applied purposefully as a model of walking in insects. Although it is not necessary to deal with leg amputation, ground irregularities or avoid high obstacles [123] in the walking motor, it is conceivable to de-sign a more insect-like motor (e.g. with six legs arranged in two rows).
In this case, bioinspiration would go one step further – not only in the means of control but also in the architectural design. Still, even for the motor in its present form it is credible to consider different inter-leg coordination mechanisms (cf. Fig.45(b). Moreover, if the motor were provided with additional sensing capabilities, other coordination rules were likely to augment its performance. For example, a load sensing capability in the legs (based on direct piezoelectric effect and voltage monitoring) together with the application of Rule5 could lead to a higher mechanical load capacity. Last but not least, the bioinspired strategy together with a sensory feedback would likely be tolerant against imperfections and wear in the mechanical components of the motor.
As far as the optimization results are concerned, several issues need to be addressed. First of all, the optimization aimed at improving the force-generating capabilities of the motor. This was done indirectly by trying to keep the instantaneous motor velocity as high as possible for the increasing levels of loading up to 20N. Other objectives like maximizing average (no-load) motor velocity (see below), reducing wear, etc. are conceivable. The ability to satisfy different optimization objectives is a soft measure of the flexibility of the proposed approach.
Besides force optimization, another optimization task was started with the objective to maximize motor velocity. The optimization problem was formulated as
arg min
δ
w1
tr(CN)<2
| {z }
logical: 0 or 1
+w2
1
¯˙
xm +w3std(x˙m), (134) where ¯˙xm is the average motor velocity over a certain distance and std(x˙m)is the standard deviation of the instantaneous motor velocity – this term is supposed to minimize large deviations from the average value for a smoother operation of the motor. The velocity-related weightsw2 andw3 were chosen as 10 and 3, respectively, so the objec-tive function preferred solutions of high average velocity and possibly low velocity variation as long as they were statically stable. Fig.53 shows the results in a diagram and Table13summarizes the optimal parameter values for both the max. force and max. velocity optimiza-tions. The diagram refers to Fig.35(a) and39(a) and shows the real velocity data for a selection of drive frequencies up to 2 kHz when the classicalsineandforcewaveforms are employed. The simulation data utilizing the bioinspired approach is shown with black and gray x-markers for the force (see previous section) and velocity maximizing optimizations, respectively. The latter result improves the no-load velocity of the motor as compared to the force maximizing strategy
15
10
5
00 500 1000 1500 2000
Drive frequency [Hz]
Velocity [mm/s]
0 100
0 1
sine max. vel.
max. force
force
Figure 53: Classical vs. bioinspired waveform generation in terms of average motor velocity when no external loading was applied. The thick lines correspond to unloaded motor velocities measured for two classical waveforms – sine and force (see sect.35). The x-symbols show simulated motor velocities for the bioinspired waveform generation when optimized for max. velocity and max. (stall) force output. In all cases the bioinspired strategy is superior to the classical one.
almost by 50 % and overpowers the classical strategy by 100 % even if compared to the fastersinewaveform. Actually, this result is very close to the theoretical limit of 15 mm/s at 2 kHz for the pairwise drive strategy.14
The superb force and velocity maximizing results raise a question of the reliability of the theoretical results from the simulation. The question can only be answered upon evaluating the reliability of the
14The theoretical limit corresponds to the case when the legs contact the drive rod interchangeably over the longest diagonal of the work area and the instantaneous leg velocity is transmitted to the drive rod under the condition of perfect stiction.
Table 13:Optimized parameter values for max. force and max. velocity optimizations.
Parameter Max. force optimization Max. velocity optimization
r1 -0.1987 -0.0337
r2 0.3000 0.3000
r∗2 0.2410 0.4000
r3 0.3710 0.3000
k1 0.5000 0.5000
k3 1.0000 1.0000
IxP0 1.5000e-6 1.5000e-6
IxP4 -2.3467e-6 -2.3467e-6 IzP21 2.3500e-6 2.3500e-6
physical motor model from chapt.3– please refer to the discussion in sect. 3.6. At this point two additional comments are to be made. In the context of motor force maximization, Fig.52(b) shows stall force limits for the bioinspired strategy over a range of drive frequencies.
In sect.3.5, it has been shown that the physical motor model with the LuGre friction model is able to account for the frequency-dependent stall force limits observed in the motor. In Fig. 52(b) frequency depen-dency can be seen as well. Here, the stall forces are higher than under the application of the classicalforcewaveform which is most welcome but they also show a tendency to increase with drive frequency which contrasts with the result from chapt. 3. Furthermore, these forces are partly higher than the static friction limit for the estimated friction coefficientµst of 0.14. Although, in dynamic operation, it is possible to achieve stall forces higher than the static limit, this result does not seem to be reliable. Supposedly, it is the effect of model limitations in emulating the dynamic friction with linear terms [see equation (86a)].
The contribution of dynamic friction in the interaction between the legs and the drive rod is not bounded in the linear approximation.
Still, this modeling decision did not prevent the physical model from explaining the experimental data and is not supposed to fundamen-tally challenge the superiority of the bioinspired force generation. In the context of velocity maximization, much higher velocities than with the classical drive approach, close to the theoretical limit for the pairwise strategy, can be observed. However, this would require the legs to travel the entire distance along the horizontal diagonal of the work area rhombus and contrasts with the leg deflection nonlinearity introduced in sect. 3.3.3. Since the physical motor model does contain the nonlinearity, another explanation is needed. Furthermore, in the bioinspired approach no velocity decrease close to the resonance re-gion of 3 kHz (cf. Fig. 39(a)) is observed. This is presumably another advantage of the bioinspired approach since the actual leg trajectories differ among legs and are devoid of pure frequency components to a larger degree than in the classical approach.
As a last comment in this section, the issue of direction change is discussed. So far in the discussion, the motor was assumed to move in one direction – forward – according to the direction of the stance trajectory. As far asWalknetis concerned, the issue of backward walking has been addressed recently and a solution proposed [176, 177]. In the walking motor, the direction change is realized classically by issuing the driving waveforms backwards. This method cannot be used in the bioinspired approach since AEP and PEP need to be swapped as well and the coordination rules adjusted (mirrored in the simplest case). However, the direction change in the motor can be realized also by swapping the driving voltages UA(i) and UB(i) for each leg (i) without the need of a change in the direction of waveform traversal. The latter solution is actually implemented in the drive
electronics of the motor and described in more detail in the next chapter.
6 F R E Q U E N C Y M A T C H I N G I N W A V E F O R M G E N E R A T I O N abstract
This chapter is concerned with the issue of generating the driving waveforms at a particular frequency. On this account, the motor-drive electronics devel-oped for this work is introduced with the focus on the waveform generating unit and the dependency between its internal register settings and the fre-quency output. An algorithm based on the solution to the B´ezout’s identity is proposed to match the desired frequency. The algorithm is evaluated and the problems related to the limitations of the target hardware indicated. In a next step, an efficient approach based on a look-up table is proposed and shown to reduce the frequency errors to less than1%. Additionally, this chapter compares the developed electronics with commercial products and introduces a motor direction change strategy based on phase swapping.