Design Of A Multi-Arm Surgical Robotic System For Dexterous . - UCLA

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Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c) 2016 by ASMEDesign of a Multi-Arm Surgical Robotic Systemfor Dexterous ManipulationitedDejan MilutinovićComputer EngineeringUniversity of California, Santa CruzSanta Cruz, CA, 95064Email:Dejan@soe.ucsc.edupyedZhi LiElectrical & Computer EngineeringDuke UniversityDurham, NC, 27708Email: zhi.li2@duke.edutNbeen dedicated to developing surgical robotic systems thatshow high levels of manipulation dexterity and precision notachievable by the surgeons’ hand, provide viewing anglesotherwise unavailable to surgeons’ views, and minimize thetrauma to the tissue surrounding the surgical site. Advancements in surgical robot technology has led to the development of new surgical techniques that would otherwise be impossible.Surgical procedures are traditionally performed by twoor more surgeons, along with staff nurses. Due to the heavycognitive load and manual demands of surgical procedures,the collaborative effort of two or more surgeons is often required. With the introduction of surgical robots into operating rooms, the dynamics between the primary and assistingsurgeons changes significantly. The primary surgeon, whocontrols the surgical robot, is immersed in a surgical consoleand is physically removed from the surgical site itself, whilethe assistant is usually located next to the patient and holdsanother set of non-robotic surgical tools. Reproducing theinteraction of two surgeons with the surgical site using surgical robotic systems requires at least four robotics arms andtwo stereo cameras rendering the surgical site. Once multiple robotic arms are introduced, several operational modesare available in which each pair of arms can be under fullhuman control or in a semi-autonomous mode (supervisorycontrol).In spite of the advantages, the introduction of multiple robotic arms into a relatively small space presents challenges. From the operational perspective, there is a needto maximize the common workspace that is accessible bythe end effectors of all four arms. This common workspaceneeds to overlap with the surgical site dictated by the patient’s internal anatomy. Increasing the common workspaceAcceptedManuscripSurgical procedures are traditionally performed by two ormore surgeons along with staff nurses: one serves as the primary surgeon and the other as his/her assistant. Introducing surgical robots into the operating room has significantlychanged the dynamics of interaction between the surgeonsand with the surgical site. In this paper, we design a surgicalrobotic system to support the collaborative operation of multiple surgeons. This Raven IV surgical robotic system hastwo pairs of articulated robotic arms with a spherical configuration, each arm holding an articulated surgical tool. Itallows two surgeons to teleoperate the Raven IV system collaboratively from two remote sites.To optimize the mechanism design of the Raven IV system, we configure the link architecture of each robotic arm,along with the position and orientation of the four bases andthe port placement with respect to the patient’s body. Theoptimization considers seven different parameters, which results in 2.3 1010 system configurations. We optimize thecommon workspace and the manipulation dexterity of eachrobotic arm. We study here the effect of each individual parameter and conduct a brute force search to find the optimalset of parameters. The parameters for the optimized configuration result in an almost circular common workspace witha radius of 150 mm, accessible to all four arms.otCoJacob RosenBionics LabMechanical & Aerospace EngineeringUniversity of California, Los AngelesLos Angeles, CA, 90095Email: rosen@seas.ucla.edu1IntroductionSurgical robots recently introduced into the operatingroom have significantly changed the way surgery is conducted. Together with the clinical breakthroughs in new surgical techniques, these technological innovations in roboticsystem development have improved the quality and outcomes of surgery. In the last decade, research efforts haveJMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use1

pyedFig. 1: Raven IV Surgical Robot System - CAD rendering of the fourRaven’s arms interacting with the patient. In the figure, most of the actuators were removed from the base of each arm to expose to the rest of thearms and the shared workspace. The workspaces are marked with transparent cones and their intersection defines the shared workspace.tNotCoThe Raven IV surgical robot system consists of two pairsof surgical robotic arms. These two pairs are mirror images of each other, which results in their symmetric kinematics. Each surgical robot arm has seven degrees of freedom (DOFs): six DOFs for positioning and orienting the endeffector and one for opening and closing the surgical tool attached to the surgical arm.Axis 3abAxis 1ippatient-robot or robot-robot collisions.Previous research efforts mainly focused on the designof port placement for cardiac procedures while using severalexisting robotic arm architectures, such as the Zeus [1, 2]or DaVinci [3, 4] or a similar, four-bar mechanism [5] inserted between the ribs. With the introduction of four roboticarms, a new optimization approach is required for designing the size and shape of the common workspace of the fourrobotic arms while ensuring the kinematic performance ofeach robotic arm. The scope of this research effort is a kinematic optimization of the surgical robotic arms in terms oftheir structural configurations, as well as their positions (portplacement) and orientations with respect to the patient.In this research, we introduce the mechanism design andoptimization of the Raven IV (Fig. 1) surgical robotic system. It has two pairs of articulated robotic arms and, therefore, supports two surgeons in collaboration using two surgical consoles that are located either next to the patient orat two remote locations. Raven IV is the second generationof Raven I [6]- [16]. The kinematic optimization of RavenI was based on the analysis of the workspace of a singlearm [15, 17]. Several major structural changes are made tominimize the footprint of the individual robotic arm including the following: (1) all the actuators located on the base ofthe robot are mounted on top of the base allowing the baseto be moved closer to the patient body; (2) the dimensionsof the actuation package are reduced; (3) the link lengths arechanged based on reported results; (4) the tensioning mechanisms of the cables are relocated in the base plate to providebetter access and solid performance; (5) a universal tool interface is designed to accept surgical robotics tools from different vendors; and (6) a unique tool with a dual joint wristis designed and incorporated into the system.In addition, we propose a method to optimize the geometry of the four robotic arms and the relative position and orientation of their bases. The cost function in our optimizationaccounts for (1) the size and shape of the common workspaceof all the arms, (2) the mechanism isotropy, and (3) the mechanism stiffness. In minimally invasive surgery, the surgicaltools designed to be attached to a surgical robotic arm are thesame as the ones used in traditional surgery. The optimization does not target a specific internal organ or anatomicalstructure, but is instead based on sizes of patient and animal models. Our method is proposed for the optimization ofthe Raven IV surgical robotic system, but can be generallyapplied to the optimization of a wider spectrum of similarrobotic systems.itedJournal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c)may2016leadbyto ASMElarger robotic arms, which in turn may result inscrLeft Robotic Armx2 x0y0z2z1z0 x2 x0y0z0z1Axis 2x1ManuedMethodologyWe propose a method to optimize the kinematics of theRaven IV surgical robotic arms. In this section, we presentthe forward and inverse kinematics, the Jacobian matrix, andthe cost function for the optimization. The cost function accounts for the link lengths of the spherical mechanism, theport spacing, the base orientations of the robotic arms, andthe manipulation isotropy in the common workspace.Axis 3x1Axis 1Right Robotic Arm(a) Surgical robot arm.z5Tool of the right robotic armx6x5z6Bottom of the toolptceAc2Axis 2z2x4z4x4z3z4x3Top of the tool(b) Surgical tool.Fig. 2: Reference frame of the Raven IV surgical robotic system.The base frame is located at the converging center ofthe spherical mechanism, which is formed by the first threelinks of a Raven IV arm (Fig. 2a). The Denavit-Hartenberg(DH) Parameters (see Table 1) are derived with the standardmethod defined by [18]. The derivation of the forward andJMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use2

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c)2016kinematicsby ASMEis presented in Appendix.inverseTable 1: Denavit-Hartenberg (DH) Parameters for Raven IV Arms.αiaidiθiLeft01π α00θ1 (t)Robot12 β00 θ2 (t)(1,3)23000π/2 θ3 (t)34 π/20d4 (t)045π/2a50π/2 θ556 π/200π/2 θ601π α00π θ1 (t)0y12 β00θ2 (t)(2,4)23000π/2 π θ3 (t)34 π/20d4 (t)045 π/2a50π/2 θ556 π/200π/2 θ6-400xot100d4 [0, 250] mm [ 86 , 86 ] [ 86 , 86 ]θ6tNθ2 [20 , 140 ]0z [0 , 90 ]300200θ3 [ 86 , 86 ]θ50-200ipθ1-200(a) Unit: mm.RobotRange400200CoRight200itedipyedi 1zRobot3002001000-100-200-300scr-100edManuThe design of the surgical tools follows the genericgeometry of a minimally-invasive surgical tool. Thus,our method focuses on optimizing the shape of commonworkspace and the manipulability in it, and will determinethe geometry of the first two links and the relative positionsof the bases of the four Raven arms with respect to eachother.The Common Workspace and the Reference PlaneThe common workspace of our surgical system is theintersection of the workspaces of all the four Raven arms.Fig. 3 depicts the arrangement of the four Raven arms withrespect to each other. The gray bars represent the bases ofthe arms, while the magenta and the cyan bars represent thefirst and the second links of each arm, respectively. The common workspace of the four Raven IV arms is 3-dimensional.When optimizing the mechanical design of the system, wedefine a reference 2D plane, which is 150 mm below theplane that includes the ports of the four surgical arms. Typically, the surgical tools are inserted half way into the patientwhen the tool tips are operating in the reference plane. Sincethe surgical tools frequently operate in the reference plane,we decide to optimize the geometry of the projection of the3D common workspace on this plane, as well as the manipulability within the projected area. In the following sections,Accept2.1Reference Plane-200-300-300-200-1000x100200300(b) Unit: mm.Fig. 3: The common workspace projected onto the reference plane: (a) 3Dview; (b) projection onto the x-z plane. For each Raven IV arm, the gray barrepresents its base. The magenta and the cyan bars represent the first andthe second links, respectively.we will refer this area as the common workspace for simplicity.2.2Area-Circumference RatioWe want to optimize the shape of the commonworkspace in addition to maximizing its size. The optimizedcommon workspace should be a circular area as possible, sothat the surgical tools are given free space to move uniformlyin any direction. Here we define a variable ς, which is the ratio between the area and its circumference, to collectivelyevaluate the area and shape of the common workspace (seeEq. (1)):JMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use3

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c) 2016 by ASMEς AreaCircum f erence(1)To evaluate the isotropy of a Raven IV arm, we analytically derive the Jacobian matrix using the velocity propagation method. The angular and the linear velocities arepropagated iteratively from frame ı to frame ı 1 as:According to the isoperimetric inequality, the circle hasthe largest possible area among all the shapes with the samecircumference. The area-circumference ratio of a circle ςc isproportional to its radius r:(2)100 200 200Red Common WorkspaceGreen Reachable by Left Arm OnlyBlue Reachable by Right Arm Only02000 100 200400Red Common WorkspaceGreen Reachable by Left Arm OnlyBlue Reachable by Right Arm Only 2000200x (mm)x (mm)(a)(b)tN 100Note that for a prismatic joint, θ̇i 1 0 in Eq. (4), andfor a revolute joint, d i 1 0 in Eq. (5).The Raven IV arm is structured such that the positioningof the surgical tool tip in a three-dimensional (3D) workspaceonly depends on the first 3 DOFs. The remaining 4 DOFsdictate the tool tip orientation and, therefore, do not affectthe mechanism’s kinematic manipulability. As a result, theanalytical derivation of the Jacobian takes into account thefirst 3 DOFs (i.e., θ1 , θ2 and d4 ) which determine the positionof the surgical tool. The irrelevant DOFs, including θ3 , α4 ,θ5 and θ6 , are set to zeros.According to the velocity propagation method, the angular velocity of the tool’s wrist for the left arm is:400ip0ManuscrFig. 4: Example of two typical common workspaces of two Raven armsconstructed for two different link lengths defined by α and β : (a) two-armconfiguration defined by the link lengths α 65 , and β 15 resulting inς 2.23; (b) two-arm configuration defined by the link lengths α 65 ,β 80 resulting in ς 4.48.ptedFig. 4 shows two common workspaces of two Ravenarms, resulting from different link lengths. The commonworkspace depicted in Fig. 4b (with ς 4.48) has thepreferred shape compared to the workspace illustrated inFig. 4a.Mechanism IsotropyIsotropy measures the kinematic manipulability of theconfiguration of a mechanism. Its value ranges between 0 to1. A mechanism is mechanically locked at the configurationwhere the isotropy is 0, losing one or more degrees of freedom. At a configuration where the isotropy is 1, the mechanism is able to move equally in all directions and, therefore,has the best mapping between the joint space and the endeffector space. The isotropy is computed as one over thecondition number of the Jacobian matrix J (Eq. (3)).Acce2.3Iso (5)Co200100iii vi 1 i 1i R( ωi Pi 1 vi ) di 2 Ẑi 1(4)ot200y (mm)y (mm)Practically, the common workspace has an amorphicshape that cannot be analytically expressed. However, maximizing ς will result in the common workspace that is as closeto a circle as possible .itedi 1rπr2 2πr 2iωi 1 i 1i R ωi θ̇i 2 Ẑi 1pyedς i 11Condition number o f J(3) c2 cβ sα θ̇1 sβ cα θ̇1 sβ θ̇23 v3 s2 sα θ̇1c2 sβ sα θ̇1 cβ cα θ̇1 cβ θ̇2 (6)and for the right arm is: c2 cβ sα θ̇1 sβ cα θ̇1 sβ θ̇23 v3 s2 sα θ̇1c2 sβ sα θ̇1 cβ cα θ̇1 cβ θ̇2(7)The linear velocities of the tool’s wrist are the same forboth left and right arms, which are: 03v3 0 d 4(8)Therefore, the analytically derive Jacobian matrix forthe left arm is: c2 cβ sα sβ cα sβ 03s2 sα0 0 J c2 sβ sα cβ cα cβ 1(9)JMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use4

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c)by rightASMEand2016for thearm is:Axis 3200(10)Axis 2-500fz2000fyywx0-200500yWorld Coordinate FramexwpyedfxFig. 5: Parameters for the optimization of the common workspace (unit:mm).ResultsIn this section, we use a brute force method to searchin the whole parameter space for the parameter values thatmaximize the value of the cost function. We also study howeach individual parameter affects the cost function.tNotCo33.1Overall OptimizationA brute force search in the parameter ranges and withthe resolutions listed in Table 2 was conducted to maximizethe cost function Cmax from expression Eq. (11). The searchexplored the total of 2.304 1010 parameter combinations,each of them representing a specific configuration of the fourrobotic arms. The configuration that maximizes the costfunction is depicted in Fig. 6a. This configuration resultedinto the largest circular common workspace shared by thefour arms as depicted in Fig. 6b) with an approximate radiusof 150 mm.Fig. 7, Fig. 10, Fig. 11 and Fig. 14 show trends of Cmaxwith respect to the parameters. According to Fig. 11, thelargest Cmax value is for max φx and min φz . For all other optimization parameters, the largest Cmax value is in the middleof the parameter ranges. Table 2 shows the parameter ranges,resolutions, and preferred values of our optimization usingbrute force method, with an optimal Cmax . To find an evenbetter Cmax and its corresponding parameter values, we conduct another brute force search in the neighborhood of theoptimal parameter value of α, β, φy , bx , by and Isomin withrefined resolutions (Cmax 533.01 when bx 90 mm).C max{ς · Iso · Isomin}α3 β3ed(α,β,φx ,φy ,φz ,bx ,by )ManuscripCost FunctionThe common workspace is optimized taking into account four goals. The first two are to maximize (1) thesum of the isotropy across the entire common workspace( Iso), and to minimize (2) the isotropy (Isomin ) of the common workspace. We also want to maximize (3) the AreaCircumference ratio (ς) given bounded isotropy values. Finally, we want to maximize (4) the stiffness of the mechanism to reduce the end effector position and orientation errors due to link deformations. In a spherical geometry of themechanism, the axes of the first three links intersect in a single point, which defines its remote center. The kinematics ofthe mechanism is independent of the radius of the sphere. Asa result, the link lengths of the spherical mechanism are measured by angles, while the radius of a spherical mechanismdetermines the space around the point where the surgical toolis inserted into the patient’s body.With the above considerations, we define the followingcost function to optimize the mechanical design and configuration of the Raven IV surgical system:Axis 1bybx-200As shown in equations (9) and (10), the analytical Jacobian matrix has a unit vector corresponding to the prismatic joint along the z-axis of Frame 4. Thus, the mechanismisotropy of a Raven IV arm depends only on the 2 2 top leftsub-matrix of the Jacobian, denoted as 3 Js .2.4ba0ited 00 1z (c2 cβ sα sβ cα ) sβ3s2 sα0J c2 sβ sα cβ cα cβ(11)AcceptIn Eq. (11), Iso denotes the sum of the actual isotropy ofthe points in the common workspace and Isomin denotes theminimum isotropy required in the common workspace. Thedenominator α3 β3 describes our goal regarding the maximization of the structure stiffness which is inversely proportional to the cube of the link lengths.To summarize, the cost function Eq. (11) maximization computes the following parameters: (1) the link lengthsof the first two links α (the angle between the Axis 1 andAxis 2) and β (the angle between Axis 2 and Axis 3); (2)the base orientation of the arms denoted by φx , φy and φz andmeasured by the rotations about the axes of the world coordinate frame Xw , Yw and Zw , respectively; (3) the port spacingbx and by , which are the horizontal distances between thebases of the Raven IV arms; and (4) the minimum isotropyrequired in the workspace denoted by Isomin .3.2Link LengthGiven the spherical shape of the mechanism, the lengthsof the first two links are expressed as two angles, α andβ. These two link lengths are fixed in the design process,whereas other parameters of the Raven robotic arms can beadjusted as part of setting up the system. The size of theworkspace of a single Raven arm is maximized when α andβ are 90 . However, for the rigidity of the mechanism, weJMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use5

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c)2016by ASMETable2: Parameterranges and preferred values for the optimization of theRaven IV surgical robotic system.200Optimal valueResolutionα[5 , 90 ]85 20 β[5 , 90 ]65 20 200φx[ 20 , 20 ]20 10 200φy[ 20 , 20 ]10 10 0φz[ 20 , 20 ] 20 10 200ybx[50, 200] (mm)100 (mm)50 (mm)by[50, 200] (mm)50 (mm)50 (mm)Isomin[0.1, 0.9]0.50.20400ited2000 200xpyed(a) Unit: mm.200Cmax 526.3338 for Isomin 0.51000Isotropy PerformanceLimiting the minimal acceptable value of the isotropyIsomin has a significant effect on the common workspace optimization result. The Jacobian matrices derived in forwardkinematics (see equations (9) and (10)) have three variables,including θ1 (the shoulder joint angle), θ2 (the elbow jointangle) and d4 (the tool shaft displacement). However, as depicted in Fig. 8a, the plot of the isotropy as a function of θ1and θ2 indicates that the isotropy of the Raven robotic armmechanism varies only with θ2 . In Fig. 8, we choose thedifferent Isomin in the common workspace to show that theθ2 value range shrinks as Isomin increases, regardless of armconfiguration and link length.We further find that Isomin affects the shape of the common workspace, the optimal link lengths and the maximumof the cost function. Fig. 9 depicts the area-circumferenceratio ς as a function of link lengths α and β for differentIsomin . Fig. 10 further shows that Cmax varies with Isomin andis maximal when Isomin 0.5.ot 200Red CW of Four Raven ArmsGrey CW of Two Raven ArmsOthers CW of One Raven Arm 300 400 200ip3.3 100tNgenerally prefer shorter link lengths. Fig. 7 depicts the costfunction value Cmax for the optimal configuration, while αand β are varied. The figure shows that for α, β [0 , 90 ],the unction Cmax has the largest value when α 85 andβ 65 .CoResultzRange02004003.4AcceptRobot Base OrientationThe base orientation of each Raven arm is determinedby three rotation angles in the world coordinate system. Therotation angles about the Xw , Yw and Zw axes are denoted byφx , φy and φz , respectively. A mirror image axial symmetry is assumed for the rotations with respect to all the axesand the following text refers to the top right Raven arm (firstquadrant) in Fig. 13a.Fig. 11 shows Cmax as a function of the base orientationin each individual axis, φx , φy , φz [ 20 , 20 ]. When vary-Fig. 6: Optimal configuration of the Raven IV surgical robot four armsfollowing a brute force search (a) Relative position and orientation of thesystem bases (b) optimized workspace.50080400β (deg)edManuscr(b) Unit: mm.603004020010020204060α (deg)80Fig. 7: Cmax as a function of the first two link lengths α and β.ing one of the angles φx , φy , or φz , the rest of them are setto zeros. In Fig. 11, Cmax monotonously increases with φx ,monotonously decreases with φz and it reaches its maximumfor φy 10 . The diagram shows that Cmax is most sensitiveto the change in the base rotation about the x-axis and leastsensitive to the change in the base rotation about the z-axis.In Fig. 12, we plot Cmax as a function of various combinations of base orientations in three perpendicular planes.Fig. 13 shows the top, front, and side views of the four RavenJMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use6

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c) 2016 by ASMEIsomin 0.100.8Iso0.6Isomin 0.300.60.4Isomin 0.700.2Isomin 0.900.400500.202015010050θ406080θ22(a)100120φz0 20Fig. 8: The representative plot of the mechanism isotropy as a functionof θ1 and θ2 for the first two link lengths α 55 and β 40 : (a) themechanism isotropy of the Raven arm as a function of θ1 and θ2 , showingthat the isotropy does not depend on θ1 ; (b) the mechanism isotropy of theRaven arm as a function of θ2 , showing that the minimal required workspaceisotropy Isomin limits the range for θ2 50(b)100100150bx (mm)200Fig. 14: Performance criteria Cmax as a function of port spacing along thetwo orthogonal directions bx and by .scripFig. 9: Isomin affects the optimized shape of the common workspace depicted by the area-circumference ratio ς as a function of link lengths: (a)when Isomin 0 then ςmax 6.64, and the optimal link lengths are α 80 and β 40 ; (b) when Isomin 0.5 then ςmax 6.55, and the optimal linklengths are α 70 , β 35 .20tN20150ot2010Co60by (mm)800φ (deg)empirical data of port placement in minimally invasive surgical applications.β6β80 10Fig. 11: Effect of base orientation (φx , φy , and φz ).20062φy140(b)40φx200ited100 0θ1400Isomin 1ed00minceptFig. 10: Cmax varies with Isomin .3.5AcIV arms for the optimal base orientation, i.e., φx 20 ,φy 10 , and φz 20 .Port SpacingFig. 14 depicts Cmax as a function of port spacing andshows that it monotonically decreases as the distance between the ports along the x-axis increases, while it reachesits maximum when the distance between the ports along they-axis is 100 mm. As a result, the expected benefit is maximized by separating the port locations 50 mm along the xaxis and 100 mm along the y-axis. This result coincides withConclusions and DiscussionProviding a couple of surgeons the level of access, manipulability, dexterity of the surgical site, as well as the visual views of it via robotic technology requires at least fourrobotic arms and two stereo cameras rendering the surgicalsite. The core of this research was to optimize the design offour surgical robotic arms to maximize the shared workspacewhile both maximizing the manipulatable factors and stiffness, and minimizing their footprint. Given the generic nature of the surgical robotic system, its design did not targetany specific anatomical structures or surgical procedures.The design parameters of the system can be divided intotwo groups (1) design parameters that are fixed following thefabrication of the robotic arms, i.e., angular link lengths, and(2) design parameters that are changeable at any point duringthe operation of the system, i.e., positions and orientationsof the individual robotic arms, as well as the relationship between them, i.e., spacing between the bases and the relativeorientation to each other and the surgical site.The cost function for optimizing the design accountsfor geometry kinematics and stiffness parameters. The effect of each parameter was studied individually followed bythe brute force search across the range of all the parameters.The effects of the individual parameters on the isotropy, linklengths and base orientation are as follows.JMR-15-1292, Zhi LiDownloaded From: me.org/ on 09/13/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use7

Journal of Mechanisms and Robotics. Received October 07, 2015;Accepted manuscript posted July 14, 2016. doi:10.1115/1.4034143Copyright (c) 2016 by ASME0200 1040010φz (deg)3003000200 1010020500500104000300z40010φy (deg)20500φ (deg)20200 10100100 20 200φx (deg) 20 20200φx (deg) 20 2000φy (deg)(b)20(c)ited(a)20pyedFig. 12: Cmax is plotted as a function of various base orientations (φx , φy , and φz ).3002002000 200 2000200x 2000200400tNx400(a) Top view.0 100 200ot 200Coz1000zy200(b) Front view. 2000y200(c) Side view.scripFig. 13: The top, front and side views of the four Raven IV arms (unit: mm).AcceptedManuIsotropy: The analytical derivation of the system showsthat the mechanism isotropy performance of a Raven armdepends on a 2 2 sub-matrix of the 3 3 Jacobian matrixfor the end-effector positioning (i.e., θ1 , θ2 and d4 ) oncethe Jacobian matrix is expressed in the coordinate of thetool’s shaft. Given the spherical shape of the mechanism,the isotropy is a function only of the elbow joint. The maximal and minimal values are functions of the two link lengths.Bounding the mechanism isotropy ensures high performanceof the entire system. An increase of the minimum acceptablevalue of the isotropy leads to a smaller common workspace.However, the overall performance criteria is maximized oncethe minimal isotropy is set to 0.5.Link Lengths: The first two links of the mechanism wereoptimized. Given the spherical geometry of the mechanism,the link lengths are expressed as angles. The kinematics ofthe mechanism is independent of the sphere’s radius. The radius is set to provide sufficient space to encapsulate the MISport. Setting the angles of the first two links to be 90 eachallows to position the end effector at the tip of the tool inserted along the radius anywhere in the sphere. However,there are two major disadvantages in setting the link angularlength to this value. First, the longer the link, the more flexible the mechanism is. Second, if the link angular length islonger, there is a higher chance of collision between the surgical robotic arms and the body of the patient. Optimizingthe mechanism for link angular length shows that as the linklength increases, the performance criterion improves; however, the best performance is accomplished when the linklengths are set to α 85 and β 65 . Setting the minimal isotropy to a value of 0.5 eliminates some combinat

(a) Surgical robot arm. Tool of the right robotic arm z 4 x 4 z 5 z x 5 6 x 6 z 4 x 4 x 3 z 3 Bottom of the tool Top of the tool (b) Surgical tool. Fig. 2: Reference frame of the Raven IV surgical robotic system. The base frame is located at the converging center of the spherical mechanism, which is formed by the first three links of a Raven .

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