Characterisation of Frenet-Serret and Bishop motions with applications to needle steering

Frenet-Serret and Bishop rigid-body motions have many potential applications in robotics, graphics and computer-aided design. In order to study these motions, new characterisations in terms of their velocity twists are derived. This is extended to general motions based on any moving frame to a space curve. Furthermore, it is shown that any such general moving frame motion is the product of a Frenet-Serret motion with a rotation about the tangent vector. These ideas are applied to a simple model of needle steering. A simple kinematic model of the path of the needle is derived. It is then shown that this leads to Frenet-Serret motions of the needle tip but with constant curvature. Finally, some remarks about curves with constant curvature are made. © Cambridge University Press 2013.

[1]  Fopke Klok.  Two moving coordinate frames for sweeping along a 3D trajectory , 1986, Comput. Aided Geom. Des..

[2]  Allison M. Okamura,et al.  Planning for Steerable Bevel-tip Needle Insertion Through 2D Soft Tissue with Obstacles , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[3]  Jin Seob Kim,et al.  Nonholonomic Modeling of Needle Steering , 2006, Int. J. Robotics Res..

[4]  Azriel Rosenfeld,et al.  Egomotion analysis based on the Frenet-Serret motion model , 2005, International Journal of Computer Vision.

[5]  Jon M. Selig.  Curves of stationary acceleration in SE(3) , 2007, IMA J. Math. Control. Inf..

[6]  Christoph Engelhardt,et al.  Closed Space Curves of Constant Curvature Consisting of Arcs of Circular Helices , 1998 .

[7]  J. Monterde.  Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion , 2009, Comput. Aided Geom. Des..

[8]  Bahram Ravani,et al.  Curves with rational Frenet-Serret motion , 1997, Comput. Aided Geom. Des..

[9]  Seth Earl Elliott,et al.  The theory of screws , 2022 .

[10]  Rida T. Farouki,et al.  Helical polynomial curves and double Pythagorean hodographs I. Quaternion and Hopf map representations , 2009, J. Symb. Comput..

[11]  K. H. Hunt,et al.  Geometry of screw systems1Screws: Genesis and geometry , 1990 .

[12]  E. Salkowski.  Zur Transformation von Raumkurven , 1909 .

[13]  Ali Meghdari,et al.  Spatial rational motions based on rational frenet-serret curves , 2004, 2004 IEEE International Conference on Systems, Man and Cybernetics (IEEE Cat. No.04CH37583).

[14]  Peter Donelan,et al.  On the Hierarchy of screw systems , 1993 .

[15]  R. Bishop.  There is More than One Way to Frame a Curve , 1975 .

[16]  Jin Seob Kim,et al.  Diffusion-Based Motion Planning for a Nonholonomic Flexible Needle Model , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[17]  S. Shankar Sastry,et al.  Three-dimensional Motion Planning Algorithms for Steerable Needles Using Inverse Kinematics , 2010, Int. J. Robotics Res..