The kinematics model of the seedling pick-up mechanism was established by considering one side of the mechanism as an example [
27‐
31]. The motion diagram of the seedling pick-up mechanism is shown in Figure
2, and the rectangular coordinate system
Oxy was established. Points
O,
M1, and
O1 are the rotation centers of the planetary carrier, the middle gear, and the planetary gear, respectively. Point
H is the cusp of the seedling pick-up arm. Points
P and
Q are the meshing points between the middle gear and the sun gear and between the middle gear and the planetary gear, respectively.
φ0 is the initial installation angle of the seedling pick-up mechanism,
δ0 is the planetary carrier corner (acute angle between lines
M1O and
M1O1), and
α0 is the installation angle of the seedling pick-up arm.
β is the central angle of the pitch curve of the toothed part of the incomplete eccentric circular gear.
α is the center angle corresponding to the partial pitch curve of the middle non-circular gear meshing with the incomplete eccentric circular gear.
θ is the initial rotation angle of the incomplete eccentric circular gear.
R1 and
R2 are the distances from the meshing point
P between the sun gear and the middle gear to the rotation centers
O and
M1, respectively.
\( R^{\prime}_{1} \) and
\( R^{\prime}_{2} \) are the radius vectors of the meshing pitch curves of the incomplete non-circular gears 2 and 3, respectively.
R21 and
R3 are distances from the meshing point
Q between the middle gear and the planetary gear to rotation centers
M1 and
O1, respectively.
3.2.1 Displacement Model of Seedling Pick-Up Mechanism
When the planetary carrier rotates counterclockwise by the angle
\(\varphi_{1}\) from the initial position of the mechanism shown in Figure
2(a), the rotation angles of the middle gear and the planetary gear relative to the planetary carrier are
\(\varphi_{2}\) and
\(\varphi_{3}\), respectively, as shown in Figure
2(b).
(1)
Angular-displacement analysis of the middle gear relative to the planetary carrier
The drive of the middle gear and the sun gear is a combined non-circular gear transmission. The transmission of the two pairs of gears should satisfy the requirement that the pitch-curve perimeters of the meshing parts are equal during meshing. Then, the analysis and solution of the angular displacement of the middle gear relative to the planetary carrier can be divided to two cases.
When the planetary carrier rotates by the angle
\(\varphi_{1}\) from −
\(\varphi_{0}\) to
β −
\(\varphi_{0}\), the middle non-circular gear 4 meshes with the incomplete eccentric circular gear 1. Thus, the following equation is obtained:
$$ \varphi_{2}\, =\,\int_{{ - \varphi_{0} }}^{{\varphi_{1} }} {\frac{{R_{1} (\varphi_{1} )}}{{R_{2} (\varphi_{2} )}}\text{d} \varphi_{1} } , $$
(1)
where
$$ R_{1} \text{(}\varphi_{1} \text{)} = {{e}}\cos (\varphi_{1} + \theta ) + \sqrt {R^{2} - {{e}}^{2} \sin^{2} (\varphi_{1} + \theta )} , $$
(2)
$$ R_{2} \left( {\varphi_{2} } \right) = L - R_{1} \left( {\varphi_{1} } \right). $$
(3)
Here, R and e represent the radius and eccentric distance, respectively, of the incomplete eccentric circle, and L represents the center distance between the middle gear and the sun gear.
Using Eqs. (
1) and (
3), the following equation is obtained:
$$ \alpha { = }\int_{{ - \varphi_{0} }}^{{\beta - \varphi_{0} }} {\frac{{R_{1} (\varphi_{1} )}}{{L - R_{1} (\varphi_{1} )}}} \text{d} \varphi_{1} . $$
(4)
According to Eq. (
4), the center distance
L is calculated using numerical integration after the parameters
β,
α,
R,
e, and
θ are determined.
To ensure the completeness of the middle non-circular gear and the smoothness and continuity of the periodic pitch curve, the pitch-curve equation for the middle non-circular gear 4 that does not mesh with the sun gear is
$$ R_{2} \text{(}\varphi_{2} \text{)}\, = \,b_{0} + b_{1} \varphi_{2} + b_{2} \varphi_{2}^{2} {\text{ + b}}_{3} \varphi_{2}^{3} ,\;\;\;\;\alpha \le \varphi_{2} \le 2\uppi\text{.} $$
(5)
In summary, the pitch-curve equation for the middle non-circular gear is
$$ R_{2} (\varphi_{2} ) = \left\{ \begin{array}{ll} & L - R_{1} (\varphi_{1} ), \quad \quad \quad \quad \quad \quad - \varphi_{0} \le \varphi_{1} \le \beta - \varphi_{0} , \hfill \\ & b_{0} + b_{1} \varphi_{2} + b_{2} \varphi_{2}^{2} {{ + b}}_{3} \varphi_{2}^{3} , \quad \alpha \le \varphi_{2} \le 2\uppi\text{.} \hfill \\ \end{array} \right. $$
(6)
In Eq. (
6), the coefficients
b0,
b1,
b2, and
b3 are calculated according to the requirements of the smoothness and continuity of the periodic pitch curve of the middle non-circular gear.
When the planetary carrier rotates by the angle
\(\varphi_{1}\) from
β −
\(\varphi_{0}\) to 2π −
\(\varphi_{0}\), the incomplete non-circular gears 2 and 3 mesh with each other, and the following equation is obtained:
$$ \varphi_{2} { = }\int_{{\beta - \varphi_{0} }}^{{\varphi_{1} }} {\frac{{R^{\prime}_{1} (\varphi_{1} )}}{{R^{\prime}_{2} (\varphi_{2} )}}\text{d} \varphi_{1} } , $$
(7)
where
\( R^{\prime}_{2} (\varphi_{2} ) = L - R^{\prime}_{1} \left( {\varphi_{1} } \right). \)
The transmission-ratio function of the incomplete non-circular gears 2 and 3 is
$$ i_{2} (\varphi_{1} ) = \frac{{R^{\prime}_{2} (\varphi_{2} )}}{{R^{\prime}_{1} (\varphi_{1} )}}. $$
(8)
Here, the transmission-ratio function is constructed as follows:
$$ {{i}}_{2} (\varphi_{1} ) = a_{0} + a_{1} \sin \varphi_{1} + a_{2} \cos \varphi_{1} . $$
(9)
In Eq. (
9), the transmission-ratio coefficients
a1,
a2 and
a3 can be determined according to the continuity condition of the transmission ratio of the two pairs of gears.
According to Eqs. (
8) and (
9), the pitch-curve equation for the transmission of the incomplete non-circular gears 2 and 3 is
$$ \left\{ \begin{aligned} R^{\prime}_{1} = \frac{{Li_{2} (\varphi_{1} )}}{{1{ + }i_{2} (\varphi_{1} )}}, \hfill \\ R^{\prime}_{2} = \frac{L}{{1{ + }i_{2} (\varphi_{1} )}}. \hfill \\ \end{aligned} \right. $$
(10)
(2)
Angular-displacement analysis of the planetary gear relative to the planetary carrier
When the middle gear meshes with the planetary gear, the two gears should satisfy the requirement that the pitch-curve perimeters of their meshing parts are equal. Then, the following equation is obtained:
$$ \varphi_{3} = \int_{0}^{{\varphi_{2} }} {\frac{{R_{21} (\varphi_{1} )}}{{R_{3} (\varphi_{3} )}}} \text{d} \varphi_{2} , $$
(11)
where
$$ R_{21} (\varphi_{1} ) = R_{2} (\varphi_{2} +\uppi - \delta_{0} ),\;\;\;R_{3} (\varphi_{3} ) = a - R_{21} (\varphi_{1} ). $$
Here, a is the distance between the centers of the middle gear and the planetary gear.
When the planetary carrier rotates by the angle
\( \varphi_{1} \) from zero to 2π, the middle gear and the planetary gear correspondingly rotate by the angles
\( \varphi_{2} \;\;{\text{and}}\;\;\varphi_{3} \), respectively, from zero to 2π. Thus, the following equation is obtained:
$$ 2\uppi = \int_{0}^{{2\uppi}} {\frac{{R_{21} }}{{a - R_{21} }}} \text{d} \varphi_{2} . $$
(12)
\( R_{21} (\varphi_{1} ) \) can be approximately calculated using
\( R_{2} \)\( \left( {\varphi_{2} } \right) \) according to the equations below Eq. (
11). The distance
a between the centers of the middle gear and the planetary gear can be calculated using the numerical-integration method according to Eq. (
12) [
23]. After
\( R_{21} (\varphi_{1} ) \) and
a are calculated,
\( \varphi_{3} \) can be calculated using Eq. (
11).
The relative displacement of the cusp
H of the seedling pick-up arm is
$$ \left\{ \begin{array}{ll} X_{H} = L\cos (\varphi_{0} + \varphi_{1} ) + {{a}}\cos (\varphi_{0} + \varphi_{1} + \delta_{0} ) \hfill \\ \;\; + S\cos (\varphi_{0} + \varphi_{1} + \delta_{0} + \alpha_{0} + \varphi_{3} ), \hfill \\ Y_{H} = L\sin (\varphi_{0} + \varphi_{1} ) + {{a}}\sin (\varphi_{0} + \varphi_{1} + \delta_{0} ) \hfill \\ \;\; + S\sin (\varphi_{0} + \varphi_{1} + \delta_{0} + \alpha_{0} + \varphi_{3} ), \hfill \\ \end{array} \right. $$
(13)
where
S is the distance from the cusp of the seedling pick-up arm to the rotation center of the planetary gear.