# Pacific Lamprey Inspired Climbing

Brian Van Stratum  Kourosh Shoele  Jonathan E. Clark  Department of Mechanical
Engineering, FAMU-FSU College of Engineering, Tallahassee, FL, 32310, USA

###### Abstract

Snakes and their bio-inspired robot counterparts have demonstrated locomotion
on a wide range of terrains. However, dynamic vertical climbing is one
locomotion strategy that has received little attention in the existing snake
robotics literature. We demonstrate a new scansorial gait and robot inspired
by the locomotion of the Pacific Lamprey. This new gait allows a robot to
steer while climbing on flat, near-vertical surfaces. A reduced-order model is
developed and used to explore the relationship between body actuation and
vertical and lateral motions of the robot. Trident, the new wall climbing
lamprey-inspired robot, demonstrates dynamic climbing on flat vertical
surfaces with a peak net vertical stride displacement of 4.1 cm per step.
Actuating at 1.3 Hz, Trident attains a vertical climbing speed of 4.8 cm/s
(0.09 Bl/s) at specific resistance of 8.3. Trident can also traverse laterally
at 9 cm/s (0.17 Bl/s). Moreover, Trident is able to make 14% longer strides
than the Pacific Lamprey when climbing vertically. The computational and
experimental results demonstrate that a lamprey-inspired climbing gait coupled
with appropriate attachment is a useful climbing strategy for snake robots
climbing near vertical surfaces with limited push points.

##  I INTRODUCTION

Snakes, limbless lizards and other long slender organisms are incredibly
adaptable to various terrains. Incorporating diverse locomotion gaits or
modes, they are able to move on or through media as diverse as sand, water,
trees, and even aerial environments [1, 2, 3, 4]. Starting with Hirose’s
Active Cord Mechanisms, roboticists have utilized inspiration from snakes in
their designs for almost 50 years[5]. The snake’s mobility in various terrain
and their long slender body morphology have interested researchers in
designing bio-inspired robots for challenging applications in search and
rescue, surveillance, and exploration operations [6, 7]. These snake-like
robots can navigate a diverse range of environments such as sandy slopes [8]
was well as in the water [9] and vegetated areas [10]. Like their biological
counterparts, they can push against obstacles in their environment for
locomotion[11, 12]. Snake robots also exhibit amphibious multi-modal
functionality [13, 14] as well as the ability to climb cracks, ladders, and
poles. Still, they are not yet designed for climbing shear walls with
performance comparable to their legged counterparts. [15, 16, 17, 10].

Figure 1:  L. tridentata or the Pacific Lamprey climbs vertical surfaces by
compressing and extending its body. The main characteristics of their dynamic
motion are represented using a reduced-order model. Which is employed to
design and steer Trident Climber. L. tridentata have an oral disk and buccal
funnel for generating adhesion Trident Climber’s analogous feature is a
microspine array.

Legged wall-climbing robots are oftentimes designed with the help of reduced-
order dynamic models which enables such robots to reach increased speed,
agility, efficiency, and become robust to disturbances [18, 19, 20]. On the
other hand, much less focus is put on dynamic locomotion in terrestrial snake
robots. Snakes in general, and robots moving in a snake-like way, experience
low inertial forces during terrestrial locomotion [21]. As a result, their
performance is typically dependent only on the instantaneous state of the
robot, and kinematic or geometric control approaches are effective. Notable
exceptions to this generalization occur when studying locomotion in the air of
the gliding snake Chrsopelea paradisi [22, 23] and numerical exploration of
sinus lifting, sidewinding, and lateral undulation modes for snake robots
[24].

Similarly, it is common to make a quasi-static assumption to model snake
robots while climbing. Inchworm-inspired climbing locomotion utilizing
suction-based adhesion has shown to be effective for traversing cylindrical
and glass domains [25]. Another approach for climbing cylindrical obstacles is
to utilize the many degrees of freedom of snake robots to bend around a pole
forming the snake body into a helix while rolling on its longitudinal axis
[10, 26, 27]. Shift control is another snake robot control method used for
climbing which employs predetermined bending shapes designed for structured
environmental features such as ladder rungs or regularly spaced holds. These
contrived shapes propagate posteriorly along a climbing snake-like robot’s
body[5, 16]. On the other hand, nature provides an example of dynamic and
agile snake-like climbing in the Pacific Lamprey.

The Pacific Lamprey or  Lampetra tridentata, is a fish with a unique capacity
for climbing. tridentata holds on to vertical surfaces with its oral disk, and
moves its body, propagating body curvature from anterior to posterior in an
anguilliform swimming motion. Kemp, Tsuzaki and Mosser documented the
kinematics of the climbing of the Pacific Lamprey [28]. They report that
Lamprey climbing involves, “intermittent bouts of activity where compression
of the body alternates with rapid straitening resulting in vertical progress”.
This type of body actuation allows the lamprey to progressively climb up
vertical barriers such as dams, weirs, and waterfalls over wet, sharp, and
smooth surfaces. Moreover, lampreys use this same body actuation to move
laterally while climbing. A Numerical simulation by Zhu and Kemp later
investigated the climbing of the Pacific Lamprey as a continuous flexible
beam[29]. They discovered that kinematic body undulation reported by Kemp is
associated with optimal climbing efficiency.

We note that current snake robots are incapable of climbing steep, smooth
surfaces in a lamprey-like way. We hypothesize that a feedforward contraction
and expansion control policy can produce bio-inspired climbing locomotion in a
simple multi-link segmented robot. We explore this hypothesis with a new robot
designed using a reduced-order dynamic model. A special attachment mechanism
is designed to mimic the unique purchase-keeping function of L. tridentata’s
head using a microspine array. Figure 1 illustrates our design process moving
from the biological inspiration of L. tridentata to the robot Trident Climber
using the reduced-order serial linkage model.

In section II of this paper, we present our hybrid dynamic mathematical and
numerical model of Lamprey and show how insights from the model were used to
design the Trident robot. We explain our experimental method In section V. In
Section VI we present results from the experimental and numerical studies. In
section VII we apply these results and demonstrate steering a lamprey-like
climbing robot. We also demonstrate that Trident performs favorably when
compared with the animal exemplar. Section VIII finishes with some conclusions
and considers future work.

Figure 2: When lampreys are climbing a mean body compression ratio of 12% has
been measured. At this compression, the body of the lamprey shows between one
and two curvature waves (one wave is shown in the figure.) We propose to
minimally approximate a single body curvature wave with a three-piece serial
linkage. Joint angle $\gamma$ is a corresponding measure of compression for a
three-piece serial linkage (when both joint angles are held $180^{o}$ out of
phase.)

##  II Reduced Order Modeling

A computationally tractable model that captures salient aspects of the
dynamics allows insights from physics to be encoded into a robot design and
provides knowledge of animal locomotion. It has been shown that the climbing
of cockroaches and geckos can be modeled with a simplified climbing model or
template [30]. In the spirit of these models, we propose to model the
continuum body of the lamprey using serial planar linkages. As shown in fig.
2, three links are chosen because it is the minimum needed to reproduce the
curvature wave observed in climbing lamprey. Reducing this continuum system to
a small number of degrees of freedom allows a detailed study of the robot’s
performance with a limited number of actuators.

Figure 3: Mechanism diagram depicting the analytical model for L. tridentata
inspired climbing. Conventional definitions for inertial link angles $\theta$
and relative joint angle $\gamma$ are shown. Reaction forces ($F_{x}$ and
$F_{y}$) at the head are not present during the flight phase.

Figure 3 shows a schematic of the computational model used in our study. The
lamprey head is modeled as a revolute joint with no actuation torque. Body
muscle actuation is represented by torques applied at the second and third
revolute joints. The parameters of this model are tabulated in section IV. We
model lamprey climbing in two separate phases: an attached phase (stance
phase) when its oral disk is applying suction force on the surface and a
flight phase when it is moving only under the influence of gravity. In order
to model these two phases, a hybrid dynamic model is proposed.

###  II.1 Stance

The Newton-Euler method is used to derive the dynamic equations of motion and
to calculate the internal forces required for robot design and dynamic
equation transition. Equations for the $ith$ link in series are given by
Equation 1.

$\displaystyle Fx_{i}-Fx_{i+1}-\text{sgn}(\dot{x}_{i})\,C_{x}$
$\displaystyle=m\,\ddot{x_{i}}$ (1)
$\displaystyle-m\,g\,\cos(\alpha)+Fy_{i}-Fy_{i+1}-\text{sgn}(\dot{y}_{i})\,C_{y}$
$\displaystyle=m\,\ddot{y_{i}}$
$\displaystyle\frac{L}{2}\left(Fy_{i}+Fy_{i+1}-Fx_{i}-Fx_{i+1}\right)+\tau_{i-1}-\tau_{i}$
$\displaystyle=\frac{k_{isf}}{12}m\,L^{2}\,\ddot{\theta}_{i}$

Where $Fx_{i}$ and $Fy_{i}$ are the reaction forces from the revolute joints
in the horizontal and vertical directions respectively, $k_{isf}$ is a
unitless scaling factor to account for gearbox inertia. $C_{x}$ and $C_{y}$
are environmental drag forces in $x$ and $y$ directions. Reaction forces and
torques from the next distal link are absent for the last link and thus
$Fx_{i+1}$, $Fy_{i+1}$ and $\tau_{i}$ are zero for the last link. For the
stance phase the Cartesian positions ($x_{1-3}$, $y_{1-3}$) and speeds
($\dot{x}_{1-3}$, $\dot{y}_{1-3}$) of links are expressed in terms of the
generalized coordinates and speeds of the system state $\theta_{1-3}$ and
$\dot{\theta}_{1-3}$, respectively. Here link one is the head link and link
three is the posterior link. The system of nine equations can be obtained for
$\ddot{\theta}_{1-3},\,Fx_{1-3}$ and $Fy_{1-3}$ as a function of the system
states ($\theta_{1-3}$ and $\dot{\theta}_{1-3}$) and input torques
$\tau_{1-2}$. The final governing equation can be written as,

$\mathbf{M}(\bm{\theta})\begin{bmatrix}\ddot{\bm{\theta}}\\\
\mathbf{f}\end{bmatrix}=\mathbf{C}(\bm{\theta})\,\dot{\bm{\theta}}^{2}+\mathbf{G}(\bm{\theta},\dot{\bm{\theta}})+\mathbf{U}\mathcal{T}$
(2)

where

$\begin{array}[]{cccc}\bm{\theta}=\begin{bmatrix}\theta_{1}\\\ \theta_{2}\\\
\theta_{3}\end{bmatrix}&\dot{\bm{\theta}}^{2}=\begin{bmatrix}{\dot{\theta}_{1}}^{2}\\\
{\dot{\theta}_{2}}^{2}\\\ {\dot{\theta}_{3}}^{2}\\\
\end{bmatrix}&\mathbf{f}=\begin{bmatrix}Fx_{1}\\\ Fy_{1}\\\ Fx_{2}\\\
Fy_{2}\\\ Fx_{3}\\\ Fy_{3}\\\
\end{bmatrix}&\mathcal{T}=\begin{bmatrix}\tau_{1}\\\ \tau_{2}\\\
\end{bmatrix}\end{array}$

while

$Fy_{1}>0.$

Here, $\mathbf{M}$ is a $9\times 9$ matrix as a function of the robot
geometry. The matrix $\mathbf{C}$ is a $9\times 3$ is a nonlinear function of
position and accounts for the coriolis forces. The symbol
$\dot{\bm{\theta}}^{2}$ is a $3\times 1$ vector of the element-wise squares of
the time derivatives of $\bm{\theta}$, and $\mathbf{G}$ is a $9\times 1$
vector represents the effect of gravity and environmental forces acting on the
linkages. $\mathbf{U}$ is a $3\times 2$ input matrix with $-1$ on the main
diagonal and $1$ on the subdiagonal and zeros everywhere else. These equations
apply while $Fy_{1}$ is positive.

###  II.2 Flight

During the flight phase, the head is free to translate as well as rotate, and
the dynamics of the serial linkage can be described with five position states
and five velocity states instead of the previous three and three,
respectively. Similarly, four reaction forces are required instead of the
previous six. The coupled ODEs with nine equations and nine unknowns can be
expressed as:

$\mathbf{M}(\bm{\theta})\begin{bmatrix}\ddot{\bm{\theta}}\\\ \ddot{x}\\\
\ddot{y}\\\
\mathbf{f}\end{bmatrix}=\mathbf{C}(\bm{\theta})\,\dot{\bm{\theta}}^{2}+\mathbf{G}(\bm{\theta},\dot{\bm{\theta}},\dot{x},\dot{y})+\mathbf{U}\mathcal{T}$
(3)

where $\mathbf{f}=\begin{bmatrix}Fx_{2}&Fy_{2}&Fx_{3}&Fy_{3}\end{bmatrix}^{T}$

and ($x$,$y$) is the Cartesian position of the first link. These equations
apply while the head’s vertical speed is positive in the inertial reference
frame.

###  II.3 Transitions

The conditions that result in transitions between the flight and stance phases
are functions of the states. Our model assumes lift-off occurs when the $y$
component of the head reaction force becomes zero.

When this occurs, the system states at the end of the stance phase are mapped
into the initial stages of the flight phase according to the following
transform,

$\displaystyle\small\begin{pmatrix}\theta_{1fi}\\\ \dot{\theta}_{1fi}\\\
\theta_{2fi}\\\ \dot{\theta}_{2fi}\\\ \theta_{3fi}\\\ \dot{\theta}_{3fi}\\\
x\\\ \dot{x}\\\ y\\\ \dot{y}\\\ \end{pmatrix}=\begin{pmatrix}1&0&&\dots&0\\\
0&\ddots&&&\\\ \vdots&&&&\vdots\\\ &&&\ddots&0\\\ 0&\dots&&0&1\\\
0&\dots&&&0\\\ 0&-\frac{L}{2}\,\text{s}(\theta_{1})&0&\dots&0\\\
0&\dots&&&0\\\ 0&\frac{L}{2}\,\text{c}(\theta_{1})&0&\dots&0\\\
\end{pmatrix}\begin{pmatrix}\theta_{1s}\\\ \dot{\theta}_{1s}\\\ \theta_{2s}\\\
\dot{\theta}_{2s}\\\ \theta_{3s}\\\ \dot{\theta}_{3s}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\\\
\frac{L}{2}\text{c}(\theta_{1})\\\ 0\\\ \frac{L}{2}\text{s}(\theta_{1})\\\
0\\\ \end{pmatrix}$ (4)

where $c(\theta)$ and $s(\theta)$ stand for $\cos$ and $\sin$ respectively. In
Equation 4 the vector on the left-hand side is the initial conditions for the
flight phase states and the vector containing $\theta_{1-3}$ and
$\dot{\theta}_{1-3}$ on the right-hand side is the states at the end of the
stance phase.

We base our model on the passive attachment directional adhesion mechanism
resembling geckos and cockroaches which achieve good performance in other
dynamic climbing studies. Previous studies show that such a passive attachment
can be modeled by assuming the negative spine velocity defines the moment of
sticking [31]. As such, the exit condition for the flight phase is based on
the relative velocity of the microspine array at the head of the serial
linkage given by the following equation for the vertical velocity of the head:

$\dot{y}_{head}=\dot{y}+\frac{L}{2}\cos(\theta_{1}-\pi)\,\dot{\theta}_{1}>0$
(5)

In summary, climber dynamics are governed by Equation 2 while the head force
is positive; then, the stance states are mapped to the flight states by the
transformation given in Equation 4. Finally, the climber motion is governed by
Equation 3 until the vertical head velocity reduces to zero when the flight
phase ends. The resulting hybrid dynamic motion governed by the stance and
flight phases constitutes a single jump.

###  II.4 Control Policy

Hirose’s early snake robot control policies prescribed feedforward or open-
loop joint angle profiles generating snake-like motion he called serpanoid
curves [32]. Later, Hatton et al [26] developed methods for generating joint
space trajectories or gaits that generate desired locomotion shapes using
“keyframes”. Hatton’s process produced the helix shape that enables tree
climbing and other scrpted gaits. Later improvements on this approach allowed
for climbing on changing pole diameters and delicate structures [27]. In the
dynamic climbing and running community, the earliest successes were
demonstrated with a feedforward control policy[33, 34]. In this work, we first
explore a gait with a feedforward control policy of the joints angle that
mimics a lamprey’s neural ’clock’ [35]. Our control policy parameterizes
Trident Climbers’ shape changes by specifying a compression speed, extension
speed, compression magnitude, and rest time. For the three-link model employed
here, we fully specify a control policy by describing the commanded joint
speed in compression $\omega_{C}$, the commanded joint speed in extension
$\omega_{E}$, an amount of compression $\gamma_{max}$ and a rest time $t_{r}$.
This study reduces the parameter space by setting the second joint angle to
always be $180^{o}$ out of phase with the first.

Figure 4: Triangular-shaped joint actuation pulse. These relative angles are
the joint angles in Trident’s joint space.

Given the desired joint angle profile in figure 4, we use a proportional
derivative feedback control law based on the actual relative joint angles and
compute the joint actuation torques [$\tau_{c1}\,\tau_{c2}$] as

$\begin{split}\begin{pmatrix}\tau_{c1}\\\
\tau_{c2}\end{pmatrix}=&K_{p}\begin{pmatrix}\phantom{-}\gamma_{c}-\gamma_{1}\\\
-\gamma_{c}-\gamma_{2}\end{pmatrix}+K_{d}\,\begin{pmatrix}\phantom{-}\omega_{C/E}-\dot{\gamma}_{1}\\\
-\omega_{C/E}-\dot{\gamma}_{2}\end{pmatrix}\end{split}$ (6)

The calculated joint torques $\tau_{c1}$ and $\tau_{c2}$ are subjected to the
following motor model that limits the commanded joint torque to a saturation
limit given by the linear motor model,

$\tau_{max}=\left\\{\begin{array}[]{llcc}\frac{v}{v_{m}}\tau_{ST},&-\infty<\dot{\gamma}\leq
0\\\
-\frac{\tau_{ST}}{\omega_{NL}}\,\dot{\gamma}+\frac{v}{v_{m}}\tau_{ST},&0<\dot{\gamma}\leq\frac{v}{v_{m}}\omega_{NL}\\\
0,&\frac{v}{v_{m}}\omega_{NL}<\dot{\gamma}<\infty\end{array}\right.$ $\hskip
17.07164pt\tau_{min}=\left\\{\begin{array}[]{llcc}0,&-\infty<\dot{\gamma}\leq-\frac{v}{v_{m}}\omega_{NL}\\\
-\frac{\tau_{ST}}{\omega_{NL}}\,\dot{\gamma}-\frac{v}{v_{m}}\tau_{ST},&-\frac{v}{v_{m}}\omega_{NL}<\dot{\gamma}\leq
0\\\ -\frac{v}{v_{m}}\tau_{ST},&0<\dot{\gamma}<\infty\end{array}\right.$

where the parameters of the motor model are given in table 2. Other motor and
control policy parameters were discovered through the numerical optimization
in section III.

##  III Numerical Study

The hybrid dynamic system composed of equations 2 and 3 was solved using
Matlab’s ODE45 algorithm, and the mathematical transformation in equation 4
was implemented using the events parameter. This numerical study was first
used to find an optimal robot morphology and controller. The optimization
problem is stated in equation 7. Here we consider the jump height as a
function of the design vector $X$. The design Vector $X$ includes Trident
Climber’s link length $L$, joint trajectory parameters:
$\omega_{C},\,\omega_{E}\text{ and }\gamma_{max}$ and PD control gains
$K_{p}\text{ and }K_{d}$.

$\displaystyle\underset{X}{\text{minimize}}$ $\displaystyle-\Delta\,y(X)$ (7)

where

$X=\begin{bmatrix}L\\\ \gamma_{max}\\\ \omega_{C}\\\ \omega_{E}\\\ K_{p}\\\
K_{d}\\\ \end{bmatrix}$

For a given stall torque and no-load speed associated with three different
available motors, a single shooting approach utilizing Matlab’s fminsearch was
used to find a locally optimal design vector $X$. The design parameters for
three available motors as well as the predicted jump height are reported in
Table 1. The 75:1 reduction ratio produces superior performance, and the two
equivalent power motors are shown with corresponding optimized design vectors
for sensitivity comparison. These simulations were performed drag free
($C_{x}=C_{y}=0$) and thus the resulting jump heights are optimistic relative
to what will be presented in the experimental section VI.

A numerical parameter variational study was also conducted in order to
visualize the climbing performance of this jumper, we systematically explore a
wide range of the compression speed $\omega_{C}$ and the maximum actuation
joint angle $\gamma_{max}$. Other fixed parameters of this test are given in
table 2. The contour plots of vertical and lateral displacement, as well as
efficiency, are shown as contour plots in section VI. The parametric study is
done for $5^{o}$ increments of the joint angle and $70$ deg/s increments of
the compression speed, resulting in 880 data points. Each individual
simulation result is shown as a gray dot in section VI.

motor |  $L$ $[mm]$ |  $\omega_{C}$ $[\frac{deg}{s}]$ |  $\omega_{E}$ $[\frac{deg}{s}]$ |  $\gamma_{max}$ $[\frac{deg}{s}]$ |  $K_{p}$ |  $K_{d}$ |  $\Delta y$ $[mm]$
---|---|---|---|---|---|---|---
50:1 | 89 | 718 | 682 | $125^{o}$ | 1.6 | .02 | 55
75:1 | 182 | 1580 | 900 | $140^{o}$ | 1.0 | 0.1 | 122
100:1 | 141 | 1324 | 718 | $132^{o}$ | 1.1 | .05 | 93
Table 1: Design vectors that maximize jumping height for three different motor
choices. Of the motors considered, these gearing choices bracket the optimal
choice.

##  IV Robot Platform – Trident Climber

The robotic platform Trident Climber, shown in figure 5, was constructed in
order to test and validate the three-link model and demonstrate a lamprey-
inspired gait in hardware. Trident Climber is composed of three 3D-printed ABS
segments connected and driven by 2 DC gearmotors. Link lengths were chosen to
optimize the jump height as described in section III.

Figure 5: Trident Climber Hardware Diagram Table 2: Jumper Model Parameters Parameter | Symbol | Value | Units
---|---|---|---
Mass per link | m | 54 | $g$
Acceleration due to gravity | g | 9.81 | $ms^{-2}$
Climbing wall angle | $\alpha$ | $20$ | deg
Link length | $L$ | 182 | $mm$
Coulomb drag force in $Y$ per link | $C_{y}$ | 0.45 | $N$
Coulomb drag force in $X$ per link | $C_{x}$ | 0.08 | $N$
Rotational Inertial Scale Factor111$K_{isf}$ was experimentally determined by tuning model to match the experimental data | $k_{isf}$ | 2 | -
Motor voltage | $v$ | 11 | Volts
Motor spec voltage | $v_{m}$ | 6 | Volts
Motor no-load speed | $\omega_{NL}$ | 410 | RPM
Stall torque | $\tau_{ST}$ | 127 | $N\,mm$

These optimizations suggest that Pololu’s dual shaft High Power Micro Metal
Gear Motor with a 75:1 gear head (part no: 2361) and a 182 mm link length are
the best hardware choices. A hall effect magnetic incremental encoder is
attached to the motor rotor output shaft, and the output of the gearhead is
connected to an aluminum coupler fixed to the bottom of the adjacent forward
link. The combination 75:1 speed-reducing gearhead and the twelve count per
revolution encoder provide relative joint angle feedback at about 910 counts
per revolution. Both joint’s relative joint angle $\gamma$ are capable of $\pm
90^{o}$ of rotation measured from the previous link’s centerline.

Trident Climber is equipped with eight microspines [36] forming an array at
the tip of its head analogous to L. tridentata’s oral disk and buccal funnel.
Microspines are built by a shaped deposition manufacturing [37] technique of
sequential material removal and addition. In this way, a composite array of
compliant hooks is made that is highly effective at engaging surface
asperities in a climbing substrate[34].

Communications are facilitated by an Adafruit Bluefruit LE SPI Friend (part
number 2633). Trident Climber is controlled by a PJRC Teensy 3.5
microcontroller. The Teensy logs data as well as generating the joint angle
profiles, reading incremental encoders, and sending PWM control signals to the
motor drivers based on a 1kHz digital implementation of the control policy in
Equation 6.

System power is supplied by three LiPo batteries connected in series, forming
a nominal 11.5 volt bus. This voltage is regulated to five volts for the
microcontroller, encoders, and logic signal level supply and delivered
unregulated to the driving motors. Trident Climber is equipped with a Pololu
current sensor (part no. 4041) for logging motor current. Trident Climber
equipped with batteries and microspines weighs 162 gm.

##  V Experimental Procedure

Figure 6: Comparison of experimental data and simulation shape kinematics in
the shape space during one jump at compression speed of 1000 deg/s, extension
speed of 500 deg/s, and maximum relative joint angle of $90^{o}$. The first
joint angle defining the head-to-middle section angle $\gamma_{1}$ is shown in
green achieving approximately $90^{o}$ and the second distal joint angle
$\gamma_{2}$ defining the angle between the middle and tail pieces is shown
achieving approximately $-90^{o}$. Here the experimental data trace is the
average for 29 jumps.

###  V.1 Model Calibration

Figure 5 shows reflective markers for logging the head link’s position. A
Vicon motion capture system with four calibrated cameras is employed to record
the position data at 100 Hz. Preliminary testing revealed that drag plays a
significant role in the dynamics of this form of climbing at this mass scale.
In order to account for drag and spine release cost, Trident Climber was
placed on its climbing surface and a pulley system was used to add weight
incrementally to determine drag force in the $y$ direction. The value of this
drag force is reported on a per link basis in table 2 as $C_{y}$. Lateral drag
was similarly measured and reported in table 2 as $C_{x}$. Here we assume that
lateral motion in each link’s local coordinate frame, can be reasonably mapped
to the global $x$ direction and local link vertical motion is primarily in the
$y$. In table 2 it can be seen that $C_{y}$ is much greater than $C_{x}$, 0.48
vs 0.08 respectively. This disparity is due to the force required to make the
microspines release. This spine release cost is not present for motion in the
lateral direction. Moreover, the scaling factor $k_{isf}$ to account for the
gearbox reflected inertia was found to be $2.0$.

###  V.2 Hardware Experiments

In this study, we consider climbing behaviors that allow vertical and lateral
motion. We quantify the performance of the climbing by evaluating the net
vertical displacement, the net lateral displacement, and energetic efficiency.
For each experiment the robot was commanded to jump 29 times, 15 right and 14
left. Experimental data of each jump was post-processed such that the vertical
and lateral displacement was evaluated separately. The time-series data from
the Vicon and the onboard joint encoder and current log were binned such that
the mean and standard deviation of the peak jump could be determined for each
experiment. Sample experimental results of this internal shape data for a 29
jump experiment is compared to the three-link model numerical results in Fig.
6. Here we see the correspondence between the simulation shape variables
($\gamma_{1},\gamma_{2}$) and the shape variables as measured by Trident’s
incremental encoders. Fig. 6 also depicts the simulated transition from stance
to flight phases, shown with the dashed and dotted-dashed lines, respectively.
An example of global reference frame data from the Vicon system is shown in
Fig. 7. This plot gives the time histories for Trident’s binned joint angles
during one experiment.

Efficiency is computed by comparing the ratio of work done by the motors at
the output shaft to the increase in potential energy of Trident Climber. The
work done by the motor is calculated by time integrating the instantaneous
power of each motor. This is done using the measurements from the joint
encoder and motor current according to Equation 8.

$\eta=\frac{m\,g\,\Delta
y}{\int_{0}^{t_{flight}}P_{motor}\,dt}=\frac{m\,g\,\Delta
y}{k_{T}\int_{0}^{t_{flight}}\dot{\gamma}_{1}\,i_{1}+\dot{\gamma}_{2}\,i_{2}\,dt}$
(8)

where $\dot{\gamma}$ is the motor speed and $k_{T}$ is the motor torque
constant.

Figure 7: Example of post-processed Vicon Data from a single jumping
experiment at compression speed $\omega_{C}$ of 1000 deg/s and maximum
relative joint angle of $\gamma_{max}$of $60^{o}$. This data point is 8(c,f,i)
at the $60^{o}$ operating point. Data is sampled at 100 samples per second and
binned allowing the mean and standard deviation to be found for each sample
time after the beginning of the jump.

##  VI Finding parameters for peak climbing – Simulation and Experimental
Results

###  VI.1 Parameter Variation Study

The outside columns of figure 8 show comparisons of the experimental data with
error bars reflecting one standard deviation of data as in figure 7. For
instance, the data represented in figure 7 is a single data point in the cyan
trace at $\gamma_{max}=60^{o}$ in subfigure 8(c).

Figure 8: Results of 57 jumping experiments compared against simulation
showing results of varying the maximum commanded relative joint angle
$\gamma_{max}$ and the command compression speed $\omega_{C}$. From top to
bottom, the Rows of the figure represent vertical displacement, Lateral
displacement, and locomotion efficiency. Left-most column depicts the effect
of compression speed $\omega_{C}$ by holding $\gamma_{max}$ constant.
Simulation is also compared against experimental data. The middle column shows
simulated contour plots with color-coded dashed lines showing slice locations.
The rightmost column plots the effect of $\gamma_{max}$ on the performance
metrics of interest for several compression speed $\omega_{C}$ values. Gray
dots represent a simulated data point.

Fig. 8(a) shows the effect of compression speed on vertical displacement and
each data set (single color) is a horizontal cross-section of Fig. 8(b). The
jump height is positively correlated with the compression speed until around
$\omega_{c}\approx 1000\,deg/s$, when it begins to decrease with further
increase of $\omega_{c}$. Experimental observations and simulation results
both display the same location of peak jump height and a slight dependence on
the max joint angle $\gamma_{max}$. However, the simulation model
overestimates the jump height by about three millimeters.

Fig. 8(c) shows the effect of the maximum commanded joint angle on the
vertical displacement of a single jump and each curve is a vertical cross-
section of Fig. 8(b). We observe that within the joint angle constraint of
Trident Climber, vertical displacement is positively and linearly correlated
with the max joint angle. Similar to what is observed in Fig. 8(a) the model
over-predicts jump height by about three millimeters for most cases, except
for low compression speeds. Peak vertical climbing displacement of 47mm $\pm
5$ occurs for a commanded compression speed of $\approx 1000deg/s$ and
$\gamma_{max}$ of $90^{o}$.

Fig. 8(d-f) shows the lateral displacement of a single jump. Similar to
vertical jump displacement, there is a peak compression speed, $\approx 1000$
deg/s, with slight dependence on joint angle. Also, the lateral displacement
has a positive linear correlation with joint angle. Unlike vertical
displacement, the model underpredicts the lateral distance by about 10 to 15
millimeters in most cases.

Efficiency is shown in Fig. 8(g-i). In 8(h) we observe that a peak efficiency
of $14\%$ occurs at $\gamma_{max}=90^{o}$ and $\omega_{C}=1000$ deg/s. Fig.
8(i) reports efficiency based on the $\Delta y_{max}$ (see fig. 7.) As such,
this number is the same output as the Numerical study. In order to make a fair
comparison of Trident’s single jump data to continuous climbing and to other
robots we consider the net vertical displacement (see fig. 7.) Trident Climber
slips some distance, typically about 5 mm, as its microspines engage the
carpet. Consequently, the $14\%$ reported is the max efficiency that can be
attained if the spines attach immediately at the apex. As such, the net
efficiency for the peak efficiency operating point is $12\%$ or a specific
resistance of 8.3.

##  VII Locomotion Strategies

Figure 9: a.) Contour plot showing the vertical-to-lateral displacement ratio
in the control parameter space. b.) Feedforward joint angle trajectories that
cause vertical right and left locomotion. Note that Fig. 8 vertical
displacement is at peak. there is a displacement cost for reattaching on the
order five to 10 mm thus $\gamma_{max}$ on the order of 50o results in
negligible net vertical progress. Figure 10: Steering trajectory for Trident
Climber. Trident Climber steers around an obstacle by first applying 10
lateral left actuation pulses as depicted in Fig. 9(b) Lateral Right with a
$\gamma_{max}$ of 50o and compression speed $\omega_{C}$ of -2000 deg/s.
Similarly, vertical climbing is achieved by actuating as in Fig. 9(b) Vertical
with 31 alternating sign compressions with $\gamma_{max}$ of 90o and
$\omega_{C}$ of 1000 deg/s. Finally, Trident Climber traverses left with 10
pulses the same as vertical right but with positive compressions resulting in
leftward jumps. Note that positive compression speeds result in the top two
links convex left and result in left jumps; vice versa for negative
compression speeds.

###  VII.1 Steering

After determining the control parameter space that relates to climbing
performance, we consider steering in the context of vertical climbing. Figure
9 is generated by computing the vertical-to-lateral displacement ratio for the
parameter variational study. This plot reveals how the actuation parameters
($\gamma_{max}$, $\omega_{C}$) affect steering on Trident. The actuation in
the stared regimes can be used to steer Trident Climber, resulting in net
motion that is either vertical or mostly lateral. The yellow contour labeled
“Vertical Regime” represents strides that have the largest vertical component.
So if vertical progress is desired, and high lateral deviation is acceptable,
the “Vertical Regime” is the optimal operating point. This vertical jumping
regime is characterized by a large joint angle $90^{o}$ and the commanded
compression speed is appropriately matched to the characteristics of the motor
($\approx 1000$ deg/s for our motor). To achieve mostly net vertical
locomotion the actuation ($\omega_{C}$) must alternate sign since consecutive
steps alternate displacement in the lateral direction. As such, two jump steps
form a stride with a $y$ displacement of two times the vertical step
displacements reported in Fig. 8.

If, on the other hand, the objective is to achieve lateral motion, then
Trident can be actuated in the region labeled as “Lateral Regime Left/Right.”
Here the jumping is characterized by a low joint angle and a compression speed
slightly faster than what is optimal for vertical climbing. For the marked
case, the compression speed is $\approx 2000$ deg/s. Considering that the
entire blue contour with a ratio between 0.3 and 0.4 has a similar
displacement ratio, priority is given to the higher compression speed since it
results in faster cycle speeds. However, considerations other than speed might
lead one to choose the regime on the left side of the blue contour. As shown
in Fig. 8(c), the peak vertical displacement for this particular case
$(\omega_{C}=2000\,deg/s\,,\,\gamma_{max}=50^{o})$ is about 10 mm, accounting
for spine engagement displacement cost results in close to zero net lateral
displacement and 25 mm lateral jumps.

Another key result is that the body shape in the stance phase predicts the
jump direction. By the convention in Fig. 3, positive compression speeds on
the first motor $\dot{\gamma}_{1}$ result in shapes where the first two links
are deflected with their convex sides pointing left and as a result, the net
lateral motion is to the left. For example, the jumper in Fig. 3 is poised to
jump left. Conversely, negative compression speeds, result in rightward
lateral jumps.

By incorporating insights about locomotion direction and the ratio of vertical
to lateral displacement into our feedforward control scheme, we were able to
pick a proper pulse profile, similar to the one depicted in Fig. 9(b) to steer
Trident Climber. Note that vertical locomotion utilizes a pulse with
compression speed of alternating sign. The lateral motion utilizes small, fast
pulses with a compression speed corresponding to the desired lateral
displacement Fig. 10 shows Trident Climber demonstrating the utility of these
actuation pulses for steering around an obstacle.

###  VII.2 Mechanism of Performance

The principle of impulse and momentum explains the positive correlation
between max joint angle, vertical jump, and lateral motion. Impulse and
momentum theory states that the change in the vertical velocity of Trident is
proportional to the integral of the vertical head reaction force with respect
to time.

$\Delta\,\vec{V}\propto\int\vec{F}(t)\,dt$ (9)

Athletic jumps thoroughly saturate the motor and limit the head reaction
force. Considering equation 9 and noting that though the value of $\vec{F}(t)$
is invariant due to saturation at any $t$, increasing the duration of the
force profile will still increase the impulse and the vertical velocity.
Conversely, When commanded speeds are outside the motor’s limit, the head
reaction force has a much shorter time interval to develop a vertical impulse.
For this reason, peak performance observed in terms of height, efficiency, and
to a lesser degree, lateral travel is due to a good matching of command pulse
to motor power output and link length. Furthermore if maximizing vertical step
displacement is the primary goal, consideration should be given to maximizing
the range of joint motion.

The experimental results and the model predictions agree well except for at
low compression speeds. The poor correlation between the experiments and the
model in this region is presumably because the energetic cost of detachment is
significant compared to the kinetic energy gained during the stance phase;
therefore, the revolute joint assumption used to model the head attachment
does not represent the dynamics of these cases.

Generally speaking, the stance-to-flight transition occurs when commanded
compression ends and commanded extension begins. In other words, the stance
phase is roughly equivalent to when the body is compressing and a flight phase
begins when the actual max joint angle is reached and the extension phase
begins.

###  VII.3 Comparison to Biological Exemplar

Kemp and Mosser documented that L. tridentata climbs with a mean stride length
of 3.59 cm [28]. Trident Climber achieves a step height of $4.1\pm 0.25$ cm in
the vertical climbing regime, which is about $14\%$ better than the mean cycle
step height published for L. tridentata.

Considering Fig. 2, and applying the law of cosines to the triangles generated
by shortening the overall length of the three links we can write down the
relationship between sinuosity $\Omega$ and $\gamma_{max}$ given as

$\gamma_{max}(\Omega)=\pi-\arccos\left\\{\frac{5}{4}-\frac{9\Omega^{2}}{4}\right\\}$
(10)

Kemp et al publish that L. tridentata climb with a sinuosity of 88%. Equation
10 gives $\gamma_{max}(0.88)\approx 60^{o}$ suggesting that Trident Climber’s
compression is comparable to observed compression in  L. tridentata.
Previously, Zhu and Mosser suggested that lampreys climb in such a way as to
optimize their climbing efficiency, and as can be seen in Fig. 8(h),
$\gamma_{max}=60^{o}$ is indeed close to an efficiency peak for Trident
Climber.

When we compare our compression and extension speeds of Trident Climber with
previously published data from L. tridentata, Kemp et al [28] show the
compression time of L. tridentata is about 160 ms and this range is similar
when looking at other videos of Lampreys climbing. Keeping the Froude number
constant and scaling to Trident Climbers length scale, this corresponds to 450
deg/s compression speed [38]. Figure 8(h) shows the efficiency of this
location ($\gamma_{max}=60^{o},\omega_{C}=450\,deg/s$) is approximately 10%.
The efficiency of this operating point is within 25% of the model’s peak. This
minor discrepancy is perhaps due to L. tridentata being force limited in its
stance phase, muscles or oral disc holding power, or attachment dynamics.

We could not compare the lateral displacement performance between Trident
Climber and its biological counterpart as, to our knowledge, no prior work has
quantified the lateral motion by L. tridentata while climbing.

##  VIII Conclusion and future work

We demonstrate a lamprey-inspired climbing gait using a simple control policy
for climbing near-vertical surfaces both in simulation and in hardware. We
envision that this attachment and body actuation profile will allow other
snake robots to climb on certain steep terrain with dynamically similar
performance.

We use a Newton-Euler model for a three-link pendulum to design a lamprey-
inspired robot and validate the model with experimental data. We find that the
resulting motion gives qualitatively similar motion to the animal exemplar L.
tridentata and produces an efficiency peak performance for body actuation
geometry very close to the animal and previous continuum models. These
numerical and experimental findings show that a three-link pendulum can
function as a reduced-order model for lamprey-inspired climbing.

Many bio-inspired robots fail to perform as fast and agile as their animal
counterparts. Trident climber achieved peak 4.7 cm strides for single jump
tests and continual climbing speeds of 4.8 cm/s

Both the single jump and continual climbing performance of Trident exceeds the
mean performance of the animal exemplar climbing at similar frequencies even
though Trident is approximately $75\%$ of the mean length of animals studied
[28].

Trident achieves a specific resistance of 8 whereas other dynamic climbing
robots have achieved as low as 5 and better [20]. Such a reduction in
efficiency tends to make lamprey-inspired locomotion less compelling for only
a climbing task. However, given that this 60% reduction in efficiency comes
with a long slender body morphology with the various strengths of its body’s
small cross-sectional area and is known to be able to swim [39], a snake robot
morphology should be given strong consideration for a multi-modal robot that
must swim and climb. This inclination to long slender morphology is bolstered
by noting that the joint actuation is essentially the same control policy
applied in a pulsed way for climbing and a continuous way for swimming.

In terms of future work, Trident requires a head design that moves in the
dorso-ventral plane to facilitate attachment when climbing wall angles beyond
70o. Additionally, a suction-based design could allow for climbing on smooth
surfaces such as ship hulls. Moreover, modeling and experimentation show that
the stride length of lamprey-inspired climbing is positively correlated up to
a $90^{o}$ joint angle. The simulations indicate that this trend continues up
to $\gamma_{max}=140^{o}$; therefore, future lamprey-inspired climbers should
consider ways to achieve more flexibility for better performance. The simple
control policy employed in this study demonstrates the stability of this
locomotion, but utilizing trajectory optimization would allow for faster, more
efficient gaits. Based on this modeling, we hypothesize that Lampreys
coordinate the buccal funnel such that it releases at its body’s max
compression and reactivates at the apex of head flight. Some experiments with
L. tridentata with synchronized video and force plate data are needed to
confirm this hypothesis.

## Acknowledgment

This work was funded by the National Science Foundation Expanding Frontiers in
Research and Innovation Program, grant number 1935278. Special thanks to Ann
Grote of the US Fish and Wildlife Service for the lamprey footage included in
the online Media. Thanks to Max Austin for helpful conversations regarding
hardware as well as matters regarding bioinspiration. Thanks to Hamza Asif and
Ash Chase for their help with climbing experiments.

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# The mystery of Carleson frames

Ole Christensen, Marzieh Hasannasab, Friedrich Philipp, Diana Stoeva

###### Abstract

In 2016 Aldroubi et al. constructed the first class of frames having the form
$\\{T^{k}\varphi\\}_{k=0}^{\infty}$ for a bounded linear operator on the
underlying Hilbert space. In this paper we show that a subclass of these
frames has a number of additional remarkable features that have not been
identified for any other frames in the literature. Most importantly, the
subfamily obtained by selecting each $N$th element from the frame is itself a
frame, regardless of the choice of $N\in\mathbb{N}.$ Furthermore, the frame
property is kept upon removal of an arbitrarily finite number of elements.

##  1 Introduction

Let $\mathbb{D}$ denote the open unit disc in the complex plane. Recall that a
sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ is said to
satisfy the Carleson condition if

$\displaystyle\displaystyle\inf_{n\in\mathbb{N}}\prod_{k\neq
n}\frac{|\lambda_{k}-\lambda_{n}|}{|1-\overline{\lambda_{k}}\,\lambda_{n}|}>0.$
(1.1)

In [1] Aldroubi et al. proved the following result:

###### Theorem 1.1

Assume that $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ satisfies the
Carleson condition, and let $\\{e_{k}\\}_{k=1}^{\infty}$ denote an orthonormal
basis for a separable Hilbert space ${\cal H}.$ Consider the bounded linear
operator $T:{\cal H}\to{\cal H}$ given by
$Te_{k}:=\lambda_{k}e_{k},\,k\in\mathbb{N},$ and let
$\\{m_{k}\\}_{k=1}^{\infty}$ be any scalar sequence such that
$0<C_{1}\leq|m_{k}|\leq C_{2}<\infty.$ Then, with

$\displaystyle\varphi:=\sum_{k=1}^{\infty}m_{k}\sqrt{1-|\lambda_{k}|^{2}}\,e_{k},$
(1.2)

the family $\\{T^{k}\varphi\\}_{k=0~{}}^{\infty}$ is a frame for ${\cal H}.$

For the sake of easy reference we will call the frames in Theorem 1.1 for
Carleson frames. The construction in Theorem 1.1 provides the first examples
of frames that are given as the orbit of a bounded operator. Except for the
Riesz bases (see [6]) this class still forms the only explicitly known example
of such frames. The purpose of the current paper is to show that under a weak
extra condition, such frames have a number of important features that have not
been identified for any other frames in the literature. Most importantly, the
subsequence of $\\{T^{k}\varphi\\}_{k=0~{}}^{\infty}$ formed by selecting each
$N$th element of the sequence is itself a frame, regardless of the choice of
$N\in\mathbb{N}.$

Our frame results are based on a number of manipulations on the Carleson
condition. For this reason we first collect what is known about the Carleson
condition in the literature.

###### Proposition 1.2

Given a sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ of
distinct numbers, the following hold:

  * (i)

[9] If

$\displaystyle\exists\,c\in(0,1)\mbox{ such that
}\frac{1-|\lambda_{k+1}|}{1-|\lambda_{k}|}\leq c<1,\quad\forall
k\in\mathbb{N},$ (1.3)

then $\\{\lambda_{k}\\}_{k=1}^{\infty}$ satisfies the Carleson condition.

  * (ii)

[9] If $\\{\lambda_{k}\\}_{k=1}^{\infty}$ is positive and increasing, then the
condition (1.3) is also necessary for $\\{\lambda_{k}\\}_{k=1}^{\infty}$ to
satisfy the Carleson condition.

  * (iii)

[10] If $\\{\lambda_{k}\\}_{k=1}^{\infty}$ satisfies the Carleson condition,
then $\lim_{k\to\infty}|\lambda_{k}|=1.$

  * (iv)

[3] If there is some $n\in\mathbb{N}$ such that $\\{\lambda_{k}\\}_{k\geq n}$
satisfies the Carleson condition, then also $\\{\lambda_{k}\\}_{k=1}^{\infty}$
satisfies the Carleson condition.

##  2 Frame subsequences of the Carleson frames

In this section we will consider a subclass of the Carleson frames in Theorem
1.1; more precisely, we will add a weak assumption on the sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}.$ The resulting class of
Carleson frames have the remarkable properties mentioned in the abstract.

###### Theorem 2.1

Let $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ and assume that
$\\{|\lambda_{k}|\\}_{k=1}^{\infty}$ is a strictly increasing sequence which
satisfies the Carleson condition. Let $\\{e_{k}\\}_{k=1}^{\infty}$ denote an
orthonormal basis for a separable Hilbert space ${\cal H},$ consider the
bounded linear operator $T:{\cal H}\to{\cal H}$ given by
$Te_{k}:=\lambda_{k}e_{k},\,k\in\mathbb{N},$ and let
$\\{m_{k}\\}_{k=1}^{\infty}$ be any scalar sequence such that
$0<C_{1}\leq|m_{k}|\leq C_{2}<\infty.$ Then, with

$\displaystyle\varphi:=\sum_{k=1}^{\infty}m_{k}\sqrt{1-|\lambda_{k}|^{2}}\,e_{k},$
(2.1)

the following hold:

  * (i)

The family $\\{T^{Nk}\varphi\\}_{k=0}^{\infty}$ is a frame for ${\cal H}$ for
any $N\in\mathbb{N}.$

  * (ii)

If $\lambda_{1}\neq 0,$ the family $\\{T^{Nk}\varphi\\}_{k=K}^{\infty}$ is a
frame for ${\cal H}$ for any $N,K\in\mathbb{N}.$

  * (iii)

If $\lambda_{1}\neq 0,$ the family $\\{T^{j+Nk}\varphi\\}_{k=0}^{\infty}$ is a
frame for ${\cal H}$ for any $N\in\mathbb{N}$ and any
$j\in\\{0,1,\dots,N-1\\}.$

Proof. As the first step of the proof we will show that the assumptions imply
that $\\{\lambda_{k}^{N}\\}_{k=1}^{\infty}$ satisfies the Carleson condition
for all $N\in\mathbb{N}.$ First, let $N=1$. By assumption the sequence
$\\{|\lambda_{k}|\\}_{k=1}^{\infty}$ consists of distinct numbers, so
Proposition 1.2 (ii) implies that (1.3) holds. Using now Proposition 1.2 (i)
on the sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}$, it follows that
$\\{\lambda_{k}\\}_{k=1}^{\infty}$ also satisfies the Carleson condition, as
claimed. Now, fix an arbitrary $N\in\mathbb{N}$, $N>1$. Then

$\displaystyle\frac{1-|\lambda_{k+1}^{N}|}{1-|\lambda_{k}^{N}|}=\frac{1-|\lambda_{k+1}|}{1-|\lambda_{k}|}\cdot\frac{1+|\lambda_{k+1}|+\ldots+|\lambda_{k+1}|^{N-1}}{1+|\lambda_{k}|+\ldots+|\lambda_{k}|^{N-1}}.$

As in the first part of the proof, take $c$ which satisfies (1.3) and choose
any $\varepsilon\in(0,\frac{1-c}{c}).$ Using Proposition 1.2 (iii), we see
that

$\lim_{k\to\infty}\frac{1+|\lambda_{k+1}|+\ldots+|\lambda_{k+1}|^{N-1}}{1+|\lambda_{k}|+\ldots+|\lambda_{k}|^{N-1}}=1$

and therefore there is $K_{0}\in\mathbb{N}$ so that

$\frac{1+|\lambda_{k+1}|+\ldots+|\lambda_{k+1}|^{N-1}}{1+|\lambda_{k}|+\ldots+|\lambda_{k}|^{N-1}}<1+\varepsilon\
\mbox{ for }k>K_{0}.$

Then for $k>K_{0}$ we have that

$\frac{1-|\lambda_{k+1}|^{N}}{1-|\lambda_{k}|^{N}}<c(1+\varepsilon)<1.$

By Proposition 1.2 (i), the sequence
$\\{\lambda_{k}^{N}\\}_{k=K_{0}+1}^{\infty}$ satisfies the Carleson condition
and hence $\\{\lambda_{k}^{N}\\}_{k=1}^{\infty}$ satisfies the Carleson
condition by Proposition 1.2 (iv).

By Theorem 1.1 we know that for the function $\varphi$ in (2.1),
$\\{T^{k}\varphi\\}_{k=0}^{\infty}$ is a frame for ${\cal H}.$ Now consider
$N\in\mathbb{N},\,N\geq 2.$ Then the bounded operator $T^{N}$ is given by
$T^{N}e_{k}=\lambda_{k}^{N}e_{k},\,k\in\mathbb{N}.$ Since
$\\{\lambda_{k}^{N}\\}_{k=1}^{\infty}$ satisfies the Carleson condition,
Theorem 1.1 shows that $\\{T^{Nk}\widetilde{\varphi}\\}_{k=0}^{\infty}$ is a
frame for ${\cal H}$ whenever
$\widetilde{\varphi}=\sum_{k=1}^{\infty}\widetilde{m_{k}}\sqrt{1-|\lambda_{k}^{N}|^{2}}e_{k}$
for some coefficients $\widetilde{m_{k}}$ that are bounded below and above. We
aim at taking

$\displaystyle\widetilde{m_{k}}=m_{k}\sqrt{\frac{1-|\lambda_{k}|^{2}}{1-|\lambda_{k}^{N}|^{2}}};$

note that this choice actually implies that $\widetilde{\varphi}=\varphi.$
Now,

$\displaystyle\widetilde{m_{k}}=m_{k}\sqrt{\frac{1-|\lambda_{k}|^{2}}{1-|\lambda_{k}^{N}|^{2}}}$
$\displaystyle=$ $\displaystyle
m_{k}\sqrt{\frac{1-|\lambda_{k}|}{1-|\lambda_{k}|}\cdot\frac{1+|\lambda_{k}|}{1+|\lambda_{k}|+\cdots+|\lambda_{k}|^{2N-1}}}$
$\displaystyle=$ $\displaystyle
m_{k}\sqrt{\frac{1+|\lambda_{k}|}{1+|\lambda_{k}|+\cdots+|\lambda_{k}|^{2N-1}}}.$

Using that $0\leq|\lambda_{k}|\leq 1$ and that $C_{1}\leq|m_{k}|\leq C_{2},$
we see that $C_{1}(2N)^{-1/2}\leq|\widetilde{m_{k}}|\leq C_{2}.$ Thus
$\widetilde{m_{k}}$ is indeed bounded below and above, i.e., we have proved
that $\\{T^{Nk}\varphi\\}_{k=0}^{\infty}$ is a frame as stated in (i).

In order to prove (ii), fix any $N,K\in\mathbb{N},$ and note that

$\displaystyle\\{T^{Nk}\varphi\\}_{k=K}^{\infty}=\\{T^{NK}T^{Nk}\varphi\\}_{k=0}^{\infty}.$

Since $1\geq|\lambda_{k}|\geq|\lambda_{1}|>0$ for all $k\in\mathbb{N},$ the
bounded operator $T$ is surjective, and hence $T^{NK}$ is bounded and
surjective; thus the result in (ii) follows immediately from (i) and the well-
known fact that the image of a frame under a bounded surjective operator
itself forms a frame. The result in (iii) follows from the same argument
applied to the operator $T^{j}.$ $\square$

###### Example 2.2

As a concrete example of a sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}$
satisfying all the assumptions in Theorem 2.1 , one can take
$\\{\lambda_{k}\\}_{k=1}^{\infty}=\\{1-\alpha^{-k}\\}_{k=1}^{\infty}$, for any
$\alpha>1.$ $\square$

Note that the assumptions in Theorem 2.1 are on the absolute values
$|\lambda_{k}|.$ The following example shows that
$\\{\lambda_{k}\\}_{k=1}^{\infty}$ satisfying the Carleson condition does not
imply that $\\{\lambda_{k}^{N}\\}_{k=1}^{\infty}$ satisfies the Carleson
condition for $N>1.$

###### Example 2.3

Consider a sequence $\\{\mu_{k}\\}_{k=1}^{\infty}$ that satisfies the Carleson
condition. Take a number $q\in(0,1)$ such that $\mu_{k}\neq q$ and
$\mu_{k}\neq-q$ for every $k\in\mathbb{N}$. By Proposition 1.2 (iv), the
sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}:=\\{q,-q,\mu_{1},\mu_{2},\mu_{3},\ldots\\}$
also satisfies the Carleson condition, but the sequence
$\\{\lambda_{k}^{2}\\}_{k=1}^{\infty}$ does not.

Theorem 2.1 shows that for any $N\in\mathbb{N},$ the considered class of
Carleson frames can be considered as the union of $N$ frames, namely,

$\displaystyle\\{T^{k}\varphi\\}_{k=0}^{\infty}=\bigcup_{j=0}^{N-1}\\{T^{Nk+j}\varphi\\}_{k=0}^{\infty}.$
(2.2)

We will now show if we assume that
$\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$ consist of positive
scalars, then “one can weave the elements from these frames from a
sufficiently large index” (see Theorem 2.5 for a precise statement). For this
purpose we need the subsequent lemma.

###### Lemma 2.4

Let $\\{T^{k}\varphi\\}_{k=0}^{\infty}$ be a Carleson frame as in Theorem 1.1,
for which the sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$
consists of positive and strictly increasing scalars, and fix any
$N\in\mathbb{N}.$ Then, for any choice of
$j_{k}\in\\{0,1,\dots,N-1\\},\,k\in\mathbb{N}_{0},$

$\displaystyle\sum_{k=0}^{\infty}||T^{Nk}\varphi-T^{Nk+j_{k}}\varphi||^{2}<\infty.$
(2.3)

Proof. By direct calculation, and using the upper bound on the sequence
$\\{m_{k}\\}_{k=0}^{\infty}$ in (1.2)

$\displaystyle\sum_{k=0}^{\infty}||T^{Nk}\varphi-T^{Nk+j_{k}}\varphi||^{2}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{n=1}^{\infty}\left|m_{n}\lambda_{n}^{Nk}\sqrt{1-\lambda_{n}^{2}}-m_{n}\lambda_{n}^{Nk+j_{k}}\sqrt{1-\lambda_{n}^{2}}\right|^{2}$
$\displaystyle\leq C_{2}^{2}$
$\displaystyle\sum_{n=1}^{\infty}(1-\lambda_{n}^{2})\sum_{k=0}^{\infty}\left(\lambda_{n}^{2N}\right)^{k}\left(1-\lambda_{n}^{j_{k}}\right)^{2}.$

Using that $0<1-\lambda_{n}^{j_{k}}\leq 1-\lambda_{n}^{N}\leq
1-\lambda_{n}^{2N},$ this implies that

$\displaystyle\sum_{k=0}^{\infty}||T^{kN}\varphi-T^{kN+j_{k}}\varphi||^{2}$
$\displaystyle\leq$ $\displaystyle
C_{2}^{2}\sum_{n=1}^{\infty}(1-\lambda_{n}^{2})(1-\lambda_{n}^{N})^{2}\frac{1}{1-\lambda_{n}^{2N}}$
$\displaystyle\leq$ $\displaystyle
C_{2}^{2}\sum_{n=1}^{\infty}(1-\lambda_{n}^{2})\leq\frac{C_{2}^{2}}{C_{1}^{2}}||\varphi||^{2}<\infty,$

as desired. $\square$

###### Theorem 2.5

Let $\\{T^{k}\varphi\\}_{k=0}^{\infty}$ be a Carleson frame as in Theorem 1.1,
for which the sequence $\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{D}$
consists of positive and strictly increasing scalars, and fix any
$N\in\mathbb{N}.$ Then, for any choice of
$j_{k}\in\\{0,1,\dots,N-1\\},\,k\in\mathbb{N}_{0},$ there exists
$J\in\mathbb{N}_{0}$ such that the set

$\displaystyle\\{T^{Nk}\varphi\\}_{k=0}^{J-1}\cup\\{T^{Nk+j_{k}}\varphi\\}_{k=J}^{\infty}$
(2.4)

is a frame.

Proof. Fix $N\in\mathbb{N};$ by Theorem 2.1 we know that
$\\{T^{Nk}\varphi\\}_{k=0}^{\infty}$ is a frame for ${\cal H}.$ Let $A$ denote
a lower frame bound, and fix any choice of $j_{k}\in\\{0,1,\dots,J-1\\};$
then, by Lemma 2.4, we can choose $J\in\mathbb{N}_{0}$ such that

$\displaystyle\sum_{k=J}^{\infty}||T^{Nk}\varphi-T^{Nk+j_{k}}\varphi||^{2}<A.$

By standard perturbation results in frame theory (see, e.g., page 565 in [4]),
this implies that in the frame $\\{T^{Nk}\varphi\\}_{k=0}^{\infty}$ we can
replace the elements $\\{T^{Nk}\varphi\\}_{k=J}^{\infty}$ by
$\\{T^{Nk+j_{k}}\varphi\\}_{k=J}^{\infty}$ while the frame property is kept,
as desired. $\square$

Formulated in the terminology of weaving frames (see [2]), Theorem 2.5 says
that if we fix a sufficiently high number $J$ of the first elements of the
frame $\\{T^{Nk}\varphi\\}_{k=0}^{\infty},$ we can weave the “tail frames”
$\\{T^{Nk+j}\\}_{k=J+1}^{\infty},\,j=0,\dots,N-1,$ and keep the frame
property.

We conjecture that the very particular properties of the frames in Theorem 2.1
imply that indeed the frames $\\{T^{Nk+j}\\}_{k=0}^{\infty},\,j=0,\dots,N-1,$
are weaving for any $N=2,3,....;$ that is, we conjecture that Theorem 2.5
holds with $J=0.$ Note that for $N=2,$ this would mean that the frames
$\\{T^{2k}\varphi\\}_{k=0}^{\infty}$ and $\\{TT^{2k}\varphi\\}_{k=0}^{\infty}$
are weaving. Conditions for this to hold are provided in [2]; however, they
are formulated in terms of the frame bounds, and the available frame bound
estimates (see [7]) are not strong enough to give the desired conclusion.

The “subsampling property” in Theorem 2.1 (i) is a very special property for
the considered frames, and has not been identified for any other frame in the
literature. Let us illustrate this by showing that it can not hold for a
number of important classes of frames.

###### Example 2.6

For $a,b>0$ and $g\in L^{2}(\mathbb{R}),$ let $E_{mb}T_{na}g(x):=e^{2\pi
imbx}g(x-na),$ and assume that $\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is a
Gabor frame for $L^{2}(\mathbb{R}).$ Since a necessary condition for such a
family to be a frame is that $ab\leq 1,$ it is clear that “subsamples”
$\\{E_{mb}T_{nNa}g\\}_{m,n\in\mathbb{Z}}$ or
$\\{E_{mNb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ never can be frames for arbitrary
large values of $N\in\mathbb{N}.$ $\square$

###### Example 2.7

For $a>1,b>0,$ and $\psi\in L^{2}(\mathbb{R}),$ let
$D_{a^{j}}T_{kb}\psi(x):=a^{j/2}\psi(a^{j}x-kb),$ and consider the wavelet
system $\\{D_{a^{j}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}.$ In contrast to the
Gabor case, there exist wavelet frames for arbitrary choices of the parameters
$a>1,b>0,$ see [8]. That is, assuming that
$\\{D_{a^{j}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ is a frame in general does not
exclude that $\\{D_{a^{Nj}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ can be a frame,
even for large values of $N\in\mathbb{N}.$ However, for the frame
constructions in the literature, there is a connection between the choice of
parameters $a,b$ and the corresponding function $\psi,$ i.e., changing the
parameters in a wavelet frame will in general also force a change in the
generating function (see [5] for an illustration of this).

For the case of a band limited frame
$\\{D_{a^{j}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ we can exclude the possibility
that $\\{D_{a^{Nj}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ is a frame for large
values of $N\in\mathbb{N}.$ To be precise, assume that
$\\{D_{a^{j}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ is a frame with bounds $A,B;$
then it is well-known that

$\displaystyle
bA\leq\sum_{j\in\mathbb{Z}}|\widehat{\psi}(a^{j}\gamma)|^{2}\leq
bB,\,a.e.\,\gamma\in\mathbb{R}.$ (2.5)

Assume now that $\text{supp}\,\widehat{\psi}\subset[-D,-C]\cup[C,D]$ for some
$C,D>0.$ Then, considering $\gamma\in[C/4,C/2],$ it is clear that for
sufficiently large values of $N\in\mathbb{N},$ we have
$\sum_{j\in\mathbb{Z}}|\widehat{\psi}(a^{Nj}\gamma)|^{2}=0,$ i.e.,
$\\{D_{a^{Nj}}T_{kb}\psi\\}_{j,k\in\mathbb{Z}}$ can not be a frame. $\square$

To illustrate the issue further, we finally prove that despite the fact that
“arbitrarily sparse” subfamilies of the Carleson frames in Theorem 2.1 still
form frames, no frame can have the property that arbitrary infinite
subfamilies keep the frame property.

###### Proposition 2.8

No frame $\\{f_{k}\\}_{k=1}^{\infty}$ for an infinite-dimensional Hilbert
space ${\cal H}$ has the property that every infinite subsequence
$\\{f_{k}\\}_{k\in I}$ is a frame for ${\cal H}.$

Proof. Let $\\{f_{k}\\}_{k=1}^{\infty}$ denote an arbitrary frame for ${\cal
H},$ and let $\\{e_{j}\\}_{j=1}^{\infty}$ denote an orthonormal basis for
${\cal H}.$ We will inductively construct an infinite family
$\\{g_{\ell}\\}_{\ell=1}^{\infty}\subset\\{f_{k}\\}_{k=1}^{\infty}$ which is
not a frame. First, since $\\{f_{k}\\}_{k=1}^{\infty}$ is a Bessel sequence,
choose $N_{1}\in\mathbb{N}$ such that

$\displaystyle\sum_{k=N_{1}}^{\infty}|\langle e_{1},f_{k}\rangle|^{2}\leq 1,$

and let $g_{1}:=f_{N_{1}}.$ Thus, the lower frame bound (if it exists) for
$\\{f_{k}\\}_{k=N_{1}}^{\infty}$ is at most $1.$ Now, choose $j_{1}>1$ such
that $|\langle e_{j_{1}},f_{N_{1}}\rangle|^{2}\leq 2^{-2},$ choose
$N_{2}>N_{1}$ such that

$\displaystyle\sum_{k=N_{2}}^{\infty}|\langle e_{j_{1}},f_{k}\rangle|^{2}\leq
2^{-2},$

and let $g_{2}:=f_{N_{2}}.$ Then

$\displaystyle|\langle
e_{j_{1}},f_{N_{1}}\rangle|^{2}+\sum_{k=N_{2}}^{\infty}|\langle
e_{j_{1}},f_{k}\rangle|^{2}\leq 2^{-1},$

i.e., the lower frame bound (if it exists) for
$\\{f_{N_{1}}\\}\cup\\{f_{k}\\}_{k=N_{2}}^{\infty}$ is at most $2^{-1}.$

Now, choose $j_{2}>j_{1}$ such that $|\langle
e_{j_{2}},f_{N_{1}}\rangle|^{2}+|\langle e_{j_{2}},f_{N_{2}}\rangle|^{2}\leq
2^{-3},$ choose $N_{3}>N_{2}$ such that

$\displaystyle\sum_{k=N_{3}}^{\infty}|\langle e_{j_{2}},f_{k}\rangle|^{2}\leq
2^{-3},$

and let $g_{3}:=f_{N_{3}}.$ Then

$\displaystyle|\langle e_{j_{2}},f_{N_{1}}\rangle|^{2}+|\langle
e_{j_{2}},f_{N_{2}}\rangle|^{2}+\sum_{k=N_{3}}^{\infty}|\langle
e_{j_{2}},f_{k}\rangle|^{2}\leq 2^{-2},$

i.e., the lower frame bound (if it exists) for
$\\{f_{N_{1}}\\}\cup\\{f_{N_{2}}\\}\cup\\{f_{k}\\}_{k=N_{3}}^{\infty}$ is at
most $2^{-2}.$

Inductively, having constructed $g_{1},\dots,g_{\ell}$ for some
$\ell\in\mathbb{N}$ (as well as the scalars $j_{1},\dots,j_{\ell-1}$ and
$N_{1},\dots,N_{\ell}$), choose $j_{\ell}>j_{\ell-1}$ such that $|\langle
e_{j_{\ell}},f_{N_{1}}\rangle|^{2}+|\langle
e_{j_{\ell}},f_{N_{2}}\rangle|^{2}+\cdots+|\langle
e_{j_{\ell}},f_{N_{\ell}}\rangle|^{2}\leq 2^{-\ell-1},$ choose
$N_{\ell+1}>N_{\ell}$ such that

$\displaystyle\sum_{k=N_{\ell+1}}^{\infty}|\langle
e_{j_{\ell}},f_{k}\rangle|^{2}\leq 2^{-\ell-1},$

and let $g_{\ell+1}:=f_{N_{\ell+1}}.$ Then

$\displaystyle|\langle e_{j_{\ell}},f_{N_{1}}\rangle|^{2}+|\langle
e_{j_{\ell}},f_{N_{2}}\rangle|^{2}+\cdots+|\langle
e_{j_{\ell}},f_{N_{\ell}}\rangle|^{2}+\sum_{k=N_{\ell+1}}^{\infty}|\langle
e_{j_{\ell}},f_{k}\rangle|^{2}\leq 2^{-\ell},$

i.e., the lower frame bound (if it exists) for
$\\{f_{N_{1}}\\}\cup\\{f_{N_{2}}\\}\cup\cdots\cup\\{f_{N_{\ell}}\\}\cup\\{f_{k}\\}_{k=N_{\ell+1}}^{\infty}$
is at most $2^{-\ell}.$ By construction, $\\{g_{j}\\}_{j=1}^{\infty}$ is a
subsequence of
$\\{f_{N_{1}}\\}\cup\\{f_{N_{2}}\\}\cdots\cup\\{f_{N_{\ell}}\\}\cup\\{f_{k}\\}_{k=N_{\ell+1}}^{\infty}$
for any $\ell\in\mathbb{N},$ and thus no strictly positive lower bound exists.
$\square$

Acknowledgment: F. Philipp was funded by the Carl Zeiss Foundation within the
project DeepTurb–Deep Learning in and from Turbulence. He was further
supported by the Free State of Thuringia and the German Federal Ministry of
Education and Research (BMBF) within the project THInKI–Thüringer
Hochschulinitiative für KI im Studium. D. Stoeva acknowledges support from the
Austrian Science Fund (FWF) through Project P 35846-N “Challenges in Frame
Multiplier Theory”, grant DOI 10.55776/P35846.

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Ole Christensen, Department of Applied Mathematics and Computer Science,
Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
Email<EMAIL_ADDRESS>
Marzieh Hasannasab, Department of Applied Mathematics and Computer Science,
Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
Email<EMAIL_ADDRESS>
Friedrich Philipp, Technische Universität Ilmenau, Institute of Mathematics,
Weimarer Straße 25, D-98693 Ilmenau, Germany
Email<EMAIL_ADDRESS>
Diana Stoeva, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-
Platz 1, Vienna 1090, Austria
E-mail address<EMAIL_ADDRESS>