[Paper Review] Forward and Inverse Kinematics of a Single Section Inextensible Continuum Arm

Ali Nazari, Diego Castro+1 more|arXiv (Cornell University)|Jul 15, 2019

Soft Robotics and Applications20 references4 citations

TL;DR

This paper proposes a novel inextensible continuum arm using a rigid-chain backbone and pneumatic muscle actuators to enable high payload capacity and decoupled stiffness-position control. It derives closed-form forward and inverse kinematics using a constant-curvature parametric model, validated via spiral path simulation showing reliable muscle length tracking within physical constraints.

ABSTRACT

Continuum arms, such as trunk and tentacle robots, lie between the two extremities of rigid and soft robots and promise to capture the best of both worlds in terms of manipulability, dexterity, and compliance. This paper proposes a new kinematic model for a novel constant-length continuum robot that incorporates both soft and rigid elements. In contrast to traditional pneumatically actuated, variable-length continuum arms, the proposed design utilizes a hyper-redundant rigid chain to provide extra structural strength. The proposed model introduces a reduced-order mapping to account for mechanical constraints arising from the rigid-linked chain to derive a closed-form curve parametric model. The model is numerically evaluated and the results show that the derived model is reliable.

Motivation & Objective

• Address the limitations of traditional pneumatically actuated continuum arms in payload capacity and structural compliance.
• Overcome the lack of independent control over tip position and stiffness in existing designs.
• Develop a kinematic model that enables closed-form inverse kinematics for a single-section inextensible continuum arm.
• Enable reliable path following with physical feasibility constraints using a reduced-order parametric model.
• Facilitate future development of multi-section arms for human-friendly and high-load manipulation tasks.

Proposed method

• Design a continuum arm with a 3D-printed rigid-chain backbone and three antagonistic pneumatic muscle actuators (PMAs).
• Model the arm using a constant-curvature parametric approach with curve parameters θ (orientation) and φ (curvature extent).
• Derive forward kinematics using parametric equations relating muscle lengths l₂, l₃ to spatial tip position (Px, Py, Pz).
• Develop closed-form inverse kinematics by solving trigonometric identities from position equations to compute φ and θ from tip coordinates.
• Use a possibility map of muscle length combinations to validate inverse kinematic results against mechanical feasibility.
• Simulate a 3D spiral path in task space and track muscle length and curve parameter evolution using the derived inverse kinematic model.

Experimental results

Research questions

• RQ1 How can a continuum arm achieve both high structural stiffness and compliance through a hybrid rigid-soft design?
• RQ2 Can a closed-form inverse kinematic solution be derived for a single-section inextensible continuum arm with mechanical constraints from a rigid backbone?
• RQ3 To what extent does the inverse kinematic model produce physically feasible muscle length combinations during complex 3D path following?
• RQ4 How does the inclusion of a rigid-chain backbone enable decoupled control of stiffness and tip position compared to traditional variable-length arms?
• RQ5 What is the accuracy and reliability of the parametric kinematic model in simulating complex, non-planar motion trajectories?

Key findings

• The closed-form inverse kinematics solution successfully computes curve parameters and muscle lengths for a 3D spiral path, demonstrating numerical feasibility.
• Muscle length profiles during path following remained within the physically allowable range as confirmed by the possibility map in Figure 10.
• The derived model achieved reliable trajectory tracking with consistent convergence, indicating robustness for real-time control applications.
• The parametric model accurately captured the spatial position and orientation of the arm tip using only the actual degrees of freedom (muscle length changes).
• The use of a rigid-chain backbone enabled structural integrity and higher payload capacity while maintaining the dexterity of soft continuum arms.
• The simulation results confirmed that the model supports decoupled stiffness and position control, a key enabler for human-robot interaction and object manipulation tasks.