cs.ROOct 10, 2026

PC7 Model: Dynamics of Expanding and Contracting Soft Robots using Piece-wise Constant 7D Deformation Based on Similarity Transformation Group

Authors: Yamato Kudo, Taiki Ishigaki, Ko Ayusawa, Ko Yamamoto

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Abstract

This paper proposes a novel soft robot model named piece-wise constant 7D deformation (PC7) model that can represent expanding and contracting deformation of, for example, artificial muscle actuator. In the proposed model, we newly define the 7D deformation vector based on SIM(3) - 3D similarity transformation group - whereas a 6D deformation vector was defined for Cosserat rod based on SE(3) - the special Euclid group - in the piece-wise constant strain (PCS) model proposed in the previous study. The additional dimension of the 7D deformation vector represents expanding and contracting deformation, which comes from the scaling parameter in SIM(3). Common mathematical properties between SE(3) and SIM(3) allow us to directly extend the kinematics and dynamics of the PCS model to those of the PC7 model. In particular, we derive the dynamical equation of the PC7 model based on Euler-Poincare equation in a mathematically precise manner. We validate the proposed model through dynamical simulations assuming a thin cylinder structure or McKibben-type artificial muscle actuator.

Figures & tables

Fig. 1 : Schematic diagram of (a) Cosserat rod and (b) microstretch rod [ 20 ] . In the Cosserat rod, the configuration of miscrostructre is represented by SE(3), and a 6D deformation \mbox{\boldmath\xi}_{\mathfrak{se}} is defined by se(3)\mathfrak{se}(3) [ 9 ] . In the microstretch rod, the configuration can be represented by SIM(3), and we can newly define a 7D deformation \mbox{\boldmath\xi}_{\mathfrak{sim}} by sim(3)\mathfrak{sim}(3) , as proposed in this study.
Fig. 2 : Cross section of the stretch miscrostructure in a microstretch rod. Without loss of generality, the zz -axis of the local frame can be set aligned to the tangent direction of ss .
Fig. 3 : Piece-wise constant 7D deformation (PC7) model.
Fig. 4 : Cross section of a thin-walled cylinder.
Fig. 5 : Parameters of a McKibben artificial muscle actuator.
ParameterValue
Number of segments  n~n3
s1,s2,s2s_{1},s_{2},s_{2}0.1,0.2,0.3 0.1,0.2,0.3~ m
Density  ρ~\rho1.1×103 1.1\times 10^{3}~ kg ⋅\cdot m -3
Initial radius  R0~R_{0}0.015 0.015~ m
Initial thickness  w0~w_{0}1.5×10−3 1.5\times 10^{-3}~ m
Young’s modulus  E~E110 110~ kPa
Table I : Parameter setting of thin-walled cylinder assumed in the simulation
Fig. 6 : Results of dynamical simulation of the PC7 model assuming a thin cylinder made of silicone rubber.
 ParameterValue
 Number of segments  n~n3
 s1,s2,s3s_{1},s_{2},s_{3}0.01,0.28,0.01 0.01,0.28,0.01~ m
RubberDensity  ρ~\rho1.1×103 1.1\times 10^{3}~ kg ⋅\cdot m -3
Initial radius  R0~R_{0}7×10−3 7\times 10^{-3}~ m
Initial thickness  w0~w_{0}0.7×10−3 0.7\times 10^{-3}~ m
Young’s modulus  E~E1.62 1.62~ MPa
Table II : Parameter Setting of McKibben-type Artificial Muscle Actuator assumed in the Simulation
Fig. 7 : Results of dynamical simulation of the PC7 model assuming a Mckibben-type artificial muscle actuator
Fig. 8 : Relationships between the contraction ratio and contraction force for an initial weave angle is 20 degrees.
Fig. 9 : Snapshots of the finite element analysis. Before transformation on the left and after transformation on the right.
DeformationPressureChou et al. [ 26 ]PC7 ModelFEM
process[MPa]   
Contruction0.5136.0269120.8651N/A
 0.3111.138484.906656.9392
 0.190.026769.658831.9533
Extension0.592.637046.5258N/A
 0.373.313131.5580142.4767
Table III : RMSE results in the condition ϕ0=\phi_{0}= 20 deg
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