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Peng et al. Soft Sci. 2025, 5, 38 https://dx.doi.org/10.20517/ss.2025.31 Page 9 of 19

Table 2. Parameters for simulation and experiments
Parameter Value Parameter Value
m d 5.04 g m b 0.6 g
l 120 mm r 0.4 mm
-8 3 -7 2
v r 6.03 × 10 m A 5.03 × 10 m
4
ρ 9.35 × 10 kg/m 3 E r 77.59 GPa
L 8 mm ψ 120°
0

where (P , P , P ) is the simulated result of the tip position; Δe , Δe and Δe are the absolute errors between
z
y
y
x
z
x
the simulation and experiment on the X, Y and Z axes, respectively. The simulation accurately predicts the
deflection with a variety of known external loading conditions at the tip. The average error ΔE increases
from 0.92 to 1.42 mm as the load increases from 0 to 30 g. Detailed information is listed in Table 3.
The SMA springs contract and expand the tendons to bend the continuum robot. However, the actual
bending moment applied to the continuum robot is less than the moment calculated using Equation (7) due
to the friction between the tendons and disks. As shown in Figure 4A, the modified force can be expressed
as

(10)

where F is the force measured by the load cell.
L

As shown in Figure 4B, the actual bending shape (black dots) of the robot was reconstructed from
displacement data measured at eight connection points between disks along the backbone. The modeled
bending shape (blue line) from Equation (10) demonstrates significantly higher accuracy using the modified
tension-based forward kinematics model compared to the unmodified approach. For instance, the average
positional errors (ΔE) for point P were 3.34 mm with tension modification vs. 11.65 mm without
modification.

As shown in Figure 4C, a 3D FEM model was developed to predict the continuum robot’s bending shape,
with disk material modeled as plastic and backbone material as superelastic SMA. The robot’s bending in
the XOZ plane was simulated with SMA2 actuated by input forces ranging from 0 to 4 N. These FEM results
were compared against the proposed forward kinematics model. Both models show close agreement with
experimental data when accounting for weight and material factors. For quantitative comparison, the
average positional errors (ΔE) at point P between simulations and experiments were calculated. Figure 4D
illustrates that ∆E increases with input force from 0 to 4 N, reaching maximum values of 16 mm for FEM vs.
4.3 mm for the proposed model. Consequently, the proposed model demonstrates superior shape prediction
accuracy. This performance difference primarily stems from the FEM model’s inability to account for
tendon-disk friction. Additionally, FEM required 30 min per simulation case and cannot provide inverse
kinematics solutions, whereas the proposed method computes both forward and inverse kinematics in
approximately 100 ms. In summary, the proposed method offers greater suitability for continuum robot
shape sensing and control compared to FEM.

Shape sensing based on forward kinematics and experimental verification
As mentioned above, the shapes of the continuum robot can be predicted using the Cosserat model based
on the tensions of three tendons. To further verify the model, various experiments were conducted using a