Dimensional synthesis of motion generation of a spatial RCCC mechanism

Abstract

A dimensional synthesis method for a spatial RCCC rigid-body guidance mechanism that satisfies non-periodic design requirements is proposed. By solving the 21-dimensional spatial RCCC mechanism motion generation problem in five steps, the dimension of the numerical atlas database is effectively reduced. The installation angle parameters, link twists, link lengths and link offset are matched using the established numerical atlas database. The mechanism geometric parameters obtained from the database are then optimized using a genetic algorithm. On this basis, the equations for calculating the actual sizes and installation position parameters of the objective mechanism are established; thus, the spatial RCCC mechanism motion generation with non-periodic design requirements is realized. Since there are fewer independent variables in each step of the proposed reduced dimensional synthesis method, the computational burden is smaller than existing numerical atlas method. Avoiding the solution of complex nonlinear equations, the method is therefore not limited by the number of prescribed positions compared with the analytical method. Moreover, since the problem of synchronous optimization of input angles and geometric parameters is avoided, the efficiency is far higher than optimization method. Two examples are presented to demonstrate the efficacy and accuracy of the proposed theory.

Appendices

Appendix 1

Based on Eq. (11), the projection of point P_tⁿ (x_Ptⁿ, y_Ptⁿ, and z_Ptⁿ) of rigid-body line P_tⁿQ_tⁿ in complex plane x′O′y′ can be expressed as:

$$x_{Pt}^{n} + iy_{Pt}^{n} = - \left[ {r_{P} \sin \alpha_{Px} \cos \left( {\theta_{2}^{n} + \alpha_{Pyz} } \right) + a_{12} } \right]e^{{i\theta_{1}^{n} }} - i\left[ {\left( {r_{P} \cos \alpha_{Px} - S_{2}^{n} } \right)\sin \alpha_{12} + r_{P} \sin \alpha_{Px} \sin \left( {\theta_{2}^{n} + \alpha_{Pyz} } \right)\cos \alpha_{12} } \right]e^{{i\theta_{1}^{n} }} + O_{x} + iO_{y}$$

(29)

According to Eq. (29), to eliminate installation position parameters O_x and O_y, the difference between the projection of point P_t^M of the rigid-body line P_t^MQ_t^M and that of point P_t^M+1 of the rigid-body line P_t ^M+1Q_t ^M+1 in the complex plane x′O′y′ can be expressed as:

$${\mathbf{F}}^{M} = x_{Pt}^{M + 1} + iy_{Pt}^{M + 1} - \left( {x_{Pt}^{M} + iy_{Pt}^{M} } \right).$$

(30)

Then, the vector F^M is rotated corresponding Δθ₁^M (Δθ₁^M = θ₁^M+1-θ₁^M and M = 1, 2, …, N-2) around the z-axis.

$${\mathbf{F^{\prime}}}^{M + 1} = \frac{{x_{Pt}^{M + 2} + iy_{Pt}^{M + 2} - \left( {x_{Pt}^{M + 1} + iy_{Pt}^{M + 1} } \right)}}{{e^{{i\Delta \theta_{1}^{M} }} }}.$$

(31)

According to Eqs. (30) and (31), we can obtain:

$${\mathbf{F}}^{M} - {\mathbf{F^{\prime}}}^{M + 1} = x_{Pt}^{M + 1} + iy_{Pt}^{M + 1} - \left( {x_{Pt}^{M} + iy_{Pt}^{M} } \right) - \frac{{x_{Pt}^{M + 2} + iy_{Pt}^{M + 2} - \left( {x_{Pt}^{M + 1} + iy_{Pt}^{M + 1} } \right)}}{{e^{{i\Delta \theta_{1}^{M} }} }}$$

(32)

Substituting Eq. (29) into (32) gives:

$$\begin{aligned} {\mathbf{F}}^{M} - {\mathbf{F^{\prime}}}^{M + 1} = & - r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \cos \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]e^{{i\theta_{1}^{M + 1} }} + r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]e^{{i\theta_{1}^{M} }} \\ + i\left\{ {\left( {S_{2}^{M + 1} - S_{2}^{M + 2} } \right)\sin \alpha_{12} - r_{P} \sin \alpha_{Px} \cos \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]} \right\}e^{{i\theta_{1}^{M + 1} }} \\ - i\left\{ {\left( {S_{2}^{M} - S_{2}^{M + 1} } \right)\sin \alpha_{12} - r_{P} \sin \alpha_{Px} \cos \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]} \right\}e^{{i\theta_{1}^{M} }} \\ \end{aligned}$$

(33)

According to Eq. (23), the relative displacement of the coupler link Γ^j* can be expressed as:

$$\Gamma^{j * } = S_{2}^{M} - S_{2}^{M + 1} = - \frac{{z_{Pt}^{M} - z_{Pt}^{M + 1} + r_{P} \sin \alpha_{Px} \sin \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]}}{{\cos \alpha_{12} }}\;{\text{and}}$$

(34)

$$S_{2}^{M} - S_{2}^{M + 1} = - \frac{{z_{Pt}^{M} - z_{Pt}^{M + 1} + r_{P} \sin \alpha_{Px} \sin \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]}}{{\cos \alpha_{12} }}$$

(35)

$$S_{2}^{M + 1} - S_{2}^{M + 2} = - \frac{{z_{Pt}^{M + 1} - z_{Pt}^{M + 2} + r_{P} \sin \alpha_{Px} \sin \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]}}{{\cos \alpha_{12} }}.$$

(36)

Substituting Eqs. (35) and (36) into (33) gives:

$$\begin{gathered} {\mathbf{F}}^{M} - {\mathbf{F^{\prime}}}^{M + 1} + {\text{i}}\left( {z_{Pf}^{M + 1} - z_{Pf}^{M + 2} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M + 1} }} - {\text{i}}\left( {z_{Pf}^{M} - z_{Pf}^{M + 1} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M} }} = - r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \cos \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]e^{{i\theta_{1}^{M + 1} }} \hfill \\ + r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]e^{{i\theta_{1}^{M} }} \hfill \\ + i\left\{ { - r_{P} \sin \alpha_{Px} \sin \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]\tan \alpha_{12} - r_{P} \sin \alpha_{Px} \cos \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 2} + \alpha_{Pyz} } \right)} \right]} \right\}e^{{i\theta_{1}^{M + 1} }} \hfill \\ - i\left\{ { - r_{P} \sin \alpha_{Px} \sin \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]\tan \alpha_{12} - r_{P} \sin \alpha_{Px} \cos \alpha_{12} \left[ {\sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) - \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)} \right]} \right\}e^{{i\theta_{1}^{M} }} \hfill \\ \end{gathered}$$

(37)

According to Eq. (37), there are two unknowns in equation r_Psinα_Px and α_Pyz. To find α_Pyz, the following can be obtained:

$$\omega = \frac{{{\mathbf{F}}^{M} - {\mathbf{F^{\prime}}}^{M + 1} + {\text{i}}\left( {z_{Pf}^{M + 1} - z_{Pf}^{M + 2} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M + 1} }} - {\text{i}}\left( {z_{Pf}^{M} - z_{Pf}^{M + 1} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M} }} }}{{{\mathbf{F}}^{M + 1} - {\mathbf{F^{\prime}}}^{M + 2} + {\text{i}}\left( {z_{Pf}^{M + 2} - z_{Pf}^{M + 3} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M + 2} }} - {\text{i}}\left( {z_{Pf}^{M + 1} - z_{Pf}^{M + 2} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M + 1} }} }}.$$

(38)

According to the known parameters, only α_Pyz is unknown in Eq. (38). For direct expression, the objective mechanism size parameter α_Pyz can be expressed as:

$$\alpha_{Pyz} = - \arctan \left( {\frac{{\omega \tau_{1} - \upsilon_{1} }}{{\omega \tau_{2} - \upsilon_{2} }}} \right),$$

(39)

where

$$\upsilon_{1} = e^{{i\theta_{1}^{M + 1} }} \left[ {\left( {\cos \theta_{2}^{M + 2} - \cos \theta_{2}^{M + 1} } \right) + h\left( {\sin \theta_{2}^{M + 2} - \sin \theta_{2}^{M + 1} } \right)} \right] + e^{{i\theta_{1}^{M} }} \left[ {\left( {\cos \theta_{2}^{M} - \cos \theta_{2}^{M + 1} } \right) + h\left( {\sin \theta_{2}^{M} - \sin \theta_{2}^{M + 1} } \right)} \right],$$

$$\upsilon_{2} = e^{{i\theta_{1}^{M + 1} }} \left[ {\left( {\sin \theta_{2}^{M + 1} - \sin \theta_{2}^{M + 2} } \right) + h\left( {\cos \theta_{2}^{M + 2} - \cos \theta_{2}^{M + 1} } \right)} \right] + e^{{i\theta_{1}^{M} }} \left[ {\left( {\sin \theta_{2}^{M + 1} - \sin \theta_{2}^{M} } \right) + h\left( {\cos \theta_{2}^{M} - \cos \theta_{2}^{M + 1} } \right)} \right],$$

$$\tau_{1} = e^{{i\theta_{1}^{M + 2} }} \left[ {\left( {\cos \theta_{2}^{M + 3} - \cos \theta_{2}^{M + 2} } \right) + h\left( {\sin \theta_{2}^{M + 3} - \sin \theta_{2}^{M + 2} } \right)} \right] + e^{{i\theta_{1}^{M + 1} }} \left[ {\left( {\cos \theta_{2}^{M + 1} - \cos \theta_{2}^{M + 2} } \right) + h\left( {\sin \theta_{2}^{M + 1} - \sin \theta_{2}^{M + 2} } \right)} \right],$$

$$\tau_{2} = e^{{i\theta_{1}^{M + 2} }} \left[ {\left( {\sin \theta_{2}^{M + 2} - \sin \theta_{2}^{M + 3} } \right) + h\left( {\cos \theta_{2}^{M + 3} - \cos \theta_{2}^{M + 2} } \right)} \right] + e^{{i\theta_{1}^{M + 1} }} \left[ {\left( {\sin \theta_{2}^{M + 2} - \sin \theta_{2}^{M + 1} } \right) + h\left( {\cos \theta_{2}^{M + 1} - \cos \theta_{2}^{M + 2} } \right)} \right]\;\;{\text{and}}$$

$h = i\sin \alpha_{12} \tan \alpha_{12} + i\cos \alpha_{12} .$

Similarly, α_Qyz can be determined. Substituting α_Pyz into Eq. (37), r_Psinα_Px can be expressed as:

$$r_{P} \sin \alpha_{Px} = \frac{{{\mathbf{F}}^{M} - {\mathbf{F^{\prime}}}^{M + 1} + {\text{i}}\left( {z_{Pt}^{M + 1} - z_{Pt}^{M + 2} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M + 1} }} - {\text{i}}\left( {z_{Pt}^{M} - z_{Pt}^{M + 1} } \right)\tan \alpha_{12} e^{{i\theta_{1}^{M} }} }}{{\beta_{1} + \beta_{2} }},$$

(40)

where

$$\beta_{1} = \left[ {\varpi_{1} \left( {\cos \theta_{2}^{M} - \cos \theta_{2}^{M + 1} } \right) - \varpi_{2} \left( {\sin \theta_{2}^{M} - \sin \theta_{2}^{M + 1} } \right)} \right]e^{{i\theta_{1}^{M} }} ,$$

$$\beta_{2} = \left[ { - \varpi_{1} \left( {\cos \theta_{2}^{M + 1} - \cos \theta_{2}^{M + 2} } \right) + \varpi_{2} \left( {\sin \theta_{2}^{M + 1} - \sin \theta_{2}^{M + 2} } \right)} \right]e^{{i\theta_{1}^{M + 1} }} ,$$

$$\varpi_{1} = \cos \alpha_{Pyz} + \left( {i\sin \alpha_{12} \tan \alpha_{12} + i\cos \alpha_{12} } \right)\sin \alpha_{Pyz} \;\;{\text{and}}$$

$$\varpi_{2} = \sin \alpha_{Pyz} - \left( {i\sin \alpha_{12} \tan \alpha_{12} + i\cos \alpha_{12} } \right)\cos \alpha_{Pyz} .$$

Similarly, r_Qsinα_Qx can be determined.

Appendix 2

Substituting the obtained mechanism size parameters r_Psinα_Px and α_Pyz into Eq. (30) gives:

$$\begin{aligned} {\mathbf{F}}^{M} = & - \left[ {r_{P} \sin \alpha_{Px} \cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) + a_{12} } \right]e^{{i\theta_{1}^{M + 1} }} - i\left[ {\left( {r_{P} \cos \alpha_{Px} - S_{2}^{M + 1} } \right)\sin \alpha_{12} + r_{P} \sin \alpha_{Px} \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)\cos \alpha_{12} } \right]e^{{i\theta_{1}^{M + 1} }} \\ + \left[ {r_{P} \sin \alpha_{Px} \cos \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) + a_{12} } \right]e^{{i\theta_{1}^{M} }} + i\left[ {\left( {r_{P} \cos \alpha_{Px} - S_{2}^{M} } \right)\sin \alpha_{12} + r_{P} \sin \alpha_{Px} \sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right)\cos \alpha_{12} } \right]e^{{i\theta_{1}^{M} }} \\ \end{aligned}$$

(41)

Based on Eq. (41), r_Pcosα_Px can be solved as follows:

$$r_{P} \cos \alpha_{Px} = \frac{{{\mathbf{F}}^{M} + \phi_{1} + \phi_{2} }}{{\left[ {e^{{i\theta_{1}^{M} }} - e^{{i\theta_{1}^{M + 1} }} } \right]i\sin \alpha_{12} }},$$

(42)

where

$$\phi_{1} = \left[ {r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right) + \sin \left( {\theta_{2}^{M + 1} + \alpha_{Pyz} } \right)i\cos \alpha_{12} } \right] + a_{12} - S_{2}^{M + 1} i\sin \alpha_{12} } \right]e^{{i\theta_{1}^{M + 1} }} \;\;{\text{and}}$$

$$\phi_{2} = - \left[ {r_{P} \sin \alpha_{Px} \left[ {\cos \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right) + \sin \left( {\theta_{2}^{M} + \alpha_{Pyz} } \right)i\cos \alpha_{12} } \right] + a_{12} - S_{2}^{M} i\sin \alpha_{12} } \right]e^{{i\theta_{1}^{M} }} .$$

Similarly, r_Qcosα_Qx can be solved. r_P, r_Q, α_Px and α_Qx can be determined according to Eqs. (40) and (42).

According to Eqs. (11) and (12), the basic dimensional parameters of the mechanism and the installation angle parameters are obtained; then, the installation position parameters of the objective mechanism spatial RCCC can be represented as follows:

$$O_{x} = - [r_{P} \sin \alpha_{Px} \cos (\theta_{2}^{n} + \alpha_{Pyz} ) + a_{12} ]\cos \theta_{1}^{n} + [(r_{P} \cos \alpha_{Px} - S_{2}^{n} )\sin \alpha_{12} + r_{P} \sin \alpha_{Px} \cos \alpha_{12} \sin (\theta_{2}^{n} + \alpha_{Pyz} )]\sin \theta_{1}^{n} - x_{Pt}^{n} ,$$

(43)

$$O_{y} = - [r_{P} \sin \alpha_{Px} \cos (\theta_{2}^{n} + \alpha_{Pyz} ) + a_{12} ]\sin \theta_{1}^{n} - [(r_{P} \cos \alpha_{Px} - S_{2}^{n} )\sin \alpha_{12} + r_{P} \sin \alpha_{Px} \cos \alpha_{12} \sin (\theta_{2}^{n} + \alpha_{Pyz} )]\cos \theta_{1}^{n} - y_{Pt}^{n} \;\;{\text{and}}$$

(44)

$$O_{z} = \left( {r_{P} \cos \alpha_{Px} - S_{2}^{n} } \right)\cos \alpha_{12} - r_{P} \sin \alpha_{Px} \sin \left( {\theta_{2}^{n} + \alpha_{Pyz} } \right)\sin \alpha_{12} - S_{1} - z_{Pt}^{n} .$$

(45)

Appendix 3

See Fig. 

Fig. 7

The flowchart of reduced dimensional synthesis method

7.