-TG
1 2 3 4
L3
L4 L1
L2
Fp
Fl
vx C x, z D U M xm
xm
LS WR
PEP x, z
PEP x, z
PEP x, z
PEP x, z
L3
L4 L1
L2
L3
L4 L1
L2
L3
L4 L1
L2
vx
x, z
swing
stance swing
stance swing
stance swing
stance
Figure 49: Top-level block diagram of the bioinspired waveform generator.
The controller C generates a set of(x,z)-coordinates (trajectories) depending on the drive velocityvx and leg coordination rules.
The trajectories are mapped to a set of drive voltages U by the driver D and fed into the motor M. For the choice of architecture from Fig.45(c) the controller consists of four trajectory generators (TG), leg state (LS) and Walknet rules (WR) modules. Note that the coordination rules affect only the directly neighboring legs.
penalizing the objective function during optimization (see below). Fur-thermore, since the focus of this work is put on the application of the walking motor as a force generator (see especially chapt.7-8), the nat-ural objective of the optimization is to maximize its force-generating capabilities. According to Table11, there are altogether 9 unknown parameters to be optimized. Despite this moderate dimensionality, the optimization landscape is abundant in terms of local minima and thus an evolutionary algorithm [166] was chosen as in chapt.3. Also the optimization procedure was set up in a similar way to the one in sect. 3.5(see Fig.35(b)). For this purpose, the already optimized mo-tor model from sect.3.5was used. This time the bioinspired waveform generator according to the implementation from the previous section was utilized and each leg was driven independently.10 As a drive frequency, a constant value of vmaxx was chosen since the maximal drive velocity is most critical in terms of stability when the swing velocity is set to maximum as well.11 In order to provide means of force-generation maximization, the motor was loaded increasingly in a stepwise manner. 12 linearly spaced loads Fl,i between 0 and 20 N were applied to the drive rod and the corresponding motor veloci-ties ˙xm,i for i ∈ {1, . . . , 12} logged. The optimization problem was formulated in terms of the average motor velocity as follows
arg min
δ
O(δ) =arg min
δ
w1
tr(CN)<2
| {z }
logical: 0 or 1
+ 1
1
12(∑ix˙m,i), ( 133)
whereδ is the vector of unknown to-be-optimized parameters, tr(CN) is the trace of the contact matrix defined in (87) and w1 is a weight term of 10000 for penalizing the contact condition of less than two legs. The initial state of all legs was at AEP (P0) in order to avoid random penalties for a possibly good parameter choice due to a disadvantageous initial condition12. After a fixed number of 100 iterations, the optimization task was stopped. The optimized set of parameters was then used to evaluate the performance of the bioinspired drive strategy. First, the issue of stability was addressed.
Fig.50(a) shows drive rod contacts with particular legs (in stance) over several walking periods for three different drive frequencies. Even for the maximal drive frequency of 3 kHz, at least two legs support the drive rod. In this case, the contact patterns resemble the pairwise drive strategy with alternating contacts between legs1&3and2&4. For lower drive frequencies, the amount of legs being in stance at the
preload distribution and non-resonant operation, the movement of the drive rod inx is stable as long as it is supported at two or more distinct points.
10This can easily be done in the simulation but is not possible in the real motor due to the hardwired pairwise driving strategy; cf. sect.5.5.
11This follows from the fact that more legs are in stance for low drive velocities resulting in an increased overall stability.
12For a random initial condition, an (otherwise) optimal solution could be rejected if less than two legs contacted the drive rod at the initial stage due to the high penalty.
Leg numberLeg numberLeg number 4
3 2 1
4 3 2 1
4 3 2 1
0.1 ms
0.2 ms
2 ms
0.1 ms
0.2 ms
2 ms
time time
3 kHz
1 kHz
100 Hz
0.4 2
1 0.4
2
1 0.4
2
1
Leg / drive rod contacts Commanded leg elevations
z trajectory [µm]z trajectory [µm]z trajectory [µm]
3 kHz
1 kHz
100 Hz
(a) (b)
Figure 50: Velocity-dependent generation of gaits. The black bars in (a) indi-cate contact periods between the legs and the drive rod. Typical insect gait patterns known from the biology [61] albeit with a much higher frequency bandwith can be observed. For low drive velocities, all legs tend to contact the drive rod with only short swing periods repositioning one leg at a time. For higher veloc-ities, the stance and swing phase durations converge until they become approximately equal. At this point, alternating contacts between the drive rod and two leg pairs known from classical control approaches can be recognized. In (b) the corresponding commanded leg elevation trajectories are shown (leg1– solid,2– dashed,3– dash-dotted and4– dotted line).
same time increases. At 100 Hz a metachronal or wave gait [82,149] can be observed in which all legs are in stance most of the time and a series of return strokes propagates occasionally among them. These velocity-dependent patterns are two extremes of a continuum of gaits with a varying duty cycle β [82] as known from the biology. Since drive rod contact depends on leg elevation, ztrajectories are shown additionally to the different gait patterns in Fig.50(b). For the wave gait, the trajectories for different legs fall close together. Also note the different curve traversal velocities during the swing and stance phases.
The results of optimization are promising. Not only in terms of stability but also in terms of force generation. Fig. 51illustrates these results with help of a load force vs. motor velocity diagram recorded
0 1 2 3 4 5 6 7 8 9 10
0 2 4 6 8 10 12 14 16
Velocity [mm/s]
Load force [N]
swing-in region inset stall forces
0 1 2
8 10 12 14 16
1 kHz force (classical) 1 kHz force (bioinspir
ed)
Figure 51: Load force vs. motor velocity characteristics with the bioinspired waveform generation optimized for force compared to the classi-cal drive strategy with theforcewaveform. The x-symbols show simulated velocities at increasing load force levels as long as the commanded direction of motor motion could be maintained. The results are shown for5exemplary drive frequencies (100,250,500, 1000and1500Hz). Thick gray curves correspond to motor load characteristics measured at250,500and1000Hz for the classical force waveform (see also Fig.35). The bioinspired strategy is superior to the classical one in terms of both the velocity and stall force limits. The dashed region to the left shows high frequency transient phenomena (see Fig.52). The inset to the right shows the stall force limits with an increased vertical resolution.
with the optimized set of parameters. Gray curves are the real motor characteristics known from Fig.35(b). x-markers represent the sim-ulation data for velocities above 0 mm/s and are grouped together with interpolating lines according to one of the5 drive frequencies they were measured at. For the sake of clear depiction, only two load curves at 1 kHz are labeled – the real curve for the classical force wave-form and the simulated curve for the bioinspired approach. The latter curve is clearly superior to the former one. Both the drive velocity and stall force limit are about 50 % higher in the bioinspired approach as compared to the classical drive strategy. The stall force limits for the bioinspired drive approach are illustrated additionally in Fig.52(b) for a larger number of drive frequencies. While higher stall force limits could be expected from the increased amount of legs in contact with the drive rod and the optimization objective, the superior veloc-ity performance is an additional and most welcome gain. This gain can only partly be attributed to a longer step size in the bioinspired solution (cf. Fig.25(a) and52(a)). For a more detailed discussion on the reliability of simulation results, the reader is referred to the next section. Another interesting phenomenon in the optimization results
(a) (b)
12 13 14 15 16 17 18
110 500 1000 1500 2000
Frequency [Hz]
-2 -1 0 1 2
0.4 1.4
0.81.0 1.2 1.6
0.6 1.8 0.4 1.4
0.81.0 1.2 1.6
0.6 1.8
10 Hz 1.5 kHz
deflection [µm]
elevation [µm]elevation [µm] Stall force [N]
1 1
2
2...
3 4
5 6 7...
static friction limit AEP
AEP Fp = 100 N
µst = 0.14
Figure 52: Transient (swing-in) phenomena and stall force limits in the force optimized bioinspired waveform generation. (a) shows the initial leg trajectory (1, dashed line) starting from AEP and the sub-sequent trajectories (2,3,4etc.) up to the point when a stable movement pattern could be observed (number followed by an ellipsis). For high drive velocities, several movement cycles pre-ceded the stable pattern aka swing-in period. (b) stall force limits shown for11exemplary drive frequencies between10and2000 Hz. The static friction borderline, corresponding to the friction coefficient from sect.3.4.2, is shown with a dashed line.
is highlighted in the dashed region to the left of Fig.51. For high drive frequencies (in the figure 10 and 1.5 kHz), the no-load veloc-ity is smaller than the velocveloc-ity at the first load level of approx. 2 N.
This phenomenon is termedswing-in.13 It is explained graphically in Fig.52(a). For low drive frequencies, the paths followed by the legs do not differ significantly between the initial and the ongoing trajectories.
However, as the drive frequency increases the emergence of a steady coordination between the legs is delayed. For 1.5 kHz it takes about7 walking cycles until a steady trajectory can be observed. Thus, in a strict sense, the transient (swing-in) region of Fig.51is not a part of the load characteristics of the motor but only a transient phenomenon.
However, no further effort was put into correcting the load charac-teristics due to the illustrative purpose of the swing-in region and because its suppression was not an objective during the optimization.
In sect.5.2 it has been already mentioned that the emergence of a stable walk depends on the per se redundant leg coordination rules but that their relative importance has not been sufficiently quantified yet.
13In analogy to the GermanEinschwing-Verhalten.