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Analysis of Contact Stability and Contact Safety of a Robotic Intravascular Cardiac Catheter under Blood Flow Disturbances
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Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.more » « less
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Abstract The disturbances caused by the blood flow and tissue surface motions are major concerns during the motion planning of an intracardiac robotic catheter. Maintaining a stable and safe contact on the desired ablation point is essential for achieving effective lesions during the ablation procedure. In this paper, a probabilistic formulation of the contact stability and the contact safety for intravascular cardiac catheters under the blood flow and surface motion disturbances is presented. Probabilistic contact stability and contact safety metrics, employing a sample-based representation of the blood flow velocity distribution and the heart motion trajectory, are introduced. Finally, the contact stability and safety for an magnetic resonance imaging-actuated robotic catheter under main pulmonary artery blood flow disturbances and left ventricle surface motion disturbances are analyzed in simulation as example scenarios.more » « less
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It is known that any contact $$3$$-manifold can be obtained by rationally contact Dehn surgery along a Legendrian link $$L$$ in the standard tight contact $$3$$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $$L$$ describing a given contact $$3$$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $$3$$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $$S^1\times S^2$$, the Poincar\'e homology sphere and the Brieskorn sphere $$\Sigma(2,3,7)$$. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $$3$$-sphere. We further obtain results for the $$3$$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.more » « less
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Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the incremental potential contact formulation [Li et al. 2020], and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.more » « less
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/IROS45743.2020.9341527
