Model-Based Error Correction for Flexible Robotic Surgical Instruments

Transcription

1 Model-Based Error Correction for Flexible Robotic Surgical Instruents Ran A. Beasle Robert D. Howe Division of Engineering and Applied Sciences Harvard Universit Abstract Robots proise to enhance iniall-invasive surger, but flexion of the thin instruent shaft introduces error into odels of the robot kineatics. Visual or electroagnetic tracking of the instruent tip provides correct forward kineatics, but uncertaint in shaft bending and port location leaves residual errors in inverse kineatics. These errors can cause incorrect otions that preclude the use of iage-guidance tools. This paper proposes a odel-based controller to correct the coanded otions. Coparison with a controller assuing a straight instruent shaft quantifies otion errors resulting fro the use of a straight shaft controller. Analsis of the flexed shaft controller shows sensitivit to shaft length, shaft stiffness, tip force, and sensor noise. I. INTRODUCTION Iage-guided robotic surger uses three-diensional edical iage data to guide instruents to specific tissue locations identified in the iage. This approach can involve either autonoous robotic control or huan-in-the-loop techniques such as augented realit and virtual fixtures [1,]. In either case, good perforance depends on accuratel placing the instruent tip at the specified iage coordinates. Unfortunatel, instruent shaft flexibilit can prove a serious liitation. For exaple, a 1N force applied to the tip of a cantilevered instruent for a coercial surgical robot (straight endoscissors for the ZEUS Surgical Robot Sste, Coputer Motion, Inc, Goleta, Calif.; stainless steel shaft 355 long, 3.35 outer diaeter, 1.47 inner diaeter) causes a 15 tip deflection, uch larger than the target anato in an procedures. Such bending has been shown to liit accurac in anial studies of iage-guided cardiothoracic procedures [, 6]. As surgeons seek to iniize instruent size to reduce daage to health tissue and access saller anatoical structures, this proble will becoe ore pronounced. In ters of robot control, instruent flexibilit alters the kineatic structure of the anipulator so that the assued relationship between the robot s joint and tip coordinates becoes inaccurate. A nuber of approaches to kineatic error correction have been developed in the robotics literature. Offline calibration attepts to precisel deterine kineatic paraeters, but does not account for online changes resulting fro interactions with unodeled environents [7, 8]. Research in flexible robot anipulators handles changes in endpoint-joint relationships during operation, but focuses on dnaic flexibilit [9, 1]. Such an approach assues known static kineatics, which is not the case for the quasi-static deforation seen in iniall invasive surger. Tracking the instruent tip with coputer vision techniques or an electroagnetic sensor solves the proble of error in the forward kineatics, as the instruent tip coordinates are directl easured. This easureent enables the use of iage guidance in robotic iniall invasive procedures. Measuring the tip position will not directl correct the inverse kineatics, unfortunatel. The inverse kineatics will still be odeled off a straight instruent shaft, and can cause divergent otions of the instruent tip even with tip tracking [11]. With tip easureents and known errors in the kineatic odel, there are convergence etrics for the error in acobian-based controllers, but these etrics do not reduce the otion errors [11, 1]. A odel-based controller is a logical approach to reducing flexion-induced errors. In this paper we develop a echanical odel of the instruent shaft deflection, based on a sipl supported bea. Using the developed odel and tip tracking, we propose a real-tie ethod to reduce the errors in the control of a surgical instruent. The etric introduced in [11] is used to quantif the convergence properties of the resulting acobian controller in the presence of easureent error. Siulations copare the resulting controller with a controller assuing a rigid instruent shaft. Sensitivit analsis shows the effect of easureent noise on the proposed controller. II. CONTROLLER DESIGN A. Flexion Model Fig. 1 diagras a robot-assisted surgical procedure in which the instruent shaft is flexing. For the purposes of the following analsis, the instruent shaft is odeled as a straight rod, connected to the robot ar at a wrist. The controller s job is to accuratel position the tip in a world coordinate frae. As in an laparoscopic and thoracoscopic procedures, we assue that the instruent is introduced into the patient s bod through an endoscopic port, a rigid cannula a few c in length that is tightl inserted into a sall incision. As the instruent tip interacts with tissue, radial tip forces generate reaction forces at the port and at the robot wrist. These forces result in bending of the instruent. The goal is then to deterine the actual kineatics of the instruent, i.e. its shape as it flexes during the procedure.

2 World Robot Port, p p Wrist, p w Patient Instruent tip, p t Fig. 1. Flexion in surgical instruent shaft due to force on tip. The Euler-Bernoulli bea bending equations were chosen to odel the instruent flexion, with an assuption of a sipl supported configuration, i.e., the wrist, port, and tip appl onl forces noral to the shaft and no oents (Fig. ). The sipl supported configuration fits the situation where just the tip of the instruent is in contact with the patient and this odel precludes coplicated shapes such as s-curves in the interest of low coplexit. With both ends of the bea on the x-axis the height of the bea in odel space ([14]) is Fp b x 3 ( x) = Lbx 1 + x a (1) 6EI L L and the slope of the bea is Fp b 3x '( x) = Lb x a () 6EI L L where x is the position along the bea, F p is the force at the port acting at a distance a along the bea fro the wrist, L is the total length of the bea, b = L-a, E is the Young s odulus, I is the cross sectional oent of inertia, and 1 x a x a =. (3) ( x a) x a F p and a are the two unknowns in this odel, so two easureents of the bea position, slope, or curvature are necessar for a solution. Additionall, easureents are necessar for registration between world space and odel space. We assue that instruent positions and orientations (slopes) are sensed near both ends: at the tip end using either electroagnetic or visual sensing, and at the wrist end using either electroagnetic or robot ar kineatic sensors (e.g., high resolution joint encoders). The easureents perit (, ) wrist odel port F p (a, (a)) instruent tip x odel (L, ) F w F t Fig.. Sipl supported bea odel of instruent shaft in odel space. calculation of the odel paraeters, and thus the acobian relating the otion of the bea s endpoints. Siilarl, the port position or wrist force, port force, or tip force could be easured directl instead of the sensors used here; these alternatives are evaluated in the Discussion section below. B. acobian Construction Fig. 3 diagras the inverse-acobian controller used in this paper and described in [11]. The preise is that the existing joint level controller accepts a desired angular configuration θ 1 d = θ + ir ( pdesired pt ) (4) where θ is the vector of current joint angles, r is the robot acobian relating joint otions to wrist otions, i is the acobian relating otions of the wrist end of the instruent shaft to otions of the tip end, p desired is the desired tip position, and p t is the current tip position. The appeal of acobian-based controllers is that the ove the instruent tip directl towards the desired position in world space, with clear safet advantages for working inside a patient s bod. Dnaics are unnecessar because surgical robots ove slowl, are highl geared, and are often in contact with viscoelastic tissues, resulting in a quasi-static sste. We assue that the robot ar can precisel position the wrist and the robot s acobian is known, so our task is to find the acobian for the instruent shaft. For clarit, we consider position control in -D here; orientation control and extension to 3-D are discussed below. The acobian of the instruent shaft is constructed b taking the partial derivative of the tip position with respect to the robot wrist position Existing robot controller Endpoint sensing τ p Ar & desired p d oint level ( i r ) -1 θ c θ d = θ+ θ θ c instruent control θ Mechanical odel p t, p w, θ t, θ w Figure 3. Block diagra of controller.

3 world odel p w θ w p p a φ (a) xt xt xw w i = (5) t t xw w where p w = (x w, w ) is the robot wrist position in world space, and p t = (x t, t ) is the instruent tip position in world space. The position of the instruent tip in world space can be written as a transforation fro odel space xt cos( ϕ ) xw = L + (6) t sin ( ϕ ) w where φ is the angle between odel space and world space (Fig. 4). To incorporate the port location, p p, which deterines the kineatic relationship between the wrist and tip, this angle can be written as 1 p w 1 ( a) ϕ = Tan Tan. (7) xp x w a Cobining (1), (5), (6), and (7), and perforing the derivative operations in (5), the acobian can be written as a function of p w, p p, a, F p, (a), and known paraeters. Now the ters p p, a, F p, and (a) ust be written as functions of the position and angle easureents at the instruent endpoints, p w, p t, θ w, and θ t. The port position in world space can be written as a transforation fro its position in odel space xp Cos( ϕ) Sin ( ϕ) a xw = + (8) p Sin ( ϕ) Cos ( ϕ ) ( a) w 1 t w ϕ = Tan. (9) xt xw Fro (1), the deflection height at the center force can be written in ters of F p and a Fp a ( a L) ( a). (1) = EI 3L F p and a can be written as functions of the orientation easureents EI ( ' ( ) ' ( L) ) Fp = (11) a a L x world Fig. 4. Relationship between world and odel fraes p t x odel θ t where ( ' + ' ) '( L) ' L L a = Tan( w ) Tan (1) = θ ϕ (13) L = θ ϕ. (14) Cobining these equations perits calculation of the desired for of the acobian, which is reproduced in the appendix. III. EVALUATION To quantif the error in this inverse acobian controller we use a etric fro [11]. The true and estiated acobians, i and Ĵ i, are cobined to create a ap fro the current position error to the position error after a otion coanded b the estiated acobian (Fig. 5) ˆ 1 G = I i. (15) i The induced Euclidean nor then finds the worst case for the resulting ratio between the new and old error lengths G G p d = ax = i pd pd t σ ( G) (16) where p d is the old error vector. If this etric is less than 1 across the workspace, then coanded otions alwas shrink the length of the position error. Therefore, values below 1 indicate onotonic position convergence (lower values signifing faster convergence), and values above 1 indicate the potential for divergence (higher values signifing faster divergence). IV. METHODS The etric explained in the previous section will be used to copare the proposed controller with a controller based on kineatics of a straight shaft. The etric will also be used to deterine the sensitivit of the proposed controller to sensor noise. A. Coparison with a Rigid Instruent Model Using the proposed etric, the controller introduced in this work can be copared with other controllers. The controller for the Zeus Surgical Sste assues a rigid instruent shaft p t,new = p tip + p actual p actual = Ĵ -1 p d p new = p desired - p t,new = (I - Ĵ -1 ) p d p desired p d = p desired - p t p t Fig. 5. Geoetr of the otions for an erroneous inverse acobian controller.

4 (Tip position error) / L, acobian error etric acobian error etric (tip position error) / L. L =.1775 F t /EI = and uses a acobian based on the lever equation xt L xp xw = t pp pw p. (17) w Without direct easureent, the port position, p p, is estiated over successive controller ccles. The port position is chosen as the intersection point of ras drawn along the instruent shaft as easured b wrist sensors. To quantif the benefits of tip tracking, the distance can be calculated between the actual tip position and the tip position estiated b the current Zeus rigid lever controller. To calculate the actual tip position, first the right triangle fored b a and (a) is solved for a (Fig. 4) p w L =.355 F t /EI = 3 - L =.355 F t /EI = Fig. 6. Position error and acobian otion increase as the instruent tip is inserted farther into the port. p p = a + a (18) with (1) substituted in for (a). Then (1), (6), and (7) are cobined to calculate the tip position. In addition, to quantif the benefits of the flexion odel, the induced Euclidean nor can be used to copare the lever acobian with the flexion acobian. The rigid instruent acobian is calculated b substituting (17) into (5). B. Sensitivit Analsis for acobian To copute the effects of sensor noise on the proposed controller, the induced Euclidean nor is used again. The flexion odel acobian is substituted into (15) as both the true and estiated acobians, the latter instance being calculated with additive sensor noise. The noise has a unifor distribution along each sensor axis, with a axiu equal to the nuber listed in the figures and a iniu equal to the negative of the sae nuber. To investigate convergence, the induced Euclidean nor is then evaluated at various insertion depths and with various agnitudes of sensor noise. For a given insertion depth, the axiu error across noise values is shown. One of the advantages of the flexion odel is an estiate of the port position, the accurac for which is unaffected b flexion. Error in port localization due to sensor noise is also calculated. V. RESULTS For all plots, the tip and wrist forces will be assued to act such that F w and F t are in the direction in odel space; axial forces in the x direction do not produce bending. Taking advantage of rotational setr about the port, all plots show the values as a function of insertion depth, r = [(x t - x p ) + ( t - p ) ] ½. A. Coparison with a Rigid Instruent Model The greater the insertion depth, the larger the error in tip position for the current Zeus controller without tip tracking (Fig. 6). Additionall, the otion error for the rigid instruent acobian is proportional to the insertion depth, as deterined b (16) with the flexion odel acobian as the true acobian. B. Sensitivit Analsis for acobian Noise in the position and orientation easureents degrades the perforance of the flexion odel acobian. To deonstrate these effects, (16) was calculated across the tip workspace, using the flexion acobian with noise in each easureent for the estiated acobian and without easureent noise for the true acobian. The following plots show the worst (axiu) values across all noise directions. In the presence of sensor noise, the otion errors fro the flexion odel acobian are sallest when the instruent is inserted approxiatel 7% of its length (Fig. 7). The otions becoe divergent as either the tip or the wrist ove close to the port. Decreasing the sensor noise reduces the controller s otion errors. The estiated paraeters, F p and a, can also be exained. Since F p and the structural rigidit, EI, alwas enter the bea bending equations together, the ter F p /EI was estiated. This ter can be thought of as a noralized force. F p /EI can be incorrect b orders of agnitude outside of the convergent area due to the easureent noise, but iproves as approaches.5 (Fig. 8). The error in estiating the port position decreases as the instruent tip oves further fro the port (Fig. 9). Increasing the instruent length, L, or the noralized force, F t /EI, increases the convergence of the controller, siilar to decreasing sensor noise (Fig. 1). Increasing these paraeters increases flexion for a constant tip force. For a given level of sensor noise, increasing the flexion would increase the signal-to-noise ratio, explaining the siilarit between Figs. 7 and 1. VI. DISCUSSION Tracking the instruent tip in iniall invasive surger avoids errors in the forward kineatics and allows a ore accurate kineatic odel for otion control. The odelbased controller presented here for flexible instruent shafts can reduce otion error in coparison with controllers assuing rigid links. The sipl supported bea odels the

5 acobian error etric Noise scaling.,..,.1.1,..1,.1 Fp/EI estiation error (%) Noise scaling.,..,.1.1,..1,.1 situation in which the onl forces on the instruent are at the wrist, port, and tip, and there are no oents. Barring sensor noise, the flexion odel coands accurate otions in this situation whereas the rigid instruent odel results in otion errors. To copare the sensor noise agnitudes used in this paper to a coercial product, the published RMS accurac of the Libert agnetic tracking sste is.76 position and.15 degrees orientation or about.1l and.6 radians (Polheus Technologies, Colchester, VT). For this level of noise with L =.355 and F t /EI = 3 -, the insertion depth should be constrained to.55 < <.85 to ensure onotonicall convergent otions (Fig. 7). Iproved signal processing or a tracking technolog with less noise would relax these bounds on insertion depth. pp - pp,estiated /L Fig. 7. The otions coanded b the flexion based acobian becoe less accurate as sensor noise increases. Values greater than 1 signif divergent otions. The first scaling nuber is the position error scaling, with respect to L. The second nuber is the angular error (rad). L =.355. F t /EI = Noise scaling.,..,.1.1,..1, Fig. 9. The error in port position estiation decreases as insertion depth increases. The first scaling nuber is the position error scaling, with respect to L. The second nuber is the angular error (rad). L =.355. F t /EI = Fig. 8. In the presence of sensor noise, F p /EI is best estiated in the center of the workspace. The first scaling nuber is the position error scaling, with respect to L. The second nuber is the angular error (rad). L =.355. F t /EI = 3 -. As shown in Fig. 6, the error in tip position estiation can be on the order of centieters if a bent instruent shaft is assued to be straight. Such an error agnitude is larger than the sizes of an surgical targets [, 6]. Tip tracking easures the actual position of the tip, allowing effective iage guidance. In this paper, the positions and orientations of the instruent s endpoints are used to solve for the flexion odel paraeters. Other easureent possibilities exist, however, including the port position and orientation, and forces/torques on the wrist, port, and tip. Of these possibilities, easuring the force and torque at the robot wrist is the ost feasible. Accuratel easuring the position and orientation of the instruent shaft at the port is difficult due to the uncertain relationship between the phsical port and an contact points between the port and the instruent shaft. Siilarl, the difficult of easuring the forces exerted on the port, or the acobian error etric L = b, Ft/EI = c L = b, Ft/EI = 1.5 c L = 1.5 b, Ft/EI = c Fig. 1. The otions coanded b the flexion based acobian becoe ore accurate as either L increases or the tip force increases. b =.355. c = 3 -. The noise agnitude is.l for the position easureents and. rad for the angle easureents. Values greater than 1 signif divergent otions.

6 instruent tip, obstruct direct easureent of one of the odel paraeters, F p. Construction of a force sensor sall enough to fit through the port, or of a port incorporating a force sensor, would allow this direct easureent. The acobian in this paper onl handles otions in the plane containing the instruent, but is easil extended to 3-D. If the robot exerts a force on the instruent that is perpendicular to the plane of bending, torque will be generated at the port position. The entire plane of bending will then rotate about a line that passes through the port position. Orientation control can be achieved b calculating the acobian relating the change of orientation of the tip to the change in position of the wrist. Assuing infinitesial otions, oving the wrist does not change the shape of the bea, so the onl change in orientation is caused b the changing registration between the odel and world spaces. The coluns of the positional acobian atrix describe the tip otions resulting fro wrist otions along the world space axes, allowing calculation of the consequential change in rotation. In the interests of a better fit between the odel and the true instruent shaft shape, the sipl supported configuration can be replaced with a ore coplicated assuption of the shape. The new configuration would allow the controller to handle ore coplex instruent/environent interactions, but would contain ore unknowns and thus require correspondingl ore easureents. Those easureents could coe fro a force/torque sensor as discussed above or fro additional position and orientation easureents. APPENDIX: FLEXION ACOBIAN 1,1 1,,1, 1,1 1, i =,1, = 1+ LSin( γ ϕ) Y X ( tep tep1) + Leas tep3 = LSin ( γ ϕ) X Y( tep tep1) + Leas tep3 = LCos( γ ϕ) Y X ( tep1 tep) + Leas tep3 = 1 LCos( γ ϕ) X Y( tep1 tep) + Leas tep3 ( a) 1 γ = Tan a tep1 = 3L ( a) tep = a L tep1 ( 3 * ) tep3 = La + tep a a + a eas X = xp x Y = p w w 1 L = X + Y REFERENCES [1] T. M. Peters, Iage-guided surger: fro X-ras to virtual realit, Coputer Methods in Bioechanics and Bioedical Engineering, vol. 4, pp. 7-57,. [] S. S. Park, R. D. Howe, and D. F. Torchiana, Virtual fixtures for robotassisted iniall-invasive cardiac surger, Medical Iage Coputing and Coputer-Assisted Intervention 1, W.. Niessen and M.A. Viergever, Eds. Berlin: Springer-Verlag, Berlin, 1, pp [3].. Abbott and A. M. Okaura, Virtual fixture architectures for teleanipulation, IEEE Intl. Conf. on Robotics and Autoation, pp , 3. [4] G. Guthart and K. Salisbur, r., The IntuitiveTM telesurger sste: overview and application, IEEE Intl. Conf. on Robotics and Autoation, pp ,. [5] R. A. Beasle and R. D. Howe, Tactile tracking of arteries in robotic surger, IEEE Intl. Conf. on Robotics and Autoation, Vol. 4, pp ,. [6] E. Coste-Maniere, L. Adhai, F. Mourgues, and O. Bantiche, Optial planning of roboticall assisted heart surger: first results on the transfer precision in the operating roo, Intl.. of Robotics Research, vol. 3, no. 4-5, pp , 4. [7]. M. Hollerbach and C. W. Wapler, The calibration index and taxono for robot kineatic calibration ethods, Intl.. of Robotics Research, vol. 15, no. 6, pp , [8] Z. Roth, B. W. Mooring, B. Ravani, An overview of robot calibration, IEEE Trans. Robotics and Autoation, vol. 3, no 5, pp , [9] D. M. Rovner and R. H. Cannon, r., Experients toward on-line identification and control of a ver flexible one-link anipulator, Intl.. of Robotics Research, vol. 6, no. 4, pp. 3-19, 1987 [1] D.-S. Kwon and W.. Book, A tie-doain inverse dnaic tracking control of a single-link flexible anipulator,. of Dnaic Sstes, Measureent, and Control, vol. 116, pp. 193-, [11] R. A. Beasle, R. D. Howe, and P. E. Dupont. Kineatic Error Correction for Miniall Invasive Surgical Robots, IEEE Intl. Conf. on Robotics and Autoation, vol. 1, pp , 4. [1] C. C. Cheah, M. Hirano, S. Kawaura, and S. Arioto, Approxiate acobian control for robots with uncertain kineatics and dnaics, IEEE Trans. Robotics and Autoation, vol. 19, no. 4, pp. 69-7, 3. [13] T. Belendez, C. Neipp, and A. Belendez, Large and sall deflections of a cantilever bea, Eur.. Phs., vol. 3, pp ,. [14] T.. Lardner and R. R. Archer. Mechanics of solids: an introduction. New York: McGraw-Hill, 1994.