATH] Institut rhône-alpin des systèmes complexes, IXXI, 69007 Lyon, France Introduction The model of deterministic Threshold Boolean automata networks (called TBANs for short in the sequel) has been developped in the 1940’s by McCulloch and Pitts in MP43 as a way to represent logically the interactions between neuron... |
After a presentation of important definitions for the study in Section , new theoretical results of phase transitions are given. |
Model definitions Although this work focuses on nonlinear TBANs whose architecture is partially defined in a part of the lattice on [MATH] , let us present TBANs from the general point of view. Let [MATH] be such an arbitrary network. [MATH] is composed by [MATH] nodes interacting over time through a labelled digraph [... |
[MATH] is the set of nodes, elements of [MATH] , whose states are valued in [MATH] [MATH] when the node is inactive and [MATH] when it is active) and [MATH] is the set of arcs linking elements with each others. A TBAN is characterised by: |
an interaction matrix [MATH] of order [MATH] : it defines the structure of [MATH] and each coefficient [MATH] is the label of arc [MATH] |
of [MATH] and gives the interaction weight node [MATH] has on node [MATH] . If [MATH] is null, then [MATH] , else node [MATH] is said to be a neighbour of node [MATH] and we note [MATH] . In this case, node [MATH] is called an inducer/activator (resp. repressor/inhibitor) of node [MATH] if [MATH] |
(resp. [MATH] ); threshold vector [MATH] of dimension [MATH] : each element [MATH] is called the activation threshold of node [MATH] |
[MATH] local transition functions which define the local evolution of each of the nodes in the TBANs. The general concept of the local evolution of a node [MATH] , namely the calculation of its state at time [MATH] being given [MATH] |
and the state of any node [MATH] at time [MATH] , is the following: if the potential of [MATH] at time [MATH] i.e. , the sum of the interaction weights received from its active neighbours, is greater than (resp. not greater than) its activation threshold then its state at time [MATH] equals [MATH] |
(resp. [MATH] ). Thus, if we denote by [MATH] the state of node [MATH] at time [MATH] the local transitions functions are: [EQUATION] |
where [MATH] represents the Heaviside (or sign-step) function and is such that [MATH] An application [MATH] is called a configuration of [MATH] . In other words, the vector [MATH] is the configuration of [MATH] at time [MATH] |
In the sequel, in order to highlight the emergence of phase transitions from the dynamical behaviour of TBANs, we will give a particular attention to the notion of boundary conditions. We will explain this later. Nevertheless, since we focus on TBANs on [MATH] , let us present general definitions of the notions of cent... |
Definition 1 Let [MATH] an arbitrary digraph. The boundary of [MATH] is the set of its sources. Let [MATH] and [MATH] be two distinct vertices of a digraph [MATH] . The |
distance [MATH] is the length of the shortest path linking [MATH] to [MATH] . If there is no path from [MATH] to [MATH] [MATH] is defined as equal to |
[MATH] Definition 2 Let [MATH] an arbitrary digraph. The eccentricity [MATH] of a non isolated vertex [MATH] is the maximal distance less than [MATH] |
from [MATH] and every other vertex of [MATH] , such that [MATH] Definition 3 Let [MATH] an arbitrary digraph. The centre of [MATH] is the set of its vertices of minimal eccentricity. |
In this report, we differentiate the notions of neighbourhood and strict neighbourhood of nonlinear two-dimentional TBANs according to the following definitions. |
Definition 4 Let [MATH] be a two-dimensional TBAN on [MATH] . The neighbourhood [MATH] of node [MATH] is the set composed of nearest-neighbours nodes (i.e., nodes at distance [MATH] to [MATH] ) of [MATH] and [MATH] itself. |
Definition 5 Let [MATH] be a two-dimensional TBAN on [MATH] . The strict neighbourhood [MATH] of node [MATH] is such that [MATH] |
Let us now define the properties of isotropy and translation invariance of the two-dimensional TBANs considered. Definition 6 Let [MATH] be a two-dimensional TBAN on [MATH] [MATH] is isotropic |
if and only if: [EQUATION] Definition 7 Let [MATH] be a two-dimensional TBAN on [MATH] [MATH] is translation invariant if and only if, given [MATH] , it holds that: |
[EQUATION] As a consequence, TBANs considered in this study are symmetric i.e. , they are such that [MATH] . According to these properties of isotropy and translation invariance, it is easy to see that Definition can be applied directly to nonlinear TBANs on [MATH] . Conversely, the set of boundary obtained from the ap... |
Definition 8 The external boundary (called boundary for short), denoted by [MATH] , is the set of nodes of [MATH] at distance [MATH] (in terms of distance in [MATH] ) to at least one node of [MATH] such that: |
[EQUATION] An illustration of centre and boundary of a TBAN on [MATH] is given in Figure TBANs in the sequel are attractive i.e. , they are such that |
[MATH] and [MATH] . Note also that activation thresholds are all fixed to [MATH] and that auto-interaction potentials are always taken into account. Thus, the [MATH] ’s play the role of activation thresholds. Furthermore, as said in the introduction, nonlinearity is added in the model of TBANs considering that interact... |
[MATH] . Indeed, we consider also coalition potentials . For instance, given a node [MATH] of a TBAN [MATH] at time [MATH] whose state is not known, if we consider that the evolution of node [MATH] |
takes into account coalition of neighbours couples, the interaction potential of node [MATH] equals [MATH] , where [MATH] defines the interaction weight that the couple of active nodes [MATH] and [MATH] has on [MATH] . Remark that the [MATH] ’s correspond to thresholds (considering them separately) and that the [MATH] ... |
Let [MATH] be the temperature parameter . We give the following notations of interaction potentials for every node [MATH] of an arbitrary TBAN [MATH] to ease the reading: |
[MATH] , called singleton potential a function of the auto-interaction weight of an arbitrary node [MATH] (always taken into account); |
[MATH] , where [MATH] couple potential , a function of interaction weights received by node [MATH] from its strict nearest neighbours; |
[MATH] where [MATH] , called triple potential , a function of interaction weights received by node [MATH] from couples of its active neighbours; |
[MATH] , where [MATH] , called quadruple potential , a function of interaction weights received by node [MATH] from triples of its active neighbours; |
[MATH] , where [MATH] , called quintuple potential , a function of interaction weights received by node [MATH] from quadruples of its active neighbours. |
Definition 9 stochastic TBAN [MATH] of order [MATH] on [MATH] is a TBAN whose local transition function [MATH] calculates the probability for node [MATH] to be at state [MATH] at time [MATH] knowing the configuration projected on its neighbourhood [MATH] at time [MATH] and taking into account [MATH] -uple, |
[MATH] -uple, …, [MATH] -uple potentials, with [MATH] such that: [EQUATION] where [MATH] is the nonlinear term such that: [EQUATION] |
Remark that, in the case of TBANs of order [MATH] i.e. [MATH] ), if [MATH] tends to [MATH] , then the stochastic local transitions functions defined in are equivalent to the deterministic one defined in Equation . Before going further, let us insist that, from Definition , we derive that nonlinear TBANs studied in this... |
Theoretical approach and phase transitions Let us recall that TBANs considered in the sequel are isotropic, translation invariant, nonlinear. Moreover, we add that they are attractive. Given a stochastic TBAN [MATH] , that means that [MATH] |
3.1 Projectivity matrix Definition 10 cylinder [MATH] is a configuration x such that: [EQUATION] If [MATH] denotes the invariant measure of a stochastic TBAN [MATH] composed of [MATH] |
nodes, indexed from [MATH] to [MATH] , such that [MATH] tends to infinity, we have the following projectivity and conditional relations. Indeed, we can write |
projectivity equations such that: [EQUATION] where [MATH] is the probability to observe the configuration [MATH] . We calso write conditional equations i.e. , the Bayes formulas) such that: |
[EQUATION] where [MATH] denotes the conditional probability that state of node [MATH] equals [MATH] knowing cylinder [MATH] such that: |
[EQUATION] Consider [MATH] such that nodes of [MATH] are ordered according to the lexical order of their indices. For every subset [MATH] of [MATH] of size [MATH] , we denote by [MATH] the minimal index of nodes belonging to [MATH] Projectivity matrix |
[MATH] of order [MATH] is defined such that (i) the [MATH] first lines contain respectively the coefficients of the projectivity equations for any of the [MATH] different couple [MATH] and (ii) the last line contains the coefficients of the conditional equation [MATH] that calculates the global probability for the cent... |
[EQUATION] From this system of equations, it is easy to write: [EQUATION] where [MATH] [MATH] [MATH] , …, [MATH] , …, [MATH] , …and [MATH] .. |
Projectivity and conditional equations are in general linearly independent. However, under specific parametric conditions such as conditions of non uniqueness of the invariant measure, that is not the case. From the work of Dobrushin in Dob68b Dob68a Dob68c Dob69 |
in the framework of random fields, we derive the following definition. Definition 11 Let [MATH] be an arbitrary stochastic attractive TBAN. Let |
[MATH] (resp. [MATH] ) be a boundary of [MATH] composed of nodes whose state is fixed to [MATH] (resp. [MATH] ). The dynamical behaviour of [MATH] admits a phase transition if and only if the invariant measure of the Markov chain associated to [MATH] does not equals that of the Markov chain associated to [MATH] |
From Equations and Definition 11 , we can directly write the following proposition. Proposition 1 Given [MATH] a stochastic attractive TBAN, the nullity of the determinant of its associated projectivity matrix [MATH] is a necessary condition for [MATH] to admit a phase transition in its dynamical behaviour. |
Lemma 1 Dem81 The nullity of the determinant of a projectivity matrix is characterised by: [EQUATION] Because of our hypotheses of isotropy and translation invariance, it is interesting to note that we can use the spatial Markovian property in order to make easier solving the system of projectivity equations. The spati... |
depends only on the states of its neighbours, which allows to reduce [MATH] to the centre [MATH] strict neighbourhood, namely [MATH] . Then, it is simpler to build the associated projectivity matrix [MATH] of order [MATH] |
3.2 Results Basing our approach on Proposition , in this section, we prove the existence of parametric conditions of stochastic nonlinear TBANs that admit phase transitions. |
First, from the spatial Markovian property of TBANs and because [MATH] , the right member of the equation of Lemma can be written pairing the subsets [MATH] and |
[MATH] , namely considering that: [EQUATION] By hypothesis, nonlinear term [MATH] is symmetric and equals [MATH] . The symmetry property of the nonlinear term means that [MATH] |
Lemma 2 Given [MATH] a nonlinear TBAN of order [MATH] and [MATH] a symmetric nonlinear term such that [MATH] , we have: [EQUATION] |
Proof. Let us note [MATH] . Trivially, developing the left member of Equation by definition of nonlinear terms, we can write: [EQUATION] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.