4.1.1 The rationality analysis of the initial actuated pairs

This coordinate of 3-DoF PM with limbs of embedding structures is seen in Fig. 1d, this PM has five limb groups (The actuated pairs are R₁, R₇ and R₂):

1. limb groups. 1: R₇H₁R₆P₁R₁R₅R₃R₄ limb, P₁R₁R₅R₃R₄ limb, R₂P₂R₅P₃ limb;

2. limb groups. 2: R₇H₁R₆P₁P₃ limb, P₁R₁P₃ limb, R₂P₂R₃R₄ limb;

3. limb groups. 3: R₇H₁R₆P₁R₁R₅R₃R₄ limb, R₂P₂R₅P₃ limb;

4. limb groups. 4: R₇H₁R₆P₁R₁P₃ limb, R₂P₂R₃R₄ limb;

5. limb groups. 5: R₇H₁R₆P₁R₅R₃R₄ limb, R₁R₅R₃R₄ limb, R₂P₂R₅P₃ limb.

In limb group 1, the first limb is R₇H₁R₆P₁R₅R₃R₄ limb, the second limb is P₁R₁R₅R₃R₄ limb, the third limb is R₂P₂R₅P₃ limb. The limb twist systems of the three limbs are as follows (the actuated screws are ${\mathit{\$}}_{\mathrm{11}}^{\mathrm{1}}$, ${\mathit{\$}}_{\mathrm{21}}^{\mathrm{1}}$ and ${\mathit{\$}}_{\mathrm{31}}^{\mathrm{1}}$):

Where, is the twist associated with the jth kinematic pair in the ith limb of limb group M; ${\mathit{\$}}_{\mathrm{11}}^{\mathrm{1}}$ and ${\mathit{\$}}_{\mathrm{12}}^{\mathrm{1}}$ are in YZ plane, the axes of the two twists are parallel to Z axis; the vector $r_{\mathrm{16}}=r_{\mathrm{24}}=(x_{{\mathrm{R}}_{\mathrm{3}}}$, $y_{{\mathrm{R}}_{\mathrm{3}}}$, $z_{{\mathrm{R}}_{\mathrm{3}}})$ and $r_{\mathrm{17}}=r_{\mathrm{25}}=(x_{{\mathrm{R}}_{\mathrm{4}}},y_{{\mathrm{R}}_{\mathrm{4}}},z_{{\mathrm{R}}_{\mathrm{4}}})$, and $(x_{{\mathrm{R}}_{\mathrm{4}}},y_{{\mathrm{R}}_{\mathrm{4}}},z_{{\mathrm{R}}_{\mathrm{4}}})$ is the coordinate of the center of moving platform. The reciprocal screw systems of each limb in limb group 1 are as follows:

Where, a and b are determined by: $a{x}_{{\mathrm{R}}_{\mathrm{4}}}+b{y}_{{\mathrm{R}}_{\mathrm{4}}}=\mathrm{0}$. By Eqs. (4)–(6), the mechanism constraint screw system can be obtained as follows:

By Eq. (7), the mechanism twist system $₁:

Equation (8) demonstrates this PM has three DoF as follows: (a) Moving platform rotates around Z axis, and indicated by $₁₁; (b) Moving platform translation along guideway, namely, radial translation centered on Z axis, and indicated by $₁₂; (c) Moving platform rotation along Z axis, and indicated by $₁₃.

It can be seemed from (a)–(c), this PM do not have parasitic motion, and the DoF is correct. The remnant-freedom analyzes as follows:

Delete the actuated screws in Eqs. (1)–(3), the limb twist systems of each limb after locking the inputs are as follows:

According to Eqs. (9)–(11), the three limb reciprocal screw systems after locking the inputs are as follows:

Based on Eqs. (12)–(14), the reciprocal screws in the three limbs after locking the inputs have not increased, so the inputs in limb group 1 are irrational. The mechanism constraint screw system after locking the inputs, namely, ${\mathit{\$}}_{\mathrm{1}}^{r^{\prime }}$ is:

Equation (15) validates that, the rank of ${\mathit{\$}}_{\mathrm{1}}^{r^{\prime }}$ is less than six, so the PM has remnant-freedom at this time. Based on the same method, the limb groups 2–5 does not meet input rationality judgment condition, and the actuated pairs R₁, R₇ and R₂ are irrational.

It is worth noting that, in the analysis of limb group 2–4, this PM has remnant-freedom, but this PM appears miscalculation DoF when the limb 5 is selected. The process of analysis is as follows:

In limb group 5, the first limb is R₇H₁R₆P₁R₅R₃R₄ limb, the second limb is R₂P₂R₅P₃ limb, the third limb is R₁R₅R₃R₄ limb. The first and second limb twist systems are the same as Eqs. (1) and (3), the third limb twist system is as follows:

The mechanism constraint screw system, namely, ${\mathit{\$}}_{\mathrm{5}}^r$:

The DoF obtained from limb group 5 is:

Obviously, the number of DoF is wrong. Contrast to limb group 1, the case of this problem is that lack of prismatic pair P₁ in R₁R₅R₃R₄ limb changes the mechanism constraint screw system. Prismatic pair P₁ indicates translation along Z axis, which is synchronous motion of each component, so the twist indicates this motion must be included in each limb twist system, or one twist has same axis. Therefore, one conclusion is obtained: in order to ensure correct DoF, the twist indicates this motion must be included in each limb twist system (conclusion 1 for short). Conclusion 1 is a necessary condition to ensure correct DoF, and this conclusion can delete irrational limb group to simplify the analysis process.

4.1.2 Reasonable justification after reselected actuated pairs

When the actuated pairs are R₇, R₁ and R₂, all the limb groups do not satisfy the judging conditions, so the actuated pairs need to be reselected. According to the principle that the kinematic pairs near the fixed platform are selected as the actuated pairs, the mechanism has two sets of kinematic pairs for actuated pairs, the first group is R₁, P₁, R₂, and the second group is R₁, P₂, R₂. In order to facilitate the transformation, the second sets of kinematic pairs are selected, and the motor 1 is changed to a linear motor at this time. After reselected inputs, the PM is seen in Fig. 5a, the sketch of limbs and actuated pairs coordinate frame of equivalent architecture are seen in Fig. 5b.

Figure 5The 3-DoF PM with limbs of embedding structures after reselecting actuated pairs.

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When actuated pairs are R₁, R₇ and R₂, the limb groups are as follows: (a) Limb group I: P₁R₁P₃ limb, P₂R₂R₃R₄ limb; (b) Limb group II: P₂R₅P₁R₁P₃ limb, R₂R₃R₄ limb; (c) Limb group III: P₁R₁R₅R₃R₄ limb, R₂P₂ R₅P₁P₃ limb.

The conclusion 1 mentioned above shows that the limb group II will lead to the error of the calculation of the DoF. For the irrational limb group, the rationality of the inputs is analyzed only through the remaining limb groups. The analysis process is as follows:

In limb group I, the first limb is P₁R₁P₃ limb and the second limb is P₂R₂R₃R₄ limb, the twist system of the first limb is:

The twist system of the second limb in limb group I:

Based on Eqs. (19)–(20), the reciprocal screw systems of each limb are as follows:

Based on Eqs. (21)–(22), the mechanism twist system $_I is:

According to Eq. (23), the DoF is correct. The two limb reciprocal screw systems after locking the inputs are as follows:

In which, a₁, b₁ and b₁ are determined by the equation group $\left\{\begin{array}{c}{a}_{\mathrm{1}}{y}_{{\mathrm{R}}_{\mathrm{4}}}-b_{\mathrm{1}}{x}_{{\mathrm{R}}_{\mathrm{4}}}+c_{\mathrm{1}}=\mathrm{0}\\ a_{\mathrm{1}}(y_{{\mathrm{R}}_{\mathrm{4}}}-y_{{\mathrm{R}}_{\mathrm{3}}})+b_{\mathrm{1}}(x_{{\mathrm{R}}_{\mathrm{3}}}-x_{{\mathrm{R}}_{\mathrm{4}}})=\mathrm{0}\end{array}\right.$. The quantity of additional reciprocal screw is equal to the number of actuated pairs, indicating that the inputs of each limb are reasonable. The mechanism constraint screw system after locking inputs, namely, ${\mathit{\$}}_{\mathrm{I}}^{r^{\prime }}$ is as follows:

The rank of Eq. (26) is six, namely, this PM loses all DoFs at this time. However, Eq. (25) only demonstrates one configuration, it is needed to analyze whether there is remnant-freedom in other configurations.

Let r₁, r₂ and r₃ indicates vectors of guideway, bar 2 and bar 1, respectively. It can be obtained from Eqs. (27)–(29):

In order to avoid singularity, just meet the condition: |r₃|<|r₂|. In fact, this PM meets this condition, so dead point dose not present, namely, this PM dose not present remnant-freedom in any configuration, so the inputs are reasonable. Using the same method, the ${\mathit{\$}}_{\mathrm{III}}^{r^{\prime }}$ is as follows:

The rank of Eq. (34) is less than six, so the PM has remnant-freedom at this time. Contrast to limb group I and II, the case of this problem is excess kinematic pairs in limb group III, which leads to insufficient constraint of moving platform. Therefore, the other conclusion is obtained: in the process of input analysis, the limb group with fewer kinematic pairs should be preferred. Conclusion 1 and 2 can be called limb group selection rule.

According to analyze different inputs and limb groups (comparative analysis seen in Table 1), the reasonable inputs and limb group of 3-DoF PM with limbs of embedding structures are determined, and limb group selection rule is summarized. This research lays on a foundation for input rationality analysis for similar PM.

Table 1Comparison of different inputs.

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