## Abstract

Dual-stage actuators, which combine two actuators with different characteristics, have gained interest due to their large-range, high-resolution positioning capabilities. Control of such systems is challenging because it requires balancing the relative contributions of the individual actuators in terms of speed, range, and precision. The most common approach is to allocate effort to the actuators based on frequency but this can lead to misallocation in the case of low-frequency short-range trajectories. In this paper, the problem of trajectory allocation in dual-stage actuator systems is addressed using a recently developed range-based filter. The theoretical basis of the range-based filter is rigorously derived for the first time and insights regarding its use, specifically its reinterpretation as a speed-based filter, and its range-frequency response characteristics are presented. The new analysis not only explains the behavior of the filter clearly, but it provides a more robust strategy for incorporating range constraints in filter design for different desired trajectories.

## 1 Introduction

Conventional position actuation systems have a positional accuracy and dynamic bandwidth that are intrinsically coupled to the range of displacements that the actuator can provide, (actuator range). Designers are therefore typically faced with a choice of high-resolution, high-bandwidth actuators with relatively low range, or high-range actuators with relatively low resolution and bandwidth. Dual-stage actuation systems, however, combine two actuators with different range/resolution characteristics in order to deliver a system with both high-range and high-resolution. Although this incurs additional cost, such actuators are useful and now widely used in high-performance systems such as hard disc drives (HDDs) [1–8], advanced manufacturing/nanofabrication systems [9,10], sensing [11], and scanning probe microscopes (SPMs) [12–15].

Control of dual-stage actuator systems is a challenging problem because it requires balancing the relative contributions of the individual actuators in terms of speed, range, and precision. A variety of algorithms have been proposed for this purpose but the most common approach is to allocate effort to the actuators based on frequency [7,13,16,17]. Specifically, low-frequency trajectory components are assumed to be long-range and, thus, diverted to the long-range actuator (LRA), while high-frequency trajectory components are assumed to be short-range and are diverted to the short-range actuator (SRA). However, this can lead to misallocation in the case of low-frequency short-range trajectories which will be allocated to the LRA even though they would be better handled by the SRA due to its greater precision [18].

An alternative design approach is to allocate components of the desired trajectory to the two actuators based on the range that each actuator can accommodate, rather than on the frequency components of the signal. In a series of papers, several of the authors have developed such range-based dual-actuator designs, demonstrating improved performance in cases where conventional designs might fail [18–21]. The conceptual basis of this work is motivated by warping the horizontal axis of the signal, resulting in a “spatial domain” signal where higher-range signals appear as low-frequency waveforms, and short-range signals appear as higher-frequency waveforms. Filtering in the spatial domain, therefore, provides a way to separate trajectory components according to their range characteristics.

Although range-based filtering has proved to be effective, the theoretical connection between the conceptual framework and practical implementation has been lacking. The frequency-domain characteristics of the filter and the relationship between signal speed and range have also not been fully understood. In this work, a new speed, amplitude, and frequency-based analysis of the range-based filter is developed which provides a deeper insight into filter behavior, leading to improved strategies for range-based filter design. The paper is organized as follows. In Sec. 2, dual-actuator filtering methods are briefly reviewed and summarized, including conventional fixed-frequency allocation and new “range-based” strategies. These strategies are further analyzed in Sec. 3, where it is shown that the filter is arguably better understood as a speed-based adaptive filter, retaining the concept of frequency-based allocation, but adapting the filter cutoff frequency according to the rate of change of the input trajectory. Based on this interpretation, new and more robust strategies for incorporating range constraints in filter design are developed in Sec. 4, and the frequency and range response functions of the filter are derived. Finally, brief conclusions are given in Sec. 5.

## 2 Dual-Stage Actuator Filtering Methods

*x*(

*t*). Note that

*x*(

*t*) can have any distance units depending on the range of the actuators being used; thus, the specific units of signals presented in this paper are not specified. The signal

*x*(

*t*) must be decomposed and allocated appropriately to actuation signals

*y*

_{1}(

*t*) and

*y*

_{2}(

*t*) which are sent to the LRA and SRA, respectively. A common method is to split the command signal according to its frequency content, using complementary filters

*G*

_{1}(

*s*) and

*G*

_{2}(

*s*) to allocate low frequencies (below a bandwidth

*ω*) to the LRA and high frequencies (above this bandwidth) to the SRA. Assuming a unity-gain, first-order filter design, and using the subscripts 1 and 2 to denote the long- and short-range actuator respectively, the outputs of these complementary filters may be expressed in the Laplace domain as follows:

_{b}*x*(

*t*) into the so-called spatial domain by plotting displacement

*x*(

*σ*) as a function of

*σ*the cumulative distance traveled

This results in a sawtooth/triangular waveform with gradient ±1 regardless of the shape or frequency of the original waveform *x*(*t*). Figure 1, for example, illustrates the transformation of a multi-sinusoidal signal *x*(*t*) = *A*[0.9sin(*ωt*) + 0.1sin(5*ωt*)] into the spatial domain. Such signals are common in engineering applications. Note that sinusoids of different frequency *ω*, but same amplitude *A*, have the same cumulative displacement profile (Fig. 1). However, signals with the same frequency, but different amplitudes have different profiles in the spatial domain (Fig. 2). In particular, high amplitude waveforms have low-frequency spatial profiles, and conversely, low-amplitude waveforms have high-frequency spatial profiles.

Complementary spatial domain filters can therefore be used to filter high- and low-amplitude components and allocate them appropriately to LRA and SRA, respectively. Such filters, *G*_{1}(*s′*), *G*_{2}(*s′*), may have the same form as Eqs. (1) and (2), only with the Laplace variable s replaced by *s*′ to indicate the Laplace transform of a spatial domain, rather than time-domain, signal.

*G*

_{1}(

*s*),

*G*

_{2}(

*s*) for example, may be discretized using the Tustin approximation

*T*denotes the fixed/constant sample period and

*z*

^{−1}a unit time delay. Substituting Eqs. (4) in (1), (2) and rewriting as a pair of difference equations then gives

*n*denotes the discrete time-step. It is not immediately apparent, however, how to apply the Tustin approximation in the case of the spatial filters,

*G*

_{1}(

*s′*),

*G*

_{2}(

*s′*). For discrete signals, the warping in Eq. (3) may be approximated as

*T*in the time domain, now have a variable spatial sample interval defined by

*D*(

*n*), the distance traveled between samples. Most discretization methods assume a constant sampling interval and are therefore inapplicable in this instance. One advantage of the Tustin approximation, however, is that it is based on trapezoidal integration over a sample period and does not require all sample periods to be identical. In this case, therefore, the substitution (Eq. (4)) is replaced by

*y*(

*n*) when expressed in the time-step domain, or

*y*(

*σ*(

*n*)) when expressed in the spatial domain, are not changed by the warping of the time axis, i.e.,

*y*(

*n*) =

*y*(

*σ*(

*n*)). Substituting Eq. (8) into expressions equivalent to Eqs. (1) and (2) for

*G*

_{1}(

*s′*),

*G*

_{2}(

*s′*), and rewriting the result as a difference equation in

*y*(

*n*), rather than

*y*(

*σ*(

*n*)), then gives

Equations (9) and (10) therefore represent a convenient discretization of the spatial filtering concept which can be applied directly to the time-domain samples of desired and actual trajectories. Although the derivation of the filter was not formally presented previously, the filter itself was first introduced in Ref. [18] and has been shown to be effective in allocating signals based on the range [18–21]. In this work, the spatial cutoff frequency *ω _{b}* was chosen such that

*ω*= π/

_{b}*R*where

_{c}*R*is the desired range cutoff for the filter. The results of applying this filter with

_{c}*R*= 100 to different amplitudes/ranges of the multi-tone sinusoid considered in Figs. 1 and 2 are shown in Fig. 3(a)). When the sinusoid amplitude

_{c}*A*is significantly below the range cutoff, then all the signal is passed to the SRA and the LRA is essentially unused, (Fig. 3(a), left). The opposite behavior is observed if the sinusoid amplitude significantly exceeds the range cutoff (Fig. 3(a), right). The transition between the two occurs when the amplitude is equal to the range cutoff. In this case, the signal is split between the SRA and LRA (Fig. 3(a), center), with each actuator contributing to the final response. Note that these behaviors are independent of the frequency of the signal and (unlike conventional systems) the fact that low-frequency, low-amplitude signals are passed to the SRA provides finer control actuation for such signals. If the same signals are applied to the conventional system of Eqs. (3) and (4), as shown in Fig. 3(b)), then low-frequency signals are always passed to the LRA regardless of amplitude.

## 3 Interpretation as an Adaptive Speed-Based Filter

Although conceived as a range-based filter, the implementation defined by Eqs. (9) and (10) is arguably more easily seen as an adaptive speed-based filter. Comparing Eqs. (5) and (6) and Eqs. (9) and (10), for example, shows that both have the same functional form, only the bandwidth $\omega b\u2032$ is now adjusted dynamically according to the magnitude of changes in the commanded input

The dynamically changing bandwidth scaling factor can therefore be recognized as the difference in the spatial distance over a sample interval or the instantaneous speed of the input signal (m/s). Signals with low instantaneous speed (less than 1 m/s) and which therefore change only by a small amount over the sampling period cause the effective bandwidth *ω′ _{b}* to be lowered, thereby increasing the percentage of the signal that is allocated to the high-frequency SRA. Conversely, signals with instantaneous speeds greater than 1 m/s increase the effective bandwidth and are therefore more likely to be allocated to the LRA. Unlike the conventional system, a low-frequency but low-amplitude signal will be allocated to the SRA as desired because the speed is low. As the amplitude of the signal increases, so does the speed or derivative of the signal, increasing the bandwidth and sending more of the signal to the LRA, exactly as observed in Fig. 3(a)).

This reinterpretation of the range-based filter also provides insight into some extreme cases of filter behavior when the input waveform contains constant (i.e., zero-speed) segments. If the speed at which the signal is changing decreases to the point at which it tends to a zero-speed constant value, then the effective filter bandwidth $\omega b\u2032$ decreases to zero and all of the signal is allocated to the SRA. The output of the LRA, therefore, does not change (it receives no allocation) and neither does that of the SRA (because the input difference is zero in this case).

*A*, where the instantaneous velocity is either zero (most of the time) or a very high value (during the step transitions). Hence

*x*(

*n*) +

*x*(

*n*− 1) = 0. Likewise, for the SRA filter, Eq. (10) becomes:

The bandwidth therefore does indeed adjust to feed almost all of the input signal to the LRA, and very little to the SRA, but because the square wave is symmetric, the input to the LRA filter is zero. Neither filter therefore receives any contribution from the input during the transition, and instead, the output of both filters simply toggle their current value. The result therefore depends entirely on the initial conditions of the filters. If the square wave was not symmetric so that *x*(*n*) + *x*(*n*−1) ≠ 0, then this sensitivity to initial conditions would decay with time.

## 4 Range-Based Filter Design

*R*(= 2

*A*) of the signal. In the simplest case of a symmetric triangular trajectory, for example, the speed

*D*(

*n*)/

*T*is constant and equal to the average speed over the cycle. The latter is readily found as the distance 2

*R*traveled over one cycle divided by the cycle period 2

*π/ω*, giving

*ω*, the effective filter bandwidth defined by Eq. (11) becomes

*R*, as

_{c}The effective filter cutoff frequency is therefore equal to the frequency *ω* of the periodic input trajectory, scaled by the ratio of its range *R* to the spatial filter range cutoff *R _{c}*. If the range of the triangular waveform exceeds the cutoff range, then the frequency

*ω*of the signal falls within the effective bandwidth

*ω′*of the filter and the signal is passed to the low-frequency LRA. Conversely, if the range of the signal is smaller than the cutoff range, then the effective bandwidth of the filter is lowered. The frequency

_{b}*ω*of the signal then no longer falls within this bandwidth and is therefore passed to the high-frequency SRA. In the special case where the range of the input waveform matches the cutoff range, then the effective filter cutoff frequency becomes identical to that of the input trajectory waveform and the signal is split between the actuators.

*ω*= 0.01

_{b}*T*. Each filter has unity gain in its high-pass or low-pass band, respectively, and a gain of 1/√2 at the cutoff frequency. As expected, the response is independent of signal amplitude or range.

Plots of these functions are presented in Fig. 5, showing their dependency (or lack thereof) on both frequency and the range ratio *R*/*R _{c}*. In this case, the gains of the two filters appear almost independent of frequency, except at the limits

*ωT*=

*π*or

*ωT*= 0 when by inspection of Eqs. (20) and (21),

*G*

_{1}(

*jω*),

*G*

_{2}(

*jω*) tend to zero or unity, respectively. Instead, the filter characteristics are dominated by the range ratio. When this is low, the gain

*G*

_{2}(

*jω*) is high and

*G*

_{1}(

*jω*) is low, so most of the signal is sent to the SRA as desired. However, when the range of the signal exceeds the range cutoff, i.e., the range ratio is high, then the converse is true: the gain

*G*

_{2}(

*jω*) is low and

*G*

_{1}(

*jω*) is high, so most of the signal is sent to the LRA. If the range of the signal equals the range cutoff value, then both filters are at their crossover range “bandwidth” with a gain equal to 1/√2.

Strictly speaking, the relationships (15) and (16) or (20) and (21) hold only for triangular trajectories i.e., signals with constant speeds, but the results should hold approximately for other waveforms (e.g., single sinusoid or raised sinusoid) that have two turning points per cycle and which spend equal amounts of time increasing and decreasing, and whose maximum speeds are not significantly in excess of the average speed over a cycle. However, other waveforms with more turning points, for example, may have a different relationship between amplitude and speed. For example, consider and input signal comprising a 3-Hz sinusoidal waveform, amplitude *A* = 2, multiplied by an amplitude (*A*/2) square wave signal, as depicted in Fig. 6. The composite signal has a total range *R* = 6, but because of multiple turnarounds, travels a much larger distance in a single cycle, so the speeds are nearly 5 times that of a pure sinusoid or triangular waveform with the same overall range. If Eq. (16) is used without adjustment, then a significant part of this signal is (incorrectly) directed to the LRA (see Fig. 6, top). If, however, the range cutoff (Eq. (17)) is redefined as *R _{c}* = 2

*π*/

*ω*, then signals for this form with a range less than

_{b}*R*are now mostly passed to the SRA as indeed shown in Fig. 6 (bottom).

_{c}One should also proceed with caution in cases where there is asymmetry in the distance traveled (speed times time) in the positive and negative directions. Consider, for example, a sawtooth waveform that ramped upward with the same speed defined in Eq. (15) for the *entire* period, only stepping back to zero at the end of the cycle. The speed of the signal is sufficiently low that it will be passed to the SRA, causing the SRA to saturate even though the total distance traveled over the cycle is the same as that before, namely 2*R*. In this context, it is important to remember that the filter is ultimately a speed-based filter. No hard constraints are placed on the range which is only limited by the knowledge of the trajectory shape/properties and smart choices of the nominal filter bandwidth according to Eqs. (16) and (17).

*τ*= 1/

*ω*. Hence, from Eq. (11), the effective time constant is

_{b}Given triangular waveforms, the settling time can be found using *T _{s} =* 4

*τ′*. Consider the transient example illustrated in Fig. 7, which shows input triangular signals (

*x*) with ranges

*R*= 2 (top),

*R*= 20 (middle), and

*R*= 200 (bottom) as a solid line and the sum of the two filter outputs, i.e.,

*y*

_{1}+

*y*

_{2}, as a dashed line. Using a range cutoff of

*R*= 10 and signal frequency

_{c}*ω*= 2π rad/s, the estimated time constants and settling times and simulated settling times are shown in Table 1. As can be seen, the estimated and simulated settling times are in good agreement.

## 5 Conclusions

Conventional dual-actuator filter designs typically allocate the command signal to the individual actuators using a fixed-frequency filter. Recent “range-based” filter designs have been shown to provide better signal allocation, particularly in the case of low-frequency, low-amplitude signals. In this work, a rigorous theoretical derivation of the filter has been obtained for the first time. It has also been shown that the filter is perhaps more accurately understood as a speed-based adaptive filter, retaining the concept of frequency-based allocation, but adapting the filter cutoff frequency according to the rate of change of the input trajectory. Unlike conventional fixed-frequency dual-actuator filters, the speed-based adaptive filter allocates a low-frequency but low-amplitude signal to the SRA as desired because the speed is low. As the amplitude of the signal increases, so does the speed or derivative of the signal, increasing the bandwidth and sending more of the signal to the LRA.

This interpretation not only explains the behavior of the filter clearly, but it provides a more robust strategy for incorporating range constraints in filter design for different desired trajectories. In particular, the frequency and range response characteristics for triangular waveforms have been derived, and the same results hold approximately for other signals whose instantaneous speeds do not deviate too much from the average speed of a triangular waveform. The analysis also highlights some weaknesses of the “range-based” filter design and of conventional filtering strategies since neither case *guarantees* that range constraints will be satisfied. Future work will aim to embed hard range constraints more explicitly in the filter design process.

## Acknowledgment

The authors gratefully acknowledge financial support from the Briar Hill Foundation and from National Science Foundation (Grant Nos. CMMI 1537983 and 1537722).

## Conflict of Interest

There are no conflicts of interest.

## Nomenclature

*t*=time

*s*=continuous time Laplace transform variable

*z*=discrete time

*z*-transform variable*N*=discrete time-step

*R*=spatial range of the signal

*T*=time-domain sample period/sample interval

*R*=_{c}filter range cutoff

*x*(*t*) =desired trajectory

*y*_{1}(*t*),*y*_{2}(*t*) =actuation signals sent to LRA and SRA, respectively

*D*(*n*) =spatial domain sample interval (distance traveled between samples)

*G*_{1}(*s*),*G*_{2}(*s*) =complementary transfer functions to allocate signals to LRA and SRA, respectively

*σ*=cumulative distance traveled

*τ*=filter time constant

*ω*=_{b}filter frequency cutoff/bandwidth

## References

_{∞}Almost Disturbance Decoupling Controller and a Tracking Differentiator