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Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator Xuewen Rong1, Rui Song*,1, Hui Chai1 and Xiaolin Ma2

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Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator Xuewen Rong1, Rui Song*,1, Hui Chai1 and Xiaolin Ma2 Send Orders for Reprints to [email protected] The Open Mechanical Engineering Journal, 2014, 8, 457-461 457 Open Access Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator Xuewen Rong1, Rui Song*,1, Hui Chai1 and Xiaolin Ma2 1School of Control Science and Engineering, Shandong University, Jinan 250061, China 2School of Mechanical and Electrical Engineering, Shandong Jianzhu University, Jinan 250101, China Abstract: This paper gives a mechanism design of a six DOF hydraulically actuated manipulator firstly. Then its DH frames and link parameters are given. Next, its forward kinematic equations are derived according to homogeneous transformation method. Fourthly, the analytical solutions of its inverse kinematics are solved by given the position and posture of the end-effector simultaneously. The posture of the end-effector is given with three z-y-z Euler angles for they have obvious geometry meanings and are easy to be measured. In addition, the correctness of the inverse kinematic equations is verified in Simulink by comparing many sets of randomly produced joint variables in workspace and their corresponding inverse solutions. Keywords: DH frame, DOF, Euler angle, homogeneous transformation, hydraulic, manipulator. 1. INTRODUCTION the forward kinematic equations are given firstly. Then the analytical solutions of its inverse kinematics are solved by The traditional electric powered industrial manipulators given the position and posture of the end-effector have been used in all kinds of industrial productions. They simultaneously. are already very mature in technology and with high precision. But they are not suitable for several tasks in severe 2. MECHANISM DESIGN OF HydM environments for their electric drive mode and low power- The mechanical structure sketch of HydM is shown in mass ratio, such as underwater, construction, electric power Fig. (1). HydM has six rotary degrees of freedom (DOF) lines maintenance, etc. In order to make up for the which are all driven by hydraulic actuators. The first, the inadequacy of electric powered manipulators, some hydraulically actuated multi-joints manipulators have been fourth and the fifth joints are driven by helical rotary actuators. The second and the third joints are driven by developed in several countries. hydraulic linear cylinders. The sixth joint is driven by a Uninterrupted power supply has become indispensable miniature hydraulic motor through a worm reducer, so it can during the maintenance task of active electric power lines as a turn continuously. In fact, the end-effector is a gripper and result of today’s highly information-oriented society and its opening and closing movement is driven by a miniature increasing demand of electric utilities. The maintenance task has linear cylinder. The movement limits of each joint are listed the risk of electric shock and the danger of falling from high in Table 1. The movements of all the six joints and the grip- place. Therefore, it is necessary to realize an autonomous robot per are controlled by servo valves which are mounted on a system using electro-hydraulic manipulator because hydraulic manipulators have the advantage of electric insulation and high rotary actuator power-mass ratio [1, 2]. Dunnigan et al. [3-5] introduce several fourth joint fifth joint hydraulic manipulators working for underwater, forest and third joint construction tasks respectively. Taylor et al. [6, 7] introduce two hydraulic cylinder kinds of hydraulic manipulators for nuclear industries. Wang et al. [8] introduces a hydraulic manipulator developed in China rotary actuator but there is no application case given. Zhao Y. L. gives two hydraulic cylinder application cases of hydraulic manipulators using in electric hydraulic motor power line maintenance [9, 10]. sixth joint This paper gives the mechanical structure of a six DOF end-effector hydraulic manipulator in section 2, which is named HydM hydraulic cylinder and developed by Shandong University, China. In section 3, second joint rotary actuator hydraulic manifold *Address correspondence to this author at the School of Control Science first joint and Engineering, Shandong University, Jinan 250061, China; Tel: 0086- 531-88396813; Fax: 0086-531-88396813; E-mail: [email protected] Fig. (1). The mechanical structure sketch of HydM. 1874-155X/14 2014 Bentham Open 458 The Open Mechanical Engineering Journal, 2014, Volume 8 Rong et al. hydraulic manifold. HydM does not integrate with a p = s (a c + a c + a c − d s ) − d (s c s − c c ), hydraulic power unit, so it must be supplied hydraulic power y 1 2 2 3 23 4 234 5 234 6 1 234 5 1 5 from an external hydraulic power unit through high pressure p = a s + d + a s + a s + d c − d s s , hose. z 2 2 1 3 23 4 234 5 234 6 234 5 a3 a4 d6 where, z x y y 5 6 3 4 y5 si = sinθi , ci = cosθi , sij = sin(θi +θ j ), cij = cos(θi +θ j ), z 5 6 x3 x4 d θ a2 6 sijk = sin(θi +θ j +θk ), cijk = cos(θi +θ j +θk ). θ3 θ4 θ5 Table 1. Link parameters of HydM. z1 y2 x2 Link No. ai-1 di Joint x1 αi-1 θi Limits θ2 i (mm) (mm) Variable 1 1 0 0 d1 θ1 θ1 (-90°, 90°) d θ 1 2 0 90° 0 θ2 θ2 (0°, 120°) z0 x0 3 a2 0 0 θ3 θ3 (-125°, -5°) 4 a3 0 0 θ4 θ4 (-50°, 70°) Fig. (2). The DH frames of HydM. 5 a4 -90° d5 θ5 θ5 (-150°, -30°) 3. KINEMATICS ANALYSIS OF HydM 6 0 -90° d6 θ6 θ6 (-180°, 180°) 3.1. The DH Frames and Link Parameters of HydM According to the DH rules [11], the global frame {O0} 3.3. Inverse Kinematics of HydM and local frames {Oi} fixed to each link can be build as shown in Fig. (2). For manipulators, inverse kinematics analysis is very important since it is the base of trajectory planning and real- Then the link parameters in DH frames of each link can time control. It is the best result to obtain analytical solutions be gained as listed in Table 1. The open intervals in last of the inverse kinematic problem for it can well meet the column show the motion range of each joint. requirements of real-time control. In the case of a serial 3.2. Forward Kinematics of HydM manipulator with six degrees of freedom, it must meet one of the following two conditions to have analytical solutions of The homogeneous transformation method based DH its inverse kinematic problem. One is that it has three rules is the most commonly used method to solve the adjacent parallel joints. The other is that it has three adjacent forward kinematics of manipulators or other complex serial joints whose axes intersect at one point. HydM meets the mechanisms. According to Fig. (1) and Table 1, it is easy to first condition, so we can obtain analytical solutions of its get the homogeneous transformation matrix between two inverse kinematics. adjacent links. The HydM manipulator has six joints and seven links including the mounting base, so there have six Generally, the position and posture of the end-effector homogeneous transformation matrices. The successive are given with homogeneous transformation matrix shown as product of these matrices describes the position and posture Eq. (2). And the vector [p! , p! , p!]T gives the position while of the end-effector in global frame. And it is also the forward x y z kinematic equation of the manipulator shown as Eq. (1). the 3×3 matrix at the upper left corner of 0 T! gives the 6 posture of the end-effector in global frame {O }. " n o a p % 0 $ x x x x ' " n o a p % 0 0 1 2 3 4 5 $ n o a p ' !x !x !x !x T = T ⋅ T ⋅ T ⋅ T ⋅ T ⋅ T = $ y y y y ' (1) $ ' 6 1 2 3 4 5 6 $ ' 0 n!y o!y a!y p!y $ nz oz az pz ' T6! = $ ' (2) $ ' n o a p # 0 0 0 1 & $ !z !z !z !z ' $ ' # 0 0 0 1 & nx = c1c234c5c6 − s1s5c6 + c1s234s6 , ny = s1c234c5c6 + c1s5c6 + s1s234s6 , According to the homogeneous transformation rules, the nz = s234c5c6 − c234s6 , ox = −c1c234c5s6 + s1s5s6 + c1s234c6 , given position and posture of the end-effector described in frame {O1} can be shown with Eq. (3). Simultaneously, the o = −s c c s − c s s + s s c , o = −s c s − c c , same position and posture of the end-effector can be shown y 1 234 5 6 1 5 6 1 234 6 z 234 5 6 234 6 with Eq. (4) due to the forward kinematics. ax = −c1c234s5 − s1c5 , ay = −s1c234s5 + c1c5 , $ n!c + n!s o!c + o!s a!c + a!s p!c + p!s ' & x 1 y 1 x 1 y 1 x 1 y 1 x 1 y 1 ) & ) (3) 1 0 −1 0 −n!s + n!c −o!s + o!c −a!s + a!c − p!s + p!c az = −s234s5 , px = c1(a2c2 + a3c23 + a4c234 − d5s234 ) − d6 (c1c234s5 + s1c5 ), T T T x 1 y 1 x 1 y 1 x 1 y 1 x 1 y 1 6! = 1 ⋅ 6! = & ) & ) n!z o!z a!z p!z − d1 & ) 0 0 0 1 % ( Kinematics Analysis of a Hydraulically Actuated Manipulator The Open Mechanical Engineering Journal, 2014, Volume 8 459 1T = 1T 2 T 3T 4 T 5 T 6 2 3 4 5 6 a c a s a!z !x 1 + !y 1 " c c c + s s −c c s + s c −c s a c + a c + a c − d s − d c s % a s + a s = p! − d + a + d − d a! (18) $ 234 5 6 234 6 234 5 6 234 6 234 5 2 2 3 23 4 234 5 234 6 234 5 ' (4) 2 2 3 23 z 1 4 s 5 s 6 z $ ' 5 5 s5c6 −s5s6 c5 d6c5 = $ ' $ s234c5c6 − c234s6 −s234c5s6 − c234c6 −s234s5 a2s2 + a3s23 + a4s234 + d5c234 − d6s234s5 ' $ ' Define 0 0 0 1 # & a!c + a!s a! The Eqs.
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