Existing static and kinematic models of concentric tube robots are based on the ordinary differential equations of a static Cosserat rod. In this paper, we provide the first dynamic model for concentric tube continuum robots by adapting the partial differential equations of a dynamic Cosserat rod to describe the coupled inertial dynamics of precurved concentric tubes. This generates an initial-boundary-value problem that can capture robot vibrations over time. We solve this model numerically at high time resolutions using implicit finite differences in time and arc length. This approach is capable of resolving the high-frequency torsional dynamics that occur during unstable "snapping" motions and provides a simulation tool that can track the true robot configuration through such transitions. Further, it can track slower oscillations associated with bending and torsion as a robot interacts with tissue at real-time speeds. Experimental verification of the model shows that this wide range of effects is captured efficiently and accurately.

Mechanics-based models of thin elastic structures are prevalent in robotics research, both in soft/continuum robot modeling, and in robotic manipulation of strings, sutures, needles, and endoscopes. In all these applications, distributed loads along the device's length can affect its shape in space. Estimation of the distributed loading based on observation of the object's shape constitutes a classical mechanics inverse problem that would be useful in many applications, but this problem has received relatively little attention to date. In this paper, we propose methods to estimate distributed loads on an elastic rod using a large-deflection Cosserat-rod model and constrained nonlinear optimization. We perform experiments that illustrate the feasibility of using these methods to locate regions of high contact force along the rod, and to estimate magnitudes of the forces that are applied. Results show that overall force magnitudes and locations can be estimated with average error of 0.29 N (6.7% of average resultant magnitude) and 4 mm (2% of rod length) for complex double-bend shapes, and the shape approximation has near-zero error.

Parallel continuum robots consist of a parallel arrangement of flexible legs and are dexterous, compliant, and easily miniaturized for minimally invasive surgery. By design, parallel continuum robots exhibit large, nonlinear deformations in their legs to achieve multi-DOF end effector articulation, but excess leg bowing can limit their reachable workspace, especially for long slender designs. In this paper, we investigate a parallel continuum robot design with a passive spring backbone carrying disks that constrain the legs at intermediate points. The constraints route the legs in helical paths around the backbone and prevent large divergence of the legs, expanding the reachable workspace for slender form factors while preserving the manipulator's six degrees of freedom. We present a novel forward and inverse kinematics model, based on Cosserat rod theory, that accommodates general leg routing paths and any number of intermediate constraint disks. We also explore manipulator workspace with experiments and simulations, demonstrating that intermediate constraints expand the reachable workspace of slender parallel continuum robots.