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Page 2 of 19 Peng et al. Soft Sci. 2025, 5, 38 https://dx.doi.org/10.20517/ss.2025.31

Figure 1A, demonstrate excellent perception and response to complex environmental factors (such as an
[1-4]
object’s weight and position), allowing them to carry out sophisticated manipulation tasks . Continuum
robots, inspired by biological manipulators in nature, are a new type of bionic robot based on a continuous
[5]
backbone without joints. In contrast to discrete mechanisms with rigid links and joints , continuum robots
are inherently compliant and flexible due to their ability to elastically deform. However, their low stiffness
makes it challenging for such robots to achieve accurate bending shape sensing and control - capabilities
that are crucial for practical applications such as navigation or search and rescue operations in complex,
congested environments [6-10] .

Traditionally, embedded strain sensors have been widely used to estimate the shape of continuum robots
based on parametric curve assumptions, such as constant curvature (CC) and piecewise constant curvature
(PCC), without adequately considering material properties or actuation methods [11-13] . Flexible strain sensors
made of elastomeric materials, such as liquid metal [14,15] , carbon nanotubes [16,17] , and conductive ink
embedded in silicone [18,19] , can detect CC deformation by converting mechanical stimuli into electrical
signals . These sensors offer advantages such as low cost, small size, light weight, high sensitivity, and rapid
[20]
response speed. Furthermore, they can enable compatible and inherently safe shape sensing when integrated
into soft robots. Nevertheless, such flexible strain sensors are typically limited to detecting deformation
within a two-dimensional (2D) plane and require a substrate for integration.

Sensing approaches based on vision, electromagnetic fields (EMF), fiber Bragg gratings (FBG), and dual-
color layered structure (DCLS) sensors are also used to estimate the three-dimensional (3D) shapes of
continuum robots . However, cameras require sufficient space for hardware installation. Embedded
[21]
cameras increase the weight and volume of the continuum robot, while external cameras cannot obtain
shape data when it is obstructed by obstacles . Compared with cameras, electromagnetic sensors have a
[22]
miniature size, are easier to mount in continuum robots, and can directly provide positional and directional
[23]
information . However, electromagnetic tracking suffers from inaccuracy due to sparse pose information
from distributed sensors, as well as interference from magnetic and metallic conductive objects. FBG
sensors offer advantages such as small size, light weight, simple structure, high sensitivity, and fast response
speed. They can measure continuum deformation along the length of the robot, are immune to magnetic
fields, and thus have excellent anti-electromagnetic interference properties [24,25] . However, FBG sensors are
fragile and easily damaged, making them unsuitable for soft structures undergoing large deformations with
low stiffness. DCLS sensors can effectively monitor the bending angles and directions of joints of different
sizes without requiring additional calibration; however, external environmental temperature significantly
impacts their accuracy . Additionally, they require expensive specialized hardware equipment. In
[26]
summary, shape sensing using strain, vision, EMF, FBG, and DCLS sensors may produce significant errors
based on CC and PCC assumptions under external loads or forces.

To achieve accurate shape sensing, methods considering factors such as weight, material, and stiffness have
been widely applied in the static and dynamic modeling of continuum robots based on the finite element
method (FEM) and Cosserat rod theory, as these approaches accurately represent large deformations [27,28] .
FEM has proven effective in simulating continuum robot behavior and is suitable for modeling irregular
and nonlinear deformations. However, FEM typically requires substantial computation, making it difficult
to meet the efficiency demands of real-time kinematics applications . Methods based on Cosserat rod
[29]
theory also face limitations: (1) motor-driven actuation systems introduce noise and increase robot weight;
and (2) although cameras can provide feedback for position tracking in some robots, shape sensing and
control capabilities under unknown payloads remain inadequate. As shown in Figure 1B and C, this paper
presents a novel shape memory alloy (SMA)-actuated continuum robot capable of accurate object