# Experimental System Identification and Disturbance Observer-based Control
for a Monolithic $\mathrm{Z\theta_{x}\theta_{y}}$ Precision Positioning System

Mohammadali Ghafarian1,2, Bijan Shirinzadeh1, Ammar Al-Jodah1,3, Tilok Kumar
Das1, Tianyao Shen1 1Robotics and Mechatronics Research Laboratory (RMRL),
Department of Mechanical and Aerospace Engineering, Monash University,
Clayton, VIC 3800, Australia. 2Institute for Intelligent Systems Research and
Innovation (IISRI), Deakin University, Geelong Waurn Ponds, VIC 3216,
Australia. 3The University of Western Australia, Perth, WA 6009,
Australia.Corresponding author<EMAIL_ADDRESS>
###### Abstract

A compliant parallel micromanipulator is a mechanism in which the moving
platform is connected to the base through a number of flexural components.
Utilizing parallel-kinematics configurations and flexure joints, the
monolithic micromanipulators can achieve extremely high motion resolution and
accuracy. In this work, the focus was towards the experimental evaluation of a
3-DOF ($\mathrm{Z\theta_{x}\theta_{y}}$) monolithic flexure-based piezo-driven
micromanipulator for precise out-of-plane micro/nano positioning applications.
The monolithic structure avoids the deficiencies of non-monolithic designs
such as backlash, wear, friction, and improves the performance of
micromanipulator in terms of high resolution, accuracy, and repeatability. A
computational study was conducted to investigate and obtain the inverse
kinematics of the proposed micromanipulator. As a result of computational
analysis, the developed prototype of the micromanipulator is capable of
executing large motion range of $\pm$238.5$\mathrm{\mu m}$ $\times$
$\pm$4830.5$\mathrm{\mu rad}$ $\times$ $\pm$5486.2$\mathrm{\mu rad}$. Finally,
a sliding mode control strategy with nonlinear disturbance observer (SMC-NDO)
was designed and implemented on the proposed micromanipulator to obtain system
behaviours during experiments. The obtained results from different
experimental tests validated the fine micromanipulator’s positioning ability
and the efficiency of the control methodology for precise micro/nano
manipulation applications. The proposed micromanipulator achieved very fine
spatial and rotational resolutions of $\pm$4$\mathrm{nm}$,
$\pm$250$\mathrm{nrad}$, and $\pm$230$\mathrm{nrad}$ throughout its workspace.

###### Note to Practitioners

Piezo-actuated precision positioning systems play an increasingly important
role in the fields of micro/nano manipulation robots. They have the advantages
of fine resolution, high accuracy, fast response speed, and large output
displacement. However, such systems inherently exhibit vibration, hysteresis
behaviors, and are affected by external disturbances that could cause
oscillations and positioning errors. This study presents a robust control
methodology implemented on a 3-DOF positioning system
($\mathrm{Z\theta_{x}\theta_{y}}$), which is among the most prone system to be
affected by existing disturbances. This control methodology is designed to
improve the tracking performance in the presence of hysteresis nonlinearity,
disturbances, and modeling errors. The effectiveness of the proposed control
methodology is demonstrated by conducting a series of experiments. Due to the
ease of implementation, the developed control methodology can be applied to
other positioning systems as well.

###### Index Terms:

Sliding mode control, Nonlinear disturbance observer, Precision positioning,
Monolithic parallel manipulator, Amplification mechanism

##  I Introduction

Flexure joints have dominant superiority over traditional mechanical joints in
precision engineering including micromanipulation mechanisms. Flexure-based
parallel micromanipulators benefit from the advantages of both flexure joints
and parallel-kinematics configurations, and additionally utilize the important
characteristics of micromanipulators such as frictionless motion, absence of
mechanical play and backlash, and no need for lubrication. These features are
important and effective for various micro/nano positioning and nano-alignment
applications, and it is not surprising that flexure-based parallel micro/nano
manipulation systems stand out among others as the key element in the ultra-
precision technologies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Wei and Xu [11]
proposed a force-sensing cell microinjector based on a single-axis compliant
small-stiffness mechanism. Yong et al. [12] presented single- and dual-stage
vertical positioners for high-speed piezoelectric nanopositioning
applications. Wang et al. [13] presented a decoupled piezo-driven XY
micro/nano positioning system with a travel range of $\pm 27.7\times\pm
26.6\mathrm{\mu m^{2}}$. Tian et al. [14] presented a custom made atomic force
microscopy (AFM) which was integrated with a 3-DOF XYZ parallel-kinematics
piezo-driven micromanipulator for high-speed imaging. Cai et al. [15]
presented the design and analysis of two parallel compliant piezo-driven 3-DOF
micro/nano positioning system. Meanwhile, the inverse Bouc-Wen (BW) model was
applied as a feedforward hysteresis compensator in the feedforward/feedback
hybrid control method to compensate for the hysteresis of piezoelectric
actuators (PEAs). The proposed micromanipulator exhibited small translational
and rotational workspaces of $\pm 4.1\times\pm 5.2\times\pm 6.5\mathrm{\mu
m^{3}}$ and $\pm 112\times\pm 52.5\times\pm 48.5\mathrm{\mu rad^{3}}$,
respectively. Qin et al. [16] proposed the design of a 3-PRR (prismatic-
revolute-revolute) $\mathrm{XY\theta}$ micromanipulator. The Scott-Russell
(SR) mechanism was utilized in the proposed design to magnify the displacement
of the PEA. Dong et al. [17] presented a 6-DOF piezo-driven micromanipulator.
As a result of having no amplification mechanism, the workspace of the
micromanipulator was very limited. Yang et al. [18] demonstrated the design,
modeling, and experimental analysis of a piezo-driven $\mathrm{XY}$
micromanipulator. The experimental results illustrated that the $\mathrm{XY}$
micromanipulator had a working range of $\pm 75\times\pm 73.5\mathrm{\mu
m^{2}}$ with the resolution of $\pm 0.128\mathrm{\mu m}$ and $\pm
0.143\mathrm{\mu m}$ in the X- and Y-directions, respectively.

---
Figure 1: Monolithic $\mathrm{Z\theta_{x}\theta_{y}}$ parallel
micromanipulator

Among the flexure-based parallel micromanipulators, the compliant
$\mathrm{Z\theta_{x}\theta_{y}}$ type micro/nano positioning systems have
become the research focus due to their important advantages for in out-of-
plane positioning tasks [19, 20, 21, 22]. Qu et al. [23] presented the design,
modeling and test of a piezo-driven $\mathrm{\theta_{x}\theta_{y}}$ flexure-
based micro/nano positioning system. The experimental results indicated that
the developed micromanipulator could achieve a workspace of $\pm
515\mathrm{\mu rad}\times\pm 460\mathrm{\mu rad}$ about its two working axes
with a resolution of $\pm 0.5\mathrm{\mu rad}$. Chen et al. [24] introduced
the mechanical design, modeling and experimental tests of a large-angle
$\mathrm{Z\theta_{x}\theta_{y}}$ macromanipulator driven by four small air-gap
voice coil actuators. The proposed system could achieve rotational-motion
ranges of $\pm 41.59\mathrm{mrad}$ and $\pm 41.13\mathrm{mrad}$ in the working
axes, for which the $\mathrm{Z}$-mode frequency was $49.6\mathrm{Hz}$ and the
rotational ones were $55.45\mathrm{Hz}$ and $56.09\mathrm{Hz}$, respectively.
The motion resolution of the macromanipulator was $\pm 6.67\mathrm{\mu rad}$.
Kim et al. [25] presented an active vibration control system which was
constructed based on a non-monolithic 3-DOF $\mathrm{Z\theta_{x}\theta_{y}}$
micro/nano manipulation system with an in-plane dimension of $160\mathrm{mm}$
(diameter) and an out-of-plane height of $60\mathrm{mm}$. Cao and Chen [26]
demonstrated the development of a system identification model for a
commercially-available 3-DOF piezo-driven $\mathrm{Z\theta_{x}\theta_{y}}$
micromanipulator (P-558.TCD, Physik Instrumente). The system was driven by
four PEAs and had a motion range of $\pm 25\mathrm{\mu m}\times\pm
250\mathrm{\mu rad}\times\pm 250\mathrm{\mu rad}$. The quasi-static analysis
of a non-monolithic compliant tripod system for micro/nano positioning
applications was presented by Wei et al. [27]. The proposed micromanipulator
had an overall positioning range of $\pm 41\mathrm{\mu m}\times\pm
330\mathrm{\mu rad}\times\pm 385\mathrm{\mu rad}$. Considering the above-
mentioned studies, a compliant monolithic 3-DOF piezo-driven micromanipulator
was introduced by the authors with a larger workspace and fine resolution
capable of executing three out-of-plane motions, one translation and two
rotations. The structure of the proposed monolithic micromanipulator was
optimized completely to have a maximum working range and bandwidth frequency
higher than 100Hz [28].
Regardless of the types of micromanipulators used to perform micro/nano
manipulation tasks, an effective motion tracking control strategy can improve
the tracking performances of the system significantly. In addition,
disturbances such as cross-coupling, parametric uncertainties, etc. can
practically affect and degrade the performance of a precision positioning
system. Therefore, designing and utilizing a disturbance observer-based
control methodology to be able to estimate and compensate the effect of
disturbances for achieving high precision applications is very beneficial [29,
30, 31]. Chen et al. [32] introduced a nonlinear disturbance observer (NDO)
for robotic manipulators for various purposes such as friction compensation,
independent joint control, sensorless torque control, and fault diagnosis.
Furthermore, the global exponential stability of the proposed NDO was
guaranteed. Lau et al. [33] presented an enhanced adaptive robust disturbance
observer-based motion tracking control methodology for tracking a desired
motion trajectory in the presence of unknown or uncertain system’s parameters,
nonlinearities including hysteresis, and disturbances in the motion system.
The proposed control methodology was applied in a semi-automated hand-held ear
surgical device for the treatment of Otitis Media with Effusion (OME).

TABLE I: Mechanical and physical properties of ABS

Symbol | Quantity |  20cmValue
---|---|---
20cm$\mathrm{\nu}$ |  20cmPoisson’s ratio |  20cm$0.35$
$\mathrm{\rho}$ | Density | $0.9087\mathrm{(g/cm^{3})}$
$\mathrm{E}$ | Young’s modulus | $2200\mathrm{(MPa)}$
20cm$\mathrm{\sigma_{yield}}$ |  20cmTensile yield strength |  20cm$31\mathrm{(MPa)}$
20cm$\mathrm{\sigma_{ultimate}}$ |  |
|  20cmTensile ultimate strength |  20cm$55\mathrm{(MPa)}$

---
Figure 2: Reachable workspace of the developed micromanipulator

Motivated by the previous work [28], an experimental study of a large range
piezo-driven spatial compliant monolithic parallel
$\mathrm{Z\theta_{x}\theta_{y}}$ micro/nano manipulation system with a fine
resolution is presented in this paper. Monolithically manufactured designs are
very important for micro/nano applications and they are preferable in
comparison with the assembled manipulation structures because of elimination
of the unwanted features that affect a smooth and accurate nano-resolution
manipulation. Other advantages of the proposed micromanipulator are low
manufacturing and material cost. Nano-meter/radian resolution, large
amplification ratio, repeatability, and stability are ensured due to the
characteristics of the proposed monolithic micromanipulator. To investigate
the motion range and decouple the micromanipulator’s motions, the inverse
kinematics is obtained using FEA. The performances of the developed
micromanipulator are investigated in the real-time experiments via three
feedback control methodologies, i.e. Proportional-Integral-Derivative control
(PID), sliding mode control (SMC), and nonlinear disturbance observer-based
sliding mode control (SMC-NDO). The role of the developed NDO is to compensate
for the uncertain disturbances in the real-time experiments to achieve high
precision manipulation tasks. In the end, the experimental results, including
frequency, resolution, and several complex trajectory motion tracking analyses
are presented, and precise manipulations can be guaranteed by the developed
monolithic micromanipulator and control methodologies.

##  II Mechanical Design

As presented in Figure 1, the structure of the micromanipulator consists of a
fixed base platform, six leaf-flexure-based parallelogram mechanisms, three
flexure-based Scott-Russell amplification mechanisms, three flexure-based
spherical joint modules, and a moving platform. Two sets of leaf
parallelograms are incorporated into the input and output points of the Scott-
Russell amplification mechanism as prismatic joints. This linearizes the
motion of the Scott-Russell mechanism and increases the micromanipulator’s
stiffness. Because PEAs produce a very small displacement as a proportion of
their length, mechanical displacement amplification is inevitably required for
large displacement applications. The Scott-Russell amplification mechanism is
a well established mechanical amplifier [28, 16]. Additionally, it has the
advantage of transforming into a horizontal input to a vertical output which
is ideal for the compactness of the designed $\mathrm{Z\theta_{x}\theta_{y}}$
micromanipulator. The platform needs to generate rotational motions along two
different axes. A spherical joint, which is capable of rotating in three
different axes is adopted for the connection between the platform and the
manipulation system. In comparison with previous designs, here the PEAs are
placed outside of the micromanipulator not inside. Thus, different sizes of
PEAs can be used for the proposed monolithic design without affecting the
geometry and overall dimensions of the structural design. The overall
dimensions of the proposed design are 201mm, 180mm, 75mm. Developments in 3D
printing enable complex geometries to be easily and economically manufactured
in multiple materials. Therefore, the proposed design is fabricated using high
density and high accuracy 3D-printing from Acrylonitrile Butadiene Styrene
(ABS). The mechanical and physical properties of ABS are presented in Table I.

##  III Workspace analysis

The usable workspace is limited by material stresses. In order to maintain the
stability, repeatability, and capabilities of the micromanipulator in precise
manipulation, it is very important that the applied stress on the
micromanipulator due to the load remains less than its tensile yield strength.
Using the stress and safety factor analyses in FEA software (ANSYS), the
maximum Von-Mises stress and the minimum safety factor occur when the input
displacement of 90$\mu$m is applied to the micromanipulator as the input of
the three PEAs simultaneously. The values of maximum Von-Mises stress and
minimum safety factor corresponding to the 90$\mu$m inputs were 17.585MPa and
1.71, respectively. It is worth noting that the obtained minimum safety factor
was 1.71, as the safety factor must always be greater than 1. Therefore, more
input force/displacement could have been applied to obtain a larger workspace.
However, the value of 1.71 is considered as a lower limit for the safety
factor of the micromanipulator to avoid some deficiencies including creep,
fatigue, and mechanical failure of the micromanipulator in the experiment.
Three PEAs (Physik Instrumente P-843.60) with a maximum displacement of
90$\mu$m were assumed to be used for analysing the workspace based on the FEA.
The inverse kinematics of the micromanipulator was computed by evaluating the
motion of the system across 80 inputs, spanning the full input range, and
performing a regression on the resulting outputs. The best fit inverse
kinematics is computed as:

$\textbf{$J^{-1}$}=\leavevmode\resizebox{234.87749pt}{}{$\begin{pmatrix}-0.1909&0.0001&0.0110\\\
-0.1877&-0.0095&-0.0053\\\ -0.1875&0.0093&-0.0056\\\ \end{pmatrix}$}$ (1)

---
Figure 3: Block diagram of the proposed SMC-NDO control methodology

The resultant workspace of computational analysis is presented in Figure 2.
The values of motions along $\mathrm{Z}$, $\mathrm{\theta_{x}}$, and
$\mathrm{\theta_{y}}$ axes are $\pm$238.5$\mu$m, $\pm$4830.5$\mu$rad, and
$\pm$5486.2$\mu$rad, respectively. Since the determinant of the Jacobian
matrix is non-zero, therefore there is no singularity in the workspace of the
micromanipulator.

---
Figure 4: Schematic diagram of sensing, control, and experimental research
facility
---
a
b
c
Figure 5: Modal analysis of the micromanipulator: (a) $\mathrm{Z}$-resonant frequency (b) $\mathrm{\theta_{x}}$-resonant frequency (c) $\mathrm{\theta_{y}}$-resonant frequency |  |
---|---|---
Figure 6: System response to the applied staircase input

##  IV Controller design

The accuracy of a manipulation task is very dependent on eliminating undesired
disturbances affecting the system. Therefore, in this section, establishment
of control methodologies will be presented for the utilization in the
experimental study to overcome this challenging issue for precise micro/nano
manipulation applications.

###  IV-A Sliding Mode Control (SMC)

In general, flexure-based micromanipulators are structures comprising solid
links and flexure hinges. The monolithic construction reduces assembly errors
and guarantees high accuracy. A lumped parameter dynamic model for a flexure-
based micromanipulator can be given by:

$\displaystyle m_{q}\ddot{q}+c_{q}\dot{q}+k_{q}q=d_{q}+u_{q}$ (2)

where $q$ represents the three output motions of the micromanipulator (Z,
$\mathrm{\theta_{x}}$, $\mathrm{\theta_{y}}$), and $m_{q}$, $c_{q}$, and
$k_{q}$ are the equivalent mass, damping and stiffness in the corresponding
axis. $d_{q}$ and $u_{q}$ are the lumped disturbance and control action in a
given axis, respectively. A state-space representation of Eq. (2) can be
obtained by considering $x_{1}=q$ and $x_{2}=\dot{q}$. The tracking error
between the actual and desired motions can be introduced as follows:

$\displaystyle e=x_{1}-{{x}_{1}}_{des}$ (3)

There are two steps in the design of an SMC. The first step is designing a
sliding surface so that the plant restricted to the sliding surface has a
desired system response. This means the state variables of the plant dynamics
are constrained to satisfy another set of equations which define the so-called
switching surface. The second step is constructing a switched feedback gains
necessary to drive the plant’s state trajectory to the sliding surface. These
strategies are built on the generalized Lyapunov stability theory. In this
study, a PID sliding surface was selected to reduce the steady-state error and
improve the transient response speed. The selected sliding surface is given
by:

$\displaystyle s=\lambda_{p}e+\lambda_{i}\int_{0}^{t}edt+\lambda_{d}\dot{e}$
(4)

where $\lambda_{p}$, $\lambda_{i}$, and $\lambda_{d}$ are positive
proportional, integral, and derivative parameters to be selected,
respectively. The SMC law is intended to force the tracking error $e$ to
approach the sliding surface and then proceed to the origin along the sliding
surface. Hence, the sliding surface needs to be stable, and therefore yields
to the fact that $\dot{s}=0$. Additionally, the control action of the SMC
approach consists of two parts; equivalent control and switching control as
follows:

$\displaystyle u=u_{eq}+u_{sw}$ (5)

Thus, it can be realized that to be able to force the actual motions to
converge to the desired ones, the equivalent control and switching control
need to be as follows:

$u_{eq}=c{x}_{2}+kx_{1}+m\ddot{x}_{1_{des}}-m\frac{\lambda_{i}}{\lambda_{d}}e-m\frac{\lambda_{p}}{\lambda_{d}}\dot{e}-\hat{d}$
---
$u_{sw}=-a_{1}s-a_{2}sgn(s)$
(6)

where $a_{1}$ and $a_{2}$ are positive constants. The term $\hat{d}$ is the
estimation of the disturbance which will be derived later. The deviation
between the actual and estimated disturbance is defined by, $\tilde{d}$, and
can be given as follows:

$\tilde{d}=d-\hat{d}$ (7)

---
Figure 7: Star trajectory tracking results |  |
---|---|---
Figure 8: Experimental control tracking errors in the star trajectory

Lyapunov stability function is used to verify the design performance of the
SMC. In order to have a stable controller, it is required that $V$ must be
bounded, which means that it is important to prove $\dot{V}\leq 0$. Therefore,
the derivative of Lyapunov function is derived as follows:

$\displaystyle\dot{V}=s\dot{s}=s\\{\frac{\lambda_{d}}{m}u_{sw}+\frac{\lambda_{d}}{m}d-\frac{\lambda_{d}}{m}\hat{d}\\}=s\\{-k_{1}s$
$\displaystyle-
k_{2}sgn(s)+\frac{\lambda_{d}}{m}\tilde{d}\\}=-k_{1}s^{2}-k_{2}|s|$
$\displaystyle+\frac{\lambda_{d}}{m}\tilde{d}s\leq-
k_{1}s^{2}-|s|(k_{2}-\frac{\lambda_{d}}{m}\tilde{d})$ (8)

where $k_{1}$ and $k_{2}$ are equal to $\frac{\lambda_{d}a_{1}}{m}$ and
$\frac{\lambda_{d}a_{2}}{m}$, respectively. The stability of the designed
controller is guaranteed, if the inequality
$k_{2}\geq\frac{\lambda_{d}}{m}\tilde{d}$ is established. Furthermore, the
discontinuity of the signum function in the SMC law can cause chattering.
Therefore, in order to avoid this problem, the discontinuous signum function
is replaced by the continuous hyperbolic tangent function ($tanh$). In other
words, the function is used as an approximator of the signum function.

###  IV-B Nonlinear Disturbance Observer (NDO)

An NDO is designed to estimate the undesired disturbance $d_{q}$, introduced
in the dynamic model, Eq. (2). $d_{q}$ represents all disturbances that may be
caused by unmodeled dynamics. The objective is to design an observer such that
the estimation $\hat{d}$ yielded by the observer exponentially approaches the
disturbance $d$ under any $q$ and $\dot{q}$.
The dynamic model of the micromanipulator, Eq. (2), can be rewritten in the
general description of a form as follows:

$\dot{\textbf{x}}=f(\textbf{x})+g_{1}(\textbf{x})u+g_{2}(\textbf{x}){d}$ (9)

where x is a representation of the state variables $x_{1}$ and $x_{2}$.
Besides, functions $f$, $g_{1}$, and $g_{2}$ can also be found to be as
follows:

$g_{1}(x)=g_{2}(x)=\frac{1}{m_{q}}$
---
$f(x)=-\frac{c_{q}}{m_{q}}x_{2}-\frac{k_{q}}{m_{q}}x_{1}$
(10)

The NDO is designed according to the following formulations [34, 35]:

$\hat{d}=z+p(\textbf{x})$
---
$\dot{z}=-l(\textbf{x})g_{2}(\textbf{x})z-l(\textbf{x})\\{g_{2}(\textbf{x})p(\textbf{x})+f(\textbf{x})+g_{1}(\textbf{x})u\\}$
(11)

where $\hat{d}$ and $z$ are the estimate of the undesired disturbance and the
internal state of the nonlinear observer, respectively, and $p(\textbf{x})$ is
a nonlinear function to be designed. The nonlinear observer gain
$l(\textbf{x})$ is defined as follows:

$l(\textbf{x})=\frac{\partial p(\textbf{x})}{\partial\textbf{x}}$ (12)

It can be shown that $\hat{d}$ approaches $d$ exponentially if $p(\textbf{x})$
is chosen such that:

$\dot{\tilde{d}}+\frac{\partial
p(\textbf{x})}{\partial\textbf{x}}g_{2}(x)\tilde{d}=0$ (13)

Therefore, if $p(\textbf{x})$ is chosen as $lx_{2}$, it can be realized from
Eq. (13) that the designed NDO yields to following form in the time domain:

$\tilde{d}(t)=\tilde{d}(0)exp(-\frac{l}{m}t)$ (14)

According to Eq. (14), if $l$ is chosen to be a positive constant, when time
goes to infinity the disturbance estimation error, $\tilde{d}$, converges to
zero. Therefore, the designed NDO is globally exponentially stable.

---
Figure 9: RMRL logo trajectory tracking results |  |
---|---|---
Figure 10: Experimental control tracking errors in the RMRL logo trajectory

###  IV-C Robustness Analysis of SMC-NDO

The robustness of the proposed SMC-NDO is proven in this section. The feedback
control of the system described by Eq. (2) is robust, if control law (5) is
applied and it satisfies the conditions of:

$\lim\limits_{s\to 0^{+}}\dot{s}<0$ | , | $\lim\limits_{s\to 0^{-}}\dot{s}>0$
---|---|---
(15)

The above-mentioned conditions illustrate that if the phase trajectory around
$s=0$ points to the $s$ hyper-plane, then it will enter the sliding mode when
the hyper-plane $s=0$.
By substituting control law (5) into Eq. (2), the equation of motion of the
closed-loop system is obtained as:

$\displaystyle\ddot{q}$
$\displaystyle=\frac{1}{m_{q}}\Big{\\{}\tilde{d}_{q}-a_{1}s-a_{2}tanh(s)\Big{\\}}+\Big{\\{}\ddot{q}_{des}-\frac{\lambda_{i}}{\lambda_{d}}e-\frac{\lambda_{p}}{\lambda_{d}}\dot{e}\Big{\\}}$
(16)

Moreover, by substituting Eq. (16) into the differentiation of Eq. (4), the
dynamic sliding mode equation is obtained as:

$\displaystyle\dot{s}$
$\displaystyle=\lambda_{p}\dot{e}+\lambda_{i}e+\lambda_{d}\ddot{e}$
$\displaystyle=\lambda_{p}\dot{e}+\lambda_{i}e+\lambda_{d}(\ddot{q}-\ddot{q}_{des})$
$\displaystyle\approx\tilde{d}_{q}-a_{1}s-a_{2}tanh(s)$ (17)

Utilizing the two conditions (15), and the stability condition that was
obtained in Section III.A, ($a_{2}\geq\tilde{d}$), it is trivial that the
robustness conditions are satifisfied as following:

$\displaystyle\lim\limits_{s\to 0^{+}}\dot{s}=\tilde{d}_{q}-a_{2}tanh(s)<0$
(18) $\displaystyle\lim\limits_{s\to
0^{-}}\dot{s}=\tilde{d}_{q}-a_{2}tanh(s)>0$

Thus, the phase trajectory around $s=0$ enters the sliding mode in the hyper-
plane and $s=0$, so the PID sliding surface (4) is:

$\displaystyle s=(\lambda_{p}+\lambda_{i}s^{-1}+\lambda_{d}s)e=0$ (19)

Using Eq. (19) and having the knowledge of
$\dot{e}=\dot{x}_{2}-\dot{x}_{2_{des}}$, Eq. (9) is rewritten as following:

$x_{1}=x_{1_{des}}+e$
---
$x_{2}=x_{2_{des}}-\frac{\lambda_{p}}{\lambda_{d}}e-\frac{\lambda_{i}}{\lambda_{d}}s^{-1}e$
(20)

Thus, Eq. (20) becomes the state equation when the system enters the sliding
mode. We know that the second-order system Eq. (2) can be expressed by the
first-order state Eq. (20) and that the dynamic characteristics of the system
are independent of $d(t)$, therefore, SMC-NDO is highly robust when the system
enters the sliding mode, thereby ensuring that the system exhibits a good
dynamic response and stability.

---
Figure 11: 3D-Archimedean spiral trajectory tracking results

###  IV-D Time Convergence Stability Analysis of SMC-NDO

The time convergence of the sliding mode equation is investigated in this
section when the disturbance observer and chattering reduction methods are
used.

Lemma. [36] Assume that a continuous definite function $V(t)\geq 0$ satisfies
the subsequent differential inequality
$\dot{V}(t)+\beta_{1}V^{\alpha}(t)+\beta_{2}V(t)\leq 0$, where $\beta_{1}$,
$\beta_{2}$, and $\alpha$ are the positive constants, and the range of
$\alpha$ is between $0$ to $1$. Based on which the function $V(t)$ converges
to zero in finite time given as:

$t_{s}=\frac{1}{\beta_{2}(1-\alpha)}\ln\frac{\beta_{2}V^{(1-\alpha)}(0)+\beta_{1}}{\beta_{1}}$
(21)

Theorem. Consider system described by Eq. (2). If the surface manifold and
control law are chosen according to Eqs. (4) and (6), then sliding mode
enforcement along the sliding surface, Eq. (4), can be validated. As a result,
the system states will converge to the desired reference trajectories and
tracking errors will converge to zero in finite time.

Proof of Theorem. Consider the Lyapunov candidate functions of the form as:

$V=\frac{1}{2}s^{2}$ (22)

Taking the time derivative of Eq. (22) and utilizing Eq. (17) yield to:

$\dot{V}=s\\{\tilde{d}-a_{1}s-a_{2}tanh(s)\\}\leq-a_{1}s^{2}-a_{2}|s|$ (23)

Let $\beta_{1}=\sqrt{2}a_{2}$, $\beta_{2}=2a_{1}$, and $\alpha=\frac{1}{2}$,
then Eq. (23) can be rewritten as:

$\dot{V}(t)+\beta_{1}V^{\alpha}(t)+\beta_{2}V(t)\leq 0$ (24)

Then, according to Lemma, the differential inequality (24) yields the finite-
settling time as represented by Eq. (21), and that completes the proof.
Finally, Figure 3 provides the information on how the SMC-NDO was implemented
in the real-time experiment to achieve remarkable performances from the
micromanipulator.

TABLE II: Control parameters

Description | Parameter | Value
---|---|---
PID
PID constants | $k_{p}$ | 50
$k_{i}$ | 17.5e6
$k_{d}$ | 0
SMC
PID sliding surface constants | $\lambda_{p}$ | 50
$\lambda_{i}$ | 13e6
$\lambda_{d}$ | 100
Switching control constants | $a_{1}$ | 2.5
$a_{2}$ | 0.6
SMC-NDO
Nonlinear observer gain | $l$ | 100

|  |
---|---|---
Figure 12: Experimental control tracking errors in the 3D-Archimedean spiral
trajectory

##  V Experiments and Results

Experimental tests were carried out to investigate the mechanical and dynamic
performances of the developed flexure-based micromanipulator, as well as the
efficiency of the proposed control methodologies. The schematic diagram of the
experimental set-up is presented in Figure 4. The basic operation of the
flexure-based micromanipulator was to change the drive voltage from the
voltage control unit (amplifier module from Physik Instrumente) to the three
PEAs from Physik Instrumente (model P-843.60) to further drive the
micromanipulator. During installation of the PEAs, pre-compression forces were
applied to keep the actuators’ tip and the micromanipulator in contact. These
PEAs are multilayer PZT stacked ceramic translators capable of 90$\mu$m
displacement corresponding to a range of operating voltage from 0 to
100$\mathrm{V}$. The position of the moving platform was measured by three
capacitive sensors (Physik Instrumente D-050.00). The capacitive sensors had a
circular active area with a 4 mm radius surrounded by an annular guard ring,
with a specified working range of 50$\mu$m. The electronic interface (Physik
Instrumente E-509.C3A) to the capacitive sensors provided a distance
measurement as an analog voltage, which was recorded at the control computer
with a 16-bit analog-to-digital converter (ADC) and at the rate of
$10\mathrm{kHz}$. In order to reduce the external vibration disturbances, all
experimental tests were performed on a vibration-isolated optical table.

|  |
---|---|---
Figure 13: Hysteresis cancellation of different control methodologies TABLE
III: Summary of the numerical results of trajectory tracking and hysteresis
compensation

| Controller | Z-axis |  $\theta_{x}$-axis |  $\theta_{y}$-axis
---|---|---|---|---
Trajectory tracking
RMSE ($\mathrm{\mu m}$) |  $\%$Improvement (SMC-NDO/PID) | RMSE ($\mathrm{\mu rad}$) |  $\%$Improvement (SMC-NDO/PID) | RMSE ($\mathrm{\mu rad}$) |  $\%$Improvement (SMC-NDO/PID)
Star | PID | 0.1094 | 91.04 | 3.9298 | 90.80 | 0.3903 | 66.92
SMC | 0.1012 | 3.6336 | 0.3596
SMC-NDO | 0.0098 | 0.3615 | 0.1291
RMRL logo | PID | 0.0239 | 74.06 | 2.9472 | 87.61 | 8.8958 | 91.07
SMC | 0.0184 | 2.2886 | 6.9058
SMC-NDO | 0.0062 | 0.3652 | 0.7942
3D-Archimedean spiral | PID | 0.0819 | 96.83 | 2.7101 | 95.44 | 0.5296 | 76.25
SMC | 0.0660 | 2.1850 | 0.4345
SMC-NDO | 0.0026 | 0.1237 | 0.1258
| Hysteresis compensation
|  $\%$Hysteresis max. |  $\%$Improvement (SMC-NDO/PID $\&$ SMC) |  $\%$Hysteresis max. |  $\%$Improvement (SMC-NDO/PID $\&$ SMC) |  $\%$Hysteresis max. |  $\%$Improvement (SMC-NDO/PID $\&$ SMC)
| PID | 0.49 | 100 | 0.44 | 100 | 0.54 | 100
SMC | 0.36 | 0.38 | 0.44
SMC-NDO | 0 | 0 | 0

###  V-A System Identification

System identification of the developed micromanipulator was conducted to
investigate the dynamic characteristics. Hence, a sinusoidal sweep signal with
the frequency linearly varying from $1\mathrm{Hz}$ to $250\mathrm{Hz}$ was
applied to the PEAs input, one at a time, in order to excite the modes of
vibration of the platform. The frequency responses to these inputs are
presented in Figure 5. Resonance was observed in the response along the
$\mathrm{Z}$, $\mathrm{\theta_{x}}$, and $\mathrm{\theta_{y}}$ axes occured at
$113.4\mathrm{Hz}$, $173.1\mathrm{Hz}$, and $184.2\mathrm{Hz}$, respectively.
Finally, three continuous-time transfer functions were identified from the
frequency response along each motion axis and formulated as follows:

$TF_{z}=\frac{3.774e05}{s^{2}+58.44s+5.08e05}$ (25)

$TF_{\theta_{x}}=\frac{5.792e05}{s^{2}+59.16s+1.186e06}$ (26)

$TF_{\theta_{y}}=\frac{1.454e06}{s^{2}+76.13s+1.346e06}$ (27)

Considering the transfer functions (15)-(17), $m_{q}$, $c_{q}$, and $k_{q}$
can easily be calculated for implementing into to the SMC and SMC-NDO.
In the next stage, i.e. closed-loop experiment, the control parameters were
found by empirical rules based on observation of the response to reference and
disturbance changes. These selected control parameters for the PID, SMC, and
SMC-NDO control schemes are presented in Table II.

###  V-B Resolution characterization of the system

To test the closed-loop system positioning resolution, the staircase response
of the positioner was measured along each working axis. The magnitude of each
step was chosen to be 20nm for the translational axis and 900nrad for the
rotational axes. The obtained results from the three control techniques are
presented in Figure 6. The best resolution result was captured utilizing the
SMC-NDO control technique. The minimum resolution was found to be
8$\mathrm{nm}$, 500$\mathrm{nrad}$, and 460$\mathrm{nrad}$ along the
$\mathrm{Z}$, $\mathrm{\theta_{x}}$, and $\mathrm{\theta_{y}}$ axes,
respectively. The positioning resolution is limited by the measurement noise
in motion axes, quantization in the measurement of rotation, the resolution of
capacitive sensors, and the output resolution of the data acquisition card.

###  V-C Motion tracking characterization of the system

To further illustrate the motion abilities of the developed micromanipulator
alongside with the designed control approaches, three complex curves tracking
experiments were conducted. In which, a star, Robotics and Mechatronics
Research Laboratory (RMRL) logo, and 3D-Archimedean spiral were selected and
designed as the target tracking trajectories due to their high level of
complexities. The tracking results are presented in Figures 7 to 12.
The results of the trajectory tracking and tracking errors of the star
trajectory implementing the three proposed control schemes are presented in
Figures 7 and 8. The best tracking error was obtained by implementing SMC-NDO
and the maximum values of error were 14$\mathrm{nm}$ for the Z-axis,
700$\mathrm{nrad}$ and 494$\mathrm{nrad}$ for the $\mathrm{\theta_{x}}$ and
$\mathrm{\theta_{y}}$ axes, respectively. Figure 7 illustrates the projection
of the tracked trajectory on the ZY-plane.
The RMRL logo was designed as a symbol of the most complex trajectory to
verify the performance of the micromanipulator. The results of the trajectory
tracking and tracking errors are presented in Figures 9 and 10. The projection
of the designed trajectory on the XY-plane is presented in Figure 9. According
to the results, the micromanipulator was capable of tracking the designed logo
with high precision and accuracy. It must be noted that by using sensors with
a higher resolution, more accurate motion tracking can be achieved even with
the same control methodologies proposed in this work.
The tracking results of the 3D-Archimedean spiral are presented in Figures 11
and 12. These two figures provide the difference between the desired and
actual trajectories while implementing different control methodologies. Same
as the previous trajectories tracking, the SMC-NDO control scheme was observed
to outperform the SMC and PID controllers. Also, the SMC provided a better
tracking performance than the PID control scheme. Considering SMC-NDO, the
maximum tracking error along the Z, $\mathrm{\theta_{x}}$, and
$\mathrm{\theta_{y}}$ axes were observed to be 11$\mathrm{nm}$,
644$\mathrm{nrad}$, and 650$\mathrm{nrad}$, respectively. On the other hand,
with respect to the output displacement and rotations, there were within error
of $\%0.045$ along the Z-axis, $\%0.088$ along the $\mathrm{\theta_{x}}$-axis
and $\%0.148$ along the $\mathrm{\theta_{y}}$-axis, respectively.

###  V-D Hysteresis Compensation

Hysteresis is an undesired effect for precise manipulation using a piezo-
actuated micromanipulator. Hence, a series of sinusoidal inputs were generated
and applied to the developed micromanipulator to investigate the capability of
the proposed control methodologies for hysteresis compensation. The results
are presented in Figure 13. It can be observed that the hysteresis effect was
compensated/eliminated utilizing the SMC-NDO control methodology.

Finally, the performances of micromanipulator and control methodologies
regarding very complex motion tracking and hysteresis compensation are
summarized in Table III. According to this table, the maximum and minimum
improvements of $\%$100 and $\%$67 were achieved, respectively, by
implementing the SMC-NDO control methodology. The minimum bound of the
improvement will increase if a smooth time-transitioned trajectory is
considered to track, which is the case in many practical applications.

##  VI Conclusion

A monolithic parallel $\mathrm{Z\theta_{x}\theta_{y}}$ micromanipulator with
large range, high resolution, and high bandwidth frequency was presented. This
positioning system utilized flexure-based components to produce desired
motions in three out-of-plane directions. The linearized displacement relation
between the actuation space and the output Cartesian space of the
micromanipulator was established utilizing FEA. Hence, the micromanipulator’s
workspace was obtained implementing the inverse kinematics. Additionally,
stress level in the flexure hinges was found to be below the yield point of
the material (ABS) under the maximum input of the PEAs, thus verifying the
repeatability and stability of the developed micromanipulator for the precise
manipulation tasks. An SMC control methodology with nonlinear disturbance
observer was introduced and utilized for the experimental evaluation of the
micromanipulator. This control methodology was established to track very
complex desired motion trajectories with high accuracy. It was also designed
to accommodate system parametric uncertainties, cross-axis coupling, and
nonlinearities including unknown disturbances and the hysteresis effect in the
$\mathrm{Z\theta_{x}\theta_{y}}$ micromanipulator. The stability and
robustness of the proposed closed-loop control technique (SMC-NDO) were
proved, and the convergence of the position tracking errors to zero was
guaranteed by the established system. Furthermore, an experimental facility
was established to investigate the effectiveness of control methodologies and
micromanipulator’s high precision positioning capabilities. Based on the
recorded experimental results, the positioning system showed very fine
position and orientation resolutions of $\pm$4$\mathrm{nm}$,
$\pm$250$\mathrm{nrad}$, and $\pm$230$\mathrm{nrad}$ throughout the
micromanipulator’s motion range of $\pm$238.5$\mathrm{\mu m}$ $\times$
$\pm$4830.5$\mathrm{\mu rad}$ $\times$ $\pm$5486.2$\mathrm{\mu rad}$. Finally,
based on the root mean square error (RMSE) analysis of trajectory motion
tracking and maximum hysteresis investigation, high resolution, high accuracy,
hysteresis elimination, and consequently the effectiveness of the SMC-NDO
control methodology were established.

## Acknowledgment

This work was funded by the Australian Research Council (ARC) Discovery
Project (DP) grant, and the Australian Research Council (ARC) Linkage
Infrastructure, Equipment and Facilities (LIEF) grant.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] M. Ghafarian, B. Shirinzadeh, T. Das, A. Al-Jodah, and W. Wei, “Design of a novel parallel monolithic 6-DOF compliant micromanipulation mechanism,” in _IEEE/ASME International Conference on Advanced Intelligent Mechatronics, AIM_ , vol. 2018-July, 2018. [Online]. Available: https://doi.org/10.1109/AIM.2018.8452401
  * [2] T. Das, B. Shirinzadeh, M. Ghafarian, A. Al-Jodah, and J. Pinskier, “Characterization of a compact piezoelectric actuated microgripper based on double-stair bridge-type mechanism,” in _Journal of Micro-Bio Robotics_ , vol. 16, no. 1, pp. 79–92, 2020. [Online]. Available: https://doi.org/10.1007/s12213-020-00132-5
  * [3] A. Al-Jodah, B. Shirinzadeh, M. Ghafarian, Y. Tian, and L. Clark, “Design and analysis of a novel 3-DOF large range micropositioning mechanism,” in _2018 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)_ , pp. 991–996, 2018. [Online]. Available: https://doi.org/10.1109/AIM.2018.8452386
  * [4] Y. L. Yang, Y. D. Wei, J. Q. Lou, L. Fu, and S. Fang, “Design and control of a multi-DOF micromanipulator dedicated to multiscale micromanipulation,” _Smart Materials and Structures_ , vol. 26, no. 11, 2017. [Online]. Available: https://doi.org/10.1088/1361-665X/aa8f73
  * [5] A. Al-Jodah, B. Shirinzadeh, M. Ghafarian, T. Kumar Das, Y. Tian, and D. Zhang, “A fuzzy disturbance observer based control approach for a novel 1-DOF micropositioning mechanism,” _Mechatronics_ , vol. 65, no. October 2019, 2020. [Online]. Available: https://doi.org/10.1016/j.mechatronics.2019.102317
  * [6] M. Ghafarian, B. Shirinzadeh, A. Al-Jodah, T. K. Das, W. Wei, Y. Tian, and D. Zhang, “Design of a novel parallel monolithic 3-DOF compliant micromanipulator,” in _Proceedings of MARSS 2019: 4th International Conference on Manipulation, Automation, and Robotics at Small Scales_ , 2019. [Online]. Available: https://doi.org/10.1109/MARSS.2019.8860961
  * [7] A. Al-Jodah, B. Shirinzadeh, M. Ghafarian, T. K. Das, J. Pinskier, Y. Tian, and D. Zhang, “Modeling and a cross-coupling compensation control methodology of a large range 3-DOF micropositioner with low parasitic motions,” _Mechanism and Machine Theory_ , vol. 162, p. 104334, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0094114X21000926
  * [8] M. Ghafarian, B. Shirinzadeh, A. Al-Jodah, T. K. Das, W. Wei, Y. Tian, and D. Zhang, “An XYZ micromanipulator for precise positioning applications,” _Journal of Micro-Bio Robotics_ , vol. 16, no. 1, pp. 53–63, 2020. [Online]. Available: https://doi.org/10.1007/s12213-020-00124-5
  * [9] M. Ghafarian, B. Shirinzadeh, A. Al-Jodah, T. K. Das, W. Wei, and T. Shen, “Modeling and prototype experiment of a monolithic 3-PUU parallel micromanipulator with nano-scale accuracy,” _Smart Materials and Structures_ , vol. 29, 2020. [Online]. Available: https://doi.org/10.1088/1361-665X/ab8a6e
  * [10] T. Das, B. Shirinzadeh, M. Ghafarian, and J. Pinskier, “A flexure-based 2-DOF microgripper for handling micro-objects,” in _2018 International Conference on Manipulation, Automation and Robotics at Small Scales (MARSS)_ , pp. 1–6, 2018. [Online]. Available: https://doi.org/10.1109/MARSS.2018.8481193
  * [11] Y. Wei and Q. Xu, “Design and Testing of a New Force-Sensing Cell Microinjector Based on Small-Stiffness Compliant Mechanism,” _IEEE/ASME Transactions on Mechatronics_ , vol. 26, no. 2, pp. 818–829, 2020\. [Online]. Available: http://dx.doi.org/10.1109/tmech.2020.3003992
  * [12] Y. K. Yong, S. P. Wadikhaye, and A. J. Fleming, “High speed single- and dual-stage vertical positioners,” _Review of Scientific Instruments_ , vol. 87, no. 8, 2016. [Online]. Available: http://dx.doi.org/10.1063/1.4960080
  * [13] F. Wang, Z. Huo, C. Liang, B. Shi, Y. Tian, X. Zhao, and D. Zhang, “A Novel Actuator-Internal Micro/Nano Positioning Stage with an Arch-Shape Bridge-Type Amplifier,” _IEEE Transactions on Industrial Electronics_ , vol. 66, no. 12, pp. 9161–9172, 2019. [Online]. Available: https://doi.org/10.1109/TIE.2018.2885716
  * [14] Y. Tian, K. Cai, D. Zhang, X. Liu, F. Wang, and B. Shirinzadeh, “Development of a XYZ scanner for home-made atomic force microscope based on FPAA control,” _Mechanical Systems and Signal Processing_ , vol. 131, pp. 222–242, 2019. [Online]. Available: https://doi.org/10.1016/j.ymssp.2019.05.057
  * [15] K. Cai, Y. Tian, F. Wang, D. Zhang, X. Liu, and B. Shirinzadeh, “Design and control of a 6-degree-of-freedom precision positioning system,” _Robotics and Computer-Integrated Manufacturing_ , vol. 44, pp. 77–96, 2017\. [Online]. Available: http://dx.doi.org/10.1016/j.rcim.2016.08.005
  * [16] Y. Qin, B. Shirinzadeh, D. Zhang, and Y. Tian, “Design and Kinematics Modeling of a Novel 3-DOF Monolithic Manipulator Featuring Improved Scott-Russell Mechanisms,” _Journal of Mechanical Design_ , vol. 135, no. 10, p. 101004, 2013. [Online]. Available: https://doi.org/10.1115/1.4024979
  * [17] Y. Dong, F. Gao, and Y. Yue, “Modeling and prototype experiment of a six-DOF parallel micro-manipulator with nano-scale accuracy,” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ , vol. 229, no. 14, pp. 2611–2625, 2015. [Online]. Available: https://doi.org/10.1177/0954406214562461
  * [18] Y. Yang, Y. Wei, J. Lou, and F. Xie, “Design and analysis of a new flexure-based XY stage,” _Journal of Intelligent Material Systems and Structures_ , vol. 28, no. 17, pp. 2388–2402, 2017. [Online]. Available: https://doi.org/10.1177/1045389X17689929
  * [19] H. Kim, J. Kim, D. Ahn, and D. Gweon, “Development of a nanoprecision 3-DOF vertical positioning system with a flexure hinge,” _IEEE Transactions on Nanotechnology_ , vol. 12, no. 2, pp. 234–245, 2013. [Online]. Available: https://doi.org/10.1109/TNANO.2013.2242088
  * [20] H. J. Lee, H. C. Kim, H. Y. Kim, and D. G. Gweon, “Optimal design and experiment of a three-axis out-of-plane nano positioning stage using a new compact bridge-type displacement amplifier,” _Review of Scientific Instruments_ , vol. 84, no. 11, 2013. [Online]. Available: https://doi.org/10.1063/1.4827087
  * [21] H. S. Kim and Y. M. Cho, “Design and modeling of a novel 3-DOF precision micro-stage,” _Mechatronics_ , vol. 19, no. 5, pp. 598–608, 2009. [Online]. Available: http://dx.doi.org/10.1016/j.mechatronics.2009.01.004
  * [22] M. T. Pham, S. H. Yeo, T. J. Teo, P. Wang, and M. L. S. Nai, “Design and Optimization of a Three Degrees-of-Freedom Spatial Motion Compliant Parallel Mechanism With Fully Decoupled Motion Characteristics,” _Journal of Mechanisms and Robotics_ , vol. 11, pp. 1–8, 2019. [Online]. Available: https://doi.org/10.1115/1.4043925
  * [23] J. Qu, W. Chen, J. Zhang, and W. Chen, “A piezo-driven 2-DOF compliant micropositioning stage with remote center of motion,” _Sensors and Actuators, A: Physical_ , vol. 239, pp. 114–126, 2016. [Online]. Available: http://dx.doi.org/10.1016/j.sna.2016.01.025
  * [24] G. Chen, Y. Ding, X. Zhu, P. Liu, and H. Ding, “Design and modeling of a compliant tip-tilt-piston micropositioning stage with a large rotation range,” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ , vol. 0, no. 0, pp. 1–14, 2018\. [Online]. Available: https://doi.org/10.1177/0954406218781401
  * [25] H. S. Kim, Y. M. Cho, and J. H. Moon, “Active vibration control using a novel three-DOF precision micro-stage,” _Smart Materials and Structures_ , vol. 19, no. 5, 2010. [Online]. Available: https://doi.org/10.1088/0964-1726/19/5/055001
  * [26] Y. Cao and X. Chen, “State space system identification of 3-degree-of-freedom (DOF) piezo-actuator-driven stages with unknown configuration,” _Actuators_ , vol. 2, no. 1, pp. 1–18, 2013. [Online]. Available: https://doi.org/10.3390/act2010001
  * [27] W. Huaxian, L. Wei, L. Yufei, W. Yuqiao, and Y. Xuefeng, “Quasi-static analysis of a compliant tripod stage with plane compliant lever mechanism,” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ , vol. 231, no. 9, pp. 1639–1650, 2017. [Online]. Available: https://doi.org/10.1177/0954406215619193
  * [28] M. Ghafarian, B. Shirinzadeh, A. Al-Jodah, T. K. Das, and J. Pinskier, “FEA-based optimization of a complete structure of a monolithic z/tip/tilt micromanipulator,” _Journal of Micro-Bio Robotics_ , vol. 16, no. 1, pp. 93–110, 2020. [Online]. Available: https://doi.org/10.1007/s12213-020-00133-4
  * [29] Y. Pi and X. Wang, “Observer-based cascade control of a 6-DOF parallel hydraulic manipulator in joint space coordinate,” _Mechatronics_ , vol. 20, no. 6, pp. 648–655, 2010. [Online]. Available: http://dx.doi.org/10.1016/j.mechatronics.2010.07.002
  * [30] Y. Zhang and P. Yan, “Sliding mode disturbance observer-based adaptive integral backstepping control of a piezoelectric nano-manipulator,” _Smart Materials and Structures_ , vol. 25, no. 12, pp. 1–12, 2016. [Online]. Available: http://dx.doi.org/10.1088/0964-1726/25/12/125011
  * [31] Y. Zhang, P. Yan, and Z. Zhang, “A disturbance observer-based adaptive control approach for flexure beam nano manipulators,” _ISA Transactions_ , vol. 60, pp. 206–217, 2016. [Online]. Available: http://dx.doi.org/10.1016/j.isatra.2015.10.005
  * [32] W. H. Chen, D. J. Ballance, P. J. Gawthrop, and J. O’Reilly, “A nonlinear disturbance observer for robotic manipulators,” _IEEE Transactions on Industrial Electronics_ , vol. 47, no. 4, pp. 932–938, 2000. [Online]. Available: https://doi.org/10.1109/41.857974
  * [33] J. Y. Lau, W. Liang, and K. K. Tan, “Adaptive sliding mode enhanced disturbance observer-based control of surgical device,” _ISA Transactions_ , vol. 90, pp. 178–188, 2019. [Online]. Available: https://doi.org/10.1016/j.isatra.2018.12.048
  * [34] A. Al-Jodah, B. Shirinzadeh, M. Ghafarian, T. K. Das, Y. Tian, D. Zhang, and F. Wang, “Development and control of a large range XY$\Theta$ micropositioning stage,” _Mechatronics_ , vol. 66, no. December 2019, p. 102343, 2020. [Online]. Available: https://doi.org/10.1016/j.mechatronics.2020.102343
  * [35] A. Al-Jodah, B. Shirinzadeh, J. Pinskier, M. Ghafarian, T. K. Das, Y. Tian, and D. Zhang, “Antlion Optimized Robust Control Approach for Micropositioning Trajectory Tracking Tasks,” _IEEE Access_ , vol. 8, pp. 220 889–220 907, 2020. [Online]. Available: https://doi.org/10.1109/ACCESS.2020.3043411
  * [36] S. Mobayen, “An adaptive fast terminal sliding mode control combined with global sliding mode scheme for tracking control of uncertain nonlinear third-order systems,” _Nonlinear Dynamics_ , vol. 82, no. 1-2, pp. 599–610, 2015. [Online]. Available: http://dx.doi.org/10.1007/s11071-015-2180-4


The Quantum Union Bound made easy

Ryan O'Donnell[1]
Ramgopal Venkateswaran
Computer Science Dept., CMU.
The author ordering was randomized.

We give a short proof of Gao's Quantum Union Bound and

§ INTRODUCTION

Let $\rho \in \C^{d \times d}$ be a quantum mixed state and let $A_1, \dots, A_m \geq 0$ be (orthogonal) projectors on $\C^d$, which may be thought of as “quantum events”.
We write $\overline{A}_t = \Id - A_t$, where $\Id$ is the identity operator.
For intuition, we think of the $A_t$'s as “good” events that happen with high probability: we write

\[
{\E}_{\rho}[A_t] \coloneqq \tr(\rho A_t) = 1 - \eps_t,
\]

and hence the “bad” event $\ol{A}_t$ has $\E_{\rho}[\ol{A}_t] = \eps_t$.
Suppose we now sequentially measure $\rho$ with the two-outcome projective measurements $(\ol{A}_1, A_1)$, $(\ol{A}_2, A_2)$, …, $(\ol{A}_m, A_m)$.
For $0 \leq t \leq m$, let $\rho_t$ denote the state conditioned on outcomes $A_1, \dots, A_t$ all occurring.
The Quantum Union Bound question now asks, “What is the probability, $\Succ$,  that all $m$ “good” outcomes occur?”
We may also ask the related question of Gentle Sequential Measurement: Conditioned on all good outcomes occurring, how far is the resulting state $\rho_m$ from $\rho$ (say, in trace distance)?

For full details of the history of these questions, see the discussion in [5].
An important milestone regarding the Quantum Union Bound came from Sen [7], who established $\Fail \leq 2 \sqrt{\Loss}$, where we denote $\Fail = 1 - \Succ$ and $\Loss = {\sum}_t \eps_t$.
Subsequently, Gao [4] obtained the square of Sen's upper bound.
His results were:

(Gao's Quantum Union Bound.)  $\Fail \leq 4\Loss$. $(\heartsuit)$

(Gentle Sequential Measurement.) $\Dtr{\rho}{\rho_m} \leq \sqrt{\Loss}$. $(\diamondsuit)$

Khabbazi Oskouei, Mancini, and Wilde [5] obtained a further improvement to $(\heartsuit)$, discussed in <Ref>.
In this work we give a simple proof of a common generalization of <Ref>.
Denoting  fidelity (<Ref>) by $\Fid{{}\cdot{}}{{}\cdot{}}$, we show:

$\displaystyle 1 \leq \sqrt{\Succ} \sqrt{\Fid{\rho}{\rho_m}} + \sqrt{\Fail}\sqrt{\Loss}$.

Then to deduce <Ref> from <Ref>, we use $\Fid{\rho}{\rho_m} \leq 1$ to get

\begin{equation}
1 \leq \sqrt{\Succ} + \sqrt{\Fail}\sqrt{\Loss}
= \sqrt{1-\Fail} + \sqrt{\Fail}\sqrt{\Loss} \ \ \implies\ \  \Fail  \leq \frac{4\Loss}{(1+\Loss)^2} \leq 4 \Loss. \tag*{$(\heartsuit')$}
\end{equation}

Here $\Fail  \leq \frac{4\Loss}{(1+\Loss)^2}$ arises from solving the quadratic for $\Fail$; it assumes $\Loss \leq 1$.
One can also get $(\heartsuit')$ via AM-GM: $\tfrac12 \Fail \leq 1 - \sqrt{1-\Fail} \leq \sqrt{\Fail}\sqrt{\Loss} = \sqrt{\tfrac12\Fail}\sqrt{2\Loss} \leq \tfrac12(\tfrac12 \Fail + 2\Loss)$.

To deduce <Ref>, we apply Cauchy–Schwarz to <Ref> obtaining

\begin{equation}
1 \leq \cancelto{1}{\sqrt{\Succ + \Fail}}\ \sqrt{\Fid{\rho}{\rho_m} + \Loss} \ \ \implies\ \ %\Fid{\rho}{\rho_m} \geq 1 - \Loss \quad\implies\quad
\Infid{\rho}{\rho_m} \coloneqq 1 - \Fid{\rho}{\rho_m} \leq \Loss, \tag*{$(\diamondsuit')$}
\end{equation}

stronger than <Ref> thanks to the Fuchs–van de Graaf [3] inequality .

§.§ Notation

For two  states $\rho, \sigma \in \C^{d \times d}$, their fidelity is $\Fid{\rho}{\sigma} = \|\sqrt{\rho} \sqrt{\sigma}\|_1^2 = (\tr \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}})^2$, where we recall the Schatten $1$-norm $\|M\|_1 = \tr\sqrt{M M^\dagger} = \tr\sqrt{M^\dagger M}$.

The fidelity between two states is at most $1$; this is a consequence of the matrix Cauchy–Schwarz inequality $\|M_1 M_2\|_1^2 \leq \tr(M_1^\dagger M_1) \tr(M_2^\dagger M_2)$.

Let $M$ be a $d$-column matrix with $M^\dagger M \leq \Id$, thought of as a nondestructive measurement matrix (so that $M^\dagger M$ is one element of a POVM).
The probability of $M$ occurring when $\rho$ is measured is $\E_\rho[M^\dagger M]$, and we denote the resulting state conditioned on this outcome by
\rho | M = (M \rho M^\dagger) / \E_\rho[M^\dagger M].
(We tacitly assume the denominator is nonzero.)

We work over finite-dimensional Hilbert spaces for simplicity, but this is inessential; the proofs extend to any separable Hilbert space.

§ PROOF

For quantum states $\rho, \sigma \in \C^{d \times d}$ and $A \in \C^{d \times d}$ with $\ol{A} \geq 0$,

\[
\sqrt{\Fid{\rho}{\sigma}} \leq \sqrt{{\E}_\sigma[\textnormal{$A^\dagger A$}]} \sqrt{\Fid{\rho}{\sigma|A}} +  \sqrt{{\E}_\sigma[\ol{A}]}\sqrt{{\E}_\rho[\ol{A}]}.% \overset{\textnormal{(if $A$ is a projector)}}{=} \|A\|_\sigma \sqrt{\Fid{\rho}{\sigma|A}} +  \|\ol{A}\|_\sigma \|\ol{A}\|_\rho.
\]

We have $\sqrt{\Fid{\rho}{\sigma}}
= \|\sqrt{\rho} \sqrt{\sigma}\|_1
= \|\sqrt{\rho} (A + \ol{A}) \sqrt{\sigma}\|_1
\leq \|\sqrt{\rho} A \sqrt{\sigma}\|_1 + \|\sqrt{\rho} \ol{A} \sqrt{\sigma}\|_1$.
On one hand,

\[
\|\sqrt{\rho} A \sqrt{\sigma}\|_1 = \tr \sqrt{\sqrt{\rho} A \sigma A^\dagger \sqrt{\rho}} = \sqrt{{\E}_\sigma[A^\dagger A]} \sqrt{\Fid{\rho}{\sigma | A}}.
\]

On the other hand, by matrix Cauchy–Schwarz we have

\[
\|\sqrt{\rho} \ol{A} \sqrt{\sigma}\|_1 = \left\|\sqrt{\rho} \ol{A}^{1/2} \ol{A}^{1/2} \sqrt{\sigma}\right\|_1\leq \sqrt{{\E}_\sigma[\ol{A}]}\sqrt{{\E}_\rho[\ol{A}]}.
\]

(Remark: in <Ref> we note that in fact $\|\sqrt{\rho} \ol{A} \sqrt{\sigma}\|_1^2 = \Fid{\rho|\ol{A}^{1/2}}{\sigma|\ol{A}^{1/2}} \cdot {\E}_\sigma[\ol{A}]\cdot {\E}_\rho[\ol{A}]$.)

For a geometric interpretation with pure states, see <Ref>.  We now prove <Ref>.

For $0 \leq t \leq m$, consider the event that the “good” outcomes $A_1, \dots, A_t$  all occur.
We write $p_t$ for the probability of this event, $\rho_t$ for the state $\rho$ conditioned on this event, and $r_t = \sqrt{p_t} \sqrt{\Fid{\rho}{\rho_t}}$.
For $1 \leq t \leq m$ we write $q_t$ for the probability that $\ol{A}_t$ is the first “bad” outcome that occurs. Now

\[
r_{t-1} - r_{t}
= \sqrt{p_{t-1}}\parens*{\sqrt{\Fid{\rho}{\rho_{t-1}}} - \sqrt{{\E}_{\rho_{t-1}}[A_t]} \sqrt{\Fid{\rho}{\rho_{t-1} | A_t}}}
\leq \sqrt{p_{t-1}}\sqrt{{\E}_{\rho_{t-1}}[\ol{A}_t]}\sqrt{{\E}_{\rho}[\ol{A}_t]}
= \sqrt{q_t} \sqrt{\eps_t},
\]

where the inequality used <Ref> and $A_t^\dagger A_t = A_t$.
Summing this for $t = 1 \dots m$ yields

\[
1 - \sqrt{\Succ} \sqrt{\Fid{\rho}{\rho_m}} = r_0 - r_m \leq \sum_{t=1}^m \sqrt{q_t}\sqrt{\eps_t} \leq \sqrt{{\sum}_t q_t} \sqrt{{\sum}_{t} \eps_t} = \sqrt{\Fail}{\sqrt{\Loss}},
\]

where the last inequality is Cauchy–Schwarz.

Anshu [1] has observed that if the above proof is written using subnormalized pure states, it becomes structurally very similar to Sen's proof [7].

[1]
A. Anshu.
Quantum union bound, 2021.
<https://people.eecs.berkeley.edu/ anuraganshu/Union_bound.pdf>.

[2]
W. Dickinson and M. Salmassi.
The right right triangle on the sphere.
The College Mathematics Journal, 39(1):24–33, 2008.

[3]
C. Fuchs and J. van de Graaf.
Cryptographic distinguishability measures for quantum-mechanical
IEEE Trans. Inform. Theory, 45(4):1216–1227, 1999.

[4]
J. Gao.
Quantum union bounds for sequential projective measurements.
Physical Review A, 92(5):052331, 2015.

[5]
S. Khabbazi Oskouei, S. Mancini, and M. Wilde.
Union bound for quantum information processing.
Proceedings of the Royal Society A, 475(2221):20180612, 2019.

[6]
T. Ogawa and H. Nagaoka.
A new proof of the channel coding theorem via hypothesis testing in
quantum information theory.
In Proceedings of the International Symposium on Information
Theory, page 73. IEEE, 2002.

[7]
P. Sen.
Achieving the Han–Kobayashi inner bound for the quantum
interference channel.
In Proceedings of the International Symposium on Information
Theory, pages 736–740, 2012.

[8]
A. Winter.
Coding theorem and strong converse for quantum channels.
IEEE Transactions on Information Theory, 45(7):2481–2485,

§ ADDITIONAL COMMENTARY

§.§ Simpler proof of Gentle Sequential Measurement

We remark that if one's only goal is to prove <Ref>, the proof is even simpler.
Assuming $A$ is a projector, applying Cauchy–Schwarz to <Ref> yields

\[
\sqrt{\Fid{\rho}{\sigma}} \leq \sqrt{{\E}_\sigma[A]} \sqrt{\Fid{\rho}{\sigma|A}} +  \sqrt{{\E}_\sigma[\ol{A}]}\sqrt{{\E}_\rho[\ol{A}]} \leq \cancelto{1}{\sqrt{{\E}_\sigma[A] + {\E}_\sigma[\ol{A}]}} \  \sqrt{\Fid{\rho}{\sigma|A} + {\E}_\rho[\ol{A}]}.
\]

Squaring and rearranging yields:

If $\rho, \sigma \in \C^{d \times d}$ are states and $A \in \C^{d \times d}$ is a projector,
\Infid{\rho}{\sigma|A} \leq \Infid{\rho}{\sigma} + {\E}_\rho[\ol{A}].

Taking $\sigma = \rho_{t-1}$ and $A = A_t$ we get $\Infid{\rho}{\rho_t} \leq \Infid{\rho}{\rho_{t-1}} + \eps_t$, and hence $\Infid{\rho}{\rho_m} \leq \Loss$ by iterating.

§.§ Fidelity and conditioning

We first recall some traditional matrix notation:

If $M$ is any matrix, recall that $|M|$ denotes $\sqrt{M^\dagger M}$
, so $\|M\|_1 = \tr |M|$.

For any $M \in \C^{m \times \ell}$, $N \in \C^{\ell \times n}$, it is immediate that $|M \cdot N| = \bigl|\,|M| \cdot N\,\bigr|$.
Taking trace on both sides and using $\|X\|_1 = \|X^\dagger\|_1$, we can infer $\|M \cdot N \|_1 = \bigl\|\,|M| \cdot |N^\dagger| \,\bigr\|_1$.

Now we introduce some additional notation:

For $\rho \in \C^{d \times d}$ a quantum state and $A \in \mathbb{C}^{d \times d}$, we write $\|A\|_\rho = \sqrt{\E_\rho[A^\dagger A]}$.

If $A$ is a projector then $\|A\|^2_\rho = \E_\rho[A] = \Fid{\rho}{\rho | A}$.
(The latter formula, basically the “Gentle Measurement Lemma” [8, 6], follows just by writing the definitions and using $\sqrt{\rho} A \sqrt{\rho} \geq 0$, since $A \geq 0$.)

<Ref> may alternately be written as  $\displaystyle \sqrt{\rho | M} = \frac{|\sqrt{\rho} M^\dagger|}{||M||_{\rho}}$.

Although <Ref> looks neat as stated, we actually prefer the definition of fidelity that doesn't have the square built in (as in, e.g., the Nielsen–Chuang text).
For lack of better symbols, we introduce the following notation for it:

We write $\Fidroot{\rho}{\sigma} = \sqrt{\Fid{\rho}{\sigma}} = \|\sqrt{\rho}\sqrt{\sigma}\|_1$, and $\Infidroot{\rho}{\sigma} = \sqrt{\Infid{\rho}{\sigma}} = \sqrt{1 - \Fidroot{\rho}{\sigma}^2}$.

Now the following is an immediate consequence of <Ref> and <Ref>:

If $\rho, \sigma \in \C^{d\times d}$ are  states and $M, N \in \mathbb{C}^{d \times d}$, then $\displaystyle \Fidroot{\rho | M}{\sigma | N} = \frac{\|\sqrt{\rho} M^\dagger N \sqrt{\sigma}\|_1}{\|M\|_\rho \|N\|_\sigma}$.

This formula is quite useful.
In particular ($M = \Id$, $N = A$) it implies $\Fidroot{\rho}{\sigma | A} = \|\sqrt{\rho} A \sqrt{\sigma}\|_1/ \|A\|_\sigma$, which is identical to the first fact derived in our main <Ref>.
Note furthermore that if $A \geq 0$,

\[
\|\sqrt{\rho} A \sqrt{\sigma}\|_1 = \|\sqrt{\rho} \sqrt{A} \sqrt{A} \sqrt{\sigma}\|_1 = \Fidroot{\rho | \sqrt{A}}{\sigma | \sqrt{A}} \cdot \sqrt{{\E}_\rho[A]}\sqrt{{\E}_\sigma[A]}
\]

where we used <Ref> again ($M = N = \sqrt{A}$).
This shows the second fact derived in our main <Ref> (more precisely, it shows the “Remark” at the end, after replacing $A$ with $\ol{A}$).
Finally, putting these two implications together yields:

If $A \geq 0$, then $\displaystyle \Fidroot{\rho}{\sigma | A} \leq \frac{\sqrt{{\E}_\rho[A]}\sqrt{{\E}_\sigma[A]}}{\|A\|_\sigma}$.    If $A$ is furthermore a projector, the right-hand side simplifies to $\sqrt{{\E}_\rho[A]}$; i.e., $\Fidroot{\rho}{\sigma | A} \leq \Fidroot{\rho}{\rho | A}$.

§.§ Obtaining the bound from [5]

The proof given by Khabbazi Oskouei, Mancini, and Wilde [5] included an improvement to Gao's Quantum Union Bound: they showed that

\begin{equation*} \tag{$\clubsuit$}
\Fail^* \coloneqq \Fail - \eps_1  \leq p'\eps_1 + (p+p')\sum_{1 < t < m} \eps_t + p \eps_m
\end{equation*}

for any (positive) $p,p'$ with $1/p + 1/p' = 1$.
(Gao's bound is implied by the $p = p' = 2$ case.)
They also gave an application where it is essential that $p$ may be made arbitrarily close to $1$.
We can obtain the same bound by slightly modifying our proof of <Ref>.

In the modified proof, we simply save on the first term since we know that $q_1 = \eps_1$.

This gives

\[
1 - \sqrt{\Succ} \sqrt{\Fid{\rho}{\rho_m}} \leq \eps_1 + \sqrt{\Fail^*}\sqrt{\Loss - \eps_1}.
\]

But from <Ref> we obtain $\Fid{\rho}{\rho_m} = \Fid{\rho}{\rho_{m-1} | A_m}\leq \Fid{\rho}{\rho | A_m} = 1 - \eps_m$. Thus

\[
1 - \sqrt{1-\Fail^* -\eps_1} \sqrt{1 - \eps_m} \leq \eps_1 + \sqrt{\Fail^*}\sqrt{{\sum}_{t > 1} \eps_t}.
\]

One can solve the associated quadratic equation for $\Fail^*$ to get a sharp, but messy, bound.In particular, I am pretty sure you can get $\Fail^* \leq \frac{2(1-3L)\eps_1 + 4L + 2(1-L)\eps_m}{(1+L)^2}$ for $L = \sum_{1 < t < m} \eps_t$, and also $(\clubsuit)/(1+L)^2$
More simply, we can use AM-GM twice to get  $1 - \sqrt{1-\Fail^* - \eps_1} \sqrt{1 - \eps_m} \geq \tfrac12 (\Fail^* + \eps_1 + \eps_m)$, and

\[
\eps_1 + \sqrt{\Fail^*}\sqrt{{\sum}_{t > 1} \eps_t}
= \eps_1 + \sqrt{p^{-1}\Fail^*}\sqrt{p{\sum}_{t > 1} \eps_t}
\leq \eps_1 + \tfrac12p^{-1}\Fail^* + \tfrac12 p{\sum}_{t > 1} \eps_t.
\]

Putting these together yields $\displaystyle \Fail^* + \eps_1 + \eps_m \leq 2\eps_1 + p^{-1}\Fail^* + p{\sum}_{t > 1} \eps_t$, which yields $(\clubsuit)$ after multiplication by $p'$ and rearrangement (note that $p+p' = pp'$).

§.§ Intuition I: Bhattacharyya coefficient

A useful way of discovering results concerning quantum fidelity is via analogy with its easier-to-understand classical counterpart:

Recall that for two probability distributions $p,q$ on $[d]$, their Bhattacharyya coefficient is $\BC{p}{q} = \sum_{i=1}^d \sqrt{p_i}\sqrt{q_i} \in [0,1]$.
(This equals $\Fidroot{\mathrm{diag}(p)}{\mathrm{diag}(q)}$.)

The well-known classical analogue (indeed, consequence) of the Fuchs–van de Graaf inequality is:

The total variation distance $\dtv{p}{q} = \frac12 \|p-q\|_1$ satisfies $\dtv{p}{q} \leq \sqrt{1 - \BC{p}{q}^2}$.
(This is slightly sharper than bounding total variation distance by Hellinger distance.)

An event $A \subseteq [d]$ is the analogue of a projector, so the following  can be compared to <Ref>:

If $A \subseteq [d]$ is an event, then $p(A) = \Pr_p[A] = \E_p[1_A] = \BC{p}{p|A}^2$.

The analogue of our main <Ref> is also natural in the classical case:

If $A \subseteq [d]$, then $\displaystyle \BC{p}{q}  = \sqrt{q(A)}\,\BC{p}{q|A}+ \sqrt{q(\ol{A})}\sqrt{p(\ol{A})}$.

Since $(q|A)_i$ is $q_i/q(A)$ if $i \in A$, and is $0$ if $i \in \ol{A}$, we get

\[
\BC{p}{q} = \sum_{i \in A} \sqrt{p_i} \sqrt{q_i} + \sum_{i \in \ol{A}} \sqrt{p_i} \sqrt{q_i} = \sqrt{q(A)}\, \BC{p}{q|A} + \sum_{i \in \ol{A}} \sqrt{p_i} \sqrt{q_i},
\]

and the result follows by applying Cauchy–Schwarz to the second term.

§.§ Intuition II: Pure states and geometry

As observed by Gao [4], a purification argument immediately shows that to prove quantum union bounds, it suffices to consider pure states.
This can assist with geometric intuition, particularly if one imagines — with only mild loss of generality — that all states and projectors are real.

In this case, let $\ket{\psi_t}$ denote the unit vector in $\R^d$ obtained by conditioning on the first $t$ projective measurements succeeding.
Then if $H = H_{t+1}$ denotes the subspace onto which $A_{t+1}$ projects, the  analysis of the $(t+1)$th measurement really only depends on four vectors, namely $\Proj_{H} \ket{\psi_0}$, $\Proj_{H} \ket{\psi_t}$, $\Proj_{H^\bot} \ket{\psi_0}$, and $\Proj_{H^\bot} \ket{\psi_t}$.
So without loss of generality we may project everything into $\R^4$, with the first three vectors spanning $\R^3$.
We can then picture a globe in $\R^3$ of unit radius, with $H_{t+1}$ being  the plane of the equator, $\ket{\psi_0}$ and $\ket{\psi_{t+1}}$ lying on the globe's surface, and $\ket{\psi_t} = r \ket{\wt{\psi}_t} + \ket{\wt{\psi}_t^\bot}$ for some $\ket{\wt{\psi}_t}$ on the globe's surface, with $0 \leq r \leq 1$ and $\ket{\wt{\psi}_t^\bot}$ pointing into the fourth dimension.
For $j \in \{0,t,t+1\}$, we'll write $(\lambda_j, \phi_j)$ for the longitude/latitude of $\ket{\psi_j}$ (or $\ket{\wt{\psi}_j}$ when $j = t$).
We may assume that $\lambda_t = \lambda_{t+1} = 0$, and hence $\ket{\psi_{t+1}} = (0,0)$.
(See the left image in <Ref>.)

For $j \in \{t,t+1\}$, let us write $\Delta_j$ for the angle between $\ket{\psi_0}$ and $\ket{\psi_j}$, and also write $\wt{\Delta}_t$ for the angle between $\ket{\psi_0}$ and $\ket{\wt{\psi}_t}$ (equivalently, $r \ket{\wt{\psi}_t}$).
We claim that

\[
\cos \Delta_{t+1} = \cos \phi_0  \cos \lambda_0, \qquad \cos \wt{\Delta}_{t} = \cos \phi_t \cos \phi_0 \cos \lambda_0  + \sin \phi_t \sin \phi_0, \qquad \cos \Delta_t \leq \cos \wt{\Delta}_t.
\]

The first formula is the spherical Pythagorean Theorem applied to the  triangle with vertices $\ket{\psi_0}$, $(\lambda_0,0)$, and $\ket{\psi_{t+1}}$.
The second is the great-circle distance formula; equivalently, the spherical Cosine Law applied to the triangle formed by $\ket{\psi_0}$, the north pole (blue dot), and $\ket{\wt{\psi}_t}$.
Finally, the inequality holds because the angle, $\Delta_t$, that $\ket{\psi_0}$ makes with $\ket{\psi_t}$ is at least the angle, $\wt{\Delta}_t$, it makes with $r\ket{\wt{\psi}_t}$, since the former is equal to the latter plus a vector $\ket{\wt{\psi}_t^\bot}$ that is orthogonal to both $\ket{\psi_0}$ and $r\ket{\wt{\psi}_t}$.
Combining the above three results now yields

\begin{equation}    \label[ineq]{ineq:gao-trig}
\cos \Delta_{t} \leq \cos \phi_t  \cos \Delta_{t+1} +  \sin \phi_t \sin \phi_0,
\end{equation}

whichignoring signs is exactly the relationship derived in our main <Ref> (with $\rho$ being $\ket{\psi_0}$ and $\sigma$ being $\ket{\psi_t}$ and $A$ being projection onto $H_{t+1}$).

On the left, justifying <Ref> for pure states.  On the right, tightness for <Ref>.

§.§ Tightness

The factor of $4$ appearing in the Quantum Union Bound is tight, even in the case of one pure qubit with real amplitudes.
To see this, fix a large $m$ and then consider $\delta \to 0^+$.
Now suppose the initial state of the qubit is $\ket{0}$, and $A_t$ projects onto the line in $\R^2$ making an angle of $(-1)^t \cdot \delta$ with $\ket{0}$.
Then one hand, $\eps_t = \sin^2(\pm\delta) \sim \delta^2$ for each $t$, so $\Loss \sim m \delta^2$.
On the other hand,

\[
\Fail = 1 - (1 - \sin^2\delta)(1-\sin^2 2\delta)(1 - \sin^2 2\delta) \cdots (1 - \sin^2 2\delta) \sim (4m-3)\delta^2.
\]

From this we see that the constant “$4$” in <Ref>'s $\Fail \leq 4\Loss$ cannot be replaced by any smaller constant.

In fact, the same idea can be used to show that the refined bound denoted $(\clubsuit)$ in <Ref>

is asymptotically tight for all fixed $m \geq 2$ and $p,p'$.
To see this, let $\delta_t = a_t \delta$ for constants $a_1, \dots, a_m$, and let $A_t$ project onto the line in $\R^2$ making an angle of $(-1)^t \cdot \delta_t$ with $\ket{0}$.
Then on one hand, $\eps_t = \sin^2(\pm \delta_t) \sim a_t^2 \delta^2$, and hence the bound from $(\clubsuit)$ is

\begin{equation}
\Fail \lesssim \Bigl(a_1^2 + p' a_1^2 + (p+p')\sum_{1 < t < m} a_t^2 + p a_m^2\Bigr)\cdot \delta^2 \tag*{$(\clubsuit')$}
\end{equation}

On the other hand,

\begin{align*}
\Fail &= 1 - (1- \sin^2\delta_1)(1 - \sin^2(\delta_1 + \delta_2))(1 - \sin^2(\delta_2+\delta_3)) \cdots (1- \sin^2(\delta_{m-1} + \delta_m)) \\
&\sim \Bigl(a_1^2 + (a_1+a_2)^2 + (a_2 + a_3)^2 + \cdots + (a_{m-1} + a_m)^2\Bigr)\cdot \delta^2.
\end{align*}

But note that whenever $1/p + 1/p' = 1$, it is possible for $a_t, a_{t+1}$ to satisfy $(a_t + a_{t+1})^2 = p'a_t^2 + pa_{t+1}^2$.
(Specifically, this happens if $a_{t+1}/a_t = p'/p$.)
So if this identity is always satisfied, then $(\clubsuit')$ is indeed tight up to lower-order $O(\delta^4)$ terms.

Next we show that Gentle Sequential Measurement bounds $(\diamondsuit')$ are exactly tight (assuming $\sum_t \eps_t \leq 1$), even in the case of pure state qutrits with real amplitudes.
This also implies exact tightness of $(\diamondsuit)$, since $\Dtr{\rho}{\sigma} = \Infid{\rho}{\sigma}^{1/2}$ for pure states $\rho, \sigma$.
To show this, suppose $\ket{\psi_0}$ and $\ket{\psi_t}$ are states in $\R^3$ at angle $\Delta_t$, and let angle $\delta = \delta_{t+1}$ be given.
(One may imagine that $\sin^2 \Delta_t = \sum_{1 \leq i \leq t} \eps_t$ already, and $\sin^2 \delta = \eps_{t+1}$.)
We will show that there is a two-dimensional subspace $H_{t+1}$ (the image of $A_{t+1}$) such that: (i) $H_{t+1}$ makes an angle of $\delta$ with $\ket{\psi_0}$; (ii) the state $\ket{\psi_{t+1}}$ resulting from a successful measurement of $\ket{\psi_t}$ by $A_{t+1}$ has an angle $\Delta_{t+1}$ from $\ket{\psi_0}$ satisfying

\begin{equation}    \label{eqn:Deltat}
\sin^2 \Delta_{t+1} = \sin^2 \Delta_{t} + \sin^2 \delta.
\end{equation}

As we can arrange this for every $t$, we conclude that $(\diamondsuit')$ can be exactly tight.

It is not hard to see that to maximize $\Delta_{t+1}$, we should choose $H_{t+1}$ to ensure that the (great-circle) arc connecting $\ket{\psi_t}$ to $\ket{\psi_{t+1}}$ is orthogonal to the arc connecting $\ket{\psi_0}$ and $\ket{\psi_t}$, as in the image on the right of <Ref>.
(In that image, one might imagine that $\ket{\psi_{t}}$ could have been any point on the green dashed small circle of radius $\Delta_t$ around $\ket{\psi_0}$; to maximize $\Delta_{t+1}$ we want the arc connecting $\ket{\psi_{t}}$ to $H_{t+1}$ to be tangent to this green circle.)

Thus it remains to verify that <Ref> holds for the dark blue “Lambert (three-right-angle) quadrilateral” with corners $\ket{\psi_0}$, $\ket{\psi_t}$, $\ket{\psi_{t+1}}$, and $X$ (the state if $\ket{\psi_0}$ were successfully measured by $A_{t+1}$).
This is an elementary (though perhaps lesser-known) fact of spherical geometry.
To verify it, one may form the three pale blue reflections of the Lambert quadrilateral, giving a centrally symmetric spherical quadrilateral.
Then it is easy to verify that the triangle formed by $\ket{\psi_0}$ and the points depicted as $Y$ and $Z$ form a so-called half-sum triangle (a right right triangle, in the terminology of [2]), with the triangular angles at $\ket{\psi_0}$ and $Z$ summing to the angle at $Y$.
But then <Ref> is immediate from Dickinson and Salmassi's “Preferred Spherical Pythagorean Theorem” [2].

§.§ How we discovered our proof

The proof we gave is short enough that one might imagine just discovering it from scratch.
Alternatively, one might imagine discovering it by trying to prove the classical Union Bound while working exclusively with Bhattacharyya coefficient.
But in fact, we essentially came up with our proof by iteratively refining and unifying the original proofs of Gao and of Khabbazi Oskouei–Mancini–Wilde.
(Indeed, along the way we had a version of our proof that was roughly $\alpha$ pages long, for each real number $0.5 \leq \alpha \leq 4.0$.)

The parallels are as follows:
As noted, our main <Ref> essentially becomes the geometric equality <Ref> when reduced to the pure state case.
In turn, this is equivalent to“inequality (10)” in [4].
Gao proves this inequality in a different but straightforward fashion, and his deduction of <Ref> from it is also relatively straightforward.
(His “Lemma 1” parallels our <Ref>.)
Then like our proof, Gao's proof of <Ref> is inductive and uses <Ref> (his “(10)”), but the inequalities he invokes are significantly more complicated.
It seems that introducing our quantity “$r_t$” is important for getting a slick proof.
As for the Khabbazi Oskouei–Mancini–Wilde proof, the steps in it are all individually straightforward; however, it seems that working explicitly with fidelity, as we do, helps to get a clean proof.
Our key <Ref> may be viewed as hidden in the proof of <cit.>; one can extract it upon converting their calculational/iterative proof into an induction.

§.§ Acknowledgments

R.O. is supported by NSF grant FET-1909310 and ARO grant W911NF2110001.
We thank Mark Wilde for several remarks that improved the presentation of this paper, and we thank Anurag Anshu for observing the parallel between our proof and Sen's.

[1]
A. Anshu.
Quantum union bound, 2021.
<https://people.eecs.berkeley.edu/ anuraganshu/Union_bound.pdf>.

[2]
W. Dickinson and M. Salmassi.
The right right triangle on the sphere.
The College Mathematics Journal, 39(1):24–33, 2008.

[3]
C. Fuchs and J. van de Graaf.
Cryptographic distinguishability measures for quantum-mechanical
IEEE Trans. Inform. Theory, 45(4):1216–1227, 1999.

[4]
J. Gao.
Quantum union bounds for sequential projective measurements.
Physical Review A, 92(5):052331, 2015.

[5]
S. Khabbazi Oskouei, S. Mancini, and M. Wilde.
Union bound for quantum information processing.
Proceedings of the Royal Society A, 475(2221):20180612, 2019.

[6]
T. Ogawa and H. Nagaoka.
A new proof of the channel coding theorem via hypothesis testing in
quantum information theory.
In Proceedings of the International Symposium on Information
Theory, page 73. IEEE, 2002.

[7]
P. Sen.
Achieving the Han–Kobayashi inner bound for the quantum
interference channel.
In Proceedings of the International Symposium on Information
Theory, pages 736–740, 2012.

[8]
A. Winter.
Coding theorem and strong converse for quantum channels.
IEEE Transactions on Information Theory, 45(7):2481–2485,