IROS2019国际学术会议论文集2457

Abstract The use of continuum manipulators in surgical applications has increased recently. A continuum surgical manipulator is usually tele-operated after it is fully inserted into a patients cavity. Clearly, it is still possible to control the continuum surgical manipulator while it is not fully inserted, although the mobility might be reduced under its partially-inserted configurations. However, such a control scheme for realizing the configuration transition during the manipulators gradual insertion is missing. This paper hence proposes a novel kinematic framework of controlling a continuum surgical manipulator with configuration transitions for improved kinematic performance in teleoperation. The kinematic framework includes prioritized formulation of the Jacobian-based inverse kinematics, prediction-based constraint imposition and a set of configuration transition strategies. Both numerical simulations and experimental investigations were carried out to validate the proposed idea. I. INTRODUCTION Reduced trauma in MIS (Minimally Invasive Surgery) benefits patients in terms of less pain, quicker recovery and lower postoperative complication risks 1. The operative challenges of MIS stimulated the development of robot-assisted surgical platforms over the past decades 2-4. While most of the existing robotic surgical platforms use articulated structures, continuum mechanisms provide surgeons with alternative choices 4. Continuum structures are adopted in the designs of several surgical manipulators 5-10 for distal dexterity and inherent safety. Benefiting from the design compactness, structural compliance, proximal actuation scheme, and potentials in miniaturization, continuum surgical robots are particularly suited for SPL (Single Port Laparoscopy) and NOTES (Natural Orifice Transluminal Endoscopic Surgery). A multi-segment continuum surgical manipulator is usually fully inserted into a patients organ cavity for teleoperation afterwards. Under this working pattern, these multi-segment continuum manipulators suffer from an unreachable volume inside their workspace, as analyzed in *This work was supported in part by the National Natural Science Foundation of China (Grant No. 51722507, Grant No. 51435010 and Grant No. 91648103), and in part by the National Key R corresponding author: Kai Xu). Jiangran Zhao is with Beijing Surgerii Technology Co., Ltd, Beijing, China (e-mail: jiangran.zhao). 11. Nevertheless, a multi-segment continuum manipulator is in fact capable of being tele-operated even in a partially inserted configuration. Teleoperation in these partially inserted configurations can expand the manipulators kinematic performance, even though the mobility might be reduced in the partially inserted configurations. A kinematics framework for realizing the configuration transition during the gradual insertion of a continuum manipulator is yet missing. This paper hence proposes such a framework to handle the teleoperation and configuration transition when the continuum surgical manipulator is gradually inserted as shown in Fig. 1(a) to Fig. 1(d). The manipulator in Fig. 1 has two inextensible continuum segments with a rigid stem in between. The segment #1 is stacked on a base stem, as shown in Fig. 1(d). The rest configurations in Fig. 1(a) to Fig. 1(c) are introduced in detail in Section II. The continuum surgical manipulator is mounted onto and actuated by an actuation unit, as shown in Fig. 1(e). The actuation unit can realize rotation about its axis as well as be translated by a linear actuator for stem feeding. A trocar is fixed at the distal end of the linear actuator and guides the insertion of the manipulator. This paper is organized as follows. Section II defines the associated configurations of the manipulator. Kinematics of Configuration Transition Control of a Continuum Surgical Manipulator for Improved Kinematic Performance Shuan Zhang, Student Member, IEEE, Qi Li, Haozhe Yang, Jiangran Zhao and Kai Xu, Member, IEEE Rigid stem Segment #1 Base stem Axial rotation Trocar The 1stconfigThe 3rd config The 4thconfig Segment #2 (d) (a)(b)(c) Segment #2 The 2ndconfig Axial rotation Trocar 220 LL 220 LL= Feeding Linear Actuator Actuation Unit (e) Fig. 1. Different configurations of the continuum surgical manipulator: (a) the 1st configuration with the segment #2 partially inserted, (b) the 2nd configuration with the rigid stem partially inserted, (c) the 3rd configuration with the segment #1 partially inserted, and (d) the 4th (fully inserted) configuration; (e) the manipulators actuation scheme. The 4th config IEEE Robotics and Automation Letters (RAL) paper presented at the 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 Copyright 2019 IEEE the manipulator in different configurations is derived in Section III, while the configuration transition strategies are proposed in Section IV. Numerical and experimental validations are presented in Section V with the conclusions summarized in Section VI. II. CONFIGURATION DEFINITION The 2-segment continuum surgical manipulator in Fig. 1(d) includes two inextensible bending segments. Each segment possesses two DoFs (Degrees of Freedom). The base stem can be fed along and rotated about its axis. Thus the manipulator has 6 DoFs, excluding the actuation of its end effector (a gripper). When the manipulator is fully inserted as in Fig. 1(d), it will not be able to reach anywhere close to the trocar. Please refer to the manipulators workspace in Fig. 3(a), where an unreachable volume is shown. This unreachable volume can clearly be reached, if the manipulator can be tele-operated when it is partially inserted as shown in Fig. 1(a). Then the insertion process leads to 4 configurations as follows. Please note, when the inextensible segment is being partially inserted, the inserted portion can still bend. While the insertion continues, the feeding motion is kinematically treated as the segment length changing. In other words, when a segment is being partially inserted, it is treated as it possesses 3 DoFs: 2-DoF bending and 1-DoF length changing. The 1st Configuration (C1): only the segment #2 is being partially inserted as shown in Fig. 1(a). Here, the manipulator has 4 DoFs: axial rotation, length changing and 2-DoF bending of the segment #2. The 2nd Configuration (C2): the rigid stem is being partially inserted as in Fig. 1(b). The manipulator still has 4 DoFs: axial rotation and feeding of the rigid stem, as well as 2-DoF bending of the segment #2. The 3rd Configuration (C3): the segment #1 is being inserted as in Fig. 1(c). The manipulator has 6 DoFs: 2-DoF bending of the segment #2, as well as axial rotation, length changing and 2-DoF bending of the segment #1. The 4th Configuration (C4): the base stem is being fed as in Fig. 1(d). C4 is the configuration in which the manipulator is normally tele-operated. Workspace of the manipulator in different configurations in Fig. 3(b) shows that the workspace in one configuration overlaps with that in its adjacent configuration(s). This leads to the transition strategies between the configurations, which are detailed in Section IV, after the development of the kinematics in Section III. It is clear from Fig. 3 that, with the configuration transition control, the unreachable volume in the manipulators workspace is reduced, indicating better usage of the inherent motion capability of the continuum surgical manipulator. III. KINEMATICS The nomenclature and coordinates are defined in Section III.A, while the kinematics of a single segment is presented in Section III.B. Kinematics of the manipulator in various configurations is derived using that of a single segment as in Section III.C to Section III.G. A. Nomenclature and Coordinates The surgical manipulator includes two structurally similar continuum segments. With the coordinates attachment for the tth segment in Fig. 2(a), the coordinate attachments for the entire manipulator is shown in Fig. 2(b). The definitions are as follows, while the nomenclature is defined in Table I. The Base Ring Coordinate T tbtbtb tb = xyz is attached to the base ring of the tth segment at the center. tbz is perpendicular to the base ring and tbx is oriented to the first backbone. Bending Plane Coordinate #1 T t1t1t1 t1 = xyz shares its origin with tb. Its XY plane is aligned with the bending plane of the tth segment. Bending Plane Coordinate #2 t2tt T 22 t2= xyz is attached to the end ring of the tth segment at the ring center. Its XY plane is aligned with the bending plane. End Ring Coordinate T tetete te= xyz shares its origin with 2t. tez is perpendicular to the end ring and tex is oriented to the first backbone. The World Coordinate T www w= xyz is attached to the trocar and wz is aligned with the base stems axis. B. Kinematics of the tth Segment A single segment includes of a base ring, an end ring, several spacer rings and several backbones. The backbones are attached to the end ring. Pulling and pushing the backbones bends the segment, while the segment can also be entirely fed forward or drawn backwards. An imaginary central backbone characterizes the shape and length of the segment. Following the widely adopted constant curvature bending assumption summarized in 12, kinematics of the tth segment can be derived. The homogenous transformation matrix Bending plane t 1 ty tby 2 tte =xz tex 2 ty tey t tbx 1 tbt =zx 1 tz 2 tz t End ring Imaginary central backbone Spacer ring Base ring 2 e y 2 e x 2 e z 2 b y 2 b x 2 b z 1 e y 1 e x 1 b x 1 b y 1 b z wx wy wz (a)(b) Backbone Fig. 2. Coordinates attachement and nomenclature of (a) the tth segment and (b) the continuum surgical manipulator. relating te and tb is given as follows. 1 3 1 tbtb tbtete te = Rp T 0 (1) Where the expressions of tbRte and tbPte, involving Lt, t and t, are detailed in 13. A single segment is a 3-DoF structure during insertion and a 2-DoF structure once it is fully inserted. The instantaneous kinematics then has two sets of expressions as in Eq. (2). Derivation details can be referred to 9, 14. (3) ( 3)( 3) (3) ( ) ( ) (2) ( 2)( 2) (2) T tv ttttt t t t Tt tv tttt t L = = = J J J v x J J J (2) (3) cos)cos)sin) ) (sinh()h(h( sin(sinh()sinh(h()cos) sisin 0 n (cos) tv t tttttttt t t ttttttt t ttt t ttt L L L L L = J (3) (3) sin0cossin cos0sinsin 00cos1 tt tttt t t = J (4) Where ()(1 co)hs/ ttt =. (2) (:,1)(:,3) ( 3)( 3) tvtvtv = JJJ (5) (2) (:,1)(:,3) (3)(3) ttt = JJJ (6) C. Configuration Vectors Each configuration vector j consists of the variables that fully describe the manipulator in the corresponding configuration. A parameter may act as a variable in one configuration while becomes constant in other configurations. The variables and constants in the different configurations are summarized in Fig. 4. The only length variable in each configuration vector is the length of the segment or stem being inserted. The lengths of the fully inserted segments and stems become constants. The configuration vectors are defined as follows. (t) is used to express j to facilitate expressing the Jacobian matrices. The configuration vector of the manipulator in C4 is: 1122( )(12)(422) T T TTT ss L= (7) The configuration vector of the manipulator in C3 is: 311122(13)(22) T T TT L= (8) The manipulators configuration vector in C2 is: 222( )(22) T T TT rr L= (9) The manipulators configuration vector in C1 is: 122()223 T T T L= (10) D. Kinematics in the 4th Configuration The coordinate system attachment of the manipulator in C4 is shown in Fig. 2(b). The base stem of the manipulator can be rotated and fed along wz in w. (s) = LsT parameterizes the rotation and the feeding accordingly. The continuum segment #1 is stacked on the base stem. Thus 1b is obtained by rotating wby about szand translating w by a distance of Ls along wz. TABLE I NOMENCLATURE USED IN THE KINEMATICS MODEL Symbol Definition t Index of the segments. t = 1, 2. j Index of the configurations. j = 1, 2, 3, 4. Lt, Lt0 Inserted length and the full length of the imaginary central backbone of the tth segment. t Rotation angle from 1 tx to 2 tx about 1 tz. t Rotation angle from 1 ty to tbx along tbz. v(t), (t) Tip velocity and angular velocity of the tth segment. (t2), (t3) Configuration vectors of the tth continuum segment. (t2)= t tT when the segment is fully inserted (2-DoF segment), and (t3) = t Lt tT when it is being inserted (3-DoF segment). J(t2), J(t3) Jacobian matrices of the tth segment when it is fully or partially inserted, respectively. J(tv2), J(t2), J(tv3), J(t3) Jacobian matrices of the linear and angular velocities of the tth segment. ( )(2)( 2)ttvt =vJ or ( )(3)( 3)ttvt =vJ, while ( )(2)( 2)ttt =J or ( )(3)( 3)ttt =J. Lr, Lr0 Inserted length and the full length of the rigid stem. Ls, Ls0 Inserted length and the full length of the base stem. Axial rotation realized by the actuation unit. (r), (s) Configuration vectors of the rigid stem and the base stem: (r) = LrT and (s) = LsT j The configuration vector of the manipulator in the jth configuration. Jj The manipulators Jacobian matrix in its jth configuration. Jjv, Jj Jacobian matrices of the tip velocity and angular velocity of the manipulators end effector in the jth configuration. (b) (a) Unreachable volume Trocar + Trocar entry port Manipulator in C1 Unit: mm Manipulator in C4 Fig. 3. Translational workspace of the continuum surgical manipulator: (a) in Configuration C4, and (b) in Configurations C1 to C4. The continuum segment #2 and the segment #1 are connected by a rigid stem. 2b is obtained from 1e by a translation of distance Lr0 along 1 e z. The corresponding homogenous transformation matrix for the tip pose of the manipulator in C4 is: 112 21122 wbeb ebee w b =TTTTT (11) The instantaneous kinematics is derived as follows. 44 =xJ (12) 12(2 2) 4 3 11(1 2)2( 1 22) 21 wbww wwbbv w w b e w b = zz z pR WR J J 0R JR J (13) Where 11 12(1 2)(1 2) be ev = + WpJJ, AB C p is the position vector from the origin of frame B to the origin of frame C, expressed in frame A; p is the skew-symmetric matrix of a vector p; J (tv2) and J (t2) are obtained from Eq. (5) and Eq. (6). E. Kinematics in the 3rd Configuration The manipulator in C3 is shown in Fig. 1(c). With the segment #1 partially constrained by the trocar, the Base Ring Coordinate 1 b of the 3-DoF segment #1 is obtained from w only by a rotation of about wz. The instantaneous kinematics is derived as follows. 33 =xJ (14) 12(2 2) 1(1 3)2(22 1 2 3 ) 2 wbww e ww wbbv wbb = pR WR J J R JR J z z (15) Where 11 22(1 3)(1 3) be ev = + WpJJ; J(1v3) and J(13) are from Eq. (3) and Eq. (4). F. Kinematics in the 2nd Configuration In C2, the 2-DoF segment #2 is the only continuum segment and the manipulator now has only 4 DoFs. The homogenous transformation matrix for the tip pose of the manipulator in C2 is as follows. 2 222 wb eb w e =TTT (16) In C2, 2 b is obtained from w by a rotation of about wz and a translation of Lr along wz. The instantaneous kinematics is derived as follows. 22 =xJ (17) 2(2 2) 2 3 12 1 (2) 2 2 wbw e w wwbv wb = pR J J Rz0 zz J (18) Where J (2v2) and J (22) are obtained from Eq. (5) and Eq. (6). G. Kinematics in the 1st Configuration In the C1 configuration of the continuum manipulator, 2 b is obtained from w only by a rotation of about wz. The instantaneous kinematics is derived as follows. 11 =xJ (19) 2(2 3) 2 1 2 1 () 2 3 wbv wbw e w wb = z z pR J J R J (20) Where J(2v3) and J(23) are from Eq. (3) and Eq. (4). IV. CONFIGURATION TRANSITION The configuration vectors of the manipulator vary in these different configurations. Thus the Jacobian-based inverse kinematics approach cannot be directly used to tele-operate the manipulator from a pose in C4 to another pose in C2. A kinematics framework for configuration transition control is hence proposed, including i) a prioritized-Jacobian formulation, ii) a prediction-based constraint imposition, and iii) a set of configuration transition strategies. A. Prioritized-Jacobian Formulation In Configuration C1 and C2, the manipulator only possesses 4 DoFs and it will not always be able to reach a desired position and orientation at the same time. During teleoperation in a surgical procedure, the desired position is considered more important for the surgical manipulator to follow, while orientation errors of the surgical end effector may be tolerated if limited by the kinematic capability of the manipulator. is hence written using a prioritized-Jacobian formulation in (21) with the derivation details and explanations available in 15, assigning higher priority to the desired linear velocity. ()() jjvjvjvjjvjvjjv + =+J vIJJJJIJJJ v (21) Where Jjv and Jj are from T TT jjvj = JJJ in the jth configuration; and M+ is the pseudoinverse of a matrix M. To maintain the numerical stability, the damped least-squares formulation for the singularity robust M+ is used. B. Prediction-Based Constraint Imposition The continuum segment should always be subject to a maximal bending curvature constraint, in order to prevent the structure from being damaged by excessive bending. During the Jac