Concentric push-pull robots (CPPR) operate through the mechanical interactions of concentrically nested, laser-cut tubes with offset stiffness centers. The distal tips of the tubes are attached to each other, and relative displacement of the tube bases generates bending in the CPPR. Previous CPPR kinematic models assumed two tubes, planar shapes, no torsion, and no external loads. In this paper, we develop a new, more general CPPR model accounting for any number of tubes, describing their variable-curvature 3D shape when actuated, including the effects of torsion and external loads. To accomplish this, we employ a modified Kirchhoff rod model for each tube (with offset stiffness center) and embed the constraints of concentricity. We use an energy method to determine robot shape as a function of actuation and external loading. We experimentally validate this kinetostatic model on prototype CPPRs with two tubes and three tubes and non-constant laser-cut patterns that create variable curvature and stiffness. Experimental results agree with the model, paving the way for use of this model in design optimization, planning, and control of CPPRs.

Parallel-continuum robots combine the advantages of both parallel and continuum robotics. They offer a compromise between the inherent compliance and slenderness of continuum robots and the high precision and strength of rigid-link parallel robots. Throughout recent years there has been an increasing research interest in these novel architectures, which form closed kinematic chains that feature flexible, continuous links undergoing elastic deformations. As the number of publications in this emerging research field is steadily increasing, this survey article summarizes and reviews the state of the art in parallel-continuum robots, discussing their design and modeling. A definition and notation for parallel-continuum robots is introduced, allowing for a clear classification. In conclusion, current open research questions and possible applications for such robots are discussed.

Concentric push–pull robots (CPPR) combine the simplicity, miniaturization potential, and open lumen of concentric-tube robots with the kinematic advantages and stability of push–pull multibackbone designs. A CPPR segment is made from a pair concentric tubes with notches asymmetrically cut into their sides in opposing directions. The two tubes are attached to one another at their tips, and push–pull translation of the tube bases relative to each other changes the curvature along the length of the combined tube pair. Custom, variable-curvature shapes are possible by varying the notch parameters along the tubes. In this article, we present a planar, variable-curvature mechanics model for the actuated segment shape and a method for designing the notch pattern to achieve a desired planar, variable-curvature shape with maximal stiffness within specified strain limits. Experiments validate accuracy for various shapes, materials, and cross sections, showing that the design method achieves a variety of desired shapes. We also demonstrate a multisegment robot made from multiple tube pairs that can independently rotate and actuate, increasing the robot DOF.