Since, parallel structure is proposed by Stewart (1993),
the 6-DOF 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 real-time 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 robots dynamics and its impact on control performance
Meng et al. (2002) proposed a direct-error-compensation
method of measuring the error of a six-freedom-degree parallel mechanism CMM.
Yang et al. (2010) developed PD control with
gravity compensation for hydraulic 6-DOF 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 6-DOF parallel robot. Then the Plücker system of leg is built, considering the degree of freedom of the 6-DOF parallel robot and the coupling angle and coupling velocity are calculated by a closed-solution inverse kinematics. The performance including stability, precision and robustness of the 6-DOF 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.
With reference to Fig. 1, which represents the 6-DOF 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.
|| Model of 6-DOF parallel robot
|| 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 A1 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 X-axis and the line of A1BA which is the projection of A1B1 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 A1B1 is denoted by L1, 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 A1H1B1 is given as:
Thus the anti-screw system of leg A1H1B1 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 anti-screw system of leg A1H1B1 has a rotational motion along the line of A1B1 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.
There are two frames describing the motion of the moving plate: an inertia frame (Xa, Ya, Za) located at the center of the base plate and a body frame (Xb, Yb, Zb) located at the center of the moving plate with Zb-axis pointing outward (Fig. 1). The length vector of the ith leg is calculated as:
where, AAi is the position of the lower joint Ai in the inertia frame, BAi is the position of the upper joint Bi in the inertia frame.
|| One Leg of the 6-DOF parallel robot
where, ATB is the transformation matrix form the body frame (Xb, Yb, Zb) to the inertia frame (Xa, Ya, Za), BiB is the position of the upper joint Bi 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 (Xni, Yni, Zni) located at the joint Ai with the Zni-axis parallel to the length vector of leg and Xni-axis parallel to the rotational axis of lower gimbal in outward, the leg body frame (Xmi, Ymi, Zmi) located at the same point with Xmi-axis parallel to the rotational axis of upper gimbal as shown in Fig. 3.
where, GAmi is the installation vector of the upper gimbal in the inertia frame, GAmi = ATBGBmi, GBmi is the installation vector of the upper gimbal in the plate body frame.
where, GAni is the installation vector of the lower gimbal in the inertia frame.
There are 2 DOF between the leg fixed frame (Xni, Yni, Zni) and the leg body frame (Xmi, Ymi, Zmi): one translational motion alone Zni-axis and one rotational motion around Zni-axis. The coupling angle ψi is then calculated as:
where, ψi is positive number when XmixXni have the same direction with LAi, otherwise ψi is negative number.
The angular velocity ωsi of coupling is given as:
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 Δi is computed as:
where, Da is the screw-pitch.
Therefore, the practical length of the ith Lt is given as:
where, Δpi is the difference of the ith leg between practice and theory in the previous moving plate position.
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 6-DOF
parallel robot as the simulation model. Each leg of the 6-DOF 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
|| Rotational angle on Z-axis
|| Parameters of the tank simulator
All parameters of the 6-DOF parallel robot configuration and the initial conditions,
which are kept invariable during the simulation, refer to Table
It investigates two controlling curves (the compensation controlling curves and the un-compensation controlling curves) to test the precision of the coupling compensation. Taking the actuator velocity which makes the moving plate of rotated around Za-axis to make the sinusoidal motion which amplitude is 30° as the input of 6-DOF parallel robot model, this study analyzes performance of Tank Simulator. Without the coupling compensation, 6-DOF parallel robot generates an extra moving on the direction of Z-axis. Rotated angle with coupling compensation and without coupling compensation of Z-axis is shown in Fig. 4. Besides, Fig. 5 and Fig. 6 show the translational velocity and acceleration on Z-axis.
From the experiment results, it is obvious that the motion on command direction
according to preconceived track with accurate amplitude and period, but the
6-DOF parallel robot generate an extra moving along with Z-axis 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 6-DOF parallel robot. The variation of moving velocity on Z-axis 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 X-axis or Y-axis
as the research process, the translational error on X-axis or Y-axis decreased
|| Moving speed of Z-axis
|| Moving acceleration of Z-axis
|| Movement distance on X-axis
Movement distance on X-axis and Y-axis 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.
|| Movement distance on Y-axis
|| Photo of Tank Simulator test prototype
Therefore the lifetime of the Tank Simulator can be prolonged.
In this study, the motion of 6-DOF parallel robot is studied base on screw theory. The following conclusions can be drawn:
||Due to analysis of the legs of 6-DOF 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
||Mechanism-model 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 6-DOF 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