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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Guided self-organization can be regarded as a paradigm proposed to understand how to guide a self-organizing system towards desirable behaviors, while maintaining its non-deterministic dynamics with emergent features. It is, however, not a trivial problem to guide the self-organizing behavior of physically embodied systems like robots, as the behavioral dynamics are results of interactions among their controller, mechanical dynamics of the body, and the environment. This paper presents a guided self-organization approach for dynamic robots based on a coupling between the system mechanical dynamics with an internal control structure known as the attractor selection mechanism. The mechanism enables the robot to gracefully shift between random and deterministic behaviors, represented by a number of attractors, depending on internally generated stochastic perturbation and sensory input. The robot used in this paper is a simulated curved beam hopping robot: a system with a variety of mechanical dynamics which depends on its actuation frequencies. Despite the simplicity of the approach, it will be shown how the approach regulates the probability of the robot to reach a goal through the interplay among the sensory input, the level of inherent stochastic perturbation,

A self-organizing system is characterized by several fundamental properties: the capability to create a globally coherent pattern out of local interactions, resilience to changes in the operating enviroment, as well as non-deterministic spontaneous dynamics. On the other hand, in an engineered dynamical system, realization of a particular behavior is commonly achieved through a methodical step by step planning process with predictable outcomes. Guided self-organization (GSO) is a paradigm proposed to combine these two approaches for generating behaviors in a favorable way,

Different approaches to achieve GSO have been proposed in the last several years. Based on information theory, for example, several approaches based on predictive information, and information driven evolutionary design have been proposed [

On the other hand, studies in biological systems suggest that in particular cases, inherent stochasticity may indeed increase the performance of the systems. The phenomena where noise increases a particular performance metric of a nonlinear system are commonly referred to as stochastic resonance [

Recently, it has also been suggested that a proper level of inherent random dynamics may increase the performance of living beings as a whole by driving an appropriate behavior. Commonly, phenomena where stochasticity in an internal control structure drives the behavior of the system are observed at cellular and molecular level [

In order to understand how inherent stochastic dynamics can drive the behaviors of living beings and artificial systems like robots, it must however be noted that they are embodied systems,

The main goal of this paper is to understand how to guide self-organizing behaviors of a dynamic robot as an embodied system, by coupling the mechanical dynamics with a suitable central control structure which enables the system to balance its deterministic and stochastic dynamics. The approach taken is to use a mathematical framework known as attractor selection mechanism as the control structure [

The rest of the paper will be organized as follows: the next section will present the proposed GSO approach. Afterward, we will explain the simulated curved hopping robotic system used as a paradigmatic example of the embodied system in this paper, along with the conducted experiments and the experimental results. Finally, we will conclude and discuss related future work.

The GSO approach proposed in the manuscript is based on a coupling between the mechanical dynamics of an embodied system and a control structure known as attractor selection mechanism [

where

The potential function

The basic concept of the GSO framework presented in this manuscript is explained further in

The embodied system used as a paradigmatic example in this manuscript is a simulated curved beam hopping robot used in our previous research [

The model of the robot, along with the Cartesian axes used, is shown in _{r}_{o}_{p}_{y}_{pi=1..4}_{yi=1..4}_{pi=1..4}_{yi=1..4}_{yi}_{y2}_{pi}_{p3}_{1..4}). _{1}_{2}_{3}_{4}_{b1}_{b2}_{b3}._{i}_{g}_{x}_{y}_{z}_{x}_{y}_{z}

The model is implemented by using standard components of MATLAB Simmechanics™ [_{i}

where _{zci}

As mentioned in the Introduction, in this manuscript, we will demonstrate the proposed GSO enables the robot to perform a simple case of goal-directed locomotion. Therefore, we also build a simple model to describe the robot motion in two dimensional Cartesian space, _{i}_{i}_{i}

In order to characterize the motion in two dimensional space, we therefore use the average of
_{m}_{m}_{m}_{m}_{0}_{n}_{dm}_{Dm}_{θm}_{φm}

As in this paper we focus on enabling the robot to reach a goal in two dimensional spaces, we break the symmetry of the robot design by only use one rotating object on the right side of the robot as shown in

Knowing the frequency response of the tip alone is unfortunately not adequate to understand how the robot will behave in two dimensional spaces. We run several simulations for 10 s for several different frequencies and observe the trajectory in two dimensional spaces. _{0}_{5}_{Dm}_{θm}_{φm}_{i}_{i}_{i}

_{m}_{m}_{m}_{m}_{m}_{θm}

A similar pattern is shown by,

By comparing

Having described the ASM dynamics as well as the model and mechanical dynamics of the curved beam hopping robot, this section will explain how a GSO scheme can be realized by coupling the mechanical dynamics with the dynamics of the ASM. To be more specific, the next two sections will describe the experiment setting and results of a goal directed locomotion based on the proposed approach.

In order to understand whether and how the proposed GSO approach enables the robot to perform goal-directed locomotion, first, we couple the mechanical dynamics with the dynamis of ASM by having

where

The potential function is shown in

For the experiment, the simulation time is set as 250 (s). The simulated robot starts at (

The experimental results can be summarized by

With an approriate level of noise,

The way the different behaviors due to different noise levels affect the goal directed locomotion performance can be summarized by

To further clarify how the different behaviors shown in

It should also be noticed that because in this manuscript

Another perspective to understand the robot behavior is to see it as random behavior which is biased by sensory experience through the use of

It is also interesting to discuss how to increase the probability to reach the goal by modifying

In this paper, we present an approach to guide the self-organization of an embodied system by coupling the mechanical dynamics of the system with an internal control structure known as attractor selection mechanism (ASM). As a paradigmatic example, the embodied system used in this paper is a simulated curved beam hopping robot: a system with a variety of mechanical dynamics which solely depends on its actuation frequencies. Due to the robot’s response to different actuation frequencies, it has been shown that the mechanical dynamics will lead into two behavioral primitives: at low frequencies, the robot has a tendency to change direction at low travelling speed, while at high frequencies, the robot has tendency to keep going in the same direction at high travelling speed. However, as the frequency becomes even higher, the probability of unstable locomotion also becomes higher.

The attractor selection mechanism takes advantage of the mechanical dynamics by considering the two behavioral primitives as different attractors. By utilizing internally generated noise, the mechanism enables the robot to gracefully shift between random and deterministic behaviors, represented by the two attractors. More specifically, if the sensory input indicates that the robot is approaching its goal, the robot will behave more deterministically by having more tendencies to keep performing the current behavior. Otherwise, the robot will behave more stochastically until it finds another suitable attractor which brings it closer to the goal. It has been shown that for a particular set of mechanical design parameters and constant in the sensory feedback function, there is a suitable level of inherent stochastic perturbation,

Despite its simplicity, we have demonstrated that the proposed approach enables the robot to perform the simplest case of goal directed locomotion,

In this manuscript, we have not explored the possibility of using learning algorithms to couple the mechanical and ASM dynamics. From this perspective, it is interesting to discuss the comparison between the proposed approach, and guided-self organization in general, with classical learning approaches aiming at intelligent and animal-like behavior of robot. From a time scale perspective, it is possible to classify the commonly implemented learning approaches in robotics to evolutionary robotics, reinforcement learning and developmental robotics. Inspired by biological evolution, evolutionary robotics aims to optimize robot behavior over many iterations,

GSO is probably most related to developmental robotics and embodied intelligence concept. Developmental robotics is an emergent area of research at the intersection of robotics and developmental sciences, aiming at understanding the developmental mechanism which might lead to open-ended learning in embodied machines [

As future works, we plan to further explore the effect of different kinds of parameters, namely the mechanical design parameters, sensory feedback function parameters, level of noise, and the potential function representing the coupling between ASM dynamics and the mechanical dynamics. We will also explore the possibility of using learning algorithms to couple the mechanical and ASM dynamics, as well as to co-optimize the involved parameters in this approach to increase the performance.

This study was supported by the Swiss National Science Foundation Grant No. PP00P2123387/1, and the Swiss National Science Foundation through the National Centre of Competence in Research Robotics.

S.G.N., X.Y. and F.I. conceived and designed the experiment; S.G.N., X.Y. and Y.K. performed experiments; S.G.N., X.Y. and Y.K. wrote the paper; S.G.N., X.Y. and Y.K. analyzed the data; F.I. read and commented on the paper.

The authors declare no conflict of interest.

The basic concept of attractor selection mechanism (ASM) based guided self-organization (GSO).

The embodied system studied in the manuscript: (

The model of the two dimensional motion of the system, where
_{i}

The dominant amplitudes of the robot tip’s oscillation along different axes for different values of motor frequencies.

Trajectory examples of the robot for

The average values of the three parameters used to explain the behavior of the robot in two dimensional spaces (top) along with the extracted different behaviors for different frequency ranges (bottom): (

The potential function

Goal directed locomotion behaviors with different level,

The performance of goal directed locomotion with different level of noise.

The probability density function of the frequency and position of the robot.

Mechanical parameters of the curved beam hopping robot.

Name | Value | Name | Value |
---|---|---|---|

_{o} |
0.160 (m) | _{p1} |
2.554 (rad) |

_{r} |
0.030 (m) | _{p2} |
2.570 (deg) |

_{b1} |
0.305 (m) | _{p3} |
1.665 (deg) |

_{b2} |
0.275 (m) | _{p4} |
1.803 (deg) |

_{b3} |
0.082 (m) | _{p1} |
15 (N/rad) |

_{y1} |
0.151 (m) | _{p2}_{p4} |
13 (N/rad) |

_{y2} |
0.141 (m) | _{y1}_{y4} |
3.517 (N/rad) |

_{y3} |
0.135 (m) | 0.030 (kg) | |

_{y4} |
0.145 (m) | 0.331 (kg) |

Parameters for the Ground Contact Model (GCM).

Name | Value | Name | Value |
---|---|---|---|

_{x} |
10 (N/m) | _{x} |
10 (Ns/m) |

_{y} |
10 (N/m) | _{y} |
10 (Ns/m) |

_{z} |
10^{5} (N/m) |
_{z} |
10 (Ns/m) |