Scholarly article on topic 'Closed-Loop Planar Motion Control of a Steerable Probe With a “Programmable Bevel” Inspired by Nature'

Closed-Loop Planar Motion Control of a Steerable Probe With a “Programmable Bevel” Inspired by Nature Academic research paper on "Medical engineering"

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Academic research paper on topic "Closed-Loop Planar Motion Control of a Steerable Probe With a “Programmable Bevel” Inspired by Nature"

Closed-Loop Planar Motion Control of a Steerable Probe With a "Programmable Bevel" Inspired by Nature

Seong Young Ko, Member, IEEE, Luca Frasson, and Ferdinando Rodriguez y Baena, Member, IEEE

Abstract—Percutaneous intervention has attracted significant interest in recent years, but many of today's needles and catheters can only provide limited control of the trajectory between an entry site and soft tissue target. In order to address this fundamental shortcoming in minimally invasive surgery, we describe the first prototype of a bioinspired multipart probe that can steer along planar trajectories within a compliant medium by means of a novel "programmable bevel," where the steering angle becomes a function of the offset between interlocked probe segments. A kinematic model of the flexible probe and programmable bevel arrangement is derived. Several parameters of the kinematic model are then calibrated experimentally with a fully functional scaled-up prototype, which is 12 mm in diameter. A closed-loop control strategy with feed-forward and feedback components is then derived and implemented in vitro using an approximate linearization strategy that was first developed for car-like robots. Experimental results demonstrate satisfactory 2-D trajectory following of the prototype (0.68 mm tracking error, with 1.45 mm standard deviation) using an electromagnetic position sensor that is embedded at the tip of the probe.

Index Terms—Biologically inspired robots, closed-loop control, medical robots and systems, needle steering, nonholonomic motion planning.

I. Introduction

PERCUTANEOUS intervention has always attracted significant interest because it is performed through the skin, and as such, it has several advantages for the patient [1]. Tumor biopsy, brachytherapy, deep brain stimulation, and localized drug delivery, for instance, benefit from this operative technique to reduce tissue trauma and hospitalization time.

In order to localize a lesion, preoperative planning using computer tomography (CT) or magnetic resonance imaging (MRI) is often necessary. If the lesion is reachable through a straight path, a straight rigid needle can be used. For a safe operation,

Manuscript received October 25, 2010; revised March 14, 2011; accepted June 2, 2011. Date of publication July 29, 2011; date of current version October 6, 2011. This paper was recommended for publication by Associate Editor E. Guglielmelli and Editor W. K. Chung upon evaluation of the reviewers' comments. This work was supported by the EU-FP7 Project ROBOCAST (FP7-ICT-215190) and by the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013)/ERC Grant Agreement (258642-STING).

The authors are with the Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: s.ko@imperial.ac.uk; luca.frasson07@imperial.ac.uk; f.rodriguez@imperial.ac.uk).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TR0.2011.2159411

the location of the needle tip can be monitored by means of external markers that are mounted on the proximal end of the needle by relying on the fixed geometrical relationship between the base and needle point.

Recently, there have been efforts to introduce steerable needles in percutaneous interventions where a straight path does not seem possible or is not safe. Steerable needles are generally very flexible and can, thus, bend during tissue penetration. In principle, this feature could enable a surgeon to take a "roundabout" way to a target if "no-go" areas, for instance, important vessels, are encountered along the straight line path intersecting entry and target locations. A suitable steering strategy, however, needs to be developed to exploit the flexibility of the needle, and localization of the tip position becomes more difficult since it is no longer possible to extrapolate tip position from a base measurement.

Three main approaches to the needle steering problem have been proposed to date. By modeling the material and geometric properties of needles and their behavior in soft tissue, DiMaio and Salcudean developed a model-based trajectory planner, where needle deflection and the soft tissue's deformation are used as a means to predict tip orientation in a compliant medium [2]. For real-time simulation and path planning, Glozman and Shoham, subsequently, developed a needle steering algorithm using a simpler model for the needle and the soft tissue, where the model considered springs to predict needle-tissue interaction forces [3]. These approaches take into account the deflection of a relatively stiff needle, which can be controlled by applying a suitable combination of moments at its base.

The second approach to needle steering relies upon the concept of preloaded concentric tubes, which are able to slide with respect to each other to produce curvilinear configurations [4], [5]. By modeling the kinematics and dynamics properties of these nested segments, accurate tip motion control in 3-D has been demonstrated [6], while path tracking along curvilinear trajectories within a compliant medium (i.e., not within a lumen or cavity) has yet to be achieved.

The third and final approach to the needle steering problem exploits a thin and flexible needle with a bevel tip. In this embodiment, the asymmetry of the needle tip is exploited to produce a curved path within tissue, which can be controlled by altering the bevel angle and area, the needle material, and its cross-sectional diameter. On this basis, Webster, III, et al. proposed a kinematic model of a bendable needle with a fixed bevel tip in 3-D space [7]. Alterovitz et al. derived a

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motion-planning algorithm in the presence of measurement uncertainty to obtain better targeting accuracy for a bevel-tip flexible needle [8]. Reed et al. modeled the torsional dynamics of a flexible needle to analyze torsional behavior during needle insertion [9]. These studies focus on open-loop trajectory control that is based on a kinematic model of the probe, coupled with a probe-tissue interaction model.

Indeed, the control of flexible needles represents a current research focus, with several attempts being published in the literature that have substantially advanced the state of the art. Kallem and Cowan proposed a plane alignment control algorithm for needle steering along planar trajectories to minimize the off-plane error [10]. They utilized a stereo camera to measure the tip position of a flexible needle and implemented a full-state observer to estimate missing states (such as the rotational degrees of freedom of the needle, which could not be explicitly measured). In order to generate multiple trajectories with a single flexible probe with a fixed bevel tip, Minhas et al. [11], [12] and Wood et al. [13] proposed a duty-cycling spinning algorithm by periodically changing the orientation of the needle along its long axis by means of a base-mounted revolute actuator, trajectories with different radii could be achieved. In order to reduce drilling effects during the duty-cycle spinning of a beveled needle, Hauser et al. proposed an algorithm using variable helical paths, based on the principle that the flexible needle will generate a helical path during simultaneous rotation and insertion [14].

In the presence of significant uncertainty, for instance, that introduced by complex deformations of a soft tissue under dynamic load, closed-loop control is required to keep the needle on a predefined trajectory while being subjected to dynamic loading conditions. In closed-loop feedback control, an external sensing device, whether incorporated into the needle tip (e.g., the electromagnetic (EM) sensor that is described further in this paper) or available during the insertion process (e.g., intraoperative fluoroscopy), is required to monitor the actual tip position within the substrate. To our knowledge, only two demonstrations of the latter approach (i.e., with image-guided feedback control) are reported in the literature, while the use of an embedded sensor to steer a needle has remained an open research challenge until now. Specifically, Glozman and Shoham utilized fluoroscopic images to measure the deflection of a stiff needle (without bevel tip) when inserted into tissue [15]. In [16], a position estimator that is based on stereo camera images is used to steer a thin and flexible bevel tip needle into gelatin. An "ON-OFF" controller, which switches between "bevel-right" and "bevel-left," is complemented by a path planning module, torsion compensation, and an off-plane error minimization algorithm. These approaches have been generally successful but rely upon external sensors and complex image processing that limit the range of viable applications for this type of technology.

Recent works also demonstrated possible clinical applications of these methods. Majewicz et al. demonstrated three potential clinical applications of needle steering in ex vivo tissue simulations: ablation, biopsy, and brachytherapy [17]. Burdette et al. have integrated a two-segment concentric tube with an ultrasonic-based ablator, performing multiple thermal ablation

in ex vivo bovine liver under 3-D ultrasound monitoring. With a single penetration, by retracting and pushing only one of the nested tubes, three different tissue locations were ablated [18].

This paper describes the design, implementation, and control of a flexible multipart probe that is inspired by the ovipositor, or egg-laying channel, of certain insects, the foundations for which have previously been published [19]-[22]. The probe, which is composed of four interlocked probe segments, is able to alter its direction by means of a "programmable bevel tip," which is described here, alongside the development of a control strategy to drive the probe along planar trajectories within a compliant medium (gelatin). A small EM position sensor, which is embedded within the tip of the probe, is employed to monitor the tip position and orientation, providing the command signal for a bespoke feedback controller that is developed for the probe. A significant advantage of this system is that it is expected to be able to follow arbitrary curvilinear trajectories without discontinuities. With the ability to smoothly change the orientation of the probe's tip without the need for torsion along the long axis of the probe, thanks to the programmable bevel concept, the probe is also expected to cause less strain on the surrounding tissue, with a consequent reduction in tissue damage. Possible future target applications include keyhole neurosurgery (e.g., implantation of deep brain stimulation electrodes) and tissue biopsy of inaccessible, deep-seated regions of the body.

This paper is organized as follows. The biological inspiration for this study and the "programmable bevel" concept are outlined in Section II. Section III describes the kinematic model that is developed for the probe and bevel assembly. A bespoke control strategy for 2-D needle steering, which is built around the kinematic model described in Section III, is explained in Section IV. Section V describes the experimental results that are obtained with a first 12-mm outer diameter (OD) proof-of-concept prototype and also explains the calibration of important parameters that are related to the kinematic model. Finally, conclusion and future work are outlined in Section VI. As a point of note, this paper expands on a previous report [23] by covering the foundations of the biologically inspired probe design, the implementation of a scaled-up 12-mm prototype, and the experimental evaluation of the control strategy outlined in [23] within an artificial brain-like medium (gelatin).

II. Biological Inspiration A. Flexible Probe Inspired by Ovipositing Wasps

Ovipositing wasps [see Fig. 1(a)] and the unique approach that they employ to penetrate different kinds of substrate in order to lay eggs are the source of inspiration for the novel flexible and steerable probe described here.

The ovipositor, which is a very long, thin, flexible structure [24], [25], consists of two or more segments (valves), which are connected by means of a special dovetail mechanism, as shown in Fig. 1(b), and are able to slide with respect to each other. An inner channel is used to deliver eggs in the substrate: Some species lay eggs into the bark of wood, while parasitic species lay them into the soft tissue of hosts, such as larvae of other insects, often in a number of separate but adjacent locations.

Fig. 1. (a) Giant ichneumon wasp: Rhyssa persuasoria. Adult boring the surface of trunk infested with wood wasp larvae. Image courtesy of Boris Hrasovec, Faculty of Forestry. (b) Diagrammatic representation of oblique view of transversely cut ovipositor (modified from [26]).

Even though the ovipositor is avoided of intrinsic muscular, the multipart structure allows the reorientation of the ovipositor tip, which in turn enables it to steer within the substrate [26], [27].

Inspired by this natural design, a flexible probe that is potentially capable of 3-D steering in soft tissue is currently under development at Imperial College London. A proof-of-concept flexible probe prototype, which is composed of four segments that are connected to each other by means of an interlocking mechanism [21], is used to demonstrate probe steering. The current scaled-up prototype is not clinically viable, as a significantly smaller OD (i.e., 1-4 mm) would be required for clinical deployment. However, ongoing work on the miniaturization of the probe, the optimization of the probe's interlocking mechanism [22], and the modeling of the probe-tissue interaction forces [28] is expected to facilitate its applicability in the near future.

A unique feature of the design centers upon the relationship between the offset between interlocked probe segments and the steering angle of the probe, or "programmable bevel" concept, which provides the foundations for the control strategy described in this study.

B. Bioinspired Programmable Bevel

In the programmable bevel concept that is described in the following sections, two interlocked probe segments would be sufficient to steer the tip in a plane. The subdivision of the probe into four interlocked segments, however, stems from the need to stabilize the insertion process and enables the future extension of this study to 3-D steering. In the wider context of this study, a unique reciprocating insertion method is being investigated, where each probe segment is inserted one at a time, while the remaining, stationary segments act as "rails," helping to transfer the forward push from the back of the probe, along the long axis of the probe, to the tip. It is believed that this approach to probe insertion will minimize the amount of tissue deformation at the probe-tissue interface, with a consequent reduction in tissue damage. This hypothesis is currently under investigation and does not relate directly to the study presented here but provides justification for the four-part embodiment of the probe described next.

Fig. 2. Programmable bevel concept applied to a four-part probe. Segments I and III define tip orientation (up or down), while segments II and IV are always aligned with the lagging steering segment (i.e., segment I in the diagram).

A diagram that illustrates the key features of the four-part probe concept is illustrated in Fig. 2. Through numerous laboratory experiments, the offset between the two leading segments (i.e., segments I and III) has been shown to be related to the steering angle of the probe tip. It is believed that this phenomenon is related to the amount of unsupported length associated with the leading segment (or "steering offset" in Fig. 2), since it is likely to affect the deflection magnitude that is experienced by the segment as a result of tissue reaction forces during insertion. While an analytical description of this phenomenon is currently under investigation, Section V describes an experimental calibration process where the relationship between offset and steering angle of a prototype is found to be approximately linear, which is a finding that forms the basis of the planar motion control strategy that is presented in this paper.

Since only planar trajectories are considered, the four-part probe in Fig. 2 is modeled in 2-D only, disregarding the passive segments that simply follow the lagging steering segments (i.e., segments II and IV).

III. Kinematic Modeling

In the following modeling description, it is assumed that trajectories are defined in a plane and that the probe, which is aligned with the plane during an initial setup, is composed of two identical segments. Webster, III, et al. showed that the kinematic model of a bevel tip needle could be considered to be similar to that of a bicycle model with a fixed steering angle [7]. In the case of our probe, the steering direction can also be altered using the offset between the two segments, and the kinematic model can, thus, be considered to be similar to that of a bicycle able to steer. Fig. 3(a) shows the notation that is adopted to describe the flexible probe, while Fig. 3(b) shows the notation that is associated with a conventional bicycle model [29]. If the origin of the bicycle model is chosen at the center of the rear

Fig. 3. Notation comparison between (a) flexible probe and (b) conventional bicycle model. With the flexible probe, the curvature p changes as a function of the steering offset St, while in the conventional bicycle model, p is a function of the steering angle y. In addition, the virtual tip position of the flexible probe Pf is defined as the middle point between the two tips of the steering segments, while the corresponding point for the bicycle model Pb is chosen to coincide with the rear wheel's center point. In addition, the approach angle 9 coincides with the orientation of the approach vector of the tip in the case of the flexible probe and the orientation of the body in the case of the bicycle model. L indicates the length of the bicycle's body. Note that the steering offset St measured at the tip of the probe varies with respect to the offset S measured at the base as a function of the body configuration.

wheel Pb, its kinematic model is expressed as follows:

" x ' cos в -o -

У sin в V1 + 0

в = tan ф/L 0

-Ф- . 0 . .1.

P = f № ) = KÔt

where f(St) is empirically assumed to be a monotonically increasing function of the steering offset.

In this research, we simplify the definition of f(St) by treating the curvature p as being proportional to the steering offset, with a coefficient k (mm-2), based on the calibration experiments described in Section V-C. Therefore, the kinematic model becomes as follows:

x cos в 0-

У sin в 0

в = KÔt V1 + 0

Л- 0 1

Vt2 •

where, x, y, 9, and y indicate the x-axis coordinate, y-axis coordinate, the approach angle, and the steering angle of the bicycle model, respectively, and v1, v2, and L indicate the forward velocity of a bicycle's body, the rate of change of the steering angle, and the distance between front and rear wheels, respectively.

In (1), the relationship between the forward velocity vi and the rotational velocity 9 determines the instantaneous curvature p of a real trajectory, and it is a function of the steering angle y, i.e., p = tan(y)/L [10]. Contrary to the bicycle model, however, in our probe, the instant curvature is assumed to be a function of the steering offset St at the tip of the probe, as follows:

Fig. 4. Infinitesimal segment of the flexible probe having two segments.

The steering offset will in fact be different if measured at the base rather than the tip of the probe, because of a number of factors that are associated with this mechanism of motion, e.g., axial compressive and tensile deformation of probe segments, probe body configuration, and friction. Thus, let S describe the offset between the two main segments of the probe, as illustrated in Fig. 3, while St describes the corresponding offset, measured at the tip of the probe. Disregarding material deformation, a relationship between the two offsets that takes into account the probe's configuration can be derived as follows. Considering a flexible probe segment of infinitesimal length ds, as shown in Fig. 4, the curvature of the segment is 1/Rc, and the distance to a neutral axis for each segment is rc. The length of each segment can be expressed as follows:

h = (Rc - rc )d0 h = (Rc + rc )dd.

Given a segment with length ds, the angular difference between the two ends of the segment is d9, as in (5), shown below and the difference between the two segments becomes dS, as in (6), also shown below:

d9 = ds/Rc (5)

dS = i2 - i1 = (Rc + rc)d9 - (Rc - rc)d9

dS = 2rcd9 = ed9. (6)

By integrating both sides of (6), we can obtain a relationship between AS and the approach angle 9 as follows:

Ад = еАв = е (в - в0) = ев

In (3), v1 and vt 2 indicate the forward velocity and the changing rate of steering offset, respectively, and x and y indicate the x- and y-axis coordinates of the virtual tip position of the flexible probe Pf. Note that the subscript "t" in vt 2 indicates that here the rate of change of steering angle is based on tip measurements (i.e., St).

where £ (in millimeters) is the distance between the neutral axes of the two segments.

Assuming that the initial insertion direction for the probe is parallel to the x-axis (i.e., 00 = 0), the compensation amount is, thus, only proportional to the current tip direction of the probe. To generate the correct steering offset St at the tip of the probe, the prescribed offset imposed at the base S should, thus, be adjusted by AS, as defined in

6 = 6t + A6 6t = 6 - A6 = 6 - e0

"X" cos 0 0

y sin 0 0

0 = k (6 - e0) vi + 0

.6. 0 A.

q = G(q)v

ren x

& K (6 - e0)/coss (0)

es tan(0)

U4J y

= M (q)

3k(6 - e0)2sin 0 - ke(6 - e0)cos0 cos3 0

cos2 0

= N (q)u.

A modified kinematic model of the flexible probe that takes into account this adjustment is described in (9), where vl and v2 indicate the forward velocity and the changing rate of steering offset (with the offset applied at the base of the probe), respectively. The compensation coefficient e in (7) is determined theoretically as twice the distance from the center to the centroid of each segment [30]. e for a probe with two halves is, thus, (8R)/(Sn) and (8VdR)/(Sn) with four quarters. The kinematic model of the probe that is described in (9) is nonlinear and can be expressed in the general form

where q is a (4 x 1) vector of the flexible probe's generalized coordinates [x, y, 0, , v is a (2x 1) vector of the input velocities [v1, v2]T, and the columns gi (i = 1,2) of the (4x2) matrix G(q) are the vector fields [29].

IV. Two-Dimensional Trajectory Following A. Related Works and the Chained Form

In order to construct a feedback control strategy for the probe, we adopt an algorithm that is developed for car-like robots [29], [31]-[36], which has received significant attention over the years and is generally expressed in the chained-form representation [29], [31], [32], [36]. In this method, the bicycle model, which is similar to the one developed here for the flexible probe, can be converted into the single chained form, which has two new control inputs (u1, u2) and four new states £, £2, £3, £4) as follows:

£1 = ui £2 = U2 £3 = £2 ui

L = £3 ui. (11)

In order to obtain the relationships between the original coordinate q and the new one £, and between the original input v and the new one u, we can apply a similar approach to the one that is explained in [29]. First, by setting ^ = x and comparing (9) and (11), the relation between u1 and v1 is obtained. Then, by setting £4 = y, £3 and £2 can be obtained using the original state and, finally, u2 can be expressed using the original input. This way, the kinematic model of the probe can be converted into the chained form, as illustrated in

In (12), the function M converts the original coordinates q into the chained form's coordinates £. Similarly, (13) describes the function N, which converts the chained form's input u = [u1, u2]T back into the original input v.

once the kinematics model of the flexible probe is converted into the chained-form, feed-forward and feedback control become straightforward. First, based on a desired trajectory, feedforward or feedback control input velocities can be computed using the chained form. Then, using (13), the inputs of the chained form can be converted back into the original inputs. Sections IV-B and C reproduce many of the concepts that are described in [29], which have been included here for the sake of clarity.

B. Feed-Forward Control

Let us consider that a desired trajectory and its derivatives are given, respectively, as

Xd = xd (&(t})

yd = yd (a(t)) (14)

d dxd (&) da .

xd (t) = dtxd m = -¡^-¿1 = xd (a)"(t)

y (t) = dyd (» = ^^ = yd (a™ (15)

Here, an intermediate trajectory parameter a is used to decouple the path description from the timing information as in [29] to define the trajectory independently of the desired forward velocity. By defining £d 1 = xd and £d4 = yd, the desired feed-forward control input can be obtained, as described in

Ud1 = xd (a)a(t)

xd yd — x'dxd'y'd — 3x'dxdyd + '3x"d y'd ■

à(t). (16)

In this case, the states of the probe will be as follows:

edi xd

ed2 Wd- xd yd )/x'S

eds y'd/xd

ed4 yd

= L(xd,yd ). (17)

In our application, the value a(t) was utilized to keep the forward velocity vd 1 constant using

Vdi(t) = ±y/X 2d(t)+y2d(t)= ±y/x'2(a) + y'd2(a)&(t) &(t) = Vdi (t) \Jx'dd (a)+yf (a), whereVdi (t) > 0. (18)

C. Feedback Control

The feedback controller for trajectory tracking of the flexible probe is implemented here through approximate linearization [29]. This approach utilizes the state and input errors, which are denoted as in (19), shown below to obtain the time-varying state space form

£i £di £i i

UX j — ^U dj u* j

i = 1,..., 4 j = 1, 2.

The chained form in (11) can be represented using the non- Fig. 5. Definition of coordinate systems for raa^mg the tip of a flexible

linear error equations as follows: probe.

£1 = U1 £2 = U2

£3 = Cd2 Ud1 - £2 U = Ud 1C2 + £2 U £4 = £d3 Ud1 - £3 Щ = Ud 1 £3 + Сз U.

With the approximation £2 = £d 2 and £3 = £d3, (20) can be converted into a linear state-space representation as in

0 0 0 0 1 0-

0 0 0 0 £ + 0 1

0 Ud1(t) 0 0 Cd2(t) 0

0 0 Ud1(t) 0 L?d3 (t) 0

= A(t)£ + B(t)U

where£ = [£i £2 £3 £4]T, andu = [ui u]T. If input errors are defined as follows:

Ui = -ki £i

r k3 r k4 r

U2 = -k2£2--£3--S4 .

udi udi

The closed-loop system matrix of the probe becomes

A;(t) =

-k1 0 0 0

0 -h2 -h/Ud1 -hi/

h Cd2 Ud1 0 0

h £d3 0 Ud1 0

Equation (23) has the characteristic polynomial (24), shown below, the eigenvalues of which can be easily manipulated

det(AJ - Aci) = (X + ki)(X3 + k2X2 + k3X + k4). (24)

The overall control input to the chained form is thus

U = Ud - U.

the flexible probe. Fig. 5 depicts the coordinate frames that are adopted in this study in top-down view. Frames A, B, S, T, and F denote the coordinate frames of the EM tracking system (in which EM sensor measurements are given), the base of the flexible probe, the EM sensor, the tip of the segment that contains the sensor, and the virtual tip position of the flexible probe, respectively. All axes are defined parallel to each other at the outset.

Assuming the global frame of reference (or world coordinate system) to coincide with coordinate Frame B and provided that EM sensor position measurements are available in coordinate Frame A (i.e., A TS), the tip position in base coordinates (i.e., B TF) can be easily computed using

'TF = bTA

irp S rp T rp TS T t J-F

(atb)-1 atsstjttf

where B TF represents the 4x4 transformation matrix of Frame F with respect to Frame B.

Here, B TA is assumed to become available by means of a suitable registration process (for instance, via measurements that are obtained with a reference optical tracking system), S TT is defined as in (27), shown below, where SxT and SyT are obtained from measurements of the cross-sectional geometry of the probe, while SzT, which is affected by the chosen sensor position within the probe sensor channel (see Fig. 8), is obtained from physical measurements of the prototype, and T TF is computed using the process described in the following paragraphs.

1 0 0 0 1 0 0 0 1 0 0 0

Xj ' 1 0 0 -3.9

УТ 0 1 0 1.4

zj 0 0 1 10.9

1 0 0 0 1

D. Definition of Sensor-Specific Coordinates and Relations

This section outlines the coordinate frames that are defined and used to measure the tip position of the flexible probe. Since the probe consists of four segments, but only one is tracked by means of an EM tracking sensor to minimize the size and complexity of the probe assembly, it is necessary to compute a relationship between the sensor data and the overall tip of

As described in Section II-B, two of the four segments, namely Si and S3 in Fig. 5, are defined as steering segments. The remaining two segments, i.e., S2 and S4, are functionally passive, as they follow the lagging steering segment at all times. Thus, let us consider the situation in which there is an offset between the steering segments as shown in Fig. 6. Conforming to the notation outlined in Section III, the steering offset at the base S and the corresponding offset at the tip St can be computed as in (28) and (29), respectively, shown below, where 9 defines the angle between FZ and B z, and 4x represents the length of

segment "x"

S = ¿s3 - ¿s i

St = S - £0 = ¿s3 - ¿sl - £0.

E. Control System Overview

Fig. 7 shows the overall block diagram of the closed-loop steering algorithm that is developed for the flexible probe. It has two main control loops: the local proportional-integralderivative position controller for each probe segment and the steering controller, which implements trajectory following and consists of the feed-forward term (16) and the feedback term (22). In this research, the forward motion velocity v1 and the rate of change of steering offset v2 are arbitrarily defined as the average speed and the differential speed of the two steering segments (S3, S1) respectively, as follows:

Fig. 6. Relationship between the virtual tip of a flexible probe and the tip of the segment that contains the EM sensor.

¿S 3,d + ¿S1,. 2

V2 = ¿S 3,d — ¿S1,,

Segment lengths are calculated as in

¿S 1,d = Vi - V2/2 ¿s 3,d = V1 + V2/2

¿S 1,d = ¿S 1,ddt

Based on Fig. 6, the angular difference between frame T and frame F (i.e., yf) can be computed as in (30) and (31), shown below, where Rc is the signed radius of curvature for a given offset

\Rc \ y = St

\Rc \ yf = St/2 1 St

\p\ St \KSt \ St KS2sgn(St)

\Rc \ 2

TF can then be defined as follows:

tRf tPf

where TPF = [0 Rc(1 - cos yf) Rc sin yf ]T, and

T Rf = 0 cos yf sin yf 0 - sin yf cos yf

The rotation matrix in (32) represents a rotation about yf along the negative x-axis, and Rc has the same sign of St. Thus, Rc sin yf > 0.

In order to avoid the limiting condition of Rc becoming infinite as the steering offset approaches zero, T PF is then approximated by the first three terms of an equivalent Taylor's series for the sine and cosine functions, as in

22 • 2! 1

k4 St8

24 • 4! 26 • 6!

:4 + k4 S8

21 -1! 23 -3! 25 -5!

¿s3,d = I e.

¿S2,d = ¿S4,d = min(¿s 1,d, ¿S3,d ) where ¿d = [ ¿s 1,d ¿S2,d ¿S3,d ¿S4,d ] T.

A motion constraint is imposed on the computed output of the state controller to ensure that the command signals do not generate backward motion of any of the steering segments, as, in our probe embodiment, steering and insertion are inextricably linked (i.e., the probe tip cannot change orientation without further insertion). To ensure that the velocity of each segment remains equal to or greater than zero, the following constraint function (described with a "Constraints" block in Fig. 7) is included prior to the length conversion step

if (v2 > 2v1 )

V2 = 2v1 else if (v2 < -2v1)

V2 = -2v1. (37)

Finally, the generalized coordinates q of the flexible probe are obtained using B TF and ¿d. For practical purposes, a simple transformation is included in the computation process, as described in (38), shown below, since the orientation of Frame O in Fig. 3 is different from that of Frame B in Fig. 5:

where 0Tb =

0tbbtf

- 0 0 1 0 -

0 1 0 0

-1 0 0 0

. 0 0 0 1.

Fig. 7. Block diagram of the probe steering control strategy developed for the flexible probe. It includes a closed-loop trajectory controller, a low-level position controller for the robot actuators, and measuring blocks for the robot's state.

Fig. 8. Outer shape and cross-sectional design of a flexible probe.

The generalized coordinate, thus, becomes

atand(o2s,ois)

- ¿Ss,d - ¿Sl,d

&t&n2(b2s,bss) - ¿Ss,d - ¿Sl,d

where oij and bij

are the ith row and jth column components of O TF and B TF, respectively, and td =

S 1,d,tS2,d, (-S3,d, (-S4,d]T .

V. Experimental Validation A. Flexible Probe Prototype

The key geometrical features of the flexible probe prototype that has been developed for these experiments (length = 200 mm, OD = 12 mm) are illustrated in Fig. 8, while Fig. 9 shows a different view of the actual probe and actuation system assembly. The probe is composed of four segments that are connected by means of a dovetail mechanism, which allows sliding motion between the segments. Two of the segments are equipped

Fig. 9. Prototype of the 12-mm OD flexible probe and actuation box.

with 1.9-mm diameter hollow channels, which run along the full length of the probe: One houses the EM sensor; the other acts as a general purpose working channel, which can, for instance, be used for suction or drug delivery.

The prototype is manufactured with rapid prototyping techniques in a rubber-like material with high elasticity (TangoBlack-FullCure 970, Objet; tensile strength of 2 MPa; hardness of 61 Shore Scale A; elongation at break of 48%). The leftmost and rightmost extremities of each probe segment [see Fig. 8(a)] are made out of a more rigid plastic material (VeroWhite-FullCure 830; tensile strength of 50 MPa; hardness of 83 Shore Scale D; elongation at break of 20%) to improve the probe material toughness at stress points. The leftmost rigid part of each segment in Fig. 8(a) also features a 1.5-mm OD hole, which is used to secure the probe to a mechanical transmission cable, as shown in Fig. 9.

Each segment of the probe is controlled via a linear actuator assembly (dc motor plus lead screw arrangement: Maxon A-max22 motor, 6-W power rating, 6.77-mNm maximum continuous torque, 4.4 gear ratio, 4-mm pitch, 60% lead screw efficiency, and 28.1 N calculated thrust force) connected to a transmission link with a diameter of 1.5 mm, as shown in Fig. 9. The four linear actuators are packaged into a free-standing actuation box to improve handling and integration into the

GUI Actuation „ Brain-like EM Sensor

I rnpp r

Workstation Box Gelatine Phantom Field Generator

Fig. 10. Experimental setup.

experimental setup. Fig. 9 shows the integrated probe prototype and actuation system.

B. Experimental Setup

Fig. 10 shows the experimental setup that is used to test the performance of the closed-loop trajectory controller described in Section IV with the 12-mm OD flexible probe prototype and a gelatin sample. The linear actuators are controlled via a compactRIO embedded controller programmed in LabVIEW (National Instruments, Inc.). A LabVIEW-based graphical user interface was developed and integrated into the setup in order to program desired trajectories, monitor performance, and log key control parameters. An EM tracking sensor (Aurora 5DOF long-life sensor with 1.1 mm diameter, Northern Digital, Inc., root mean square (RMS) accuracy of 0.9 mm/0.3° [37]) was employed to measure the probe's tip position. The gelatin phantom (see Fig. 10) was prepared with 6 weight%, as used in the previous work by Minhas et al. [11], to mimic the insertion properties of biological soft tissue. A custom-made trocar, with 12.5-mm inner diameter, was also added to the setup to eliminate the possibility to buckle outside of the gelatin phantom. As a complement to the EM tracking data, all trajectories were also captured using a video camera (Sony Handycam HDR-SR10E), which is mounted on a static tripod in a top-down view arrangement.

C. Steering Offset Versus Curvature Calibration

Equation (2) describes the probe curvature as a function of the steering offset. In order to investigate this relationship, a set of experiments was performed. Ten simple insertion tests with constant steering offset S of -30, -25, -20, -15, -10, -5, 0, 10, 20, and 30 mm were performed (note that offset sign relates to the steering direction). The trajectories of the tip of the segment that contains the EM sensor (B PT) were gathered using EM tracking measurements, while the overall shape of the curved probe was captured using the video camera system and processed through a manual segmentation process that is implemented in MATLAB. Each test started at rest, with all four segments aligned and embedded 1 cm into the gelatin sample. The leading steering segment (S3 for a positive offset and Si for a negative offset) was then driven further into the tissue by the required amount. Finally, all four segments were pushed together into the sample at a constant speed of

Offset = 0 mm Offset = - 5 mm

Offset =-15 mm Offset = -25 mm

Fig. 11. Captured flexible probe trajectories for different steering offset values.

1 mm/s down to a depth of approximately 15 cm. Fig. 11 shows a few sample pictures that are captured by the camera. After each test, trajectory data that are measured by the EM tracking system for each test were first projected onto plane of best fit (by simple least squares fitting) and then fitted to a circle to find the average radius of curvature for each trajectory. Most of the trajectories and corresponding circles are plotted in Fig. 12(a), while Fig. 12(b) shows an enlarged view of the same. On the basis of these measurements, curvature values (i.e., the inverse of the radii of curvature) for each offset were computed and are plotted in Fig. 13. As can be seen, the curvature was found to be approximately proportional to the steering offset, with a coefficient k =1.85 x 10-4mm-2 (R2 = 0.976) and the maximum curvature using a 30-mm steering offset measured 0.0056 mm-1, which corresponds to a radius of curvature of 178.6 mm. The slight asymmetry in curvature values between positive and negative offsets could stem from manufacturing inaccuracies, trocar alignment errors, probe deformation, etc. This, however, is accounted for by the small but finite x-axis intercept of 3.71 mm, as illustrated in Fig. 13. While the linear approximation adopted here represents a substantial simplification and will require further research, it was found to be sufficiently accurate to produce satisfactory trajectory following, as described in Section V-D. As a point of note, these experiments were performed by penetration of a homogeneous gelatin phantom with constant speed. Although the closed-loop controller that is described in Section IV is expected to mitigate the effect of modeling uncertainties (e.g., rate dependence and heterogeneity of the substrate), a detailed study of the contribution which the material properties of the surrounding tissue have on tracking performance will be the focus of future work.

D. Trajectory Following Results

This section reports on the experimental results that are obtained with the flexible probe prototype and 2-D trajectory following controller described in Fig. 7. The parameters that are used in these experiments are summarized in Table I. The linear forward velocity was arbitrarily chosen to be 1 mm/s on

3x 10 *

Steering Offset vs. Curvature

y-0.0001 85 ( x - 3. X

—©—Measured Curvature Fitted Curvature

Steering Offset(mm) Fig. 13. Relationship between offset and curvature with linear fit.

TABLE I

Parameters Used for Trajectory Following Experiment

Parameters

Values

Ideal initial posture [.x0, yo, 00, S0]

Initial posture disturbance [x0, yo, do, S0]T Linear velocity Vdi Steering coefficient k Compensation coefficient s Time constant of low pass filter for EM sensor data q

Control gain [ki, k2, k3, k4_\

[0mm, 0mm, 0°, ôojdeai mm] [0mm, 0mm, 0°, 0 mm]T 1 mm/s 0.000185 mm"2 (8 x x 6)/(3ti) = 7.20 mm 0.1s

[0.1,0.2, 0.02, 0.001]

TABLE II

Experimental Result of Probe Steering (Errors to b Pt )

do, ideal

pe,ror (mm)

standard mro j . RMS max deviation_

Fig. 12. Fitted trajectories. (a) Best fit circles for each tested steering offset and (b) enlarged view of the trajectories achieved for offsets between —30 and 30 mm.

ExOl 150 15 17.783 0.10 0.45 0.46 1.29

Ex02 150 15 17.783 0.24 0.98 1.01 1.75

Double Ex03 150 20 23.711 0.52 0.75 0.91 1.74

bend Ex04 150 20 23.711 0.55 1.45 1.55 2.72

Ex05 150 25 29.638 0.15 1.43 1.44 2.35

Ex06 150 25 29.638 0.09 1.55 1.55 2.61

R (mm) k (mm'1) Perror (mm)

Single Ex07 250 4.29x103 21.622 2.02 1.48 2.51 5.01

bend Ex08 250 4.29xl0"3 21.622 2.23 1.34 2.60 5.44

Overall Results ofperror (mm)

1.61 5.44

*6-mm radius reflects the size of the flexible probe prototype used for validation (see Section V).

the basis of an estimate of the speed of manual insertion of a standard deep brain stimulation electrode provided by an expert neurosurgeon. The published literature [38] also recommends that the advancing or withdrawing of microelectrodes or other instruments should be not greater than 0.5 mm/s to reduce the risk of hemorrhage, which is grossly in line with the speed chosen for these experiments. In the tests, the initial generalized coordinates q are set to [0, 0, 0, 0]T. In Table I, S0,ideal indicates the steering offset for the probe. The control gains were set to [ki, k2, h, h] = [0.1, 0.2, 0.02, 0.001], with corresponding eigenvalues of [—0.1, —0.1, —0.05(1 ± ¿\/3)], on the basis of the simulated results that are presented in [23].

Eight tests were performed—six with the double-bend trajectory (40) and two with the circular trajectory (41):

Xd (o) = o

Vd (o ) = 2(i — cos( ^a)) (40)

Xd (o) = R sin (ko)

Vd(o) = R (I — cos(ko)). (41)

The parameters pertaining to the trajectory of each experiment, alongside the experimental results that are obtained, are listed in Table II. The last four columns of Table II report on the overall steering results, which are expressed as the mean, standard deviation, RMS, and maximum positional error between the probe tip and the predefined trajectory.

Fig. 14 graphically illustrates the results that are obtained from three out of the six experiments: Ex04 and Ex06 show the results that are obtained for the double-bend trajectory, and Ex07 shows one example of a single-bend trajectory. Three figures in the first row display the tip position of the flexible probe (B PF) that is estimated using (26) and (33), the tip position of the segment that contains the EM sensor (B PT), and the desired trajectory. Three figures in the second row show the positional error of the estimated tip (B PF) from the desired curvilinear trajectory. The final shapes of the flexible probe, which were captured using a video camera that is arranged in a top-down view, are shown in the last row. The green solid lines indicate the central axes of the final shapes of the probe, which were obtained by averaging the upper and lower edges at the interface between the probe and the gelatin sample. The yellow dotted line indicates the trace of the virtual probe tip, which was obtained by taking the midpoint between the two tips of the steering segments at 5-s intervals from start to end. The RMS distance errors between these two lines are 0.91, 1.79, and 1.80 mm for Ex04, Ex06, and Ex07, respectively.

One of the major factors that affect the tracking errors reported in Table II seems to stem from the interaction between the tissue and the probe, and between probe segments. In order to change the steering direction and, for instance, move left, the right segment needs to move further. In doing so, the friction between the two steering segments pushes the whole probe to the left

Fig. 14. Experimental results of probe steering. The plots in the first row describe the trajectories achieved with the virtual probe tip (B Pp) estimated using (26) and (33), the tip of the segment with embedded EM sensor (B Pt ), and desired trajectory for three example paths (i.e., Ex04, Ex06, and Ex07; left to right). The plots in the second row show the positional error from the desired trajectory. The images in the bottom row depict actual trajectories of the flexible probe. The green sold lines illustrate the center axes of the final shapes, while the yellow dotted lines indicate the trace of the virtual probe tip (B Pp) at 5-s intervals from start to end.

against the tissue. This undesired effect shifts the direction of the probe tip away from the desired trajectory, thus introducing tracking errors. Specifically, in Ex04 the probe tip trajectory is first shifted in the positive y-direction and, subsequently, in the negative y-direction, after a change in desired direction.

Another factor which has been identified to be responsible for these tracking inaccuracies relates to excessively small gains that are associated with the positional error k4. In simulation [23], the probe is not affected by the interaction between segments; thus, small control gains for the positioning error are sufficient. Conversely, in these experiments it was necessary to increase the control gains to overcome sources of uncertainty which were not considered during the kinematic modeling of the probe. From (12), it is apparent that the four states of the chained form roughly correspond to the x-axis position, the curvature of the trajectory, the direction of the trajectory, and the y-axis position of the probe tip, respectively. As in (22), the gains that are utilized in these experiments favor the tracking of the curvature and not the y-axis position tracking because [k2, k3, k4 ] = [0.2, 0.02, 0.001]. However, a series of simulation runs revealed limitations in the viable control gain range because of the constraint that is described in (37), which affects tracking convergence when the gains or the positional offset are large, as the constraint restricts the maximum changing rate of steering (since offset cannot change on the spot, i.e., with the simultaneous forward motion of one steering segments and backward motion of the other). To highlight this limitation, simulation

experiments without constraint (37) have shown that the probe can track the desired path robustly under a much wider range of initial conditions, but these results have not been reported here for brevity. Thus, the control strategy implemented will need to be refined to improve the changing rate of steering without breaking the constraint.

Limitations in the viable control gain range may also explain why the tracking errors for the single-bend trajectories (i.e., Ex07 and Ex08) are larger than those for double bends, which seems counterintuitive. Since the feed-forward input does not vary significantly during the single-bend experiments, the tracking error is caused mainly by the small gains of the feedback controller. However, in the double-bend experiments, the feedforward input changes significantly halfway into the trajectory (i.e., x « 75 mm), which causes the corresponding tracking error to reduce: The friction between segments pushes the whole body back as the steering offset gradually changes sign.

Table III summarizes how these experimental results compare with those previously reported in the literature. Even though the OD of the probe and the material properties of the probe and the substrate will affect performance, the minimum radius of curvature and the accuracy that is achieved in our experiments are broadly in line with those obtained with other steering methods. Moreover, miniaturization of the current flexible probe prototype will potentially result in improved steering performance.

In addition, steering of our flexible probe only relies on the steering offset, which in turn only depends on the forward

TABLE III

Tracking Results Placed in the Context of Previous Work

Reference Method Min. Radius of Curvature (mm) Max. Curvature (mm"1) Accuracy (mm) Specimen Measuring Method

Webster 2006 [7] Open-loop 222.7 0.0045 _* Tissue phantom (simulated muscle ballistic test media) Camera

Minhas 2007 [11] Duty-cycle (open-loop) 51.6 0.019 - Brain phantom (gelatin) Camera

Read 2008 [16] Needle flipping"* 61.0 0.016 - Tissue phantom (plastisol) Camera

Minhas 2009 [12] Duty-cycle (manual) - - 1.80±1.33 Human Cadaver brain C-arm fluoroscopy

Wood 2010 [13] Our results Duty-cycle (closed-loop) Programmable Bevel Tip (closed-loop) 333.3 178.6 0.003 0.0056 0.71 0.68±1.45 Kidney phantom (medical training model) Brain phantom (gelatin) Camera EM Sensor

* Data were not reported.

**Needling flipping indicates the "ON-OFF" controller, which switches between "bevel-right" and "bevel-left".

motion of each segment. Therefore, contrary to other research [11]—[13], [16], the control strategy that is outlined here does not require rotation of the flexible probe around the long center axis. When compared with the drilling effect of a rotating needle [39] or with the discontinuity that is introduced by flipping a flexible needle with a prebent tip [16], the proposed control approach is expected to cause less strain to the surrounding tissue and, in the case of successful prototype miniaturization, less tissue damage.

VI. CONCLUSION AND FUTURE WORK

This paper has described on-going research into the control of a bioinspired flexible probe. Based on the characteristics of certain ovipositing wasps, a novel flexible probe with multiple segments, which are connected to each other by means of a special interlocking mechanism, was manufactured, and the concept of a "programmable bevel" was proposed to steer the probe in a predefined direction: The offset between probe segments determines the steering direction of the tip thanks to a set of bevels and the amount of unsupported length that is associated with the leading segment. A kinematic model of the flexible probe was derived, based on its similarity to that of car-like robots. In this research, the curvature of the probe is assumed to be a function of the steering offset, which is shown empirically to be proportional to the steering offset through a simple set of experimental trials in gelatin. A compensation algorithm to account for differences in steering offset between base and tip coordinates was also proposed. Finally, a closed-loop control strategy, which utilizes both feed-forward and feedback components, was implemented for 2-D trajectory following using an approximate linearization technique. In vitro experimental results demonstrate that both the kinematic model and the control method perform as expected.

Although these early experimental results are promising, research to date offers significant scope for future work. Specifically, further studies are planned on the miniaturization of the current prototype down to a clinically acceptable size (e.g., mm OD for neurosurgery, ^4 mm OD for breast biopsy [40], and drainage of liver abscesses [41]), a further reduction to the minimum radius of curvature and an investigation into the tip forces required for neurosurgery. It is worth noting that the size of the working channel that results from a reduction in OD may

limit the range of applications (e.g., drug delivery) for which a miniaturized device would be suitable. However, it is expected that a 2-3 mm OD probe would be functionally similar to existing 25 gauge needles (i.e., with an inner diameter of 0.26 mm) and that further optimization of the design (e.g., location of the working channel and number of independent segments) would allow us to further increase the size of the working channel for a given OD size. In addition, tip positional tracking will also be challenging, as space for the position sensor will reduce with scale. However, the smallest EM sensor that is currently available is 0.5 mm in diameter, and connecting wires are thinner than the sensor itself [37]. Thus, it is expected that EM-based position tracking of a miniaturized probe will be possible, although alternatives (e.g., intraoperative imaging that is based on MRI and fluoroscopy) will also be considered. The inclusion of an additional sheath or cannula, as suggested in [17], will also be considered as a means to further enhance the functionality of the probe. The manufacturing of a scaled-down device will, thus, be challenging, but early experience of extrusion with a medical grade silicon blend is promising. While an analysis of the performance of this further prototype falls outside the scope of this paper, it suggests that, although taxing, miniaturization of the probe is indeed possible.

Optimization of the closed-loop control approach described here is necessary to reduce tracking error and improve performance. This optimization may include further tuning of the control gains in light of existing constraints, which may result in changes to the algorithm which overcome the limitations on the changing rate of the steering angle (or offset). In the current implementation, we used the EM sensor as ground truth. However, it will be necessary to assess the performance of the flexible probe as a whole, including the effect of any sensor-induced inaccuracies. This study will be complemented by a detailed analysis of the rate- and substrate-dependent behavior of the probe. In addition, we currently assume that there is no friction between probe segments and that the probe is very flexible, while being stiff in compression and in tension. However, experiments to date on the prototype show that such assumptions may be overoptimistic, which means that further improvements to the kinematic model are needed. As a method to reduce the interaction forces between the probe and the tissue, a reciprocal insertion process, where each segment is inserted further into

the tissue in turn, will also be considered as this approach has been shown to reduce the risk of buckling in insects [25]. Finally, both the kinematic model and control schemes described here will be extended to 3-D. With the ability to actuate all four segments independently, combinations of simultaneous steering offsets in the two orthogonal planes and their effect on tip orientation will be explored. Development of a suitable 3-D kinematic model and an extension of the current controller to 3-D are, thus, planned.

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Seong Young Ko (M'10) received the B.S., M.S., and the Ph.D. degrees in mechanical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2000, 2002, and 2008, respectively.

He was a Visiting Researcher with the Department of Electrical Engineering, University of Washington, for 6 months from 2005 to 2006. In 2008, he was a Postdoctoral Researcher with the Department of Electrical Engineering, KAIST. Since 2009, he has been a Research Associate with the Mechatronics in Medicine Laboratory, Department of Mechanical Engineering, Imperial College London, London, U.K. His research interests include medical robotics, humanrobot interaction, and intelligent control.

Ferdinando Rodriguez y Baena (M'09) received the M.Eng. (First-Class Hons.) degree in mechatronics and manufacturing systems engineering from King's College London, London, U.K., in 2000 and the Ph.D. degree in medical robotics from the Department of Mechanical Engineering, Imperial College London, in 2004.

He is currently a Senior Lecturer with the Department of Mechanical Engineering, Imperial College London, where he leads the Mechatronics in Medicine Laboratory. His current research interests include the application of mechatronic systems to medicine, in the specific areas of clinical training, diagnostics, and surgical intervention.

Luca Frasson received the B.S. and M.S. degrees in biomedical engineering from Politecnico di Milano, Milano, Italy, in 2004 and 2006, respectively, and the Ph.D. degree in medical robotics from the Department of Mechanical Engineering, Imperial College London, London, U.K., in 2010.

He is currently a Patent Consultant in Milano, Italy. His research interests include medical robotics for neurosurgery, biomimetics, and wearable sensors.