# PowerLine Unmanned Surfer (PLUS):
Shape Adaptive Dynamics Development and Control Design

Ujjval Patel  Francis Phillips  Todd Henry  and Imraan Faruque U. N. Patel,
and I. A. Faruque are with Oklahoma State University, Stillwater, OK 74078 USA
(e-mail<EMAIL_ADDRESS>i.faruque@okstate.edu). F. R. Phillips, and T. C.
Henry are with DEVCOM Army Research Laboratory, 800 Park Offices Drive,
Durham, NC 27709 USA (e-mail<EMAIL_ADDRESS>,todd.c.henry2.civ@army.mil).The Army Research Lab partially supported this
work under ARL-R-WMRD-300124. Copyright by the authors May 2024; this work may
be under consideration for publication and copyright may be transferred
without notice, after which this version may no longer be accessible.

###### Abstract

This paper introduces the powerline unmanned surfer (PLUS) concept to extend
the limited endurance of fixed wing unmanned aerial vehicles (UAVs) via in-
flight energy harvesting from overhead electrical distribution power lines,
and develops the flight dynamics and control framework to support centimeter-
scale longitudinal powerline frequency tracking. The dynamics framework models
the UAV’s shape adaptive structure, aerodynamic forces, and control inputs,
and applies the coupled flight mechanics framework to low clearance tracking
of powerline contours. This study develops a “trajectory to shape-adaptive
UAV” controller design approach for longitudinal powerline tracking through
local spatial frequency matching. The frequency-matching approach dynamically
regulates the aircraft modes’ frequency to the powerline catenary spatial
frequency using a generalized parameter linearization approach. Performance is
assessed on an example UAV implementing camber and thickness morphing by
quantifying clearance distance from neighborhood to high voltage powerline
environments and across span and chord combinations. This approach achieves
alternating periods of low-clearance tracking and antiphase oscillation, with
34% of the powerline having tracking error less than 1m. Airfoil thickness
increase has a stronger effect over Phugoid mode wavelength than thickness
decrease. Wingspan, chord, and powerline parameter sensitivities are
quantified, providing a foundation for near-field long-distance powerline
tracking and energy harvesting.

##  1 Introduction

Electric propulsion unmanned aerial vehicles (UAVs) have become integral in
applications such as surveillance and payload delivery, yet their endurance
and range are limited by the energy constraints of batteries. Although
conventional aerial refueling offers partial solutions, its limited efficiency
and infrastructure requirements highlight the need for innovative approaches.
Traditional UAV design approaches, particularly for fixed wings, often begin
with a set of aircraft performance requirements, such as range, endurance, and
cruise speed, and later develop maneuverability limits and achievable
trajectories via turn rate and speed limitations. This approach focuses on
gross aircraft performance rather than precisely tailoring the airframe’s
dynamic maneuverability characteristics to a specific mission trajectory.
Fundamentally, the dynamically achievable trajectories are not precisely
specified as input and are an iteration or an envelope specified by its
corners rather than the natural behaviors of the aircraft.

As UAV missions become more diverse, including new prescribed flight maneuvers
and trajectories, mechanisms to incorporate maneuverability and trajectory-
following in early stage design will be helpful. Applications could include
terrain following, flight in caves, or flight in cluttered environments. In
such examples, instead of ”what are the achievable trajectories for a given
configuration?” the design for extreme behaviors asks ”what airframe
configuration(s) are able to achieve a given trajectory?” In some cases, the
integration of an additional effector, such as a morphing structure or
actuator, could provide a required envelope expansion.

This design approach provides an alternative inverse to traditional
configuration design, in which an airframe’s detailed maneuverability
characteristics is an outcome of configuration choice and sizing. The design-
for-trajectory approach instead uses mission trajectory specification as an
input to specify the airframe’s outer mold line, which may include new shape
adaptation modalities to achieve new trajectories.

This study introduces the powerline unmanned surfer (PLUS) concept as an early
example of this dynamics-integrated UAV design, beginning with mission-
specific trajectory requirements. The focus of PLUS on powerline tracking
supports UAV endurance improvements by enabling in-flight energy harvesting
from overhead electric powerlines. An integrated approach that encompasses
structure, aerodynamics, and controls allows prolonged flight and improved
mission performance.

A core challenge in the implementation of PLUS lies in the control necessary
to track the discontinuous trajectories of the powerlines while maintaining
electrical connectivity. This demands an integrated control and dynamics
framework, balancing the UAV’s adaptive structure, aerodynamic forces, and
control input. The focus of this paper is on this dynamic development aspect–
a framework to incorporate airframe flight dynamics and with shape adaptation
and assess trajectory feasibility.

This dynamics framework integrates shape adaptive features into a reference
airframe, aiming to explore the benefits and challenges of shape adaptiveness
in altitude-based powerline tracking. Camber and thickness variation examples
are used to explore the relationship between shape adaptiveness and natural
aircraft longitudinal flight modes: Phugoid and short period. The natural
flight modes of the aircraft are then dynamically tailored to match the local
spatial frequency of the hanging utility lines.

A novel trajectory-to-shape adaptive design framework for longitudinal
powerline tracking has been introduced, overcoming the limitations of
conventional airframes in matching the catenary spatial frequency of
powerlines. This framework represents a significant advance in UAV design,
enabling real-time morphing of the airfoil geometry to meet the specific
demands of powerline tracking. The open-loop control approach is applied to a
shape adaptive air vehicle simulation to quantify its performance across a
range of utility line environments.

###  1.1 Previous Work

The PLUS concept connects several areas in which UAVs have received
considerable research and development, particularly in the areas of endurance
enhancement, adaptive structures, and powerline monitoring.

####  1.1.1 UAV Endurance Enhancement

Structural adaptation to increase the control authority of fixed wing small
UAV is an idea often modeled by biological systems (Harvey et al., 2022a, b;
Cheney et al., 2020; Quinn et al., 2019). Considerable previous attention
(Seigler and Neal, 2009; Kota et al., 2003) including several previous surveys
(Barbarino et al., 2011; Seigler, 2005; Jha and Kudva, 2004). Adding to these
detailed surveys is outside the scope of this paper.

Instead, we note that the range of engineered adaptation (and consequently the
resulting trajectory) is typically less aggressive than the shape adaptations
seen in avian systems. For example, (Cheney et al., 2020) used small wing
twist and camber adjustments complemented by tail movements to maintain an
optimal trajectory. Compared to conventional UAV, avian systems have higher
degrees of articulation using smooth, active surfaces that lack control
linkages and hinges at the outer mold line. The degree to which smooth active
surfaces can be used to replace various internal/external control surfaces in
the wing and tail has been researched by (Pankonien and Inman, April 2013;
Bilgen et al., 2013). Even at the comparatively conservative trajectories seen
in previous engineered work, these adaptations have suggested a need for a new
design philosophy (Ajaj et al., 2016). As UAV missions with more demanding
trajectories are addressed as done by (Rudnick-Cohen et al., 2022) and
(Rudnick-Cohen et al., 2023), the need for a detailed “trajectory-to-
aerodynamic-shape” specification method becomes a more significant limitation.

####  1.1.2 Powerline Monitoring Technologies

Powerline monitoring has been a critical application area for UAVs. (Dietsche
et al., 2021) explored the use of multirotor UAVs for close-range powerline
inspections, emphasizing the importance of stable flight dynamics near high-
tension lines. The integration of advanced event based sensors for fault
detection in powerline infrastructure was highlighted in (Rangel et al.,
2022), underscoring the utility of UAV in preventative maintenance.

Flight in close proximity to powerlines poses specialized control challenges.
Adaptive control algorithms developed in (Muñoz-Gómez et al., 2023) for UAVs
operating in complex environments laid the foundation for precision flight
controls. However, these studies focused on the use of quadcopters for
powerline inspection and survey while maintaining a safe distance from the
powerline infrastructure. These studies did not seek to leverage powerline
structures as a means of enhancing UAS capabilities.

####  1.1.3 Powerline energy harvesting

In-flight powerline energy harvesting could be accomplished from a moving
fixed wing UAV via either a trailing contact (“shoe”) or an inductive coil.
The magnetic field of a long straight wire is

$B=\frac{\mu_{0}I}{2\pi R},$

where $I$ is the current, $R$ the distance to the wire, and
$\mu_{0}=4\pi\times 10^{-7}\text{T m/s}$ the permeability of free space. A
single 345kV powerline carrying 628A could have a 125$\mu$T magnetic field at
1m (rur, 2008). Inductive powerline energy harvesting has previously been
approached for wireless sensor networks, using air-core coil and resonant
capacitor circuits to demonstrate 1-6mW energy harvesting from a 21$\mu$T
magnetic field switching at 60Hz (Tashiro et al., 2011). These studies showed
that the number of coil turns and core material affect the harvesting
potential and estimated that energy harvesting levels at publicly acceptable
levels could be comparable to photovoltaic cell output on a cloudy day
($130\mu$W/cm3 at 200$\mu$T) (Tashiro et al., 2011; Wang and Ha, 2021).
Previous work tailored towards wireless sensor networks has shown scavenging
at single amp levels, scavenging via inductors between parallel wires
(850$\mu$W from 8.4A differential currents), and highlighted the importance of
core choice for higher power densities (100mW/cm3) (Riba et al., 2022).

A UAV in flight may not enclose the wire completely, requiring the inclusion
of a demagnetization factor. Work on non-enclosing energy harvesters has
previously computed shape optimizations for such cases, such as the bow-tie
shaped non-enclosing coil in (Yuan et al., 2015).

Research efforts have explored various approaches to enabling UAVs to interact
with power lines for energy harvesting and recharging purposes, focusing on
the development of mechanisms for physically grasping power lines. (Iversen et
al., 2021a) focused on the design and testing of a mechanical grappling
mechanism that allows UAVs to secure their connection to powerlines, providing
a stable platform for energy transfer. (vom Bögel et al., 2020) evaluated the
feasibility of a drone charging function on overhead power lines and verified
this in a test setup. The work of (Stewart et al., 2022) introduced a novel
mechanical structure attached to the nose of a light-weight fixed wing
aircarft that could extract electrical energy from high-voltage powerlines
with minimal loss and convert it to a form that could be used by UAV
batteries. (Stuhne et al., 2022)’s presented a novel electromechanical
recharging station that can be mounted on energized AC power line to charge
the drone battery wirelessly without the need to modify the electrical
infrastructure. Meanwhile, (Iversen et al., 2021b) explored the long-term
impacts of continuous powerline energy harvesting on UAV battery life and
performance, offering insights into optimal charging cycles and energy
management strategies. Finally, (Nyboe et al., 2023) presented technologies
that allow UAVs to autonomously attach and recharge from the existing DC
railway infrastructure.

####  1.1.4 Adaptive Structures for Maneuver

Fixed wing UAVs typically achieve higher range and endurance relative to
similar scale multirotor configurations that can achieve more decoupled
motions (Mulgaonkar et al., 2014). One approach to mission endurance
improvement is perching at a speed that biological systems typically do with
precise control of their velocity and position (KleinHeerenbrink et al.,
2022). (KleinHeerenbrink et al., 2022) suggested that trajectory information
in combination with aeromechanical information (pressure/strain) (Castano et
al., 16-20 June 2014; Haughn and Inman, 2022) and perception
(distance/location) could be used to optimize behavior in real time for UAV.
(Moore et al., 2014) showed that at speed perching could be done with
conventional control surfaces with (Greatwood et al., 2017), and (Fletcher et
al., 2022) using sweep control and reinforcement learning to command the
desired trajectory. Both (Greatwood et al., 2017) and (Fletcher et al., 2022)
were able to, in real-world flight tests, command an approach and flare
trajectory near a perch position with reasonable accuracy using reinforcement
learning. (Ajanic et al., 2020) developed a vehicle capable of aggressive
maneuvering by creating a wing and tail capable of adaptation.

The PLUS concept distinguishes itself from previous research studies by taking
advantage of existing powerline catenary shape, thereby extending the existing
capabilities of a fixed-wing UAV.

##  2 Methods & Approach

###  2.1 Mission space definition

A central feature of this framework is its precision in tracking the catenary
contour of powerlines through ”local spatial frequency matching.” This process
entails the agile adaptation of the UAV’s spatial frequencies to align with
the undulating spatial frequency of the powerlines. The research details a
comprehensive dynamic model, accompanied by simulations to quantify the UAV’s
longitudinal tracking performance along powerline trajectories. Figure 1
illustrates the operational mission space of the PLUS system, and highlights
technical challenges to the PLUS concept.

Figure 1: Powerline unmanned surfing concept for long range fixed wing UAV
range extension and navigation.

Figure 2 illustrates the coverage of this paper, showing the interconnection
between the flight dynamics model, powerline geometry, and sensing and
feedback structure.

Figure 2: This paper’s coverage: flight dynamics model development, powerline
geometry environment, and controller development.

###  2.2 Powerline Modeling

An example powerline catenary curve is illustrated in Fig. 3, in which the
between-pole curves are modeled as a catenary influenced by environmental
parameters (Farr et al., 1980).

The expression for a powerline catenary curve was given by

$y=\dfrac{T_{0}}{\rho gA}\cosh{\left\\{\dfrac{\rho
gA}{T_{0}}\left(x-\dfrac{L}{2}\right)\right\\}}\quad\text{for
}x\in\left[0,L\right],$ (1)

where $T_{0}$ is the tension in the cable at ambient temperature, $\rho$ is
the density of material in the transmission cable, $A$ is the cross-sectional
area of the transmission cable and $g$ is gravitational acceleration. The
local curvature of the powerline catenary $R$ can be expressed as

$R=\dfrac{T_{0}}{\rho gA}\cosh^{2}{\left\\{\dfrac{\rho
gA}{T_{0}}\left(x-\dfrac{L}{2}\right)\right\\}}.$

Figure 3: Powerline geometry used in this study (Farr et al., 1980).

####  2.2.1 Local spatial frequency of a catenary

One can find the frequency of the catenary curve based on the assumption that
the physical properties of the powerline can be estimated and that the travel
speed of is constant as it flies over the powerline. Then, the local spatial
frequency of the catenary can be defined as

$\omega_{s}(x)=\dfrac{x}{y(x)}.$ (2)

###  2.3 Aircraft control approach with shape adaptive actuation

Shape adaptive actuation alters an aircraft’s aerodynamics and associated
stability derivatives by modifying its physical geometry. Variations induced
by shape adaptiveness can be formulated as a perturbation of traditional
stability derivatives (Beard et al., 2012) by introducing a vector of
generalized time-varying shape adaptive parameters $\sigma(t)$. This approach
is further specialized for the purpose of this paper for camber or thickness
and span shape adaptation.

This section presents the definition of the linear system and a linearization
approach that incorporates adaptive dynamics of the nonlinear shape as an
explicit function of $\sigma(t)$. In state-space form, a non-linear system
model $\dot{x}=f(x,u,t)$ can be expressed as

$\dot{x}\approx A[\sigma(t)]x+B\left[\sigma(t)\right]u+W[\sigma(t)]$

with

$A[\sigma(t)]=\left.\dfrac{\partial f(x,u,t)}{\partial x}\right|_{x_{0}}\qquad
B[\sigma(t)]=\left.\dfrac{\partial f(x,u,t)}{\partial u}\right|_{u_{0}},$

where $W[\sigma(t)]$ represents higher-order nonlinear terms, and the state
matrices vary with $\sigma(t)$, a time-varying vector that may consist of
system outputs, exogenous inputs, or a combination of both. In the sections
below, example dynamics of two types of shape adaptive structures for a UAV
are discussed, and how they can be used as actuation to control a shape
adaptive aircraft is developed.

####  2.3.1 Example: Airfoil shape adaptation (Camber and Thickness)

The introduction of laterally symmetric, shape-adaptive alterations
significantly impacts the longitudinal aerodynamics, particularly the normal
force and the pitch moment stability derivatives. The traditional linearized
state-space dynamics model can then be written in terms of a generic morphing
control input $\sigma(t)$. Consistent with traditional linearized modeling,
states are defined in terms of perturbations from trim values. Shape adaptive
aircraft having independent shape and flight control, as in a linear system,
have been discussed in the literature (Wu et al., 2015). An analogous
linearization allows writing force and momentum as

$\displaystyle m\ddot{x}=$
$\displaystyle-D\cos{\alpha}+L\sin{\alpha}+T-mg\sin{\theta}$ $\displaystyle
m\ddot{w}=$ $\displaystyle-D\sin{\alpha}-L\cos{\alpha}+mg\cos{\theta}$
$\displaystyle I_{yy}\dot{q}=$ $\displaystyle M_{A}-gS\sin{\theta},$

where the lift, drag, and moment can be described as

$\displaystyle L$ $\displaystyle=0.5\rho V^{2}SC_{L}$ $\displaystyle D$
$\displaystyle=0.5\rho V^{2}SC_{D}$ $\displaystyle M_{A}$
$\displaystyle=0.5\rho V^{2}ScC_{m}.$

The aerodynamic coefficients directly affected by the camber or thickness
shape $\mu$ can be written in a $C_{[L,D,m]}(\alpha,V,\mu)$ form similar to
stability derivatives as

$\displaystyle C_{L}$
$\displaystyle=C_{L}(\alpha,V,\mu)=C_{L0}+C_{L\alpha}\alpha+C_{LV}\Delta
V/V+C_{L\mu}\mu$ $\displaystyle C_{D}$
$\displaystyle=C_{D0}+C_{D\alpha}\alpha+C_{D\alpha^{2}}\alpha^{2}+C_{DV}\Delta
V/V+C_{D\mu}\mu$ $\displaystyle C_{m}$
$\displaystyle=C_{m0}+C_{m\alpha}\alpha+C_{mV}\Delta V/V+C_{m\mu}\mu.$

####  2.3.2 Example: Variable span shape adaptation

The use of variable wingspan to improve flight performance and control
authority of high endurance, medium-altitude UAV has been investigated in
detail, such as (Ajaj et al., 2009). A linear parameter varying approach
similar to (Yue et al., 2013) is used to obtain

$\displaystyle\ddot{x}$ $\displaystyle=\dot{u}+qw-rv+g\sin{\theta}$
$\displaystyle\ddot{y}$ $\displaystyle=\dot{v}+ru-pw+g\cos{\theta}\sin{\phi}$
$\displaystyle\ddot{z}$ $\displaystyle=\dot{w}+pv-qu+g\cos{\theta}\cos{\phi}$
$\displaystyle\dot{\phi}$ $\displaystyle=p+tan\theta(q\sin{\phi}+r\cos{\phi})$
$\displaystyle\dot{\theta}$ $\displaystyle=q\cos{\phi}-r\sin{\phi}$
$\displaystyle\dot{\psi}$
$\displaystyle=(q\sin{\phi}+r\cos{\phi})\sec{\theta}$ $\displaystyle p$
$\displaystyle=\dot{\phi}-\dot{\psi}\sin{\theta}$ $\displaystyle q$
$\displaystyle=\dot{\theta}\cos{\phi}+\dot{\psi}\cos{\theta}\sin{\phi}$
$\displaystyle r$
$\displaystyle=\dot{\psi}\cos{\theta}\cos{\phi}-\dot{\theta}\sin{\phi}$
$\displaystyle\sum L$ $\displaystyle=\sum(L_{0}+\Delta
L)=\dot{p}I_{x}+p\dot{I}_{x}-\dot{q}I_{xy}-q\dot{I}_{xy}-\dot{r}I_{xz}$
$\displaystyle-r\dot{I}_{xz}-qpI_{xz}-q^{2}I_{yz}+qrI_{z}-rqI_{y}+r^{2}I_{yz}$
$\displaystyle\sum M$ $\displaystyle=\sum(M_{0}+\Delta
M)=\dot{q}I_{y}+q\dot{I}_{y}-\dot{p}I_{xy}-p\dot{I}_{xy}-\dot{r}I_{yz}$
$\displaystyle-r\dot{I}_{yz}+rqI_{x}-rqI_{x}-r^{2}I_{xz}+p^{2}I_{xz}+pqI_{yz}-rqI_{z}$
$\displaystyle\sum N$ $\displaystyle=\sum(N_{0}+\Delta
N)==\dot{r}I_{z}+r\dot{I}_{z}-\dot{p}I_{xz}-p\dot{I}_{xz}-\dot{q}I_{yz}$
$\displaystyle-q\dot{I}_{yz}-p^{2}I_{xy}+pqI_{y}-prI_{yz}-qpI_{x}+q^{2}I_{xy}+qrI_{xz}.$

Assuming the adaptation of the symmetric span shape and using $\delta y$ and
$\delta\dot{y}$ as the adaptive parameters of the shape, we arrive at

$\displaystyle\sum L$
$\displaystyle=\dot{p}I_{x0}+qr(I_{z0}-I_{y0})+2m_{s}\dot{\delta}y(y_{0}p+\delta
yp)$ (3) $\displaystyle\sum M$ $\displaystyle=\dot{q}I_{y0}+rq(I_{x0}-I_{z0})$
(4) $\displaystyle\sum N$
$\displaystyle=\dot{r}I_{z0}+pq(I_{y0}-I_{x0})+2m_{s}\dot{\delta}y(y_{0}r+\delta
yr).$ (5)

The linearized aircraft model augmented with non-linear shape adaptation
thickness and span shape $\sigma=[\mu,\delta y,\delta\dot{y}]$ can be written
in terms of normalized aerodynamic stability derivatives as

$\begin{bmatrix}\dot{u}\\\ \dot{w}\\\ \dot{q}\\\ \dot{\theta}\\\
\dot{h}\end{bmatrix}=\begin{bmatrix}X_{u}(\sigma)&X_{w}(\sigma)&X_{q}&-g\cos{\theta_{0}}&0\\\
Z_{u}(\sigma)&Z_{w}(\sigma)&u_{0}&-g\sin{\theta_{0}}&0\\\
M_{u}(\sigma)&M_{w}(\sigma)&M_{q}(\sigma)&0&0\\\ 0&0&1&0&0\\\
\cos{\theta_{0}}&\sin{\theta_{0}}&0&0&0\end{bmatrix}\begin{bmatrix}u\\\ w\\\
q\\\ \theta\\\ h\end{bmatrix}.$ (6)

####  2.3.3 Structural Actuation

Shape-adaptive aircraft must include an actuation mechanism. Servos are
commonly used to modify the airfoil shape. A second-order transfer function
with delay was used to represent the servo, and a slew-rate limit was also
included (Wada et al., 2009).

##  3 Catenary frequency mapping control design

Flight speeds between pylons are in the range of 1-3 seconds, placing them
near the Phugoid mode of a Group 1 UAV (United States. Army. Unmanned Aircraft
Systems. Center of Excellence, 2010). The Phugoid mode describes a tradeoff
between airspeed and altitude, and leveraging this natural mode may be an
especially relevant method of achieving the height variation needed for
powerline tracking. A concise control architecture can be developed using
aerodynamic shape adaptation to match the Phugoid mode’s temporal frequency to
the local powerline spatial frequency in Equation (2). The powerline’s local
spatial frequency prescribes the required aerodynamic shape for close
tracking; equivalently, the Phugoid’s spatial wavelength is modulated to match
the powerline.

To implement this frequency-matching approach, Equation (6) is linearized with
respect to the nonlinear shape adaptive parameter $\sigma(t)$ about
$\sigma_{0}=0$ to obtain

$\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t)+B_{\sigma}\sigma(t)x(t),$
$\displaystyle=\left[A+B_{\sigma}\sigma(t)\right]x(t),$ (7)

with

$\displaystyle A$
$\displaystyle=\begin{bmatrix}X_{u}&X_{w}&X_{q}&-g\cos\theta_{0}&0\\\
Z_{u}&Z_{w}&Z_{q}&-g\sin\theta_{0}&0\\\ M_{u}&M_{w}&M_{q}&0&0\\\ 0&0&1&0&0\\\
\sin\theta&-\cos\theta&0&u\cos\theta+w\sin\theta&0\end{bmatrix},$
$\displaystyle B_{\sigma}$
$\displaystyle=\begin{bmatrix}X_{u\sigma}&X_{w\sigma}&0&0&0\\\
Z_{u\sigma}&Z_{w\sigma}&0&0&0\\\ M_{u\sigma}&M_{w\sigma}&M_{q\sigma}&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{bmatrix}.$

Analyzing the long period mode from Eqn. (7), the Phugoid (temporal) frequency
is

$\omega_{t}(\sigma)=\sqrt{\dfrac{g(Z_{u}+Z_{u\sigma}(\alpha,\sigma)\sigma)}{Z_{q}}}.$
(8)

The catenary’s local spatial frequency $\omega_{s}$ can be used to find the
required shape adaptation analytically, first as a requirement on the aircraft
Phugoid frequency $\omega_{t}$, then via aerodynamic sensitivity as a
requirement on the morphing parameter $\sigma$. The spatial frequency matching
requirement is

$\omega_{s}(x)=\dfrac{1}{u_{0}}\omega_{t}(\sigma),$ (9)

or, substituting Eqn. (2) and (8) into (9),

$\dfrac{u_{0}x}{y(x)}=\sqrt{\dfrac{g(Z_{u}+Z_{u\sigma}(\sigma,\alpha)\sigma)}{Z_{q}}}.$

Solving for terms dependent on morphing $\sigma$, one has

$Z_{u\sigma}(\sigma,\alpha)\sigma=\dfrac{Z_{q}u_{0}^{2}x^{2}}{gy(x)^{2}}-Z_{u}.$

which can be simplified to

$Z_{u\sigma}(\sigma,\alpha)\sigma-\dfrac{Z_{q}u_{0}^{2}x^{2}}{gy(x)^{2}}+Z_{u}=0.$
(10)

Using the lookup table for a given set of the angle of attack $\alpha$ and the
aerodynamic parameters $Z_{u\sigma}$, Eqn. (10) can be solved using numerical
solvers such as Newton Raphson method to extract an adaptive parameter of the
analytical shape as a function of the distance traveled from the post,
provided that there is post-inflection point. The resulting feedforward
controller architecture is shown in Fig. 4.

Figure 4: Controller architecture for phugoid-catenary mapping

###  3.1 Design Simulations

To generalize airframe selection and the range of shape adaptive structures
needed, using the definition of mission space and the phugoid-catenary
matching controller for a generic reference aircraft example chosen through
trial and error in simulation that gives adequate results and using that as a
base, a trend search is performed using unconstrained nonlinear least squares-
based optimization to get us aerodynamic parameters required for a given
powerline structure.

The optimization performed is

$\hat{\sigma}(t)\in\text{argmin}_{\sigma}\sum_{i=1}^{i=N}[s_{\text{ref}}-s\left(x(t),\sigma(t_{i})\right)]^{2},$

where $s_{\text{ref}}$ is the reference powerline trajectory (here assumed
known at a given time and location) and $s(x(t),\sigma(t_{i}))$ is the actual
trajectory for the aircraft for a given time series of morphing
$\sigma(t_{i})$ and the aircraft state $x(t)$.

####  3.1.1 Simulation Case Study: Generic aircraft (Group I type)

##### Shape Adaptive Aircraft Design

This study simulated a reference aircraft that includes variations in airfoil
camber and airfoil thickness. A nominal NACA2412 airfoil capable of adapting
to the shape from NACA1412 to NACA4416 for the maximum camber location change
and from NACA2406 to NACA2416 for thickness change.

Figure 5: Shape adaptive solver in loop method of aerodynamic parameter
calculation Figure 6: Pre computed shape adaptive aerodynamic parameter
calculation

To simulate continuous shape adaptive aerodynamics, XFoil and Matlab were used
to characterize the aerodynamic parameters for the aircraft flight envelope
using two methods as illustrated in Fig. 5 and Fig. 6, where the stability
derivatives related to shape adaptive parameters are calculated in loop and
pre-calculated using prior knowledge of shape adaption limits
($\underline{\sigma},\bar{\sigma}$) where $\underline{\sigma}$ is the lower
limit of morphing and $\bar{\sigma}$ is the upper limit of the morphing
parameter.

The method results in aerodynamic stability derivatives as functions of shape
adaptive parameters. A example of Eqn. (7) maneuvering about trim is

$\begin{split}\begin{bmatrix}\dot{u}\\\ \dot{w}\\\ \dot{q}\\\ \dot{\theta}\\\
\dot{h}\end{bmatrix}=\left(\begin{bmatrix}-0.074&-0.122&0&-9.811&0\\\
-1.535&-7.457&25&0&0\\\ 2.689&-5.850&-32.945&0&0\\\ 0&0&1&0&0\\\
1&0&0&0&0\end{bmatrix}\right.+\\\ \left.\begin{bmatrix}-0.332&-2.096&0&0&0\\\
-3.960&-15.336&0&0&0\\\ 25.329&95.192&-160.32&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{bmatrix}\sigma\right)\begin{bmatrix}u\\\ w\\\ q\\\ \theta\\\
h\end{bmatrix}.\end{split}$ (11)

##### Simulation Method

To illustrate the proposed controller operation, this approach is implemented
in MATLAB, where a high-fidelity nonlinear shape adaptive aircraft is
simulated with the control input developed using a linearized shape adaptive
aircraft dynamics representation as developed above. The simulation includes a
reference power line with 30m tall towers spaced at 70m intervals.

The controller has local information about the powerline geometry to estimate
the catenary spatial frequency and full state information about the aircraft
from the dynamics to predict the Phugoid temporal frequency for phugoid-
catenary mapping; the controller is re-initialized when crossing each tower.

####  3.1.2 Numerical parameter space exploration

To optimize the design of the powerline unmanned surfer (PLUS) for diverse
operational environments, simulations to explore the interplay between
different aircraft design parameters and various powerline structures were
conducted. This step was instrumental in identifying the most effective
combinations of aircraft wingspan and chords for powerline tracking
performance across a range of powerline geometries.

The trend search involved running a series of simulations, each iterating over
a set of predefined span and chord values for the UAV. These values were
tested against multiple powerline structures, characterized by different
catenary curve parameters such as sag, tension, and span length. The primary
objective was to determine how changes in the UAV’s physical dimensions
influenced its ability to adapt and maintain efficient tracking of the
powerlines, especially in varying environmental conditions.

##### Aircraft parameters

The simulations systematically varied two UAV parameters: chord length and
wingspan. The wing’s chord (leading edge to trailing edge distance) was
varied, exploring the effects of change in chord length for group I type UAVs.
The chord lengths varied from $0.203\>\text{m}$ to $0.305\>\text{m}$ with
increment of $0.51\>\text{m}$. The wingspan (distance between the tips of the
wings) was altered in each simulation run, testing a range of typical Group I
UAVs. Wingspans ranged from $1\>\text{m}$ to $1.8\>\text{m}$ with increments
of $0.4\>\text{m}$

##### Environmental parameters

The variations were evaluated across a spectrum of powerline geometries
ranging from low voltage to ultra high voltage (¿800kV), as outlined in Table
1, to quantify the spacing and height dimensions of the pylons. Furthermore,
the impact of environmental factors on the powerlines was standardized by the
percentage of sag throughout the pylon span.

Label | Voltage, kV | Pylon geometry
---|---|---
|  | Height, m | Spacing, m
Low voltage (LV) | Up to 1 | 10 - 15 | 30 - 50
Medium voltage (MV) | 1 - 69 | 15 - 30 | 50 - 150
High voltage (HV) | 69 - 230 | 30 - 50 | 150 - 400
Extra high voltage (EHV) | 230 - 500 | 50 - 80 | 300 - 500
Ultra high voltage (UHV) | 800+ | 80+ | 500+
Table 1: Powerline class and structural parameters (ESRI U.S. Federal
Datasets, 2022).

This approach provided a mechanism to quantify the UAV’s performance across
varying environmental geometry.

##  4 Results and discussion

###  4.1 Thickness morphing experiment

A representative servo actuation system was manufactured and the
identification of the time domain system was used to determine the transfer
function from the experimental data. The experiment used the additively
manufactured airfoil shown in Fig. 7 which has a degree of freedom of
thickness actuated by the servo. A camera operating at 120 frames per second
was used to digitize the thickness variation using the MATLAB image processing
toolbox (The MathWorks Inc., 2022). The actuator system received a multistep
input command and the recorded input/output signals were used to quantify the
response time, the maximum deflection angles, and the effects of the slew rate
limit of $0.11s/60^{\circ}$ degree rotation.

Figure 7: Thickness morphing airfoil actuator used in experimental
identification

A multiple-step stimulus of $3^{\circ}$ servo deflection was used for
identification, and the identification of the time-domain system considered
three transfer function model structures: first order, second order system,
and second order with time delay. The identification results in Table 2
indicate that a 2nd order model including transport delay best describes the
response.

Parameter | Model structure
---|---
| 1st order | 2nd order
|  | Nominal | With delay
Gain | 0.01333 | 0.01333 | 0.01333
Time constant | 1.0s | - | -
Damping ratio | - | 0.919 | 0.45
Natural frequency | - | 1.0Hz | 1.0Hz
Time delay | - | - | 0.05s
Fit accuracy | 86% | 91% | 93%
Table 2: Structural actuation system identification results

The identified thickness morphing transfer function was

$Y(s)=\dfrac{0.01333}{s^{2}-1.838s+1}\left(1-e^{-0.005s}\right),$ (12)

as seen in Fig. 8.

Figure 8: Morphing data fit to a second order system with time delay

###  4.2 Simulation results

The results of the aircraft trajectory based on the implemented controller are
shown in Fig. 9. The tracking of the power line is achieved periodically in
alternating troughs, leading to low and high levels of cyclic clearance, as
shown by the shaded regions in Fig. 9. The powerline clearance shown in Fig.
10 shows that for the chosen aircraft the variation of - 3% to 6.3% of the
camber or the variation of -3.2% to 5.5% thickness shown in Fig. 12 achieves
¡1m powerline spacing for 88.7m of the 260m span, or 34% of the longitudinal
region. Fig. 11 shows the velocity deviation during the controller
implementation.

Figure 9: Powerline tracking for shape adaptive aircraft Figure 10: Error
between aircraft trajectory and powerline Figure 11: Velocity deviation for
the powerline tracking Figure 12: Continuous shape adaptive input to the
system for powerline tracking

###  4.3 Parameter space exploration

The parameter space exploration examined the sensitivity to wingspan,
powerline span, and wing chord. The proficiency requirement is quantified
using the maximum aerodynamic change required by a morphing that is found by
considering the unmorphed aerodynamic coefficient relative to the maximal
morphing seen in the trial. For example, for the case of lift coefficient
$C_{L}$, the maximal $C_{L}$ variation observed is found by parameterising
lift coefficient as a function of morphing, eg $C_{L}(\alpha,\sigma)$. Then,
we find the maximum morphing parameter value and the time at which it occurs,
eg, defining

$\displaystyle t_{m}$ $\displaystyle=\arg\max_{t}{\sigma(t)},\quad\forall
t\in[0,t_{f}]$ (13) $\displaystyle\bar{\sigma}$
$\displaystyle=\max_{t}{\sigma(t)},\quad\forall t\in[0,t_{f}]$ (14)

for a trial lasting from time $t=0$ to final time $t_{f}$, then the maximum
change in lift coefficient demanded by morphing in that trial is

$\Delta
C_{L,m}=C_{L}\left(\alpha(t_{m}),\bar{\sigma}\right)-C_{L}\left(\alpha_{0},0\right),$

where $\alpha_{0}$ denotes the trimmed angle of attack.

$\Delta C_{L,m}$ was computed for a total of 4,500 cases, covering 25 trials
for each combination of 4 pylon span distances that cover LV to EHV
powerlines, 3 airfoil chords, 3 aircraft wingspans and 5 sag level
combinations.

Figure 13: Parameter space exploration results for LV powerlines (40m pylon
span)
Figure 14: Parameter space exploration results for MV powerlines (100m pylon
span) Figure 15: Parameter space exploration results for HV powerlines (300m
pylon span) Figure 16: Parameter space exploration results for EHV powerlines
(500m pylon span)

Mean morphing-driven changes in $C_{L}$ (as quantified by $\max{|\Delta
C_{L,m}}|$) increases with decreases in either mean aerodynamic chord (MAC)
decreases or wing area, with the area effect is most visible in Fig. 16. Both
are likely related to the need for higher lift coefficients on smaller
aerodynamic surfaces.

Across Figs. 13-16, the 2% powerline sag results consistently show lower
$\max{\Delta C_{L,m}}$ requirements. This lower actuation requirement is
likely due to the 2% sag powerline wavelengths being near an integer factor of
the aircraft’s Phugoid wavelength. This effect is illustrated for the cases
$d=100$ m and $d=300$ m in Fig. 17. This lowered aerodynamic actuation when
Phugoid and aircraft wavelength is matched lends support for the frequency-
matching approach to control, and Section 4.4 analyzes this relationship.

Figure 17: Illustration of lowered aerodynamic actuation needed for powerline
wavelengths being near an integer factor of the aircraft’s (c=0.254m, b=1.4m)
Phugoid wavelength

###  4.4 Phugoid wavelength variation

Figure 18 shows the Phugoid wavelength $\lambda_{\text{ph}}$ as a function of
wing span, chord, and morphed thickness calculated using the Eqn. (15),

$\lambda_{\text{ph}}=\dfrac{u_{0}}{\omega_{t}(\sigma,b,c)}$ (15)

where, $\omega_{t}(\sigma,b,c)$ is the temporal frequency of a aircraft with
wing span $b$ and chord length $c$ at a given morphed thickness of $\sigma$
flying at a trim speed of $u_{0}$.

Figure 18: Phugoid wavelength during thickness morphing ($\sigma=\pm 14\%$
thickness) shows the authority of thickness morphing over Phugoid length
relative to span $b$ and chord $c$ variation.

Fig. 18 indicates that thickness morphing significantly affects Phugoid
wavelength, and is a viable control term for matching aircraft modes to
powerline spans. Combined with Table 1, Fig. 18 can be used to identify the
feasible powerline/airframe pairings for close-range tracking and to estimate
the required thickness morphing range.

Figure 19: Phugoid wavelength during thickness morphing ($\sigma=\pm 14\%$
thickness) shows the authority of thickness morphing over Phugoid length
relative to aspect ratio

Figure 19 shows that an increase in thickness provides more authority over the
Phugoid wavelenth than a comparable decrease in thickness. This increased
effectiveness is likely related to the higher aerodynamic sensitivity of
thicker airfoils through increased lift and drag. Support for the increase in
aerodynamic sensitivity origin is also seen in that thickness morphing has
higher authority over Phugoid wavelength when both spans and chord are high,
as visible in Fig. 18. However, thickness authority is not simply a function
of wing area–Fig. 19 also shows reduced authority at higher aspect ratios.

###  4.5 Limitations and assumptions

The open loop (feedforward) structure of this control approach does not yet
account for parameter uncertainty or measurement noise, and incorporating a
sensor-based feedback controller with a dynamic estimator could yield
improvements. The approach leverages actuation through a generalized
aerodynamic change; thus, camber or thickness and span shape is a design
choice. The finding that these modes are feasible does not indicate that other
actuation modes are infeasible; rather, alternate paths may achieve this
variation, including traditional empennage actuation for spans greater than
300m. Similarly, while higher-fidelity wind tunnel and experimental
aerodynamic studies may improve the accuracy of aerodynamic parameter
variation relative to panel methods, the frequency matching framework
introduced here remains feasible when the configuration change is able to
modulate aerodynamic force magnitude (Koreanschi et al., 2016; Galffy et al.,
2019).

##  5 Conclusion

This study presents a concise dynamic model and simulation of a shape-adaptive
aircraft capable of camber thickness and span shape designed for a challenging
longitudinal powerline tracking mission, such as inflight recharging and
perching to the powerline for extension of the flight range. The aerodynamic
parameters of the aircraft are directly dependent on the time-varying
aerodynamic forces and moments, which are a function of the change in the
shape of the wing by the shape adaptive command ($\mu$-camber or thickness
adaptive parameter). The coupled longitudinal dynamic equation of the shape
adaptive process is derived by simplifying the longitudinal dynamic responses
of the wing shape adaptive process. Quasi-steady aerodynamic assumptions are
used to numerically simulate the dynamics.

A novel approach in this study is the implementation of a frequency-matching
(or wavelength-matching) technique that dynamically adjusts the local temporal
frequencies of the aircraft’s natural modes to match the spatial frequency of
the powerline’s catenary curve. This was achieved through the generalized
linearization of the adaptive parameters of the shape. The results demonstrate
the potential for a control strategy capable of achieving alternating periods
of low clearance tracking and antiphase oscillation. Furthermore, we
integrated an open-loop control strategy that manipulates camber or thickness
to maintain less than $1$ meters of powerline clearance for more than 34% of
the line, despite the challenges of alternating coverage. This dynamics and
controller development lays a strong foundation for low clearance powerline
tracking. Additionally, our findings suggest that for group I Unmanned Aerial
Vehicles (UAVs) operating at EHV (Extra High Voltage) levels and above,
intricate shape adaptation may not be required in order to fly low clearance
energy harvesting trajectories. This insight could simplify the design and
operation of UAVs for specific applications, reducing the complexity and
enhancing the reliability of energy harvesting missions in these high voltage
environments.

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