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3D Robust Online Motion Planning for Steerable Needles in Dynamic Workspaces Using Duty-Cycled Rotation

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Abstract

This work presents a closed-loop strategy for 3D online motion planning of beveled steerable needles using duty-cycled rotation. The algorithm first selects an entry point that minimizes a multi-criteria cost function and then combines an RRT-based path planner with an intraoperative replanning algorithm and workspace feedback information to constantly update the needle inputs and adjust the trajectory. Simulations in a workspace based on a typical prostate needle steering scenario show that the algorithm is robust against disturbances and model uncertainties and can provide online trajectories to avoid obstacles even under the presence of physiological motion.

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Notes

  1. 1.

    Although we use here the expression “input,” one should notice that we do not refer to the needle actuation inputs \(\omega \) and \(\nu \), but to pure geometrical parameters used to expand the RRT tree. The insertion and velocity actuation inputs are actually a consequence of the planned path formed by the concatenation of arcs obtained from such geometrical parameters.

  2. 2.

    For videos, see the first part of Online Resource 1.

  3. 3.

    For videos, see the second part of Online Resource 1.

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Acknowledgments

This work was supported by Coordenaç\({\tilde{a}}\)o de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Author information

Correspondence to Mariana C. Bernardes.

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Supplementary material 1 (mpg 39330 KB)

Supplementary material 1 (mpg 39330 KB)

Appendix

Appendix

A quaternion \(\mathbf {h}\) consists of a real component plus an imaginary part composed of three quaternionic units \(\hat{\imath },\hat{\jmath },\hat{k}\); that is, \(\mathbf {h}=a+b\hat{\imath }+c\hat{\jmath }+d\hat{k}\), where \(a,b,c,d\in \mathbb {R}\), \(\hat{\imath }^{2}=\hat{\jmath }^{2}=\hat{k}^{2}=-1\), and \(\hat{\imath }\hat{\jmath }\hat{k}=-1\). Its conjugate is given by \(\mathbf {h^{*}}=a-b\hat{\imath }-c\hat{\jmath }-d\hat{k}\).

A rotation composed of a rotation angle \(\phi \) around the axis \(\mathbf {n}=n_{x}\hat{\imath }+n_{y}\hat{\jmath }+n_{z}\hat{k}\) is given by the quaternion \(\mathbf {r}=\cos (\phi /2)+\sin (\phi /2)\mathbf {n}\).

A translation \(\mathbf {p}\) is represented by a pure quaternion; that is, a quaternion where the real part is equal to zero. Thus, \(\mathbf {p}=p_{x}\hat{\imath }+p_{y}\hat{\jmath }+p_{z}\hat{k}\).

The rigid motion is then completely represented by the dual quaternion \(\underline{\mathbf {q}}=\mathbf {r}+\epsilon \frac{1}{2}\mathbf {p}\mathbf {r}\), where \(\epsilon \) is Clifford’s dual unit, which is nilpotent; that is, \(\epsilon \ne 0\) but \(\epsilon ^{2}=0\).

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Bernardes, M.C., Adorno, B.V., Borges, G.A. et al. 3D Robust Online Motion Planning for Steerable Needles in Dynamic Workspaces Using Duty-Cycled Rotation. J Control Autom Electr Syst 25, 216–227 (2014). https://doi.org/10.1007/s40313-013-0104-4

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Keywords

  • Medical robotics
  • Needle steering
  • Motion planning