On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot

Nội dung text: On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 222-229, DOI 10.15625/vap.2019000282 On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap Le Quy Don Technical University, No. 236, Hoang Quoc Viet Street, Cau Giay District, Vietnam E-mail: xuanbien82@yahoo.com Abstract The backbone is an elastic rod which is divided into In this article, a spatial multi-section continuum many serial arcs called sections. All sections are robot with elastic backbone is considered. The movement characterized by parameters which are length of arc, of backbone is driven by cable wires through the main. curvature and bending angle. To reduce the complexity The kinematic characteristics of the continuum robot are when analyzing kinematics, the value of the curvature is demonstrated by modeling the kinematics of its flexible assumed constant along each arc [18]. The motion of backbone. The relationship between the joint variables in continuum robots is executed by motors through the joint space and the position variables in workspace is cables. The lengths of driving cables are calculated based shown by building the kinematic equations based on on the variety of joint variables on each section and the some specific assumptions. The lengths of driving cables geometries of support disks. The properties of continuum of each section are calculated based on the results of the robots are examined where in the lengths of bending inverse kinematics problem. The geometries of support segments can be adjusted in [19]. The ability to various disks are considered while calculating the lengths of section lengths would allow continuum robots to assume driving cables. The results of this research can be used to a significantly wider range of configurations than in design the control system. existing designs, expanding their potential range of applications. The kinematic modeling and the inverse Keywords: Kinematic modeling, continuum robots, kinematic problem for a spatial continuum robot first cables-driven length, inverse kinematics type are presented based on D-H techniques in [20]. Particle Swarm Optimization is used to find the optimal 1. Introduction joint variables for the inverse kinematics of a two-section planar continuum robots in [21]. These In the field of robotics, because of their dexterity, authors then investigated model of parallel robots used continuum robots become increasingly popular and is for the computation of the link’s length in [22] to solve found in a wide range of applications such as minimally inverse kinematics. The endpoint coordinates of each invasive surgery, escape works, pipeline engineering and bending section are determined using a metaheuristic military. Continuum robots are designed based on method. Authors in [23] introduced, described and tested mimicking locomotion mechanisms of soft bodies a novel design of continuum robot which has a existing in the nature such as octopus [1], [2], [3], [4], twin-pivot compliant joint construction that minimizes [5], snake [6], and trunk of elephant [2]. Continuum the twisting around its axis. A kinematics model is robots have great potential in medicine [7], [8]. There are introduced which can be applied to a wide range of two types of continuum robot structures which are twin-pivot construction with two pairs of cables per focused in research: rigid structures and soft structures section. According to this model, the approach for that are depending on the fabricated materials of each minimising the kinematic error is developed. The length part of the robot. The first type is to use rigid parts of driving cables was considered. The inverse kinematics connected together or have parallel structures [1], [9]. for a tendon-driven continuum robot is derived based on The second type of continuum robots uses an elastic an developed open-loop controller and 3D joystick in backbone consisting of many circular sections. They are [17]. The performance of the open-loop system was controlled by secondary backbones or cable wires [10], evaluated by drawing basic geometrical shapes and [11], [12], [13], or tendon [8], [14], [15], [16], [17]. This tracking recorded errors. A tendon-driven continuum type of continuum robot is the subject of research in this robot with two joints and passive flexibility is presented paper. in [24]. The inverse kinematic problem is solved based
2. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap on the protopyte physical structure and closed-loop systems which are attached to the respective sections. controller. Authors in [9] presented a novel approaches The origin of local frame is always at the first point of of kinematic modeling of the wire-driven universal joint each section. Each axis ()OZ is tangent to the arc of continuum robot which has variable backbone hardness. i The kinematic and dynamic modeling of a planar driving each section at the origin of frame.C is the centre cables single-section continuum robot is concerned in i [13]. A comprehensive static model of driving cables curvature of each arc section i . These centres are multi-section continuum robot is investigated always on the local ()OX . The other axes are considering friction effect in [12]. The kinematics i modeling of endoscopic robot is described based on Lie determined by the right-hand rule. Define some symbols group theory [18]. for section i that: In this paper, a spatial multi-section continuum robot The backbone length is L , the curvature is  , the is considered. The kinematics modeling of this robot is 0i 0i presented through building the kinematic equations of bending angle is  , the orientation angle is  . The the backbone. The inverse kinematics problem of spatial i i three-section continuum robot is solved using lengths of driving cables when  = 0 are LL, closed-loop algorithm. The solutions of joint variables in i ii12 inverse kinematics are the input data to calculate the and L . The radiuses of arcs are RR, and R . lengths of driving cables of each section. The results of i3 ii12 i3 this research can be used to design the control system. The deviational lengths of driving cables are L , L and L . The radius of circles through 2. Kinematic modeling ii12 i3 Consider the three-section continuum robot in Fig. 1. the center of holes which are used to thread the driving Each section is driven by three cables. To reduce the cables is r . The parameters of bending arc of section i complexity of the kinematic modeling, assume that the i curvature is constant within each section. are related to each other according to formula as below iii(t )s== (t)(t);i1,2,3 (1) Where si ( t ) is the length of arc i . The position vector of the section 1 endpoint in the local frame ()OXYZ 1 is determined as T 11 r =− (1cos)0sin (2) 111  11 The position of this point in the fixed frame ()OXYZ 0 1 (1− cos11 )cos 1 1 r= T r = (1 − cos )sin (3) 01 1 1 1 1 1 1 sin   1 1 where T is the rotational matrix around ()OZ Figure 1. A spatial three-section continuum robot 1 1 Where, ()OXYZ 0 is the fixed coordinate system. The frames (OXYZ )i ; i = 1,2,3 are the local coordinate
3. On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot cossin0− cos0sin 11 11 T111= sincos0 (4) T1y = 010 (7) 001 − sin0cos 11 Similarly, the position vector of endpoint of section 2 - Rotate the new frame around axis OZ1 with angle (O3 ) in frame ()OXYZ 2 is calculated as  following the matrix in (4). T 1 11 r =− (1cos)0sin (5) The position vector of endpoint of section 2 in fixed 222  22 frame is calculated as To determine the position vector of end-effector point of r02=+ r 01 T 1 T 1y T 2 r 2 (8) section 2 (O ) in fixed frame ()O X Y Z , the steps are 3 0 Similarly, the position vector of endpoint of section 3 in executed as follow fixed frame is given as T - Rotate the frame ()O X Y Z around axis OZ with rrT=+= TT TT r xEyEzE (9) 2 2 0302112233 yy angle  , we have the rotational matrix 2 Specific coordinates of the end-effector coordinates of a spatial three-section continuum robot in ()O X Y Z can cos− sin 0 0 22 T2= sin 2 cos 2 0 (6) be calculated by using MAPLE software as 0 0 1 - Rotate the frame received around axis OY1 with angle 1 , the rotational matrix is described as (10) We can generalize for the spatial n -section continuum 3. Inverse kinematics and the changes in the lengths of driving cables robot, the position vector of end-effector in ()OXYZ 0 is given as 3.1. Inverse kinematics analyzing Assume that the path xxdd(tt ), ( ) in workspace is r0E=+ r 0( n− 1) TT 1 1 y T 2 T n − 1 T (n − 1) y T n r nE (11) given. The goal is to find the joint variables in joint space qq(tt ), ( )that reproduce the given path. Define
4. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap the joint variable vector for a n -section spatial deviation of length of the first driving cable can be continuum robot as calculated as T qqqq() t = TTT (12) =−=−=L()cosLLRRr  (18) 12 n 110111011111 11 where, q =    ;1in =  is the joint i i i i variables vector of section i . The forward kinematic equations can be given as xq= f() (13) The differential kinematics equation is described as x J= q q() (14) The Jacobian matrix Jq() with size 33n . The Figure 2 . The section 1 in ( O X Z )1 multi-section spatial continuum robot is the redundant 2 system. The inverse kinematic problem for a redundant Assume that the angle between the holes is . The robot has multiple solutions in general. Due to the 3 non-square Jacobian matrix for 3n DOFs robot, the deviation of lengths of all driving cables of section 1 basic inverse solution to (15) is obtained by using the given as * pseudoinverse J of the matrix J and the inverse =Lcos11111 r  2 solution can then be written as =−Lcos()r  12111 3 (19) qJqx= *() (15) 2 =+Lcos()r  13111 3 where, the pseudoinverse J* can be computed as in [7]. When the section 1 moves, the deviation of lengths of Open-loop solutions of joint variables through numerical three cables of section 2 is determined as integration unavoidably lead to errors in workspace. In =Lcos' r  order to overcome these drawbacks, the closed-loop 21211 2 algorithm is used based on the path error e in =−Lcos()' r  22211 (20) workspace between the desired and actual path. Consider 3 2 the location error e and its derivative e which can be ' =+Lcos()23211 r  given as 3 When the both section 1 and section 2 are driven, the exxexx=−=− ; (16) dd deviation of lengths of three cables of section 2 can be The inverse kinematic solution of redundant robot based calculated as on closed-loop algorithm is given as [8] =+Lcoscos212 112rr 22 qJqx=()(d + Kx p ( d − x ))( + IJqJqq − ()()) 0 (17) 22 =−+−Lcos()cos()rr 222 112 22 33(21) Where, K is a symmetric positive definite matrix. 22 p =+++Lcos()cos()rr 232 112 22 33 3.2. Calculate the lengths of driving cables Similarly, when all of sections are driven, the deviation Consider the cable wires of section 1 in Fig 2. The of lengths of driving cables of section 3 are given as L3131 =r cos  132 + r  cos  233 + r  cos  3 2 2 2 L =r cos(  − ) + r  cos(  − ) + r  cos(  − ) 3231 13 32 2 3 33 3 3 (22) 2 2 2 L =r cos(  + ) + r  cos(  + ) + r  cos(  + ) 3331 13 32 2 3 33 3 3 Expanding with n sections, the deviation of lengths of driving cables of section n can be shown as
5. On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot Ln1 =r n 1 cos  1 + r n  2 cos  2 + + r n  n cos  n 2 2 2 L =r cos(  − ) + r  cos(  − ) + + r  cos(  − ) n2 n 1 13 n 2 2 3 n n n 3 (23) 2 2 2 L =r cos(  + ) + r  cos(  + ) + + r  cos(  + ) n3 n 1 13 n 2 2 3 n n n 3 Fig. 3 describes the geometries of constant support varied cross-section. The nine holes on disks of section 1 disks for three sections. Each section has three driving are distributed on three different circles which have the cables. So, the support disks of section 1 have nine holes radius r r,, r . which are evenly distributed on a circle with 1 2 3 r1== r 2 r 3 . Fig. 4 shows support disks which have the Figure 3. The constant end disk structure of three sections Figure 4. The varied end disk structure of three sections radius of circles are rrrmm=== 10() . Fig. 5 3.3. Numerical simulation 123 Apply the inverse kinematic algorithm for a spatial shows the diagram to solve the inverse kinematics and three-section continuum robot with the desired path as calculate the length of driving cables of sections in SIMULINK. x=+0.6 0.2 sin( t ) ( m ) E yE =+0.1 0.5 cos( t ) ( m ) (24) z=−0.4 0.1cos( t ) ( m ) E The backbone lengths of three sections are Lm0102== Lm0.3( );0.5( ) and Lm03 = 0.7( ). The
6. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap Figure 5. The calculated diagram of the inverse kinematics and the length of driving cables in SIMULINK The desired position and velocity of end-effector are shown as Fig. 6 and Fig. 7. The simulation results are described from Fig. 8 to Fig. 11. Fig. 8 shows the curvatures of sections. The maximum curvature of sections are 2 .2 5 (m ) −1 , 2(m ) −1 and 1 .5 (m ) −1 . The large the curvature value, the smaller the radius of arc value. Fig. 9 describes the bending angles in the bending planes OX Z . The maximum bending angle of sections are 0.4(),1.7()radrad and 0 . 2 5(r ) a d . The Figure 8. The curvature values of three sections orientation angles of sections are shown in Fig. 10. The location errors values of end-effector on axes (),(OY)OX 00 and ()OZ 0 are presented in Fig. 11. Figure 9. The bending angle values of three sections Figure 6. The desired endpoint path in the workspace Figure 10. The orientation angle values of three sections Figure 7. The desired endpoint velocity in the workspace
7. On the Lengths of Driving Cables in a Spatial Multi-Section Continuum Robot Fig. 12, Fig. 13 and Fig. 14 show the length of each driving cables on sections. In the Fig. 12, the cable 1 of section 1 has the maximum length change with 3 . 5(mm ) . Fig. 13 shows that, the cable 3 of section 2 has the maximum length change with 2 1(mm ) . Similarly, the cable 1 of section 3 has the maximum value with 2 2(mm ) . Figure 11. The path error of endpoint in workspace 4. Conclusion Continuum robots commonly do not contain any actuators in the robot structure itself, making them relatively compact and lighweight. Opposite, the motion transfer problem for continuum robots is becoming a challenge. In this paper, a spatial three-section continuum robot is mentioned. The motion of continuum robots is executed by motors through the driving cables. The lengths of driving cables are calculated based on the variety of joint variables on each section and the Figure 12. The deviation of lengths of driving cables of geometries of support disks. Simulation results have first section important meaning in designing the control system and selection the actuators. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.04-2017.09. 5. References [1] G. Robinson, J. B. C. Davies (1999) Continuum robot – A state of art. Proceedings of the IEEE International Figure 13. The deviation of lengths of driving cables of conference on robotics and automation Detroit second section Michigan, USA. [2] M. W. Hannan, I. D. Walker (2003) Kinematics and the implementation of an Elephant's trunk manipulator and other continuum style robots. Journal of robotics, vol. 20, no. 2, pp. 45-63. [3] T. Zheng, D. T. Branson, G. Emanuele and D. G. Caldwell (2011) A 3D Dynamic Model for Continuum Robots Inspired by an Octopus Arm. Proceedings of the IEEE International conference on robotics and automation, Shanghai, China. [4] R, Kang, D. T. Branson, G. Emanuele, D. G. Caldwell (2012) Dynamic modeling and control of an octopus inspired multiple continuum arm robot. Computers and Figure 14. The deviation of lengths of driving cables of Mathematics with Applications, vol. 64, pp. third section 1004-1016.
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