4 G R A Y- B O X I D E N T I F I C A T I O N O F S E M I P H Y S I C A L M O T O R
D Y N A M I C S
z dynamics
OP OP
OP OP PW
PW
PW PW
PW OP FB FB OP
Model-based linearization
impact dynamics waveform
waveform operating range leg deflection influence of load friction leg elongation
operating range
x dynamics
PW FB
Figure 37: Overview of nonlinearities in the walking piezoelectric motor as identified in the physical motor model from the previous chapter and possible methods of their linearization (PW – piecewise linearization, OP – operating point linearization, FB – feedback linearization). Impact dynamics (in bold face) of the legs plays the determinant role in the switched-mode behavior of the motor.
Leg elongation nonlinearity not included in the motor model is shown in gray-shade for the sake of completeness (cf. sect.3.3.3).
can be considered a hybrid switched-mode[130,7, 4] or jump parame-ter [199] system. This consideration lies in the fact that the system is subject to a sudden change in the values of its parameters depending on the contact condition between the legs and the drive rod. Any linearization attempt has to address this issue first. The most straight-forward approach is to differentiate between4cases in which either one pair, the other, both leg pairs or no legs have contact with the drive rod; the latter being an artificial case. In this manner a piecewise system [161,54] consisting of several subsystems for each case is ob-tained. However, this increases the number of state matrices which can become dramatically high if other nonlinearities are to be split into piecewise linear components. In order to reduce the number of subsystems, several realizations of state switching are conceivable and shown in Fig.38. The states are depicted as numbered circles with the necessary conditions for being in the particular state shown next to them. State transition conditions are not shown on purpose as the actual implementation may range from a finite-state automaton to a Markov process.
Fig.38(a) shows the obvious case with the cycling one leg pair – both pairs – other leg pair ... transitions. Assuming both leg pairs behave identically and resetting their state periodically reduces the number of states by one [Fig. 38(b)]; neglecting the contact dynamics during the overlapping contact condition and using a periodic reset would yield a single state implementation [see Fig.38(c)]. The no-contact condition (floating drive rod) is ignored in all cases. Although
1 reset c(i)
1 2 3
c1 c2
c1 ^ c2
1 2
reset
c(i) c1 ^ c2
(a) (b) (c)
Figure 38: Possible realizations of state switching for the impact dynamics of the legs. (a) shows the basic realization with three states where either leg pair1(c1) only, both leg pairs (c1∧c2) or leg pair2(c2) only have contact with the drive rod. (b) two-states simplification of (a) with a reset signal. (c) a further simplification neglecting contact transition dynamics.
theoretically possible, the piecewise linearization strategy proposed in Fig.38 could not be attained in a practical control scenario due to the high switching frequency of the process (up to 3 kHz) and the non-observability of the states. The latter issue is especially prohibitive in a force control scenario when the motor is severely disturbed by external forces. However, for relatively slow nanopositioning tasks under negligible loads the above linearization strategy could prove useful in a robust control scenario.
The high switching frequency of the motor remains an obstacle for any further linearization attempt of the model from chapt.3. How-ever, before proceeding to the next section in which an alternative modeling strategy based on an experimental approach is proposed, other analytic possibilities will be sketched shortly for the sake of completeness. Please note, that the goal of the following discussion is not to arrive at a realistic control-theoretical motor model but rather to shed light on possible pitfalls related to the analytic linearization ap-proach. Focusing again on Fig.37, the behavior of the motor strongly depends on the choice of the driving waveform (see sect.3.2.2 and chapt.5) which affects the level of leg deflection, elevation, drive rod contact times, etc. and thus determines the linearization procedure of the other nonlinearities. The only sensible choice seems to be the linearization for a particular waveform choice.
The term operating range in Fig.37relates to the upper bandwidth limit of the drive frequency. It has been shown in Fig.32of sect.3.4.1 that the motor can be driven with frequencies beyond its rated opera-tion. In this case, several regions of operations between and beyond the resonance regions could be distinguished and the model split into several piecewise linear models. However, the ultrasonic mode of operation is likely to lead to further difficulties due to increased current consumption, temperature rise, etc. Last but not least, this would further increase the switching frequency of the motor. For these
reasons, choosing an operating point below 3 kHz seems to be the best solution.
The nonlinearity related to leg deflection was considered in sect.3.3.3. For many waveforms a suitable operating point could be chosen or even a piecewise model proposed. An alternative approach could be based on feedback linearization [73] but this would require the knowledge of leg states which are not observable in the real motor.
A variation of this technique is shown in chapt.7where it is used to compensate the effects of external load on motor velocity.
Last but not least, the operation of the motor is based on friction.
The nonlinear LuGre model from the previous chapter proved to be suitable to explain the friction phenomena in the walking motor (see sect.3.4.2). Operating point linearization of this model in the stiction or slippage regions are possible [74, 156]. However, these simplifications obviously neglect either the dynamic or quasi-static aspect of motor operation. Moreover, linearization for the stiction condition increases model complexity by introducing two additional model states, both of which are neither controllable or observable.
A piecewise attempt, possibly with a different friction model, seems to be better suited but it poses the non-trivial problem of mediation between the cases.
Considering the multitude of difficulties in analytic motor model linearization, a different approach has been chosen in this chapter. In the following sections, the motor is seen as a gray-box [187] whose input, output and transfer characteristics can be experimentally mea-sured but little knowledge of its internal workings is assumed. The model will be derived based on an measurement and subsequent (least square) statistical data evaluation. In sect.4.2.1the basic linear model of the unloaded case is presented. This model is extended by the nonlinear influence of mechanical load in sect.4.2.2and the dynamic transfer characteristics in sect. 4.2.3. The final model will be used in chapt.7 to develop a load force compensation strategy which linearizes the model completely and to design an explicit force controller.