Fp
Fl
M
D xm x.m D
Ff Fz
Fx
Lx
Fn Lz
Rx Rz Fp
Fl
x.r xr
(a) (b)
Figure 34: Schematic depiction of the overall motor model as (a) an intercon-nection of model components and (b) a block diagram abstraction.
Table 5:Summary of model parameters – motor dynamics.
Name Value SI unit Description
Kn 1010 N/m interface stiffness (z) Kzr 105 N/m preload spring stiffness (z) mr 0.02 kg drive rod mass
is depicted in Fig.34(a). A block diagram depiction was chosen to visualise the flow of signals between different model components in a clear manner. Also for the sake of clarity, the signals were not explicitly labeled. The exact equations governing the behavior of par-ticular blocks were defined in this section. Except of the friction block which couples thez- andx-dynamics through normal forces and has an additional input, all blocks have inputs on the left and outputs on the right-hand side with top-down numbering order. Vector-valued signals have a drop shadow. Fig.34(b) shows a black box abstraction M of this model with a driver signalD, preload Fp and tangential loadFl as inputs and motor positionxm (=xr) and velocity ˙xm(=˙xr) as outputs.
model describe the behavior of the actual PiezoLegs motor in terms of macroscopically measurable data. This section addresses both of these questions with help of three experiments which were carried out with the PiezoLegs motor. As a result, reference data as well as an optimization procedure for model parameter identification have been found. It will be shown that the proposed model explains the measured data.
The purpose of the first two experiments was to find the damping coefficientsBxl,Bzlof the legs andBxr of the drive rod as well as the nonlinear leg deflection parameters χ0 and χ1 from sect.3.3.3. For this purpose, at least two different waveforms which produce distinct maximal leg deflections had to be used. Furthermore, the PiezoLegs motor had to be driven with several drive frequencies spanning its nominal range of operation in order to identify possible damping effects. The experiment had to be carried out in a load-free condition in order to prevent additional leg deflection and slipping (stiction condition). The PiezoLegs motor was driven with 6 increasing drive frequencies f1=10 Hz, . . . ,f6=2000 Hz within its rated operational range either with the force(first) or thesinewaveform (second exper-iment). The preload force Fp in z-direction (leaf spring, see Fig.20) was set to−100 N and there was no tangential load forceFl applied in x-direction (load-free operation). For each drive frequency, the corre-sponding average motor velocity ˙xmwas computed based on drive rod position measurements with a linear encoder having the resolution of 61 nm. These velocities were used as reference values for model evaluation. Two simulations mimicking the above experiments were designed in Matlab/Simulink (The MathWorks Inc., Natick, MA, USA) and used in an optimization procedure (see below) to find values of the unknown parameters which result in a best match between the measurement and simulation. Fig.35(a) shows the simulation design.
The measured reference and the simulated velocity data is shown in the diagram on the right. The simulation data is shown for an already optimized set of parameters including the friction parameters from the previous section. A good agreement between the measured and simulated data can be seen within the full operation range of the motor. The experiments indicate linear relation between the drive frequency and motor velocity. The slight deviation from the linear trend at 2 kHz is to be attributed to the proximity of the first resonance peak of thez-dynamics at 3 kHz rather than to the damping effects as the motor can be driven at substantially higher velocities beyond the resonance region (see sect.3.4.1).
D
fD
sM
x.m -100 (preload)0 (load) f1 ... f6
1
2
force
sine
D
fM
x.m -100 (preload) 33g1 g2 g3 Fl(1,1) ... Fl(3,10)
force
Simulations 1 and 2
Simulation 3 (a)
(b) 0 2 4 6 8 10 12
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
load force Fl [N]
velocity xm [mm/s].
g1 g2 g3
3
0 500 1000 1500 2000
0 1 2 3 4 5 6 7 8
f1 f2 f3 f4 f5 f6
drive frequency f [Hz]
1 2
ref. sim.
ref. sim.
velocity xm [mm/s].
Figure 35: Three simulations used in the parameter optimization procedure and their results. (a) load-free motor was driven with theforce andsine waveforms and 6 increasing drive frequencies (f1-f6) within the nominal operation region. The measured (ref.) and simulated (sim.) data is shown for the optimized parameter set.
(b) the motor was driven with 3 increasing drive frequencies (g1-g3) for each of which an increasing load was applied to the drive rod in 10 discrete steps. The magnitude of the load was based on the linear division of the measured data into 10 discrete values for each test frequency. The actually measured raw data for different frequencies is shown as intensity-coded dots. The
As the driving principle of the motor relies on friction and the stiction condition assumed in the previous experiments is violated under tangential load, an additional experiment had to be designed in order to estimate the friction parameters which also explain dynamic friction effects under load. In particular, when tangential load is applied, a decrease in motor velocity is observed until the motor stops moving for a load approaching its stall force limit. To a first approximation, this velocity decrease is linear but becomes nonlinear for high drive frequencies and high loads due to permanent slipping of the legs. Especially for high drive frequencies a large variation in the measured velocities is observed not only between different motors but also for subsequent measurements performed with the same motor. A plausible explanation of this observation lies in the fact that friction is sensible to surface contamination and the friction coefficient may vary due to scraping effects (see next section). For this purpose, in the third experiment all measurements were carried out with 4 different motors. In particular, for each motor the position of its housing was fixed to ground and a force sensor was connected in series between one end of the movable drive rod and an extension spring connected to a fixed wall. When driven at a certain drive frequency and moving away from the wall, the drive rod caused spring elongation and thus continous increase of the tangential load Fl which was measured by the force sensor. At the same time, the velocity of the motor ˙xm was measured with help of a linear encoder as in the previous experiments. The experiment was repeated for each motor at 3 different test drive frequencies g1=250 kHz, g2=500 Hz andg3=1 kHz using theforcewaveform. The measured raw data from all experiments is shown in the diagram on the right-hand side of Fig.35(b). The data points are intensity-coded according to their drive frequency. A simulation corresponding to the above experiment was implemented in Matlab/Simulink. For practical reasons the increasing tangential load was not applied continuously but in 10 discrete steps as seen schematically in the left-hand side of Fig.35(b). The results of the simulation are depicted with help of different markers in the diagram on the right-hand side. Again, there is a good agreement between the simulation and the measurements. The proposed motor model captures the most important trends in the measured data, i.e.
the nonlinear velocity decrease for high loads and higher stall force limits for lower drive frequencies. An even better agreement could be obtained if the model would be fitted to one particular motor and a dedicated weighting scheme would be used in the optimization procedure.
The optimization procedure which was employed in this chapter is differential evolution (DE) [166]. A global optimization algorithm was used due to the diversity of local minima for an optimization problem with 10 parameters representing either not directly measurable
prop-erties of the legs or highly nonlinear friction phenomena. Differential evolution was preferred as it converges faster and with more certainty than adaptive simulated annealing [197] and requires only a few con-trol variables. 10 unknown model parameters including the damping coefficients, leg deflection nonlinearity scaling factors and LuGre fric-tion model parameters from the previous secfric-tion were optimized.
These parameters formed a parameter vector or a population member.
An initial population consisted of 70 members which were uniformly distributed over a physically plausible parameter space. The quality of each population member was evaluated by means of a simple objective function. The parameter values of the current member were assigned to the motor model and the three simulations described above were run one after another. The velocity data from each simulation was collected and compared to the reference data from the experiments.
The computed least squares error was used as the quality measure.
Although the default parametrization of the differential evolution al-gorithm was used, the optimization procedure converged to a solution after only 50 iterations – see Fig.36. The choice of parameters used in the simulations of Fig.35 and presented in Table6 was obtained after 100 iterations. Although the objective functions are abundant in local minima for the given parameter space, clusters of optimal parameter values can be identified. Fig.36(b) and (c) show this ex-emplary for the leg parameters which can not be measured directly.
Objective function values of all parameter vectors evaluated during 223 iterations are shown by means of intensity-coded dots (higher/-darker intensity codes lower objective value). Fig. 36(b) shows that in the linear oscillator model of the leg there is a stronger damping in x- than in z-dimension and that the range of possible values for Bxl is much more narrow than the range forBzlwhich does not have a strong influence on model performance. This observation has a direct correspondence to the actual design of the PiezoLegs motor in which the space between the legs is filled with a resin-like substance which influences damping inx- but not inz-dimension. In contrast, Fig.36(c) shows that the choice of parameters for the nonlinear leg deflection function from sect. 3.3.3is essential for the performance of the model. The parameters χ0 and χ1 need to have values within a clearly identifiable oval region. This can be explained by observing how the shape of the deflection function (76) changes when varying these parameters (see results in Fig.28).
100 101 102 103
100 101 102 103
Bxl vs. Bzl
χ
0 vs.χ
10.5 1 1.5 2 x 10-6 0.1
0.2 0.3 0.4 0.5
[Ns/m] [unitless]
[m/V]
[Ns/m]
cross-over factor: 0.3 weighting factor: 0.6 DE - parameters:
strategy: rand/1/exp population size: 70
0 50 100 150 200 250
objective value of best member vs. # of iterations
0 2 4 6 8 x 10-6
no further iterations necessary
(a)
(b) (c)
Figure 36: Objective function values. (a) shows the development of the objective value of the best population member over an increas-ing number of iterations. Default parametrization used in the differential evolution algorithm is indicated. (b) and (c) show the objective value of all population members as intensity-coded dots (higher/darker intensity codes lower objective value) for a choice of parameters. In (b) damping factors and in (c) nonlinear deflection parameters of the legs are considered. White markers show the choice of parameters obtained after 100 iterations and presented in Table6. Note that the parameter values in (b) are logarithmically while in (c) linearly scaled.
Table 6:Summary of unknown model parameters obtained in the optimiza-tion process after 100 iteraoptimiza-tions.
Name Value SI unit Description
Bxr 91.12 Ns/m rod damping coeff.
Bxl 299.88 Ns/m leg eff. damping (x) Bzl 101.02 Ns/m leg eff. damping (z) χ0 9.95·10−7 - deflection coeff.0 χ1 0.30 m/V deflection coeff.1 λ0 9.41·106 N/m bristle stiffness λ2 284.44 Ns/m viscous friction ν 0.89·10−2 m/s Stribeck velocity µst 0.14 - static friction coeff.
µbdn 0.10 - dyn. friction coeff.