# Lagrangian Properties and Control of Soft Robots Modeled with Discrete
Cosserat Rods.

Lekan Molu, Shaoru Chen, and Audrey Sedal  Lekan Molu and Shaoru Chen are with
Microsoft Research NYC. Audrey Sedal is with McGill University’s Department of
Mechanical Engineering. Emails: {lekanmolu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The characteristic “in-plane” bending associated with soft robots’ deformation
make them preferred over rigid robots in sophisticated manipulation and
movement tasks. Executing such motion strategies to precision in soft
deformable robots and structures is however fraught with modeling and control
challenges given their infinite degrees-of-freedom. Imposing piecewise
constant strains (PCS) across (discretized) Cosserat microsolids on the
continuum material however, their dynamics become amenable to tractable
mathematical analysis. While this PCS model handles the characteristic
difficult-to-model “in-plane” bending well, its Lagrangian properties are not
exploited for control in literature neither is there a rigorous study on the
dynamic performance of multisection deformable materials for “in-plane”
bending that guarantees steady-state convergence. In this sentiment, we first
establish the PCS model’s structural Lagrangian properties. Second, we exploit
these for control on various strain goal states. Third, we benchmark our
hypotheses against an Octopus-inspired robot arm under different constant tip
loads. These induce non-constant “in-plane” deformation and we regulate strain
states throughout the continuum in these configurations. Our numerical results
establish convergence to desired equilibrium throughout the continuum in all
of our tests. Within the bounds here set, we conjecture that our methods can
find wide adoption in the control of cable- and fluid-driven multisection soft
robotic arms; and may be extensible to the (learning-based) control of
deformable agents employed in simulated, mixed, or augmented reality.

Supplementary material — The codes for reproducing the experiments reported in
this paper are available online: https://github.com/robotsorcerer/dcm.

##  I Introduction

Soft robots are attracting wide adoption in the automation community owing to
their improved bending, torsion, configurability, and compliance properties.
These properties enable customizable solutions for assistive wearable devices
[23, 1], robot grippers [12], and mobile robots [13]. In-plane bending, as a
motion deformation strategy, is a particular advantage that serial soft arms
execute better than rigid arms. With this property, soft robot grippers can
pick up delicate objects, adopt adept nonlinear motion strategies in complex
workspaces e.g. for reaching tasks. In embodied applications such as soft
exogloves [16] and exosuits [14], they provide conformal deformation that aids
improved walking gaits, and muscle rehabilitation and they conserve energy
economy.

The piecewise constant strain (PCS) Lagrangian dynamics derived by [18] is a
reduced special Euclidean group-3 (${\mathbb{SE}}(3)$) model for compliant,
serial manipulators. Derived from Cosserat [6] rod theory, it addresses
torsion, in-plane, and out-of-plane (multi-) bending motions. The PCS model
outperforms the common piecewise constant curvature (PCC) [17] and the
constant curvature variants [8] used outside of finite element modeling
methods (FEM) [5]. Controllers deployed on the PCS model however ignore the
geometric properties of the Lagrangian dynamics [4, 22]. This makes the
control equations complicated and destroys several properties of the
Lagrangian dynamics that are useful for control tasks.

Figure 1: Schematic of the configuration of an Octopus robot arm.

In this sentiment, considering the PCS model for soft multisection arms, we
first establish the structural properties of their Lagrangian dynamics. We
then exploit these for the steady-state convergence analyses in various
control proposals (under different operating model parameters) and in
different surrounding mediums using the well-established method of Lyapunov
analysis. The Octopus robot [11], a muscular hydrostat [9] with distributed
deformation throughout its articulated arms, blends the interplay between
continuum mechanics and sensorimotor control well. Thus, we consider a single
arm of the CyberOctopus [20] (configuration shown in Fig. 1) to benchmark our
controllers on the characteristic difficult-to-model-and-control in-plane
bending deformation .

The rest of this paper is structured as follows: in §II, we briefly introduce
preliminaries relevant to the contributions machinery we present in §III.
Multivariable stabilizing feedback controllers for regulating the tip point
and strain states are established in §IV. We present numerical results in §V
and conclude the paper in §VI.

##  II Notations and Preliminaries

The strain field and strain twist vectors are respectively
$\xi\in\mathbb{R}^{6}$ and $\eta\in\mathbb{R}^{6}$. The Lie algebra of
${\mathbb{SE}}(3)$ is $\mathfrak{se}(3)$. All rotation are measured
counterclockwise in corkscrew sense. The continuum is reduced to a space curve
defined by a material abscissa $X:[0,L]$, where $L$ is the robot’s length in
the reference configuration. For a configuration
$\bm{g}(X)\in{\mathbb{SE}}(3)$, its adjoint and coadjoint representations are
respectively $\text{Ad}_{\bm{g}}$, $\text{Ad}_{\bm{g}}^{\star}$ —
parameterized by a curve, the material abscissa $X$. The corresponding adjoint
and coadjoint representation of the strain twist vector are
$\text{ad}_{\xi,\eta}$ and $\text{ad}_{\xi,\eta}^{\star}$, respectively. In
generalized coordinate, the joint vector of a soft robot is denoted
$q(\xi)=[\xi_{1}^{\top},\ldots,\xi_{n_{\xi}}^{\top}]^{\top}\in\mathbb{R}^{6{n_{\xi}}}$.

###  II-A SoRo Configuration

The inertial frame is signified by the basis triad
$(\bm{e}_{1},\bm{e}_{2},\bm{e}_{3})$ (see Fig. 1) and $\bm{g}_{r}(X)$ is the
transformation from the inertial to the manipulator’s base frame. For cable-
driven arms, the point at which actuation occurs is labeled $\bar{X}$. The
configuration matrix that parameterizes curve $X$ of length $L_{n}$ is denoted
$\bm{g}_{L_{n}}$. The cable runs through the $z$-axis in the (micro) body
frame (x-axis in the spatial frame). As shown in Fig. 1, the transformation
from the base to the inertial frame is

$\displaystyle\bm{g}_{r}=\left(\begin{array}[]{cccc}0&-1&0&0\\\ 1&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right).$ (5)

We adopt a linear viscoelastic constitutive model based on Kelvin-Voight
internal forces i.e.

$\displaystyle\mathcal{F}_{i}(X)=\Sigma(\xi-\xi_{0})+\Upsilon\dot{\xi},$ (6)

with $\Sigma$ being the constant screw stiffness matrix, $\Upsilon$ the
viscosity matrix, and the reference strain $\xi_{0}=[0,0,0,1,0,0]^{\top}$ in
the upright configuration. The jacobian associated with $X$ is $J(X)$. Owing
to paper length constraints we refer readers to Renda et al. [18] for further
definitions.

###  II-B Continuous Strain Vector and Twist Velocity Fields

Let $p(X)$ describe the position vector of a microsolid on a soft body and let
$R(X)$ denote the corresponding orientation matrix. Furthermore, let the pose
$[p(X),R(X)]$ be parameterized by a material abscissa $X\in[0,L]$ for end
nodes $0$ and $L$. Then, the robot’s C-space, parameterized by a curve
$g(X):X\rightarrow{\mathbb{SE}}(3)$, is
$g(X)=\left(\begin{array}[]{cc}R(X)&p(X)\\\ 0^{\top}&1\end{array}\right)$.
Suppose that $\varepsilon(X)\in\mathbb{R}^{3}$ and
$\gamma(X)\in\mathbb{R}^{3}$ respectively denote the linear and angular strain
components of the soft arm. Then, the arm’s strain field is a state vector
along the curve $g(X)$ defined as $\breve{\xi}(X)=g^{-1}\partial g/\partial
X\triangleq g^{-1}\partial_{x}g$, and the velocity of $g(X)$ is the twist
vector field $\breve{\eta}(X)$ defined as $\breve{\eta}(X)=g^{-1}\partial
g/\partial t\triangleq g^{-1}\partial_{t}{g}$. Read $\hat{\gamma}$: the anti-
symmetric matrix representation of $\gamma$. Read $\breve{\xi}$: the
isomorphism from $\xi\in\mathbb{R}^{6}$, to its matrix representation in
$\mathfrak{se}(3)$.

###  II-C Discrete Cosserat System: Cosserat-Constitutive PDEs

The PCS model assumes that the respective strain, $\xi_{i}$ for $n$ microsolid
sections, $\\{\mathcal{M}_{i}\\}_{i=1}^{n}$, are constant for each discretized
section of the arm. Using d’Alembert’s principle, the generalized dynamics for
the PCS model (cf. Fig. 1) under external and actuation loads admits the weak
form [18]

$\displaystyle\underbrace{\left[\int_{0}^{L_{N}}J^{T}\mathcal{M}_{a}JdX\right]}_{M(q)}\ddot{q}+\underbrace{\left[\int_{0}^{L_{N}}J^{T}\text{ad}_{J\dot{q}}^{\star}\mathcal{M}_{a}JdX\right]}_{C_{1}(q,\dot{q})}\dot{q}+$
$\displaystyle\underbrace{\left[\int_{0}^{L_{N}}J^{T}\mathcal{M}_{a}\dot{J}dX\right]}_{C_{2}(q,\dot{q})}\dot{q}+\underbrace{\left[\int_{0}^{L_{N}}J^{T}\mathcal{D}{J}\|J\dot{q}\|_{p}dX\right]}_{D(q,\dot{q})}\dot{q}$
$\displaystyle-\underbrace{(1-\rho_{f}/\rho)\left[\int_{0}^{L_{N}}J^{T}\mathcal{M}\text{Ad}_{\bm{g}}^{-1}dX\right]}_{N(q)}\text{Ad}_{\bm{g}_{r}}^{-1}\mathcal{G}-\underbrace{J(\bar{X})^{T}\mathcal{F}_{p}}_{F(q)}$
$\displaystyle-\underbrace{\int_{0}^{L_{N}}J^{T}\left[\nabla_{x}\mathcal{F}_{i}-\nabla_{x}\mathcal{F}_{a}+\text{ad}_{\eta_{n}}^{\star}\left(\mathcal{F}_{i}-\mathcal{F}_{a}\right)\right]}_{\tau(q)}dX=0,$
(7)

where
$\mathcal{F}_{i}(X)\triangleq\left(M_{i}^{T},F_{i}^{T}\right)^{T}\equiv\partial_{X}\mathfrak{U}$
is the wrench of internal forces, $\bar{\mathcal{F}}_{a}(X)$ is the
distributed wrench of actuation loads, and $\bar{\mathcal{F}}_{e}(X)$ is the
external distributed wrench of the applied forces. The screw mass inertia
matrix $\mathcal{M}(X)=\text{diag}\left(I_{x},I_{y},I_{z},A,A,A\right)\rho$
for a body density $\rho$, sectional area $A$, bending, torsion, and second
inertia operator $I_{x},I_{y},I_{z}$ respectively. In (7),
$\mathcal{M}_{a}=\mathcal{M}+\mathcal{M}_{f}$ is a lumped sum of the
microsolid mass inertia operator, $\mathcal{M}$, and that of the added mass
fluid, $\mathcal{M}_{f}$; $dX$ is the length of each section of the multi-
robot arm; $\mathcal{D}(X)$ is the drag matrix; $J(X)$ is the Jacobian
operator; $\|\cdot\|_{p}$ is the translation norm of the expression contained
therein; $\rho_{f}$ is the density of the fluid in which the material moves;
$\rho$ is the body density; $\mathcal{G}=\left[0,0,0,-9.81,0,0\right]^{T}$ is
the gravitational vector; and $\mathcal{F}_{p}$ is the applied wrench at
$\bar{X}$. In standard Newton-Euler form, equation (7), is

$\displaystyle\begin{split}M(q)\ddot{q}&+\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]\dot{q}=\tau(q)+F(q)\\\
&\qquad\qquad+N(q)\text{Ad}_{\bm{g}_{r}}^{-1}\mathcal{G}-D(q,\dot{q})\dot{q}.\end{split}$
(8)

##  III Structural Properties of the PCS Model

We now establish the Lagrangian properties.

######  Theorem 1 (Structural properties of the kinetic equation).

Equation (8) satisfies the following properties:

######  Property 1 (Positive definiteness of the Inertia Operator).

The inertia tensor $\mathcal{M}_{a}(q)$ is symmetric and positive definite. As
a result $M(q)$ is symmetric and positive definite.

###### Proof of Property 1.

The jacobian, $J$, is injective by [18, equation 20]. Thus, property 1 follows
from its definition. ∎

######  Property 2 (Boundedness of the Mass Matrix).

The mass inertial matrix $M(q)$ is uniformly bounded from below by
$m\bm{I}_{n}$ where $m$ is a positive constant.

###### Proof of Property 2.

This is a restatement of the lower boundedness of $M(q)$ for fully actuated
n-degrees of freedom manipulators [19]. ∎

######  Property 3 (Skew symmetric property).

The matrix $\dot{M}(q)-2\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]$ is
skew-symmetric.

###### Proof of Property 3.

We have by Leibniz’s rule that

$\displaystyle\dot{M}(q)$
$\displaystyle=\dfrac{d}{dt}\left(\int_{0}^{L_{N}}J^{T}\mathcal{M}_{a}JdX\right)=\int_{0}^{L_{N}}\dfrac{\partial}{\partial
t}\left(J^{T}\mathcal{M}_{a}J\right)dX$
$\displaystyle\triangleq\int_{0}^{L_{N}}\left(\dot{J}^{T}\mathcal{M}_{a}J+J^{T}\dot{\mathcal{M}}_{a}J+{J}^{T}\mathcal{M}_{a}\dot{J}\right)dX.$
(9)

Therefore, $\dot{M}(q)-2\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]$
becomes

$\displaystyle\int_{0}^{L_{N}}\left(\dot{J}^{\top}\mathcal{M}_{a}J+J^{\top}\dot{\mathcal{M}}_{a}J+{J}^{\top}\mathcal{M}_{a}\dot{J}\right)dX$
$\displaystyle-2\int_{0}^{L_{N}}\left(J^{\top}\text{ad}^{\star}_{J\dot{q}}\mathcal{M}_{a}J+J^{\top}\mathcal{M}_{a}\dot{J}\right)dX$
(10)
$\displaystyle\triangleq\int_{0}^{L_{N}}\left(\dot{J}^{\top}\mathcal{M}_{a}J+J^{\top}\dot{\mathcal{M}}_{a}J-{J}^{\top}\mathcal{M}_{a}\dot{J}\right)dX$
$\displaystyle\qquad-2\int_{0}^{L_{N}}J^{\top}\text{ad}^{\star}_{J\dot{q}}\mathcal{M}_{a}JdX.$
(11)

Similarly,
$-\left[\dot{M}(q)-2\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]\right]^{\top}$
expands as

$\displaystyle-\dot{M}^{\top}(q)+2\left[C_{1}^{\top}(q,\dot{q})+C_{2}^{\top}(q,\dot{q})\right]=$
$\displaystyle\int_{0}^{L_{N}}dX^{\top}\left(-J^{\top}\mathcal{M}_{a}\dot{J}-J^{\top}\dot{\mathcal{M}}_{a}J-\dot{J}^{\top}\mathcal{M}_{a}{J}\right)$
$\displaystyle\qquad+2\int_{0}^{L_{N}}dX^{\top}\left(J^{\top}\mathcal{M}_{a}\text{ad}_{J\dot{q}}{J}+\dot{J}^{\top}\mathcal{M}_{a}J\right)$
$\displaystyle\triangleq\int_{0}^{L_{N}}\left({J}^{\top}\mathcal{M}_{a}\dot{J}-\dot{J}^{\top}\mathcal{M}_{a}J-J^{\top}\dot{\mathcal{M}}_{a}J\right)dX$
$\displaystyle\qquad-2\int_{0}^{L_{N}}J^{\top}\text{ad}^{\star}_{J\dot{q}}\mathcal{M}_{a}JdX$
(12)

where the terms in equation (12) follow from the symmetry of the matrices that
constitute the integrands. Inspecting (11) and (12), it is easy to see that
their right hand sides verify the identity

$\displaystyle\int_{0}^{L_{N}}\left(\dot{J}^{\top}\mathcal{M}_{a}J+J^{\top}\dot{\mathcal{M}}_{a}J-{J}^{\top}\mathcal{M}_{a}\dot{J}\right)dX$
$\displaystyle\quad-2\int_{0}^{L_{N}}J^{\top}\text{ad}^{\star}_{J\dot{q}}\mathcal{M}_{a}JdX=2\int_{0}^{L_{N}}J^{\top}\text{ad}^{\star}_{J\dot{q}}\mathcal{M}_{a}JdX-$
$\displaystyle\qquad\int_{0}^{L_{N}}\left({J}^{\top}\mathcal{M}_{a}\dot{J}-\dot{J}^{\top}\mathcal{M}_{a}J-J^{\top}\dot{\mathcal{M}}_{a}J\right)dX$
(13)

or

$\displaystyle\dot{M}(q)$
$\displaystyle-2\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]=$
$\displaystyle-\left[\dot{M}(q)-2\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]\right]^{\top}.$
(14)

A fortiori, the skew symmetric property follows. ∎

######  Remark 1.

Since $\dot{M}(q)$ is symmetric (cref. (9)), another way of stating the skew-
symmetric property is to write

$\displaystyle\dot{M}(q)=C_{1}(q,\dot{q})+C_{2}(q,\dot{q})+\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})\right]^{\top}.$
(15)

Owing to the symmetry of the right-hand-side (rhs), we have
$\dot{M}=2(C_{1}+C_{2})$.

######  Property 4 (Linearity-in-the-parameters).

There exists a constant vector $\Theta\in\mathbb{R}^{l}$ and an $N\times l$
dimensional regressor function $Y(q,\dot{q},\ddot{q})\in\mathbb{R}^{N\times
l}$ such that

$\displaystyle\begin{split}M(q)\ddot{q}&+\left[C_{1}(q,\dot{q})+C_{2}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}-F(q)-\\\
&\qquad\qquad+N(q)\text{Ad}_{\bm{g}_{r}}^{-1}\mathcal{G}=Y(q,\dot{q},\ddot{q})\Theta.\end{split}$
(16)

###### Proof of Property 4.

Consider the generalized constitutive law for the full Cosserat model derived
in [2, §6.3]. The reduced Lagrangian density in $\mathfrak{se}(3)$ per unit of
deformed volume (for all configurations)111Note that the full Lagrangian
density of the soft multisection manipulator is
$L_{m}=\int_{0}^{L}\mathfrak{L}(\bm{g},\eta,\xi)dX$. for a configuration
$\mathcal{B}$ is $\mathfrak{L}=\mathfrak{T}-\mathfrak{U}$ [2] where
$\mathfrak{T},\mathfrak{U}$ respectively denote the volume’s left-reduced
kinetic and elastic potential energy densities in $\mathcal{B}$. From the
Euler-Lagrange equation, we have

$\displaystyle\tau_{n}=\dfrac{d}{dt}\dfrac{\partial\mathfrak{T}}{\partial\dot{\xi}_{n}}-\dfrac{\partial\mathfrak{T}}{\partial\xi_{n}}+\dfrac{\partial\mathfrak{U}}{\partial\mathfrak{\eta}_{n}},\quad
n=1,\ldots,N.$ (17)

Suppose that the material mid-surface crosses the microstructures
$\mathcal{M}_{i}$ which correspond to the mass center, then the kinetic energy
density per unit of deformed area and its rate of change w.r.t $\xi$ are [2]

$\displaystyle\mathfrak{T}(\xi)=\frac{1}{2}\bigg{\langle}\left(\begin{array}[]{c}\omega\\\
\nu,\end{array}\right),\left(\begin{array}[]{c}\bar{I}\omega\\\
\bar{\rho}\nu\end{array}\right)\bigg{\rangle}$ (22)
$\displaystyle\mathfrak{T}(\dot{\xi})=\frac{1}{2}\bigg{\langle}\left(\begin{array}[]{c}\dot{\omega}\\\
\dot{\nu},\end{array}\right),\left(\begin{array}[]{c}\bar{I}\dot{\omega}\\\
\bar{\rho}\dot{\nu}\end{array}\right)\bigg{\rangle}$ (27)

where $\bar{\rho}$ and $\bar{I}$ respectively denote the mass and angular
inertia density per unit volume. It follows that

$\displaystyle\partial_{\xi}\mathfrak{T}=\left(\begin{array}[]{c}\bar{I}\omega\\\
\bar{\rho}\nu\end{array}\right),\quad\partial_{\dot{\xi}}\mathfrak{T}=\left(\begin{array}[]{c}\bar{I}\dot{\omega}\\\
\bar{\rho}\dot{\nu}\end{array}\right).$ (32)

In a similar vein, the left invariant density of internal energy
$\mathfrak{U}$ is [3]

$\displaystyle\mathfrak{U}(\eta)=\langle\mathcal{F}_{int},(\eta-\eta^{d})\rangle$
(33)

for a desired $\eta^{d}$ and field of internal force constraints
$\mathcal{F}_{int}:X\in[0,L]\rightarrow\mathcal{F}_{int}(X)\in\mathfrak{se}(3)$.
The potential energy per unit of metric area of the deformed surface (assuming
that it is concentrated in the mid-surface) is [2]

$\displaystyle{\partial_{\eta}\mathfrak{U}}=\left(\begin{array}[]{c}\partial\mathfrak{U}/\partial\gamma\\\
\partial\mathfrak{U}/\partial\varepsilon\end{array}\right)-\left(\begin{array}[]{c}0\\\
\varepsilon\end{array}\right)\mathfrak{L}$ (38)

so that the Euler-Lagrange equation (17) becomes

$\displaystyle\left(\begin{array}[]{c}\bar{I}\ddot{\omega}+\dot{\bar{I}}\dot{\omega}\\\
\bar{\rho}\ddot{\nu}+\dot{\bar{\rho}}\dot{\nu}\end{array}\right)-\left(\begin{array}[]{c}\bar{I}\omega\\\
\bar{\rho}\nu\end{array}\right)+\left(\begin{array}[]{c}\partial\mathfrak{U}/\partial\gamma\\\
\partial\mathfrak{U}/\partial\varepsilon\end{array}\right)-\left(\begin{array}[]{c}0\\\
\varepsilon\end{array}\right)\mathfrak{L}.$ (47)

Observe: Under the PCS assumption, each microsolid is fixed so that the energy
density per unit section of metric volume is akin to that of a rigid body. The
kinetic and potential energies for the PCS model per section $i$ of $N$
sections becomes

$\displaystyle\mathfrak{T}=\frac{1}{2}\sum_{i=1}^{N}\bigg{\langle}\left(\begin{array}[]{c}^{i+1}\omega_{i}\\\
^{i+1}\nu_{i},\end{array}\right),\left(\begin{array}[]{c}^{i+1}\bar{I}_{i}\,\,^{i+1}\omega_{i}\\\
^{i+1}\bar{\rho}_{i}\,\,^{i+1}\nu_{i}\end{array}\right)\bigg{\rangle}$ (52)

where ${}^{i+1}\omega_{i}$ is the angular velocity of section $i+1$ in the
frame of section $i$, ${}^{i+1}\nu_{i}$ is the linear velocity of section
$i+1$ in the frame of section $i$ e.t.c. Similarly, the sectional potential
energies are

$\displaystyle\mathfrak{U}(\eta_{i})=\sum_{i=1}^{N}\langle\\{\mathcal{F}_{int}\\}_{i},(\eta_{i}-\eta_{i}^{d})\rangle.$
(53)

Thus, the kinetic and potential energy are each linear in configuration
parameters so that

$\displaystyle\mathfrak{T}=\sum_{i=1}^{N}\dfrac{\partial\mathfrak{T}}{\partial\Sigma_{i}}\Sigma_{i}=\sum_{i=1}^{N}\Gamma\mathfrak{T}_{i}\Sigma_{i},$
$\displaystyle\mathfrak{U}=\sum_{i=1}^{N}\dfrac{\partial\mathfrak{U}}{\partial\Sigma_{i}}\Sigma_{i}=\sum_{i=1}^{N}\Gamma\mathfrak{U}_{i}\Sigma_{i}$
(54)

where $\Sigma_{i}$ is an inertial parameter, $\Gamma\mathfrak{T}_{i}$ is a
function of $q,\dot{q}$ and $\Gamma\mathfrak{U}_{i}$ is a function of $q$.
Using (54) and plugging (32) and (38) into (17), we conclude that the
sectionalized piecewise Cosserat dynamics is also linear-in-the-inertial-
parameters, given as $\tau(q)=Y(q,\dot{q},\ddot{q})\Theta$, where
$Y(q,\dot{q},\ddot{q})$ is the matrix function of $q,\dot{q},\ddot{q}$ and
$\Theta$ is the matrix of parameters. ∎

##  IV Multivariable Control

Figure 2: PD computed torque control block diagram.

We will control the robot’s dynamics by exploiting the properties of §III.

###  IV-A Proportional-(Integral)-Derivative Control, Redux.

We first show that regarding the generalized torque $\tau(q)$ as a control
input, $u(q,\dot{q})$, linear feedback laws are (almost) sufficient for
attaining a desired joint configuration (without gravity compensation) in the
control law.

######  Theorem 2 (Cable-driven Actuation).

For a set of gains $K_{D}$ and $K_{p}$ which are damping and positive diagonal
matrices, the control law (without gravity/buoyancy compensation)

$\displaystyle u(q,\dot{q})=-K_{p}\tilde{q}-K_{D}\dot{q}-F(q)$ (55)

under a cable-driven actuation globally asymptotically stabilizes (GAS) system
(8), where $\tilde{q}(t)=q(t)-q^{d}$ is the joint error vector for a desired
equilibrium point in space $q^{d}$.

###### Proof.

Without gravity, the term $N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G}=0$. Let

$\breve{C}(q,\dot{q})=C_{1}(q,\dot{q})+C_{2}(q,\dot{q})$

and write (8) for an arbitrary control input $u(q)$ as

$\displaystyle\begin{split}M(q)\ddot{q}&=u(q,\dot{q})+F(q)-\left[\breve{C}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}.\end{split}$
(56)

Consider the Lyapunov candidate function

$\displaystyle
V(q)=\frac{1}{2}\dot{q}^{\top}M(q)\dot{q}+\frac{1}{2}\tilde{q}^{\top}K_{p}\tilde{q}.$
(57)

Observe: $V(q)>0\,\,\forall\,\,\\{q,q^{d}\\}\setminus(q=q^{d}),\dot{q}\neq 0$;
and $V(q)=0\,\,\forall\\{q,q^{d}\\}$ when $(q=q^{d})$ and $\dot{q}=0$ in the
joint space. Differentiating $V(q)$ yields

$\displaystyle\dot{V}(q)$
$\displaystyle=\dot{q}^{\top}M(q)\ddot{q}+\frac{1}{2}\dot{q}^{\top}\dot{M}(q)\dot{q}+\tilde{q}^{\top}K_{p}\dot{\tilde{q}},$
$\displaystyle=\dot{q}^{\top}\left(u(q,\dot{q})+F(q)-\left[\breve{C}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}\right)$
$\displaystyle\qquad+\frac{1}{2}\dot{q}^{\top}\dot{M}(q)\dot{q}+\tilde{q}^{\top}K_{p}\dot{q},$
$\displaystyle=\dot{q}^{\top}\left(u(q,\dot{q})+F(q)-\left[\breve{C}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}\right)$
$\displaystyle+\tilde{q}^{\top}K_{p}\dot{q}+\dot{q}^{\top}\left[\frac{1}{2}\left(\dot{M}(q)-2\breve{C}(q,\dot{q})\right)+\breve{C}(q,\dot{q})\right]\dot{q},$
$\displaystyle=\dot{q}^{\top}\left[u(q,\dot{q})+F(q)+K_{p}\tilde{q}-D(q,\dot{q})\dot{q}\right]$
(58)

where the last line follows from the skew symmetric property established in
Remark 1. Suppose we choose the control law (55), then

$\displaystyle\dot{V}(q)=-\dot{q}^{\top}\left[K_{D}+D(q,\dot{q})\right]\dot{q}\leq
0$ (59)

if we choose $K_{D}$ as a block diagonal matrix of positive gains since the
drag term $D(q,\dot{q})>0$. Thus, we have that the potential function is
decreasing if $q$ is non-zero. However, asymptotic stability of trajectories
as a result of the potential (Lyapunov) function candidate does not
automatically follow since $\dot{V}(t)$ is not negative at $\dot{q}=0$. That
is, it is possible for the manipulator’s joint space variable derivative
$\dot{q}=0$ when $q\neq{q}^{d}$.

To prove global asymptotic stability, suppose the domain $\Omega\subset
D\in\mathbb{R}^{6N\times 6N}$ is the compact, positively invariant domain with
respect to (8). Let $\mathcal{E}$ be the set of all $q\in\Omega$ where
$\dot{V}$ is identically zero so that (59) implies that $\dot{q}=0$ and
$\ddot{q}=0$. From (56), we must have

$\displaystyle\begin{split}M(q)\ddot{q}&+\left[\breve{C}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}=-K_{p}\tilde{q}-K_{D}\dot{q}\end{split}$
(60)

which implies that $0=-K_{p}\tilde{q}$ and hence $\tilde{q}=0$. If $\Upsilon$
is the largest invariant set of $\mathcal{E}$, then by LaSalle’s invariance
theorem [10], every solution starting in $\Omega$ approaches $\Upsilon$ as
$t\rightarrow\infty$. Whence, the equilibrium is GAS. ∎

######  Corollary 1 (Fluid-driven actuation).

If the robot is operated without cables, and is driven with a dense medium
such as pressurized air or water, then the term $F(q)=0$ so that the control
law

$\displaystyle u(q,\dot{q})=-K_{p}\tilde{q}-K_{D}\dot{q}$ (61)

globally asymptotically stabilizes the system.

###### Proof.

The proof follows easily since $F(q)=0$ in (58). ∎

######  Theorem 3 (Gravity/Buoyancy-Compensated Control).

Without gravity and/or buoyancy-compensation, there are parasitic disturbances
and unmodeled parameters in (8) where an offset in the positional error
$\tilde{q}$ at steady state may show up since we do not explicitly compensate
for gravity and buoyancy.where the PD control law is not sufficient for
driving the strain states to equilibrium. The following controller will
achieve zero steady state convergence.

$\displaystyle
u(q,\dot{q})=-K_{p}\tilde{q}-K_{D}\dot{q}-F(q)-N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G}.$
(62)

###### Proof.

From (8), write

$\displaystyle\begin{split}M(q)\ddot{q}=u(q,\dot{q})+F(q)+N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G}\\\
-\left[\breve{C}(q,\dot{q})+D(q,\dot{q})\right]\dot{q}.\end{split}$ (63)

Let us choose the Lyapunov function candidate

$\displaystyle
V(q)=\frac{1}{2}\dot{q}^{\top}M(q)\dot{q}+\frac{1}{2}\tilde{q}^{\top}K_{p}\tilde{q}+\frac{1}{2}\int_{0}^{t}{\tilde{q}^{\top}(s)K_{I}\tilde{q}ds},$
(64)

where $V(q)>0\,\,\forall\,\,\\{q,q^{d}\\}\setminus(q=q^{d}),\dot{q}\neq 0$;
$V(q)=0\,\,\forall\\{q,q^{d}\\}$ when $(q=q^{d})$, $K_{I}$ is the block
diagonal integral gain, and $\dot{q}=0$ in the joint space. Tt can be verified
that

$\displaystyle\dot{V}(q)$
$\displaystyle=\dot{q}^{\top}\left[u(q,\dot{q})+F(q)+\left\\{K_{p}+\frac{K_{I}}{s}\right\\}\tilde{q}\right.$
$\displaystyle\left.\qquad\qquad-D(q,\dot{q})\dot{q}+N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G}\right]$
(65)

where we have used the frequency domain representation of the integral
operator i.e. $1/s$. If we set

$\displaystyle u(q,\dot{q})$
$\displaystyle=-\left\\{K_{p}+\frac{K_{I}}{s}\right\\}\tilde{q}-K_{D}\dot{q}-F(q)-N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G},$
(66)

we must have

$\displaystyle\dot{V}(t)=-\dot{q}^{\top}\left[K_{D}+D(q,\dot{q})\right]q\leq
0,\text{ if }\dot{q}\neq 0$ (67)

else for $\dot{V}$ identically zero, LaSalle’s principle applies. ∎

Suppose that we choose appropriate proportional and derivative gains with the
gravity-compensation term in (66), then the integral again $K_{I}$ may be
ignored as

$\displaystyle u(q,\dot{q})$
$\displaystyle=-K_{p}\tilde{q}-K_{D}\dot{q}-F(q)-N(q)\text{Ad}_{g_{r}}^{-1}\mathcal{G}.$
(68)

The proof of the theorem follows as a result.

##  V Numerical Setup and Results

a1) Cable-driven, strain twist setpoint terrestrial control. |  a2) Fluid-actuated, strain twist setpoint terrestrial control.
---|---
|
a3) Fluid-actuated, strain twist setpoint underwater control. |  a4) Cable-driven, strain twist setpoint regulation.
|
b1) Position control with no gravity-compensation. |  b2) Gravity-compensated terrestrial position control.
|
Figure 3: (a1 – a3): Linear strain twist state regulation for a 4-section (41
microsolid pieces per section) soft arm under a $10N$ lateral tip load,
$\mathcal{F}_{p}^{y}$ that is (a1) cable-driven in a sparse medium (a2) fluid-
actuated in a sparse medium. (a3) fluid-actuated in a dense medium such as
water (i.e. with drag forces compensation). (a4) Under a miniature tip-load of
$0.2N$, a cable-driven 4-section arm finely regulates strain twists to
equilibrium over time. (b1-b2): Linear strain position regulation for a cable-
driven arm operating in air (b1) and a fluid-actuated arm illustrating the
effect of steady state errors in the absence of gravity compensation. The
horizontal axes show the number of (re)-integration time-steps per second for
the adaptive Runge-Kutta-Fehlberg integrator we utilized in computing the
controllers.  (c1)
---
(c2)
Figure 4: Gravity-based compensation position and velocity control: c1) Cable-
driven gravity-compensated position control. c2) Cable-driven gravity-
compensated velocity control.

Our goal is to regulate the strain and strain velocity states of the robot per
section under different constant tip loads despite the inevitable non-constant
loads due to gravity, external forces, and inertial forces.

###  V-A System Setup and Parameters

As seen in Fig. 1, the tip load acts on the $+y$-axis in the robot’s base
frame. We use $\mathcal{F}_{p}^{y}$ to represent the tip load acting along the
$+y$ direction in what follows. Given the geometry of the robot, we choose a
drag coefficient of $0.82$ (a Reynolds number of order $104$) for underwater
operations. We set the Young’s modulus, $E=110kPa$ and the shear viscosity
modulus to $3kPa$. The bending second inertia momenta are set to
$I_{y}=I_{z}=\pi r^{4}/4$ while the torsion second moment of inertia is set to
$I_{x}=\pi r^{4}/2$ for $r=0.1m$, the arm’s radius – uniform across sections.
The arm length is $L=0.2m$. We assume a (near-incompressible) rubber material
makes up the robot’s body and set its Poisson ratio to $0.45$; the mass is
chosen as $\mathcal{M}=\rho[I_{x},I_{y},I_{z},A,A,A]$ for a cylindrical soft
shell’s nominal density of $\rho=2,000kgm^{-3}$ as used in [18]; the cross-
sectional area $A=\pi r^{2}$ so that $I_{x}=\pi r^{4}/2$. The drag screw
stiffness matrix $D$ in (8) is a function of each section’s geometry and
hydrodynamics so that $D=-\rho_{w}\nu^{T}\nu\breve{D}\nu/|\nu|$ where
$\rho_{w}$ is the water density set to $997kg/m^{3}$, and $\breve{D}$ is the
tensor that models the geometry and hydrodynamics factors in the viscosity
model (see [18, §II.B, eq. 6]). The curvilinear abscissa, $X\in[0,L]$ was
discretized into $41$ microsolids per section.

###  V-B Discussion

We adopted the recursive articulated-body algorithm [7] and integrate the
right-hand-side of differential equations (56) and (63) using a Runge-Kutta-
Fehlberg (RKF) adaptive scheme implemented in Python and Torch [15] with
relative and absolute errors respectively set to $10^{-7}$ and $10^{-9}$. We
have found these tolerance values to be crucial for a successful numerical
integration scheme as it avoids numerical instability. Depicted on the
vertical axes of each chart of Fig. 3 are the strain positions or twists along
$+y$-direction (on the robot’s local frame) while the horizontal axes depict
the number of adaptive RKF (re-) integration steps per for every tenth of a
second. The trajectory evolution over time per discretized Cosserat section is
shown in the various “dash-dotted” lines, while the “solid blue” lines denote
the reference. The controller parameters are annotated within chart together
with the amount of constant tip load in Newtons. We see that all joint
configurations are stabilized to reference strain twist states $(\dot{q}^{d})$
for respective constant tip loads. It can also be inferred that more sections
a in the discretized Cosserat model lead to less bumpy state regulation
strategy.

In (a1–a3), the strain twists are precisely regulated to zero offset errors
despite large constant tip loads (a1-a3); and with small tip disturbances
(a4). Notably, the large value of the proportional gain causes overdamping in
the transients before convergence. However, a 20% error offset at steady state
is observed in the linear strain position regulation case (b1) in the absence
of the gravity compensation term in the control law. We attribute this to the
nature of PD Lagrangian controllers realized via computed torque methods [21].
For accurate position control with PD controllers, steady state errors can be
achieved when the gravity os compensated in the control design as (b2) shows.

##  VI Conclusion

We have presented the Lagrangian properties for soft robots under a discrete
Cosserat model. These properties were then exploited to cancel out
nonlinearities in the derived controllers for strain states regulation. Our
numerical experiments confirm the conclusions from our Lyapunov analyses.
Similar to rigid robots under PD control laws with Lagrangian dynamics, we
have observed strain position steady state offsets. Position control efficacy
are shown to improve if gravity and/or buoyancy is compensated in the control
law.

##  VII Acknowledgment

A vote of thanks to James Forbes for early manuscript feedback.

## References

  * Agarwal et al. [2016] Gunjan Agarwal, Nicolas Besuchet, Basile Audergon, and Jamie Paik.  Stretchable Materials for Robust Soft Actuators Towards Assistive Wearable Devices.  _Scientific reports_ , 6(1):34224, 2016.
  * Boyer and Renda [2017] Frederic Boyer and Federico Renda.  Poincaré’s Equations for Cosserat Media: Application to Shells.  _Journal of Nonlinear Science_ , 27(1):1–44, 2017\.  ISSN 14321467.
  * Boyer et al. [2010] Frederic Boyer, Mathieu Porez, and Alban Leroyer.  Poincaré–cosserat equations for the lighthill three-dimensional large amplitude elongated body theory: application to robotics.  _Journal of Nonlinear Science_ , 20:47–79, 2010.
  * Boyer et al. [2022] Frédéric Boyer, Vincent Lebastard, Fabien Candelier, Federico Renda, and Mazen Alamir.  Statics and Dynamics of Continuum Robots based on Cosserat Rods and Optimal Control Theories.  _IEEE Transactions on Robotics_ , 39(2):1544–1562, 2022.
  * Coevoet et al. [2017] Eulalie Coevoet, Adrien Escande, and Christian Duriez.  Optimization-based Inverse Model of Soft Robots with Contact Handling.  _IEEE Robotics and Automation Letters_ , 2(3):1413–1419, 2017.
  * Cosserat and Cosserat [1909] Eugène Maurice Pierre Cosserat and François Cosserat.  _Théorie des corps déformables_.  A. Hermann et fils, 1909.
  * Featherstone [2014] Roy Featherstone.  _Rigid Body Dynamics Algorithms_.  Springer, 2014.
  * Hannan and Walker [2003] Michael W Hannan and Ian D Walker.  Kinematics and the Implementation of an Elephant’s Trunk Manipulator and other Continuum Style Robots.  _Journal of Robotic Systems_ , 20(2):45–63, 2003.
  * Hauser et al. [2014] Helmut Hauser, Rudolf M. Füchslin, and Rolf Pfeifer, editors.  _Opinions and Outlooks on Morphological Computation_.  Hauser, Helmut and Füchslin, Rudolf M. and Pfeifer, Rolf, 2014.  ISBN 978-3-033-04515-6.
  * Khalil [2015] Hassan K Khalil.  _Nonlinear Control_ , volume 406.  Pearson New York, 2015.
  * Laschi et al. [2012] Cecilia Laschi, Matteo Cianchetti, Barbara Mazzolai, Laura Margheri, Maurizio Follador, and Paolo Dario.  Soft robot arm inspired by the octopus.  _Advanced robotics_ , 26(7):709–727, 2012.
  * Manti et al. [2015] Mariangela Manti, Taimoor Hassan, Giovanni Passetti, Nicolò D’Elia, Cecilia Laschi, and Matteo Cianchetti.  A Bioinspired Soft Robotic Gripper for Adaptable and Effective Grasping.  _Soft Robotics_ , 2(3):107–116, 2015.
  * Marchese et al. [2014] Andrew D Marchese, Cagdas D Onal, and Daniela Rus.  Autonomous Soft Robotic Fish Capable of Escape Maneuvers Using Fluidic Elastomer Actuators.  _Soft robotics_ , 1(1):75–87, 2014.
  * Nuckols et al. [2021] R. W. Nuckols, S. Lee, K. Swaminathan, D. Orzel, R. D. Howe, and C. J. Walsh.  Individualization of exosuit assistance based on measured muscle dynamics during versatile walking.  _Science Robotics_ , 6(60):eabj1362, 2021.
  * Paszke et al. [2019] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala.  Pytorch: An imperative style, high-performance deep learning library.  In _Advances in Neural Information Processing Systems 32_ , pages 8024–8035. Curran Associates, Inc., 2019.
  * Polygerinos et al. [2015] Panagiotis Polygerinos, Zheng Wang, Kevin C Galloway, Robert J Wood, and Conor J Walsh.  Soft robotic glove for combined assistance and at-home rehabilitation.  _Robotics and Autonomous Systems_ , 73:135–143, 2015.
  * Qiu et al. [2023] Ke Qiu, Jingyu Zhang, Danying Sun, Rong Xiong, Haojian Lu, and Yue Wang.  An efficient multi-solution solver for the inverse kinematics of 3-section constant-curvature robots.  _arXiv preprint arXiv:2305.01458_ , 2023.
  * Renda et al. [2018] Federico Renda, Frédéric Boyer, Jorge Dias, and Lakmal Seneviratne.  Discrete cosserat approach for multisection soft manipulator dynamics.  _IEEE Transactions on Robotics_ , 34(6):1518–1533, 2018.
  * Romero et al. [2014] José Guadalupe Romero, Romeo Ortega, and Ioannis Sarras.  A globally exponentially stable tracking controller for mechanical systems using position feedback.  _IEEE Transactions on Automatic Control_ , 60(3):818–823, 2014.
  * Shih et al. [2023] Chia-Hsien Shih, Noel Naughton, Udit Halder, Heng-Sheng Chang, Seung Hyun Kim, Rhanor Gillette, Prashant G Mehta, and Mattia Gazzola.  Hierarchical control and learning of a foraging cyberoctopus.  _Advanced Intelligent Systems_ , page 2300088, 2023.
  * Spong [2022] Mark W. Spong.  An historical perspective on the control of robotic manipulators.  _Annual Review of Control, Robotics, and Autonomous Systems_ , 5:1–31, 2022.
  * Thuruthel et al. [2018] Thomas George Thuruthel, Egidio Falotico, Federico Renda, and Cecilia Laschi.  Model-based Reinforcement Learning for Closed-loop Dynamic Control of Soft Robotic Manipulators.  _IEEE Transactions on Robotics_ , 35(1):124–134, 2018.
  * Yap et al. [2016] Hong Kai Yap, Nazir Kamaldin, Jeong Hoon Lim, Fatima A Nasrallah, James Cho Hong Goh, and Chen-Hua Yeow.  A Magnetic Resonance Compatible Soft Wearable Robotic Glove for Hand Rehabilitation and Brain Imaging.  _IEEE transactions on neural systems and rehabilitation engineering_ , 25(6):782–793, 2016.


$\displaystyle=\operatorname{P}\left\\{\frac{\sqrt{n}}{k}\sum_{t=d+1}^{n}I_{3t}\geq\varepsilon\right\\}\leq\varepsilon^{-4}\frac{n^{2}}{k^{4}}\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{3t}\Big{]}^{4}.$
(D.11)

Due to their $d$-dependence, the $I_{3t}$ are trivially $\rho$-mixing of any
rate, and thus we may again apply Lemma 2.3 of Shao (1993) (for, in his
notation, $q=4$) to deduce that

$\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{3t}\Big{]}^{4}\leq
n^{2}C\big{\\{}\operatorname{E}[I_{3t}^{2}]\big{\\}}^{2}+nC\operatorname{E}[I_{3t}^{4}].$
(D.12)

Since $I_{3t}\in\\{0,1\\}$ and, hence,
$\operatorname{E}[I_{3t}^{4}]=\operatorname{E}[I_{3t}^{2}]=\operatorname{E}[I_{3t}]$,
it suffices to compute $\operatorname{E}[I_{3t}]$. Use Lemma 1 to obtain

$\displaystyle\operatorname{E}[I_{3t}]$
$\displaystyle=\operatorname{P}\Big{\\{}U_{t}>b\Big{(}\frac{n}{k[l_{1}+2]\rho}\Big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>b\Big{(}\frac{n}{k[l_{2}+2]\rho}\Big{)}\Big{\\}}$
$\displaystyle\hskip
85.35826pt-\operatorname{P}\Big{\\{}U_{t}>b\Big{(}\frac{n}{kl_{1}\rho}\Big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>b\Big{(}\frac{n}{kl_{2}\rho}\Big{)}\Big{\\}}$
$\displaystyle=\frac{k^{2}}{n^{2}}\big{[}(l_{1}+2)(l_{2}+2)-l_{1}l_{2}\big{]}\rho^{2}$
$\displaystyle\leq C\frac{k^{2}}{n^{2}}\rho^{2}.$

Together with (D.12), this implies

$\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{3t}\Big{]}^{4}\leq
C\frac{k^{4}}{n^{2}}\rho^{4}+C\frac{k^{2}}{n}\rho^{2}.$

With (D.11), this gives

$\operatorname{P}\left\\{\sqrt{n}\big{|}M_{n}((l_{1}+2,l_{2}+2)^{\prime}\rho,\bm{0})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\right\\}\leq
C\rho^{4}+C\frac{n}{k^{2}}\rho^{2}.$

Thus, by subadditivity and using that the cardinality of
$\widetilde{\mathcal{L}}$ is of the order $\rho^{-2}$, we obtain with
Assumption 4 that

$\displaystyle\operatorname{P}\\{\sqrt{n}B_{1n}\geq\varepsilon\\}$
$\displaystyle=\operatorname{P}\Big{\\{}\cup_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\big{\\{}\sqrt{n}\big{|}M_{n}((l_{1}+2,l_{2}+2)^{\prime}\rho,\bm{0})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\big{\\}}\Big{\\}}$
$\displaystyle\leq\sum_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\operatorname{P}\Big{\\{}\sqrt{n}\big{|}M_{n}((l_{1}+2,l_{2}+2)^{\prime}\rho,\bm{0})-M_{n}((l_{1},l_{2})^{\prime}\rho,\bm{0})\big{|}\geq\varepsilon\Big{\\}}$
$\displaystyle\leq
O(\rho^{-2})\Big{[}C\rho^{4}+C\frac{n}{k^{2}}\rho^{2}\Big{]}\leq
C\rho^{2}+o(1).$

Since $\rho$ can be chosen arbitrarily small, it follows that
$B_{1n}=o_{\operatorname{P}}(n^{-1/2})$. This ends the proof. ∎

For the following lemma, recall the definition of
$B_{t}(\bm{x},\bm{\xi},\eta,\eta_{0})$ in (C.8). The proof of the lemma is in
Appendix D.2.

######  Lemma 4.

Let $\eta>0$, $\eta_{0}\in\\{-1,1\\}$. Then, for any $K>0$,

$\sup_{\bm{\xi}\in[-K,K]^{2}}\sup_{\bm{x}\in[0,K]^{2}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}(\bm{x},\bm{\xi},\eta,\eta_{0})\right|=o_{\operatorname{P}}(n^{-1/2}).$

###### Proof of Proposition 7:.

The proof is identical to that of Proposition 3 using Lemma 4 instead of Lemma
3. ∎

The next lemma, whose proof is in Appendix D.2, is required for the proof of
Proposition 8.

######  Lemma 5.

For any $0<\iota<K<\infty$, it holds that, as $n\to\infty$,

$\sup_{x\in[\iota,K]}\Bigg{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{n}\Big{[}I_{\big{\\{}\widehat{U}_{t}>b(n/[kx])\big{\\}}}-I_{\big{\\{}U_{t}>b(n/[kx])\big{\\}}}\Big{]}\Bigg{|}=o_{\operatorname{P}}(1).$

###### Proof of Proposition 8:.

Observe that for all $\xi\in\mathbb{R}$

$\displaystyle\sqrt{k}\log\Bigg{(}\frac{\widehat{U}_{(\lfloor
kx\rfloor+1)}}{b(n/[kx])}\Bigg{)}\leq\xi\quad\Longleftrightarrow\quad\widehat{U}_{(\lfloor
kx\rfloor+1)}\leq e^{\xi/\sqrt{k}}b(n/[kx])$
$\displaystyle\Longleftrightarrow\quad\sum_{t=1}^{n}I_{\big{\\{}\widehat{U}_{t}>e^{\xi/\sqrt{k}}b(n/[kx])\big{\\}}}\leq
kx$
$\displaystyle\Longleftrightarrow\quad\frac{1}{\sqrt{k}}\sum_{t=1}^{n}\Big{[}I_{\big{\\{}\widehat{U}_{t}>e^{\xi/\sqrt{k}}b(n/[kx])\big{\\}}}-\operatorname{P}\big{\\{}U_{t}>e^{\xi/\sqrt{k}}b(n/[kx])\big{\\}}\Big{]}\leq\alpha\xi
x+o(1),$ (D.13)

where the $o(1)$-term is uniform in $x\in[\iota,K]$ by Lemma 1.

Denote by $D[0,K]$ the space of real-valued functions on $[0,K]$ that are
right-continuous with limits from below (Davidson, 1994). Then, it can be
proven along similar lines as in Proposition 5 that

$\frac{1}{\sqrt{k}}\sum_{t=1}^{n}\Big{[}I_{\big{\\{}U_{t}>b(n/[kx])\big{\\}}}-\operatorname{P}\Big{\\{}U_{t}>b(n/[kx])\Big{\\}}\Big{]}\overset{d}{\longrightarrow}B(x)\qquad\text{in
}D[0,K],$

where $B(\cdot)$ denotes a standard Brownian motion; see also the proof of
Proposition 2.1 in Resnick and Stǎricǎ (1997). This implies by the continuous
mapping theorem that

$\sup_{x\in[\iota,K]}\Bigg{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{n}\Big{[}I_{\big{\\{}U_{t}>b(n/[kx])\big{\\}}}-\operatorname{P}\Big{\\{}U_{t}>b(n/[kx])\Big{\\}}\Big{]}\Bigg{|}=O_{\operatorname{P}}(1).$

From this, Lemma 5 and Assumption 1 we deduce that

$\sup_{x\in[\iota,K]}\Bigg{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{n}\Big{[}I_{\big{\\{}\widehat{U}_{t}>e^{\xi/\sqrt{k}}b(n/[kx])\big{\\}}}-\operatorname{P}\Big{\\{}U_{t}>e^{\xi/\sqrt{k}}b(n/[kx])\Big{\\}}\Big{]}\Bigg{|}=O_{\operatorname{P}}(1).$

Combining this with (D.13), the conclusion follows. ∎

###  D.2 Proofs of Lemmas 4–5

###### Proof of Lemma 4:.

It suffices to consider the supremum over $\bm{\xi}\in[-K,0]^{2}$; the cases
where $\bm{\xi}$ lies in the other quadrants can be dealt with similarly.
Using the same grid $\mathcal{L}$ as in the proof of Proposition 6, we get
that

$\displaystyle\sup_{\bm{\xi}\in[-K,0]^{2}}$
$\displaystyle\sup_{\bm{x}\in[0,K]^{2}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}(\bm{x},\bm{\xi},\eta,\eta_{0})\right|\leq\sup_{\bm{x}\in[0,K]^{2}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}(\bm{x},-(K,K)^{\prime},\eta,\eta_{0})\right|$
$\displaystyle\leq\max_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}((l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0})\right|$
$\displaystyle\hskip
56.9055pt+\sup_{|\bm{x}_{1}-\bm{x}_{2}|\leq\rho}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}(\bm{x}_{1},-(K,K)^{\prime},\eta,\eta_{0})-B_{t}(\bm{x}_{2},-(K,K)^{\prime},\eta,\eta_{0})\right|$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{2n}+B_{2n}.$

By subadditivity and Lemma 3, we obtain that

$\displaystyle\operatorname{P}\\{\sqrt{n}A_{2n}\geq\varepsilon\\}$
$\displaystyle=\operatorname{P}\Bigg{\\{}\bigcup_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\Big{\\{}\sqrt{n}\Big{|}\frac{1}{k}\sum_{t=d+1}^{n}B_{t}\big{(}(l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}\big{)}\Big{|}\geq\varepsilon\Big{\\}}\Bigg{\\}}$
$\displaystyle\leq\sum_{(l_{1},l_{2})^{\prime}\in\mathcal{L}}\operatorname{P}\Bigg{\\{}\sqrt{n}\Big{|}\frac{1}{k}\sum_{t=d+1}^{n}B_{t}\big{(}(l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}\big{)}\Big{|}\geq\varepsilon\Bigg{\\}}=o(1).$

Thus, $A_{2n}=o_{\operatorname{P}}(n^{-1/2})$.

To show that $B_{2n}=o_{\operatorname{P}}(n^{-1/2})$, use a monotonicity
argument to deduce that

$B_{2n}\leq\max_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\left|\frac{1}{k}\sum_{t=d+1}^{n}B_{t}\big{(}(l_{1}+2,l_{2}+2)^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}\big{)}-B_{t}\big{(}(l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}\big{)}\right|,$

where $\widetilde{\mathcal{L}}$ is again defined as in the proof of
Proposition 6. Let $\eta_{0}=1$; the case $\eta_{0}=-1$ can be treated
analogously. Define $w_{t}$ and $\nu=\nu(\varepsilon_{0})>0$ as in the proof
of Lemma 3. In particular, if $w_{t}=1$ then

$1-\nu/2<(1+s_{t})^{-1}\leq 1\qquad\text{and}\qquad
1-\nu/2<(1+s_{t-d})^{-1}\leq 1.$

For sufficiently large $n$, such that
$(1-\nu/2)<\big{[}1-e^{K/\sqrt{k}}\varepsilon_{0}/b(n/[kK])\big{]}\leq 1$,

$\displaystyle
w_{t}\big{|}B_{t}\big{(}(l_{1}+2,l_{2}+2)^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}=1\big{)}-B_{t}\big{(}(l_{1},l_{2})^{\prime}\rho,-(K,K)^{\prime},\eta,\eta_{0}=1\big{)}\big{|}$
$\displaystyle=w_{t}\Big{|}I_{\Big{\\{}U_{t}>e^{-K/\sqrt{k}}(1+s_{t})^{-1}\big{(}1-e^{K/\sqrt{k}}\frac{m_{t}}{b(n/[k(l_{1}+2)\rho])}\big{)}b\big{(}\frac{n}{k[l_{1}+2]\rho}\big{)},}$
$\hskip
42.67912pt{}_{U_{t-d}>e^{-K/\sqrt{k}}(1+s_{t-d})^{-1}\big{(}1-e^{K/\sqrt{k}}\frac{m_{t-d}}{b(n/[k(l_{2}+2)\rho])}\big{)}b\big{(}\frac{n}{k[l_{2}+2]\rho}\big{)}\Big{\\}}}$
$\displaystyle\hskip 239.00298pt-
I_{\Big{\\{}U_{t}>e^{-K/\sqrt{k}}b\big{(}\frac{n}{k[l_{1}+2]\rho}\big{)},\
U_{t-d}>e^{-K/\sqrt{k}}b\big{(}\frac{n}{k[l_{2}+2]\rho}\big{)}\Big{\\}}}$
$\displaystyle\hskip 14.22636pt-
I_{\Big{\\{}U_{t}>e^{-K/\sqrt{k}}(1+s_{t})^{-1}\big{(}1-e^{K/\sqrt{k}}\frac{m_{t}}{b(n/[kl_{1}\rho])}\big{)}b\big{(}\frac{n}{kl_{1}\rho}\big{)},}$
$\hskip
42.67912pt{}_{U_{t-d}>e^{-K/\sqrt{k}}(1+s_{t-d})^{-1}\big{(}1-e^{K/\sqrt{k}}\frac{m_{t-d}}{b(n/[kl_{2}\rho])}\big{)}b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}}+I_{\Big{\\{}U_{t}>e^{-K/\sqrt{k}}b\big{(}\frac{n}{kl_{1}\rho}\big{)},\
U_{t-d}>e^{-K/\sqrt{k}}b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}}\Big{|}$
$\displaystyle\leq\Big{|}I_{\Big{\\{}U_{t}>(1-\nu)b\big{(}\frac{n}{k[l_{1}+2]\rho}\big{)},\
U_{t-d}>(1-\nu)b\big{(}\frac{n}{k[l_{2}+2]\rho}\big{)}\Big{\\}}}-I_{\Big{\\{}U_{t}>b\big{(}\frac{n}{k[l_{1}+2]\rho}\big{)},\
U_{t-d}>b\big{(}\frac{n}{k[l_{2}+2]\rho}\big{)}\Big{\\}}}\Big{|}$
$\displaystyle\hskip
28.45274pt+\Big{|}I_{\Big{\\{}U_{t}>(1-\nu)b\big{(}\frac{n}{kl_{1}\rho}\big{)},\
U_{t-d}>(1-\nu)b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}}-I_{\Big{\\{}U_{t}>b\big{(}\frac{n}{kl_{1}\rho}\big{)},\
U_{t-d}>b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}}\Big{|}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}I_{4t}(l_{1},l_{2})+I_{5t}(l_{1},l_{2}).$

Since $w_{\ell_{n}}=\ldots=w_{n}=1$ w.p.a. 1 as $n\to\infty$, to prove
$B_{2n}=o_{\operatorname{P}}(n^{-1/2})$, it suffices to show that

$\max_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\frac{1}{k}\sum_{t=d+1}^{n}I_{jt}(l_{1},l_{2})=o_{\operatorname{P}}(n^{-1/2})\qquad(j=4,5).$
(D.14)

We only show this claim for $j=5$, as that for $j=4$ can be proven along
similar lines. By subadditivity and Markov’s inequality,

$\displaystyle\operatorname{P}\Bigg{\\{}\sqrt{n}\max_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\frac{1}{k}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\geq\varepsilon\Bigg{\\}}$
$\displaystyle=\operatorname{P}\Bigg{\\{}\bigcup_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\Big{\\{}\frac{\sqrt{n}}{k}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\geq\varepsilon\Big{\\}}\Bigg{\\}}$
$\displaystyle=\sum_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\operatorname{P}\Big{\\{}\frac{\sqrt{n}}{k}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\geq\varepsilon\Big{\\}}$
$\displaystyle\leq\sum_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\varepsilon^{-4}\frac{n^{2}}{k^{4}}\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\Big{]}^{4}.$

Use Lemma 1 to conclude that for sufficiently small $\nu>0$,

$\displaystyle\operatorname{E}[I_{5t}(l_{1},l_{2})]$
$\displaystyle=\operatorname{P}\Big{\\{}U_{t}>(1-\nu)b\big{(}\frac{n}{kl_{1}\rho}\big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>(1-\nu)b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}$
$\displaystyle\hskip
142.26378pt-\operatorname{P}\Big{\\{}U_{t}>b\big{(}\frac{n}{kl_{1}\rho}\big{)}\Big{\\}}\operatorname{P}\Big{\\{}U_{t-d}>b\big{(}\frac{n}{kl_{2}\rho}\big{)}\Big{\\}}$
$\displaystyle=\frac{k^{2}}{n^{2}}\rho^{2}l_{1}l_{2}\Big{[}1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]}^{2}-\frac{k^{2}}{n^{2}}\rho^{2}l_{1}l_{2}$ $\displaystyle\leq
C\frac{k^{2}}{n^{2}}\rho^{2}l_{1}l_{2}\Big{[}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)\Big{]}$ $\displaystyle\leq C\frac{k^{2}}{n^{2}}\rho^{2}\nu.$

Arguing similarly as after (D.12), we obtain

$\operatorname{E}\Big{[}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\Big{]}^{4}\leq
C\frac{k^{4}}{n^{2}}\rho^{4}\nu+C\frac{k^{2}}{n}\rho^{2}\nu.$

Using Assumption 4 and that the cardinality of $\widetilde{\mathcal{L}}$ is of
the order $\rho^{-2}$, we get from this that

$\operatorname{P}\Bigg{\\{}\sqrt{n}\max_{(l_{1},l_{2})^{\prime}\in\widetilde{\mathcal{L}}}\frac{1}{k}\sum_{t=d+1}^{n}I_{5t}(l_{1},l_{2})\geq\varepsilon\Bigg{\\}}\leq
C\rho^{2}\nu+C\frac{n}{k^{2}}\nu\leq C\rho^{2}\nu,$

which can be made arbitrarily small by a suitable choice of $\nu$. This
establishes (D.14), concluding the proof. ∎

The following lemmas are required for the proof of Lemma 5. Set

$\displaystyle C_{t}(x,\bm{\theta})$
$\displaystyle=I_{\big{\\{}\widehat{U}_{t}(\bm{\theta})>b(n/[kx])\big{\\}}},$
$\displaystyle C_{t}(x)$
$\displaystyle=I_{\big{\\{}U_{t}>b(n/[kx])\big{\\}}},$ $\displaystyle
C_{t}(x,\eta,\eta_{0})$
$\displaystyle=I_{\left\\{U_{t}[1+\eta_{0}s_{t}]+\eta_{0}m_{t}>b(n/[kx])\right\\}},\qquad\eta_{0}\in\\{-1,1\\},$
$\displaystyle D_{t}(x,\eta,\eta_{0})$
$\displaystyle=C_{t}(x,\eta,\eta_{0})-C_{t}(x),$

where $m_{t}=m_{n,t}(\eta)\geq 0$ and $s_{t}=s_{n,t}(\eta)\geq 0$ are from
Assumption 3.

######  Lemma 6.

Let $\eta>0$. Then, w.p.a. 1, as $n\to\infty$,

$C_{t}(x,\eta,-1)\leq C_{t}(x,\bm{\theta})\leq C_{t}(x,\eta,1)$

for all $\bm{\theta}\in N_{n}(\eta)$ and $t=\ell_{n},\ldots,n$.

###### Proof.

The proof closely resembles that of Lemma 2 and, hence, is omitted. ∎

######  Lemma 7.

Let $\eta>0$ and $\eta_{0}\in\\{-1,1\\}$. Then, for any $0<\iota<K<\infty$,

$\sup_{x\in[\iota,K]}\Bigg{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{n}D_{t}(x,\eta,\eta_{0})\Bigg{|}=o_{\operatorname{P}}(1).$

###### Proof.

The outline of the proof is similar to that of Lemma 3. Let
$\ell_{n}\to\infty$ with $\ell_{n}=o(\sqrt{k})$. Consider the case
$\eta_{0}=1$; the case $\eta_{0}=-1$ can be dealt with similarly. Define

$v_{t}=I_{\left\\{m_{t}\leq\varepsilon_{0},\
s_{t}\leq\varepsilon_{0}\right\\}},\qquad\varepsilon_{0}>0.$

If $v_{t}=1$, there exists $\nu=\nu(\varepsilon_{0})>0$, such that for $s_{t}$
from Assumption 3 it holds that

$1-\nu/2<(1+s_{t})^{-1}\leq 1.$

Then, for sufficiently large $n$, such that
$1-\nu/2<1-\varepsilon_{0}/b(n/[kx])\leq 1$ for all $x\in[\iota,K]$, we can
bound

$\displaystyle\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}v_{t}D_{t}(x,\eta,\eta_{0}=1)\Big{|}$
$\displaystyle=\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}v_{t}\big{[}I_{\left\\{U_{t}>(1+s_{t})^{-1}(1-m_{t}/b(n/[kx]))b(n/[kx])\right\\}}-I_{\left\\{U_{t}>b(n/[kx])\right\\}}\big{]}$
$\displaystyle\leq\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}\big{[}I_{\left\\{U_{t}>(1-\nu)b(n/[kx])\right\\}}-I_{\left\\{U_{t}>b(n/[kx])\right\\}}\big{]}$
$\displaystyle=\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{\left\\{U_{t}\in\big{(}(1-\nu)b(n/[kx]),\
b(n/[kx])\big{]}\right\\}}$ $\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{6t}(x).$

For $\rho>0$, define $\mathcal{M}=\\{l\in\mathbb{N}_{0}\
\mathrel{\mathop{\mathchar 58\relax}}\ l\in[0,K/\rho]\\}$. Then,

$\displaystyle\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{6t}(x)$
$\displaystyle=\max_{l\in\mathcal{M}}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)+\sup_{|x_{1}-x_{2}|\leq\rho}\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}[I_{6t}(x_{1})-I_{6t}(x_{2})]\Big{|}$
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{3n}+B_{3n}.$

Using in turn subadditivity, Markov’s inequality, and Lemma 2.3 of Shao
(1993), we get that

$\displaystyle\operatorname{P}\\{A_{3n}>\varepsilon\\}$
$\displaystyle\leq\sum_{l\in\mathcal{M}}\operatorname{P}\Big{\\{}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)>\varepsilon\Big{\\}}$
$\displaystyle\leq\sum_{l\in\mathcal{M}}\frac{1}{k^{2}}\operatorname{E}\Bigg{[}\sum_{t=\ell_{n}}^{n}I_{6t}(l\rho)\Bigg{]}^{4}$
$\displaystyle\leq\sum_{l\in\mathcal{M}}\frac{C}{k^{2}}\Big{\\{}n^{2}\big{\\{}\operatorname{E}[I_{6t}^{2}(l\rho)]\big{\\}}^{2}+n\operatorname{E}[I_{6t}^{4}(l\rho)]\Big{\\}}.$
(D.15)

Since $I_{6t}(l\rho)\in\\{0,1\\}$, Lemma 1 implies

$\displaystyle\operatorname{E}[I_{6t}^{4}(l\rho)]$
$\displaystyle=\operatorname{E}[I_{6t}^{2}(l\rho)]=\operatorname{E}[I_{6t}(l\rho)]$
$\displaystyle=\operatorname{P}\Big{\\{}U_{t}\in\big{(}(1-\nu)b(n/[kl\rho]),\
b(n/[kl\rho])\big{]}\Big{\\}}$
$\displaystyle=\operatorname{P}\Big{\\{}U_{t}>(1-\nu)b(n/[kl\rho])\Big{\\}}-\operatorname{P}\Big{\\{}U_{t}>b(n/[kl\rho])\Big{\\}}$
$\displaystyle=\frac{kl\rho}{n}\Big{\\{}1+\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
o(1)-1\Big{\\}}$
$\displaystyle=\frac{kl\rho}{n}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+\nu
l\rho o\Big{(}\frac{k}{n}\Big{)}$ (D.16) $\displaystyle\leq
C\nu\rho\frac{k}{n}.$

Plugging this into (D.15) gives

$\displaystyle\operatorname{P}\\{A_{3n}>\varepsilon\\}$
$\displaystyle\leq\sum_{l\in\mathcal{M}}C\left\\{\frac{n^{2}}{k^{2}}[\nu\rho]^{2}\frac{k^{2}}{n^{2}}+\frac{n}{k^{2}}\nu\rho\frac{k}{n}\right\\}$
$\displaystyle\leq C\left\\{\nu^{2}\rho^{2}+\frac{\nu}{k}\rho\right\\},$

which can be made arbitrarily small by a suitable choice of $\nu$. Thus,
$A_{3n}=o_{\operatorname{P}}(1)$.

To show $B_{3n}=o_{\operatorname{P}}(1)$, define

$\widetilde{\mathcal{M}}=\\{l\in\mathbb{N}_{0}\ \mathrel{\mathop{\mathchar
58\relax}}\ [l,l+2]\subset[0,K/\rho]\\}.$

Use a monotonicity argument to deduce that

$B_{3n}\leq\max_{l\in\widetilde{\mathcal{M}}}\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}\big{[}I_{6t}([l+2]\rho)-I_{6t}(l\rho)\big{]}\Big{|}=\mathrel{\mathop{\mathchar
58\relax}}\max_{l\in\widetilde{\mathcal{M}}}\Big{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{n}I_{7t}(l,\rho)\Big{|}.$

We get by Markov’s inequality that

$\operatorname{P}\Bigg{\\{}\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big{|}>\varepsilon\Bigg{\\}}\leq\frac{\varepsilon^{-4}}{k^{2}}\operatorname{E}\Big{[}\sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big{]}^{4}.$
(D.17)

Since, by (D.16),

$\displaystyle\operatorname{E}\Big{[}I_{7t}(l,\rho)\Big{]}$
$\displaystyle=2\rho\frac{k}{n}\frac{\alpha\nu}{(1+\widetilde{\nu})^{1+\alpha}}+2\rho\nu
o(k/n)\leq C\rho\nu\frac{k}{n},$

we obtain from Lemma 2.3 of Shao (1993) that

$\displaystyle\operatorname{E}\Big{[}\sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big{]}^{4}$
$\displaystyle\leq
C\left\\{n^{2}\big{\\{}\operatorname{E}[I_{7t}(l,\rho)]^{2}\big{\\}}^{2}+n\operatorname{E}[I_{7t}(l,\rho)]^{4}\right\\}$
$\displaystyle\leq
C\Big{\\{}n^{2}\Big{[}\rho\nu\frac{k}{n}\Big{]}^{2}+n\Big{[}C\rho\nu\frac{k}{n}\Big{]}\Big{\\}}.$

Thus, the right-hand side of (D.17) can be bounded by
$C\rho^{2}\nu^{2}+\frac{C}{k}\rho\nu$. Using this, we obtain

$\displaystyle\operatorname{P}\\{B_{3n}>\varepsilon\\}$
$\displaystyle\leq\sum_{l\in\widetilde{\mathcal{M}}}\operatorname{P}\Big{\\{}\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}I_{7t}(l,\rho)\Big{|}>\varepsilon\Big{\\}}$
$\displaystyle=\sum_{l\in\widetilde{\mathcal{M}}}\Big{[}C\rho^{2}\nu^{2}+\frac{C}{k}\rho\nu\Big{]}$
$\displaystyle\leq C\\{\rho\nu^{2}+\nu/k\\},$

where we have used in the final step that the cardinality of
$\widetilde{\mathcal{M}}$ is of the order $1/\rho$. This proves

$\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}v_{t}D_{t}(x,\eta,\eta_{0})\Big{|}=o_{\operatorname{P}}(1),$

since $\nu>0$ can be chosen arbitrarily small. By Assumption 3, we have
$v_{\ell_{n}}=\ldots=v_{n}=1$ w.p.a. 1, as $n\to\infty$. Hence, the above
display also implies that

$\Big{|}\frac{1}{\sqrt{k}}\sum_{t=\ell_{n}}^{n}D_{t}(x,\eta,\eta_{0})\Big{|}=o_{\operatorname{P}}(1).$

Moreover, by boundedness of the $D_{t}(x,\eta,\eta_{0})$, we easily get
$\Big{|}\frac{1}{\sqrt{k}}\sum_{t=1}^{\ell_{n}-1}D_{t}(x,\eta,\eta_{0})\Big{|}=O(\ell_{n}/\sqrt{k})=o(1)$.
Putting these two results together, the conclusion follows. ∎

Now, we are in a position to prove Lemma 5.

###### Proof of Lemma 5:.

The proof is again similar to that of Proposition 3, where now we use Lemma 6
(instead of Lemma 2) and Lemma 7 (instead of Lemma 3). ∎

##  Appendix E Simulation Results for Varying $D$

In light of the simulation results in Section 3, we pick $k=\lfloor 0.11\cdot
n^{0.99}\rfloor$. We do so to focus here on the sensitivity of our tests with
respect to the number of included lags $D\in\\{1,\ldots,10\\}$. We reconsider
the models of Section 3 with misspecified volatility dynamics (in Section E.1)
and with misspecified innovations (in Section E.2).

###  E.1 Misspecified Volatility

$n$ | Test | $\alpha$ | $D$
---|---|---|---
|  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
500 | $\mathcal{P}_{n}^{(D)}$ | 1% | 0.3 | 0.5 | 0.6 | 0.5 | 0.5 | 0.6 | 0.6 | 0.5 | 0.4 | 0.4
|  | 5% | 2.6 | 2.5 | 2.5 | 2.5 | 2.2 | 2.3 | 2.1 | 1.9 | 1.8 | 1.7
|  | 10% | 8.1 | 5.4 | 5.3 | 4.8 | 5.0 | 4.9 | 4.5 | 4.2 | 3.8 | 3.7
| $\mathcal{F}_{n}^{(D)}$ | 1% | 0.5 | 0.4 | 0.4 | 0.3 | 0.3 | 0.4 | 0.3 | 0.2 | 0.2 | 0.2
|  | 5% | 2.0 | 1.7 | 1.8 | 1.8 | 1.6 | 1.6 | 1.4 | 1.2 | 1.0 | 0.8
|  | 10% | 4.6 | 3.8 | 3.4 | 3.4 | 3.1 | 3.2 | 2.9 | 2.5 | 2.1 | 1.9
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 1.2 | 1.7 | 2.3 | 2.8 | 3.0 | 3.2 | 3.4 | 3.5 | 3.4 | 3.5
|  | 5% | 3.2 | 3.9 | 4.8 | 5.3 | 5.8 | 6.1 | 6.5 | 6.6 | 6.9 | 6.9
|  | 10% | 5.7 | 6.1 | 7.3 | 7.4 | 8.2 | 8.5 | 9.0 | 9.4 | 9.3 | 9.8
1000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 0.8 | 1.0 | 0.9 | 0.9 | 0.7 | 0.5 | 0.5 | 0.6 | 0.5 | 0.5
|  | 5% | 4.8 | 3.4 | 3.5 | 3.3 | 3.1 | 2.8 | 2.7 | 2.5 | 2.2 | 2.0
|  | 10% | 9.7 | 7.0 | 6.5 | 6.1 | 5.9 | 5.7 | 5.4 | 4.9 | 4.4 | 4.0
| $\mathcal{F}_{n}^{(D)}$ | 1% | 0.7 | 0.7 | 0.7 | 0.6 | 0.5 | 0.5 | 0.6 | 0.5 | 0.4 | 0.4
|  | 5% | 3.5 | 2.9 | 2.9 | 2.2 | 2.0 | 2.0 | 1.8 | 1.4 | 1.4 | 1.3
|  | 10% | 6.9 | 5.4 | 5.2 | 4.4 | 4.2 | 3.9 | 3.7 | 3.4 | 2.7 | 2.4
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 1.3 | 2.0 | 2.5 | 2.5 | 2.9 | 3.0 | 3.3 | 3.7 | 3.7 | 3.7
|  | 5% | 2.7 | 4.0 | 5.0 | 5.3 | 5.7 | 5.7 | 6.2 | 6.5 | 6.8 | 6.5
|  | 10% | 4.8 | 6.4 | 7.2 | 7.6 | 8.3 | 8.5 | 8.6 | 8.7 | 9.1 | 9.3
2000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 0.9 | 1.0 | 1.1 | 1.1 | 1.0 | 0.9 | 1.0 | 0.9 | 0.8 | 0.7
|  | 5% | 4.2 | 4.0 | 3.8 | 3.6 | 3.4 | 3.0 | 2.9 | 2.8 | 2.5 | 2.6
|  | 10% | 7.9 | 7.7 | 7.3 | 6.9 | 6.5 | 6.2 | 5.8 | 5.4 | 5.1 | 5.2
| $\mathcal{F}_{n}^{(D)}$ | 1% | 1.0 | 1.3 | 1.3 | 1.3 | 1.1 | 1.2 | 1.0 | 0.9 | 0.8 | 0.8
|  | 5% | 3.9 | 3.5 | 3.6 | 3.5 | 3.2 | 3.0 | 2.7 | 2.3 | 2.2 | 2.1
|  | 10% | 7.4 | 6.8 | 6.2 | 6.3 | 5.8 | 5.1 | 4.6 | 4.5 | 4.1 | 3.7
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 1.6 | 2.2 | 3.0 | 3.3 | 3.5 | 3.8 | 3.9 | 4.1 | 4.3 | 4.2
|  | 5% | 3.2 | 4.3 | 5.2 | 5.8 | 6.0 | 6.4 | 6.7 | 7.0 | 7.4 | 7.5
|  | 10% | 4.9 | 6.0 | 7.3 | 7.8 | 8.4 | 8.6 | 9.0 | 9.3 | 9.7 | 9.9
Table 3: Size in % of tests based on $\mathcal{P}_{n}^{(D)}$,
$\mathcal{F}_{n}^{(D)}$ and $\operatorname{LB}_{n}^{(D)}$ for significance
levels $\alpha\in\\{1\%,5\%,10\%\\}$ and $D=1,\ldots,10$. Results are for
model (4) with
$\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{s}=(0.046,0.027,0.092,0.843,0)^{\prime}$.

Reconsider the setup of Section 3.1 in the main paper. The only difference is
that we now consider a fixed $k=\lfloor 0.11\cdot n^{0.99}\rfloor$ and varying
$D\in\\{1,\ldots,10\\}$. When $\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{s}$ in
(4), the results correspond to size, which is displayed in Table 3. As a
benchmark, we have again included the $\operatorname{LB}_{n}^{(D)}$-test with
the estimation correction of Carbon and Francq (2011). We see that,
particularly for small sample sizes, our tests tend to be undersized. Yet, the
size distortions substantially decrease with increasing $n$. Except for $D=1$,
size is reasonably stable across different choices of the number $D$ of
included lags.

$n$ | Test | $\alpha$ | $D$
---|---|---|---
|  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
500 | $\mathcal{P}_{n}^{(D)}$ | 1% | 3.8 | 8.3 | 11.0 | 13.0 | 14.3 | 14.7 | 14.8 | 14.9 | 14.9 | 14.7
|  | 5% | 12.1 | 15.6 | 20.1 | 22.2 | 23.9 | 25.0 | 24.9 | 25.0 | 24.6 | 24.0
|  | 10% | 20.7 | 21.7 | 26.8 | 30.6 | 31.9 | 32.7 | 32.9 | 32.4 | 31.6 | 31.1
| $\mathcal{F}_{n}^{(D)}$ | 1% | 5.3 | 8.5 | 11.5 | 13.8 | 15.3 | 15.9 | 15.9 | 15.6 | 15.5 | 15.1
|  | 5% | 11.2 | 16.4 | 20.6 | 23.7 | 25.2 | 26.1 | 26.1 | 25.8 | 25.0 | 24.3
|  | 10% | 17.0 | 23.4 | 28.0 | 31.0 | 32.8 | 33.5 | 33.4 | 33.5 | 32.7 | 31.6
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 4.3 | 8.0 | 10.6 | 12.5 | 14.1 | 14.8 | 15.1 | 15.4 | 15.9 | 15.5
|  | 5% | 7.3 | 12.3 | 15.2 | 17.6 | 19.5 | 20.8 | 20.9 | 20.8 | 21.1 | 20.9
|  | 10% | 9.6 | 15.1 | 18.9 | 21.2 | 22.9 | 24.1 | 24.5 | 24.7 | 24.7 | 24.2
1000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 9.5 | 13.8 | 19.9 | 24.0 | 27.4 | 29.8 | 31.2 | 31.4 | 31.6 | 31.0
|  | 5% | 19.1 | 27.0 | 34.1 | 39.0 | 43.5 | 46.2 | 47.0 | 47.3 | 47.6 | 46.9
|  | 10% | 27.2 | 34.1 | 43.0 | 49.1 | 53.3 | 55.6 | 56.5 | 56.8 | 56.8 | 56.6
| $\mathcal{F}_{n}^{(D)}$ | 1% | 10.0 | 16.9 | 23.9 | 29.9 | 33.7 | 35.9 | 37.6 | 37.7 | 37.4 | 37.1
|  | 5% | 20.0 | 30.4 | 38.7 | 44.9 | 49.4 | 52.3 | 53.2 | 53.9 | 53.9 | 53.4
|  | 10% | 27.5 | 39.0 | 48.5 | 55.5 | 59.2 | 61.1 | 62.1 | 62.2 | 62.3 | 61.8
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 5.9 | 10.3 | 13.7 | 15.6 | 17.3 | 18.3 | 18.8 | 19.2 | 19.1 | 19.0
|  | 5% | 9.7 | 15.2 | 19.4 | 22.4 | 23.6 | 25.0 | 25.7 | 25.9 | 26.0 | 25.3
|  | 10% | 12.4 | 18.7 | 23.3 | 26.1 | 27.7 | 28.8 | 29.2 | 29.7 | 29.4 | 29.1
2000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 14.2 | 27.3 | 39.8 | 49.0 | 55.8 | 60.5 | 62.9 | 64.0 | 64.6 | 65.5
|  | 5% | 26.7 | 44.4 | 58.1 | 67.1 | 73.1 | 76.3 | 78.2 | 79.2 | 79.5 | 79.5
|  | 10% | 33.0 | 54.4 | 67.4 | 75.3 | 80.9 | 83.3 | 84.7 | 85.6 | 85.9 | 85.2
| $\mathcal{F}_{n}^{(D)}$ | 1% | 18.8 | 35.1 | 50.2 | 61.4 | 68.6 | 72.5 | 74.7 | 75.5 | 76.2 | 76.3
|  | 5% | 31.5 | 52.3 | 67.7 | 77.2 | 82.5 | 85.4 | 87.2 | 87.2 | 87.4 | 87.1
|  | 10% | 41.6 | 61.8 | 76.3 | 84.2 | 88.6 | 90.4 | 91.2 | 92.0 | 92.1 | 91.6
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 06.7 | 12.9 | 16.2 | 18.2 | 19.7 | 20.4 | 21.0 | 21.1 | 21.2 | 20.6
|  | 5% | 10.4 | 18.2 | 22.4 | 24.2 | 25.3 | 26.0 | 26.7 | 26.9 | 26.9 | 26.5
|  | 10% | 13.3 | 21.9 | 26.6 | 28.5 | 29.8 | 29.9 | 30.2 | 30.5 | 30.1 | 29.7
Table 4: Power in % of tests based on $\mathcal{P}_{n}^{(D)}$,
$\mathcal{F}_{n}^{(D)}$ and $\operatorname{LB}_{n}^{(D)}$ for significance
levels $\alpha\in\\{1\%,5\%,10\%\\}$ and $D=1,\ldots,10$. Results are for
model (4) with
$\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{p}=(0.046,0.027,0.092,0.843,0.089)^{\prime}$.

Comparing our $\mathcal{P}_{n}^{(D)}$\- and $\mathcal{F}_{n}^{(D)}$-test with
the Ljung–Box test, we find that all tests are comparable in their sensitivity
to the choice of $D$. Furthermore, while our tests may have inferior size at
the 5%- and 10%-level for small samples, size tends to be better overall for
$n=2000$. E.g., for the $\operatorname{LB}_{n}^{(D)}$-test, the empirical
rejection frequencies at the 1%-level vary between 1.6% an 4.3% for $n=2000$,
while the fluctuations for the $\mathcal{F}_{n}^{(D)}$-test are much smaller
(size between 0.8% and 1.3%). The fact that size of our tests improves more
for increasing $n$ than for the Ljung–Box test is as expected, because our
tests are based on the tail copula, thus, requiring reasonably large samples.

Figure 6: From top to bottom: $Y_{t}$; $z_{t}$, $\sigma_{t}$ (black) and
$\widehat{\sigma}_{t}$ (red, dashed); $\sigma_{t}-\widehat{\sigma}_{t}$;
$\widehat{\varepsilon}_{t}$.

Now, we turn to a comparison of power by simulating from (4) with
$\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{p}$. Table 4 displays the results.
As expected, power increases for all tests the larger the sample. For almost
all sample sizes and choices of $D$, our tests are more likely than the
Ljung–Box test to signal a misspecified model. This difference in power
increases markedly in $n$, because the power increase in $n$ is rather modest
for $\operatorname{LB}_{n}^{(D)}$. Table 4 reveals larger differences in
empirical rejection frequencies across different $D$ than Table 3.
Irrespective of the sample size, the power of our tests reaches the maximum
for $D\approx 7$. Comparing $\mathcal{P}_{n}^{(D)}$ and
$\mathcal{F}_{n}^{(D)}$, we find that clear differences in power favoring
$\mathcal{F}_{n}^{(D)}$ only emerge in large samples.

Figure 7: Top: Estimates $\widehat{\rho}_{n}^{(d)}$ of the autocorrelation at
lag $d$; bottom: Estimates $\widehat{\Lambda}_{n}^{(d)}(1,1)$.

Finally, we illustrate the difference between $\operatorname{LB}_{n}^{(D)}$
and our tests now for a representative sample from (4) with
$\bm{\theta}^{\circ}=\bm{\theta}^{\circ}_{p}$ and $n=500$. Figure 6 shows the
time series $Y_{t}$. True volatility
$\sigma_{t}=\sigma_{t}(\bm{\theta}^{\circ})$ and estimated volatility
$\widehat{\sigma}_{t}$ (estimated based on the misspecified APARCH(1,1) model)
are shown in the middle panel, with the differences
$\sigma_{t}-\widehat{\sigma}_{t}$ plotted below. While volatility seems to be
captured well by the misspecified model during calm phases, this is not the
case in turbulent periods. This, in turn, induces some serial dependence only
in the extreme regions of the distribution of $\widehat{\varepsilon}_{t}$.

To see this more clearly, consider Figure 7. It shows lag $d$-estimates of the
squared residual autocorrelations, $\widehat{\rho}_{n}^{(d)}$, and the tail
copula, $\widehat{\Lambda}_{n}^{(d)}(1,1)$.777We have excluded the estimates
$\widehat{\rho}_{n}^{(d)}=\widehat{\Lambda}_{n}^{(d)}(1,1)=1$ for $d=0$ to
zoom in on the relevant autocorrelations in Figure 7. The pointwise
95%-confidence intervals are indicated by the red dashed lines in both panels.
None of the autocorrelations are significant at the 5%-level, yet the
$\widehat{\Lambda}_{n}^{(d)}$ suggest two significant lags of the tail copula.
Thus, while the linear dependence in the body of the distribution appears to
be negligible, the misspecified volatility estimates nonetheless induce some
serial extremal dependence in the filtered residuals.

###  E.2 Misspecified Innovations

$n$ | Test | $\alpha$ | $D$
---|---|---|---
|  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
500 | $\mathcal{P}_{n}^{(D)}$ | 1% | 1.4 | 2.9 | 3.2 | 3.4 | 3.6 | 3.6 | 3.5 | 3.4 | 3.5 | 3.3
|  | 5% | 5.6 | 5.7 | 6.2 | 6.4 | 6.6 | 6.7 | 6.7 | 6.5 | 6.5 | 6.4
|  | 10% | 12.0 | 9.3 | 9.7 | 9.8 | 9.8 | 10.0 | 10.0 | 9.5 | 9.3 | 9.1
| $\mathcal{F}_{n}^{(D)}$ | 1% | 2.4 | 3.2 | 3.9 | 4.1 | 4.2 | 4.2 | 4.1 | 4.1 | 4.0 | 3.8
|  | 5% | 5.8 | 6.2 | 6.3 | 6.4 | 6.6 | 6.6 | 6.7 | 6.5 | 6.4 | 6.0
|  | 10% | 9.4 | 9.7 | 9.3 | 9.3 | 9.2 | 8.7 | 8.8 | 8.4 | 8.4 | 8.0
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 3.2 | 5.2 | 5.5 | 5.8 | 5.9 | 5.6 | 5.5 | 5.7 | 5.7 | 5.3
|  | 5% | 6.0 | 8.3 | 8.6 | 9.1 | 8.8 | 8.7 | 8.4 | 8.3 | 8.5 | 7.9
|  | 10% | 8.3 | 10.7 | 11.4 | 11.3 | 11.3 | 10.7 | 10.4 | 10.4 | 10.5 | 9.9
1000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 1.8 | 1.9 | 2.3 | 2.5 | 2.7 | 2.5 | 2.6 | 2.6 | 2.5 | 2.4
|  | 5% | 6.1 | 4.8 | 5.5 | 5.4 | 5.5 | 5.5 | 5.6 | 5.4 | 5.5 | 5.4
|  | 10% | 10.9 | 9.1 | 8.5 | 8.9 | 8.5 | 8.6 | 8.7 | 8.8 | 8.6 | 8.3
| $\mathcal{F}_{n}^{(D)}$ | 1% | 1.9 | 2.4 | 2.6 | 2.7 | 2.8 | 2.8 | 2.7 | 2.7 | 2.7 | 2.6
|  | 5% | 5.0 | 5.5 | 5.6 | 5.3 | 5.1 | 4.7 | 4.8 | 4.8 | 4.9 | 4.8
|  | 10% | 9.0 | 9.0 | 8.9 | 8.3 | 7.9 | 7.2 | 7.1 | 7.1 | 7.2 | 6.8
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 2.3 | 4.4 | 5.1 | 5.2 | 4.7 | 4.5 | 4.6 | 4.7 | 5.2 | 5.1
|  | 5% | 4.2 | 7.3 | 7.7 | 7.3 | 6.3 | 6.3 | 6.7 | 6.8 | 7.3 | 7.3
|  | 10% | 5.7 | 9.3 | 9.7 | 8.8 | 8.2 | 7.8 | 8.1 | 8.4 | 9.0 | 9.1
2000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 1.1 | 1.6 | 1.7 | 1.7 | 1.7 | 1.5 | 1.6 | 1.7 | 1.7 | 1.5
|  | 5% | 4.6 | 5.2 | 5.0 | 4.7 | 4.5 | 4.4 | 4.2 | 4.1 | 4.0 | 4.0
|  | 10% | 9.7 | 9.2 | 9.1 | 8.7 | 8.3 | 7.5 | 7.6 | 7.4 | 7.2 | 7.0
| $\mathcal{F}_{n}^{(D)}$ | 1% | 1.8 | 2.0 | 2.0 | 2.0 | 2.1 | 2.0 | 1.9 | 1.9 | 1.8 | 1.8
|  | 5% | 5.9 | 5.6 | 5.3 | 4.8 | 5.0 | 4.4 | 4.2 | 4.1 | 4.0 | 3.7
|  | 10% | 10.2 | 9.4 | 9.2 | 8.2 | 7.8 | 7.3 | 6.8 | 6.3 | 6.3 | 5.9
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 1.3 | 3.6 | 4.9 | 4.6 | 4.4 | 3.8 | 3.9 | 3.8 | 4.2 | 4.1
|  | 5% | 2.8 | 5.6 | 7.3 | 6.7 | 6.3 | 5.4 | 5.8 | 5.6 | 6.2 | 6.1
|  | 10% | 4.1 | 7.2 | 8.9 | 8.5 | 7.7 | 6.7 | 7.0 | 7.0 | 7.6 | 7.5
Table 5:  Size in % of tests based on $\mathcal{P}_{n}^{(D)}$,
$\mathcal{F}_{n}^{(D)}$ and $\operatorname{LB}_{n}^{(D)}$ for significance
levels $\alpha\in\\{1\%,5\%,10\%\\}$ and $D=1,\ldots,10$. Results are for
model (5) with $(a_{1},b_{1},c_{1})^{\prime}=(-3,0,0)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,0,0)^{\prime}$.

Now, we reconsider the setup of Section 3.2 in the main paper with fixed
$k=\lfloor 0.11\cdot n^{0.99}\rfloor$ and varying $D\in\\{1,\ldots,10\\}$. We
first report size, where $(a_{1},b_{1},c_{1})^{\prime}=(-3,0,0)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,0,0)^{\prime}$ for model (5). Table 5
displays the tests’ size, together with that of the estimation effects-
corrected Ljung–Box test (Francq and Zakoïan, 2010, Theorem 8.2). Except
perhaps for small samples and $\alpha=1\%$, the size of our tests is very
good, and generally better than that of the Ljung–Box test. Moreover, the size
of our tests is also very stable across different $D$.

$n$ | Test | $\alpha$ | $D$
---|---|---|---
|  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
500 | $\mathcal{P}_{n}^{(D)}$ | 1% | 47.2 | 51.8 | 50.8 | 48.6 | 47.0 | 45.0 | 44.0 | 42.6 | 40.8 | 39.7
|  | 5% | 63.2 | 61.8 | 61.5 | 59.0 | 56.8 | 55.5 | 53.8 | 52.8 | 51.2 | 50.1
|  | 10% | 71.0 | 67.6 | 66.6 | 65.3 | 63.6 | 61.7 | 59.9 | 58.4 | 57.1 | 55.8
| $\mathcal{F}_{n}^{(D)}$ | 1% | 55.9 | 56.9 | 55.1 | 53.1 | 50.9 | 49.1 | 47.6 | 46.2 | 44.7 | 43.0
|  | 5% | 66.6 | 67.0 | 65.2 | 62.9 | 60.7 | 59.0 | 57.3 | 55.5 | 54.1 | 52.6
|  | 10% | 72.0 | 72.3 | 70.7 | 68.7 | 66.2 | 64.4 | 62.8 | 61.1 | 59.7 | 58.2
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 13.5 | 15.5 | 15.0 | 14.1 | 13.1 | 12.4 | 11.3 | 10.3 | 9.9 | 9.8
|  | 5% | 16.9 | 18.0 | 17.0 | 16.3 | 15.4 | 14.7 | 13.7 | 12.9 | 12.3 | 12.2
|  | 10% | 19.7 | 19.7 | 18.4 | 17.2 | 16.6 | 15.9 | 14.9 | 14.3 | 14.0 | 13.8
1000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 67.5 | 68.0 | 67.9 | 66.2 | 64.6 | 62.5 | 60.7 | 58.8 | 57.4 | 56.2
|  | 5% | 77.1 | 78.4 | 77.8 | 76.2 | 74.8 | 73.1 | 71.7 | 70.5 | 69.4 | 67.9
|  | 10% | 81.6 | 82.5 | 82.5 | 81.4 | 79.7 | 78.4 | 76.9 | 75.8 | 74.7 | 73.7
| $\mathcal{F}_{n}^{(D)}$ | 1% | 72.5 | 74.7 | 73.4 | 71.5 | 70.0 | 68.2 | 66.3 | 64.9 | 63.4 | 62.2
|  | 5% | 80.5 | 82.3 | 81.8 | 80.7 | 78.9 | 77.5 | 75.7 | 74.5 | 73.5 | 72.6
|  | 10% | 83.6 | 85.6 | 85.3 | 84.6 | 83.3 | 81.6 | 80.5 | 79.1 | 78.0 | 76.8
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 6.2 | 7.5 | 7.7 | 7.7 | 7.5 | 7.2 | 7.1 | 6.9 | 6.7 | 6.6
|  | 5% | 8.1 | 9.0 | 8.9 | 8.9 | 8.7 | 8.5 | 8.3 | 8.1 | 7.9 | 7.7
|  | 10% | 9.3 | 9.8 | 9.7 | 9.4 | 9.2 | 9.2 | 9.0 | 8.8 | 8.7 | 8.5
2000 | $\mathcal{P}_{n}^{(D)}$ | 1% | 84.9 | 87.8 | 88.0 | 87.1 | 86.2 | 85.5 | 84.8 | 83.8 | 82.7 | 81.9
|  | 5% | 90.6 | 92.6 | 92.4 | 92.4 | 91.7 | 90.8 | 90.1 | 89.4 | 88.9 | 88.4
|  | 10% | 91.9 | 94.4 | 94.6 | 94.3 | 93.8 | 93.5 | 92.5 | 92.2 | 91.6 | 90.8
| $\mathcal{F}_{n}^{(D)}$ | 1% | 88.9 | 90.6 | 90.6 | 90.1 | 89.4 | 88.4 | 87.9 | 87.4 | 86.9 | 86.7
|  | 5% | 92.5 | 93.9 | 94.1 | 93.9 | 93.4 | 92.6 | 92.2 | 92.0 | 91.4 | 90.9
|  | 10% | 93.9 | 95.6 | 95.6 | 95.4 | 95.3 | 94.5 | 93.9 | 93.8 | 93.5 | 93.0
| $\operatorname{LB}_{n}^{(D)}$ | 1% | 1.8 | 2.4 | 2.5 | 2.3 | 2.4 | 2.5 | 2.7 | 2.7 | 2.8 | 3.0
|  | 5% | 2.5 | 2.9 | 3.2 | 3.1 | 3.1 | 3.0 | 3.1 | 3.1 | 3.2 | 3.6
|  | 10% | 3.0 | 3.7 | 3.6 | 3.5 | 3.5 | 3.4 | 3.6 | 3.5 | 3.5 | 3.9
Table 6: Power in % of tests based on $\mathcal{P}_{n}^{(D)}$,
$\mathcal{F}_{n}^{(D)}$ and $\operatorname{LB}_{n}^{(D)}$ for significance
levels $\alpha\in\\{1\%,5\%,10\%\\}$ and $D=1,\ldots,10$. Results are for
model (5) with $(a_{1},b_{1},c_{1})^{\prime}=(-3,-6,0.6)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,-2,0.6)^{\prime}$.

Now, we turn to a comparison of power by simulating from (5) with
$(a_{1},b_{1},c_{1})^{\prime}=(-3,-6,0.6)^{\prime}$ and
$(a_{2},b_{2},c_{2})^{\prime}=(-1,-2,0.6)^{\prime}$. Table 6 displays the
results. Once more our tests are more likely than the Ljung–Box test to signal
a misspecified model for all sample sizes and choices of $D$. This difference
in power increases markedly in $n$, because the
$\operatorname{LB}_{n}^{(D)}$-test even loses power for increasing $n$. Table
6 reveals larger differences in empirical rejection frequencies across
different $D$ than Table 3. There is a tendency for power to decrease for
larger $D$. Yet, that power decrease is only small up until $D=5$. Comparing
$\mathcal{P}_{n}^{(D)}$ and $\mathcal{F}_{n}^{(D)}$, we find that there are
only minor differences favoring $\mathcal{F}_{n}^{(D)}$.

Summing up the results of Appendices E.1 and E.2, we recommend the
$\mathcal{F}_{n}^{(D)}$-test with $D=5$, due to its good size and power. By
virtue of its simplicity, we can, however, also recommend the
$\mathcal{P}_{n}^{(D)}$-based test as a viable alternative. Although the
choice $D=5$ leads to good results in both simulation setups, we recommend—as
is common practice for other tests—to also report results for other choices of
$D$.