Abstract Objective To study the interpolation algorithm of parallel processing machine based on Stewart mechanism. Methods The idea of ​​common time-division interpolation algorithm and the inverse kinematics model of parallel machining mechanism were used to solve the relationship between the tool trajectory and six telescopic rods. An interpolation algorithm was derived and an object-oriented programming method was used to simulate the method. Software, verifying the availability of the interpolation algorithm through computer simulation. Results and conclusions The simulation results for the machining process of spherical and hyperboloid parts show that the proposed interpolation algorithm is correct and practical, and provides a new feasible interpolation algorithm for the parallel processing machine.
Key words Stewart mechanism parallel processing machine inverse kinematics interpolation algorithm trajectory planning
Interpolation Algorithm of Parallel Connected
Machine Tool Based on Stewart Mechanism
Zhang Junliang Zhang Jianmin Hao Juan Jiang Ming Wang Xinyi
(School of Mechanical Engineering and Automation, Beijing Institute of Technology, Beijing 100081)
Abstract Aim To research the interpolation algorithm of parallel connected machine tool based on stewart mechanism.Methods The common idea of ​​time-discrete interpolation algorithm and the inverse kinematics model of parallel connected machine tool were used to evaluate the relation between tool motion path and six telescopic Legs and to derive one kind of interpolation algorithm.Object oriented program design method was adopted to work out the simulation software and the availability of the interpolation algorithm was testified by computer simulation.Results and Conclusion Simulation results of machining process of sphere parts and one- Sheet-hyperboloid parts show that the interpolation algorithm proposed is correct and practical, providing a new and desirable interpolation algorithm for parallel connected machine tool.
Key words stewart mechanism; parallel connected machine tool; inverse kinematics; interpolation algorithm; path planning
The parallel processing machine based on the Stewart mechanism is a novel and promising machine tool. Compared with the ordinary numerical control machine tool has several remarkable characteristics: The cutting force is borne by 6 axles, and only the tension or the pressure, therefore the machine tool deformation is small, the bearing capacity is strong; The mechanism rigidity is good, the movement part light weight, therefore the inertia is small , It can perform higher acceleration and deceleration movements; without guide rails, it can eliminate the influence of general geometric errors and wear, etc. on the accuracy, so it can obtain high processing accuracy.
Due to solving the inverse kinematics [1,2] of the Stewart mechanism (according to the position and attitude of the end tool, solving the value of each joint variable) is relatively simple and consistent with the position control concept of the mechanism. Therefore, the inverse kinematics model of the parallel processing machine can be built using this feature.
1 parallel processing machine inverse kinematics
The inverse kinematics of the parallel processing machine consists of two parts, namely the inverse kinematics of the Stewart parallel mechanism and the relationship between the tool pose and the pose of the parallel platform.
The schematic diagram of the Stewart organization is shown in Figure 1. The coordinate of the center O′ of the movable platform and the point Pi of the spherical hinge center of the movable platform in the fixed coordinate system is (xO, yO, zO)T, (xi, yi, zi)T, and the Pi point is in the dynamic coordinate system. The coordinates are (xPi, yPi, zPi). If the coordinates of the center bi of the fixed platform in the fixed coordinate system are (di, ei, fi) (i=1, 2,..., 6), the following formula holds:
[xi yi zi]T=T[xPi yPi zPi]T+[xO yO zO]T, (1)
Fig.1 Structure of the Stewart mechanism
Where T is a rotation transformation matrix expressed by Euler angles α, β, γ,
(2) Derived from the distance formula between two points
In the dynamic coordinate system fixed with the moving platform, the relative coordinates of the tool tip point Pt are (xPt, yPt, zPt) = (0, 0, -lt), and lt is the distance from the tool tip to O'. then
[xO yO zO]T=T[xt yt zt]T+[T13 T23 T33]Tlt. (4)
Let the unit vector of the tool axis be nx, ny, nz, and the cosine of the direction of the coordinate axes of the tool and fixed coordinate system be θx, θy, θz.
Because the last column of the rotation matrix (2) is exactly the unit vector of the z-axis of the dynamic coordinate system, it also coincides with the unit vector of the tool axis.
α=±arcsin(cosθx/sinθz), β=±θz. (5)
When ny ≥ 0, equation (5) takes a negative sign; when ny<0, equation (5) takes a positive sign. When Euler angle γ = -α, the moving platform does not rotate around the z-axis of the rigid coordinate system. This reduces the amount of computation and largely avoids the interference between the six telescopic rods.
2 Time-slicing interpolation algorithm based on inverse kinematics of parallel processing machine
The time division method, data sampling interpolation method [3], is to subdivide the entire period of processing a straight line or a circle orphan into a number of equal time intervals, called the unit time interval (or interpolation period). The interpolation calculation is performed once every unit time interval, and the feed amount of each coordinate axis during this time interval is calculated, and the processing is performed until the machining end point is calculated. The core of this algorithm is to find the geometric relationship between tool movement trajectory and six telescopic rods.
2.1 Linear interpolation algorithm
After selecting the interpolation cycle of the CNC and determining the feedrate, the contour step f is then determined. Therefore, as long as the position and attitude of the tool at the end of each interpolation period are found, the six-bar increments can be calculated using the inverse kinematics formula. The schematic diagram of linear interpolation is shown in Fig. 2 .
Figure 2 Linear interpolation
Set a straight line 
The angle with the z-axis is γ, and the angle between its projection and the x-axis is α, then the contour step f ( 
The projections on the three coordinate axes are fsinγcosα, fsinγsinα, fcosγ. Let B coordinate be (xB, yB, zB), then the coordinate of point C is
xC=xB+fsinγcosα, yC=yB+fsinγsinα,zC=zB+fcosγ. (6)
Knife has a fixed attitude and a variable attitude given two ways: fixed attitude mode does not need to calculate the rotation matrix, the amount of calculation is small; variable attitude mode requires the end of the interpolation cycle tool attitude, calculate the rotation matrix, a large amount of calculation. With the position of the tool, you can use the inverse kinematics formula to find the expansion and contraction increments of 6 telescopic rods [4].
2.2 Circular interpolation algorithm
Circular interpolation also requires the coordinates of the tool nose and the attitude of the tool at the end of the interpolation cycle. Circular interpolation diagram shown in Figure 3. Set the plane of the arc over the z-axis, and the center of the circle coincides with the origin O of the coordinate. The contour step f ( 
And 
,) The projections on the three coordinate axes are fcos(γ+Δγ/2)cosα, fcos(γ+Δγ/2)sinα, -fsin(γ+Δγ/2), and Δγ=f/r (r is a circle Arc radius).
Fig. 3 circular interpolation
Let the coordinates of point B be (xB, yB, zB), then the coordinates of point C are
(7)
The machining of the space surface (ball-end milling cutter as an example) generally selects the attitude of the tool (tool axis) perpendicular to the tangent plane of the curved surface, and determines the tool attitude according to the surface equation. After the position of the tool is determined, inverse kinematics is used to find the telescopic increments of the six telescopic rods.
Because the interpolation algorithm includes the inverse kinematics operation of the Stewart mechanism, the amount of calculation of the linear interpolation and the circular interpolation is larger than that of the linear interpolation and circular interpolation of the numerical control machine tool. This shows that the mechanical structure of the machine based on the spatial parallel mechanism is very simple, and the mathematical analysis is very complex [5,6].
3 imitation
The emulator uses an object-oriented programming approach and is written in Visual C++ MFC. The purpose is to verify the correctness of the inverse kinematics model and the interpolation algorithm of the mechanism. The arc interpolation algorithm was tested on spherical parts and the linear interpolation algorithm was tested on single-leaf hyperboloid parts. Figure 4 shows the block diagram of the 6-pole interpolation algorithm. The parameters of the Stewart mechanism are: the radius of the moving platform is 400mm, the radius of the fixed platform is 500mm, the ratio of length to short sides of both platforms is 0.365, the maximum length of the telescopic shaft is 1500mm, the shortest length is 600mm, and the diameter of the shaft is 40mm. For spherical parts, the radius is 100mm, the angle is 60°, and the position is 980mm. The tool length is 30mm and the diameter is 5mm. For single-leaf hyperboloids, the radius is 100mm, the angle is 20°, and the position is 980mm. The tool length is 200mm and the diameter is 5mm. Through simulation, the correctness of the interpolation algorithm based on inverse kinematics model of parallel processing machine was verified.
Figure 4 interpolation algorithm flow chart
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