Kinematics analysis of a ship-based stabilized platform

Xue-leiWANG

，Bin ZHANG2，Zhi-xin FENG1，Shi-xia LV1

（1 Beijing Polytechnic，Beijing 100176，China）

（2 China Agricultural University，Beijing 100083，China）

Abstract：This paper presented a stabilized platform with five UPSactive limbs and one RRS passive limb based on the characteristics of the shipmovement.First，the stabilized platform’s prototype is presented and the inverse position formulas are derived.Then the inverse/forward velocities，inverse/forward accelerations formulas of the stabilized platform are derived.Third，examples for solving the length，velocity and acceleration of the active limbs are derived and the result shows that the proposed mechanism has good kinematic characteristic.

Key words：Stabilized platform，Active limbs，Inverse position，Kinematics

1 Introduction

Stabilized platforms isolate the disturbances through real-time online detection and servo compensation systems［1-3］，so that they can provide a relatively stable working environment for the equipments，which will have great significance for both national defense security and economic construction.

It has been noted that the parallelmechanisms have advantages of high stiffness， small errors， simple structures，good dynamic performances and easy to control［4-8］，it is an idealmodel for large-scale and multi-axis stabilized platform.Therefore，Zhejiang University proposed a stabilization and automatic tracking system which is based on a novel 4TPS-PS PM with 4-DOF，including translation along one coordinate axis and rotations around three coordinate axes［9］.

Themarine environment changes frequently causing multidimensional swings to transport shipswhichmove over the sea［10］，such as the roll，pitch，yaw，surge，sway and heave，but the motion of sway has small impact on the safety of ships.Focusing on the demand of the stabilized platform，a novel parallel mechanism with 5 DOFs is needed.The 5-DOF parallelmechanisms have attracted much attention because of their advantages of low-cost，easy manufacture，simple structure，less driver and suitability for various occasions，etc［11-14］.

Therefore Ding Zehua［11］proposed a novel hybrid mechanism with 5-DOFwhich was composed of a fully decoupled parallel mechanism with three moving degrees of freedom and a seriesmechanism with two rotational degrees of freedom.Guo Qiang［12］analyzed the kinematics and workspace of a 4 SPRR-SPR 5-DOF parallelmechanism and showed that the mechanism has good application prospects in cutting and grinding parts of complex space surface.Zhang Canguo［13］presented a novel 5-DOF parallelmanipulator with compound spherical joints and its DOF and characteristics are analyzed，besides，the observation method is adopted to judge the constrained force/torques in the sub chains.Wang Chenyu［14］proposed a novel 5-DOF parallelmechanism for the special case of 5-DOF and its kinematics is analyzed.

Focusing on the above，a novel 5-DOF parallel stabilized platform with three rotations and two translations is presented according to the requirements of the transportation.The stabilized platform is given in Fig.1.Hence，this paper focuses on the analysis of the kinematic of the proposed stabilized platform.

Fig.1 A ship and a stabiIized p Iatform

2 The 5-DOF p latform

The 5-DOF stabilized platform is composed of a moving platform，a fixed base and five UPS-type active limbsli（i=1，2，…，5），and one RRS-type passive limbl6，see Fig.2（a）.The coordinate systemO-XYZando-xyzare fixed on the fixed base atOand themoving platform atoseparately in Fig.2（b）.

Fig.2 The 5-DOF paraIIe IstabiIized p Iatform

3 Inverse disp lacem en t kinem atics of the stabilized p latform

Mechanism displacement analysis is to obtain the position relationship between the input components and the output component through the geometric relationship between the components of the mechanism.The inverse analysis ofmechanism position is to solve the input component parameters of mechanism when the output component parameters are known.Themotion control of the robot is closely related to the inverse kinematics solution.The motion parameters of the active part can be obtained by the inverse kinematics solution，which lays a foundation for themotion control of the robot.

The position ofAi（i=1，2，…，5）inO-XYZ，Bi（i=1，2，…，5）ino-xyz，andBi（i=1，2，…，5）inO-XYZcan be expressed as：

Where，P=（x，y，z）is position vector ofo-xyzinOXYZ，andmatrixis a rotation transformationmatrix fromo-xyztoO-XYZandis formed by three Euler rotations of（XY1Z2），is derived as below：

The length of each active limb can be expressed as

At the initial time，suppose pointoofo-xyzinOXYZis（0，0，H），α=0，β=0，γ=0，hX=0 andhZ=0，P=hX，0，H+hZ）.Substituting equations（1）-（4）into equation（5）and the formulas for solving the lengthli（i=1，2，…，5）of each active limbs can be derived as follows：

4 Inverse/forward velocity and acceleration

The active limbs’length ofl1，l2，l3，l4，l5are equations aboutα，β，γ，hXandhZ，supposeli（α，β，γ，hX，hZ）（i=1，2，…，5）are the lengths of each active limbs，Then

4.1 Inverse/Forward Velocity

Differentiating equation（7）with respect to time and the inverse velocity of active limbs can be derived as below

Suppose

The inverse velocity can be expressed as

And the forward velocity can be expressed as

4.2 Inverse/Forward Acceleration

The inverse acceleration of the mechanism is derived by differentiating equation（9）with respect to time，which can be expressed as bellow

Suppose

Then the inverse acceleration ocan be derived as

From equation（9）to（12），the inverse/forward acceleration can be expressed as below

Where，is a 5×5×5 matrix.

5 Analytic Solved Exam p les

In the stabilized platform，supposera=400 mm，rb=100 mm，H=800 mm，when thefive independent pose parameters（α，β，γ，hX，hZ）versus timetis given，the length，velocity and acceleration of five active limbs can be solved.Suppose

The length，velocity and acceleration of the five active limbs are shown in Fig.3 to Fig.5.The results show that the displacement，velocity and acceleration of the active limbs are varied smoothly.It implies that the proposed stabilized platform has good characteristics of kinematics.

Fig.3 The Iength of active Iegs

Fig.4 The ve Iocity of ac tive Iegs

Fig.5 The acce Ieration of active Iegs

6 Conclusions

This paper presents a novel 5-DOF parallel shipbased stabilized platform based on the characteristics of shipmotions.Thewhole structure is simple and the interference among the active limbs and moving platform can be avoided easily.

The inverse displacement，inverse/forward velocities，inverse/forward accelerations of the 5-DOF stabilized platform has been derived and the formulas are concise.The Jacobian matrix and Hessian matrix are deduced at the same time.Example for solving the length，velocity and acceleration of the active limbs are given，we can know from the results that the kinematic characteristic of the proposed mechanism is good.

7 Acknow ledgm ents

The authors would like to thank the reviewers for their critical reviews and suggestions to improve the clarity of this article.They would also like to thank themanaging editor for the works on themanuscript.