Simulation of 6-DOF Parallel Robot for Coupling Compensation Method

INTRODUCTION

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 robot’s 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.

COUPLING ANALYSIS

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.

Fig. 1:	Model of 6-DOF 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 A₁ contains two orthogonal rotational motion which are given in Plücker coordinates by:

(1)

(2)

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 A₁B_A which is the projection of A₁B₁ is denoted by β. Thus, the translational motion of screw joint is given in Plücker coordinates by:

(3)

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:

(4)

(5)

(6)

(7)

(8)

(9)

Therefore, the screw system of leg A₁H₁B₁ is given as:

(10)

Thus the anti-screw system of leg A₁H₁B₁ has 1 DOF which is denoted by:

(11)

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.

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:

(12)

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 6-DOF parallel robot

(13)

where, ^AT_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.

(14)

(15)

where, G^A_mi is the installation vector of the upper gimbal in the inertia frame, G^A_mi = ^AT_BG^B_mi, G^B_mi is the installation vector of the upper gimbal in the plate body frame.

(16)

(17)

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:

(18)

where, ψ_i is positive number when X_mixX_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:

(19)

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:

(20)

where, Da is the screw-pitch.

Therefore, the practical length of the ith L_t is given as:

(21)

where, Δ_pi is the difference of the ith leg between practice and theory in the previous moving plate position.