CHAPTER VII CONCLUSION AND FUTURE DIRECTIONS
7.3 Control of a Flexible Bevel-Tip Steerable Needle
Flexible bevel-tip steerable needles have generalized much attention recently in minimally invasive surgical procedures. These needles can be used in many complex surgical procedures such as biopsy, regional anesthesia, prostate cancer bratherapy etc.
These needles have many advantages over the straight stiff needles such as the ability to access many critical locations inside the tissue while avoiding sensitive organs like nerves, blood vessels etc. In this research we have designed a feedback controller to drive the needle tip to a 3D target location inside a deformable tissue. We presented the necessary and sufficient condition for controllability and accessibility. A continuous- discrete extended Kalman filter was designed and implemented to estimate the states from the measurement and process noise. We also developed a robust controller that works in the presence of modeling uncertainty of the needle curvature. We performed open loop needle insertion into the artificial tissue phantom and measured the path that the needle followed while cutting through the tissue. Then we fit a circle through those data points to obtain the curvature of the needle, which is used in the controller.
Extensive simulation results demonstrated the efficacy of the proposed controller. The controller behaved well in the presence of measurement noise and in the presence of modeling uncertainty. We also tested our controller with a grid of 1500 target points. We verified our controller with experiments in an artificial tissue phantom and bovine liver.
Experimental results demonstrated convergence of the needle tip to the desired targets with average error of 1.34 mm in 9 experimental insertions to various targets in the artificial phantom and of 2.53 mm in 5 experimental insertions in bovine liver.
In the future we intend to combine our controller with 3D path planners. We believe it will be possible to do this in a straightforward manner by sampling a planned trajectory and controlling sequentially to points along it. Depending on the computational efficiency of the planner, it may be possible to re-plan during insertion, as has been done in prior planar work [143]. Switching between a current and new plan is expected to be straightforward with our controller, since it requires only a new desired point. We also note that in less challenging clinical scenarios where no obstacles are present, our controller may be used directly without a planner. We look forward to future clinical studies employing bevel steering under 3D control.
APPENDIX A
PASSIVITY THEORY
Passivity Theory
Passivity is an abstract formulation of the idea of energy dissipation and passivity theorem, which is based on the input-output point of view, deals with stability problem defined for linear as well as nonlinear systems. A passive system, therefore, cannot generate energy that guarantees stable behavior of the system. A system is said to be passive if
0 )
(
0
t Pd Eit
E (A.1)
where, Eidenotes the initial energy of the system and E(t)is the total energy of the system at time t. Prepresents the net power at input and output ports. If we consider the initial energy to be zero then we will get a well known energy equation of the form
0 )
(
0
t Pd
T dt
E u y (A.2)
where, uand yrepresent system input and output vectors, respectively. In our system, they are usually causal pair of force, f and velocity, v. For a 2-port network, the equation changes to:
t Pd t f vd f v d t
E
0 0
0 )) ( ) ( ) ( ) ( ( )
( (A.3)
where, )f( is the controller force at time , )vd( is the desired velocity of the target and )v( is the measured robotic finger joint velocity. Signs should be chosen carefully for all forces and velocities so that their product is positive when power enters into the system port. Equation (A.3) signifies that a passive system must not generate energy by itself but it can only dissipate the input energy. We assume that the plant is passive.
Therefore, we could guarantee the stability of the system by making the controller passive since the passivity gives the sufficient condition for stability.
Status of a Network Port
Consider a single port network system representing the conjugate variables that define the power flow into the network as shown in Figure A.1. The conjugate variables are force
)
(f and velocity (v).
Figure A.1: One-port network system representing components.
The status of a port can be defined as the product of the conjugate variables. A port is passive if the sign of the product is positive (i.e., f.v0), that means energy is flowing into the network system. A port is active if the sign is negative (i.e., f.v0), that means energy is flowing out of the network system. These conditions are shown schematically
in Figure A.2. Based on the status of the port one can easily calculate the net power flow into the network and which gives the PO for one-port network.
Figure A.2: (a) Energy flow into the network system and (b) energy flow out of the network system based on the sign of the product between force and velocity at a port.
PO may or may not be negative at a particular time depending on operating conditions of the one-port element’s dynamics. However, if it is negative at any time then the one-port may be contributing to instability. Knowing the exact amount of the generated energy, a time-varying damping element can be designed to dissipate only the required amount of energy. This element is called PC. The PC can take the form of a dissipative element in a series or parallel configuration depending on the input causality [104]. Figure A.3 shows the series configuration of the PC for a one-port network system where is an adjustable damping elements at the port. Choice of configuration depends on input/output causality of model underlying each port.
(a)
(b)