Mechanical Design and Kinematics Analysis of a Hydraulically Actuated Manipulator

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.


INTRODUCTION
The traditional electric powered industrial manipulators have been used in all kinds of industrial productions.They are already very mature in technology and with high precision.But they are not suitable for several tasks in severe environments for their electric drive mode and low powermass ratio, such as underwater, construction, electric power lines maintenance, etc.In order to make up for the inadequacy of electric powered manipulators, some hydraulically actuated multi-joints manipulators have been developed in several countries.
Uninterrupted power supply has become indispensable during the maintenance task of active electric power lines as a result of today's highly information-oriented society and increasing demand of electric utilities.The maintenance task has the risk of electric shock and the danger of falling from high place.Therefore, it is necessary to realize an autonomous robot system using electro-hydraulic manipulator because hydraulic manipulators have the advantage of electric insulation and high power-mass ratio [1,2].Dunnigan et al. [3][4][5] introduce several hydraulic manipulators working for underwater, forest and construction tasks respectively.Taylor et al. [6,7] introduce two kinds of hydraulic manipulators for nuclear industries.Wang et al. [8] introduces a hydraulic manipulator developed in China but there is no application case given.Zhao Y. L. gives two application cases of hydraulic manipulators using in electric power line maintenance [9,10].This paper gives the mechanical structure of a six DOF hydraulic manipulator in section 2, which is named HydM and developed by Shandong University, China.In section 3, *Address correspondence to this author at the School of Control Science and Engineering, Shandong University, Jinan 250061, China; Tel: 0086-531-88396813; Fax: 0086-531-88396813; E-mail: rsong@sdu.edu.cn the forward kinematic equations are given firstly.Then the analytical solutions of its inverse kinematics are solved by given the position and posture of the end-effector simultaneously.

MECHANISM DESIGN OF HydM
The mechanical structure sketch of HydM is shown in Fig. (1).HydM has six rotary degrees of freedom (DOF) which are all driven by hydraulic actuators.The first, the fourth and the fifth joints are driven by helical rotary actuators.The second and the third joints are driven by hydraulic linear cylinders.The sixth joint is driven by a miniature hydraulic motor through a worm reducer, so it can turn continuously.In fact, the end-effector is a gripper and its opening and closing movement is driven by a miniature linear cylinder.The movement limits of each joint are listed in Table 1.The movements of all the six joints and the gripper are controlled by servo valves which are mounted on a hydraulic manifold.HydM does not integrate with a hydraulic power unit, so it must be supplied hydraulic power from an external hydraulic power unit through high pressure hose.

The DH Frames and Link Parameters of HydM
According to the DH rules [11], the global frame {O 0 } and local frames {O i } fixed to each link can be build as shown in Fig. (2).
Then the link parameters in DH frames of each link can be gained as listed in Table 1.The open intervals in last column show the motion range of each joint.

Forward Kinematics of HydM
The homogeneous transformation method based DH rules is the most commonly used method to solve the forward kinematics of manipulators or other complex serial mechanisms.According to Fig. (1) and Table 1, it is easy to get the homogeneous transformation matrix between two adjacent links.The HydM manipulator has six joints and seven links including the mounting base, so there have six homogeneous transformation matrices.The successive product of these matrices describes the position and posture of the end-effector in global frame.And it is also the forward kinematic equation of the manipulator shown as Eq. ( 1). where,

Inverse Kinematics of HydM
For manipulators, inverse kinematics analysis is very important since it is the base of trajectory planning and realtime control.It is the best result to obtain analytical solutions of the inverse kinematic problem for it can well meet the requirements of real-time control.In the case of a serial manipulator with six degrees of freedom, it must meet one of the following two conditions to have analytical solutions of its inverse kinematic problem.One is that it has three adjacent parallel joints.The other is that it has three adjacent joints whose axes intersect at one point.HydM meets the first condition, so we can obtain analytical solutions of its inverse kinematics.
Generally, the position and posture of the end-effector are given with homogeneous transformation matrix shown as Eq.(2).And the vector [ ! p x , ! p y , ! p z ] T gives the position while the 3×3 matrix at the upper left corner of 0 !T 6 gives the posture of the end-effector in global frame {O 0 }.
According to the homogeneous transformation rules, the given position and posture of the end-effector described in frame {O 1 } can be shown with Eq. (3).Simultaneously, the same position and posture of the end-effector can be shown with Eq. ( 4) due to the forward kinematics.
According to the homogeneous transformation method, the position and posture of the end-effector described in frame {O 3 } can be derived as the following matrix: From Fig.
(3), it can be concluded that the frame {O 6 } is transformed from the original posture duplicated with the global frame {O 0 } through three rotation transformations.The first one makes {O 6 } to rotate α around axis z 6 to { !O 6 }.