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Research Papers

# Run-to-Run Optimization Control Within Exact Inverse Framework for Scan TrackingOPEN ACCESS

[+] Author and Article Information
Ivan L. Yeoh

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195
e-mail: ivanyeoh@uw.edu

Per G. Reinhall, Martin C. Berg, Eric J. Seibel

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195

Howard J. Chizeck

Department of Electrical Engineering,
University of Washington,
Seattle, WA 98195

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received June 7, 2016; final manuscript received February 11, 2017; published online June 5, 2017. Assoc. Editor: Maurizio Porfiri.

J. Dyn. Sys., Meas., Control 139(9), 091011 (Jun 05, 2017) (12 pages) Paper No: DS-16-1296; doi: 10.1115/1.4036231 History: Received June 07, 2016; Revised February 11, 2017

## Abstract

A run-to-run optimization controller uses a reduced set of measurement parameters, in comparison to more general feedback controllers, to converge to the best control point for a repetitive process. A new run-to-run optimization controller is presented for the scanning fiber device used for image acquisition and display. This controller utilizes very sparse measurements to estimate a system energy measure and updates the input parameterizations iteratively within a feedforward with exact-inversion framework. Analysis, simulation, and experimental investigations on the scanning fiber device demonstrate improved scan accuracy over previous methods and automatic controller adaptation to changing operating temperature. A specific application example and quantitative error analyses are provided of a scanning fiber endoscope that maintains high image quality continuously across a 20 °C temperature rise without interruption of the 56 Hz video.

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## Introduction

The conventional approach to use a fixed array of millions of electro-optical transducing elements in imaging and display devices is improved upon by the single-element scanning fiber device. For imaging, current technologies like charge-coupled device (CCD) and complementary metal–oxide–semiconductor (CMOS) employ large arrays of photosensitive cells that convert light into electrical charge; for display, technologies like liquid crystal display (LCD) and organic light-emitting diode (OLED) use microscopic arrays of light emitting cells. In both applications, these grids are composed of tens of millions of electro-optical elements. Besides the intrinsic manufacturing cost and complexity, the electro-optical element array approach trades miniaturization for higher resolution: to acquire high-definition images, a large number (and subsequently a wide area) of imaging cells are needed.

At the University of Washington, Seattle, WA, a novel imaging/display solution has been developed where a single optical element—an optical fiber—is spatially scanned by a miniature piezoelectric-tube [1,2]. This method enables dramatically compact devices such as the scanning fiber endoscope (SFE) [1]. Within the SFE, a red–green–blue laser beam is steered as the single waveguiding optical fiber is actuated by a piezoelectric-tube. Miniature lenses focus the laser light into a small spot on the object plane (e.g., respiratory track or surgical cavity). This laser spot is rapidly scanned across the field by precise electromechanical actuation, while the reflected light is registered to generate images at video rate (30–60 fps). The entire scan engine is contained within a 1.2 mm-diameter by 9 mm-long rigid probe, while the thin electrical wires and optical fibers run through a 2 m-long flexible tether. This unprecedented form factor makes possible highly noninvasive and laser-based medical procedures [35], small-footprint robotic vision sensors [6], new tools for manufacturing quality control [7], and potentially many more unexplored applications.

The use of mechanical scanning shrinks the device dimensions but places stringent requirements for miniature displacement sensors and precise control systems. Because a single element is spatially scanned to (in a sense) simulate an array of optical units, the accuracy and precision with which the optical fiber is positioned over time is crucial for high-quality, distortion-free images. Space is limited within the 1.2 mm-diameter probe that houses a 0.8 mm-diameter lens assembly, 0.45 mm-diameter piezoelectric-tube, and other components. The 11.5 kHz scan frequency also imposes a high bandwidth requirement on potential sensing modalities. Consequently, until recently, no integrated displacement sensor was available to measure the deflections of the optical fiber during operation. As a result, control systems developed for the scanning fiber endoscope have all been open-loop methods which could not adapt to changing operating conditions while in use, e.g., inside a contracting or warming gastrointestinal track during surgery.

In SFE prototypes currently being used for clinical tests and recently published [8,9], the calibrated open-loop control method presented by Kundrat et al. [10,11] is the De facto approach: Prior to use, the device is coupled to a large external optical position detector which provides measurement of the laser spot position. The device is then manually calibrated by a trained operator using feedback from the optical position sensor. Finally, the device is uncoupled from the measurement system and used in endoscopic procedures, where operating conditions can be drastically different compared to the calibration chamber. The hybrid controller presented in Ref. [12] uses a similar two-step strategy where the device is first made to track a reference input well within a calibration chamber fitted with an optical detector, then operates open-loop using the stored control signal. If online measurement of the optical fiber deflection is available, analysis, simulation, and experimental data [13,14] have shown that precise scan control is possible. In Ref. [15], Smithwick et al. presented a real-time feedback controller that would regulate measurement-updated error states, which achieved ½ pixel error over 250 × 250 images. However, since an external optical position detector was needed, that controller could not be applied in practice.

Recently, the authors demonstrated for the first time in the scanning fiber endoscope, use of its single miniature piezoelectric-tube for both actuation and also collocated sensing [16], which has enabled exploration and implementation of new online and adaptive control strategies [17]. Use of the piezoelectric tube (previously functioning only as an actuator) as also a sensor is made possible by a self-sensing circuit described in Ref. [17]. This simple electronics-retrofitting process means that no bulk is added to the SFE probe, maintaining its slim form factor while providing measurements of the optical fiber deflection during use with changing operating conditions. A new adaptive feedforward controller using the piezoelectric self-sensing measurements was shown to maintain 2–3 times better scan accuracy compared to the de facto open-loop method in experiments simulating changing in-use structural stress [17].

In the most recently developed controller, a 40 ms system identification procedure (during which the sensing signal is processed) is automatically performed whenever the system should update its control signal. That system identification repeatedly modifies the dynamics model of the scanner, which is used to calculate an exact-feedforward input to achieve a given desired trajectory. With that innovation, the SFE is now able to autocalibrate to maintain high image quality while in use.

Nonetheless, there are a number of limitations with that most recent controller in Ref. [17] that this article will improve upon by presenting a novel run-to-run optimization controller. (i) First, each time the controller updates, a 40 ms interval results during which no images are channeled. Though for infrequent updates this brief blanking may be unnoticed, the interval could be reduced or eliminated. (ii) Second, each update of the control point is an independent calculation: the controller reidentifies the system dynamics de novo. No error information from the performance of previous control signals is used to refine or inform the subsequent updates. (iii) Finally and relatedly, there is no feedback loop designed to converge the system to zero tracking error. As mentioned earlier, since each system identification instance is independent, there is a certain level of stochastic error at each update, which could be regulated by a learning controller. These improvements in performance at the 40 ms timescale and at near-pixel-level spatial precision may be critical in advanced endoscopic medical procedures that integrate laser ablation with laser-based video rate imaging [5,18], which has been proposed for destroying early cancer [1].

In this publication, we introduce and demonstrate a run-to-run controller formulated on a feedforward with exact model-inversion framework that can progressively optimize to the best-achievable tracking trajectory. Section 2 will review run-to-run control and establish why run-to-run control is suitable with the SFE. Section 3 will detail a novel formulation of run-to-run control within the context of exact inversion and prove the convexity of the formulation to guarantee convergence to a global performance optimum. Section 4 will analyze and verify the proposed run-to-run optimization controller with various simulations. Sections 5 and 6 will describe the implementation and experiments carried out with our new control design. Finally, Secs. 7 and 8 will discuss our findings and concludes the article.

## Run-to-Run Control

Design of a control system for the SFE should be informed by the specific qualities of its operation. (i) First, the reference trajectory is periodic and constant frame-to-frame. Ideally, the optical fiber should start from rest, spiral outward then inward to scan a circular image field, returning to its initial resting state. This trajectory and timing should be exactly repeated for each image frame for consistent undistorted images. (ii) Second, exogenous disturbances to the system are on a much slower timescale than the periodic trajectory. At video rates, the period of each trajectory is on the order of tens of milliseconds (e.g., 16 ms). Temperature and structural stress changes during operation are typically on the order of seconds. Thus, the system disturbance is slower than the operating timescale by 50× or more.

These properties make learning-type control methods suitable for our problem. We briefly summarize the three main classes of learning controllers. Repetitive control is applied to track periodic references or to reject periodic exogenous disturbances [19]. Repetitive control is a continuous-operation controller, where a time-delay block within the controller acts as the internal model capable of generating arbitrary periodic signals.

Repetitive control is designed to handle periodic references and disturbances similar to the control types described later, but it requires continuous measurement, which does not allow batch data processing and is typically formulated in the frequency domain [19]. These restrictions make repetitive control not the most suitable one for our periodic trajectory-tracking application compared to other learning-type controllers.

Iterative Learning Control is another class of controllers designed to track periodic trajectories. With iterative learning control, the problem is formulated in explicit periods $k=1,2,3…$ with fixed duration $T$. The control input at each iteration period is modified by the tracking error achieved in the previous iteration [19] Display Formula

(1)$u(t,k)=u(t,k−1)+Ke(t,k−1)e(t,k−1)=y(t,k−1)−ydesired(t,k−1)$

$u(t,k)$ is the control input profile over $t$ for iteration $k$, $e(t,k−1)$ is the error profile over $t$ for iteration $k−1$, and $K$ is the learning gain. Notice that (i) the control input at $t$ is informed by the error at exactly $t$ (instead of $t−Δt)$ of the previous run and (ii) measurements to calculate the error over the entire period $t∈[0,T]$ are required. The former observation explains why iterative learning control can outperform feedback controllers with inherent error-to-input delay if the disturbance is periodic or slowly changing. The latter observation means that iterative learning control can only be applied where continuous measurement or estimation of the tracking error is available.

With the SFE, recent introduction of piezoelectric self-sensing allows for continuous observation of scan deflection. However, the quality of the measurement signal is dependent on accurate tuning or balancing of the sensing circuit, which can be difficult to achieve [20,21]. In Ref. [17], we tackled circuit bridge imbalance by modeling the imperfection. In this paper, we explore another control strategy (run-to-run) that relaxes the requirement for continuous measurements while still regulating the error over iterations.

Run-to-run control is a learning control class that can utilize sparse or noncontinuous sampling of the plant output or states to iteratively improve upon the control input. Generally, a linear (in parameters) regression model is used to describe the plant [19,22,23] Display Formula

(2)$z(k)=Av(k)+b(k)+ε$

$z(k)∈ℝm$ is a vector of system measurements, either direct (e.g., sparse output measurements during a run) or indirect (e.g., output-measurement-derived quality metrics). $v(k)∈ℝn$ is a vector of input parameters. The control input $u(t)$ is typically a time-profile signal, but it is finitely parameterized by $v(k)$, e.g., a switching or step signal is defined by a set of amplitudes and transition times. $A∈ℝmxn$ is a matrix relating the input parameters to the output metrics and $b(k)∈ℝm$ a vector describing disturbances or drifts to the system that the input should compensate for run-to-run. $ε∈ℝm$ denotes the random within-run disturbances that run-to-run control typically cannot reject.

The most basic run-to-run controllers attempt to estimate $b(k)$ to give the input that will achieve the desired output profile $z*(k)$. The exponentially weighted moving average (EWMA) filter method iteratively estimates Display Formula

(3)$b(k)=λ[z(k−1)−Av(k−1)]+(1−λ)b(k−1)$

To calculate the control parameters (and consequently the control input) for each iteration Display Formula

(4)$v(k)=A−1[z*(k)−b(k)]$

An improvement to EWMA relaxes the assumption that matrix $A$ is fixed, and uses adaptive control methods to iteratively estimate both $A(k)$ and $b(k)$ [22] Display Formula

(5)$K(k)=P(k−1)φ(k)[λ+φT(k)P(k−1)φ(k)]−1P(k)=[I−K(k)φT(k)]P(k−1)λθ(k)=θ(k−1)+K(k)[z(k)−φT(k)θ(k−1)]$

where $θ(k)=[A(k)b(k)]T$ is the vector of the current estimate of $A(k)$ and $b(k)$. $φ(k)=[v(k)1]T$ is the current regressor vector such that $z(k)=φT(k)θ(k)$.

Another approach to run-to-run control is to treat the periodic tracking problem as an iterative optimization procedure, termed run-to-run optimization in this paper. In essence, each period or run represents an empirical integration of the system equations that can produce measurements of, e.g., the state paths, terminal cost, and objective function. At the conclusion of each run, the control effort can be updated using these measurements and by applying techniques from numerical optimization with the goal of converging to the optimal cost value.

Run-to-run optimization as defined above is a broad category that can be further classified based on how the optimization updates are implemented [24]. Model-explicit optimization uses a model of the system dynamics to compute the local gradient based on current measurements [24,25]. In Ref. [26], a nominal model of a chemical process was used together with continuous measurements to approximate the gradient and direction of improvement to optimize production. Model-explicit optimization has a number of drawbacks including requirement for reasonably accurate model and high computational cost in calculating optimal policy [24].

In contrast, model-implicit optimization schemes do not rely on model-based gradients to compute the next-iteration control inputs. In general, the input is parameterized by a lower dimension vector $π$, such that the control input is a function of time and the parameterization values $π$ : $u(π,t)$. A simple model-implicit approach, termed evolutionary optimization [24,27], sequentially perturbs $π$ while registering the change in a performance index denoted here as $J$, in order to empirically determine the local gradient and hence the direction of improvement.

Further, if we parameterize or define measures $I$ of the system response, such that the output is a function of time and the parameterization values $I$ : $y(I,t)$, then the cost function or performance index of the system is also a function of this parameterization: $J(I)$. The problem statement is then Display Formula

(6)

$u(π,t)$ and $y(I,t)$ may also need to satisfy various constraints, and $H{}$ encapsulates the plant dynamics. $(π*,I*)$ represents the optimal point. Thus, there is a mapping $M$ between the input parameters $π$ and output parameters $I$Display Formula

(7)$I=M(π)$

References [24,25], and [27] suggest that $I$ be chosen as constructs or parameters with target values that are necessary for optimality, e.g., $I$ as the terminal values for output levels which are constrained to be 0 at $tfinal$ for an optimal solution. This configuration is designed such that as $I$ is regulated toward its target values, the overall system will then converge to the optimal solution. In Ref. [27], the mapping $M$ was linearized about the operating point, and a feedback-iterative update law was proposed Display Formula

(8)$π(k)=π(k−1)+G[I(k−1)−I*]$

At each run $k=1,2,3…$, the input parameterization is updated by the measurement-derived metrics $I$ of the previous run. In Refs. [27,28], a run-to-run algorithm of the form in Eq. (8) was used to optimize insulin dosages for diabetic patients. Figure 1 presents a block diagram of the general run-to-run optimization algorithm.

There are a number of limitations with the run-to-run optimization implementations previously reported, but inspired by these investigations we formulate a run-to-run optimization method more applicable to the scanning fiber control. The drawbacks of the model-explicit run-to-run optimization include the need for continuous measurement and intensive computation [26]. The optimality-condition parameterization approach in Refs. [24,25], and [27] is instructive, but many assumptions and simplifications, e.g., regulating only the terminal constraints or assuming nearly linear mapping $M$, are required for practical implementation. Furthermore, most of the theory was developed for chemical or biological process control, where the dynamics of the system are slower and the control inputs are usually parameterized by arcs and transition times [27]. In our present application, we are dealing with a resonant mechanical system at micro- to millisecond timescales, and we wish to track or scan arbitrary trajectories. In Sec. 3, we present a new run-to-run optimization control tailored for periodic feedforward systems using exact inversion.

## Run-to-Run Optimization Within Exact Inverse Framework

Run-to-run optimization implementations usually need to be tailored to the specific application; in this article, we present a run-to-run optimization formulation designed for the SFE that should be applicable to many similar mechanical scan-tracking problems.

As typified in Refs. [22,23,26], and [28], practical implementations of run-to-run optimization are usually modified or tuned (e.g., model simplification, selection or interpretation of output parameters, and heuristics for nominal solutions) to specific applications—in these examples, they are anticoagulant dosing, polysilicon etching, thermal processing, and insulin dosing, respectively. This is in contrast to feedback or proportional-integral-derivative (PID) controllers that are more readily applied to a broad range of problems. One explanation is that because run-to-run optimization operates on a reduced set of parameters, much freedom is available to the control system designer to specify the input parameterization $π$ and the sparse output measures $I$.

###### Problem Formulation.

The electromechanical system we wish to control is the SFE scan engine, as shown in Fig. 2. Specifically, we wish to precisely control the trajectory of the optical fiber tip over time, such that the laser beam emanating from the fiber tip is accurately steered to capture images at video rates (30–60 fps). The transverse deflections of the piezoelectric-tube plus optical fiber structure span a two-dimensional (2D) space. In Ref. [11], the two-dimensional scans were shown to be decomposable into two closely linear, orthogonal axes or vibration systems called the eigendirections. In Ref. [17], the authors presented a piezoelectric self-sensing method where the mode dynamics of the structure along each eigendirection could be identified.

To achieve large structural deflection (and hence field-of-view), the scan engine is actuated close to its first mode resonant frequency, and the damping of the structure is minimized (high Q). Given such operating conditions, the contributions of the higher-order modes may be discounted. Then, from the system identification methods demonstrated in Ref. [17], we are able to empirically isolate a transfer function $G(s)$ for the first mode dynamics Display Formula

(9)$Y(s)=G(s)U(s)$

Given a periodic desired trajectory $yd(t)$ or $Yd(s)$, exact inversion was used in Ref. [17] to achieve tracking with mean error of 1–2% of maximum amplitude. With exact inversion, the control input is directly calculated using the identified system transfer function Display Formula

(10)$Uff(s)=G−1(s)Yd(s)$

In the previous work, at each controller update, the system transfer function $G(s)$ was reidentified and the exact inverse input $uff(t)$ recalculated. The contribution of this publication is a run-to-run optimization method that will instead iteratively refine $G(s)$ and $uff(t)$ to converge to improved tracking accuracy.

###### Trajectory Error—Modeling Error Relationship.

Since the measureable variable is the directly related system output or the achieved trajectory, we begin by tying the trajectory error to the modeling error, which is important when applying system inversion.

Let the true system be defined as Display Formula

(11)$G0(s):U(s)→Y(s)or Y(s)=G0(s)U(s)$

with U(s) being the input, $Y(s)$ the output, and $G0(s)$ the true plant.

Consider the most general desired output trajectory $y(t)=δ(t)$ (a desired trajectory may be discretely approximated by a pulse train or a sequence of varying-amplitude impulses). The feedforward input corresponding to the unit impulse is Display Formula

(12)$G0(s):uff0(t)→δ(t)G0(s):Uff0(s)→11=G0(s)Uff0(s)Uff0(s)=G0−1(s)$

with uff0(t) being the feedforward input.

Introduce a modeling error when calculating exact inverse input Display Formula

(13)$Uffe(s)=G0−1(s)+ΔGinv(s)$

with Ginv(s) being the error in modeling the system inverse and $Uffe(s)$ the error-contaminated feedforward input.

The achieved trajectory $ye(t)$ using the error-contaminated control input Display Formula

(14)$G0(s):Uffe(s)→Ye(s)Ye(s)=[G0−1(s)+ΔGinv(s)]G0(s)=1+ΔGinv(s)G0(s)≡1+Ge(s)ye(t)=ℓ−1{1+Ge(s)}=δ(t)+ge(t)$

The trajectory error is $ge(t),$ defined as $Ge(s)≡G0(s)ΔGinv(s)$, i.e., the response of the true system to a “spurious input” $ΔGinv(s)$ due to modeling error.

###### Interpreting System Energy.

Next, we propose a unique parameterization of the system output by estimating the total system mechanical energy. This sparse mapping is then applied for run-to-run optimization and is shown to produce a convex cost surface.

Interpret the plant as a system of particles with total energy given by the sum of potential and kinetic energies. Given some state-space realization describing the trajectory Display Formula

(15)$y(t)=CeAtx(0)+∫0tCeA(t−τ)Bu(τ)dτ+Du(t)$

where the states $x(t)$ represent the displacement and velocities of the particles. Consider $t≥0+$ where $uffe(t)=0$ [see Note 1] Display Formula

(16)$Output:ye(t)=ge(t)=CeAtx(0+)$

Displacements vector: $d(t)=CdeAtx(0+)$

Velocities vector: $v(t)=d˙(t)=CdAeAtx(0+)≡CveAtx(0+)$Display Formula

(17)$Energy:Ee(t)=K[eAtx(0+)].2+M[eAtx(0+)].2$

$K$, $M$ are some constants related to the stiffness and mass vectors, $[…].2$ represents elementwise squaring.

Note 1: For systems with no zeros, from Eq. (13), errors in estimating the pole positions give $ΔGinv(s)=P(s)$, a polynomial in $s$, resulting in . This is the case considered further.

For more general systems, consider that with stable inversion $uffe(t)→0,t→∞$ or , then the approximation that the input is zero can be used after $t≥tf$.

Note that this analysis only considers amplitude- and time-bounded periodic desired trajectories and stable plants such that ; ; $uffe(t)→0|t→tbound$.

###### Modeling Error—Residual Energy Relationship Produces Convex Problem.

With the residual energy measure given by Eq. (17), we now relate that error measure to the modeling error and show that the residual-energy versus model-error problem is convex.

Define a measure of modeling error with the integral Display Formula

(18)$∫−∞∞|Ge(f)|2df$

We can relate modeling error to time-domain response using Parseval's theorem Display Formula

(19)$∫−∞∞|Ge(f)|2df=∫−∞∞|ge(t)|2dt≅∫0+∞ge(t)2dt$

$ge(t)$ is real-valued and is the causal response to $uffe(t)$ [Note 1].

From Eq. (16)Display Formula

(20)$∫0+∞ge(t)2dt=∫0+∞[CeAtx(0+)]2dt$

Notice that in Eq. (17), energy $Ee(t)$ is a convex function of $x(0+)$, optimal point at 0. ($eAt$ nonvarying for a given $G0(s).)$ In Eq. (20), modeling error is also a convex function of $x(0+)$, and optimal point is also at zero. Thus, if we solve for optimality using Eq. (17), we will also find the optimal point for Eq. (20).

In addition, Eq. (17) is a convex function of $x(0+)$ at any time $tS≥0+$Display Formula

(21)$Ee(tS)=K[eAtSx(0+)].2+M[eAtSx(0+)].2$

Thus, optimizing using a single sample of $x(0+)$ (per period of repeating trajectory) is sufficient to find the optimal input.

###### Defining Suitable Parameter Space.

Though Eqs. (20) and 21 are convex in $x(0+)$, we do not directly adjust the state $x(0+)$. Instead, we adjust the estimated inverse system model $ΔGinv(s)$ via a parameterization. This parameterization must be chosen such that convexity or quasi-convexity is preserved.

As an example, consider the case where $G0−1(s)$ is a polynomial in $s$. Then, the model error $ΔGinv(s)=a0+a1s+a2s2+...$ is also a polynomial in $s$ parameterized by its coefficients.

Now, Display Formula

(22)$Uffe(s)=G0−1(s)+a0+a1s+a2s2+⋯uffe(t)=L−1{G0−1(s)}+a0δ(t)+a1δ˙(t)+a2δ¨(t)+⋯$

Solving for Display Formula

(23)$x(0+)=CeAtx(0)+∫00+CeA(t−τ)Buffe(τ)dτ=CeAtx(0)+∫00+CeA(t−τ)B[L−1{G0−1(s)}+a0δ(τ)+a1δ˙(τ)+a2δ¨(τ)+⋯]dτ=CeAtx(0)+∫00+CeA(t−τ)B[L−1{G0−1(s)}]dτ+a0∫00+CeA(t−τ)B[δ(τ)]dτ+a1∫00+CeA(t−τ)B[δ˙(τ)]dτ+a2∫00+CeA(t−τ)B[δ¨(τ)]dτ+⋯$

Equation (23) shows that $x(0+)$ is affine in the coefficients Hence, by composition principles, Eqs. (17), (20), and (21) are also convex in the coefficients

## Simulation Example

In this section, we simulate the proposed run-to-run optimization controller on a simple numerical example to illustrate the algorithm and controller performance.

Given a system representing a mass–spring–damper (or identified mode dynamics) of the form Display Formula

(24)$x˙=[01−2.25−0.03][xx˙]+[00.1]uy=[10]x$

Figure 3(a) shows the desired amplitude- and time-bounded trajectory (solid trace) where $yd(t≥200)=0$. Using the exact inverse method described in Refs. [17] and [29], the feedforward input is calculated. Equation (24) system response is simulated for exact inversion without (dashed trace) and with (dash-dotted trace) modeling error $ΔGinv$. Without modeling error, the achieved response matches the desired trajectory. With modeling error, the system response deviates from desired and there are residual oscillations after $t≥200$.

The system in Eq. (24) represents a common case where displacement measurements (continuous or infrequent) of a mechanical system are available as the output. For such a system, the energy definition is straightforward, e.g., Display Formula

(25)$E(t)=Kx(t).2+Mx(t).2E(t)=[2.250][x(t)x˙(t)].2+[01][x(t)x˙(t)].2$

The constant vectors $K$ and $M$ represent the nominal stiffness and mass of the particle system in Eq. (24). Figure 3(b) plots the defined energy measure $E(t)$ of the system over time for cases without (lower trace) and with (upper trace) modeling error. The black vertical line delineates when the input $u(t)$ and desired trajectory $yd(t)$ are 0 for $t≥(tf=200)$. Without modeling error, the energy measure is zero after $t≥200$ as expected, but with modeling error the energy measure is nonzero for $t≥200$ but decays to zero since the system is stable and dissipative.

The transfer function of the system in Eq. (24) can be cast into the general form Display Formula

(26)$G(s)=1s2+as+b$

As discussed with Eq. (23), we take $a$ and $b$ to be the parameterization for the input $uff(t)$ and predict the energy measure to be convex over this space. Figure 4 plots the energy measure $E(t)$ with a single sample at time $t=200$ for different values of error in the model parameters $Δa$ and $Δb$. Figure 4 demonstrates that the energy surface is indeed convex with no local minima at, e.g., large $Δa$ but negative $Δb$, and that the optimum point is at zero modeling error $Δa=Δb=0$. This single-sample measurement convexity in Fig. 4 means that, by measuring the system just once per run, we will be able to quantify the optimality of the current model and identify a global performance optimum.

###### Energy Estimation Error.

Modeling error in terms of $Δa$ and $Δb$ is quantified and minimized via run-to-run optimization as illustrated in Fig. 4. However, the estimate for system energy requires the constants $K$ and $M$ in Eq. (25) to be inferred from the nominal plant. Expecting perfect knowledge of $K=K0$ and $M=M0$ would be unreasonable since if the true model were known, optimization would not be necessary. To explore the effect of energy estimation error, we simulate the energy surface with errors in $K$ and $M$. Figures 5(a) and 5(b) show the energy surface for 500% error in the constants $K$ and $M$, respectively. These simulations suggest that convexity and global optimality are maintained, but as the estimation error increases, the relative gradient on parts of the surface decreases, which may be interpreted as reduced certainty in the direction of improvement as the estimate deteriorates.

These simulations with example parameter values were presented to emphasize the convexity of the problem, theoretical optimality, and the robustness of the proposed algorithm.

## Experiment Methodology

We report an implementation and experimental validation of the proposed run-to-run optimization controller for a scan engine. Figure 6 shows the experimental setup. The scan engine, comprising a PZT-5A piezoelectric-tube of 0.45 mm outer diameter, 4 mm length, and a cantilevered optical fiber of 0.08 mm outer diameter and 2.27 mm length, is housed within a cylindrical electric heater with a thermistor for temperature sensing. The heater and the thermistor are connected to a computer for active temperature control to investigate the effects of changing thermal operating conditions (e.g., within a human body) on scan accuracy.

The scan engine is connected to a piezoelectric self-sensing circuit as detailed in Ref. [17] and to the computer for sensing and actuation. The self-sensing circuit is a form of capacitive bridge that senses and amplifies the piezoelectrically generated voltage as the piezoelectric-tube deforms, providing a measurement of the deflection of the scanner without need for additional sensors [17]. The use of the piezoelectric-tube as a miniature sensor maintains the small form-factor of the endoscope, and the run-to-run optimization algorithm introduced in this paper will be shown to maintain high scan accuracy over changing device temperature.

Finally, an optical position sensor (DL-20, OSI Optoelectronics) tracks the laser spot scanned by the optical fiber to provide direct measurements of the optical fiber deflections. The optical sensor data is used for verification and is not part of the control system. Data acquisition, temperature control, and run-to-run optimization control were all implemented in LabVIEW code.

The desired scan trajectory along eigendirection 1, $r1(t)$, is a ramping and collapsing (triangle-modulated) sinusoid waveform with scan frequency at $ω0=11.315 kHz$ to operate close to the first mode mechanical resonance Display Formula

(27)

To capture images, a 2D scan is spanned by actuation in both eigendirection 1, $r1(t)$ and eigendirection 2, $r2(t)$. The trajectory $r2(t)$ is exactly the same as $r1(t)$ except that $cos (ω0t)$ replaces the sinusoid. $T0$ is the period corresponding to $ω0$ and $A0$ is the maximum scan amplitude. $r(t)$ is periodic, with 180 imaging spirals and 20 resting spirals [10]. The frame rate is 56 fps, producing 360-pixel-diagonal circular images.

In this investigation, we use evolutionary optimization [24,27] to converge to the optimum point. Between each imaging frame, an energy estimate as exampled in Eq. (25) is calculated for different perturbed model parameter values. Over time, a plot of the energy surface is constructed and the convex optimum is directly selected as the new control point.

Two experiments were performed. In experiment 1, the goal was to demonstrate the new run-to-run controller fine-tuning or optimizing the scan accuracy, compared to the adaptive-feedforward control previously described in Ref. [17]. At temperatures 30 °C, 40 °C, and 50 °C, the adaptive-feedforward method was used to find an initial control input. Then, the run-to-run optimization was activated to converge to a better control point. The scan error and any improvements in accuracy were recorded. Five trials were run at each temperature: The adaptive-feedforward controller reinitializes during each trial, while the new run-to-run controller refines its output between each trial.

Experiment 2 was designed to demonstrate the run-to-run controller operating continuously to maintain scan accuracy across changing operating temperature. Starting with an initial control input at 30 °C, the temperature was raised to 40 °C then to 50 °C, with the run-to-run controller constantly attempting to recalibrate. Scan error with and without run-to-run optimization was recorded.

For quantifying the error of a 2D circular scan, we separate the positional error into normalized radial and phase/tangential components as in Ref. [17]. The squared-error time profile and mean-squared-error of an entire scan are recorded at different operating temperatures.

## Results and Interpretation

Experiments 1 and 2 were conducted as detailed in Sec. 5. Figure 7 shows an experimentally constructed energy surface as generated by the LabVIEW code. The predicted convex topology is experimentally verified. Also, the minimum point in Fig. 7 is noticeably off-center, meaning that the optimization algorithm will converge to a new operating point to the lower right of the parameter space.

###### Experiment 1: Comparing New Run-to-Run Optimization Versus Earlier Adaptive Feed-Forward Method.

At three different operating temperatures, the control input was first initialized by the adaptive feed-forward method [17]. Then, run-to-run control was used to further optimize the control input. This procedure was repeated five times at each test temperature. At 50 °C, Figs. 8(a) and 7(b) show 2D plots of the achieved scan trajectory with the adaptive feed-forward initialization, and then after run-to-run optimization, respectively. Qualitatively, there is improvement in tracking of the targeted 2D trajectory. Figures 8(c) and 8(d) show quantitatively the squared error in the radial and phase/tangential components. There is clearly a reduction in the radial error after run-to-run optimization at this operating temperature.

Figure 9 charts the radial and phase/tangential MSE achieved by the adaptive feedforward [17] and the new run-to-run optimization approach, over changing operating temperature. Figure 9 shows the accuracy over five trials and the average error of both control methods at each operating temperature. Figure 9 shows that run-to-run optimization detectably improves the scan accuracy (10–25% improvement of MSE in radial and tangential phase) beyond that of the earlier adaptive feed-forward approach.

###### Experiment 2: Performance of Run-to-Run Controller in Tracking Temperature Changes.

In the second experiment, the control input was only initialized at 30 °C. Then, just run-to-run optimization was used to maintain scan accuracy as the operating temperature was increased in steps to 40 °C and then to 50 °C. At 50 °C, Figs. 10(a) and 10(b) show 2D plots of the achieved scan trajectory with open-loop control [11] and with run-to-run optimization, respectively. Without any compensation, the trajectory does not track back to the center of the scan in Fig. 10(a), while run-to-run optimization maintains the full scan as in Fig. 10(b). Figures 10(c) and 10(d) show quantitatively the squared error in the radial and phase/tangential components. There is much lower error using run-to-run optimization particularly at the start and end (related to 2D center) of the scan.

Figure 11 charts the radial and phase/tangential MSE achieved by open-loop control [11] and run-to-run optimization over changing operating temperature. In this experiment, the control input is not reinitialized at each temperature—the run-to-run controller attempts to recalibrate itself. Figure 11 clearly shows run-to-run optimization outperforming the de facto open loop controller [11] by continually updating its optimal point. The final temperature change to 50 °C produced a 40% reduction in radial MSE and 5× reduction in phase/tangential MSE for the run-to-run versus the open-loop methods.

###### Image Quality.

To visually assess the achieved image quality, laser light was modulated through the optical fiber during each frame-scan to project target images as in Fig. 12(a). The projected images were recorded using a hand-held camera. At 50 °C, Fig. 12(b) shows the distorted images using open-loop control [11] initialized only at 30 °C. Figure 12(c) shows improved image quality at 50 °C with the adaptive feedforward controller introduced in Ref. [17]. Unintentional perspective distortion is present due to the hand-held camera not being centered above the scan, but qualitatively the line sections have become much straighter. Figure 12(d) shows the image quality further achieved by the run-to-run optimization controller presented in this paper. With the run-to-run controller, the image accuracy is refined in the central regions and the controller is able to self-adjust to the +20 °C change in operating temperature.

## Discussion

From the experimental results, we find that measurement noise manifests as nonsmoothness in the empirically constructed energy estimate surface as seen in Fig. 7. Figure 13 illustrates the effect of measurement noise with a simulation plot. We find that the convexity of the surface is maintained, but jaggedness is introduced. A small level of uncertainty is also introduced to the optimum point. Reduction in accuracy is expected with increasing levels of measurement noise, but may be reduced by averaging over a number of samples or periods at the cost of slower convergence.

The optical position detector used to quantify scan accuracy was a dual lateral-effect detector that produces two analog voltage signals corresponding to the X- and Y- laser spot position on its square detector. The voltage response of the detector may not be fully linear across the field, thus the detector can introduce its own distortion to the measurement. In addition, uncleanness of the detector surface and multiple reflections of the laser beam can distort the data. In Figs. 8(a) and 8(b) and Figs. 10(a) and 10(b), a smudging effect is observed on the top-left of the scan that is likely due to optical distortion instead of scan error; this may have artificially increased the recorded error quantifications. Improvements to the experimental setup may include precalibrating the optical position detector with known laser spot positions.

The run-to-run optimization implemented in this work requires measurement or estimation of the particle displacement $x(t)$ and velocity $x˙(t)$ as in Eq. (25). Since only displacement is directly measured, we approximate $x˙(t)$ by taking the discrete central difference Display Formula

(28)$x˙̂(t)=x(t+Δt)−x(t−Δt)2Δt$

For the achieved experimental results, this approximation is shown to be adequate. Thus, theoretically, only three displacement samples, , $x(t)$, and $x(t−Δt)$, are needed to estimate the displacement and velocity states per-period for run-to-run optimization.

In this reported implementation, 20 resting spirals follow each 180 imaging spirals (180:20, or 90% imaging duty cycle) as in Eq. (27). This compares to 180:64 (74% imaging duty cycle)1 for the de facto open-loop control in Ref. [11] and 180:36 (83% imaging duty cycle) for the adaptive-feedforward-only method previously developed in Ref. [17]. Though we have greatly reduced the fiber resting time, we could not reduce it to just three sample-times as theoretically possible with our run-to-run formulation due to unmodeled higher-order modes affecting the energy estimate. Future work will investigate including higher-order vibration modes into the controller formulation.

Most run-to-run optimization formulations [24,27], even application-tailored implementations [28,30] do not guarantee convexity of the problem or global optimality. An advantage of the run-to-run optimization framework we have demonstrated is the existence and convergence to a globally optimal performance-point.

However, a limitation with the proposed run-to-run optimization controller is the lack of gradient information from a single measurement. Instead, we rely on evolutionary optimization [27], where multiple measurements with different input parameters are taken to empirically estimate the direction of improvement. For demonstration, we brute-force constructed the entire energy surface and then move to the optimum point. This operation takes up to 2.29 s to converge to a new optimum because the entire 8 × 8 energy surface is explored for both eigendirections. The duration is a function of the frame rate (56 Hz) and the number of runs (128) required. This run-to-run algorithm can be accelerated by, e.g., finding the local gradient using a 2 × 2 search, estimating a step size, and then moving to a new 2 × 2 gradient search. Nonetheless, the device is fully operational during the entire optimization duration since there is no need for, e.g., a 40 ms probing signal as in Ref. [17].

The demonstrated improvement in SFE image stability compared to current open-loop operation under changing temperature conditions (Figs. 10 and 11) illustrates the importance of continuous adaptive control, especially in the center of the field-of-view. The SFE can use simultaneous coaxial delivery of low-power laser light for imaging and high-power laser light for therapy—the image captured by the endoscope is used to target the laser treatment. Here, precision (i.e., repeatability between image capture and targeted laser scans) to the limit of the image resolution for laser irradiation of tissue will be critical in integrating laser ablation. Figure 14 recasts the error between expected and achieved scan trajectories in Euclidean or pixel-distance. Figure 14 demonstrates that with changing temperature, the run-to-run controller maintains the scan within 0.23 pixels precision, while the open-loop scan varies by more than 2 pixels. This improvement in scan stability may be crucial for effective and safe treatment when laser ablation rapidly increases local tissue temperature during a surgical procedure.

Finally, the results of this paper show scan adaptation over changing temperature. Presently, the temperature of the probe is stabilized using a microheater and thermistor, contributing to electrical power drain. In the future, the requirement for temperature control and monitoring within the SFE may be eliminated from the fiber scanner as online adaptive control is implemented using the more advanced run-to-run optimization.

## Conclusion

This work has presented a new run-to-run optimization controller designed for precision control of the scanning fiber system. This controller utilizes measurements from piezoelectric self-sensing made possible by recent advances [16] and improves upon previous control methods [17] by iteratively converging to an optimal control point. The run-to-run optimization was derived for feedforward with exact-inversion control and shown in simulation and experiment to have a global optimum even with errors in the energy estimation model. The empirical evolutionary optimization approach of this work has benefits of robustness to modeling error and ease of implementation, but converges slower than more direct gradient methods. Future work may explore supplementing the gradient estimation by additional assumptions about the system dynamics. The contribution of this work to the scanning fiber device application is significant since it leads to improved image quality and self-calibration during use, for example, in surgical procedures where operating conditions greatly vary and cannot be precalibrated.

## Acknowledgements

This work was supported in part by NSF Grant No. CBET-1351110, NIH Grant No. R01 EB016457, and the Department of Mechanical Engineering. The authors also thank Mr. Rich Johnston, Mr. Dave Melville, and Dr. Brian Schowengerdt for support and technical assistance in the development of scanning fiber devices.

## References

Lee, C. M. , Engelbrecht, C. J. , Soper, T. D. , Helmchen, F. , and Seibel, E. J. , 2010, “ Scanning Fiber Endoscopy With Highly Flexible, 1-mm Catheterscopes for Wide-Field, Full-Color Imaging,” J. Biophotonics, 3(5–6), pp. 385–407. [PubMed]
Bryant, R. C. , Seibel, E. J. , Lee, C. M. , and Schroder, K. E. , 2004, “ Low-Cost Wearable Low-Vision Aid Using a Handmade Retinal Light-Scanning Microdisplay,” J. Soc. Inf. Disp., 12(4), pp. 397–404.
Yang, C. , Hou, V . W. , Girard, E. J. , Nelson, L. Y. , and Seibel, E. J. , 2014, “ Target-to-Background Enhancement in Multispectral Endoscopy With Background Autofluorescence Mitigation for Quantitative Molecular Imaging,” J. Biomed. Opt., 19(7), p. 076014.
Soper, T. D. , Porter, M. P. , and Seibel, E. J. , 2012, “ Surface Mosaics of the Bladder Reconstructed From Endoscopic Video for Automated Surveillance,” IEEE Trans. Biomed. Eng., 59(6), pp. 1670–1680. [PubMed]
Woldetensae, M. H. , Kirshenbaum, M. R. , Kramer, G. M. , Zhang, L. , and Seibel, E. J. , 2013, “ Fluorescence Image-Guided Photodynamic Therapy of Cancer Cells Using a Scanning Fiber Endoscope,” Proc. SPIE, 8576, p. 85760L.
Gong, Y. , Hu, D. , Hannaford, B. , and Seibel, E. J. , 2014, “ Accurate Three-Dimensional Virtual Reconstruction of Surgical Field Using Calibrated Trajectories of an Image-Guided Medical Robot,” J. Med. Imaging, 1(3), p. 035002.
Gong, Y. , Johnston, R. S. , Melville, C. D. , and Seibel, E. J. , 2015, “ Axial-Stereo 3-D Optical Metrology for Inner Profile of Pipes Using a Scanning Laser Endoscope,” Int. J. Optomechatronics, 9(3), pp. 238–247. [PubMed]
McVeigh, P. Z. , Sacho, R. , Weersink, R. A. , Pereira, V . M. , Kucharczyk, W. , Seibel, E. J. , Wilson, B. C. , and Krings, T. , 2014, “ High-Resolution Angioscopic Imaging During Endovascular Neurosurgery,” Neurosurgery, 75(2), pp. 171–180. [PubMed]
Templeton, A. W. , Webb, K. , Hwang, J. H. , Seibel, E. J. , and Saunders, M. , 2014, “ Scanning Fiber Endoscopy: A Novel Platform for Cholangioscopy,” Gastrointest. Endos., 79(6), pp. 1000–1001.
Kundrat, M. J. , Reinhall, P. G. , and Seibel, E. J. , 2011, “ Method to Achieve High Frame Rates in a Scanning Fiber Endoscope,” ASME J. Med. Devices, 5(3), p. 034501.
Kundrat, M. J. , Reinhall, P. G. , Lee, C. M. , and Seibel, E. J. , 2011, “ High Performance Open Loop Control of Scanning With a Small Cylindrical Cantilever Beam,” J. Sound Vib., 330(8), pp. 1762–1771. [PubMed]
Smithwick, Q. Y. , Vagners, J. , Johnston, R. S. , and Seibel, E. J. , 2010, “ A Hybrid Nonlinear Adaptive Tracking Controller for a Resonating Fiber Microscanner,” ASME J. Dyn. Syst. Meas. Control, 132(1), p. 011001.
Smithwick, Q. Y. , Seibel, E. J. , Reinhall, P. G. , and Vagners, J. , 2001, “ Control Aspects of the Single-Fiber Scanning Endoscope,” Proc. SPIE, 4253, pp. 176–188.
Smithwick, Q. Y. , Vagners, J. , Reinhall, P. G. , and Seibel, E. J. , 2003, “ 54.3: Modeling and Control of the Resonant Fiber Scanner for Laser Scanning Display or Acquisition,” SID Symposium Digest of Technical Papers, Baltimore, Maryland, May 21–23, Blackwell Publishing Ltd, Hoboken, NJ, Vol. 34, pp. 1455–1457.
Smithwick, Q. Y. , Vagners, J. , Reinhall, P. G. , and Seibel, E. J. , 2006, “ An Error Space Controller for a Resonating Fiber Scanner: Simulation and Implementation,” ASME J. Dyn. Syst. Meas. Control, 128(4), pp. 899–913.
Yeoh, I. L. , Reinhall, P. G. , Berg, M. C. , and Seibel, E. J. , 2015, “ Self-Contained Image Recalibration in a Scanning Fiber Endoscope Using Piezoelectric Sensing,” ASME J. Med. Devices, 9(1), p. 011004.
Yeoh, I . L. , Reinhall, P. G. , Berg, M. C. , Chizeck, H. J. , and Seibel, E. J. , 2016, “ Electro-Mechanical Modeling and Adaptive Feedforward Control of a Self-Sensing Scanning Fiber Endoscope,” ASME J. Dyn. Syst. Meas. Control, 138(10), p. 101006.
Hu, D. , Gong, Y. , Hannaford, B. , and Seibel, E. J. , 2015, “ Semi-Autonomous Simulated Brain Tumor Ablation With RAVENII Surgical Robot Using Behavior Tree,” 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, Washington, May 26–30, pp. 3868–3875.
Wang, Y. , Gao, F. , and Doyle, F. J. , 2009, “ Survey on Iterative Learning Control, Repetitive Control, and Run-to-Run Control,” J. Process Control, 19(10), pp. 1589–1600.
Hagood, N. W. , and Anderson, E. H. , 1992, “ Simultaneous Sensing and Actuation Using Piezoelectric Materials,” Proc. SPIE, 1543, pp. 409–421.
Simmers, G. E. , Hodgkins, J. R. , Mascarenas, D. D. , Park, G. , and Sohn, H. , 2004, “ Improved Piezoelectric Self-Sensing Actuation,” J. Intell. Mater. Syst. Struct., 15(12), pp. 941–953.
Good, R. , Hahn, J. , Edison, T. , and Qin, S. J. , 2002, “ Drug Dosage Adjustment Via Run-to-Run Control,” IEEE American Control Conference (ACC), Anchorage, Alaska, May 8–10, Vol. 5, pp. 4044–4049.
Butler, S. W. , and Stefani, J. A. , 1994, “ Supervisory Run-to-Run Control of Polysilicon Gate Etch Using In Situ Ellipsometry,” IEEE Trans. Semiconductor Manuf., 7(2), pp. 193–201.
Srinivasan, B. , Bonvin, D. , Visser, E. , and Palanki, S. , 2003, “ Dynamic Optimization of Batch Processes—II: Role of Measurements in Handling Uncertainty,” Comput. Chem. Eng., 27(1), pp. 27–44.
Srinivasan, B. , Palanki, S. , and Bonvin, D. , 2003, “ Dynamic Optimization of Batch Processes—I: Characterization of the Nominal Solution,” Comput. Chem. Eng., 27(1), pp. 1–26.
Zafiriou, E. , Adomaitis, R. A. , and Gattu, G. , 1995, “ An Approach to Run-to-Run Control for Rapid Thermal Processing,” IEEE American Control Conference (ACC), Seattle, WA, June 21–23, Vol. 2, pp. 1286–1288.
Srinivasan, B. , Primus, C. J. , Bonvin, D. , and Ricker, N. L. , 2001, “ Run-to-Run Optimization Via Control of Generalized Constraints,” Control Eng. Pract., 9(8), pp. 911–919.
Owens, C. , Zisser, H. , Jovanovic, L. , Srinivasan, B. , Bonvin, D. , and Doyle, F. J., III , 2006, “ Run-to-Run Control of Blood Glucose Concentrations for People With Type 1 Diabetes Mellitus,” IEEE Trans. Biomed. Eng., 53(6), pp. 996–1005. [PubMed]
Clayton, G. M. , Tien, S. , Leang, K. K. , Zou, Q. , and Devasia, S. , 2009, “ A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM,” ASME J. Dyn. Syst. Meas. Control, 131(6), p. 061101.
Zisser, H. , Jovanovic, L. , Doyle, F., III , Ospina, P. , and Owens, C. , 2005, “ Run-to-Run Control of Meal-Related Insulin Dosing,” Diabetes Technol. Ther., 7(1), pp. 48–57. [PubMed]
View article in PDF format.

## References

Lee, C. M. , Engelbrecht, C. J. , Soper, T. D. , Helmchen, F. , and Seibel, E. J. , 2010, “ Scanning Fiber Endoscopy With Highly Flexible, 1-mm Catheterscopes for Wide-Field, Full-Color Imaging,” J. Biophotonics, 3(5–6), pp. 385–407. [PubMed]
Bryant, R. C. , Seibel, E. J. , Lee, C. M. , and Schroder, K. E. , 2004, “ Low-Cost Wearable Low-Vision Aid Using a Handmade Retinal Light-Scanning Microdisplay,” J. Soc. Inf. Disp., 12(4), pp. 397–404.
Yang, C. , Hou, V . W. , Girard, E. J. , Nelson, L. Y. , and Seibel, E. J. , 2014, “ Target-to-Background Enhancement in Multispectral Endoscopy With Background Autofluorescence Mitigation for Quantitative Molecular Imaging,” J. Biomed. Opt., 19(7), p. 076014.
Soper, T. D. , Porter, M. P. , and Seibel, E. J. , 2012, “ Surface Mosaics of the Bladder Reconstructed From Endoscopic Video for Automated Surveillance,” IEEE Trans. Biomed. Eng., 59(6), pp. 1670–1680. [PubMed]
Woldetensae, M. H. , Kirshenbaum, M. R. , Kramer, G. M. , Zhang, L. , and Seibel, E. J. , 2013, “ Fluorescence Image-Guided Photodynamic Therapy of Cancer Cells Using a Scanning Fiber Endoscope,” Proc. SPIE, 8576, p. 85760L.
Gong, Y. , Hu, D. , Hannaford, B. , and Seibel, E. J. , 2014, “ Accurate Three-Dimensional Virtual Reconstruction of Surgical Field Using Calibrated Trajectories of an Image-Guided Medical Robot,” J. Med. Imaging, 1(3), p. 035002.
Gong, Y. , Johnston, R. S. , Melville, C. D. , and Seibel, E. J. , 2015, “ Axial-Stereo 3-D Optical Metrology for Inner Profile of Pipes Using a Scanning Laser Endoscope,” Int. J. Optomechatronics, 9(3), pp. 238–247. [PubMed]
McVeigh, P. Z. , Sacho, R. , Weersink, R. A. , Pereira, V . M. , Kucharczyk, W. , Seibel, E. J. , Wilson, B. C. , and Krings, T. , 2014, “ High-Resolution Angioscopic Imaging During Endovascular Neurosurgery,” Neurosurgery, 75(2), pp. 171–180. [PubMed]
Templeton, A. W. , Webb, K. , Hwang, J. H. , Seibel, E. J. , and Saunders, M. , 2014, “ Scanning Fiber Endoscopy: A Novel Platform for Cholangioscopy,” Gastrointest. Endos., 79(6), pp. 1000–1001.
Kundrat, M. J. , Reinhall, P. G. , and Seibel, E. J. , 2011, “ Method to Achieve High Frame Rates in a Scanning Fiber Endoscope,” ASME J. Med. Devices, 5(3), p. 034501.
Kundrat, M. J. , Reinhall, P. G. , Lee, C. M. , and Seibel, E. J. , 2011, “ High Performance Open Loop Control of Scanning With a Small Cylindrical Cantilever Beam,” J. Sound Vib., 330(8), pp. 1762–1771. [PubMed]
Smithwick, Q. Y. , Vagners, J. , Johnston, R. S. , and Seibel, E. J. , 2010, “ A Hybrid Nonlinear Adaptive Tracking Controller for a Resonating Fiber Microscanner,” ASME J. Dyn. Syst. Meas. Control, 132(1), p. 011001.
Smithwick, Q. Y. , Seibel, E. J. , Reinhall, P. G. , and Vagners, J. , 2001, “ Control Aspects of the Single-Fiber Scanning Endoscope,” Proc. SPIE, 4253, pp. 176–188.
Smithwick, Q. Y. , Vagners, J. , Reinhall, P. G. , and Seibel, E. J. , 2003, “ 54.3: Modeling and Control of the Resonant Fiber Scanner for Laser Scanning Display or Acquisition,” SID Symposium Digest of Technical Papers, Baltimore, Maryland, May 21–23, Blackwell Publishing Ltd, Hoboken, NJ, Vol. 34, pp. 1455–1457.
Smithwick, Q. Y. , Vagners, J. , Reinhall, P. G. , and Seibel, E. J. , 2006, “ An Error Space Controller for a Resonating Fiber Scanner: Simulation and Implementation,” ASME J. Dyn. Syst. Meas. Control, 128(4), pp. 899–913.
Yeoh, I. L. , Reinhall, P. G. , Berg, M. C. , and Seibel, E. J. , 2015, “ Self-Contained Image Recalibration in a Scanning Fiber Endoscope Using Piezoelectric Sensing,” ASME J. Med. Devices, 9(1), p. 011004.
Yeoh, I . L. , Reinhall, P. G. , Berg, M. C. , Chizeck, H. J. , and Seibel, E. J. , 2016, “ Electro-Mechanical Modeling and Adaptive Feedforward Control of a Self-Sensing Scanning Fiber Endoscope,” ASME J. Dyn. Syst. Meas. Control, 138(10), p. 101006.
Hu, D. , Gong, Y. , Hannaford, B. , and Seibel, E. J. , 2015, “ Semi-Autonomous Simulated Brain Tumor Ablation With RAVENII Surgical Robot Using Behavior Tree,” 2015 IEEE International Conference on Robotics and Automation (ICRA), Seattle, Washington, May 26–30, pp. 3868–3875.
Wang, Y. , Gao, F. , and Doyle, F. J. , 2009, “ Survey on Iterative Learning Control, Repetitive Control, and Run-to-Run Control,” J. Process Control, 19(10), pp. 1589–1600.
Hagood, N. W. , and Anderson, E. H. , 1992, “ Simultaneous Sensing and Actuation Using Piezoelectric Materials,” Proc. SPIE, 1543, pp. 409–421.
Simmers, G. E. , Hodgkins, J. R. , Mascarenas, D. D. , Park, G. , and Sohn, H. , 2004, “ Improved Piezoelectric Self-Sensing Actuation,” J. Intell. Mater. Syst. Struct., 15(12), pp. 941–953.
Good, R. , Hahn, J. , Edison, T. , and Qin, S. J. , 2002, “ Drug Dosage Adjustment Via Run-to-Run Control,” IEEE American Control Conference (ACC), Anchorage, Alaska, May 8–10, Vol. 5, pp. 4044–4049.
Butler, S. W. , and Stefani, J. A. , 1994, “ Supervisory Run-to-Run Control of Polysilicon Gate Etch Using In Situ Ellipsometry,” IEEE Trans. Semiconductor Manuf., 7(2), pp. 193–201.
Srinivasan, B. , Bonvin, D. , Visser, E. , and Palanki, S. , 2003, “ Dynamic Optimization of Batch Processes—II: Role of Measurements in Handling Uncertainty,” Comput. Chem. Eng., 27(1), pp. 27–44.
Srinivasan, B. , Palanki, S. , and Bonvin, D. , 2003, “ Dynamic Optimization of Batch Processes—I: Characterization of the Nominal Solution,” Comput. Chem. Eng., 27(1), pp. 1–26.
Zafiriou, E. , Adomaitis, R. A. , and Gattu, G. , 1995, “ An Approach to Run-to-Run Control for Rapid Thermal Processing,” IEEE American Control Conference (ACC), Seattle, WA, June 21–23, Vol. 2, pp. 1286–1288.
Srinivasan, B. , Primus, C. J. , Bonvin, D. , and Ricker, N. L. , 2001, “ Run-to-Run Optimization Via Control of Generalized Constraints,” Control Eng. Pract., 9(8), pp. 911–919.
Owens, C. , Zisser, H. , Jovanovic, L. , Srinivasan, B. , Bonvin, D. , and Doyle, F. J., III , 2006, “ Run-to-Run Control of Blood Glucose Concentrations for People With Type 1 Diabetes Mellitus,” IEEE Trans. Biomed. Eng., 53(6), pp. 996–1005. [PubMed]
Clayton, G. M. , Tien, S. , Leang, K. K. , Zou, Q. , and Devasia, S. , 2009, “ A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM,” ASME J. Dyn. Syst. Meas. Control, 131(6), p. 061101.
Zisser, H. , Jovanovic, L. , Doyle, F., III , Ospina, P. , and Owens, C. , 2005, “ Run-to-Run Control of Meal-Related Insulin Dosing,” Diabetes Technol. Ther., 7(1), pp. 48–57. [PubMed]

## Figures

Fig. 1

The general run-to-run optimization algorithm. Highlighted blocks are avenues for control system design.

Fig. 2

The SFE scan engine consisting of a piezoelectric-tube and a cantilevered optical fiber [1]

Fig. 3

(a) Desired and achieved trajectories for cases without and with modeling error and (b) energy measure over time for cases without and with modeling error

Fig. 4

Convex energy surface E(t=200) parameterized by two-variable modeling error

Fig. 5

Convex energy surface for (a) K=5K0 and M=M0 (left) and (b) K=K0 and M=5M0 (right)

Fig. 6

Experimental setup

Fig. 7

Experimentally constructed energy surface

Fig. 8

At 50 °C: (a) 2D scan result with feedforward control, (b) 2D scan result with run-to-run optimization, (c) normalized squared radial error compared between feed-forward and run-to-run result, and (d) phase/tangential squared error compared between feed-forward and run-to-run result

Fig. 9

Radial and phase/tangential mean-squared-error (MSE) over different operating temperatures, achieved by feedforward method versus run-to-run optimization: (a) radial MSE and (b) tangential MSE

Fig. 10

At 50 °C: (a) 2D scan result with open-loop control, (b) 2D scan result with run-to-run optimization, (c) normalized squared radial error compared between open-loop and run-to-run result, and (d) phase/tangential squared error compared between open-loop and run-to-run result

Fig. 11

Radial and phase/tangential MSE over different operating temperatures, achieved by open-loop method versus run-to-run optimization: (a) radial MSE and (b) tangential MSE

Fig. 12

(a) Target image to be laser-projected, (b) open-loop control result at 50 °C, (c) adaptive feedforward result at 50 °C, and (d) run-to-run optimized result at 50 °C

Fig. 13

Simulation of energy surface with 1% amplitude measurement noise

Fig. 14

Data from experiment 2 recast as pixel/Euclidean MSE over different operating temperatures, achieved by open-loop versus new run-to-run optimization

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