## 3 Experiments  
The robot that we have chosen for the experiments is called \"skidding snake\". It consists of ten massive spheres, each with a diameter of \(0.17\) length units. The spheres are connected to each other by a ball-and-socket joint, see Figure 2(a) such that each of the spheres can rotate passively like in a string of beads. Only one sphere, the head element, is driven by forces parallel to the ground which may be understood as a force vector \(y_{t}\in\mathbb{R}^{2}\) of arbitrary direction (see Figure 2(b)). The sensor values are the velocities of the head relative to the ground, _i.e._ we measure a vector \(x_{t}=\left(x_{1}^{t},x_{2}^{t}\right)^{T}\) where time is written as an upper index when indicating the components of a vector. Sensor and motor values are normalized to the range \([-1,1]\). One time step is \(0.01\) seconds. The robot is placed in a rectangular arena of \(10\)x\(10\) length units surrounded by walls. In a physical realization the head may be a robot with omnidirectional drive, see e.g. [11]. The robot is underactuated in the sense that the force acting on the head is not sufficient to drag the tail into arbitrary directions due to a considerable friction of the spheres on the ground. Instead only more or less lateral motions of the head are possible. Nevertheless, as we will see in the simulations, the controller can excite whole-body motions by exploiting inertia and gyroscopic effects. These physical effects are found by the self-organizing controller as a means for accumulating sufficient energy in order to move the full body in spite of underactuation. The experiment starts with model and controller parameters (matrices \(a\) and \(c\)) initialized as unit matrices and the adaptive thresholds \(h\) set to zero. In the initial phase of the experiment, random forces applied to the head element lead to small lateral motions of the head swaying the tail occasionally passively around. The sensor values \(x\) and controller outputs \(y\) are fluctuating around zero, as a consequence of these movements, see Figure 2. Therefore, the parameter changes \(\Delta c_{ij}\) in Equation 11 are dominated by the driving term \(\zeta_{i}v_{j}\) and the off-diagonal matrix elements of \(c\) increase (see Fig 3 top) so that the reaction of the head to its sensor values becomes more and more sensitive. The amplitude of the controller output increases due to the amplification of the noise. There is still no coherent reaction of the body to these random signals which is seen from the fact that the sensor values keep fluctuating around zero during the first \(18000\) steps. With increasing sensitivity, the controller also starts to respond to the swaying of the body and manages to amplify the latter. This process is self-organizing since the emerging rotational whole-body motions respond to the controller in a much more predictable way. Thus, in the long run the controller supports these motions of all elements. The body is seen at first meandering and eventually becoming engaged in a rotational mode where the body becomes stiff like a stick due to the gyroscopic effects of the rapidly rotating spheres of the body, see Figure 4. The diagram in Figure 5 shows the sensory and motor space, where transitions to temporary stable limit cycles represent the rotational mode of the skidding snake. Videos of the behavior are available at [12]. In the present experiment the emergence of coordinated behavior occurs from step \(18000\) corresponding to about three minutes simulated real time. By then the sensor values increase largely and indicate that the controller has achieved the capability of exciting the system. Due to the rapid increase of the sensor values the controller neurons are transferred to near the saturation region, see Figure 2 bottom. This leads to an increase of the anti-Hebbian term in the parameter dynamics Equation 11. In this way the controller parameters are decreased again and the system leaves the saturation region as part of the large-scale behavioral cycle.