论文2G精华版本 ICRA 2018 files 2471

Toward Motion Coordination Control and Design Optimization for Dual Arm Concentric Tube Continuum Robots M Taha Chikhaoui Member IEEE Josephine Granna Student Member IEEE Julia Starke Student Member IEEE and Jessica Burgner Kahrs Member IEEE Abstract Dual arm continuum robots have been considered mainly for teleoperation where human perception and cognition permitted coordination and collision free motions This paper describes theoretical investigations on automation of dual arm robots constituted of two concentric tube continuum manip ulators An optimization algorithm is developed in order to improve triangulation ability of the robot and thus enhance the arms collaborative operation This a priori knowledge provides design directives in order to fulfi ll integration reachability and collaboration requirements Further automatic control is assigned to perform online safe collaboration tasks Our initial exploration is validated with numerical simulations using robot designs based on the optimization algorithm output The control algorithm based on the relative Jacobian and Cosserat rod modeling performs simultaneously with less than 1 of the total robot s length of accuracy for both relative end effector distance control and trajectory tracking I INTRODUCTION Research in continuum robotics achieved breakthrough results in the recent years particularly in medical appli cations Their miniaturization complex shape reachability and versatile applicability allowed an increasing number of research projects to consider continuum robots for medical diagnosis and or minimally invasive surgery 1 The smallest among these are concentric tube continuum robots CTCRs with outer diameters smaller than 0 8 mm Typically these are constructed of Nitinol alloy of nickel and titanium tubes nested one inside the other and actuated in translation and rotation Not only can they access confi ned spaces within the human body they can also integrate surgical or diagnosis tools through their free lumen hence without increasing the overall diameter Construction and assembly of the active part of CTCRs is relatively simple in contrast to tendon driven or pneumatically actuated continuum robots among other structures Thus their design is more adaptable to navigate through paths in particular anatomies or to operate in workspaces of specifi c surgical sites 2 Without loss of generality automation of selected robotic tasks or at least routine sub taks can provide numerous advantages in the context of medical applications 3 These include overcoming the inter and intra variability of human motion precision ensure repeatability of the most common This work was supported by the German Research Foundation Emmy Noether Research Grant BU 2935 1 1 These authors equally contributed to the manuscript The authors are with the Laboratory for Continuum Robotics Leib nizUniversitaetHannover Appelstr 11 30167Hanover Germany chikhaoui lkr uni hannover de Robot B Robot A Fig 1 Concept of dual arm concentric tube continuum robots A and B within an endoscope shifted by a distance at their base b and which end effectors e can perform simultaneous tasks The solid pink line describes a desired trajectory assigned by the user to robot A while maintaining a constant relative distance between end effectors dashed gray line The solid black line denotes the resultant trajectory of the end effector of robot B eTA B matrices represent the mapping between the individual robot base and its end effector The green arrows represent each tube s actuation rotation and translation tasks save time and concentration of highly qualifi ed sur geons to execute more complex tasks which would typically demand cognition quick decisions and in depth analysis of composed situations Further multiple tools are required to operate in endoscopic and laparoscopic surgeries with a large variety of aims ranging from navigation to operation through diagnosis These goals can be achieved by introducing mul tiple arms in the same robotic structure In principle using multiple robotic arms can allow for co operation execution of parallel tasks robotic manipulation as well as subdivision of complex operations into simulta neous simpler tasks without redeploying the robot several times to exchange a tool for instance In this context dual arm robots were fi rst introduced to standard serial robots in industrial and humanoid robotics A Dual Arm Standard Serial Robots Prior work on dual arm standard serial robots included a study of each individual arm s manipulability in order to increase their co operation ability 4 The dual arm robot was further considered as a single structure operating on a common virtual object The feasibility of this approach was proven to solve for task confl icts that would arise if the arms IEEE Robotics and Automation Letters RAL paper presented at the 2018 IEEE International Conference on Robotics and Automation ICRA May 21 25 2018 Brisbane Australia were controlled separately 5 Motion coordination is more consistent with the use of dual arm formulation as the overall redundancy of the robot increases 6 With the appropriate redundancy resolution scheme dual arm robots can be used to execute tasks with different levels of priority For example secondary tasks include joint space stabilization validated for specifi c tasks operated by a humanoid robot 6 and a mobile humanoid robot in co manipulation with a human 7 These examples established the basics of dual arm conven tional robots and represent the starting point of this paper B Dual Arm Continuum Robots In continuum robotics dual arm robots were investi gated for single port access as part of minimally inva sive surgery They were mainly utilized in teleoperation scenarios Tendon driven continuum robots were used in bimanual teleoperation with pick and place and knot tying tasks 8 Further applications have shown suturing tasks and grape peeling using two tendon driven arms 9 In terms of integration two tendon driven continuum robots were successfully incorporated in an endoscope 8 CTCR arms were also combined in a hand held endoscope and were considered in prostate operations 10 12 This prototype which has an outer diameter of 8 3 mm permitted a better coordination between tissue retraction and holmium laser cutting Further dual arm CTCR teleoperation concerned better anatomical reachability in endonasal skull base surgery with an artifi cial tumor removal 13 14 C Assumptions and Contributions While the aforementioned examples show different appli cations of dual arm continuum robots see Fig 1 control of the end effectors is decoupled and relies exclusively on the user s decision based on visual feedback and personal estimation of the completion of tasks Hence only quali tative data were provided to evaluate the effi ciency of this approach Prior to utilization of a CTCR for a surgical application the design parameters have to be selected a priori As the design parameter space is diverse several researchers proposed the use of optimization algorithms e g to achieve a desired workspace 15 or to consider stability 16 or task and anatomical constraints 17 18 In the context of en doscopic dual arm continuum robots the design parameters of a CTCR were optimized such that they suit the reachable fi eld of view by the camera in 12 Despite 15 algorithms employed utilize non linear optimization methods and did not investigate the applicability of global optimization meth ods for parameter optimization of CTCR In this paper we present a preliminary study toward motion coordination and design optimization of dual arm CTCRs In the following we restrict the study to contactless tasks with the dual arm robot hence only free space motion is considered We present a global optimization method which optimizes the parameters of two collaborating robots for the fi rst time and further develop a novel objective which aims to increase the number of collaborative confi gurations with the robot arms The approach developed hereby aims to overcome the dependency on the user perception for end effector collabo ration with i offl ine optimization of the robot design and ii online improvement with feedback control The goal is to avoid end effector collision and to ensure completion of tasks in a quantifi ed manner To the best of our knowledge this paper investigates the theoretical foundations for motion coordination of dual arm continuum robots for the fi rst time Further an optimization algorithm is developed which allows to design collaborative dual arm robots that provide a better triangulation and operation on common tasks We optimize the dual arm parameters such that they lead to a maximum number of collaborative confi gurations of the two robotic arms and utilize a particle swarm optimization algorithm II MODELING Both the optimization algorithm and the control law are model based Hence the model of the dual arm CTCR is developed in this section The fi rst part describes the model of a single continuum arm and the second part details the kinematic chain of the dual arm robot A Kinematic Modeling of a Single Continuum Arm The kinematic model and the Jacobian matrix computa tions are based on prior developments 19 A Cosserat rod model is used hereby which has shown mean tip error of about 1 of the robot s length 20 As described in Fig 1 each tube i 1 Nt Nt number of tubes is subject to two degrees of actuation applied at its base a rotation iand a translation i In total a CTCR arm is actuated by q Rn where n 2Nt and provides m 6 degrees of freedom DoF at the end effector level if Nt 3 Each component tube is constituted by a straight length Lsat the proximal part toward the robot base and a curved length Lcat the distal part with a constant curvature outer and inner diameters outand in respectively Material parameters include the Young s modulus E and the Poisson s ratio If no external loads are applied on the robot the deformation of the CTCR is defi ned by T0 T s y 1 y0 f s y T q 2 where T T s SE 3 is the mapping between a robot s base s 0 and end effector arc length s se 3 the Lie algebra of SE 3 is the body frame twist It contains transverse strains elongation strain bending and torsion information 0 is the derivative with respect to s and maps from R6to se 3 y is a set of variables composed by a set of unknown components yu including torsional and bending strains and known components yk including the initial pose The following is based on the initial boundary conditions b at the base such as no displacement at the robot base and tangent robot backbone with bZA B axis Using Eqs 1 and 2 yu 0 yu 0 3 T 0 H yu 0 q 4 yk 0 h yu 0 q 5 where H describes the forward kinematics and h defi nes the evolution of the known components yk one can defi ne the Jacobian matrix eJ at the end effector e based on this initial value problem eJ s Eq s Eu s B uBq 6 where Eq udenotes variations in T with respect to those in q and yu 0 respectively and Bq udenotes variations in b with respect to those in q and yu 0 respectively with Eqi T qiT 1 Eu i T yu iT 1 Bq b q and Bu b yu 0 The superscript is the Moore Penrose pseudo inverse and is the inverse operation of Note that for this step e J can be computed via fi nite differences In the following the Jacobian matrices J are all defi ned in the end effector frame and the superscript e is omitted B Dual Arm Relative Jacobian Using the kinematics of a single arm one can derive the dual arm robot kinematics Assume a closed chain consti tuted of two continuum arms A and B interacting at their end effectors with a virtual rigid object The relative Jacobian matrix JAB Rm nA nB based on each individual Jacobian matrix JA Rm nAand JB Rm nB see 21 is JAB A e B e Ae A bJA Ae B b JB 7 where R6 6 is the adjoint transformation defi ned by R03 3 03 3R based on the rotation matrix R SO 3 of the full transformation T as described in Fig 1 and I3 rAB 03 3I3 is the wrench transformation matrix where I3 R3 3is the identity matrix Note that rAB R3 3is the skew symmetric matrix of the end effectors relative position vector rAB Thus the relative velocity VAB R6 is defi ned with respect to the end effector of robot A and can be written as VAB JAB qAB JAB qA qB 8 where qAB RnA nBis the joint velocity of the dual arm continuum robot Eq 8 allows for a direct mapping from the joint ve locities of each continuum arm to the relative velocity between their end effectors Regarding the dual arm robot this formulation overcomes the problem of rank defi ciency of a given continuum arm In the case the latter s Jacobian rank decreases below 6 the redundancy of the dual arm robot 12 DoF ensures singularity free motion of the relative end effector positioning task which only requires 3 DoF 3 7 mm 2 8 mm 12 2 mm S 5 mm Instrument channel B Instrument channel A Objective lens Auxiliary water channel a Robot ARobot B b Fig 2 a Endoscope active length cross section dimensions b Triangu lation between the two end effectors with angle Without loss of generality the virtual kinematic chain transformation simplifi es in this paper case as the robot arms access with the same orientation from the same entry point assuming a lateral shift S described in Fig 2a Hence the transformation matrix bTAB A bRB b rAB 01 31 R4 4 between the arm bases is defi ned by the rotation AbRB b I3 and the translation brAB S 00 T III DESIGN OPTIMIZATON The design optimization aims to determine optimal tube parameters for the two robotic arms such that they can work collaboratively on a medical task We consider two of the same robot designs for robot A and B for study and fabrication convenience with three tubes each The optimization algorithm determines optimal tube curvatures i and curved lengths Lcifor both arms The straight length Lsi is fi xed with respect to the commonly used CTCR designs The tube diameters and the entry point distance are driven by the commercial dual channel endoscope designs Typically the channel diameters vary between 2 and 4 mm with a shift distance S 5 mm as described in Fig 2a showing example dimensions from Olympus Dual Channel Gastroscope GIF 2TH180 A Robotic Collaboration Robots A and B are required to work collaboratively on a medical task We seek in a robotic design which enables a certain triangulation of the two end effectors as well as a certain distance between them for task deployment Our medical collaborators advised a triangulation angle of 90 180 as depicted in Fig 2b The triangulation angle is computed by determining the angle between the two tangent curvature vectors eZA B at both end effectors Thus the aim of the optimization algorithm is to search for two robotic arm designs which result in a maximum amount of collaborative confi gurations with the specifi ed triangula tion angle Further we constrain the Euclidean end effector distance between the two robots below a predefi ned threshold to enable collaborative manipulation during a certain task B Optimization Algorithm As this is a diverse optimization problem with several parameters to be optimized we utilize a particle swarm optimization PSO algorithm as originally proposed in 22 Algorithm 1 PSO INPUT patient dataset DEFINEG P w c1 c2 INITIALIZExk vk for t 1 G do C DETERMINE COST xtk ptbk gt b SELECT GUIDES x t k p t 1 bk gt 1 b C for k 1 P do xt 1 k vt 1 k UPDATE xtk vt k p t bk g t b w c1 c2 xt 1 k ENFORCE CONSTRAINTS xtk end end RETURNgt b TABLE I TUBE DESIGN PARAMETERS Tube 123 out in mm 1 21 0 671 92 1 282 74 2 08 E GPa 606060 0 30 30 3 Ls mm 15010050 Lc mm 0 50 0 50 0 50 Lopt c mm 30 719 523 8 mm 1 0 0 05 0 0 05 0 0 05 opt mm 1 0 0480 0170 014 Particle swarm optimization fi nds an optimal solution within a multi dimensional space and is especially suited for com plex optimization algorithms The algorithm searches for a global optimum and is adaptable for multiple objectives and parameters in the future We selected the proposed algorithm as it builds on previous work presented in 23 The swarm or population consists of P particles where each particle k defi nes a feasible solution within the search space Within each generation t of the swarm each particle is defi ned by its current position xt k and velocity vt k The velocity vt k infl uences the position of a particle at time t 1 and is governed by two guides which are the personal best solution of a particle pt bk and the global best solution gt b over all particles found so far The positions of all particles change within the search space until the maximum number of generations G has been reached The global best solution then defi nes the output of the algorithm and in this case the optimized design parameters of the robotic arms The outline of the PSO algorithm is illustrated in Algorithm 1 and further explained in the following steps 1 Initialization Each parameter of the particle s position xkat generation t 0 is initialized based on a random dis tribution within the parameter space In this case a particle xt k is defi ned by the curved lengths Lciand curvatures i of each tube respectively They are initialized regarding the constraints in Table I 2 Particle Cost Each particle solution is assigned a cost C which defi nes the quality of the parameter set A particle s cost is defi ned in this case by the number of collaborative confi gurations To maximize this number we fi rst generate the workspace of each CTCR by random sampling of the joint space Uniformly distributed random samples j are created for each set of confi guration parameters Fig 3 Collaborative confi gurations of the optimized robot design with an optimal triangulation and end effector distance for collaborative tasks qA Bi 1 i 1 i T By utilizing the forward kinematics model the space curves of the two robots are then determined for all con fi guration samples j We compute the triangulation angle for all confi guration permutations of robot A and B The overall number of collaborative confi gurations with 90 180 then def