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Page 4 of 19 Peng et al. Soft Sci. 2025, 5, 38 https://dx.doi.org/10.20517/ss.2025.31

Table 1. Mechanical parameters of the robot
Superelastic SMA Young’s modulus 56 GPa
Length 120 mm
Diameter 0.8 mm
SMA springs Diameter 3 mm
Length 15 mm
Maximum output displacement 85 mm
Maximum output force 7 N
SMA wire diameter 0.6 mm
Tension sensor Rated voltage 5-10 VDC
Range 0-500 g
Weight 4.9 g

SMA: Shape memory alloy.

Figure 2. (A) 3D model of the continuum robot; (B) Adaptivity experiment with SMA. 3D: Three-dimensional; SMA: shape memory
alloy.

Shape sensing based on tension and Cosserat rod theory
Cosserat rod theory
The CC model, due to the approximation of circular curves, cannot accurately be used to derive a robot
model with the assumption of no gravity or no external loads. Here, we use Cosserat rod theory to model
the bending shape of a soft finger, which is affected by its body weight [33,34] . Figure 3A shows a Frenet–Serret
coordinate frame of a soft finger, which is a one-dimensional rod. In this figure, O–XYZ is a global frame,
which is stationary with respect to the base; o–xyz is a local frame located at any point in the rod. Each point
of the rod can be parameterized by its unstretched length, here represented by the variable s. The position of
any point s in the global frame can be expressed as p(s), while the infinitesimal element in the local frame
can be expressed as p (s). In accordance with Cosserat rod theory, a rod can be parameterized by its
l
centerline curve in space with a three-element vector p(s) ∈ R to represent the location of a point on the
3
rod and a matrix R(s) ∈ SO(3) to specify the orientation of a local frame with respect to the global
coordinate frame.