CHAPTER VI ROBOT-ASSISTED FLEXIBLE NEEDLE INSERTION
6.10 Results and Discussions
6.10.2 Numerical Simulation Results
Figure 6.13: Simulated kinematic model matches closely with the experimental data. The blue dotted line is the experimental data and red circle is the simulated data.
controller gains are chosen as k1=0.1, k2=0.1 and k3=0.1. Before simulating the controller’s response to points both within and outside its practical workspace under these conditions, we first describe the practical workspace.
Practical Workspace:
As mentioned previously, the kinematic needle system has been shown to be theoretically controllable in the sense that it is possible to reach any point in SE(3), given a sufficiently long arc length. However, in practice, the maximum arc length will be restricted because the needle will be traveling through the human body rather than an infinite medium of soft tissue. Thus the practical workspace is the roughly trumpet-shaped space illustrated in Figure 6.14, defined by the needle’s maximum curvature and a given maximum arc length. In Figure 6.14 we illustrate the workspace for a maximum curvature of
1 . 12 /
1
cm-1 and an arc length of 13 cm, which matches the practical workspace of our experimental apparatus described in Section 6.10.3.
Figure 6.14: Practical workspace boundary for a needle with curvature 1/12.1 cm-1 and a
Control to Points within the Practical Workspace:
To test the ability of our controller to access points in a practical workspace, we considered a tissue medium of dimensions 10 cm x 10 cm x 20 cm, with the needle entering at the center of one of the square 10 cm x 10 cm faces. This point was assigned (0, 0, 0, 0, 0, 0) when the needle was aligned with the long axis of the tissue medium. We selected 1500 random 3D targets within the 10 cm x 10 cm x 8 cm volume bounded by the face of the tissue medium furthest from the needle’s starting position, and defined the practical workspace as being bounded by a curvature of 1/12.1 cm-1 with an arc length sufficient to reach to the far end of the tissue medium.
For each of the 1500 target points we simulated insertion of the needle under closed- loop control, and with the level of uncertainty mentioned previously (a standard deviation of 3 mm in each position variable and of 1.5o in each orientation variable) in initial conditions, and also in sensor readings. Each insertion was stopped as soon as the tip error (as defined by the difference in Kalman filter position values and the desired target position) stopped decreasing and began increasing.
In our simulations, 100% of targets were reached to an accuracy of less than 1 mm, and average error was 0.08 mm. Several example trajectories within our simulation data set as shown in Figure 6.15.
Figure 6.15: Several selected example trajectories taken from the simulation dataset of 1500 insertions to randomly selected target points.
Points not Directly (With Short Arc Lengths) Accessible:
Since practical needle trajectories are limited to relatively short arc lengths (the needle is not allowed to pass the target and “loop back” to reach it), we evaluated what our controller would do for points outside its (limited arc length) workspace, as illustrated in Figure 6.16. Here, we simulate four different target points, which require a curvature higher than the needle’s maximum (1/12.1 cm-1) to reach. The important observation here that despite not being physically able to reach the desired point, the controller does not go unstable, but rather approaches the desired point to the extent that it is able to reach. If we allow the time to go on longer, then the needle will loop back to reach the desired target.
Such a situation is shown in Figure 6.17.
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20 -6
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Figure 6.16: Performance of the controller when for targets not accessible with short arc lengths. Targets: (a) (20, 20, 20) cm; (b) (-18, 18, 20) cm; (c) (20, -18, 18) cm; (d) (-18, - 20, 20) cm. The initial condition in all cases are (0, 0, 0, 3o, 5o, 5o).
Figure 6.17: Performance of the controller for accessing an extreme target. Target: (3, 6, 6) cm. The initial condition is (0, 0, 0, 3o, 5o, 5o).
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-40 -20 0 20 -10 0 10 20 30
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Needle entry point
Illustration of Robust Control Benefit:
Now we will present the control system robustness with parameter uncertainty. The uncertainty includes the changes in the curvature value, which is greatly affected by the variation of tissue properties and the properties of the needle. To demonstrate the performance of the feedback controller, two simulations are conducted, (i) the system model contains parameter uncertainties but the controller does not consider them, and (ii) the robust control method is used to control the system with parameter model uncertainties. For this case, we increase the parameter by 15% with the same controller gains as before.
In the first case, the controller described by Equation 6.30 is used without consideration of the additional control term due to the presence of model parameter uncertainty. The positioning operation of the needle tip to the desired target location is shown in Figure 6.18 (black line). It is noticed that the convergence of the needle tip position error to zero is not made, although the needle tip reaches a point close to the target. The presence of modeling uncertainty affects the performance of accessing the target location and hence it results in the needle tip placement error as can be seen in Figure 6.18. Therefore, to maintain or speed up the efficiency of the control action the error should be suppressed during the motion.
In the second case, the robust feedback control described by Equation 6.33 is used to control the needle tip position of the deformable object with the presence of modeling uncertainty. The needle tip is positioned to the desired target location successfully as shown in Figure 6.18 (blue line). Comparing the needle tip placement errors with and without consideration of robust control for the system with model parameter
uncertainties, the robust control method can improve the performance of the system with model parameter uncertainties.
Figure 6.18: Accessing a target located at (25, 25, 25) cm in the presence of modeling uncertainty with robust controller (blue line) and without robust controller (black line).
6.10.3 Experimental Results