# A Novel Model for Layer Jamming-based Continuum Robots

Bowen Yi†, Yeman Fan†, and Dikai Liu This work was supported in part by the
Australian Research Council (ARC) Discovery Project under Grant DP200102497,
and by the Robotics Institute, University of Technology Sydney, Australia.The
authors are with the Robotics Institute, University of Technology Sydney, NSW
2007, Australia. Email:{bowen.yi<EMAIL_ADDRESS><EMAIL_ADDRESS>Yi and Y. Fan contributed equally to
this work. (Corresponding author: Y. Fan)

###### Abstract

Continuum robots with variable stiffness have gained wide popularity in the
last decade. Layer jamming (LJ) has emerged as a simple and efficient
technique to achieve tunable stiffness for continuum robots. Despite its
merits, the development of a control-oriented dynamical model tailored for
this specific class of robots remains an open problem in the literature. This
paper aims to present the first solution, to the best of our knowledge, to
close the gap. We propose an energy-based model that is integrated with the
LuGre frictional model for LJ-based continuum robots. Then, we take a
comprehensive theoretical analysis for this model, focusing on two fundamental
characteristics of LJ-based continuum robots: shape locking and adjustable
stiffness. To validate the modeling approach and theoretical results, a series
of experiments using our OctRobot-I continuum robotic platform was conducted.
The results show that the proposed model is capable of interpreting and
predicting the dynamical behaviors in LJ-based continuum robots.

##  I Introduction

Continuum robots can be used in many applications due to their inherent
flexibility and light weight. When interacting with the environment or humans,
there is a need to actively change the dynamical response of the robots,
particularly mechanical impedance. Indeed, numerous continuum robots have
integrated variable stiffness techniques within their design, allowing for
flexible soft motion or rigid resistance and greatly expanding their range of
applications [16, 4].

Among various stiffening techniques, jamming approaches have shown great
success in adjustable stiffness continuum robots with rapid reversible
responses [5, 19, 15]. They can be broadly classified into fiber, granular,
and layer jamming. Notably, layer jamming (LJ), the concept of which was
originally proposed in [12, 13], has received particular attention due to
light weight and compactness. It utilizes thin plastic or paper layers as its
jamming flaps. For LJ-based continuum robots, there is an airtight pneumatic
chamber in which a series of overlapping layers are installed to cover the
robot spine or wrapped as the robot body. This mechanism exploits the friction
between layers that can be controlled by external pressure via a vacuum, and
provides a large range of controllable stiffness.

Feedback control is one of the most important topics in the field of continuum
robotics, though still in infancy. Over the past few years, _model-based_
approaches have gained resurgence since more experimental evidence has shown
that feedback approaches are robust to approximations for continuum robot
dynamics [7, 9]. However, as figured out in [17], the modeling of LJ-based
continuum robots with variable stiffness has not been well addressed yet.
There are some recent works on analytical or computational models to
characterize the mechanism of stiffness variation in LJ-based continuum robots
[17, 22, 13]. However, to the best of the authors’ knowledge, there is no
_control-oriented_ model in the literature that approximates their dynamical
behaviors.

In this paper, we aim to close the above-mentioned gap by proposing a novel
model for a class of LJ-based, tendon-driven continuum robots. It integrates
the energy-based modeling technique and the LuGre frictional model [1]. The
overall model is in port-Hamiltonian form with the vacuum pressure gradient as
an additional control input. We theoretically prove the model’s ability to
illustrate the important phenomena of shape locking and adjustable stiffness.
Besides, we present an analytical relation between stiffness and negative
pressure.

Figure 1: Schematic of a layer-jamming structure in continuum robots

Notation. All functions and mappings are assumed to be $C^{2}$-continuous.
$I_{n}$ is the $n\times n$ identity matrix, $0_{n\times s}$ is an $n\times s$
matrix of zeros, and
$\mathbf{1}_{n}:=\mbox{col}(1,\ldots,1)\in\mathbb{R}^{n}$. For
$x\in\mathbb{R}^{n}$, $S\in\mathbb{R}^{n\times n}$, $S=S^{\top}>0$, we denote
the Euclidean norm $|x|^{2}:=x^{\top}x$, and the weighted–norm
$\|x\|^{2}_{S}:=x^{\top}Sx$. Given a function $f:\mathbb{R}^{n}\to\mathbb{R}$,
we define the differential operators $\nabla f:=(\frac{\partial f}{\partial
x})^{\top},\;\nabla_{x_{i}}f:=(\frac{\partial f}{\partial x_{i}})^{\top},$
where $x_{i}\in\mathbb{R}^{p}$ is an element of the vector $x$. The set
$\bar{n}$ is defined as $\bar{n}:=\\{1,\ldots,n\\}$. We use
$\mbox{diag}\\{x_{i}\\}~{}(i\in\bar{n})$ to represent the diagonal matrix
$\mbox{diag}\\{x_{1},\ldots,x_{n}\\}$, and define the set
$B_{\varepsilon}({\cal X}):=\\{x\in\mathbb{R}^{n}:\inf_{y\in{\cal
X}}|x-y|\leq\varepsilon\\}$ for a given set ${\cal X}\in\mathbb{R}^{n}$. When
clear from the context, the arguments of the functions may be omitted.

##  II Dynamic Modeling

###  II-A Preliminary of Jamming-free Model

In our previous work [21], we consider the control-oriented modeling of a
class of underactuated tendon-driven continuum robots. A high-dimensional
rigid link model is used to approximate the dynamical behavior of continuum
robots as follows:

$\begin{bmatrix}\dot{q}\\\
~{}\dot{p}~{}\end{bmatrix}=\begin{bmatrix}~{}0_{n\times n}&I_{n}\\\
-I_{n}&-D(q)~{}\end{bmatrix}\begin{bmatrix}\nabla_{q}H\\\
~{}\nabla_{p}H~{}\end{bmatrix}+\begin{bmatrix}0_{n}\\\
~{}G(q)u~{}\end{bmatrix}$ (1)

with the configuration variable $q\in{\cal X}\subset\mathbb{R}^{n}$, the
generalized momentum $p\in\mathbb{R}^{n}$, the input matrix
$G:\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$, the damping matrix
$D(q)\in\mathbb{R}^{n\times n}_{\succeq 0}$, and the tension input
$u\in\mathbb{R}^{m}$. The total energy is characterized by the Hamiltonian as

$H(q,p)=\frac{1}{2}p^{\top}M^{-1}(q)p+U(q),$ (2)

where $M:\mathbb{R}^{n}\to\mathbb{R}^{n\times n}_{\succ 0}$ is the positive
definite inertial matrix satisfying $m_{1}I\preceq M(q)\preceq m_{2}I$ for
some $m_{2}\geq m_{1}>0$, and the potential energy function $U(q)$ contains
the gravitational part $U_{\tt G}$ and the elastic part $U_{\tt E}$ that are
functions of $q$, i.e.

$U(q)=U_{\tt G}(q)+U_{\tt E}(q).$ (3)

It is shown in [21] that these functions can be modeled as

$\displaystyle U_{\tt G}$ $\displaystyle~{}=~{}\alpha_{1}[1-\cos(q_{\Sigma})]$
(4) $\displaystyle U_{\tt E}$ $\displaystyle~{}=~{}\frac{1}{2}q^{\top}\Lambda
q+U_{0}$

with the diagonal matrix
$\Lambda:=\mbox{diag}\\{\alpha_{2},\ldots,\alpha_{2}\\}$, some positive scalar
$U_{0}$ and $q_{\Sigma}:=\sum_{i\in\bar{n}}q_{i}$. Note that $\alpha_{1}$ and
$\alpha_{2}$ are some elastic and gravitational coefficients, respectively. We
refer the interested reader to [21] for more details about the robotic
structure and its modeling procedure.

###  II-B Friction Model for Layer Jamming

In continuum robots, the layer jamming technique provides a lightweight and
rapid response approach to adjust the robots’ stiffness [17, 3, 10]. In these
robots, layer jamming – consisting of a laminate of flexible strips or sheets
– is installed throughout the continuum robot’s body, and winded up into a
tube sheath, as illustrated in Fig. 1. In [21], the robot is operated in the
mode that behaves highly compliant. Meanwhile, the jamming sheath forms an
enclosed structure in which we may apply a negative pressure $-u_{\tt P}\leq
0$ (relative to the atmospheric pressure) [8]. As a result, friction between
strips or sheets would increase dramatically, thus changing the robot
stiffness and dissipating energy [17].

As described above, the pressure value $u_{\tt P}\in\mathbb{R}_{\geq 0}$ can
be adjusted online and viewed as an additional input that changes the robotic
dynamics. One of the main objectives of this paper is to propose a control-
oriented dynamical model for LJ-based continuum robots. For control purposes,
it is useful to have simple models that describe the essential properties of
continuum robots with layer jamming; in particular, when the pressure $u_{\tt
P}=0$, the model to be obtained should degrade into the LJ-free model in
Section II-A.

First, we discuss the dependence of the plant parameters $\alpha_{i}$
($i=1,2$) and the function $D(q)$ on $u_{\tt P}$. We make the following
assumptions.

###### Assumption 1

During the variation process of $u_{\tt P}$, the continuum robot satisfies

  * (a)

The mass change of air in the airtight membrane is negligible. Hence, the
gravitational parameter $\alpha_{1}$ is constant and thus _independent_ of the
pressure $u_{\tt P}$.

  * (b)

The elastic coefficient $\alpha_{2}>0$ is constant.

###### Assumption 2

The energy dissipation of the robot is only derived from the lumped friction
torque with $D(q)=0$.

Under the above assumptions, the model (1) of the LJ-based continuum robot can
be compactly written as

$\Sigma_{r}:\quad\left\\{\begin{aligned} \dot{x}&=J\nabla
H(x)+G_{r}(x)u-G_{f}\tau_{f}\\\ v&=G_{f}^{\top}\nabla
H(x)\end{aligned}\right.$ (5)

with the new variable $x:=\mbox{col}(q,p)$ and

$\displaystyle G_{r}$ $\displaystyle:=\begin{bmatrix}~{}0_{n\times m}~{}\\\
G(q)\end{bmatrix},~{}$ $\displaystyle G_{f}$
$\displaystyle:=\begin{bmatrix}~{}0_{n\times n}~{}\\\ I_{n}\end{bmatrix},~{}$
$\displaystyle J$ $\displaystyle:=\begin{bmatrix}~{}0_{n\times n}&I_{n}\\\
-I_{n}&0_{n\times n}~{}\end{bmatrix},$

where $\tau_{f}\in\mathbb{R}^{n}$ is the lumped frictional torque acting in
the links. If we view the friction $\tau_{f}$ as the “input”, then the
_passive output_ $v\in\mathbb{R}^{n}$ is, indeed, the generalized velocity
[18, 20], i.e.,

$v=M^{-1}(q)p.$ (6)

The jamming phenomenon is due to the distributed friction along the layers,
and the remaining task boils down to studying the modeling of the frictional
effects from $\tau_{f}$ and its interconnection to the system $\Sigma_{r}$.

To take this behavior into account in the model, we consider the LuGre
friction model which was proposed in [6]. It is a _dynamical_ model capable of
describing many frictional properties, such as zero slip displacement (a.k.a.
micromotion), slick-slip motion, invariance, state boundedness, and passivity
[1].

Before presenting the LuGre model, we make the following assumption about the
(lumped) normal force $F_{n}>0$ between the surfaces.

###### Assumption 3

The pressure along the layer is uniformly distributed with the value $(-u_{\tt
P})$ and is proportional to the lumped normal force, i.e., $F_{n}\propto
u_{\tt P}$.

To facilitate the following analysis, we adopt the port-Hamiltonian form of
the LuGre model [14]:

$\Sigma_{z}:\quad\left\\{\begin{aligned} \dot{z}&=-R_{z}(v)\nabla
H_{z}(z)+[{\cal N}(v)-{\cal P}(v)]v\\\ \tau_{f}&=[{\cal N}(v)+{\cal
P}(v)]^{\top}\nabla H_{z}(z)+Sv,\end{aligned}\right.$ (7)

where $z\in\mathbb{R}^{n}$ represents the virtual bristle deflection at each
joint, $v\in\mathbb{R}^{n}$ is the input – the relative generalized velocity
of the surfaces in contact given by (6), and the output
$\tau_{f}\in\mathbb{R}^{n}$ is the frictional torque. The mappings in
$\Sigma_{z}$ include the virtual bristle potential energy

$H_{z}(z)=\frac{1}{2}\sigma_{0}u_{\tt P}|z|^{2},$ (8)

the damping matrix

$\displaystyle R_{z}(v)$
$\displaystyle~{}=~{}\mbox{diag}(\beta_{1}(v),\ldots,\beta_{n}(v))$ (9)
$\displaystyle\beta_{i}(v)$ $\displaystyle~{}:=~{}{|v_{i}|\over u_{\tt
P}\rho(v_{i})},\quad i\in\bar{n}$

the state-modulation matrices

$\displaystyle{\cal N}(v)$
$\displaystyle~{}:=~{}I_{n}-\frac{1}{2}\sigma_{1}u_{\tt P}R_{z}(v)$ (10)
$\displaystyle{\cal P}(v)$ $\displaystyle~{}:=~{}-\frac{1}{2}\sigma_{1}u_{\tt
P}R_{z}(v)$ $\displaystyle S$
$\displaystyle~{}:=~{}(\sigma_{1}+\sigma_{2})u_{\tt P}I_{n},$

and the function

$\rho(v_{i})=\mu_{C}+(\mu_{S}-\mu_{C})\exp\left(-\left|{v_{i}\over
v_{s}}\right|^{\sigma_{3}}\right).$ (11)

Physical meanings of coefficients in the above model are summarized in Table
I. The interested reader may refer to [6, 2, 14] for additional details.

TABLE I: List of Coefficients in the LuGre Model $\mu_{S}$ | Stiction friction coefficient
---|---
$\mu_{C}$ | Coulomb friction
$v_{s}$ | Stribeck velocity
$\sigma_{0}$ | Bristle stiffness coefficient
$\sigma_{1}$ | Bristle damping coefficient
$\sigma_{2}$ | Viscous friction coefficient
$\sigma_{3}$ | Curve parameter (further tuning of the Stribeck effects)

###### Remark 1

The model $\Sigma_{z}$ is well-posed for all $u_{\tt P}\geq 0$ even though
$u_{\tt P}$ appears in the denominator of the function $\beta_{i}$ in (9).
There is due to the product $R_{z}(v)\nabla H_{z}$ in the dynamics and $\nabla
H_{z}$ being linear in $u_{\tt P}$. If the pressure $u_{\tt P}=0$, we have
$\tau_{f}=0$ for which we roughly regard zero friction injected to the robotic
mechanical dynamics $\Sigma_{r}$. The friction torque $\tau_{f}$ at the
steady-state stage becomes $\tau_{f}^{\tt
ss}=[\mbox{diag}\\{\rho(v_{i})\\}\mbox{sign}(v)+\sigma_{2}v]u_{\tt P}$, with
$\mbox{sign}(v):=\mbox{col}(\mbox{sign}(v_{1}),\ldots,\mbox{sign}(v_{n}))$
collecting the signs of $v_{i}$.

###### Remark 2

The LuGre model has the boundedness property for the internal state, i.e., the
set ${\cal
E}_{z}:=\\{z\in\mathbb{R}^{n}:~{}|z|\leq{\mu_{S}\over\sigma_{0}}\\}$ [1]. The
input-output pair $(v,\tau_{f})$ satisfies the particularly appealing
passivity property

$\int_{0}^{t}v^{\top}(s)\tau_{f}(s)ds\geq H_{z}(z(t))-H_{z}(z(0)),\quad\forall
t\geq 0$

if the coefficients satisfy some inequality constraints [2].

###  II-C Variable Stiffness Model with Layer Jaming

The overall dynamical model for the LJ-based continuum robots is the negative
interconnection of $\Sigma_{r}$ and $\Sigma_{z}$. For convenience, we define
the full systems state as

$\chi:=\mbox{col}(q,p,z)\in\mathbb{R}^{3n}.$ (12)

Its dynamics can be compactly written in port-Hamiltonian form as [14]

$\dot{\chi}=[{\cal J}-{\cal R}]\nabla{\cal H}+{\cal G}(\chi)u$ (13)

with the total Hamiltonian

$\displaystyle{\cal H}(\chi,u_{\tt P})$
$\displaystyle~{}:=~{}H(q,p)+H_{z}(z,u_{\tt P})$ (14)
$\displaystyle~{}=~{}\underbrace{\frac{1}{2}p^{\top}M^{-1}(q)p}_{\small\mbox{kinematic
energy}}+\underbrace{\frac{1}{2}\sigma_{0}u_{\tt
P}|z|^{2}+U(q)}_{\small\mbox{potential energy}}$

and the matrices

$\displaystyle{\cal J}(\chi,u_{\tt P})$
$\displaystyle~{}:=~{}\begin{bmatrix}J&-G_{f}{\cal N}^{\top}~{}\vspace{0cm}\\\
~{}{\cal N}G_{f}^{\top}&0_{n\times n}\end{bmatrix}$ (15) $\displaystyle{\cal
R}(\chi,u_{\tt P})$
$\displaystyle~{}:=~{}\begin{bmatrix}~{}G_{f}S(v)G_{f}^{\top}&G_{f}{\cal
P}^{\top}~{}\vspace{0cm}\\\ {\cal P}^{\top}G_{f}^{\top}&R_{z}\end{bmatrix}$
$\displaystyle{\cal G}(\chi)$
$\displaystyle~{}:=~{}\begin{bmatrix}G_{r}^{\top}&0_{n\times
m}^{\top}\end{bmatrix}^{\top}.$

Note that ${\cal N},{\cal P}$ and $S$ are linear functions of the pressure
$u_{\tt P}$. The overall model has an $(m+1)$-dimensional input

$u_{\chi}=\begin{bmatrix}~{}u~{}\\\ ~{}u_{\tt P}~{}\end{bmatrix}$

with all the elements non-negative.

###### Remark 3

The damping matrix ${\cal R}$ can be expanded as ${\cal
R}=\mbox{diag}(0_{n\times n},{\cal R}_{22})$ with

${\cal R}_{22}:=\begin{bmatrix}(\sigma_{1}+\sigma_{2})u_{\tt
P}I_{n}&-\frac{1}{2}\sigma_{1}u_{\tt P}R_{z}(v)\vspace{.2cm}\\\
-\frac{1}{2}\sigma_{1}u_{\tt P}R_{z}^{\top}(v)&R_{z}(v)\end{bmatrix}.$

Clearly, the positive semidefinitenss of ${\cal R}$ is equivalent to

$\sigma_{1}+\sigma_{2}-{|v_{i}|\over 4\rho(v_{i})}\geq 0,\quad i\in\bar{n}.$
(16)

For any coefficient, a small $|v|$ can always guarantee (16), thus making
${\cal R}$ qualified as a _damping_ matrix.

##  III Interpretation to Key Phenomena

In this section, we theoretically verify that the model proposed in Section
II-C can interpret the two phenomena in LJ-based continuum robots – shape
locking and tunable stiffness.

###  III-A Shape Locking

Shape locking is one of the important capabilities of LJ structures when
applied to continuum robots [22, 17, 19]. Tensions along the cables can change
the robot’s configuration from its undeformed shape; when a vacuum with
negative pressure $(-u_{\tt P})$ is applied before the release of tension
actuation, the continuum robot is able to preserve its current shape. This
phenomenon is known as shape locking. In this subsection, we aim to illustrate
that shape locking can be characterized by the proposed model. First, we
formulate its mathematical definition as follows.

###### Definition 1

(Shape Locking) Consider the LJ-based continuum robotic model with zero input
$u$. If the deformed configuration $\bar{q}\neq 0_{3}$ guarantees the set
${\cal E}_{\tt SL}:=\\{(q,p,z)\in\mathbb{R}^{3n}:q=\bar{q},~{}p=0_{3}\\}$
under $u_{\tt P}>0$ forward invariant, i.e.,

$\chi(0)\in{\cal E}_{\tt
SL}~{}\implies~{}[~{}\dot{q}(t)=0,~{}\dot{p}(t)=0,~{}\forall t\geq 0~{}],$
(17)

then we call this invariance as shape locking.

The following proposition gives a theoretical analysis of the shape-locking
phenomenon using the proposed model. Immediately after its proof, we will show
an intuitive illustration.

###### Proposition 1

Consider the LJ-based continuum robot model (13) without external input, i.e.,
$u=0_{m}$. For arbitrary configuration $q_{a}\in\mathbb{R}^{n}$ and a
_constant_ pressure $u_{\tt P}>0$,

  * (a)

There exists a vector $z_{a}\in\mathbb{R}^{n}$ such that $(q_{a},0_{n},z_{a})$
is an equilibrium;

  * (b)

The equilibria manifold

${\cal M}:=\\{(q,p,z)\in\mathbb{R}^{3n}:p=0,~{}\nabla U(q)=\sigma_{0}u_{\tt
P}z\\}$

is locally asymptotically stable.

###### Proof:

First, let us verify the existence of $z_{a}$ such that $(q_{a},0_{n},z_{a})$
is an equilibrium. From (6), $p=0$ implies the velocity $v=0$, thus

$\dot{q}=\nabla_{p}H=M^{-1}(q)p=0.$

The dynamics of $z$ is given by

$\dot{z}=-R_{z}(0)\nabla H_{z}+[{\cal N}-{\cal P}]v\Big{|}_{v=0}=0,$

where we have used the fact $R_{z}(0)=0$ from (9). For the momentum, we have
the following:

$\displaystyle\dot{p}~{}={}$ $\displaystyle-{\partial\over\partial
q}\left\\{\frac{1}{2}p^{\top}M^{-1}(q)p\right\\}-\nabla U(q_{a})+Sv$ (18)
$\displaystyle+[{\cal N}+{\cal P}]\nabla H_{z}\Big{|}_{p=0}$
$\displaystyle~{}={}$ $\displaystyle-\nabla U(q_{a})+[I_{n}-\sigma_{1}u_{\tt
P}R_{z}(v)]\sigma_{0}u_{\tt P}z$ $\displaystyle~{}={}$ $\displaystyle-\nabla
U(q_{a})+\left(I_{n}-\sigma_{1}\mbox{diag}\left\\{{|v_{i}|\over\rho(v_{i})}\right\\}\right)\bigg{|}_{v=0}\sigma_{0}u_{\tt
P}z$ $\displaystyle~{}={}$ $\displaystyle-\nabla U(q_{a})+\sigma_{0}u_{\tt
P}z.$

Hence, for any non-zero $u_{\tt P}$, the point
$\chi_{\star}:=\mbox{col}(q_{a},0_{n},z_{a})$ with

$z_{a}:={1\over\sigma_{0}u_{\tt P}}\nabla U(q_{a})$ (19)

is an equilibrium.

The next step of the proof is to show the local asymptotic stability of the
manifold ${\cal M}$. Calculating the time derivative of the overall
Hamiltonian, it yields for $\chi\in B_{\varepsilon}({\cal M})$ with a small
$\varepsilon>0$,

$\displaystyle\dot{\cal H}$ $\displaystyle~{}=~{}-[\nabla{\cal H}(\chi,u_{\tt
P})]^{\top}{\cal R}(\chi,u_{\tt P})\nabla{\cal H}(\chi,u_{\tt P})$ (20)
$\displaystyle~{}\leq~{}-\left\|\begin{bmatrix}\nabla_{p}{\cal H}\\\
\nabla_{z}{\cal H}\end{bmatrix}\right\|_{{\cal R}_{22}}^{2}$
$\displaystyle~{}\leq~{}0,$

in which we have used the fact that in $B_{\varepsilon}({\cal M})$ the matrix
${\cal R}$ is positive semidefinite from Remark 3. Thus, in the neighborhood
of the manifold ${\cal M}$, the system is Lyapunov stable. In the set

$\\{\chi\in\mathbb{R}^{3n}:\|\mbox{col}(\nabla_{p}{\cal H},\nabla_{z}{\cal
H})\|_{{\cal R}_{22}}=0\\},$ (21)

it should verify

$\displaystyle(\sigma_{1}+\sigma_{2})M^{-1}(q)p-\frac{1}{2}\sigma_{0}\sigma_{1}u_{\tt
P}R_{z}(v)z$ $\displaystyle~{}=~{}0$ (22)
$\displaystyle-\frac{1}{2}\sigma_{1}R_{z}(v)M^{-1}(q)p+\sigma_{0}R_{z}(v)z$
$\displaystyle~{}=~{}0.$ (23)

Let us first consider (23). There are two possible cases:

  * •

case (i): $R_{z}(v)=0$ (or equivalently $p=0$).

  * •

case (ii): For some $j\in\bar{n}$, $\beta_{j}(v)\neq 0$, and thus

$M^{-1}(q)p=2{\sigma_{0}\over\sigma_{1}}z.$ (24)

For case (i), the trajectory verifies $p(t)\equiv 0$, thus

$\dot{p}=-\nabla U(q)+\sigma_{0}u_{\tt P}z=0,$

which is exactly the manifold ${\cal M}$. For case (ii), we substitute (24)
into (22), resulting in

$4(\sigma_{1}+\sigma_{2})z=\sigma_{1}^{2}\beta_{j}(v)u_{\tt P}z.$ (25)

There are two sub-cases: case (ii-1) $z=0$ and case (ii-2) $z\neq 0$. For case
(ii-1), the trajectory should guarantee $z\equiv 0$ and thus

$\displaystyle\dot{z}$ $\displaystyle~{}=~{}-{\cal R}_{z}(v)\nabla
H_{z}(0)+[{\cal N}-{\cal P}]v\Big{|}_{v\neq 0}$ $\displaystyle~{}=~{}[{\cal
N}-{\cal P}]v\Big{|}_{v\neq 0}=0.$

Since ${\cal N}-{\cal P}=I_{n}$, it contradicts with $v\neq 0$ in case (ii).
Thus, there is no feasible trajectory. For case (ii-2), the equation (25) can
be rewritten as

$\sigma_{1}+\sigma_{2}=\sigma_{1}^{2}{|v_{j}|\over 4\rho(v_{j})}.$ (26)

Note that $\lim_{|v|\to 0}\rho(v_{j})=\mu_{C}$. For given coefficients
$\sigma_{1},\sigma_{2}$, the equation (26) does not admit any feasible
solution for a sufficiently small $\varepsilon>0$. Therefore, the only
feasible solutions in $B_{\varepsilon}({\cal M})$ are all on the equilibria
manifold ${\cal M}$.

The system is time invariant since we consider constant pressure $u_{\tt P}$.
As we have shown above, ${\cal M}$ is the largest invariant set in the
neighborhood $B_{\varepsilon}({\cal M})\subset\mathbb{R}^{3n}$. Applying the
LaSalle’s invariance principle [11, Sec. 3], the manifold ${\cal M}$ is
locally asymptotically stable. $\triangleleft$

###### Remark 4

The above proposition shows that

  * (i)

If the initial condition $\chi(0)$ starts from any configuration $q_{a}$ and
zero momentum $p(0)=0$, we may always find a virtual bristle vector $z_{a}$
such that the system trajectory maintains at the initial values over time, and
we also note ${\cal M}\subset{\cal E}_{\tt SL}$. In this way, it achieves
shape locking.

  * (ii)

A more realistic scenario is that the continuum robot achieves deformation
with the tension input $u\in\mathbb{R}^{m}$; then we apply a vacuum and
release the actuator. Once the tension release is completed, the initial
condition is given by $\chi(0)=(q(0),0_{3},0_{3})$ instead of
$(q_{a},0_{3},z_{a})$. Proposition 1(b) shows the _local_ asymptotic stability
of the manifold ${\cal M}$, which means if the initial distance

$d(\chi(0),{\cal M}):=\inf_{\chi^{\prime}\in{\cal
M}}|\chi^{\prime}-\chi(0)|<\varepsilon_{0}$ (27)

is small, the trajectory ultimately converges to equilibrium
$(q_{a},0_{3},z_{a})\in{\cal M}$.

  * (iii)

From (ii), the convergence only happens when $\varepsilon_{0}>0$ is small.
Note that the vector $z_{a}$ is parameterized as
$z_{a}={1\over\sigma_{0}u_{\tt P}}\nabla U(q_{a})$. Thus, a large value of
$u_{\tt P}$ can impose the initial condition $(q(0),0_{3},0_{3})$ in a small
neighborhood of ${\cal M}$; see Fig. 2 for an intuitive illustration.
_Physically, it means that a large pressure value $u_{\tt P}$ is capable of
achieving shape locking. _

  * (iv)

The above item shows that after releasing the actuation, the system will
change from the initial configuration $(q(0),0_{3},0_{3})$ to the new
equilibrium $(q_{a},0_{3},z_{a})$, and it will be closed to each other with a
high pressure $u_{\tt P}$. It means when the continuum robot changes from
flexible to stiff, we may observe a tiny positional change that has been
experimentally verified in [4, Sec. III-B].

Figure 2: An illustration of the initial condition and the equilibria manifold
${\cal M}$. For a given initial condition $(q(0),0_{3},0_{3})$, a larger
$u_{\tt p,1}$ implies a smaller distance from $\chi(0)$ to ${\cal M}$, thus
$\chi(0)$ located in its domain of attraction; a smaller $u_{\tt p,2}$ may
cause the initial condition outside the domain of attraction, failing to
achieve shape locking.

###  III-B Adjustable Open-loop Stiffness

The open-loop equilibrium $\chi_{\star}:=(q_{\star},p_{\star},z_{\star})$ is
the origin. In the stiffness analysis, we assume that there is an external
torque $\tau_{\tt ext}$ acting on the dynamics of $p$, i.e., the dynamics with
$u=0$ becomes

$\dot{\bar{\chi}}=[{\cal J}-{\cal R}]\nabla{\cal H}+G_{0}\tau_{\tt ext}.$ (28)

with $G_{0}=\mbox{col}(0_{3\times 3},I_{3},0_{3\times 3})$, under which there
is a shifted equilibrium $\bar{\chi}:=\mbox{col}(\bar{q},0,\bar{z})$.

###### Definition 2

(Stiffness) Assume that we can find a positive semidefinite matrix
$K\in\mathbb{R}^{3\times 3}$ such that

$\tau_{\tt ext}:=K(\bar{q}-q_{\star})$ (29)

solves (28)-(29). When taking $\bar{q}\to q_{\star}$ and $\bar{z}\to
z_{\star}$, if the limit of $K$ exists, we call $K$ the overall stiffness.

We are now in position to present the open-loop stiffness of the proposed LJ-
based continuum robotic model.

###### Proposition 2

Consider the LJ-based continuum robotic model (13). Its overall stiffness in
the sense of Definition 2 at the open-loop equilibrium $\chi_{\star}$ is given
by

$\displaystyle K=\alpha_{1}\mathbf{1}_{n\times n}+[\alpha_{2}+\sigma_{0}u_{\tt
P}]I_{n},$ (30)

with $\mathbf{1}_{n\times n}\in\mathbb{R}^{n\times n}$ an all elements ones.

###### Proof:

Let us consider a tiny displacement $(\delta q,\delta
z)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ around $(q_{\star},z_{\star})$, i.e.,

$q=q_{\star}+\delta q,\quad z=z_{\star}+\delta z.$ (31)

For ease of analysis, we rewrite the model in an Euler-Lagrangian form

$\displaystyle\dot{M}(q)\ddot{q}+C(q,\dot{q})\dot{q}+\nabla U(q)$
$\displaystyle~{}=~{}\tau_{\tt ext}-\tau_{f}$ (32) $\displaystyle\dot{z}$
$\displaystyle~{}=~{}-\sigma_{0}\mbox{diag}\left\\{{|\dot{q}_{i}|\over\rho(\dot{q}_{i})}\right\\}z+\dot{q}$
$\displaystyle\tau_{f}$
$\displaystyle~{}=~{}(\sigma_{0}z+\sigma_{1}\dot{z}+\sigma_{2}\dot{q})u_{\tt
P},$

with zero initial condition, in which $C(q,\dot{q})$ is the Coriolis and
Centrifugal term [18].

Linearizing the dynamics (32) around $q_{\star}=0,\dot{q}_{\star}=0$ and
$z_{\star}=0$ and invoking (31), we obtain the model

$\displaystyle M_{\star}\delta\ddot{q}+[\sigma_{1}u_{\tt
P}]\delta\dot{q}+[\nabla^{2}U(q_{\star})+$ $\displaystyle\sigma_{0}u_{\tt
P}I_{3}]\delta q$ (33) $\displaystyle=\tau_{\tt ext}+\mathcal{O}(\delta
q^{2})$

with $M_{\star}:=M(q_{\star})$ and high-order remainder term ${\cal O}(\delta
q^{2})$, in which we have used the facts

$C(q_{\star},0)=0,\quad\nabla U(q_{\star})=0.$

Since

$\displaystyle\sigma_{1}u_{\tt P}$ $\displaystyle>0$
$\displaystyle\nabla^{2}U(q_{\star})+\sigma_{0}u_{\tt P}I_{3}$
$\displaystyle\succ 0,$

the linearized dynamics (33) is exponentially stable at equilibrium

$\delta q=[\nabla^{2}U(q_{\star})+\sigma_{0}u_{\tt P}I_{3}]^{-1}\tau_{\tt
ext}+\mathcal{O}(\delta q^{2}).$

By taking $|\delta q|\to 0$, the algebraic equation (29) is obtained with $K$
given by

$K=\nabla^{2}U(q_{\star})+\sigma_{0}u_{\tt P}I_{3}.$

Substituting the function $U$ in (3) into the above, we obtain (30) and
complete the proof. $\triangleleft$

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$55$$60$$65$$70$$75$$80$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$Pressure
$u_{\tt P}$ [kPa]Stiffness [N/mm]$\begin{aligned} K_{\tt T}&=0.0046u_{\tt
P}+0.3068\\\ R^{2}_{s}&=0.8083\end{aligned}$ Figure 3: Relation between
$u_{\tt P}$ and the transverse stiffness from the experiments in which we used
the jamming sheath with 5 layers. (“$\times$” shows the mean values; color
band represents the $\pm$1 standard deviation.) Figure 4: Photos showing
sequence of the shape-locking experiments: (a) Phase 1: Initial configuration
without $u$; (b) Phase 2: Drive to the bending configuration 60∘ via tendon
force $u$; (c) Phase 3: Vacuum to $u_{\tt P}=30$ kPa with motor-driven
retained; (d) Phase 4: Vacuum retained and tendon released $u=0$. (Photos were
taken in the steady state of each phase.) Phase 1Phase 2Phase 3Phase
4$-4$$-2$$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$5$$10$$15$$20$$25$Time
[s]Force $u_{1}$ [N] Figure 5: Trajectory of the tendon force $u_{1}$ in the
stiffness experiment. Figure 6: Overlay photos of the shape-locking
phenomenon. Left: displacement of 9.2mm with $u_{\tt P}=30$ kPa; Right:
displacement of 3.8mm with $u_{\tt P}=80$ kPa. (“ $\CIRCLE$” and “$\CIRCLE$”
are used to mark a fixed point on the robot body; one-side contours are also
highlighted in the figure.)

##  IV Experiments

In this section, we verify the theoretical analysis in Section III on our
continuum robotic platform OctRobot-I. Although the overall stiffness matrix
$K\succ 0$ cannot be measured directly, we are able to detect the transverse
stiffness $K_{\tt T}\in\mathbb{R}_{\geq 0}$ in the end-effector around the
open-loop equilibrium $q_{\star}$ of the continuum robot. We used the same
testing setup and approach as in [21, Sec. VI-B], and repeated three times
under the same conditions. The open-loop transverse stiffness under different
negative pressures $(-u_{\tt P})$ is plotted in Fig. 3. The coefficient of
determination $R_{s}^{2}$ is 0.8083 showing good linearity with respect to the
value $u_{\tt P}$. This verifies the results in Section III-B.

The second experiment was designed to verify the results about shape-locking.
The robot was initialized from the open-loop configuration $q_{\star}=0$
(Phase 1), and then driven to the bending angle of 60∘ via tendon (Phase 2).
When the system kept at the steady-state stage, we vacuumed and kept the
jamming layer sheath to a negative pressure of -30 kPa (Phase 3), and then
released the tendons (Phase 4). During this process, the sequence photos are
presented in Fig. 4, and the force $u_{1}$ was tunned as the trajectory in
Fig. 5. Note that we use the time interval $[-5,0]$ s to denote the initial
status before starting the motor drive. As illustrated in Figs. 4(c)-(d), it
achieved shape locking after applying a negative pressure ($-u_{\tt P}$). To
clearly show the shape-locking phenomenon, Fig. 6 illustrates the overlay
photos of Phases 3 and 4 with two different $u_{\tt P}$ ($30$ and $80$ kPa).
It can be observed tiny positional changes as theoretically predicted in
Remark 4(iv) – a larger $u_{\tt P}$ yielded a smaller displacement (3.8mm of
80 kPa and 9.2mm of 30 kPa).

##  V Conclusion

In this paper, we have presented a novel dynamical model for layer jamming-
based continuum robots, which integrates the energy-based modeling approach
and the LuGre frictional model. In terms of the proposed model, we
theoretically analyze its dynamical behavior and show its usefulness in
interpreting the two important phenomena (i.e., shape locking and adjustable
stiffness) in this kind of robots with quantitative results. These have been
experimental verified on our robotic platform.

The motivation of this work is to propose a _control-oriented_ model, and
naturally our future work will center on feedback controller synthesis using
the proposed model. Another important direction is to revisit Assumption 3
regarding the relation between the pressure and lumped normal force. This
would be helpful to improve the fitting accuracy on the relation between robot
stiffness and negative pressure.

## References

  * [1] K. J. Aström and C. Canudas-De-Wit.  Revisiting the LuGre friction model.  IEEE Control Syst. Mag., 28(6):101–114, 2008.
  * [2] N. Barahanov and R. Ortega.  Necessary and sufficient conditions for passivity of the LuGre friction model.  IEEE Trans. Autom. Control, 45(4):830–832, 2000.
  * [3] W. H. Choi, S. Kim, D. Lee, and D. Shin.  Soft, multi-DoF, variable stiffness mechanism using layer jamming for wearable robots.  IEEE Robot. Autom. Lett., 4(3):2539–2546, 2019.
  * [4] A. B. Clark and N. Rojas.  Assessing the performance of variable stiffness continuum structures of large diameter.  IEEE Robot. Autom. Lett., 4(3):2455–2462, 2019.
  * [5] A. B. Clark and N. Rojas.  Malleable robots: Reconfigurable robotic arms with continuum links of variable stiffness.  IEEE Trans. Robot., 38(6):3832–3849, 2022.
  * [6] C. C. De Wit, H. Olsson, K. J. Astrom, and P. Lischinsky.  A new model for control of systems with friction.  IEEE Trans. Autom. Control, 40(3):419–425, 1995.
  * [7] C. Della Santina, C. Duriez, and D. Rus.  Model-based control of soft robots: A survey of the state of the art and open challenges.  IEEE Control Syst. Mag., 43(3):30–65, 2023.
  * [8] Y. Fan, D. Liu, and L. Ye.  A novel continuum robot with stiffness variation capability using layer jamming: Design, modeling, and validation.  IEEE Access, 10:130253–130263, 2022.
  * [9] E. Franco and A. Garriga-Casanovas.  Energy-shaping control of soft continuum manipulators with in-plane disturbances.  Int. J. Robot. Res., 40(1):236–255, 2021.
  * [10] S. Jadhav, M. R. A. Majit, B. Shih, J. P. Schulze, and M. T. Tolley.  Variable stiffness devices using fiber jamming for application in soft robotics and wearable haptics.  Soft Robot., 9(1):173–186, 2022.
  * [11] H. K. Khalil.  Nonlinear Systems.  Prentice Hall, 3 edition, 2001.
  * [12] Y.-J. Kim, S. Cheng, S. Kim, and K. Iagnemma.  Design of a tubular snake-like manipulator with stiffening capability by layer jamming.  In IEEE Int. Conf. Intell. Robots Syst., pages 4251–4256. IEEE, 2012.
  * [13] Y.-J. Kim, S. Cheng, S. Kim, and K. Iagnemma.  A novel layer jamming mechanism with tunable stiffness capability for minimally invasive surgery.  IEEE Trans. Robot., 29(4):1031–1042, 2013.
  * [14] J. Koopman, D. Jeltsema, and M. Verhaegen.  Port-Hamiltonian description and analysis of the LuGre friction model.  Simul. Model. Pract. Theory., 19(3):959–968, 2011.
  * [15] M. Langer, E. Amanov, and J. Burgner-Kahrs.  Stiffening sheaths for continuum robots.  Soft Robot., 5(3):291–303, 2018.
  * [16] Y. S. Narang, A. Degirmenci, J. J. Vlassak, and R. D. Howe.  Transforming the dynamic response of robotic structures and systems through laminar jamming.  IEEE Robot. Autom. Lett., 3(2):688–695, 2017.
  * [17] Y. S. Narang, J. J. Vlassak, and R. D. Howe.  Mechanically versatile soft machines through laminar jamming.  Adv. Funct. Mater., 28(17):1–9, 2018.  Art. no. 1707136.
  * [18] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez.  Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications.  Springer, 1998.
  * [19] J. L. C. Santiago, I. S. Godage, P. Gonthina, and I. D. Walker.  Soft robots and kangaroo tails: Modulating compliance in continuum structures through mechanical layer jamming.  Soft Robot., 3(2):54–63, 2016.
  * [20] A. van der Schaft.  $L_{2}$-Gain and Passivity Techniques in Nonlinear Control.  Springer, 3rd edition, 2017.
  * [21] B. Yi, Y. Fan, D. Liu, and J. G. Romero.  Simultaneous position-and-stiffness control of underactuated antagonistic tendon-driven continuum robots.  ArXiv Preprint, 2023.  (arXiv:2306.03865).
  * [22] Y. Zhao et al. A soft continuum robot, with a large variable-stiffness range, based on jamming.  Bioinspir. Biomim., 14(6), 2019.  Art. no. 066007.