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

Yamato Kudo; Taiki Ishigaki; Ko Ayusawa; Ko Yamamoto

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

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 $\mathfrak{se}(3)$ [ 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 $\mathfrak{sim}(3)$ , as proposed in this study. — 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 $\mathfrak{se}(3)$ [ 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 $\mathfrak{sim}(3)$ , as proposed in this study.

Fig. 2 : Cross section of the stretch miscrostructure in a microstretch rod. Without loss of generality, the $z$ -axis of the local frame can be set aligned to the tangent direction of $s$ .

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.

Parameter	Value
Number of segments $~n$	3
$s_{1},s_{2},s_{2}$	$0.1,0.2,0.3~$ m
Density $~\rho$	$1.1\times 10^{3}~$ kg $\cdot$ m -3
Initial radius $~R_{0}$	$0.015~$ m
Initial thickness $~w_{0}$	$1.5\times 10^{-3}~$ m
Young’s modulus $~E$	$110~$ kPa

Table I : Parameter setting of thin-walled cylinder assumed in the simulation

Panel of Figure 7 — Fig. 6 : Results of dynamical simulation of the PC7 model assuming a thin cylinder made of silicone rubber.

	Parameter	Value
	Number of segments $~n$	3
	$s_{1},s_{2},s_{3}$	$0.01,0.28,0.01~$ m
Rubber	Density $~\rho$	$1.1\times 10^{3}~$ kg $\cdot$ m -3
	Initial radius $~R_{0}$	$7\times 10^{-3}~$ m
	Initial thickness $~w_{0}$	$0.7\times 10^{-3}~$ m
	Young’s modulus $~E$	$1.62~$ MPa

Table II : Parameter Setting of McKibben-type Artificial Muscle Actuator assumed in the Simulation

Panel of Figure 9 — 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.

Deformation	Pressure	Chou et al. [ 26 ]	PC7 Model	FEM
process	[MPa]
Contruction	0.5	136.0269	120.8651	N/A
	0.3	111.1384	84.9066	56.9392
	0.1	90.0267	69.6588	31.9533
Extension	0.5	92.6370	46.5258	N/A
	0.3	73.3131	31.5580	142.4767

Table III : RMSE results in the condition

\phi_{0}=

20 deg

Soft Robotics

Soft robotics studies robots with compliant bodies and actuators that deform to move, sense, and interact with uncertain environments.

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Soft Robotics Lie Group Methods Continuum Robot Control

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This paper introduces a general Lie group framework for modeling continuum soft robots, employing Cosserat rod theory combined with cumulative parameterization on the Lie group SE(3). This novel approach addresses limitations present in current strain-based and configuration-based methods by providing geometric local control and eliminating unit quaternion constraints. The paper derives unified analytical expressions for kinematics, statics, and dynamics, including recursive Jacobian computations and an energy-conserving integrator suitable for real-time simulation and control. Additionally, the framework is extended to handle complex robotic structures, including segmented, branched, nested, and rigid-soft composite configurations, facilitating a modular and unified modeling strategy. The effectiveness, generality, and computational efficiency of the proposed methodology are demonstrated through various scenarios, including large-deformation rods, concentric tube robots, parallel robots, cable-driven robots, and articulated fingers. This work enhances modeling flexibility and numerical performance, providing an improved toolset for designing, simulating, and controlling soft robotic systems.

Abstract

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