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1 ball gear attitude adjustment mechanism motion analysis The ball gear mechanism installation method is as shown in Figure 1, the middle ball gear is fixed, the concave ball gear can be moved by the tie rod to carry out the spatial yaw motion. According to the structural characteristics of the ball gear mechanism, the following coordinate system is established. As shown in 2, the fixed coordinate system S1 (x1, y1, z1) is fixed to the middle ball gear 1, the cross 1 coordinate system Sc (xc, yc, zc), the tie frame (referred to as the tie rod) coordinate system St (xt, yt, zt), the concave spherical gear 2 (planetary wheel) fixed coordinate system S2 (x2, y2, z2).
The tie position is described by the connection of the two ball gears. The control tie rod rotates around the xc axis and the y1 axis, and the planetary ball gear can realize 360 ​​omnidirectional yaw motion under the driving thereof. To describe the yaw motion of the mechanism, two parameters are defined: the yaw axis n and the azimuth angle, as shown. The yaw axis is always in the x1y1 plane, and the spatial orientation of the tie rod at a certain moment can be seen as being rotated around a fixed yaw axis n in the yaw plane.
2 The meshing model and the meshing model of the ball-gear mechanism of the pointing error model 21 ball gear mechanism satisfy the conjugate meshing motion relationship between the two ball gears. Therefore, the input and output of the mechanism have a certain motion relationship, and the mechanism outputs The yaw motion of the shaft can be achieved by the rotation of the tie rod around the cross 1 (or the cross around the active ball gear 1). For example, the yaw angle of the tie rod is 2, the output shaft yaw angle is p; the azimuth movement of the output shaft of the mechanism can be It is realized by the rotation of the tie rod around the polar axis (z1 axis) of the ball gear 1, such as the azimuth angle of the middle tie rod and the output shaft. The above motion process can be described by a series of coordinate transformations.
When the cross 1 is turned y around the y1 axis of the ball gear 1 and the tie bar is rotated from the position coincident with the zc axis about the z1 axis to the position shown therein, that is, coincident with the zt axis, the zt axis can be expressed in the coordinate system S1 as: Zt=sinycostsinysintcosy1.(1) The yaw axis vector can be obtained by multiplying the z1 axis and the zt axis by n=z1zt. (2) Set the number of teeth, modulus and pressure angle of the two ball gears to be the same, then the two ball gears are in any yaw The gear ratio in the plane is 1, reference [3], the angle between the tie rod and the z1 axis (ideally y) can be obtained by calculating the angle between the center line of the two ball gears and the z1 axis:
=12p=arccosO1O2z1|O1O2||z1|.
(3) The transformation matrix of the planetary gear system S2 to the fixed system S1 can be realized by first shifting the fixed coordinate system S1 to the O2 point, then turning around the x1 axis to obtain the coordinate system S20, and then rotating the coordinate system S20 around the axis. When the vector n is turned to p, the gear 2 fixed coordinate system S2 is obtained. The coordinate transformation matrix M12=M120M202 from the coordinate system S2 to the coordinate system S1, (4) wherein the coordinate transformation matrix M120 from the coordinate system S20 to the coordinate system S1 =10xO20-10yO20-1zO201, (xO2, yO2, zO2) represents the position of the O2 point in the fixed system S1, that is, the vector O1O2. Let the direction angles of the axis vector n in the fixed system S1 be n, n and n, respectively, because the vector n is perpendicular to the z1 axis, so n = 90. The coordinate transformation matrix M202 = M202 (n from the coordinate system S2 to the coordinate system S20) , n, 90, p) = cos2n (1-cosp) cospcosncosn (1-cosp) cosnsinp0cosncosn (1-cosp) cos2n (1-cos) cosp-cosnsinp0-cosnsinpcosnsinpcosp01. Thus the teeth of the gear 2 in the fixed system S1 are obtained The surface equation is R(2)1=M12R2, (5) where R2 is the tooth profile surface equation of the ball gear 2.
It is known from the characteristics of the tooth profile of the ball gear that the inner side of the ring tooth is a saddle face and the outer side is a convex face. When the two ball gears are meshed and transmitted, the outer side of one ball gear and the inner side of the other ball gear form a pair of conjugate curved surfaces, and the contact feature is a point contact between a convex surface and a saddle surface. According to the tooth surface contact analysis method, the position vector and the unit normal vector of the gear 1 and the gear 2 should be expressed in the same coordinate system S1, and the common contact points of the two tooth surfaces should be the same point in the fixed coordinate system S1, and the two teeth The unit normals of the faces should be collinear with each other. When meshing, the two tooth faces should meet the following equation at the contact point:
R1-R(2)1=0, (6)n1n(2)1=0.
(7) Since |n1|=|n(2)1|=1, the nonlinear equations composed of equations (6) and (7) are actually composed of five equations, containing unknowns u1,1,x, y, u2, 2 and p, where u1, 1, u2 and 2 are the tooth profile parameters of the two ball gears, respectively. Given the input angle x, y, the other five unknowns can be solved by solving the nonlinear equations to determine a contact point of the two tooth faces.
The pointing error model of the 22-ball gear mechanism is selected in the coordinate system established by 2, the x1y1 plane is selected to be parallel to the horizontal plane, and the y1 axis is oriented as the reference direction (north-north direction). Since the mechanism has two rotational degrees of freedom, the output shaft can be The yaw motion is performed within a spatial orientation of 360 about the z1 axis. The spatial orientation of the output shaft is described by the available yaw angle p and azimuth. The pointing accuracy error of the mechanism can be expressed by the pointing error of the output axis, that is, the yaw angle error and the azimuth error are expressed as p(t, y) = p(t, y) - p(t, y), (8) (t, y) = (t, y) - (t, y), (9) where: p, p, and respectively represent the yaw angle and azimuth of the actual and ideal output axes; t and y represent the input angles of the mechanism, respectively.
3 Influence of installation error on mechanism pointing accuracy analysis The omnidirectional yaw motion of the ball gear attitude adjustment mechanism in space is realized by two mutually perpendicular series movements of the rotation of the fixed ball and the rotation of the tie rod around the cross 1 by the cross 1 of.
The assembly error of each component during assembly will have a certain impact on the pointing accuracy of the mechanism. The following is a specific pointing accuracy analysis of the mechanism by establishing the installation error model of each component.
The influence of the installation error of the cross 1 on the pointing accuracy of the mechanism is as shown in Fig. 1. The installation error existing between the cross 1 and the fixed ball gear is composed of two parts: 1) the center distance caused by the cross 1 and the center of the ball gear 1 do not coincide Error C; 2) vertical axis misalignment error c of the axis of rotation of the cross 1 (yc axis) and the y1 axis of the fixed ball gear, as shown in 3, respectively. It should be noted that the horizontal axial misalignment error need not be considered, because the ball gear is a rotating component, and there is no such error as long as the appropriate initial coordinate system position is selected. In the figure, Sp(xp, yp, zp) and Sa(xa, ya, za) are auxiliary coordinate systems, and other coordinate systems are set the same. When the tie bar moves to the middle position in the same way, according to the coordinate transformation relationship in the figure, the transformation matrix Mpc=MpaMac from the cross coordinate system Sc to the fixed system Sp can be obtained, (10) where Mpa and Mac are respectively From Sa to Sp, the Sc to Sa transformation matrix can be obtained by the DH coordinate method. Thus, the zc axis can be expressed as: zc=Mpczp. (11) and then the formula for calculating the angle c1 between the zc axis and the z1 axis is c1=arccoszcz1|zc||z1|.(12) according to FIG. For the motion relationship, the zt axis can be expressed as zt=sinc1cos(tc)sinc1sin(tc)cosc11.
(13) The calculation formula of the yaw axis vector n is the same as (2). Let C denote the theoretical center distance of the two ball gears. According to the relationship in the figure, the following vector relationship can be obtained: OpO2=Czt, O1O2=O1Op OpO2.
(14) Then, it can be calculated according to the formula (3), but due to the installation error, p and the relationship in the formula (3) are not satisfied, let p be an unknown amount, and then obtain the coordinates from the coordinate system S2 to the fixed system S1. The transformation is the same as (4), and then by solving the nonlinear equations (6) and (7), p can be obtained, and the representation of the mechanism output axis z2 in the fixed system S1 is further obtained, and finally according to equations (8) and (9). ), the pointing error of the output shaft can be calculated.
Effect of 32-tie frame mounting error on mechanism pointing accuracy The mounting error between the tie rod and the cross 1 consists of horizontal and vertical axis misalignment errors, such as h and v, respectively.
Sv(xv, yv, zv) and Sh(xh, yh, zh) are auxiliary coordinate systems, and the settings of other coordinate systems are the same. When the tie rod is moved to the neutral position in the same manner, the zt axis can be expressed as zt=sin(y v)costsin(y v)sintcos(y v)1.
(15) The yaw axis vector n can be obtained according to the formula (2). Similarly, due to the installation error, p and the relationship in equation (3) are not satisfied. As with the step of section 31, p can be calculated by solving the nonlinear equations, thereby calculating the pointing error of the ball gear mechanism.
33 cross 1 and the tie rod have the effect of the installation error on the pointing accuracy of the mechanism. This situation is actually the superposition of the installation error of the cross 1 and the tie rod in the previous 31 and 32 sections, not only the center distance error, the mechanism is in two There is an axis misalignment error on the mutually perpendicular axes of rotation (the middle yc axis and the xt axis), causing the xt axis of the tie rod to be not completely perpendicular to the yc axis of the cross, thereby having a certain influence on the pointing accuracy of the mechanism. The installation error of the cross 1 is the same as C and c in the middle, and the installation error of the tie rod is the same as h and v. When the tie rod transmits motion in the same way, combining the motion transformation relationship of the 3 and the medium coordinate system, the zt axis can be calculated as zt=sin(c1 v)cos(tc)sin(c1 v)sin(tc)cos (c1 v)1.
(16) Similarly, the pointing error of the mechanism can be calculated according to the same steps.
Depending on the motion characteristics of the mechanism, the mounting error between the tie rod and the cross 2 and the cross 2 and the planetary ball gear can be eliminated by the initial mounting position of the adjustment mechanism.
4 Conclusions 1) As a new type of spaceborne antenna attitude adjustment mechanism, the ball gear attitude adjustment mechanism has the characteristics of light weight, compact structure and large carrying capacity, and has strong practical value in the aerospace field.
2) The installation error has an important influence on the pointing accuracy of the ball gear attitude adjustment mechanism. The installation error of different components has different effects on the yaw angle error and the azimuth error. The research conclusion provides important theoretical guidance for the practical application of the ball gear attitude adjustment mechanism.