# Self-Reconfigurable Soft-Rigid Mobile Agent with Variable Stiffness and
Adaptive Morphology

Luiza Labazanova, , Shuang Peng, Liuming Qiu, Hoi-Yin Lee,
Thrishantha Nanayakkara, , and David Navarro-Alarcon This work is supported in
part by the Research Grants Council (RGC) of Hong Kong under grants 14203917
and 15212721, and in part by the 2019/20 Belt and Road Scholarship (Research
Postgraduate).L. Labazanova, L. Qiu, H. Lee, and D. Navarro-Alarcon are with
the Faculty of Engineering, The Hong Kong Polytechnic University, HK (e-mail:
<EMAIL_ADDRESS>dna@ieee.org).S. Peng
is with the Department of Aerospace and Mechanical Engineering, University of
Southern California, USA. (e-mail: shuangpe@usc.edu).T. Nanayakkara is with
the Dyson School of Design Engineering, Imperial College London, UK (e-mail:
t.nanayakkara@imperial.ac.uk).

###### Abstract

In this paper, we propose a novel design of a hybrid mobile robot with
controllable stiffness and deformable shape. Compared to conventional mobile
agents, our system can switch between rigid and compliant phases by
solidifying or melting Field’s metal in its structure and, thus, alter its
shape through the motion of its active components. In the soft state, the
robot’s main body can bend into circular arcs, which enables it to conform to
surrounding curved objects. This variable geometry of the robot creates new
motion modes which cannot be described by standard (i.e., fixed geometry)
models. To this end, we develop a unified mathematical model that captures the
differential kinematics of both rigid and soft states. An optimised control
strategy is further proposed to select the most appropriate phase states and
motion modes needed to reach a target pose-shape configuration. The
performance of our new method is validated with numerical simulations and
experiments conducted on a prototype system. The simulation source code is
available at the GitHub repository.

###### Index Terms:

Hybrid soft-rigid robots, shape control, variable stiffness, adaptive
morphology, mobile agents.

##  I Introduction

Autonomous mobile robots are one of the most prominent and well-established
concepts in robotics. Self-contained design coupled with intelligent control
systems allows mobile robots to perform a large variety of tasks. As such,
they have been utilised in many industrial and service applications, including
construction, conveyance, surface exploration, surveillance, entertainment,
and others [1].

Despite their advantages, mobile robots face considerable limitations common
to most conventional robots. Constant morphology and rigid structures
undermine their ability to adapt to surroundings, handle delicate objects and
safely interact with humans. Soft robotics can solve these issues [2];
however, it complicates (and possibly limits) the system’s mobility and
control. Existing soft robots operate via cumbersome actuators, lack high
force capacity, and their flexible electronics are not developed enough to
provide untethered design [3].

In this work, we aim to improve the performance of mobile robotic agents by
leveraging paradigms of both conventional and soft robotics. We introduce a
hybrid system with adaptive morphology that consists of a pair of wheeled
locomotion units (LU’s) connected by a variable-stiffness fibre (VSF).
Sections with low-melting-point alloy (LMPA) within VSF serve as toggles
between constant-shape and deformable modes. When the whole fibre is rigid,
the robot acts as a conventional mobile robot. When LMPA is molten, VSF
sections become soft; thus, a robot can reshape itself and conform to objects
with various curved geometries [4, 5]. After the solidification of the fibre,
the structure (which is now rigid) maintains a modified shape. LU’s carry
power and electronics, which provides autonomy to a mobile system. Omni-wheels
at each unit enable the robot to undergo versatile deformations, as well as to
move freely in a rigid state at any VSF configuration.

This new class of hybrid systems with adaptive morphology is primarily
inspired by nature [6]. Biological species exploit similar variable shape
principles to gain additional functionalities and facilitate locomotion in
dynamic environments [7]. These benefits promoted research efforts in
developing a new class of highly reconfigurable systems, including mobile
robots [8, 9, 10]. For example, Kim et al. applied variable morphology to
extend the locomotion capabilities of Whegs [11], which enables a robot to
traverse uneven surfaces. On flat surfaces, they are transformed into round
wheels to facilitate high-speed motion. Similar idea lies behind robots that
fold their whole body into the ball to roll on a plane [12, 13]. Another
solution was demonstrated by Nygaard et al. [14], in which a quadruped robot
changed its legs length on the fly; A terrain-adaption algorithm allows this
system to adjust its configuration to optimise energy efficiency and fit into
unstructured environments.

Several works have exploited variable morphology to create multi-functional
devices such as self-reconfigurable robots [15]. However, most of these
systems utilise rigid modules only. Many researchers have also worked towards
self-foldable programmable matter with soft modules [16, 17], but they
typically lack mobility. Karimi et al. [18] presented one of the pioneer
studies in hybrid self-reconfigurable systems. They developed a loop-shaped
robot composed of rigid mobile units linked by a flexible membrane that
becomes stiff by jamming its particles with an untethered vacuum system. This
solution demonstrated promising results in locomotion and object handling,
yet, its fixed number of active units may limit the system’s scalability.

This paper presents a new class of robots that can be used in numerous
applications as a single robot or as a unit of a self-reconfigurable modular
system and it will be called in a sequel a ”Self-Reconfigurable Soft-Rigid
Agent” (2SRA). Self-reconfigurable refers to the capability of our robot to
reconfigure its shape and its potential to be a part of a self-reconfigurable
modular system. The original contribution of this work is twofold: (i) The
development and experimental validation of a novel mobile robotic system with
phase transition capabilities; (ii) The derivation of a mathematical model
that captures the hybrid soft-rigid behaviour of the system and its locomotion
properties.

The rest of this paper is organised as follows: Sec. II describes the robot’s
design, Sec. III derives the mathematical models, Sec. IV proposes the control
strategy, Sec. V presents the experiments, and Sec. VI gives final
conclusions.

##  II Materials and Methods

###  II-A Variable-Stiffness Fibre

Variable-stiffness fibre (VSF) aims to alter the configuration and degrees of
freedom of a 2SRA. For omnidirectional motion, the fibre must be stiff and
capable of maintaining its shape. In the soft state, it should easily deform
under the bending torque produced by the locomotion units. Therefore, the
compliant features of the VSF are designed to have high flexibility and low
elasticity.

We adopted a method presented by Tonazzini et al.[19] to implement stiffness
variability in the fibre. Desired mechanical features are achieved by
integrating a low melting point alloy (LMPA) and a Joule heater for melting
the alloy into the soft tube. Due to relatively low melting temperatures
(47–62 ∘C) and large solid/liquid phase change, LMPAs became popular in
applications where controllable stiffness is required [20]. Among LMPAs, we
chose Field’s metal (composition by weight: 32.5% bismuth, 51% indium, and
16.5% tin). It is characterised by high absolute stiffness when solid (Young
modulus is more than 3GPa), a melting point around 62 ∘C and low viscosity[21,
22]. Unlike several other alloys, Field’s metal does not contain toxic
elements; therefore, it is easy and safe to utilise.

Figures 1(a, b) show the VSF and its inner structure. The fibre is designed as
a serial chain of Field’s metal VSS (short for the variable-stiffness segment)
with plastic links on both ends. This approach decreases fabrication
complexity and increases robustness since separately fabricated modular
segments can be easily replaced if they fail. For experiments, we fabricated a
fibre with three links and two VSS. Each plastic tube and VS segment are 30 mm
and 40 mm long, respectively. Heaters are fabricated by coiling an enamelled
copper wire (0.1 mm in diameter) around the silicone rod (5 mm in diameter). A
silicone tube (internal diameter is 7 mm; external diameter is 8 mm) coaxial
to the inner core is used to encapsulate the whole device. The Field’s metal
is distributed between the tube and the rod. NTC thermistors (MF52AT) embedded
in both fibre segments measure the metal’s temperature and provide feedback
for stiffness control. Plastic links closing the ends of the device prevent
metal leakage when it is in the liquid state and protect wires from external
impact.

At room temperature, the metal layer and the entire fibre are rigid. A soft
state of the fibre is achieved by applying current (0.7-0.9 A) to the heater
and melting the alloy. Separated heaters enable fibre segments to undergo
phase transition independently, which provides greater deformation diversity.

###  II-B Locomotion Units

We considered two main requirements when developing locomotion units:

  1. a)

In a soft state, LU’s must be able to actuate each end of the variable-
stiffness fibre and bend segments into arcs with the desired curvature.

  2. b)

In a rigid state, LU’s must be able to maintain a 3 DoF motion of a 2SR agent
on a plane with a given VSF configuration.

Figure 1: Variable-stiffness fibre (VSF) with two segments: (a) Perspective
view; (b) Cross-sectional view. (c) Locomotion unit (LU). (b) Assembled 2SRA.

To meet these conditions, LU’s are designed as a nonholonomic robot (Fig.1(c))
with two omni-wheels (radius of 10 mm) placed perpendicular to each other on
the adjacent unit sides. Their rotation in the same direction induces the
unit’s rotation, while the counter rotation promotes the linear motion along
the diagonal between the wheels. A castor wheel in the opposite corner from
the omni-wheels is used to stabilise the unit. Steel bars are placed inside
the LU frame to increase the unit weight, which helps to improve the wheels’
contact with the surface and to resist the restoring force produced by
deformed VS fibre. When VSF is rigid, units and the bridge create a single
agent with four omni-wheels, see Fig.1(d). Their mutual displacement
guarantees that the 2SR robot stays omnidirectional regardless of the fibre
shape.

A custom controller board with STM32F446RET6 microcontroller is designed to
control brushless DC motors driving omni-wheels and actuate VSSs. AS5600
magnetic encoders increase the accuracy of motors’ speed control. PWM output
with analogue signal captured from the thermistor as feedback is used to
control heaters. A wireless serial board provides communication with the
server. All electronics are encapsulated in the 3d-printed plastic frame.
($46\times 46\times 78$ mm).

##  III Modelling

Integrating entirely different natures of rigid and soft bodies in one robot
significantly expands its capabilities; however, it becomes challenging to
derive a unified model. This section introduces a hybrid strategy that
combines 2SRA configurations in both rigid and soft states and allows to
switch between different motion modes in a simple manner. This model is
adapted for a robot with two variable-stiffness segments, but it can be
customised for any fibre design.

###  III-A Analysis of the System’s Hybrid Kinematics

Due to the 2SRA dual nature, we conduct kinematic analysis separately for
rigid and soft states and then combine those pieces in one model. To simplify
calculations, we assume fibre segments are inextensible and strictly bend into
circular arcs with constant curvature. Also, we consider that motion of the
units in the soft state is independent of each other.

Figure 2(b) demonstrates a scenario when one fibre segment is soft, and the
2SR agent can change its shape. Locomotion units can move together or one at a
time. According to schematics in 2(a), omni-wheel indexing starts from the
left side wheel of LU1 and follows clockwise. Robot’s body frame $\\{b_{0}\\}$
is attached to the middle point of the middle plastic link. Two other frames
$\\{b_{1}\\}$ and $\\{b_{2}\\}$ are attached to the VSS ends. The angles
between these frames and the wheel frames $\\{w_{i}\\}$
$\left(i\in[1..4]\right)$ are $\beta_{1}=\frac{\pi}{2}$, $\beta_{2}=0$,
$\beta_{3}=-\frac{\pi}{2}$, and $\beta_{4}=\pi$ with the former two wheels
associated with $\\{b_{1}\\}$ and the latter two wheels associated with
$\\{b_{2}\\}$.

The biggest contribution to the fibre bending is provided by the side wheels
with axes of rotation parallel to adjacent links. Velocities tangential to
bending trajectory correspond to the maximum torque and thus are energy
efficient. On the other hand, the motion of the other wheels promotes free rod
deformation, which is challenging to perform since it requires high forces
that are not feasible with the current design. Therefore, we neglect these
wheels during VSS bending and keep their rotational speed at zero:
$\omega_{1}=\frac{1}{\rho_{\omega}}v_{1}$,
$\omega_{3}=-\frac{1}{\rho_{\omega}}v_{2}$, and $\omega_{2}=\omega_{4}=0$,
where $\omega_{i}$ is angular velocity of the $i^{th}$ wheel; $v_{1}$ and
$v_{2}$ are velocities of the first and second units traversing some curved
path; and $\rho_{\omega}$ is the wheel radius. In matrix representation, this
relationship is defined as
$\boldsymbol{\omega}=\mathbf{V}^{s}\boldsymbol{\upsilon}^{s}$:

$\mathbf{V}^{s}=\frac{1}{\rho_{\omega}}\begin{bmatrix}1&0&0&0\\\ 0&0&-1&0\\\
\end{bmatrix}^{\intercal},$ (1)

where $\mathbf{V}^{s}$ is the wheels configuration matrix when 2SR agent is in
a soft state; $\boldsymbol{\omega}=[\omega_{1}...\omega_{4}]^{\intercal}$ and
$\boldsymbol{\upsilon}^{s}=[v_{1},v_{2}]^{\intercal}$.

In the second scenario, the fibre is rigid, and 2SRA behaves like a
conventional mobile robot. A remarkable feature of this state is that
omnidirectional motion can be achieved at any fibre configuration since the
relative position of omni-wheels ensures three degrees of freedom. For
omnidirectional robots with four wheels, a generalised wheel model is defined
as:

$\omega_{i}=\begin{bmatrix}\frac{1}{\rho_{i}}&\frac{t\gamma_{i}}{\rho_{i}}\end{bmatrix}\begin{bmatrix}\text{c}\psi_{i}&\text{s}\psi_{i}\\\
-\text{s}\psi_{i}&\text{c}\psi_{i}\end{bmatrix}\begin{bmatrix}1&0&-\tensor[^{b_{0}}]{y}{{}_{w_{i}}}\\\
0&1&\tensor[^{b_{0}}]{x}{{}_{w_{i}}}\end{bmatrix}\begin{bmatrix}u_{0}\\\
v_{0}\\\ r_{0}\end{bmatrix},$ (2)

where $\omega_{i}$ is the $i^{th}$ wheel driving speed; $\rho_{i}$ is the
radius of the $i^{th}$ wheel; $\gamma_{i}$ is an angle between the roller axis
and the $\hat{x}_{i}$ axis of the wheel frame $\\{w_{i}\\}$; $\psi_{i}$ is an
angle between the body frame $\\{b_{0}\\}$ and $\\{w_{i}\\}$;
$\tensor[^{b_{0}}]{x}{{}_{w_{i}}}$ and $\tensor[^{b_{0}}]{y}{{}_{w_{i}}}$ are
position coordinates of $\\{w_{i}\\}$ with reference to $\\{b_{0}\\}$;
$u_{0}$, $v_{0}$, and $r_{0}$ are forward, lateral, and angular velocities of
the robot. By $\text{s}\psi_{i}$ and $\text{c}\psi_{i}$ we denote sine and
cosine of the angle $\psi_{i}$. Since all wheels have the same radius
$\rho_{\omega}$, and $\gamma_{i}$ for omni wheels is zero, equation (2) can be
rewritten in the form of
$\boldsymbol{\omega}=\mathbf{V}^{r}\boldsymbol{\upsilon}^{r}$:

$\mathbf{V}^{r}_{i}=\begin{bmatrix}\text{c}\psi_{i}&\text{s}\psi_{i}&\tensor[^{b_{0}}]{x}{{}_{w_{i}}}\text{s}\psi_{i}-\tensor[^{b_{0}}]{y}{{}_{w_{i}}}\text{c}\psi_{i}\end{bmatrix}/\rho_{\omega},$
(3)

where $\mathbf{V}^{r}_{i}$ is the $i^{th}$ row of a configuration matrix
$\mathbf{V}^{r}$ when a 2SR agent is in a rigid state and
$\boldsymbol{\upsilon}^{r}=[u_{0},v_{0},r_{0}]^{\intercal}$. Variables
$\psi_{i}$, $\tensor[^{b_{0}}]{x}{{}_{w_{i}}}$ and
$\tensor[^{b_{0}}]{y}{{}_{w_{i}}}$ depend on the fibre shape and thus to be
recalculated accordingly. Analysis of the relationship between them and
segments curvatures are given in the next section.

Figure 2: (a) Schematic representation of a 2SR robot with two variable-
stiffness segments (VSS). (b) Reconfiguration of a 2SR robot when one of its
segments is soft. The motion of a locomotion unit (LU) adjacent to the rigid
segment results in a displacement of the body frame $\\{b_{0i}\\}$. (c)
Geometrical features of constant curvature approximation for a VSS arc.

Combination of both approaches in
$\boldsymbol{\omega}=\mathbf{V}\boldsymbol{\upsilon}$ gives a vector of input
velocities
$\boldsymbol{\upsilon}=(\boldsymbol{\upsilon}^{s},\boldsymbol{\upsilon}^{r})$,
in which either $v_{1}$ and $v_{2}$ or $u_{0}$, $v_{0}$ and $r_{0}$ are non-
zero. This condition depends on the segments’ stiffness. To represent the
stiffness of the $j^{th}$ VSS, we introduce a boolean variable
$\varsigma_{j}$, which is equal to zero when the segment is rigid and one when
it is soft. Thus, we obtain a unified wheels configuration matrix:

$\mathbf{V}_{i}=\begin{bmatrix}\sigma(i=1)&-\sigma(i=3)&\overline{\sigma}\text{c}\psi_{i}&\overline{\sigma}\text{s}\psi_{i}&\overline{\sigma}\tau_{i}\end{bmatrix}/\rho_{\omega},$
(4)

where $\sigma=\varsigma_{1}\vee\varsigma_{2}$ and
$\tau_{i}=\tensor[^{b_{0}}]{x}{{}_{w_{i}}}\text{s}\psi_{i}-\tensor[^{b_{0}}]{y}{{}_{w_{i}}}\text{c}\psi_{i}$.
Expression in parentheses is a conditional statement, $\vee$ is a logical
disjunction, and notation $\overline{A}$ is a logical negation.

###  III-B Wheels Configuration

For the robots with fixed geometry, matrix $\mathbf{V}$ from (4) is constant.
However, a 2SRA is capable of changing its shape, which entails the variation
of the wheels’ coordinates relative to the body frame. Therefore, the issue to
address is determination of the $i^{th}$ wheel coordinates
$\tensor[^{b_{0}}]{\mathbf{q}}{{}_{w_{i}}}=[\tensor[^{b_{0}}]{x}{{}_{w_{i}}},\tensor[^{b_{0}}]{y}{{}_{w_{i}}}]^{\intercal}$
and $\psi_{i}$ for any robot configuration. Coordinates of the wheels with
reference to the associated fibre ends are always constant and can be
expressed as:

$\tensor[^{b}]{\mathbf{Q}}{{}_{w}}=\begin{bmatrix}-h_{1}&-h_{2}&h_{1}&h_{2}\\\
0&h_{3}&0&-h_{3}\end{bmatrix},$ (5)

where $h_{1}=\frac{2l_{1}+2a+d}{2}$, $h_{2}=\frac{2l_{1}+a}{2}$, and
$h_{3}=\frac{a+d}{2}$; $a$ is the block side length, $d$ is the wheel
thickness, and $l_{1}$ is the length of the plastic links at the ends of the
fibre. Positional coordinates w. r. t. the robot body frame $\\{b_{0}\\}$ are
further derived through a homogeneous transformation

$\begin{bmatrix}\tensor*[^{b_{0}}]{\mathbf{q}}{{}^{{\intercal}}_{w_{i}}}&1\end{bmatrix}^{\intercal}=\tensor[^{b_{0}}]{\mathbf{T}}{{}_{b_{j}}}\begin{bmatrix}\tensor*[^{b_{j}}]{\mathbf{q}}{{}^{{\intercal}}_{w_{i}}}&1\end{bmatrix}^{\intercal},$
(6)

where $\tensor[^{b_{j}}]{\mathbf{q}}{{}_{w_{i}}}$ is the $i^{th}$ column of
$\tensor[^{b}]{\mathbf{Q}}{{}_{w}}$ and $\\{b_{j}\\}$ denotes the fibre end
associated to the $i^{th}$ wheel. Thus, the solution to the problem is reduced
to mapping frames $\\{b_{j}\\}$ to the frame $\\{b_{0}\\}$, which depends on
the VSS arc curve.

One of the approaches to describe the curve and position of the points along
its length was suggested by Marchese et al.[23]. A soft arm is described as a
sequence of circular arcs of signed curvature $\kappa$. With given starting
orientation $\theta_{0}$, the position of any point along the arm is expressed
as:

$\begin{gathered}x_{j}(p)=x_{j-1}\left(L_{j-1}\right)+\int_{0}^{p}\text{c}\left(\kappa_{j}p^{\prime}+\theta_{0}\right)dp^{\prime}\\\
y_{j}(p)=y_{j-1}\left(L_{j-1}\right)+\int_{0}^{p}\text{s}\left(\kappa_{j}p^{\prime}+\theta_{0}\right)dp^{\prime},\end{gathered}$
(7)

where $j$ denotes the $j^{th}$ arm segment; $L_{j}$ is the length of the
$j^{th}$ segment; $p$ is the position along the segment. Since all VS segments
are inextensible, and their length $l$ remains constant during robot
deformation, we derive a transformation matrix from this model as follows:

$\tensor[^{b_{0}}]{\mathbf{T}}{{}_{b_{j}}}=\begin{bmatrix}\text{c}\alpha_{j}&\pm\text{s}\alpha_{j}&\mp\left(\frac{l_{0}}{2}+\frac{\text{s}\alpha_{j}}{\kappa_{j}}\right)\\\
\mp\text{s}\alpha_{j}&\text{c}\alpha_{j}&\frac{1-\text{c}\alpha_{j}}{\kappa_{j}}\\\
0&0&1\end{bmatrix},$ (8)

where $l_{0}$ is the length of the middle link, $j$ is the segment index, and
$\alpha_{j}=\kappa_{j}l$ is the degree of curvature, see Fig. 2(c). First sign
in $\pm$ or $\mp$ corresponds to $j=1$ and the second one to $j=2$.
Accordingly, we find an orientation of the wheel frame:

$\psi_{i}=\phi_{b_{0}}\mp\alpha_{j}+\beta_{i},$ (9)

where $\phi_{b_{0}}$ is orientation of the body frame $\\{b_{0}\\}$ w.r.t. to
the global frame.

###  III-C LU Paths Estimation

Figure 3: Shape deformation of a 2SR robot and corresponding logarithmic
spirals when: (a), (b) one soft VSS is bent by an adjacent LU; (c), (d) one
soft VSS is bent by the opposite LU; (e), (f) both segments are soft.

The further analysis involves estimating paths traced by LU’s when either of
the VS segments is soft. We consider segment deformation as an arc of a
constant length and curvature. Assuming that VSS can bend into a full circle
(though it is not feasible with a real robot), we can see that its tip traces
a curve, which reminds a piece of a logarithmic spiral, see Fig. 3. Matlab was
used to fit the spiral to the segment arc points according to the following
curve representation:

$\rho=ae^{b\theta},\quad x=\rho\text{c}\theta,\quad y=\rho\text{s}\theta,$
(10)

where $\theta$ is the spiral angle from $x$-axis; $\rho$ is the spiral radius
at an angle $\theta$; and $a$ and $b$ are arbitrary constants.

We distinguish three general motion modes at a soft state: one segment is
soft, and the adjacent unit is moving; one segment is soft, and the opposite
unit is moving; both segments are soft. Graphs in Fig. 3 demonstrate that each
mode corresponds to a differently shaped logarithmic spiral; therefore, we
first analyze each spiral and then combine their features into a unified
kinematic model.

TABLE I: Parameters of logarithmic spirals Spiral |  $a/l$ |  $b$ |  $c_{x}/l$ |  $c_{y}/l$
---|---|---|---|---
I |  2.325 |  $\pm 0.3165$ |  -0.1223 |  0.1782
II |  3.3041 |  $\pm 0.083$ |  0.1988 |  0.1640
III |  2.4471 |  $\pm 0.2229$ |  -0.2722 |  0.3949

The first considered case ($\varsigma_{1}=0$, $\varsigma_{2}=1$, $v_{1}=0$,
and $v_{2}\neq 0$) is illustrated in Fig. 3(a, b). By varying the fibre length
$l$ from 1 to 100 cm in Matlab we determined that spiral parameter $b$,
relationship between $l$ and $a$, and relationship between $l$ and spiral
centre $\\{c\\}$ are constant. Parameters of all three spirals are listed in
Table I. The sign of $b$ depends on curvature $\kappa$; if it is positive,
then $b$ is negative and vice versa. In a 2SRA, the change of curvature
$\kappa$ linearly depends on the change of $\theta$; therefore, we approximate
their relationship by mapping their ranges
$-\frac{\pi}{3}\leq\theta_{1}\leq\frac{7\pi}{3}$ and
$-\frac{2\pi}{l}\leq\kappa_{2}\leq\frac{2\pi}{l}$:

$\kappa_{j}=3\left(\theta_{1}-\pi\right)/(2l).$ (11)

Motion of the unit adjacent to a rigid VSS ($\varsigma_{1}=0$,
$\varsigma_{2}=1$, $v_{1}\neq 0$, and $v_{2}=0$) resembles a logarithmic
spiral even more clearly, see Fig. 3(c, d). Here the absolute value of $b$ is
sensitive to the ratio of the lengths of soft and rigid items. We applied
fibre custom values used during fabrication. Relationship between spiral angle
and segment curvature with given ranges $-\pi\leq\theta_{2}\leq 3\pi$ and
$-\frac{2\pi}{l}\leq\kappa_{2}\leq\frac{2\pi}{l}$ is:

$\kappa_{j}=\left(\theta_{2}-\pi\right)/l.$ (12)

The last case involves a 2SRA deformation with both segments being soft
($\varsigma_{1}=1$, $\varsigma_{2}=1$, $v_{1}=0$, and $v_{2}\neq 0$), see Fig.
3(e). Bending spreads uniformly along the fibre; thus, segment curvatures
change equally. According to the corresponding logarithmic spiral (Fig. 3(f)),
units meet each other faster since segments’ curvatures are summed up.
Therefore, we limit each VSS to bend into a half-circle so that
$-\frac{\pi}{3}\leq\theta_{3}\leq\frac{7\pi}{3}$ and
$-\frac{\pi}{l}\leq\kappa\leq\frac{\pi}{l}$:

$\kappa_{j}=3\left(\theta_{3}-{\pi}\right)/(4l).$ (13)

###  III-D Unified Jacobian Matrix

Configuration of a 2SR agent is fully defined by five generalised coordinates
$\mathbf{q}=[x_{b_{0}},y_{b_{0}},\phi_{b_{0}},\kappa_{1},\kappa_{2}]^{\intercal}$,
where $x_{b_{0}}$, $y_{b_{0}}$ and $\phi_{b_{0}}$ are position coordinates and
orientation of the body frame $\\{b_{0}\\}$ w.r.t. to the global frame, and
$\kappa_{j}$ is the curvature of the $j^{th}$ VSS. Then, the forward
kinematics of a 2SR agent is described as:

$\mathbf{\dot{q}}=\mathbf{J}\boldsymbol{\upsilon},$ (14)

where $\boldsymbol{\upsilon}=\mathbf{V^{\dagger}}\boldsymbol{\omega}$ and
$\mathbf{J}$ is a Jacobian matrix. $\mathbf{V^{\dagger}}$ is the Moore-Penrose
pseudo inverse of a configuration matrix $\mathbf{V}$ from (4).

In order to find a unified Jacobian matrix, we apply a state-centric approach
as in the previous sections. We divide a Jacobian matrix into two parts:
$\mathbf{J}=[\mathbf{J}^{s},\mathbf{J}^{r}]\in\mathbb{R}^{5\times 5}$, where
$\mathbf{J}^{s}$ and $\mathbf{J}^{r}$ are Jacobian matrices that correspond to
the soft and rigid states, respectively. $\mathbf{J}^{r}\in\mathbb{R}^{5\times
3}$ consists of a rotation matrix $R_{z}(\phi_{b_{0}})$ and two zero rows. The
matrix $\mathbf{J}^{s}\in\mathbb{R}^{5\times 2}$ maps velocity input commands
into $\mathbf{q}$ coordinates according to the logarithmic spirals traced by
locomotion units. Spirals are assigned serial numbers according to the order
described in the previous section.

VSS curvature can be determined through the spiral angle $\theta$, which rate
of change is proportional to the speed of the object tracing the spiral:
$\dot{\theta}=\frac{1}{\rho\left(\theta\right)}v$. By substituting the angle
$\theta_{k}$ by the curvature equations from (11)-(13), we receive a
relationship between curvature change and the LU velocity $v$:

$\begin{gathered}\dot{\kappa}_{j}=\frac{m_{k}}{l\rho_{k}(\kappa_{j})}v,\end{gathered}$
(15)

where $k=[1..3]$ denotes the serial number of the spiral,
$\mathbf{m}=\left[\frac{3}{2},1,\frac{3}{4}\right]^{\intercal}$ is the set of
constants that correspond to the $k^{th}$ spiral, and
$\rho_{k}(\kappa_{j})=a_{k}\text{exp}\left({b_{k}(\frac{\alpha_{j}}{m_{k}}+\pi)}\right)$.
Further, we denote the proportionality coefficient as
$K_{k,j}=\frac{m_{k}}{l\rho_{k}(\kappa_{j})}$.

The next step is to find the body frame coordinates. In case I, the middle
plastic link is stationary, thus
$\dot{x}_{b_{0}}=\dot{y}_{b_{0}}=\dot{\phi}_{b_{0}}=0$. States II and III
involve the motion of the middle link with a body frame attached to it. It’s
orientation is equal to the orientation of the tangent at the VSS arc point
closest to the body frame, which is calculated as
$\phi_{b_{0}}=\phi_{b_{j}}\pm\alpha_{j}$, see Fig. 2(c). It implies that the
rate of change of the robot orientation linearly depends on the rate of
curvature change:

$\dot{\phi}_{b_{0}}=\frac{m_{k}}{\rho_{k}(\kappa_{j})}v.$ (16)

We denote the proportionality coefficient as
$\Phi_{k,j}=\frac{m_{j}}{\rho_{j}(\kappa)}$.

Figure 4: (a) The relationship between 2SR robot coordinates and the geometry
of the logarithmic spiral I. (b) 2SRA control strategy.

Schematic representation of 2SRA geometry during shape deformation in Fig.
4(a) demonstrates the relationship between fibre frames and logarithmic spiral
I. The connection point _G_ between VSS and the middle link lies on the spiral
and its position w.r.t. the spiral centre $\\{c\\}$ is found according to
(10). Since the position of the point _F_ attached to the body frame origin is
fixed relative to _G_ , we find:

$\tensor[^{c}]{\emph{F}}{}=\begin{bmatrix}\frac{l_{0}c\alpha_{j}}{2}+\rho_{1}(\kappa_{j})c\frac{\alpha_{j}}{m_{1}}&\frac{l_{0}s\alpha_{j}}{2}+\rho_{1}(\kappa_{j})s\frac{\alpha_{j}}{m_{1}}\end{bmatrix}^{\intercal}.$
(17)

We can express frames $\\{b_{j}\\}$ through $\\{b_{0}\\}$ and $\\{c\\}$. Thus,
position coordinates of the point _F_ with respect to the global frame
$\\{o\\}$ can be expressed as a series of kinematic transformations as:
$\tensor[^{o}]{\emph{F}}{}=\tensor[^{o}]{\mathbf{T}}{{}_{b_{0}}}\tensor[^{b_{0}}]{\mathbf{T}}{{}_{b_{j}}}\tensor[^{b_{j}}]{\mathbf{T}}{{}_{c}}\tensor[^{c}]{\emph{F}}{}$,
which partial derivatives with respect to $\kappa_{j}$ provides the
relationship between the rate of change of the robot position coordinates, the
LU’s velocities $v_{j}$, and their logarithmic trajectories:

$x_{b_{0}}=\frac{\partial\tensor[^{o}]{\emph{F}}{{}_{x}}}{\partial\kappa_{j}}\frac{\partial\kappa_{j}}{\partial
t},\quad
y_{b_{0}}=\frac{\partial\tensor[^{o}]{\emph{F}}{{}_{y}}}{\partial\kappa_{j}}\frac{\partial\kappa_{j}}{\partial
t}.$ (18)

Denoting
$\boldsymbol{\Delta}_{k,j}=\frac{\partial\tensor[^{o}]{\emph{F}}{}}{\partial\kappa_{j}}K_{k,j}$,
$s_{1}=\overline{\varsigma_{1}}\varsigma_{2}$,
$s_{2}=\varsigma_{1}\overline{\varsigma_{2}}$,
$s_{3}=\varsigma_{1}\varsigma_{2}$, and
$\mathbf{C}_{k_{1},j}^{k_{2},k_{3}}=[\boldsymbol{\Delta}_{k_{1},j}^{\intercal},(-1)^{j+1}\Phi_{k_{1},j},K_{k_{2},1},K_{k_{3},2}]^{\intercal}$
we get:

$\mathbf{J}^{s}=\begin{bmatrix}s_{1}&-s_{1}&s_{2}&s_{1}\\\
s_{2}&s_{2}&s_{2}&s_{1}\end{bmatrix}^{\intercal}\circ\begin{bmatrix}\mathbf{C}_{2,2}^{1,2}\\\
\mathbf{C}_{2,1}^{2,1}\end{bmatrix}^{\intercal}+s_{3}\begin{bmatrix}\mathbf{C}_{3,2}^{3,3}\\\
\mathbf{C}_{3,1}^{3,3}\end{bmatrix}^{\intercal},$ (19)

where $\circ$ denotes the element-wise matrix multiplication.

##  IV Controller Design

A distinctive feature of a 2SR agent is the real-time transition between rigid
and soft states, which extends locomotion capabilities and increases the
robot’s performance in conforming to the surroundings. This quality is
exploited by a single-agent control strategy designed for scenarios with no
obstacles between the initial and target configurations. A block diagram in
Fig. 4(b) shows that the strategy comprises three main components: Motion
Planner, Wheel Drive, and Stiffness Controller.

At any given moment, a 2SR agent can switch between four locomotion modes:
omnidirectional motion and motion along three logarithmic spirals. A Motion
Planner aims to select a sequence of these modes most suitable for the given
task. Possible configurations of the VSF are defined by a set
$\boldsymbol{\varsigma}=[[0,0],[0,1],[1,0],[1,1]]^{\intercal}$. Each iteration
of the trajectory formation calculates unified Jacobian matrices corresponding
to the elements of $\boldsymbol{\varsigma}$. They are further supplied to the
inverse kinematics equations. Among the Euclidean distances between the target
configuration and the set of potential configurations, we choose the smallest
one. However, in this approach, we might get solutions involving frequent
changes in segments’ stiffness. Phase transitions of the Field’s metal take
time; therefore, these situations are highly undesirable. We avoid this issue
by maintaining established stiffness values as long as the robot drives
towards the target. Once an intermediate configuration stops updating, the
locomotion mode is changed to the appropriate option. The Motion Planner
pseudocode is given in Algorithm 1.

Input: Initial configuration $q_{0}$, target configuration $q_{t}$

Output: List of stiffness values $\Omega$, list of velocities $\mathcal{V}$

1 $\varsigma\leftarrow$ set of possible VSF stiffness values

2 $\Omega,\mathcal{V}\leftarrow$ empty lists

3 while  _$||q\textbf{}-q_{t}|| >0$_ do

4 for  _$i=1$ to $4$_ do

5 $J\leftarrow$ hybridJacobian$\left(q_{0},q,\varsigma[i]\right)$

$\upsilon[i]\leftarrow J^{-1}\lambda(q_{t}-q)$

// $\lambda$ \- feedback gain

6 $\dot{q}\leftarrow J\upsilon[i]$

7 $q^{*}[i]\leftarrow q+\dot{q}dt$

8 $\Delta q^{*}\leftarrow||q^{*}-q_{t}||$

9 $i^{*}\leftarrow\underset{i}{\mathrm{arg\,min}}\,\Delta q^{*}$

10 if  _$i^{*}\neq i_{prev}$_ then

11 if  _$||q-q^{*}[i_{prev}]|| >$ 0_ then

12 $i^{*}\leftarrow i_{prev}$

13

14

15 $q\leftarrow q^{*}[i^{*}]$

16 Add $\varsigma[i^{*}]$ to $\Omega$; Add $\upsilon[i^{*}]$ to $\mathcal{V}$

Algorithm 1 2SR Agent Motion Planner

A sequence of 2SRA velocities returned by the Motion Planner is converted into
the wheels’ velocities by Wheel Drive defined by the model derived in (1)-(7).
A sequence of VSSs’ stiffness values is mapped to the temperature values that
are required to either melt the Field’s metal (taken with margin) for the soft
state or to cool it down for the rigid state:
$T_{j}(\varsigma_{j})=65^{\circ}$ if $\varsigma_{j}=1$ else $25^{\circ}$.

The stiffness controller maintains the melting point when it is requested and
notifies the main controller when the required temperature has been achieved.
A PI controller regulates the current applied to the heater and receives
feedback from the sensor. Temperature is measured via an NTC MF52AT
thermistor. We constrain its operating range to $[0^{\circ}...85^{\circ}]$ to
avoid overheating the copper coil. This range is mapped to voltage output
$[1.1,\ldots,3.3]$ V.

##  V Results

###  V-A Experimental Setup and Simulation

Figure 5(a) shows an experimental set-up, which involved a 2SR agent with
three Aruco markers attached to the locomotion units and the central VSF link,
RealSence camera, thermal camera and a server. The coordinates of the VSF
marker corresponded to the position coordinates of the robot $b_{0}$ frame,
while the LU markers were used to estimate the segments’ curvature with the
assumption that it is constant along the arc. The agent’s motion was kept
within the tracked boundaries defined by Aruco markers, which, with the other
markers, were captured by the RealSense camera. A thermal camera was used to
analyse VSS temperature.

Figure 5: (a) Experimental setup. (b) Simulation results: difference between
configuration trajectories generated by Forward Kinematics (FK) and Motion
Planner (MP).

Before conducting experiments, we passed our model through a simulator
deployed in a Python environment. Plots in 5(b) compare the forward kinematics
(FK) and Motion Planner (MP) trajectories. We can see that the configuration
change in FK is linear, while MP generates a slower start but reduces the
Euclidean distance between initial and target configurations faster. It
happens because the Algorithm 1 works similarly to the gradient descent, which
results in a slight discrepancy between the positional coordinates of the
target configuration and the configuration reached by the robot upon deforming
into the desired shape. Therefore, 2SRA has to switch stiffness and motion
modes to fit the target completely. Within 100 simulations, a robot exploited
at most four motion modes, in which the last one was always rigid.

Figure 6: (a) Five different scenarios of the pose-shape formation when a
robot has a task to reach some target configuration. The top row shows the
behaviour of a 2SR agent in the simulator, and the bottom row is the robot’s
real-case motion/deformation. Blue colour corresponds to a soft segment; red
colour denotes a rigid segment. A faded image shows a target, while the bright
one is a configuration within a current trajectory. (b) Thermal image of a
variable-stiffness segment in a soft state. The bight-coloured area shows the
high-temperature zone corresponding to the molten LMPA inside the segment’s
shell. (c) Contours and a median curve of the bent segment extracted from a
thermal image. (d) Comparison of the median curve, its smoothed version and
the approximated curve with constant curvature along its length.

###  V-B Curvature Estimation

The proposed hybrid model uses a constant curvature (CC) approximation of the
fibre segments. We evaluated the actual curvature distribution along the
segment to validate a CC assumption. One segment was maintained rigid and
straight while its adjacent locomotion unit was fixed on the plane. Another
segment was bent in both directions with a degree of curvature varying from
$[-\pi...\pi]$. The hot area of the soft VSS was extracted from the images
captured by the thermal camera, see Fig. 6(b) and further processed for
curvature analysis. Binary image in Fig. 6(c) shows the segment contours and
median curve constructed via the OpenCV library. Figure 6(d) compares the
estimated, smoothed and approximated segment arcs. Finally, calculating the
gradient in each point of the smoothed median curve returned a corresponding
array of curvatures. Test trials were repeated five times.

Results of the experiment presented in figure 7(d) show that the constant
curvature and the mean curvature of the median curves extracted from the
thermal images are quite similar at the small values, and the difference
between them increases with larger bending. The overall mean square error is
41.32, and the average standard deviation is 12.93. According to the obtained
results, CC approximation adequately represents the segment curvature.

###  V-C Pose-Shape Formation

Pose-Shape Formation tasks involve reaching a given target with simultaneous
motion on the plane and bending the segments. We implemented a control
strategy provided in Section IV and conducted five experiments with different
initial and target configurations. They were obtained by manually deforming
the robot, placing it in the desired spot and measuring the corresponding
configuration through a RealSense camera. The same inputs were tested in the
simulator to assess the errors occurring in the real-case scenario.

All trials differ in VSF stiffness values, motion modes and execution time.
Figure 6(a) demonstrates the execution process of the tasks and corresponding
simulations. In both scenarios, a 2SRA shows a similar movement pattern. At
each moment of the robot motion/deformation, we assessed a reference
configuration selected by a Motion Planner and an estimated configuration
measured through camera. The comparison of the trajectories obtained during
the 3rd experiment and its simulation is provided in Fig. 7(a). The best fit
is displayed for the curvature and orientation coordinates. As was mentioned
in Sec. V-A, a discrepancy in the position trajectories occurs since the
algorithm aims to minimize the Euclidean distance $\Delta q=||q-q_{t}||$ in
the fastest way, which preferences the curvature over position coordinates.
Therefore, the robot fits $x_{b_{0}}$ and $y_{b_{0}}$ the last. Figure 7(c)
shows how the robot approaches the target. We stopped the motion as soon as
$\Delta q$ hit a given threshold instead of trying to reach zero, which helped
avoiding frequent stiffness alternation. Bar plot in Fig. 7(b) summarises the
results of all experiment cases by calculating the Dynamic Time Warping (DTW)
distance between the simulated and experimental configuration trajectories.
These errors are not significant, and a 2SR robot is capable of approaching
the target under the proposed hybrid model.

Figure 7: (a) Comparison of the configuration trajectories obtained during the
Pose-Shape Formation experiment (3rd case) and the corresponding simulation.
Reference is the configuration returned by a Motion Planner and further
supplied to the robot; Estimated value is the configuration measured via
camera and Aruco markers. (b) Quantified difference (DTW distance) between the
configuration trajectories from the Pose-Shape Formation experiments and
corresponding simulations. (c) Minimization of the error $\Delta q$ during
task execution. (d) Comparison of the VSS constant curvature (CC)
approximation and mean curvature (MC) of the segment arc extracted from
thermal images. Solid coloured area demonstrates curvature distribution along
the arc.

###  V-D Applications

Figure 8: 2SR agent performing objects handling tasks: (a) conforming to an
object with a cylindrical shape. (b) gripping and moving a cube on the plane.

The adaptive nature of a 2SR agent makes it applicable to a large variety of
tasks. One of its main advantages is that a robot can conform to objects with
different shapes. In Fig. 8(a), a 2SR agent approaches a cylindrical object
and bends one of its segments until it fits the given shape. Another example
shown in Fig. 8(b) demonstrates the object manipulation by a 2SRA. Execution
of this task runs as follows: a robot approaches the target, makes both
segments soft, grasps the cube and switches back to the rigid state. After
fibre solidification, a robot is capable of moving freely on a plane while
handling the cube.

These are a few examples of how a 2SRA can be used. Potential applications are
not limited to any specific area since, unlike other mobile robots, a 2SR
agent aims to combine the best features of conventional and soft robots with
adaptive morphology. Comparison of a 2SRA with the state of the art is given
in Table II.

TABLE II: Comparison of a 2SR agent with the state of art |  [10] |  [12] |  [18] |  2SRA
---|---|---|---|---
Shape shifting |  ✓ |  ✓ |  ✓ |  ✓
Conforming to surroundings |  ✓ |  ✗ |  ✓ |  ✓
Variable stiffness |  ✗ |  ✗ |  ✓ |  ✓
Variable motion modes |  ✓ |  ✓ |  ✗ |  ✓
High-speed locomotion |  ✓ |  ✓ |  ✗ |  ✓
Untethered actuation |  ✗ |  ✓ |  ✓ |  ✓

##  VI Conclusion

The paper presented a new type of mobile robot that can behave either as a
conventional rigid mobile agent or as a hybrid soft robot. It consists of two
locomotion units connected by a serial variable-stiffness bridge. VSF enables
the transition between rigid and soft states and the robot morphing. LU’s
carry power and electronics, perform 2d motion and actuate the fibre. A
unified hybrid model is developed to describe the behaviour scenarios of a 2SR
agent in both states. Shape deformation and unit motion when some or all VSF
segments are soft are described based on approximated logarithmic spirals
traced by the units. The proposed design and model were tested in simulation
and physical experiments.

Experiments in reaching the target and handling the objects demonstrated
promising results. A 2SRA benefits from the ability to adapt to the
surroundings and chooses the most appropriate stiffness and motion mode to
execute the task. In future work, we will focus on reducing the phase
transitioning time and developing a modular system that contains multiple 2SR
agents. It will significantly increase their performance as a part of the
swarm and expand the area of their applications.

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