Kinematics Analyzing of a Spatial Multi-Section Continuum Robot

Nội dung text: Kinematics Analyzing of 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. 215-221, DOI 10.15625/vap.2019000281 Kinematics Analyzing of a Spatial Multi-Section Continuum Robot Duong Xuan Bien1, Chu Anh My1, Nguyen Van Cong1, Do Tien Lap1, and Tran Van Hieu2 1Le Quy Don Technical University, No. 236, Hoang Quoc Viet Street, Cau Giay District, Vietnam E-mail: xuanbien82@yahoo.com 2Engineering Intermediate School-General Technical Department Abstract of the spatial continuum robot has at least two DOFs. Over the past several years, a new class of robots, known The number of DOF is much larger than the minimum as soft robots has been studied by a number of researchers. One required to perform the task in the workspace. This of the types of the soft robots is continuum robot which has means the spatial continuum robot is a redundant system. high degree of freedom or continuous, backbone structures. In Although redundant characteristic helps robot moving this article, a spatial multi-section continuum robot with elastic backbone are considered. The forward and inverse kinematics smoothly and flexibly. This residual movement creates problem of a spatial three-section continuum robot are solved as many difficulties in kinematics and dynamics modeling. the illustrative example. Forward kinematics is the first step In particular, the first difficulty is the inverse kinematics towards solving the inverse kinematics and dynamics problem. problem. The kinematic modeling and the inverse Inverse kinematics problem plays important role in designing kinematic problem are presented for a spatial continuum the control system for robots. robot based on D-H techniques [18]. Inverse kinematics is solved by using the optimal algorithm SQP for a class Keywords: Kinematic modeling, continuum robots, forward kinematics, inverse kinematics. of continuum bionic handling arm [19]. The modeling approach is inspired from a model of a hyper-redundant backbone-based manipulator. Newton Rapson iterative 1. Introduction and Damp Least Square method are used to solve inverse Soft and continuum robots driven by tendons or kinematic directly using forward kinematic model [20]. cables have wide-ranging applications. Smooth and A heuristic approach to iteratively solve the inverse complex motion, continuum robots can change the shape kinematics problem of a continuum robot is proposed and compatibility when interacting with human. They [21]. Some simulation results show that the algorithm is have great potential in medicine [1], [2]. highly effective in computing with different topologies. A continuum robot which uses an elastic backbone In this paper, a spatial multi-section continuum robot consisting of three circular sections is shown in Fig. 1. It is considered. The kinematics modeling of this robot is is controlled by secondary backbones or driving cables presented through building the kinematic equations of [3], [4], [5], [6], or tendon [7], [8], [9], [10], [11]. the backbone. Based on these equations, the forward and inverse kinematics problem of spatial three-section continuum robot are analyzed. The inverse kinematics is solved using the Jacobian method with pseudoinverse matrix combining the closed-loop algorithm at the velocity level. The results of this research can be used for considering the other continuum robot such as varied multi-section, more numbers of degree of freedom and build the dynamics equations and design the control Figure 1. A spatial three-section continuum robot system. The kinematic and dynamic equations of continuum 2. Kinematic modeling robots are built by using the Hamilton equations [12], Elliptical integration [13], Cosserat beam theory [3], [7], Firstly, considering the two-section continuum robot as [14], [15], [16], Virtual power [8] or Ritz-Galerkin [17]. Fig. 2. Assume that the curvature is constant within each The continuum robots have complex structure and section. Where, ()OXYZ is the fixed coordinate many degree of freedom (DOFs). For example, a section 0
2. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap, and Tran Van Hieu system. The ()O XYZ and ()O X Y Z are the local 1 1 2 (1cos)cos−   11 coordinate systems which are attached to the section 1 1 1 and section 2, respectively. The origin of local frame is rT011111 r==− P (1cos)sin (3) 1 always at the first point of each section. Axes ()OZ 1 1 sin 1  1 and ()OZ 2 are tangent to the arc of the corresponding where T1 is the rotational matrix around ()OZ 1 sections. The contact point is the origin of frame. C1 cossin0− 11 T sincos0 and C2 are centre of curvatures of arcs section 1 and 2. 111=  (4) 001 These centres are always on the local ()OX and 1 Similarly, the position vector of end-effector point (point ()OX 2 . The other axes are determined by the right-hand E) in frame ()O X Y Z 2 is calculated as rule. T 11 r =− (1cos)0sin (5) 222E  22 To determine the position vector of end-effector point in fixed frame ()O X Y Z 0 , the steps are executed as follow - Rotate the frame ()OXYZ 2 around axis OZ2 with angle 2 , we have the rotational matrix cos− sin 0 22 T2= sin 2 cos 2 0 (6) 0 0 1 - Rotate the frame received around axis OY with angle Figure. 2. A two-section backbone of a spatial continuum robot 1 The parameters of bending arc are related to each other  , the rotational matrix is described as according to formula as below 1 (t )s== (t)(t);i1,2,3 (1) cos0sin iii 11 T1Y = 010 (7) Where s ( t ) is the length of arc i . The position vector − sin0cos i 11 of the section 1 endpoint (point P) in the local frame - Rotate the new frame around axis OZ1 with angle 1 ()OXYZ is determined as 1 follow the matrix in (4). T The position vector of end-effector in fixed frame is 11 r111P =− (1 cos ) 0sin (2) calculated as  11 T rrTT=+= T r rrr (8) 02011 1 2 02020202yxyz The position of this point in the fixed frame ()OXYZ 0 Specific coordinates can be shown as
3. Kinematics Analyzing of a Spatial Multi-Section Continuum Robot 111 r0211112122121x =−+−−+(1cos)cos(coscoscossinsin)(1cos)sinsinco s 122 111 r0211112122121y =−+−−+(1cos)sin(cossincoscossin)(1cos)sinsinsi n (9) 122 11 1 r021122z =−− sinsincos(1cos) + cossin12 12 2 With the similar analysing, the end-effector coordinates of a spatial three-section continuum robot in ()O X Y Z 0 can be determined as (10) Consider a spatial n -section continuum robot, the 3.1. Forward kinematics analyzing Forward kinematics is the first step towards solving the position vector of end-effector in ()OXYZ is given as 0 inverse kinematics and dynamics problem. The lengths of three sections sequence are rrTT00(1)1Enynyn=+ 121 TTTT nE−−− (n 1) r (11) LmLmLm123===0.3();0.5();0.7() and 3. Kinematic analyzing Lm= 0.7() . The joint variables of sections are given In this section, the forward and inverse kinematics 03 problem of spatial three-section continuum robot are as (12). Fig. 3, Fig. 4 and Fig. 5 show the values of analyzed. curvature of three sections, respectively. Sectiont1 :1 mt 0.2= sin( radt +== ) (rad );0.5 cos(−1 ) ( ); sin( ) ( ) 111 222 Sectiont2 :0.8 mt 0.1sin(= radt +== ) rad ( );0.5 cos(−1 ) ( ); 0.5 sin( ) ( ) (12) 222 333 Sectiont3 :0.5 mt 0.1sin(= rad +== ) ( );0.3 cos(−1 ) ( ); 0.3 si n(t ) ( rad ) 333 444 Figure 3. The curvature values of three sections Figure 4. The bending angle values of three sections
4. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap, and Tran Van Hieu the joint variable vector for a n -section spatial continuum robot as T qqqq() t = TTT (13) 12 n where, q ==  ;1in is the joint iiii variables vector of section i . The forward kinematic equations can be given as xq= f () (14) The differential kinematics equation is described as Figure 5. The orientation angle values of three sections x J= q q () Fig. 6 shows the diagram to solve forward kinematics (15) problem for continuum robots. The results are the positions and velocity of the endpoint which are The Jacobian matrix Jq() with size 33 n is described in Fig. 7 and Fig. 8. presented JJJJJ 111213141(3) n Jq() = JJJJJ212223242(3) n (16) JJJJJ 313233343(3) n The multi-section spatial continuum robot is the Figure 6. Solving forward kinematics problem in SIMULINK redundant system. The inverse kinematic problem for a redundant robot has multiple solutions in general. Due to the non-square Jacobian matrix for 3n DOFs robot, the basic inverse solution to (15) is obtained by using the pseudoinverse J* of the matrix J and the inverse solution can then be written as [22] qJqx= *() (17) where, the pseudoinverse J* can be computed as Figure 7. The endpoint position in workspace JJJJ*1= TT()− (18) A common method of including the null space in a solution is the formulation in [22], [23] and the general inverse solution can be described as qJq=+− xIJq()(() J ()) qq 0 (19) where, I is the unit matrix with size 33nn and q0 is the initial joint vector. Open-loop solutions of joint variables through numerical integration unavoidably lead to errors in workspace [22]. In order to overcome these drawbacks, the closed-loop algorithm is used based on Figure 8. The endpoint velocity in workspace the path error e in workspace between the desired and 3.2. Inverse kinematics analyzing actual path. Consider the location error e and its derivative e which can be given as Assume that the path xxdd(tt ), ( ) in workspace is e= xdd − x; e = x − x (20) given. The goal is to find the joint variables in joint The generalized closed-loop inverse kinematic algorithm space qq(tt ), ( ) that reproduce the given path. Define can be expressed by [22], [23]
5. Kinematics Analyzing of a Spatial Multi-Section Continuum Robot * algorithm is given as qJqxKxx=+−()(())dpd (21) qJqxKxxIJqJ=+−+− ()(())(()()) qq (23) Combining (15) and (21) together, there will be dpd 0 Perform the algorithm in MATLAB/SIMULINK e K+= e 0 (22) p environment as Fig. 9. where, Kp is a symmetric positive definite matrix. Combining (19) and (22) together, the inverse kinematic solution of redundant robot based on closed-loop Figure. 9. Inverse kinematic solution in MATLAB/SIMULINK Apply the inverse kinematic algorithm for a spatial three-section continuum robot with the desired path as xtm=+0.60.2 sin( )() E ytmE =+0.10.5 cos( )() (24) ztm=−0.40.1cos( )() E The desired position and velocity of end-effector are shown as Fig. 10 and Fig. 11. The simulation results are described from Fig. 12 to Fig. 15. Fig. 12 shows the curvatures of sections. The maximum curvature of Figure 10. The desired endpoint path in the workspace sections sequence are 2.25()m−1 , 2(m−1 ) and 1.5(m−1 ) . The larges the curvature value, the smaller the radius of arc value. Fig. 13 describes the bending angle in the bending plane OXZ , respectively. The maximum bending angle of sections are 0.4(),1.7()radrad and 0.25()rad . The orientation angles of sections are shown in Fig. 14. The location errors values of end-effector on axes (OX )00 ,(OY) and ()OZ 0 are presented in Fig. Figure 11. The desired endpoint velocity in the workspace 15.
6. Duong Xuan Bien, Chu Anh My, Nguyen Van Cong, Do Tien Lap, and Tran Van Hieu analyzing kinematic modeling of the spatial multi-section continuum robots based on some specific assumptions. The forward and inverse kinematics problem of a spatial three-section continuum robot are solved as the illustrative example. The simulation results of inverse kinematic applying closed-loop algorithm for a three-section spatial continuum robot show high efficiency and small errors. The results of this research can be generalized for other continuum robot such as varied multi-section robot, more numbers of degree of freedom and build the dynamics equations and design Figure 12. The curvature values of three sections the control system. 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] B. K. Jessica, D. C. Rucker (2015) Continuum Robots Figure 13. The bending angle values of three sections for Medical Applications: A Survey. Transactions on robotics, DOI: 10.1109/TRO.2015.2489500 [2] M. B. Wooten, I. D. Walker (2018) Vine-Inspired Continuum Tendril Robots and Circumnutations. Robotics, vol. 7, no. 58, pp. 2-16. [3] F. Renda, M Giorelli, M. Calisti, M Cianchetti, C. Laschi (2014) Dynamic Model of a Multibending Soft Robot Arm Driven by Cables. IEEE Trasactions on robotics, pp. 1-14. [4] Y. Liu, J. Chen, J. Liu (2018) Nonlinear mechanics of flexible cables in space robotic arms subject to Figure 14. The orientation angle values of three sections complex physical environment. Nonlinear dynamic. [5] Y. Han, L. Zhou, W. Xu (2019) A comprehensive static model of cable-driven multi-section continuum robots considering friction effect. Mechanism and Machine Theory, vol. 135, pp. 130-149. [6] Amouri, C. Mahfoudi, S. Djeffal (2019) Kinematic a nd Dynamic Modeling and S imulation Analysis of a cable-driven continuum robot. Computational Methods and Experimental Testing in Mechanical Engineering, pp. 27-37. [7] D. C. Rucker, R. J. Webster (2011) Statics and Dynamics of Continuum Robots with General Tendon Figure 15. The path error of endpoint in workspace Routing and External Loading. Transactions on robotics, vol. 27, no. 6, pp. 1033-1044. 4. Conclusion [8] S. R. William, P. B. Tzvi (2014) Continuum Robot The motion of continuum robots is generated Dynamics Utilizing the Principle of Virtual Power. through the bending of the robot over a given section Transactions on robotics, vol. 30, no. 1, pp. 275-287. unlike traditional robots where the motion occurs in [9] L. Zheng, W. Liao, R. Hongliang, Y. Haoyong (2017) discrete location, i.e., joints. This paper considers Kinematic comparison of surgical tendon-driven
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