The paddle controller
Trace 3: paddler with the type-2 thrust controller with active gain control.
4.5 Concluding remarks
4.5.1 Overview of a paddle/thrust controller
In this chapter an example was presented of an oscillating neural network that can serve as a paddle controller. It is loosely based on a CPG that performs a very similar task: generating wing movements in the locust. In both cases the problem at hand is to control the flapping of an appendage in such a way that the flapping appendage generates a thrust that is a monotonic function of the command signal driving the CPG.In order to be of any use for locomotion, a model of a locomotion controller should contain an additional stage which specifies how the action generated by the locomotion controller generates the forces that actually move the whole system. In the case of the paddle controller, a model was developed that specifies the mapping of paddle movement to propulsive thrust. 4.5.1 a The basal oscillator network The basal network which underlies the controller consists of two mutually inhibiting identical half-centres, each of which consists of an interneurone (or a pool of interneurones)Ithat is self-inhibited through a delay; a leaky integrator T. The I interneurones project onto the motor neurones ( and ).
The two half-centres which make up the oscillator are in fact elementary oscillators themselves: small networks that produce an oscillation when presented with a non-zero input. These elementary oscillators project onto the two output motor neurones of the oscillator. Mutual inhibition between the two elementary oscillators takes place between the
Iinterneurones: it serves to ensure that the oscillations in and are in counter phase.
Because of this fact the difference of the output of the motor neurones oscillates smoothly at half the frequency at which the motor neurones oscillate. If the two outputs of the CPG are projected onto a pair of antagonistic muscles, the difference in the outputs is a measure for the net force excerted by this muscle pair.
The self-inhibition ofIthroughTin the two (left and right) elementary oscillators can be either a normal, subtractive inhibition, or a shunting inhibition. A CPG with the former type of inhibition has been termed a type-1 oscillator, with the latter type of inhibition a type-2
oscillator. The type-2 oscillator needs an additional mutual inhibitory connection between theTneurones in order to be able to generate a stable, non-decaying oscillation.
The synaptic weight on the mutual inhibition between theIinterneurones regulates the frequency and type of the generated oscillation. Weights larger than 1 result in a lower frequency of oscillation and a switching mode of oscillating. The motor neurones show an increasing tendency to alternating, almost tonic, activity in either the one or the other. Weights smaller than 1 result in an increased frequency and (already for very small differences
114 CHAPTER4. THE PADDLE CONTROLLER
from 1) a decay of the oscillation into two identical tonic levels of activation in the motor neurones.
4.5.1 b Gain control to regulate the behaviour of the CPG In order to ensure that the CPG responds fast enough to changes in its input, a control loop is added to the basal network. This control loop measures the current bounds between which the output oscillates, compares them to the bounds which were learnt to correspond to the current input, and sets a gain factor to scale the output of the two motor neurones. The scaled output innervates the levator and depressor antagonistic muscle-pair of the paddle.
The gain control network is smart in the sense that it does not bluntly scale both antag- onistic outputs. Such no-nonsense scaling might be costly in terms of energy and wear and tear on various body parts, especially when a much larger output were required. Instead only the output of the motor neurone with the largest output is scaled with the calculated gain factor; the gain factor to the other motor neurone is decreased. Like this the innervation of the levator muscle is increased when a larger amplitude is requiered in the upward stroke; the innervation to the depressor muscle is increased less or even decreased (and vice versa of course).
Both types of network can be driven into a state of tonic activity by a strong positive transient in the input: when this happens the firing frequencies in the two output channels display very small oscillations around a non-zero bias which follows the input. The average sum of the outputs does retain its normal value however, indicating that the oscillations in the two motor neurones are still identical, in counterphase and symmetric around the bias. The tonic activity drops only very slowly, restoring the normal oscillatory behaviour; it can not be corrected by the gain control loop since this system bases its judgement upon the sum of the output of the motor neurones.
In the type-2 network tonic activity can be eliminated by using the gain control loop as a reset for theTneurones. The calculated gain factor is projected onto theTneurones in such a way that their output is decreased when the generated oscillation is too small and increased when the oscillation is too large. Since a strong positive transient is followed by a short period during which the amplitude of the oscillation is too large, this additional connection helps to prevent tonic activity due to a temporary larger self-inhibition.
Altering the output of the T neurones (i.e. the level of self-inhibition) has an effect on the frequency of the generated oscillation too. A larger self-inhibition results in a higher frequency because the self-inhibited neurone switches itself off faster. Similarily a lower self-inhibition results in a lower frequency.
The projection from the gain control loop to the T neurones therefore increases the fre- quency of oscillations that are persistently too high (e.g. because the gain control does not function properly or fast enough); the frequency of oscillations that are persistently too low is decreased. When the CPG is in the switching mode these side-effects change the shape of the oscillation even more into that of a block wave: overshoots following a switch are decreased and shortened, undershoots preceding the next switch are increased and lengthened.
4.5.1 c A model for generating thrust from paddling Any body that moves at a certain speed while immersed in some medium experiences a drag that is proportional to its area, drag coefficient, the density of the medium, and the square
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