[Paper Review] On the Modelling of Soft-robots as Quasi-Continuum Lagrangian Dynamical Systems with Well-posed Input Matrix

Ernesto Olguín-Díaz, Christian A. Trejo-Ramos+2 more|arXiv (Cornell University)|Dec 6, 2018

Soft Robotics and Applications5 references4 citations

TL;DR

This paper proposes a quasi-continuum Lagrangian dynamical model for braided continuum soft-robots with a well-posed input matrix, incorporating a scalar varying mass density field to capture coordinate-dependent inertial effects and a non-uniform center of mass. The model ensures passivity via skew-symmetric Coriolis forces and enables affine control input structure, enabling application of rigid-robot control tools to soft-robot systems.

ABSTRACT

In this paper, considering a braided continuum soft-robot, whose radial deformation is constrained but elongation is assumed, a quasi-Lagrangian model is proposed that meets the Lagrangian models properties, including a well-posed input matrix. Actuation is considered throughout three inner pressure cambers, and torsional effects are neglected. The closed-form analytical model is obtained using a scalar varying mass density field, previously neglected in the literature, which produces on one hand a varying center of mass, which generally does not lay in the backbone curve, and one the other hand a coordinate-dependent inertial tensor. The Lagrangian approach enforces the basic skew symmetric property, thus exhibiting passivity. The advantage of dealing with all these effects together display the following distinct features: extit{i)} the Lagrangian soft-robot dynamic model is similar to the Lagrangian rigid-robot case; extit{ii)} the non-linear system is affine in the control input; extit{iii)} the continuum deformable body stands for a segment of constant curvature, when interconnected with other segments of different constant curvature each, would leads to a quasi-continuum $n$-segments variable curvature soft-robot, yet preserving the aforementioned previous features of one segment. edc{Representative simulations and videos are shown, in open- and closed-loop.

Motivation & Objective

• To develop a Lagrangian dynamic model for soft-robots that preserves structural properties similar to rigid-robot models, such as passivity and skew-symmetric Coriolis terms.
• To address the lack of a well-posed input matrix in existing soft-robot models by incorporating pneumatic pressure inputs via virtual work and power equivalence.
• To introduce a scalar varying mass density field, which enables a coordinate-dependent inertial tensor and a non-backbone-centered center of mass, previously neglected in the literature.
• To establish a closed-form analytical model that is affine in control input and suitable for model-based control design.
• To extend the applicability of established rigid-robot control methodologies—such as passivity-based and variable structure control—to soft-robot systems.

Proposed method

• The model is derived using a Lagrangian formulation with a scalar varying mass density field, which affects both the center of mass and the inertial tensor.
• Kinematics are based on constant curvature segments with generalized coordinates: length (l), azimuthal curvature (ψ), and curvature magnitude (κ), forming a 3D configuration space.
• The virtual work principle is applied to map pneumatic pressure inputs from chamber forces to generalized coordinates, yielding an affine control input structure.
• The input matrix B(q_e) is derived as a configuration-dependent mapping using Jacobians of pressure application points and rotation matrices, ensuring well-posedness.
• The quasi-Lagrangian model is constructed via power equivalence and kinematic transformation, preserving the skew-symmetric property of the Coriolis matrix.
• The resulting dynamic equation includes mass, Coriolis, gravity, damping, and stiffness terms, with passivity guaranteed by the skew-symmetry of C_ξ.

Experimental results

Research questions

• RQ1 Can a soft-robot dynamic model be formulated in a Lagrangian framework that preserves the structural properties of rigid-robot models, such as passivity and skew-symmetric Coriolis terms?
• RQ2 How can a well-posed input matrix be constructed for soft-robots actuated by internal pressure, ensuring affine dependence on control inputs?
• RQ3 What is the impact of a spatially varying mass density field on the center of mass and inertial tensor in soft-robot dynamics?
• RQ4 To what extent can established rigid-robot control tools be adapted to soft-robot systems using a well-structured Lagrangian model?
• RQ5 How does the inclusion of a non-uniform center of mass affect the dynamic behavior and control design of soft-robots?

Key findings

• The proposed model introduces a scalar varying mass density field, which results in a non-uniform center of mass that does not necessarily lie on the backbone curve.
• The inertial tensor becomes coordinate-dependent due to the spatial variation of mass density, enabling more accurate dynamic representation.
• The input matrix B(q_e) is explicitly derived and shown to be well-posed, ensuring affine dependence on pressure inputs p1, p2, p3.
• The Coriolis matrix C_ξ satisfies the skew-symmetric condition C_ξ + C_ξ^T = Ṁ_ξ, guaranteeing passivity in the system.
• The dynamic model is affine in control input and structurally similar to rigid-robot Lagrangian models, enabling reuse of established control design tools.
• Numerical simulations based on a physical prototype confirm the expected behavior, validating the model's accuracy and stability.