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Research Papers

Approximating Markov Chain Approach to Optimal Feedback Control of a Flexible Needle

[+] Author and Article Information
Javad Sovizi

Department of Mechanical
and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: javadsov@buffalo.edu

Suren Kumar

Department of Mechanical
and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: surenkum@buffalo.edu

Venkat Krovi

Department of Mechanical
and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: vkrovi@buffalo.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 7, 2015; final manuscript received May 9, 2016; published online July 13, 2016. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 138(11), 111006 (Jul 13, 2016) (7 pages) Paper No: DS-15-1496; doi: 10.1115/1.4033834 History: Received October 07, 2015; Revised May 09, 2016

We present a computationally efficient approach for the intra-operative update of the feedback control policy for the steerable needle in the presence of the motion uncertainty. The solution to dynamic programming (DP) equations, to obtain the optimal control policy, is difficult or intractable for nonlinear problems such as steering flexible needle in soft tissue. We use the method of approximating Markov chain to approximate the continuous (and controlled) process with its discrete and locally consistent counterpart. This provides the ground to examine the linear programming (LP) approach to solve the imposed DP problem that significantly reduces the computational demand. A concrete example of the two-dimensional (2D) needle steering is considered to investigate the effectiveness of the LP method for both deterministic and stochastic systems. We compare the performance of the LP-based policy with the results obtained through more computationally demanding algorithm, iterative policy space approximation. Finally, the reliability of the LP-based policy dealing with motion and parametric uncertainties as well as the effect of insertion point/angle on the probability of success is investigated.

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Figures

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Fig. 1

Needle kinematics nomenclature

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Fig. 3

The needle motion for three different initial conditions generated by using (a) iterative policy approximation and (b) LP approach. Needle tip angle is assumed to be zero in all three initial conditions.

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Fig. 2

Control policies corresponding to the iterative policy approximation technique ((a)–(c)) and LP approach ((d)–(f))

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Fig. 4

KS densities of the needle tip positions when steering the needle from different initial conditions. The needle tip angle is assumed to be zero for all four insertion points.

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Fig. 5

Probability of success versus initial (insertion) condition (a) θ = −45 deg, (b) θ = 0 deg, and (c) θ = 45 deg

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Fig. 6

Effect of the parametric uncertainty on the probability of success (a) θ = −45 deg, (b) θ = 0 deg, and (c) θ = 45 deg

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