Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms

Nội dung text: Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (2019), pp. 141 – 155 DOI: KINEMATIC AND DYNAMIC ANALYSIS OF A SERIAL MANIPULATOR WITH LOCAL CLOSED LOOP MECHANISMS Chu Anh My∗, Vu Minh Hoan Le Quy Don University of Technology, Hanoi, Vietnam ∗E-mail: mychuanh@yahoo.com Received: 05 September 2018 / Published online: 01 April 2019 Abstract. This paper addresses the kinematic and dynamic modelling and analysis for a robot arm consisting of two hydraulic cylinders driving two revolute joints of the arm. The two cylinders and relevant links of the robot constitute two local closed kinematic chains added to the main robot mechanism. Therefore, the number of the generalized coordinates of the mechanical system is increased, and the mathematical modelling is more complex that requires a formulation of constraint equations with respect to the local closed chains. By using the Lagrangian formulation with Lagrangian Multipliers, the dynamic equations are first derived with respect to all extended generalized coordinates. Then a compact form of the dynamic equations is yielded by canceling the Multipliers. Since the obtained dynamic equations are expressed in terms of independent generalized coordinates which are selected according to active joint variables of the arm, the equations could be best suitable for control law design and implementation. The simulation of the forward and inverse kinematics and dynamics of the arm demonstrates the motion behavior of the robot system. Keywords: hydraulic robot; robot kinematics; robot dynamics; local closed mechanism. 1. INTRODUCTION Most of industrial manipulators commonly used in industries are usually actuated by electric motors, such as the welding robots, the assembly robots, etc. The use of elec- tric motors actuating active robot joints possesses several advantages: easy to control, high positioning accuracy, and high flexibility. However, if a manipulator is designed to operate in a large workspace with high loading capability, the use of electric motors for the design could lead to a very heavy architecture of the robot. Counterweights could be added to balance to shaking forces. In that case, hydraulic cylinders driving robot joints is often used for the design. The presence of hydraulic cylinders increases the stiff- ness of robot structure so that the robot is capable of handling heavy parts in a larger operational space. Moreover, the counterweights could be avoided since the cylinders actuating revolute joints play a role of auxiliary links appended to the main structure. c 2019 Vietnam Academy of Science and Technology
2. 142 Chu Anh My, Vu Minh Hoan Though there will be advantages when using hydraulic cylinders for robot designs, the presence of cylinders in a robot architecture involves complex procedures for the math- ematical modelling, analysis and control. The addition of cylinders to the conventional serial kinematic chain of robot arm architecture could constitute local closed kinematic chains within the entire robot mechanism. Issues related to the hybrid serial–parallel fea- ture of the robot structure, the geometry, the mass and the inertia of cylinders must be taken into account. In the literature, numerous works have been carried out to investigate several as- pects related to the dynamic modeling and analysis of serial manipulators and paral- lel robots [1–21]. The fundamentals of kinematic and dynamic modelling and analysis of serial manipulators can be found in [1,2], where Denavit-Hartenberg approach was mostly used for the kinematic modelling and D’Alembert–Lagrange Formulation for the dynamic modelling. As for more complex robotic systems, there has been a number of researches dealing with different issues related to the kinematics and the dynamics as well. The research presented in [3] studied the kinematic and dynamic modelling for closed chain manipulators, [4] addressed algorithms for the dynamic analysis of serial robots having a large number of joints, [5] investigated the inverse kinematics and dy- namics of the redundant robots, [6] studied the dynamics of mobile serial manipulators. In parallel with researches concerning with the serial robot dynamics, a massive number of researches related to the dynamics and control of parallel robots has been addressed as well such as publications [7–21]. The researches [8,10,13] investigated methods for the inverse and forward dynamic modelling and analysis of the 3-PRS type parallel manipu- lators. The Screw theory was used in [9], and the matrix approach was employed in [11] for the dynamics and control of the parallel robots. A general solution to the problem of dynamic modelling and analysis of parallel robots was presented in [12]. For the issue of control law design and development, [14] investigated a model-based technique, [15–17] addressed the robust control algorithms, whereas the sliding mode control was designed in [18]. The earlier foundation for control law designed for the parallel robots one can find in [19], in which the computed torque technique was used as for those of the common serial manipulators. The control law design for a 6-DOF hexapod robot was presented in [20], and a technique to improve the control quality for parallel robots was investigated in [21]. In recent years, there have been several attempts to investigate the dynamic mod- elling and analysis of the hybrid serial–parallel robots [22–31]. Ibrahim & Khalil [22] studied recursive solutions for obtaining the inverse and direct dynamic models of hy- brid robots that are constructed by serially connected non-redundant parallel modules. Pisla et al. [23] addressed geometric and kinematic models of a surgical hybrid robot used for camera and active instruments positioning. Tanev [24] investigated a kinematic analy- sis of a hybrid parallel–serial robot manipulator which consists of two serially connected parallel mechanisms. Each mechanism has three degrees of freedom. Xu et al. [25] pre- sented a particular hybrid manipulator for computer controlled ultra-precision freeform polishing. This hybrid manipulator is composed of a 3-DOF parallel module, a 2-DOF serial module and a turntable providing a redundant DOF. Zeng & Fang [26] studied a method of structural synthesis and analysis of serial–parallel hybrid mechanisms with
3. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 143 spatial multi-loop kinematic chains based on the displacement group theory. Zeng & Ehmann [27] presented a design method for parallel hybrid-loop manipulators, which is based on the constrained motion properties of related spatial over constrained linkages and general parallel mechanisms. Zhang & Gao [28] investigated a method for perfor- mance optimization of a 5-DOF compliant hybrid parallel robot micromanipulator. The issues related to kinematics modelling and analysis for the robots were also addressed in [29–31]. Though there has been a number of researches dealing with the dynamic modelling and analysis of different robot architectures, little attention has been paid to robots actu- ated by hydraulic actuators. Issues related to the mathematical modelling of manipula- tors which have local closed kinematic chains constituted by adding hydraulic cylinders seem to be overlooked. In this paper, the dynamic modelling and analysis of such a robot arm having two hydraulic cylinders constituting two local closed chains appended to a conventional arm mechanism are addressed. First, the kinematic equations and constraints equations are derived, by using the notation of Denavit–Hartenberg. Second, Lagrange’s formulation is used to establish differential equations of motion, with the use of Lagrangian multipli- ers. Finally, based on the formulated DAEs, a technique of Lagrangian multipliers can- cellation is employed to obtain ODEs describing the dynamics of the entire robot system. Finally, the dynamic analysis is carried out with the help of the symbolic and numeric computing Maple and Matlab softwares. As a result, the proposed investigation could be used further for the purposes of dynamic simulation and control law design for a class of hydraulic architecture robots. 2. KINEMATICS Let’s consider a robot in Fig 1. Different from the conventional robot arm architec- ture, the considering manipulator has two hydraulic cylinders. The first cylinder AC is to drive the revolute joint B (joint variable θ2), and the second one EG drives the joint F (joint variable θ5). Due to the use of the cylinders, two closed kinematic chains are constituted: ABCD and EFGH. The robot have three DOFs and three active joints. The first active joint is a revolute joint (joint 1) with joint variable θ1. The second and the last active joints are prismatic joints (with the joint variables d4 and d7) because of the relative motion of the two parts of the hydraulic cylinders. Therefore, among eight joint variables θ1, d4, d7, θ2, θ3, θ4, θ5 and θ6 (see Fig. 1), only three joint variables θ1, d4 and d7 are independent. In Fig.1, Oxyz is defined as a reference frame, and OExEyEzE as an end effector frame, of which the origin locates at point E. Other local coordinate systems attached to links are defined by using Denavit–Hartenberg (DH) notations. l1, l10, l11, l2, l21, l22, l23, l3, l5, l51, l6 are geometric parameters of links. The main mechanism of the arm includes three links: link 1, link 2, and link 5. The first local closed chain (ABCDA) involves link 1, link 2, link 3 and link 4. The second local closed chain (KFGHK) relates to link 2, link 5, link 6 and link 7. Based on all the coordinates in Fig.1 which are defined with respect to the DH convention, the parameters of the robot links are presented in the following tables.
4. 144Kinematic and dynamic analysis Chuof a Anhserial My, manipulator Vu Minh Hoan with local closed loop mechanisms 3 Fig 1. A hybrid serial-parallel robot arm Fig. 1. A hybrid serial-parallel robot arm In Fig. 1, O0x0y0z0 is defined as a reference frame, and OExEyEzE as an end effector frame, of which the origin locates at point E. Other local coordinate systems attached to links are defined by using Table 1. The DH parameters re- Table 2. The DH parameters related to the main mecha- Denavit – Hartenberg (DH) notations. lllllllllll,,,,,,,,,, are geometric parameters of lated to the main mechanism 1 10 11 2 21 22 23 3 5 51nism 6 links. The main mechanism of the arm includes three links: link 1, link 2, and link 5. The first local closed chain (ABCDA) involves link 1, link 2, link 3 and link 4. The second local closed chain (KFGHK) i relatesθi to linkdi 2, linkai 5, linkα 6i and link 7. Basedi θoni all dthei coordinatesai αi in Fig. 1i whichθi ared idefinedai withα i respect to the DH convention, the parameters of the robot links are presented in the following tables. 1 θ 1 Tablel 11. The0 DH parametersπ/2 related3’ to theθ3 main0 mechanisml3 0 1’ 0 0 0 −π/2 2 θ 2 0 l2 0 3 π/2 0 0 π/2 1 0 l11 0 π/2 5 θ5 0 l5 i 0 4 0 d4 0 −π/2 2 θ2 0 l22 0 1 1 l1 0 / 2 2 0 l 0 Table 3. The DH parameters2 2 related the local closed chain KFGHK 5 5 0 l5 0 i θ d a α i θ d a α Table 2.i The DH parametersi i related to thei local closed chain ABCDAi i i i 6’ θ6 0 l6 0 2 0 0 l21 0 i i 6 π/2 0 0 π/2 5 θ5 0 l51 0 3’ 3 0 l3 0 1’ 0 0 0 / 2 7 0 d7 0 −π/2 3 / 2 0 0 / 2 1 0 l11 0 / 2 4 0 d4 0 / 2 2 2 0 l22 0 Based on the parameters in Tabs.1–3, the transformation matrices describing the motion of any two successive coordinate systems can be derived accordingly. For the Table 3. The DH parameters related the local closed chain KFGHK main mechanism, the cumulative matrix H0E is computed as follows   cos θ1 cos (θ2 + θ5) − cos θ1 sin (θ2 + θ5) sin θ1 − cos θ1 (l5 sin θ2 sin θ5 − l5 cos θ5 cos θ2 − l2 cos θ2)  sin θ1 cos (θ2 + θ5) − sin θ1 sin (θ2 + θ5) − cos θ1 − sin θ1 (l5 sin θ2 sin θ5 − l5 cos θ5 cos θ2 − l2 cos θ2)  H0E =   .  sin (θ2 + θ5) cos (θ2 + θ5) 0 l5 sin θ2 cos θ5 + l5 cos θ2 sin θ5 + l2 sin θ2 + l1  0 0 0 1 (1)
5. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 145 The matrix H0E describes the kinematics of motion of one active joint and two pas- sive joints. In other words, H0E characterizes the kinematic relationship of the end link of the arm with the use of only one independent joint variable θ1 and other two dependent joint variables θ2 and θ5. In order to determine the kinematic relationship with all independent joint variables, the constraints raised due to the local closed chains need to be taken into account. 2.1. Constraint equations Consider the first local closed chain (ABCDA). In order to write constraint equations regarding of this closed feature of the mechanism, a virtual cut at the joint C is made, so that transformation matrices written with respect to two routines ABC and ADC, respec- tively, are yielded as   cos θ2 − sin θ2 0 l22 cos θ2  sin θ cos θ 0 l sin θ + l  H =  2 2 22 2 11  , (2) ABC  0 0 1 0  0 0 0 1   − sin θ3 − cos θ3 0 d4 cos θ3 + l3 cos θ3  cos θ − sin θ 0 d sin θ + l sin θ  H =  3 3 4 3 3 3  . (3) ADC  0 0 1 0  0 0 0 1   l22 cos θ2 1 1 In the first routine, the position C of the point C is C =  l22 sin θ2 + l11 , and in 0   d4 cos θ3 + l3 cos θ3 2 the second one, the point C is calculated as C =  d4 sin θ3 + l3 sin θ3 . Therefore, two 0 independent constraint equations involved in this closed chain are determined as f1 = l22cosθ2 − (d4 + l3) cosθ3 = 0, (4) f2 = l22sinθ2 − (d4 + l3) sinθ3 + l11 = 0, (l + d )2 + (l )2 − (l )2 sin (θ ) = 3 4 11 22 , 3 2 (l + d ) l 3 4 11 (5) 2 2 2 (l3 + d4) − l11 − l22 sin (θ2) = . 2l11l22 Similarly, the other two constraint equations involved in the second local closed chain KFGHK are obtained as f3 = l51 cos θ5 + l21 − d7 cos θ6 − l6 cos θ6 = 0, (6) f4 = l51 sin θ5 − d7 sin θ6 − l6 sin θ6 = 0,
6. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 5 Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 5 f1 l 22cos 2 d 4 l 3 cos  3 = 0 (4)   f 2 f1 l 22sinl22cos 2 2 d 4 d 4l 3 sin l 3  3cos + l 11 3 = 0 0 (4) f2 l 22sin 2 d 4 l 3 sin 3 + l 11 0 2 2 2 l3 d 4 l 11 l 22 sin  2 2 2 3 l 2 dl d l l l 3 34 4 11 11 22 sin 3 (5) 2 2 l 2 d 2 l 3 4 11 l3 d 4 l 11 l 22 (5) sin  2 2 2 2 l d l l 32l11 4 l 22 11 22 sin 2 2l l Similarly, the other two constraint equations involved 11in 22 the second local closed chain KFGHK are obtainedSimilarly, as the other two constraint equations involved in the second local closed chain KFGHK are obtained as f3 l 51cos 5 l 21 d 7 cos  6 l 6 cos  6 0 f lcos l d cos l cos  0 (6) 3 51 5 21 7 6 6 6 f 4 l 51sin 5 d 7 sin  6 l 6 sin  6 0 (6) f4 l 51sin 5 d 7sin 6 l 6 sin  6 0 146 Chu Anh2 My, Vu 2 Minh 2Hoan l6 d 7 2 l 51 2 l 21 2 cos  l d l l 5 6 7 51 21 cos 5 2 l l 51 21 2 l51 l 21 (7) 2 2 2 2 2 (7) l d(l + 2d l) − 2 l(l ) − 2 (l ) 6= l 76 d 517 l 21 51 l 21 cos cos 6 ( θ 5) 6 7 51 21 , cos 6 2 l l 2 d(l )(l ) 212 l 6 l 7 51 d 21 21 6 7 (7) (l + d )2 − (l )2 + (l )2 cos (θ ) = 6 7 51 21 . 6 2 (l )(l + d ) ForwardForward kinematic kinematicss simulation simulation 21 6 7 2.2. Forward kinematics simulation Based on Eqs. (1,5,7), with the inputs given as  t t , d t 0.1 0.005 t and Based on Eqs. (1,5,7), with the inputs given as1 1 t t4 , d4 t π0.1 0.005 t and Based on Eqs. (1), (5) and (7), with the inputs given18 18 as θ1 (t) = × t, d4 (t) = 0.1 + T 18T dd7 tt 0.10.1 0.02 0.02 t ,t , the the position position of of the the end end effector effector E ExEEE t x y t t y z t t zand t theand velocity the velocity T 0.0057 × t and d7 (t) = 0.1 + 0.02 × t, the position of the end E effectorEE E = [xE (t) yE (t) zE (t)] T T T EEand x x the tt y velocity y t t z z t E t˙ can=can [bex˙ be numerically( tnumerically) y˙ (t) zcomputed˙ computed(t)] can and andshown be numericallyshown in the in following the following computed Figs. Figs.(2,3), and (2,3),and shown Fig. and Fig. in the EEEE EE E E E 4,4,following respectivelyrespectively Figs . 2,3 and4, respectively . 2 2 1.8 1.8 ] m [ ] 1.6 E m z [m] z [ z 1.6 , z E y 1.4 , xE,yE,zE [m] xE,yE,zE E 1.4 x 1.2 1.2 0.1 -0.3 0.1 0 -0.3 0 -0.1 -0.2 -0.1 -0.2 -0.2 -0.1 -0.2 -0.3 -0.1 -0.3 x [m] -0.4 0 y [m] x [m] -0.4 0 y [m] T 6 Fig 2. E x t y t zMy Anh t T Chu, Hoan Minh Vu Fig 3. A trajectory of E in the workspace Fig. 2. E = [x (tE) y (tEE) z (t )]T Fig. 3. A trajectory of E in the workspace Fig 2. E Ex t yE t zE t Fig 3. A trajectory of E in the workspace EEE dxE,dyE,dzE [m/s] dxE,dyE,dzE T Fig 4. E x t y t z t E EE T Fig. 4. E˙ = [x˙E (t) y˙E (t) z˙E (t)] Inverse kinematics simulation T Given a trajectory of the end-effector in the work space, E x t y t z t , the joint E EE variables are calculated as follows: 2 1/2 1 arcsinAA / 1 2 2 (8) d4 2l11 l 22 sin 2 l 11 l 22 l 3 , d 2l l cos l2 l2 l 7 51 21 5 51 21 6 2 2 2 2 y zM l1 l 2 l 5 cos( ) M C 5 2 2l2 l 5 B2 l2 l 5 (9) mn ' ' 2 2 2 2 2 2 sin(2 ) ; m n n p m p n2 p2 2 yE A 2 where A ; B 2 ; m zE l1 ; n ( l5 C l 2 ); p l5 1 C xE 1 A Figs. 5 and 6 show the values of 1 t , d4 t , d 7 t , 2 t and 5 t computed with respect to a trajectory given as
7. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 147 2.3. Inverse kinematics simulation T Given a trajectory of the end-effector in the work space, E = [xE (t) yE (t) zE (t)] , the joint variables are calculated as follows  2
   1/2 θ1 = arcsin A/ 1 + A , q 2 2 d4 = 2l11l22 sin θ2 + l11 + l22 − l3, (8) q 2 2 d7 = 2l51l21 cos θ5 + l51 + l21 − l6, y2 (z − l )2 − l2 − l2 cos(θ ) = M + M 1 2 5 = C, 5 2l l B2 2l l 2 5 √ 2 5 (9) mn ± ∆0 sin(θ ) = ; ∆0 = m2n2 − n2 + p2
   m2 − p2
   , 2 (n2 + p2) s y A2 p = E = = ( − ) = ( + ) = − 2 where A ; B 2 ; m zE l1 ; n l5C l2 ; p l5 1 C . KinematicKi nematicand dynamicxE and dynamic analysis1 analysis+ ofA a serial of a serialmanipulator manipulator with localwith local closed closed loop loop mechanisms mechanisms 7 7 Figs.5 and6 show the values of θ1 (t) , d4 (t) , d7 (t), θ2 (t) and θ5 (t) computed with respect2 to a trajectory given as 2 2 1 1 t t 2 55 t 2 t2 t 200 t 2 2 1 1 t t 55 t 2 t t 200 t 2 x E cos 92160000 t 100 10000 cos 2560000 t 25 2000 x E cos 92160000 t 100 10000 cos 2560000 t 25 2000 8000 28000 1600 2 32 1600 64 32 64 18 18 1280000012800000 18 18 r r   2     2 +    2 1 2 1 t t 55 πt 2 t (t 200) πt 2 x E = + 2 + − cos 92160000 − (t + 100) −210000 2 − cos 2560000 − (t + 25) − 2000 2 , 2 1 1 1t 1 t t55 t 55 t t 2 2 t t t200 t 200 t t 2 2 y y 8000 2 1600 32 sin 64 sin 9216000018 92160000 t 100t 100 10000 10000 12800000sinsin18 2560000 2560000 t 25t 25 2000 2000 E E 8000 28000 1600 2 32 16002 64 32 64 18 18  r 1280000012800000 18 18 r 1 1 t t 55 πt  2 2 t (t + 200) πt  2 2 yE = + + − sin 92160000 − (t + 100) − 10000 − sin 2560000 − (t + 25) − 2000 , 8000 2 1600 32 64 2 18 2 128000004 4 3 18 3 2 2 1 1 2 2 2 2 2 2 t t t t 377t377t 23 t 23 t 7 7 z z 2560000r 2560000 t 25 t 2000 25 2000 92160000r 92160000 t 100 t 100 10000 10000 E E 1  2  2 t4 t3 377t2 23t 7 12800000z =12800000 2560000 − (t + 25)2 − 2000 92160000 − (t + 100)2 − 10000 +1280000012800000+ 51200 51200+ 512000 512000− 2560+ 2560. 10 10 E 12800000 12800000 51200 512000 2560 10 3 3 Teta T2eta 2 Teta T5eta 5 2.5 2.5 2 2 1.5 1.5 1 1 Teta1[rad],d4,d7 [dm] Teta1[rad],d4,d7 Teta1[rad],d4,d7 [dm] Teta1[rad],d4,d7 0.5 0.5 0 0 0 5 10 15 0 5 times [s] 10 15 times [s] Fig 5. The curves  t , d t and d t Fig 6. The curves  t and  t 1( ) 4 ( ) 7 ( ) ( )2 ( 5) Fig 5.Fig. The 5. curves The curves 1 t ,θd14 t t ,andd4 t d7and t d7 t Fig.Fig 6 .6. The The curves curves θ22 tt andand θ5 tt To validate the inverse kinematic computation, a numerical experiment is carried out. The robot configurations at t = 0 sec and t = 10 sec are shown in Figs.7 and8. In Fig 7. The robot configuration at t = 0 Fig 8. The robot configuration at t = 10sec Fig 7. The robot configuration at t = 0 Fig 8. The robot configuration at t = 10sec
8. KinematicKinematic and and dynamic dynamic analysis analysis of aof serial a serial manipulator manipulator with with local local closed closed loop loop mechanisms mechanisms 7 7 2 2 1 1 1 1t t t t 55 55 t t 2222t tt t200 200 t t 2 2 2 2 x x cos cos 92160000 92160000 t 100t 100 10000 10000 cos cos 2560000 2560000 t t 25 25 2000 2000 E E 80008000 2 2 1600 1600 32 32 64 64 18 18 12800000 12800000 18 18 2 2 1 1 1 1t t t t 55 55 t t 2 2 2 2 t tt t200 200 t t 2 2 2 2 y y sinsin 92160000 92160000 t 100 t 100 10000 10000 sinsin 2560000 2560000 t t 25 25 2000 2000 E E 80008000 2 2 1600 1600 32 32 64 64 18 18 1280000012800000 18 18 44 3 3 2 2 112222 2 2 2 2 tt t t 377 377tt 23 23 t t 7 7 z z 25600002560000 t 25t 25 2000 2000 92160000 92160000 t 100 t 100 10000 10000 E 12800000E 12800000 12800000 12800000 51200 51200 512000 512000 2560 2560 10 10 3 3 TetaTeta 2 2 TetaTeta 5 5 2.5 2.5 ] ] 2 2 m m d d [ [ 7 7 d d , , 4 4 1.5 1.5 d d , , ] ] d d a a r r [ [ 1 1 a a 1 1 t t e e T T 0.5 0.5 0 0 0 0 5 5 10 10 1515 timestimes [s] [s] Fig Fig5. The 5. The curves curves 1 t 1, t d 4, d t 4 and t and d 7 d t 7 t FigFig 6. 6The. The curves curves  2 2t tand and  5  5 t t 148 Chu Anh My, Vu Minh Hoan FigFig 7. The 7. The robot robot configuration configuration at tat = t 0= 0 FigFig 8. 8The. The robot robot configuration configuration at at t =t = 10s 10secec Fig. 7. The robot configuration at t = 0 Fig. 8. The robot configuration at t = 10 sec Figs.7 and8, the two revolute joints θ2 and θ5 are driven with two hydraulic cylinders which represent the active joints d4 and d7. All the parameters of the robot are shown in Figs.7 and8 as well. 3. DYNAMICS Let’s denote T s = [θ1d4d7θ2θ3θ5θ6] is the vector of redundant generalized coordinates; T q = [θ1d4d7] is the vector of independent generalized coordinates; T z = [θ2θ3θ5θ6] is the vector of dependent generalized coordinates;  T f = f1 f2 f3 f4 is the vector of constraint equations; ∂f Φ = is the Jacobian matrix of constraint equations; s ∂s  T λ = λ1 λ2 λ3 λ4 is the vector of Lagrangian Multipliers;  T τ = τ1 F4 F7 0 0 0 0 is the vector of applied torque/forces. By using the Lagrangian formulation, the system of equations of motion including constraint equations for the robot can be written as T M (s, t) ¨s + C (s, ˙s, t) ˙s + g (s, t) + Φs λ = τ (t) , (10) f (s, t) = 0. (11) The global mass matrix can be computed as [1] 7   M (s, t) = m JT J + JT A I ATJ , (12) ∑ i Ti Ti Ri i i i Ri i=1 where mi and Ii are the mass and the inertia of body i, respectively; JTi and JRi are trans- lational and rotational Jacobian matrices of body i, accordingly.
9. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 149 The angular velocity of the links are calculated as follows:    ˙   ˙  0 θ2sinθ1 θ3sinθ1  ˙   ˙  ω1 =  0  , ω2 =  −θ2cosθ1  , ω3 = ω4 =  −θ3cosθ1  , θ˙ 1 θ˙1 θ˙1 
    
    sin θ1 θ˙2 + θ˙5 sin θ1 θ˙2 + θ˙6  ˙ ˙
     ˙ ˙
    ω5 =  − cos θ1 θ2 + θ5  , ω6 = ω7 =  − cos θ1 θ2 + θ6  . θ˙1 θ˙1 The vectors of the mass center of the links are determined as follows:  1   1  l2 cos θ1 cos θ2 l3 cos θ1 cos θ3    2   2  0      1   1  rC1 =  0 , rC2 =  l2 sin θ1 cos θ2 , rC3 =  l3 sin θ1 cos θ3 , l /2  2   2  1  1   1  l sin θ + l l sin θ + l 2 2 2 1 2 3 3 10     l4   l3 + d4 − cos θ1 cos θ3 1  2  l5 cos θ1 cos (θ2 + θ5) + l2 cos θ1 cos θ2    2         l4   1  rC4 =  l3 + d4 − sin θ1 cos θ3  , rC5 =  l5 sin θ1 cos (θ2 + θ5) + l2 sin θ1 cos θ2 ,  2   2       1   l4  l + d − sin θ + l l5 sin (θ2 + θ5) + l2 sin θ2 + l1 3 4 2 3 10 2  1  l6 cos θ1 cos (θ2 + θ6) + l23 cos θ1 cos θ2  2     1  rC6 =  l6 sin θ1 cos (θ2 + θ6) + l23 sin θ1 cos θ2 ,  2   1  l sin (θ + θ ) + l sin θ + l 2 6 2 6 23 2 1   l   l + d − 7 cos θ cos (θ + θ ) + l cos θ cos θ  6 7 2 1 2 6 23 1 2         l7  rC7 =  l6 + d7 − sin θ1 cos (θ2 + θ6) + l23 sin θ1 cos θ2  .  2    l    l + d − 7 sin (θ + θ ) + l sin θ + l  6 7 2 2 6 23 2 1 The matrix of Coriolis and Centrifugal terms is calculated as [1] ∂M (s, t) 1  ∂M (s, t) T C (s, ˙s, t) = (E ⊗ ˙s) − (˙s ⊗ E ) , (13) ∂s n 2 ∂s n
10. 150 Chu Anh My, Vu Minh Hoan The potential energy of the system can be calculated as follows 1  1   1    l   = m gl + m g l sin θ + l + m g l sin θ + l + m g l + l + d − 4 sin θ ∏ 2 1 1 2 2 2 2 1 3 2 3 3 10 4 10 3 4 2 3  1   1  + m g l + l sin θ + l sin (θ + θ ) + m g l + l sin θ + l sin (θ + θ ) 5 1 2 2 2 5 2 5 6 1 23 2 2 6 2 6   l   + m g l + l sin θ + l + d − 7 sin (θ + θ ) . 7 1 23 2 6 7 2 2 6 Based on the calculation of the total potential energy of the entire system, the terms related to the gravity effect is calculated as  ∂Π T g (s, t) = . (14) ∂s In order to analyze the dynamic behavior of the robot system, the DAEs (10, 11) need to be transformed in a way that the Multipliers are cancelled. ∂f Rewrite Φ = in the following form s ∂s  ∂f ∂f  Φ =  Φ Φ  = . (15) s q z ∂q ∂z Let’s consider the following expression  T T T  −1 R = E, −Φq Φz . (16) Hence T T R Φs = 0. (17) Note that   0 0 0 0  − cos θ − sin θ 0 0   3 3   0 0 − cos θ6 − sin θ6  T   Φ =  −l22 sin θ2 l22 cos θ2 0 0  , s    l3 sin θ3 + d4 sin θ3 −l3 cos θ3 − d4 cos θ3 0 0     0 0 −l51 sin θ5 l51 cos θ5  0 0 l6 sin θ6 + d7 sin θ6 −l6 cos θ6 − d7 cos θ6 (18)
11. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 151  1 0 0   0 1 0     0 0 1     1   0 − 0   l22 sin (θ2 − θ3)   ( − )   cos θ2 θ3  R =  0 − 0  , (19)  (d4 + l3) sin (θ2 − θ3)   1     0 0   l51 sin (θ6 − θ5)   cos (θ − θ )   0 0 6 5   (d7 + l6) sin (θ6 − θ5)  ˙s = R ˙q, (20) ¨s = R ¨q + R˙ ˙q. (21) Substituting (17), (20), (21) into (10) yields M¯ ¨q + C¯ ˙q + Gq¯ = τq, (22) T T
    T where M¯ = R M (s, t) R, C¯ = R M (s, t) R˙ + C (s, ˙s, t) R , G¯ = R g (s, t) , and τq = RTτ (t). T Eq. (22) is expressed in term of independent generalized coordinates, q = [θ1d4d7] . Notice that all the formulations above are implemented and demonstrated in Maple envi- ronment. The following section shows the numeric solutions of the forward and inverse dynamic issues. 3.1. Forward dynamics simulation The dynamical parameters of the robot system are given in Tab.4 Table 4. The parameters of the robot Center of gravity Inertia Link Mass xC yC zC Ixx Iyy Izz Ixy Iyz Izx 1 0 lC1 0 m1 I1x I1y I1z 0 0 0 2 lC2 0 0 m2 I2x I2y I2z 0 0 0 3 0 0 lC3 m3 I3x I3y I3z 0 0 0 4 0 lC4 0 m4 I4x I4y I4z 0 0 0 5 lC5 0 0 m5 I5x I5y I5z 0 0 0 6 0 0 lC6 m6 I6x I6y I6z 0 0 0 7 0 lC7 0 m7 I7x I7y I7z 0 0 0 l1 = 0.7 m; l10 = 0.4 m; l11 = 0.3 m; l2 = 0.6 m; l20 = 0.2 m; l21 = 0.4 m; l22 = 0.4 m; l23 = 0.2 m; l3 = 0.4 m; l4 = 0.4 m; l5 = 1.2 m; l51 = 0.8 m; l52 = 0.4 m; l6 = 0.4 m; l7 = 0.4 m.
12. 152 Chu Anh My, Vu Minh Hoan We assume that lC1 = l1/2 = 0.35 m; lC2 = l2/2 = 0.3 m; lC3 = l3/2 = 0.2 m; lC5 = l5/2 5π 5π = 0.6 m; l = l /2 = 0.2 m; l = l /2 = 0.2 m; l = l /2 = 0.2 m; − ≤ θ ≤ ; C6 6 C4 4 C7 7 6 1 6 0.1 ≤ d4 < l4; 0.1 ≤ d7 < l7, m1 = 80 kg; m2 = 60 kg; m3 = 20 kg; m4 = 10 kg; m5 = 50 kg; m6 = 20 kg; m7 = 10 kg. 2 2 2 2 2 I1x = m1l1/12; I1z = m1l1/12; I2x = 0; I2y = m2l2/12; I2z = m2l2/12; I3x = m3l3/12; 2 2 2 2 I3y = m3l3/12; I3z = 0; I4x = m4l4/12; I4y = 0; I4z = m4l4/12; I5x = 0; I5y = m5l5/12; I5z = 2 2 2 2 2 m5l5/12; I6x = m6l6/12; I6y = m6l6/12; I6z = 0; I7x = m7l7/12; I7y = 0; I7z = m7l7/12. The applied torque/forces are given as τ1 (t) = 1.5 × sin (2t) , F4 (t) = 50 (20 + t) , F7 (t) = 30 (20 − t) . Fig.9 shows the time evolution of θ1 (t), d4 (t) and d7 (t). 0.4 teta1 0.35 d4 d7 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 times [s] Fig. 9. The time evolution of θ1 (t), d4 (t) and d7 (t) 3.2. Inverse dynamics simulation To demonstrate the inverse dynamic analysis, two cases of simulation are consid- ered. The inputs of the simulation are given as π × t θ (t) = , d (t) = 0.1 + 0.005 × t, and d (t) = 0.1 + 0.02 × t. 1 18 4 7 For the first case, the mass of the link 3, 4, 6 and 7 equals to zero. In the second case, the mass of the link 3, 4, 6 and 7 are given as m3 = 20 kg, m4 = 10 kg, m6 = 20 kg and m7 = 10 kg. Figs. 10–12 show the results of the inverse dynamic analysis. The “black” curves are the time evolution of the computed torque/forces corresponding to the first case, while the “gray” ones are for the second case.
13. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 13 0.4 teta1 0.35 d4 d7 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 times [s] Fig 9. The time evolution of 1 t , d4 t and d7 t Inverse dynamics simulation To demonstrate the inverse dynamic analysis, two cases of simulation are considered. The inputs of the simulation are given as t  t , d( t ) 0.1 0.005 t , and d( t ) 0.1 0.02 t . 1 18 4 7 For the first case, the mass of the link 3, 4, 6 and 7 equals to zero. In the second case, the mass of the link 3, 4, 6 and 7 are given as m =20 kg, m =10 kg, m =20 kg and m =10 kg. 3 4 6 7 Figs 10, 11 and 12 show the results of the inverse dynamic analysis. The “red” curves are the time evolution of the computed torque/forces corresponding to the first case, while the “green” ones are for the second case. Kinematic and dynamic analysis of a serial manipulator with local closed loop mechanisms 153 -0.006-0.006 1900 M1M1 F4 1 1 1800 1 M1M1 F4 -0.008-0.008 2 2 2 1700 -0.01-0.01 1600 1500 -0.012-0.012 1400 -0.01-0.0144 F4 [N] 1300 -0.016-0.016 1200 1100 -0.018-0.018 1000 -0.02-0.02 900 00 55 1010 1515 0 5 10 15 timestimes [s] [s] times [s] 14 My Anh Chu, Hoan Minh Vu Fig 10. 1()t Fig 11. F4 () t Fig. 10. τ1(t) Fig. 11. F4(t) 1100 F7 1 F7 1000 2 900 800 700 600 500 400 0 5 10 15 times [s] Fig 12. F4 () t It can be seen that there is a considerable changeFig. 12. ofF 7the(t )computed torque/forces when considering the mass of the hydraulic cylinders in the dynamic model of the robot system. It can be seen that there is a considerable change of the computed torque/forces when4. consideringCONCLUSION the mass of the hydraulic cylinders in the dynamic model of the robot system.The kinematic and dynamic equations for a particular type of robot have been formulated. It has shown that when the mass and inertia of a hydraulic cylinder driving a revolution joint of a robot are considered, such the cylinder and relevant links of4. the CONCLUSION robot constitute a local closed mechanism appended to the mainThe robotkinematic architecture. and dynamic By taking equations into account for this a particular feature type of of the robot hydraulic have robot, been the for- kinematic and dynamic modelling and analysis of the robot are more accurate. This could help to design mulated.more efficient It has and shown more effective that when control the laws mass for the and arm. inertia of a hydraulic cylinder driving a revolution joint of a robot are considered, such the cylinder and relevant links of the robotAcknowledgment: constitute a localThis closedresearch mechanismis funded by Vietnam appended National to the Foundation main robotfor Science architecture. and By takingTechnology into account Development this particular (NAFOSTED) feature under ofgrant the number hydraulic 107.04-2017.09. robot, the kinematic and dy- namic modelling and analysis of the robot are more accurate. This could help to design more efficient and more effective control laws for the arm. REFERENCES [1] Nguyen V Khang, and Chu A My. Fundamentals of industrial robots, Text Book. Vietnam Education Publisher, 2011 (in Vietnamese). [2] Craig J J. Introduction to Robotics: Mechanics and Control, 3rd Edn., Pearson Education, Upper Saddle River, NJ, USA, 2005. [3] Murray J and Lovell G. Dynamic modeling of closed-chain robotic manipulators and implications for trajectory control. IEEE T. Robotic. Autom. 1989; 5: 522–528. [4] Mata V, Provenzano S, Valero F and Cuadrado J I. Serialrobot dynamics algorithms for moderately large numbers of joints. Mech. Mach. Theory. 2002; 37: 739–755. [5] Tran H N. Inverse kinematic, dynamics and sliding mode control of redundant manipulator. PhD thesis. Institute of Mechanics, Vietnam Academy of Science and Technology. 2010 (In Vietnamese) [6] Chu A My, et al. Mechanical design and dynamics modelling of RoPC robot. Proceedings of International Symposium on Robotics and Mechatronics, Hanoi, Vietnam. Sept 2009; 92-96. [7] Merlet J P. Parallel robots. London Kluwer Academic Publishers, 2000.
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