IROS2019国际学术会议论文集 1362

Abstract Continuum robots composed of elastic backbones have a broad application prospect in the narrow and restricted environment because they overcome the disadvantages of traditional articulated robots such as being bulky and inflexible Statics plays an important role in the planning and control of the continuum robot composed of the elastic backbone Pseudo Rigid Body PRB theory has shown great potential in the description of flexible body statics The PRB 3R model accurately describes the large deformation of the flexible body and has high computational efficiency However PRB 3R models mostly focus on the planar static modeling and there are few applications in three dimensional 3D statics In this paper a 3D static modeling method of cable driven continuum robot based on PRB 3R theory is proposed By introducing the equilibrium constraint equations of resultant force moment and bending plane normal of the elastic backbone the state of the continuum robot is determined The 3D static equations established by the proposed method take into account the comprehensive effects of the elastic force external force gravity and friction A static verification experiment system of the cable driven continuum robot is designed to verify the proposed method The accuracy of the proposed method is verified by comparison with experimental data The maximum position error between simulation and experimental results is 7 6 I INTRODUCTION In disaster relief nuclear and radiation equipment repair toxic waste sampling minimally invasive surgery and other situations due to the small space and facing greater risk it is not suitable for people or traditional robot to work in these environments Therefore the continuum robot with a slender body and a high degree of freedom became an important choice 1 2 Since the backbone of the continuum robot is composed of elastic materials it has high adaptability to move and can avoid complex obstacles For example a continuum robot for surgery could perform minimally invasive surgery 3 greatly reducing the patient s wound size On the other hand it could reach the surgical areas that could not be reached by traditional surgical instruments 4 It is necessary to establish mathematic models before bringing continuum robots into practical applications Kinematic models of continuum robots were formulated in Research supported in part by the China Postdoctoral Science Foundation under Grant 2018M631473 in part by the National Natural Science Foundation of China under Grant 61673239 in part by the Guangdong Natural Science Foundation under Grant 2018A030310679 and in part by the Basic Research Program of Shenzhen under Grant JCYJ20170412171459177 and Grant JCYJ20160428182227081 Shaoping Huang Deshan Meng corresponding author and Xueqian Wang corresponding author are with the Graduate School at Shenzhen Tsinghua University 518055 Shenzhen China emails meng deshan and wang xq Bin Liang and Weining Lu are with the Department of Automation Tsinghua University 100084 Beijing China many previous researches 5 6 The piecewise constant curvature kinematics was widely used in the continuum manipulator 6 but the actual model was often not constant curvature 7 To establish an accurate model the effects of external forces friction and elastic force should be considered 8 9 The general theories of elasticity and energy minimization method were widely used to deal with the static problems of continuum manipulators 10 The Cosserat rod theory was used to represent the manipulator as a curve in space with relevant elasticity actuation and gravitational forces considered for a cable driven continuum robot 11 D C Rucker et al had used the Kirchhoff rod theory to build the static model of concentric tube continuum manipulators 12 The method based on the Bernoulli Euler beam model 10 was mainly used in the concentric tube continuum manipulators and the simple plane bending situation But the equations of Bernoulli Euler beam model contained the integral term which resulted in inefficient calculations and the inability to extract parameters for specific analysis 13 The principle of virtual power based on constant curvature was used in cable driven continuum manipulators to discuss the control methods of friction and driving force 14 Many researchers have carried out more detailed studies on various aspects of cable driven continuum manipulators Rucker 9 12 used the principle of virtual power to analyze the external loading but did not consider the influence of friction which led to the primary error of his model Rone 15 discussed the frictional effects of a multi segment cable driven continuum manipulator and the coupling effects between two segments However the external loading is not considered and the coupling effects are not qualitatively discussed in 15 Yuan proposed a method of analyzing the workspace of a cable driven continuum manipulator by using static analysis which takes into account the internal cable tension the external payload and the gravity force 16 Yuan also proposed a static model of continuum robots with considering multi section structure 3D deformation interactions with the external environment and the internal actuating friction 16 Moreover a PRB 3R model for a steerable ablation catheter was established to achieve the goal of implementing hybrid force position control of the catheter tip during the ablation procedure in two dimensional 2D space 17 A general 3D PRB model was established for magnetic resonance imaging actuated catheters 18 Despite the previous researchers efforts PRB models are rarely used in continuum robots That is because the PRB 3R model with high accuracy can only be used for 2D plane calculation before this paper On the other hand parameters need to be optimized in the general form of PRB which is still a great challenge in simplifying the model of continuum robots Nowadays most continuum robots work in 3D space so it is necessary to extend the PRB 3R model to 3D space for A 3D Static Modeling Method and Experimental Verification of Continuum Robots Based on Pseudo Rigid Body Theory Shaoping Huang Deshan Meng Member IEEE Xueqian Wang Bin Liang and Weining Lu 2019 IEEE RSJ International Conference on Intelligent Robots and Systems IROS Macau China November 4 8 2019 978 1 7281 4003 2 19 31 00 2019 IEEE4672 modeling At the same time the modeling of statics is an important aspect of continuum robots Some researchers have conducted preliminary research on the influence of friction on the statics of cable driven continuum manipulators 16 19 20 Based on the PRB 3R theory proposed by Su 21 the static model of a multi segment cable driven continuum manipulator was established in 2D space 22 In this paper a 3D static modeling method for continuum manipulators is established based on the PRB 3R theory The remainder of this paper is organized as follows The modeling of the cable driven continuum robot is formulated in Section II with kinematics and static analysis The experimental verification and simulation results are presented in Section III Section IV summarizes the results and describes future work Cable j j 1 2 m Cable j j m 1 m 2 2m Cable j j s 1 m 1 s 1 m 2 s m Segment 1 Disk 1 Disk n 1 Segment 2 Disk n 1 Disk 2n 1 Segment s Disk s 1 n 1 Disk s n 1 Fig 1 The schematic of a typical cable driven continuum robot II MODELING A Geometric Structure of Continuum Robots The schematic of a typical cable driven continuum robot is shown in Fig 1 The ellipses represent supporting disks the blue line represents the backbone of the continuum robot and the thin lines below represent the driving cables of each segment As shown in the figure the continuum robot has a total of s segments each segment has n subsegments and each segment is driven by m driving cables Every two disks and the backbone between them are defined as a subsegment of the continuum robot The cable holes are evenly distributed on the circle of the same radius on these disks The driving cables that control the jth segment will be fixed to the j n 1 th disk so there will be fewer cables to pass to the next segment The following sections will analyze the model based on the following assumptions Compressive and shear loads are neglected due to the incompressibility of the modeled elastic backbone compared with its bending and twist The backbone has linear super elasticity and meets the Hook s law Gravitational loading will be applied at each disk s center of mass and the mass of the backbone is neglected The deformation of the backbone can be simulated by the PRB 3R theory The assumptions 1 and 2 had been adopted in most previous researches of this type of continuum robot and had been proved to be feasible 6 14 23 Assumption 3 has been used in the previous work 22 which had sufficient accuracy in 2D space B The Pseudo Rigid Body 3R Model A PRB 1R model is enough to analyze a simple deformation where a constant force or torque is applied at the end of the beam However the direction and magnitude of the driving forces of the cable driven continuum robot are changing during the work A PRB 3R model was proposed by Su 21 to solve the static model of a flexible beam The error was less than 1 even the beam had a large deformation 21 In this theory a beam with a length of l is equivalent to four rigid links joined by three torsion springs The torques at the three torsion springs are given by 1 T 2 30 x y F l F l M J 1 where 1 2 and 3 are the torques of the three torsion springs JT is the matrix which consists of the deflection angle of three torsion springs and Fx Fy and M0 are the tip loads This PRB 3R model was previously developed in 22 to describe the static load displacement relationship of the cable driven continuum robot in 2D space As shown in Fig 2 a PRB 3R model in 3D space is established The difference between 3D and 2D is that the angle i shown in Fig 2c needs to be solved We will discuss it in Section II D Besides to get the static equilibrium equation the cable forces external loads and gravitational loads will be used to calculate the moment on the right side of 2 The frictional force and pressure between cable and cable hold will be used to calculate the 4673 1Ai 1 j Oi 2Ai 1 j 3Ai 1 j Oi 1 1Ai j 3Ai j 2Ai j xi OG xG yG zG zi 1 yi 1 xi 1 2Fi j 3Fi j1Fi j a Gi L Pi 1 Pi 2 Pi 3 Oi b Oi Oi 1 xi ni c 0L 1L 1i 2i 3i 2L 3L i Fig 2 The geometric relationship of the continuum robot a The definition of the coordination system of ith subsegment b The 3D PRB 3R model of ith subsegment c The normal vector and deflection angle of the deflection plane cable tensions The static equation for each torsion spring is established as follows cableext MMM g i ki ki ki k 2 where i 1 2 n s is the number of subsegment and k 1 2 3 is the number of the torsion spring in the Subsegment i M cable i k M ext i k and M g i k are the equivalent moment of the cable forces the external loads and the gravitational loads applied to the torsion spring respectively When the cable tensions and external force are given combining with i the equivalent PRB angles can be calculated by 2 Therefore both the configuration and the end pose of the continuum robot can be calculated C Kinematic Analysis The geometric relationship of the continuum robot is shown in Fig 2 For the sake of concision the figure demonstrates the ith subsegment with three driving cables kAi j represents the kth rope in the driving cables driving the jth segment in the ith subsegment represents the deflection angle relative to the xG axis of the first driving cable which drives the first segment Oi represents the center of the coordinate system of the ith subsegment The definition of the coordinate system is shown in the Fig 2 a where the coordinate system of the base Disk 1 coincides with the global coordinate system OG xG yG zG In Fig 2 b Pi j and i j are positions and angles of the three torsion springs respectively j represents the characteristic radius factor of the PRB 3R model Fig 2 c shows that the normal vector and deflection angle of the deflection plane are ni and i respectively Firstly we need to establish the coordinate transformation relationship from the coordinate system Oi to the coordinate system Oi 1 We define the rotation matrix R i rotating around axis z by an angle of i as shown in 3 cossin00 sincos00 0010 0001 R ii ii i 3 The transformation matrix along with the four links in Fig 2 b can be written as follows cos0sinsin 0100 sin0coscos 0001 T i ji jji j i j i ji jji j L L 4 where j 0 1 2 3 and i 0 0 Therefore the transformation matrix from the coordinate system Oi to the coordinate system Oi 1 can be expressed as follows 1 0 1 2 3 T RT T T T R i iiiiiii 5 The general expression of the coordinates of the cable holes on Disk 1 is established as follows 1 cos sin 0 A k j rr 6 where 2 j 1 m s 2 k 1 m Then the coordinates of the cable holes on i 1 th subsegment can be derived from ith subsegment as follows 11 T kik i jii j AA 7 D Static Analysis The structure of each subsegment of the continuum robot is the same Thus once the static equations of any subsegment are obtained the static formulation of the whole robot can be obtained by recursion As shown in Fig 2a the direction of the cable force can be expressed by the cable holes between the two disks Therefore the direction of the cable force can be defined as follows 11 p kkkkk i ji ji ji ji j AAAA 8 The cable force can be expressed as follows Fp kkk i ji ji j f 9 4674 where kfi j represents the magnitude of the cable force When the friction is taken into account the cable force will be changed if the subsegment is bent 22 24 The cable force can be recursively calculated from the base to the tip as shown below 1 1 kk i jij q kk iji j ffe 10 where kqi j represents the direction of the frictional force The value of kqi j is either 1 or 1 The history status of driving cables will decide the direction of the frictional force 20 is the coefficient of friction between the cable and the cable hole In particular when i 0 kf0 j is the initial force of the driving cable That is the input of the continuum robot k i 1 j is the angle between kAi 2 j kAi 1 j and kAi 1 j kAi 1 j To get the final static equation we map the applied forces of each subsegment to the center of the distal disk at the end of this subsegment Firstly we calculate the effect of the cable force All the cables of each segment will pass through the first segment and the second segment will no longer be applied by the cables of the first segment and so on Therefore we can obtain the formula of the resultant cable force and torque applied to the center of the distal disk of the ith subsegment as follows 1 FF ms Ck ii j kji n 11 11 1 AMFO ms Ckk ii j k i j ij i n 12 Secondly we define that the external force is acting on point E and the coordinates of point E can also be obtained by using kinematics relationship Therefore the external force applied to the center of the distal disk of the ith subsegment are shown as follows ee FF i 13 1 ee MFOE ii 14 Thirdly only the gravity of the disks from the ith disk to the end of the continuum robot will affect the ith subsegment Thus the gravitational loading applied to the center of the distal disk of the ith subsegment are shown as follows 1 g 1 FG s n ij j i 15 1 g 1 1 MGOO s n ijj j i i 16 Finally all the loadings applied to the center of the distal disk of the ith subsegment are shown as follows Totaleg FFFF C iiii 17 Totaleg MMMM C iiii 18 On the other hand we need to calculate the deflection angle i From Assumption 1 we can conclude that the backbone of a subsegment will keep in a plane when the continuum robot at equilibrium Based on this equilibrium condition the directions of M Total i and F Total i need to satisfy the following two conditions 1 M Total i should be parallel to the normal vector of the deflection plane 2 F Total i should be perpendicular to the normal vector of the deflection plane Then we obtain the constraint equations as follows Total 0 n M ii 19 Total 2 n F ii 20 Let us assume the spring torques are proportional to the PRB angles i e iji j K 21 where Kj is the spring stiffness Substituting 17 18 and 21 into 2 yields the static equations of the continuum robot as follows TotalTotal 1 FPOM ji jii jii K 22 Here we contribute a simplified static equation which does not have integration or differential terms and can be solved easily The static model has been implemented in MATLAB using the fsolve function to solve 19 20 and 22 III EXPERIMENTAL VALIDATION A Experimental Setup Cameras One Subsegment Marker Cable 1 Backbone xG yG zG Cable 2 Cable 3 Fig 3 The devices for experimental validation Cables are tensioned using hanging weights TABLE I THE PARAMETERS OF THE CONTINUUM ROBOT PROTOTYPE SHOWN IN FIG 3 Symbol Property Value E Young s Modulus 6 79 1010 Pa I Moment of Inertia 4 83 10 12 m4 d Routing Hole Radius 0 04 m g Gravitational Acceleration 9 785 m s2 L Backbone Length 0 25 m md Mass of the disk 4 79 10 2 kg 4675 TABLE II THE PARAMETERS OF PRB 3R MODEL Symbol Value K1 3 25EI L K2 2 84EI L K3 2 95EI L 0 0 125 1 0 35 2 0 0388 3 0 136 A simple experiment is designed to verify the model We designed a one subsegment continuum robot with two disks DSM Somos Resin and a backbone Super elastic Nickel Titanium Alloy between them Three driving cables ASTM 304 are tensioned using hanging weights in the actuation module The accuracy of the model is verified by verifying the direction angle of the bending plane and the tip s coordinate The experimental devices are shown in Fig 3 A subsegment is fixed on the desk and a marker is placed at the tip The coordinates of the marker are collected by the 3D camera system NaturalPoint Inc DBA OptiTrack with a specified accuracy of 0 20 mm RMS The camera system has been calibrated before the experiment and the global coordinate system centered on the base as shown in the Fig 3 is established The precision of the vision measuring syst