The goal of motion planning is to obtain a set of control actions that guides the needle to a predetermined target while avoiding obstacles and satisfying probable user-defined criteria. To explore a model-based control approach, first we need to model tissue-needle interac- tion. For this purpose, a discrete version of the bicycle-like model initially developed by Webster et al. [5] and widely used in other work (e.g., [10–12]) is proposed. Torsion com- pensation will be added to this model in section 6.5. Note that choosing a rigid material to represent tissue and ignoring distributed friction may not be realistic [5]; however, it is acceptable for preliminary experiments and to study the effect of needle deflection dur- ing steering without the complexities arising from tissue behavior. It is also presumed in this work that the tip position is measured by an imaging modality or an electromagnetic tracker, and the workspace boundaries and the obstacle arrangement are known parameters. Therefore, this path planner is only concerned with steering the needle inside this imaging plane. We believe that this simplified scheme will contribute in better understanding of the modeling and control issues in more generic path-following cases.
It is assumed that the rotation of the needle at the base leads to reorientation of the tip and hence control of the trajectory of the needle. This is the method based on which the needle is manipulated inside the tissue. One DOF actuated at the needle base is a pure insertion of length δ while the second DOF is a pure tip rotation of 180o. Under this condition
and regardless of the insertion velocity, the needle moves along an arc of approximately constant radius ρ in the direction of the bevel. A 180o rotation of the needle causes the
bevel at the tip to point in the opposite direction of the preceding arc of deflection. Thus, the entire motion of the needle is fully described by the motion of the needle tip and a number of circular segments with the radiusρ. The value of ρis dependent on the tissue- needle interaction properties, which includes tissue elasticity, needle geometry, friction, and clamping forces during insertion. If one can determine the aforementioned turning points inside the tissue, the planning problem becomes straightforward since the rest of the actions are merely pure insertions. Moreover, it should be noted that the proposed control action that is a bang-bang strategy is applied at discrete time intervals using digital controllers. Therefore, investigating a proper strategy for discretization is studied as a part of the modeling. In this regard, the scheme proposed in [10, 11] is outlined next.
One control action is a pure insertion of lengthδand the other one is180orotation followed
by an insertion of the same length. Tracking circular segments, the needle state at instant k is defined as Sk = (xk yk θk bk)T where the tip position tk = (xk yk)T and the tip orientation angle θk are rounded values to the nearest points on corresponding projection planes. In this representation, we overlay a control circle with radius ρon a∆-grid plane consisting of horizontal and vertical∆spacing. Each control circle is also divided intoNr discrete arcs of the same length ofδwhere,
δ = 2πρ
Nr =ρα
∆and αare spatial and angular resolutions, respectively. ∆×[∆δ]is the insertion length projected on horizontal or vertical axes in each step. At each position on the ∆-grid net- work, the needle may be in any of theNrorientation states and in any of the two clockwise or counter-clockwise directions. Each direction corresponds to two control circles that are tangent to each other at tk. The bevel direction orbk is a binary variable which keeps the history of the previous arc and switches at turning points. The tip angle orθkis the tangent angle of the current control circle. From an initial value, the tip angle increments or decre- ments byαdepending on the instantaneous control action and the current circular segment. Fig. 6.1 demonstrates the projected needle motion on the imaging plane through segmented arcs. Consequently, this selection results in a model withndiscrete states as given by,
n = 4πρxmaxymax ∆2δ
where xmax and ymax are the depth and the height of the imaging plane. Accordingly, a better resolution implies a larger discrete model which results in more computational burden demanded by the motion planner.
As mentioned earlier, the tip curvature may be deflected from its intended path as a result of unpredictable tissue-needle interaction, local tissue displacements, system noise or any other unknown artifacts. To incorporate any motion artifacts stemming from this complex behavior,θk and the value ofρhave to be updated during the operation. Since the value of ρ seems to be more deterministic in this case, it is set to a fixed value. It is worth noting that in some cases, applying a constantρfor the entire operation may not be feasible unless tissue properties and needle geometry remain consistent. Also, more deviation may be introduced when a rotation is accompanied by an insertion rather than just a pure insertion.
Figure 6.1:Circular segments projected on the∆-grid network.
Incorporating all these complex uncertainties has been ignored in the current work.
Finally, deflection from circular segments was modeled using a discrete Gaussian distribu- tion as defined in (6.1). In this definition,βi (1≤ i≤ 3)specifies statistical properties of the tip deviation, and at the end of each iteration, correction is made by simply adding β˜k toθk. Here,β˜k = 0corresponds to the deterministic motion with no deviation.
P{β˜k=mα}= β1 if m= 0 β2 if m=±1 β3 = 12(1−β1−2β2) if m=±2 (6.1)