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.

Soft and continuum robots driven by tendons or cables have wide-ranging applications, and many mechanics-based models for their behavior have been proposed. In this paper, we address the unsolved problem of predicting robot deflection and stiffness with respect to environmental loads where the axial displacements of the tendon ends are held constant. We first solve this problem analytically for a tendon-embedded Euler-Bernoulli beam. Nondimensionalized equations and plots describe how tendon stretch and routing path affect the robot's output stiffness at any point. These analytical results enable stiffness analysis of candidate robot designs without extensive computational simulations. Insights gained through this analysis include the ability to increase robot stiffness by using converging tendon paths. Generalizing to large deflections in three dimensions (3-D), we extend a previous nonlinear Cosserat-rod-based model for tendon-driven robots to handle prescribed tendon displacements, tendon stretch, pretension, and slack. We then provide additional dimensionless plots in the actuated case for loads in 3-D. The analytical formulas and numerically computed model are experimentally validated on a prototype robot with good agreement.

Parallel continuum robots (PCRs) combine the compactness, simplicity, and compliance of continuum robots with the precision and strength of rigid-link parallel robots. In this paper, we provide a generalized Cosserat-rod-based kinetostatic model framework that accommodates various joint types and problem formulations (e.g., forward and inverse kinematics under loads, and deflection-based and actuation-based force sensing) useful for simulation and control. Linearization of this general model provides the manipulator Jacobian, end-effector compliance, input stiffness, and wrench reflectivity matrices, which allow us to examine the effect of design parameters on dexterity, force application, and force-sensing ability. Using ellipsoids based on the matrices, we provide a set of design simulations and graphically depict the relationships between pose, actuation, and forces. We further provide a nondimensional analysis of the compliance of PCRs. Finally, we experimentally demonstrate and validate actuation-based force sensing on a prototype six-degree-of-freedom PCR, demonstrating 3-D force sensing with a median magnitude and a directional error of 0.23 N (8% of actual load) and 12°, respectively.

The large dynamic deflections of continuum robots, soft robots, and slender elastic objects can be accurately modeled with classical rod theories in nonlinear elasticity. In this paper, we propose a real-time computational approach for solving the partial differential equations of a dynamic Kirchhoff rod. Our approach is based on implicit time discretization of the Kirchhoff equations and subsequent solution of the resulting continuous spatial boundary value problem at each time step. This modular approach can exhibit low numerical damping, handle arbitrarily large time steps, and provide an accurate, high-order representation of the rod shape in steady-state. We experimentally validated the method by capturing footage of a dynamic rod with a high speed camera and comparing this experimental data with simulations using the proposed approach. Soft-real-time performance is achieved, and the relationship between time step and real-time performance is explored in a plot.

Classic theories in nonlinear elasticity have increasingly been used to obtain accurate and efficient models for continuum robots and other elastic structures. Numerically computed solutions of these models typically satisfy the first-order conditions necessary for equilibrium, but do not provide any information about the elastic stability of the solution. The inability to detect or avoid physically unstable model solutions poses a major hindrance to reliable model-based simulation, planning, design, and control. In this paper, we adapt results from optimal control to determine the stability of Kirchhoff rods and Cosserat rods subject to general end constraints, including coupled multirod models which describe parallel continuum robots. We formulate a sufficient condition for the stability of a solution, a numerical test for evaluating this condition, and a heuristic stability metric. We verify that our numerical stability test agrees with the classical results for the buckling of single columns with various end constraints and for multicolumn frames. We then validate our approach experimentally on a six degree-of-freedom parallel continuum robot.

Parallel continuum robots have the potential to provide multi-degree-of-freedom articulation using a structure that is simple, compact, compliant, and highly scalable. These characteristics may be useful in micromanipulation, endoscopic robotic-assisted surgery, and human-robot interaction. Our prior work formulated a kinematic model which treats a parallel continuum robot as a set of multiple Cosserat rods with coupled boundary conditions. In this paper, we detail methods for the efficient numerical solution of this model at rates that enable real-time interactive simulation, motion planning, design optimization, and control. Exploitation of the model structure enables a significant reduction in the number of integrations required to evaluate the boundary value Jacobian matrix used in a shooting method. Our approach is used to teleoperate a prototype robot using real-time inverse kinematics solutions, and simulation tests show that inverse kinematics solutions are consistently computed at rates of several kilohertz using standard desktop computing hardware.