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2.2 Vesicle geometry, connectivity and networking The three dimensional (3D) vesicle connection opportunities and complex internal wave reactions represent a rich computation substrate. Such depth raises difficulties when attempting to manually explore computation modalities. To reduce the complexity to a level where m... |
In a previous study we have shown that logic circuits can be created with uniform discs arranged in hexagonal networks , hexagonal packing being the most efficient method of sphere (disc) packing. Further opportunities to modulate wave fragment behaviour are presented when disc size, connection angle and connection eff... |
2.3 Computer simulation experiments Increasing the relative disk size can be used not only to allow space for wave fragment collisions (Fig. a) but other effects are also apparent. Figure illustrates the front development of the same wave through progressively smaller terminating discs. In the larger discs the wave fra... |
Wave fragments cannot survive when the fragment mass drops below some critical level and this is evident when comparing progressively smaller aperture sizes with fixed size discs. In our system and with a disc radius of 28 units the critical level surrounds an aperture gap of 4 units. Below that fragments do propagate ... |
. We have found that using a narrow beam aperture in orthogonal networks where wave fragments do not normally propagate into connected perpendicular discs particularly useful in creating simple logic gates (Sect. 3.1 ). Furthermore, diode junctions can be created when networks of narrow (type J1) and broadband (type J2... |
further explorations in this work rely on combining just the two types, narrow (J1) and broadband (J2) (Fig. c). Results 3.1 Elementary logic gates |
Electronic logical gates form the building blocks of more complex digital circuitry forming the foundations of complex high level components such as microprocessors. Although we do not envisage creating traditional vonn Neumann architecture microprocessors in BZ vesicles, the ability to create simple logic gates with B... |
capable of such processing. Logic gates and composite circuits of logic gates have been created several times before using the BZ substrate, for example |
. Here we illustrate a selection of key gates can be created using nothing other than interconnected BZ discs. Figure illustrates the operation of the most elementary of gates the inverter (NOT gate). The circuit operation starts with the simultaneous application of the circuit input (left most disc) in conjunction wit... |
The operation of an AND gate and the inversion, the NAND gate are shown in Fig. & Fig. . The result of a wave collision in the NOT gate was exploited to deflect and extinguish the source wave into the disc edge, whereas in the AND gate the collision between the two inputs results in 2 perpendicular fragments, one of wh... |
A NAND gate can created by combining the NOT gate and the AND gate Fig. . NAND gates are known as universal gates since all other gates can be created from arrangements of NAND gates alone. The NOT gate (Fig. ) is integrated below the AND gate in the lower row (Fig. ) where the activity of a horizontal source signal in... |
The OR gate is used to detect the presence of one or more signals. A logical ‘1’ on any input results in an output (Fig. 10 ). Common amongst all these gates, the output value of a logical ‘1’ or ‘0’ as indicated by the presence or absence of wave is only valid at a specific point in the development and in these instan... |
The XOR gate is used to signal a difference between signals, producing an output when inputs alternate regardless of the composition of the difference. Figure 12 illustrates the BZ disc implementation along with the inversion NXOR in Fig. 11 |
3.2 Half adder The half adder is a sub-system used in binary addition circuits. The half adder adds two binary digits and when connected with another half adder creates a full 1 bit adder. One bit adders can then in turn be connected together to make [MATH] bit adders (Fig. 13 ). A half adder can be constructed from a ... |
A 1 bit half adder created from BZ discs can also be constructed from connecting a BZ disc AND gate and XOR gate (Sect. 3.1 ). Figure 14 shows the BZ disc conjunction for the half adder circuit. The input [MATH] |
needs to be repeated on the other side of input [MATH] in order for this circuit to work. This is necessary in order to overcome the signal passing problem , a universal problem for systems where signals propagate along specific planular channels. There are two ways to overcome this problem, either add identity to the ... |
illustrates one such temporal separation strategy, where signal [MATH] passes over signal [MATH] but becomes shifted in time. The circuit operates with two types of apertures, one that creates a narrow beam (type J1) wave and another that creates a broad beam (type J2) wave. Signals [MATH] [MATH] travel from bottom to ... |
[MATH] on the right. The signal [MATH] is split at the junction to the first disc and a fragment [MATH] travels horizontally towards |
[MATH] . Meanwhile [MATH] is already traversing the first disc and has progressed into the final disc before [MATH] crosses the [MATH] path allowing [MATH] to cross [MATH] . A time shift [MATH] now exists between |
[MATH] and [MATH] so any further processing between [MATH] and [MATH] must therefore delay [MATH] by [MATH] . This strategy relies on allowing sufficient time for the refractory tail of signal [MATH] to have a negligible effect on [MATH] . If the signals are not sufficiently separated then [MATH] will extinguish [MATH]... |
Venturing into 3 dimensions (3D) resolves the signal passing problem all together, allowing signals to be routed vertically. At this stage only 2 dimensional (2D) structures of discs have been explored, but these are approximations of our target computation node, a 3D BZ vesicle. In this current 2D perspective, overcom... |
Another specific solution for the half adder circuit which removes the need to repeat one of the inputs is possible if all the signal modulation techniques are exploited; disc connection geometry, disc size and aperture efficacy (Sect. 2.2 ). Figure 17 demonstrates a half adder design where most of the processing occur... |
3.3 Memory cells Memory is an essential facet of both adaptive behaviour in Nature and in synthetic computation. It permits animals and machines to build an internal state independent from the current external world state. In this section we present a simple 1 bit volatile read write memory cell constructed entirely wi... |
in so much that the existence or absence of a rotating wave represents the setting or resetting of 1 bit of information. When two BZ waves progress in opposite directions around an enclosed channel, loop or ring of connected discs, then at some point the two opposing wave fronts will meet and are always mutually annihi... |
(Fig. 19 b). Furthermore the rotating wave can be terminated by the injection of another asynchronous wave rotating in the opposite direction (Fig. 19 .c). Opposing inputs into a loop are analogous to a memory set or reset Reading the state of the cell without changing the state can be achieved by connecting another ou... |
The loop and a unidirectional gate (diode) are the two key constructions of this type of memory cell. Unidirectional gates in BZ media have previously been created by exploiting asymmetric geometries or chemistry on either side of a barrier |
. An alternative design is possible however using discs connected with different apertures. Figure 18 illustrates a diode constructed from a right angle junction connected by a broad band (type J2) aperture to a vertical column and by a narrow beam (type J1) aperture to a horizontal row. Signal flow is only possible fr... |
and other functions. In practice fine control of wave diffusion is however difficult to achieve and hence we have restricted our choice between just two types. |
As the rotating wave progresses around the loop in the memory cell illustrated in Fig. 19 , the opposing input cell also inadvertently becomes an output cell. This may be undesirable in some designs but can be easily resolved by adding another pair of diode junctions to the circuit. Figure 20 shows such a design, where... |
Discussion Our research is an exploratory component within a collaborative project that aims to create a lipid encapsulated BZ vesicle and organise those vesicles into a functional network. The lipid membrane and the non-linear oscillatory nature of the BZ medium encodes some of the features apparent in biological info... |
Another analog between Natural processing and vesicles is the relationship to artificial Life. The cell is the building block of all known life on Earth. Mechanistic explanation for the genesis of Life and the cell remain elusive, but a key aspect of cell morphology is the concept of a cell wall and the ability to sepa... |
. One theory (amongst many) is a role for lipids in the spontaneous formation of simple cells and hence the development of a separate entity different from the surrounding environment. Whether spontaneous lipid cell formation played a role in early Life remains to be seen, but the principle of an enclosing membrane to ... |
. Computer simulations of such behaviour of similar simple processing units, known as ‘Cellular Automata’ (CA) has been extensively studied |
and can lead to interesting Life like behaviour The exploration in the computation modalities of BZ encapsulated vesicles is promising then on (at least) two levels. The macro scale of organised behaviour (classification of this work) and the small scale emergent oscillatory behaviour analogous to cellular automata. Th... |
Future work Experiments are currently in progress to replicate these simulation results in real chemistry (an example of the AND gate is shown in the appendix (Sect. )). Our goal is to explore computational modalities of interconnected discs and vesicles. In doing so we hope that such work will both inspire novel chemi... |
Summary Creating components, gates and circuits commonly used in the design of discrete conventional computers within geometrically constrained constructions containing sub-excitable BZ media has been extensively studied both in simulation and real chemistry. This work has shown that some of the previous circuits, logi... |
Appendix An example of some initial results from on-going laboratory work is shown in Fig. 21 a. An identical background image intensity is proportional to [MATH] and is projected onto a thin layer of silica gel containing a photo sensitive (Ru(bpy) [MATH] ) catalyst for the BZ reaction. The gel is submerged in catalys... |
Acknowledgements The work is part of the European project 248992 funded under 7th FWP (Seventh Framework Programme) FET Proactive 3: Bio-Chemistry-Based Information Technology CHEM-IT (ICT-2009.8.3). The authors wish to acknowledge the support of the EPSRC grant number EP/E016839/1 for support of Ishrat Jahan. We would... |
# Source: arxiv 1010.4999 # Title: On the Stability of Swarm Consensus Under Noisy Control # Sections: all # Downloaded: 2026-03-03T02:01:57.109997+00:00 |
On the Stability of Swarm Consensus Under Noisy Control Abstract Representation of a swarm of independent robotic agents under graph-theoretic constructs allows for more formal analysis of convergence properties. We consider the local and global convergence behavior of an [MATH] -member swarm of agents in a modified co... |
Introduction Control of a swarm of robots may be achieved in many different ways. The seminal work of Reynolds (1987) gave the first algorithmically efficient representation of a flock with very simple rules. Reynolds’ work was ground-breaking, allowing simulation of the group dynamics involving very large numbers of m... |
in the area of behavioral robotics predates Reynolds’ research, though that work was focused mostly on individual robots or robot pairs. This idea was further developed by additional research in swarm robotics by Mataric |
and Parker . Early work was quite ad-hoc, but behavioral control continues to find an increasingly rigorous mathematical framework. For variations on this topic, see, e.g., |
Swarming laws may be applied a variety of cooperative-robotics settings. Each of these settings, though, has a primary goal of achieving agreement or consensus . The specific meaning of agreement in each scenario is different, but in general the implication is that the states of each robot evolve in a coordinated way t... |
, robot rendezvous , plume localization , and robot segregation Formation control may be accomplished under several control regimes. Clearly, a centralized controller could drive robots to a desired configuration under straightforward trajectory control. It is much more difficult and interesting to develop such capabil... |
, Balch , Yamaguchi , and Antonelli Potential field functions are primarily used to balance competing objectives. Generally constructed as inverse-square functions of distance (inspired by physical models of charged particles), potential field functions yield a natural method of balancing the relative importance of suc... |
, Leonard , Olfati-Saber and Murray , Bruemmer , and Mai II Graph Representation In recent years, several researchers have explored the use of graph-theoretic concepts to provide a more abstract distributed control. Early works in graph-theoretic formation control by Olfati-Saber and Murray |
and Tanner, Jadbabaie, and Kumar showed promise. A brief discussion of graphs is in order. Graph notation varies in the literature; the following notation will be used in this paper. An undirected graph [MATH] of order [MATH] consists of a set of [MATH] nodes or vertices [MATH] and a set of connections or edges [MATH] ... |
When it is clear from the context, the subscript will be dropped, e.g., [MATH] [MATH] , and [MATH] path from node [MATH] to [MATH] is an ordered set of nodes, starting at [MATH] and ending at [MATH] , such that each consecutive pair is in [MATH] . If a path exists from every pair of nodes [MATH] in the graph, the graph... |
The degree of a node is the cardinality of [MATH] , denoted by [MATH] . The vector of degrees of the nodes of [MATH] is denoted [MATH] . If [MATH] , i.e., every node is connected to every other node except itself, then graph [MATH] is called complete. A complete graph of order [MATH] is also denoted by [MATH] |
The complement of graph [MATH] is denoted [MATH] , and is defined on the same node set as [MATH] but with all disconnected nodes in [MATH] are connected, and all connected nodes in [MATH] are disconnected. Formally, given a graph [MATH] , its complement is [MATH] |
The adjacency matrix is an important data structure in the analysis of graphs. The adjacency matrix, for an undirected graph, is a representation of the neighboring relationships among all agents. The adjacency matrix is defined as shown in Eq. ( ). |
[EQUATION] The adjacency matrix of undirected graph [MATH] is symmetrix and positive semi-definite. Most importantly though, the adjacency matrix leads to the graph Laplacian, defined as in Eq. ( ): |
[EQUATION] Note that the degree vector, [MATH] , is equivalent to the vector of row-sums of [MATH] . The eigenvalues of the graph Laplacian give information regarding the level of connectivity of the graph. |
By its definition, [MATH] will be rank deficient and will have at least one zero eigenvalue with eigenvector [MATH] . Beyond this, the total number of zero eigenvalues indicates the number of connected components of the graph. For example, [MATH] achieves maximum rank of [MATH] when the graph is fully connected, thus t... |
The values of the eigenvalues also have significance. If the set of eigenvalues, [MATH] are re-ordered by magnitude, e.g., [MATH] , then the value of [MATH] indicates the algebraic connectivity of [MATH] . As already mentioned, [MATH] if and only if the full graph is not connected. If [MATH] however, its value is an in... |
[MATH] ; in fact, the lower bound for a connected graph is [MATH] Closely related to the above definitions is the weighted graph, wherein each edge has an associated weight. Thus, the weighted adjacency matrix is defined as in Eq. ( ): |
[EQUATION] These weights may be representative of many different physical quantities, e.g., they may represent the physical distances between the nodes in Euclidean space, [MATH] , a cost (energy, time, etc.) associated with traversing the edge, etc. The weighted graph Laplacian, [MATH] , is defined in the same way, i.... |
Additional information found within the spectra of [MATH] [MATH] [MATH] , and [MATH] , are discussed in the 1988 monograph of Cvetković et al |
, as well as in . These books will be used as guides in the development of graph-based control in the proposed research. Graph-theoretic control of formations is not novel; the reported work of Tanner, Jadbabaie, and Kumar |
expounded on this topic. More recently, an extensive set of research has been conducted regarding the controllability of such swarms under graph-theoretic constructs. Mesbahi published several papers |
on controllability within a swarm for agreement , along with several related works by Hatano, Das, Rahmani, Chen, Kim, and Tan relating to the study and manipulation of the graph Laplacian and its spectra for swarm control |
. Of particular focus in the proposed research is the extension of such graph-theoretic controllability concepts to random graphs along the lines of Hatano and Mesbahi |
. The extension to the dual of controllability, observability, has additionally been explored by Mesbahi and Zelazo The key element in the works of Mesbahi is the notion of the state-dependent dynamic graph. The construct of import is a mapping, [MATH] , from the collective system state [MATH] , to the graph [MATH] |
The state-dynamic graph mapping [MATH] may be defined in many ways, provided [MATH] is a distance function. A commonly used and intuitive mapping is the Euclidean distance, |
[EQUATION] for a specified [MATH] . The 2-norm has an obvious physical meaning that is directly applicable to real connectivity problems such as wireless networking (where [MATH] indicates the communication range), or a laser rangefinder (or rangefinder pair) with full [MATH] angular coverage or an array of vision sens... |
One such coordinated goal is aggregation. For example, see Equations ( ) and ( ) for a swarming law of this type, where [MATH] is the [MATH] -dimensional state of agent [MATH] at time-step [MATH] [MATH] is the self-determined control of agent [MATH] at time-step [MATH] [MATH] is an objective function possibly independe... |
[EQUATION] [EQUATION] In order for the aggregation to be guaranteed to occur for [MATH] , the initial system graph [MATH] must be connected, i.e., |
[EQUATION] Alternatively, if [MATH] and the objective function [MATH] represents a random walk (e.g., a sample from a uniform distribution on [MATH] taken during time [MATH] ), the probability of successful agreement is |
[EQUATION] for finite [MATH] and closed space [MATH] . Note also, though, that for larger values of [MATH] , motion of clusters within the graph will decrease if the objective function, [MATH] , is truly a random walk. Thus modifications to the objective function should be made to guarantee continued motion of the swar... |
, Matarić , or Parker This research work will consider the use of graph-theoretic values, such as the degree vector [MATH] or the eigenvalues of the graph Laplacian [MATH] in manipulating the relative scaling (i.e. importance) of the swarming and objective laws. Before considering this, though, we must first consider t... |
III Proof of Consensus Stability Let [MATH] [MATH] ) be compact and convex (and so simply connected). For [MATH] [MATH] and [MATH] Let [MATH] for some [MATH] . Here, [MATH] is the vision radius . Typically, [MATH] . Note [MATH] since [MATH] |
The dynamics of the particles are given by the agreement algorithm plus a noise term [EQUATION] Because of the noise term, the particles do not converge to consensus. However, once all the particles are close to one another, they stay close. |
Proposition 1 Taking the dynamics defined by Eq. , suppose the [MATH] are i.i.d. random variables drawn from a distribution that is absolutely continuous with respected to lebesgue measure and supported on some open set [MATH] containing the origin. If [MATH] , then |
[EQUATION] Proof: [EQUATION] We now wish to show that eventually the particles must be close to one another. Lemma 1 Taking the dynamics defined by Eq. , suppose the [MATH] are i.i.d. random variables drawn from a distribution that is absolutely continuous with respected to lebesgue measure and supported on some open s... |
[EQUATION] Proof: Suppose not. Then there is an open [MATH] , some agent [MATH] , and some time [MATH] such that [MATH] for all [MATH] Take [MATH] such that [MATH] contains the ball of radius [MATH] centered at the origin. Define the sets [MATH] , and let |
[EQUATION] be the center of agent [MATH] ’s communication group at each time [MATH] . For [MATH] , if [MATH] , then [MATH] since [MATH] is non-empty and open and the [MATH] are drawn from a measure absolutely continuous with lebesgue measure. So we must have [MATH] for all [MATH] ; i.e., [MATH] for [MATH] Now, each [MA... |
Theorem 1 If [MATH] , there is a random time [MATH] such that [EQUATION] for all [MATH] Proof: Take any ball of radius [MATH] [MATH] . By the lemma, there is a positive constant [MATH] and a deterministic time [MATH] such that [MATH] for all [MATH] Then set |
[EQUATION] The [MATH] are Bernoulli random variables indexed by [MATH] and [MATH] Almost surely, there is [MATH] such that [EQUATION] |
Thus set [MATH] and apply Proposition 1. III-A Discussion The theorem takes advantage of the random walk induced by the noise. In the appropriate product space, there is a unique Lyapunov attractor, and eventually the dynamics will find it. Significantly, noisy perturbations considered are quite general, so it’s possib... |
The convexity of the space [MATH] is never used in the proof. It is needed so that the averages in the agreement algorithm are well-defined. If we’re careful about defining line of sight and which agents participate in the averaging, the assumption can be relaxed to a space that is compact with open interior. This allo... |
IV Probabilistic Connectivity Experience with real systems quickly reveals that sensing and communication ranges are rarely deterministic with range. Intuitively, likelihood of successful communication or detection is a function of range, but is more accurately modeled as a decreasing probability function. |
Intuitively, the resolution of most spatial sensors, e.g. cameras or scanning laser rangefinders, is defined in angular space. Thus the spatial resolution decreases with distance; correspondingly, the minimum detectable feature size increases with distance. Based on this observation, we posit that the detection probabi... |
The development of network technologies leading to the World Wide Web have created a robust system for communication. When networks are wired, communication can generally be assumed to be reliable to within a small amount of timeline jitter. However, with mobile roboitcs when the communication must be performed wireles... |
With these observations, we generalize the connectivity to the mathematically more tractable (with respect to continuity of derivatives) decreasing exponential function. The most basic probabilistic connectivity function we will consider is |
[EQUATION] where [EQUATION] and [MATH] is a shaping constant. Clearly though, this function is non-zero for [MATH] , thus the asymptotic convergence on a non-infinite domain can be easily seen. This is not realistic, as there must be a finite cut-off for these connectivity functions. Thus we define a maximum range, [MA... |
[EQUATION] Simulation Evolution of swarms under the influence of the averaging and probabilistic edge-formation described above have been investigated in simulation. A practical issue observed during these simulations is that the averaging effect of the consensus rule results in a decrease in the large-scale random mot... |
In response, a navigation rule is introduced where in the magnitude of the random perturbation is a function of the individual agent’s degree. With this control, individual agents in the interior of a cluster exhibit a larger magnitude of random motion, such that the cluster averaging will not filter the motion. Additi... |
VI Planned Experiments For verification of the methods described in this paper, an experimental setup with real robots is employed. The arena for robot exploration is an area that is roughly [MATH] [MATH] . The agents in this experiment are Create mobile robots from iRobot. A motion capture video system comprised of tw... |
These robots are clearly governed by dynamics that are more complex than single-integrator particles. Experiments will be conducted utilizing real on-board cameras and scanning laser rangefinders (see |
) to ascertain the limitations of this controller under these dynamics, as well as to understand the rate of convergence under these considerations. Additionally, the probabilistic functions proposed in this paper will be empirically verified. Specifically, the shapes of the sensing functions in two (or more) dimension... |
VII Conclusion Swarm agreement under certain probabilistic connectivity limitations has been proven to be asymptotically convergent for bounded spaces. A controller has been developed that conforms to the assumptions in the proof, and simulations verify the convergence under such a controller. Planned experiments will ... |
VIII ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the Army Research Office under grant number W911NF-08-1-0106. |
# Source: arxiv 1011.3912 # Title: Artificial Hormone Reaction Networks: Towards Higher Evolvability in Evolutionary Multi-Modular Robotics # Sections: all # Downloaded: 2026-03-03T01:55:55.598823+00:00 |
Artificial Hormone Reaction Networks: Towards Higher Evolvability in Evolutionary Multi-Modular Robotics Abstract The semi-automatic or automatic synthesis of robot controller software is both desirable and challenging. Synthesis of rather simple behaviors such as collision avoidance by applying artificial evolution ha... |
Introduction The (semi-)automatic synthesis of robot controllers with artificial evolution belongs to the software section of evolutionary robotics Cliff et al., 1993 . The main challenge in this field is the curse of complexity because an increase in the difficulty of the desired behavior results in a significantly su... |
Concerning the problem of finding appropriate controller designs a pleasant trend can be observed in recent literature. The most prominent candidate is presumably the HyperNEAT design Stanley et al., 2009 Clune et al., 2009 . It is based on artificial neural networks (ANN) but combines the ‘search for appropriate netwo... |
In this paper, we analyze a controller design called Artificial Homeostatic Hormone Systems (AHHS) that is based on hormones only and was introduced before Hamann et al., 2010 Schmickl et al., 2010 Schmickl and Crailsheim, 2009 Stradner et al., 2010 2009 . AHHS is a reaction-diffusion approach. Sensory stimuli are conv... |
The desired main application of AHHS is multi-modular robotics SYMBRION, 2010 REPLICATOR, 2010 . In this field, autonomous robotic modules are studied, that are able to physically connect to each other, and can also establish a communication and energy connection. Hence, they form a super-robot called ‘organism’, that ... |
or Murata et al., 2008 . Therefore, the underlying idea of diffusion in our reaction-diffusion system is that hormones diffuse from robot module to robot module and establish a low-level communication. Following our maxim of trying to reach a maximum of plasticity we use identical controllers in each module independent... |
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