A General Lie-Group Framework for Continuum Soft Robot Modeling

Lingxiao Xun; Benoît Rosa; Jérôme Szewczyk; Brahim Tamadazte

cs.ROMar 9, 2026

A General Lie-Group Framework for Continuum Soft Robot Modeling

Authors: Lingxiao Xun, Benoît Rosa, Jérôme Szewczyk, Brahim Tamadazte

Organizations: Sorbonne Université, CNRS UMR 7222, Inserm U1150, ISIR, F-75005, Paris, France. · Université de Strasbourg, CNRS, INSERM, ICube, UMR7357, Strasbourg, France.

Abstract

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.

Figures & tables

Symbol	Unit	Definition
$\dot{}$	–	Derivative with respect to time
′	–	Derivative with respect to space
$\widehat{}$	–	Converts $\mathbb{R}^{6}$ in $\mathfrak{se}(3)$
$s$	m	Reference arclength coordinate
$t$	s	Time
$p(s,t)$	m	Position vector of cross-section

Table 1: Notation and definitions.

Figure 1: Schematic diagram of the soft manipulator.

Panel of Figure 2 — Figure 2: Bspline and Hermite curve on $SE(3)$ manifold.

Panel of Figure 4 — Figure 4: Cumulative basis functions.

Parameterization	Original basis $B_{i}(\cdot)$	Cumulative basis $\displaystyle\bar{B}_{i}(\cdot)=\sum_{j=i}^{n}B_{j}(\cdot)$
Power (monomial)	$B_{i}(s)=s^{\,i}$	$\displaystyle\bar{B}_{i}(s)=\sum_{j=i}^{n}s^{\,j}=\dfrac{s^{\,i}-s^{\,n+1}}{1-s}\hskip 9.24994pt(s\neq 1)$
B - spline (degree $k$ )	$B_{i}(s)=N_{i,k}(s)$	$\displaystyle\bar{B}_{i}(s)=\sum_{j=i}^{n}N_{j,k}(s)$
Cubic Hermite † (Bézier)	$B_{i}(s)=N_{i,3}(s)$	$\displaystyle\bar{B}_{i}(s)=\sum_{j=i}^{3}N_{j,3}(s)$

Table 2: Original and cumulative basis functions for several nonlinear parameterizations

Figure 5: The variation of the kinematic Jacobian matrix concerning the arc-length coordinate $s$ . The left figure shows the Jacobian matrix obtained using the Piecewise Linear Strain (PLS) [ 28 ] parameterization method. In contrast, the right figure shows the Jacobian matrix obtained using the proposed Lie group B-spline parameterization method.

Figure 6: Schematic of a tree-structured robot. Each ellipse represents a soft or rigid body segment (e.g., $\text{body}_{i}$ ), and the small circles represent joints that allow rotational movement. The $\text{body}_{i}$ acts as a branching point, connecting to multiple child links $\text{body}_{i+1}$ and $\text{body}_{i+2}$ , illustrating a typical tree structure.

Figure 7: Three types of connections. (a) Direct rigid connection; (b) Rigid body insertion; (c) Movable joint connection.

Figure 8: Relative motion modeling in concentric tube robots. (a) General case: Rod 1 and Rod 2 are connected via a relative motion $g^{r}(s)\in\mathrm{SE}(3)$ , which includes both translation and rotation. (b) Concentric configuration: The rods are constrained to have no relative translation, resulting in a pure rotational relationship $R^{r}(s)\in\mathrm{SO}(3)$ along their shared centerline.

Figure 9: Schematic of a closed-loop robotic structure. Each ellipse represents a rigid body (e.g., $\text{body}_{i}$ ), and the small circles denote rotational joints. Unlike tree-structured robots, this configuration forms a closed kinematic loop: starting from $\text{body}_{i}$ , connections propagate through $\text{body}_{i+1}$ , $\text{body}_{i+3}$ , and $\text{body}_{i+2}$ , and return to $\text{body}_{i}$ . Such closed-chain mechanisms are common in parallel robots and constrained articulated systems.

Constraint Type	Constraint Function [1]	Jacobiant w.r.t. $g_{1}$ [2]	Jacobiant w.r.t. $g_{2}$ [3]
Fixed constraint	$b=g_{2}\boxminus g_{1}$	$\nabla_{g_{1}}b=\mathrm{d}\tau_{b}^{-1}\,\mathrm{Ad}_{\tau(b)}^{-1}$	$\nabla_{g_{2}}b=\mathrm{d}\tau_{b}^{-1}$
Articulated constraint	$b=P_{1}(g_{2}\boxminus g_{1})$	$\nabla_{g_{1}}b=P_{1}\,\mathrm{d}\tau_{b}^{-1}\,\mathrm{Ad}_{\tau(b)}^{-1}$	$\nabla_{g_{2}}b=P_{1}\,\mathrm{d}\tau_{b}^{-1}$
Sliding constraint	$b=P_{2}(g_{2}\boxminus g_{1})$	$\nabla_{g_{1}}b=P_{2}\,\mathrm{d}\tau_{b}^{-1}\,\mathrm{Ad}_{\tau(b)}^{-1}$	$\nabla_{g_{2}}b=P_{2}\,\mathrm{d}\tau_{b}^{-1}$

Table 3: Constraint functions and their Jacobians for different constraint types

Figure 10: Integration flowchat. The discrete Euler–Lagrange equation is solved to determine the next configuration based on the previous and current ones. This process is iteratively applied to compute the entire trajectory.

Figure 11: Workflow of the proposed modeling method. The process includes control point parameterization, kinematics computation, connection updates, system assembly, and static/dynamic solving.

Parameter	Value / Unit
Length	1.0 m
Front end radius	3.0 cm
Rear end radius	1.5 cm
Young’s modulus	$2.0\times 10^{5}$ Pa
Possion’s ratio	0.45

Table 4: Physical Parameters

Figure 12: Comparison of deformation results between the proposed method and the ground truth. Blue rods represent the results from the proposed method, while gray rods correspond to the ground truth. In the upper panel, an upward vertical force is applied at the right end of the rods, increasing from 0 N to 1 N in increments of 0.25 N. In the lower panel, a moment about the negative $y$ -axis is applied at the same location, increasing from 0 Nm to 0.2 Nm in increments of 0.05 Nm.

Figure 13: Error between the simulation results and the ground truth along the length of the soft rod under different end-tip loads.

Figure 14: Convergence behavior of the static solver. The plot shows the residual norm $\|{r}\|$ over iterations during the static equilibrium computation.

Figure 15: Convergence of the configuration error with increasing number of control points under B-spline basis functions of order 2 to 5.

Figure 16: Initial equilibrium configuration of the soft rod. The rod is fixed at its left end and subject to a 10 N upward force at the right end, resulting in a curved static equilibrium shape. This configuration is used as the initial condition for subsequent free oscillation analysis.

Parameter	Value / Unit
Length	1.0 m
Front end radius	5.0 cm
Rear end radius	3.0 cm
Young’s modulus	$2.0\times 10^{6}$ Pa
Possion’s ratio	0.45
Viscosity modulus	$0$ Pa $\cdot$ s

Table 5: Physical Parameters

Figure 17: Time evolution of the rod tip position along the ${z}$ -axis. After the external force is removed, the rod undergoes sustained vertical oscillation.

Figure 18: Total energy evolution of the system over time. The total energy is computed as the sum of kinetic and potential energy. The upper plot shows simulation results obtained using the implicit Euler integrator, while the lower plot presents results using the proposed symplectic integrator.

Figure 19: Deformation of the cable-driven soft manipulator. The left panel shows the initial configuration of the soft rod, with all cables untensioned and the structure in a straight, undeformed state. The right panel illustrates the deformed configuration resulting from the actuation of the cable. Three cables (red line) run through the vertices of the triangular plates, applying equal and opposite tension between the corresponding vertices of adjacent plates.

Parameter	Value / Unit
Length	1.0 m
Front end radius	4.0 cm
Rear end radius	2.0 cm
Young’s modulus	$1.0\times 10^{6}$ Pa
Possion’s ratio	0.45
Viscosity modulus	$10$ Pa $\cdot$ s

Table 6: Physical Parameters

Figure 20: The tree structure of the cable-driven soft manipulator. The cable-driven robot consists of soft elements and rigid triangular plates connected in series. Each pair of adjacent rigid plates is linked by a soft segment, with the connections between them modeled as rigid joints.

Figure 21: Concentric tube robot .The concentric tube robot consists of three tubes sharing a common central axis, with the third tube held stationary and fixed to the world coordinate frame.

Parameter	1st tube	2nd tube	3rd tube
Length (m)	0.6	0.4	0.05
Radius (mm)	2.5	5	50
Young’s modulus (GPa)	2	2	10
Possion’s ratio	0.45	0.45	0.45
Pre-curvature (1/m)	6	4	0

Table 7: Physical Parameters

Figure 22: The concentric structure of the three-tube concentric robot. The first (innermost) tube, serving as the main tube, possesses independent degrees of freedom. The remaining tubes are subordinate, with their degrees of freedom determined by the configuration of the main tube and their relative rotations with respect to it.

Figure 23: Representative deformation stages of the concentric tube robot. $\beta$ denotes the distance from the first tube’s rotation tip to the bending entry point, and $l$ is the overlap length of the bent sections. Initially, the tubes remain stable. At $t=4.16$ s, the snapping event induces instability, followed by a return to stability at $t=4.35$ s.

Figure 24: S-curve. Comparison between the simulation results obtained using the proposed method and the analytical solution.

Figure 25: Chiral structure under compressive loading. The left image shows the initial unloaded state, in which straight vertical rods are symmetrically arranged between three horizontal circular plates. These rods are stress-free and maintain parallel alignment without deformation. The right image illustrates the deformed state under vertical compressive loading. The rods exhibit significant twist-buckling behavior, undergoing compression, bending, and torsion simultaneously.

Parameter	Value / Unit
Length	0.8 m
Radius	4.0 cm
Young’s modulus	$1.0\times 10^{5}$ Pa
Possion’s ratio	0.45

Table 8: Physical Parameters

Figure 26: Structure of the simulated parallel soft–rigid mechanism and its Jacobian sparsity pattern. The left panel illustrates the topological model of a parallel mechanism composed of rigid bodies ( $R_{1}$ – $R_{3}$ ) connected by soft rods ( $S_{1}$ – $S_{8}$ ). Nodes represent components, while edges represent geometric constraints between them. The right panel shows the sparsity pattern of the Jacobian matrix derived from the static KKT system, which includes both the Hessian and the Jacobians of all constraints. The structured sparsity reflects the modular and localized nature of the hybrid soft–rigid system.

Figure 27: Strain–force response under different initial torques. The plot illustrates the vertical compressive force versus strain relationship for chiral structures with varying initial torques (0.1 Nm, 0.5 Nm, 1 Nm, and 2 Nm). Lower initial torque results in a stiffer response with rapid force increase, while higher torques induce more pronounced twist-buckling and bending, allowing for greater strain accommodation and enhanced energy absorption performance.

Figure 28: Deformation of a cable-driven soft finger. The left panel shows the initial undeformed configuration of the soft finger, where the rigid bone segments (yellow) are aligned and the soft joint rods (blue) remain straight with untensioned cables (red line). In the right panel, tensile forces applied to the embedded cables induce bending at the joints, causing the finger to flex.

Figure 29: Geometric design of the soft joint. The shape of the soft joint is parameterized using a cubic B-spline curve defined by five control points (frame). The left panel shows the B-spline curve (blue) along with the control polygon (dashed orange) and control points (frame), while the right panel displays the corresponding 3D revolved geometry used to construct the soft joint.

Parameter	Value / Unit
Thickness	1 mm
Width	20 mm
Young’s modulus	$5$ GPa

Table 9: Physical Parameters

Figure 30: Trajectory of the finger’s tip with the different soft joint combination. The labels ① and ② indicate two distinct natural shapes of the soft joints. Each subfigure corresponds to a different joint combination in the robotic finger.

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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Modeling soft robot dynamics is challenging due to their continuum structure and typically nonlinear dynamics. Creating models based on first-order principles is typically time-demanding, and their expressiveness is limited, whereas data-driven models lack interpretability and physical consistency. This work aims to overcome these challenges by introducing a port-Hamiltonian Gaussian Process Regression framework for learning and simulating the dynamics of planar, rod-like soft robots. In detail, the proposed model integrates Cosserat rod theory and Hamiltonian physics with data-driven inference to preserve the system's energy structure while accurately learning the rod dynamics. Numerical simulations show that we can achieve accurate and energy-consistent representations of a rod-like soft robot, showing the potential for a robust and interpretable pathway for modeling complex continuum mechanics.

Abstract

Soft Robotics

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