The Sensorimotor Loop as a Dynamical System: How Regular Motion Primitives May Emerge from Self-Organized Limit Cycles

Abstract

We investigate the sensorimotor loop of simple robots simulated within the LPZRobots environment from the point of view of dynamical systems theory. For a robot with a cylindrical shaped body and an actuator controlled by a single proprioceptual neuron, we find various types of periodic motions in terms of stable limit cycles. These are selforganized in the sense that the dynamics of the actuator kicks in only, for a certain range of parameters, when the barrel is already rolling, stopping otherwise. The stability of the resulting rolling motions terminates generally, as a function of the control parameters, at points where fold bifurcations of limit cycles occur. We find that several branches of motion types exist for the same parameters, in terms of the relative frequencies of the barrel and of the actuator, having each their respective basins of attractions in terms of initial conditions. For low drivings stable limit cycles describing periodic and drifting back-and-forth motions are found additionally. These modes allow to generate symmetry breaking explorative behavior purely by the timing of an otherwise neutral signal with respect to the cyclic back-and-forth motion of the robot.

Closed-loop Robots Driven by Short-Term Synaptic Plasticity: Emergent Explorative vs. Limit-Cycle Locomotion

“…There is no objective function (Gros, 2014 ), such as a maximal velocity, to be optimized. This implies that the quantitative features of the individual motion patterns change smoothly within their respective stability regions, and that one can identify the observed regular movement patters as stable limit cycles in the sensorimotor loop (Sándor et al, 2015 ). Fast switching between motion primitives would be possible by a putative overarching controller, since more than one limit cycle may be stable for given synaptic weights w 0 and z 0 .…”

“…The dynamics of the robot takes place in a phase space combining the internal variables, of both body and controller, with the ones of the environment. The stability regions of the individual limit cycles presented in Figure 3 will therefore be bounded, generically, by a suitable bifurcation, such as a supercritical Hopf bifurcation or a fold bifurcation of limit cycles (Gros, 2015 ; Sándor et al, 2015 ). Alternatively, a transition to chaos may occur.…”

“…Distinct principles such as predictive information (Ay et al, 2008 ), surprise minimization (Friston, 2011 ), chaos control (Steingrube et al, 2010 ), empowerment (Salge et al, 2014 ), homeokinesis (Der and Martius, 2012 ), cheap design (Montúfar et al, 2015 ), and curiosity (Frank et al, 2014 ) have been studied in this context. Behavior, resulting from guided self organization (Prokopenko, 2009 ) or autonomous adaption (Chiel and Beer, 1997 ), may be generated in addition through suitable synaptic (Der and Martius, 2015 ; Der, 2016 ) and intrinsic (Sándor et al, 2015 ) plasticity rules.…”

When the goal is to generate a series of activities: A self-organized simulated robot arm

“…Such a self-organized behavior can be generated both at the level of motion primitives, in case of robotic locomotion [10], and on the level of action selection [27], as demonstrated here. The resulting behavior is robust within a wide range of parameters, as it does not require precise fine tuning, which simplifies the selection of an adequate parameter set with, e.g., machine learning techniques.…”

“…It is well known that gaits and other regular muscle contractions, like breathing [6], are induced in many cases by central pattern generators [7,8], even though it is currently controversial whether this is the case for biped locomotion [9], viz for human walking. Abstracting from animal models, one may ask conversely to which extent compliant locomotion may be generated via self-organizing principles [10], that is in the absence of top-down control in the form of a central pattern generator. One talks in this context of 'embodiment' [11], when part of the computation generating locomotion is carried out by the elasto-mechanical properties of the constituting body [12].…”