INTRODUCTION
Since, parallel structure is proposed by Stewart (1993),
the 6DOF parallel robot is one of the most popular cueing simulator mechanisms
for its remarkable advantage over serial mechanisms (Huang
et al., 2005; Yang et al., 2002), where
a moving plate is connected to a base plate by six legs. The character of each
leg influences the smoothness, accuracy and realtime of the moving plate. This
perspective attracts a lot of research on error analysis and compensation.
Abdellatif and Heimann (2006) and Abdellatif
et al. (2007) opened the discussion on the influence of Passive Joint
Friction (PJF) in robot’s dynamics and its impact on control performance
Meng et al. (2002) proposed a directerrorcompensation
method of measuring the error of a sixfreedomdegree parallel mechanism CMM.
Yang et al. (2010) developed PD control with
gravity compensation for hydraulic 6DOF parallel manipulator. Cleary
and Arai (1991) analyzed the influence of moving plate by the length of
each of the six legs. Wang et al. (2007) studied
the coupling characteristics of large hydraulic Stewart Platform. Li
et al. (2009) compensates the interference of Stewart Platform based
on inverse dynamic model. Besides, researchers often use software to simulate
the moving of parallel robot (Ma et al., 2008;
Iqbal et al., 2008; Liu
et al., 2009). But there are few documents studied on the phenomenon
of the coupling.
In this study, a coupling compensation is developed to improve the control performance including steady and moving precision via compensating each leg coupling errors. This study begins with a screw theory to analyze the motion characters of 6DOF parallel robot. Then the Plücker system of leg is built, considering the degree of freedom of the 6DOF parallel robot and the coupling angle and coupling velocity are calculated by a closedsolution inverse kinematics. The performance including stability, precision and robustness of the 6DOF parallel robot with coupling compensation is analysed in theory and simulation. The controller with coupling compensation is used to Tank Simulator to prolong its lifetime.
COUPLING ANALYSIS
With reference to Fig. 1, which represents the 6DOF parallel
robot structure here considered, it can be observed that the moving plate and
the base plate are combined with six legs which consist of upper gimbal, screw
joint and lower gimbal.

Fig. 1: 
Model of 6DOF parallel robot 

Fig. 2: 
Coordinate system of leg A1H1B1 
One of the combined legs is taken to analyze its coupling based on screw theory.
It supposes that the coupling does not exist in each screw joint of legs, which means the screw joint, as same as translational joint, has only one translational motion. The frame (X, Y, Z) located at the center of the lower gimbal is shown in Fig. 2. The lower gimbal A_{1} contains two orthogonal rotational motion which are given in Plücker coordinates by:
The angle between the line of translational motion and the base plate is denoted by a, the angle between the Xaxis and the line of A_{1}B_{A }which is the projection of A_{1}B_{1} is denoted by β. Thus, the translational motion of screw joint is given in Plücker coordinates by:
The upper gimbal also contains two orthogonal rotational motions. Length of the leg A_{1}B_{1} is denoted by L_{1}, γ stands for the angle between the line of the upper gimbal center joint and the opposing lower gimbal center joint. Thus, the two orthogonal rotational motion of the upper gimbal can be calculated in Plücker coordinates as follows:
Therefore, the screw system of leg A_{1}H_{1}B_{1} is given as:
Thus the antiscrew system of leg A_{1}H_{1}B_{1} has 1 DOF which is denoted by:
where, P = 0, Q = 0. L, M, N can be expressed by R.
In the same manner, each of the leg has one constraint motion, the degrees of freedom of the moving plate is no more than 6. The result does not match with actual situation. While the antiscrew system of leg A_{1}H_{1}B_{1} has a rotational motion along the line of A_{1}B_{1} when R = 0. It supposes that coupling exits in legs moving, the constraint of leg is released. Thus the moving plate has three translational motions and three rotational motions. Therefore, the coupling exits in each leg indeed.
COUPLING COMPENSATION
There are two frames describing the motion of the moving plate: an inertia frame (X_{a}, Y_{a}, Z_{a}) located at the center of the base plate and a body frame (X_{b}, Y_{b}, Z_{b}) located at the center of the moving plate with Z_{b}axis pointing outward (Fig. 1). The length vector of the ith leg is calculated as:
where, A^{A}_{i} is the position of the lower joint A_{i} in the inertia frame, B^{A}_{i} is the position of the upper joint B_{i} in the inertia frame.

Fig. 3: 
One Leg of the 6DOF parallel robot 
where, ^{A}T_{B} is the transformation matrix form the body frame (X_{b}, Y_{b}, Z_{b}) to the inertia frame (X_{a}, Y_{a}, Z_{a}), B_{i}^{B} is the position of the upper joint B_{i} in the plate body frame.
The leg can rotate around the axis of gimbal, while the upper part of the leg is sliding inside the lower part by an actuating force. This motion is considered by two frames: a leg fixed frame (X_{ni}, Y_{ni}, Z_{ni}) located at the joint A_{i} with the Z_{ni}axis parallel to the length vector of leg and X_{ni}axis parallel to the rotational axis of lower gimbal in outward, the leg body frame (X_{mi}, Y_{mi}, Z_{mi}) located at the same point with X_{mi}axis parallel to the rotational axis of upper gimbal as shown in Fig. 3.
where, G^{A}_{mi} is the installation vector of the upper gimbal in the inertia frame, G^{A}_{mi} = ^{A}T_{B}G^{B}_{mi}, G^{B}_{mi} is the installation vector of the upper gimbal in the plate body frame.
where, G^{A}_{ni} is the installation vector of the lower gimbal in the inertia frame.
There are 2 DOF between the leg fixed frame (X_{ni}, Y_{ni}, Z_{ni}) and the leg body frame (X_{mi}, Y_{mi}, Z_{mi}): one translational motion alone Z_{ni}axis and one rotational motion around Z_{ni}axis. The coupling angle ψ_{i }is then calculated as:
where, ψ_{i} is positive number when X_{mi}xX_{ni} have the same direction with L^{A}_{i}, otherwise ψ_{i} is negative number.
The angular velocity ω_{si} of coupling is given as:
where,
J is the Jacobian matrix of the general velocity of the moving plate to the velocity of the upper attachment points.
The difference of the ith leg between practice and theory Δ_{i} is computed as:
where, Da is the screwpitch.
Therefore, the practical length of the ith L_{t} is given as:
where, Δ_{pi} is the difference of the ith leg between practice and theory in the previous moving plate position.
RESULTS
Model simulation is a simple and effective to verify proposed method or system
that many researchers used (Aqel, 2006; Eker
et al., 2002). This study takes Tank Simulator which base on 6DOF
parallel robot as the simulation model. Each leg of the 6DOF parallel robot
has an upper part sliding inside a lower part to imitate the physical feeling
of driving a Tank for the three translational motions (surge, sway and heave)
and the three rotational motions (pitch, roll and yaw). Coupling compensation
has been used in order to guarantee against numerical problems in the solution
process.

Fig. 4: 
Rotational angle on Zaxis 
Table 1: 
Parameters of the tank simulator 

All parameters of the 6DOF parallel robot configuration and the initial conditions,
which are kept invariable during the simulation, refer to Table
1.
It investigates two controlling curves (the compensation controlling curves and the uncompensation controlling curves) to test the precision of the coupling compensation. Taking the actuator velocity which makes the moving plate of rotated around Z_{a}axis to make the sinusoidal motion which amplitude is 30° as the input of 6DOF parallel robot model, this study analyzes performance of Tank Simulator. Without the coupling compensation, 6DOF parallel robot generates an extra moving on the direction of Zaxis. Rotated angle with coupling compensation and without coupling compensation of Zaxis is shown in Fig. 4. Besides, Fig. 5 and Fig. 6 show the translational velocity and acceleration on Zaxis.
From the experiment results, it is obvious that the motion on command direction
according to preconceived track with accurate amplitude and period, but the
6DOF parallel robot generate an extra moving along with Zaxis which can causes
vibration of Tank Simulator that can not neglect. All coordinate values, especially
the Z coordinate value, decreased dramatically after the coupling compensation
of 6DOF parallel robot. The variation of moving velocity on Zaxis after the
compensation was tends to zero; acceleration has decreased from 225 mm sec^{2}
to almost zero. In the same manner, taking the rotation of Xaxis or Yaxis
as the research process, the translational error on Xaxis or Yaxis decreased
dramatically.

Fig. 5: 
Moving speed of Zaxis 

Fig. 6: 
Moving acceleration of Zaxis 

Fig. 7: 
Movement distance on Xaxis 
Movement distance on Xaxis and Yaxis are shown as Fig. 7
and Fig. 8. After considering the coupling compensation of
the controller, the vibration of Tank Simulator (Fig. 9) has
decreased and the stability has improved.

Fig. 8: 
Movement distance on Yaxis 

Fig. 9: 
Photo of Tank Simulator test prototype 
Therefore the lifetime of the Tank Simulator can be prolonged.
CONCLUSIONS
In this study, the motion of 6DOF parallel robot is studied base on screw theory. The following conclusions can be drawn:
• 
Due to analysis of the legs of 6DOF parallel robot model,
there exists coupling between the six degrees of freedom. In this study,
it builds the coupling velocity and acceleration. Besides, the coupling
compensation is proposed to optimize the moving track 
• 
Mechanismmodel combined motion method which is presented
in this study considers the influence of the motion controller and actuator
that can improves the reliability and authenticity of 6DOF parallel robot 
• 
Experimental results show that the precision can be increased
by the coupling compensation. It eliminates jitter in the motion that can
prolong life of Tank Simulator 