Keywords

1 Introduction

The SCARA parallel robot can achieve 1 rotation and 3 translations and has been used widely in industries such as sorting, packaging and assembling. The SCARA serial robot first developed by Hiroshi [1] has 4 dofs. It has a cantilever design. So it features low rigidity, low effective load and high arm inertia. It is very hard for its end effector to move at high speed.

The SCARA parallel robot FlexPicker developed by ABB Corporation [2] has one output platform and one sub-platform (see Fig. 10). It is obtained by adding a RUPU branch to the original Delta parallel robot [3]. The SCARA parallel robot Par4 developed by Pierrot [4–6] has one output platform and 2 sub-platforms (see Fig. 9a). Relative motion between the two sub-platforms is converted to rotation of the output platform. This robot features high flexibility, high speed and high rigidity. Based on Par4 robot, Adept Corporation developed the SCARA parallel robot Quattro, which boasts the highest operation speed [7–9]. Some innovative work has also been done by Huang Tian on this type of SCARA parallel robot [10–13]. The SCARA parallel robot X4 has one output platform and 4 parallel-connected branches (see Fig. 7a). It features simple structure and can realize a rotation within the range of ±90° [14].

Obviously, the most important part of the above SCARA parallel robots is the 3T1R PM involved. In this paper, topological structure synthesis of 3T1R PM is discussed systematically using the method based on POC equations [15–23] in order to provide the readers with more comprehensive understanding about the topological structure of 3T1R PM and to obtain some more new 3T1R PMs with potential practical use.

The topological structure synthesis method based on POC equations introduced in this paper is the further development of our early work on topological structure synthesis of PMs and now becomes a systematic and easy-to-follow method for topological structure synthesis of PMs due to some new achievements of this paper, such as: (a) general method for topological structure synthesis of simple branches based on POC equation for serial mechanisms; (b) general method for topological structure synthesis of complex branches based on replacement of sub-SOCs in a SOC branch by topologically equivalent sub-PMs; (c) general method for determination of geometrical conditions for assembling branches between the fixed platform and the moving platform; and (d) general criteria for selection of driving pairs.

2 Theoretical Bases

2.1 POC Equation for Serial Mechanism

The POC equation for serial mechanism [15, 16, 21, 23] is

$$ M_{S} \text{ = }\bigcup\nolimits_{i = 1}^{m} {M_{{J_{i} }} } \text{ = }\bigcup\nolimits_{j = 1}^{k} {M_{{sub - SOC_{j} }} } $$
(1)

where, \( M_{S} \)—POC set of the end link relative to the frame link, m—number of kinematic pairs, \( M_{{J_{i} }} \)—POC set of the ith kinematic pair (Table 1), \( M_{{sub - SOC_{j} }} \)—POC set of the jth sub-SOC (12 sub-SOCs containing R and P pairs only are listed in Table 2).

Table 1. POC set of kinematic pair
Full size table
Table 2. POC set of sub-SOC
Full size table

“Union” operation rules for Eq. (1) include 8 linear operation rules and 2 nonlinear criteria [21, 23].

2.2 POC Equation for PM

The POC equation for PM [15, 16, 21, 23] is

$$ M_{Pa} \text{ = } \cap_{j = 1}^{(v + 1)} M_{{b_{j} }} $$
(2)

where, \( M_{Pa} \)—POC set of the moving platform, \( M_{{b_{j} }} \)—POC set of the end link in the jth branch, \( \nu \)—number of independent loop.

“Intersection” operation rules for Eq. (2) include 12 linear operation rules and 2 nonlinear criteria [21, 23].

According to Eq. (2), there is

$$ M_{{b_{\text{j}} }} \supseteq M_{Pa} $$
(3)

2.3 DOF Formula

  1. (1)

    The DOF formula [22] is

    $$ F = \sum\limits_{i = 1}^{m} {f_{i} } - \sum\limits_{j = 1}^{\nu } {\xi_{{L_{j} }} } $$
    (4a)
    $$ \xi_{{L_{j} }} \text{ = }\dim .\left\{ {\left( {\bigcap\nolimits_{j = 1}^{j} {M_{{b_{j} }} } } \right)\bigcup {M_{{b_{(j + 1)} }} } } \right\} $$
    (4b)

    where, F—DOF of mechanism, f i —DOF of the ith kinematic pair, \( \xi_{{L_{j} }} \)—number of independent displacement equations for the jth independent loop, \( M_{{b_{j} }} \)—POC set of the end link in the jth branch.

  2. (2)

    Criteria for driving pair selection [22]

    For the mechanism with DOF = F, select and lock (make the two links connected by a kinematic pair into one integral link) F kinematic pairs. Only if DOF of the obtained new mechanism is 0, the selected F kinematic pairs can all be used as driving pairs simultaneously.

3 Method for Topological Structure Synthesis of Branch

3.1 Method for Topological Structure Synthesis of Simple Branch

The branch containing no loop is called simple branch. It is also referred to as SOC branch. Main steps of SOC branch topological structure synthesis include:

  • Step 1. Determine POC set and DOF of SOC branch

    1. (1)

      Determine POC set of SOC branch \( (M_{S} ) \) according to Eq. (3).

    2. (2)

      Determine DOF of SOC branch according to dimension of \( M_{S} \).

  • Step 2. Determine kinematic pair combination scheme of SOC branch

    According to Eq. (1), the number of kinematic pairs of SOC branch containing only R and P pairs shall meet the following requirements:

    1. (1)

      DOF of SOC branch

      $$ F = m_{R} + m_{P} $$
      (5)

      where, \( F \) – DOF of branch, \( m_{R} \) – number of R pair, \( m_{P} \) – number of P pair.

    2. (2)

      Range of R pair number

      $$ m_{R} \ge \dim \{ M_{S} (r)\} $$
      (6)

      where, \( \dim \{ M_{S} (r)\} \) – number of independent rotation elements in \( M_{S} \).

    3. (3)

      Range of P pair number

      $$ m_{P} \le \begin{array}{*{20}c} {\dim \{ M_{S} (t)\} } \\ \end{array} $$
      (7)

      where, \( \dim \{ M_{S} (t)\} \) – number of independent translation elements in \( M_{S} \).

  • Step 3. Determine sub-SOCs contained in the SOC branch

    With \( M_{S} \) and the kinematic pair combination scheme being known, determine the sub-SOCs contained in SOC branch according to Eq. (1), Tables 1 and 2. Connect these sub-SOCs in tandem and obtain the desired SOC branch.

  • Step 4. Check POC set of the obtained SOC branch

    For each SOC branch obtained in step 3, check whether its POC set complies with the design requirement according to Eq. (1).

Example 4.1.

Topological structure synthesis of SOC branch with 4 DOFs (3T1R)

  • Step 1. Basic functions of the SOC branch

    1. (1)

      There is \( M_{S} = \left[ \begin{aligned} t^{3} \hfill \\ r^{1} \hfill \\ \end{aligned} \right] \). Select an arbitrary point on the end link as base point o’.

    2. (2)

      F = dim{MS} = 4.

  • Step 2. Determine kinematic pair combination scheme of SOC branch

    Since \( \begin{array}{*{20}c} {\dim \{ M_{S} (r)\} } \\ \end{array} = 1 \) and \( \begin{array}{*{20}c} {\dim \{ M_{S} (t)\} } \\ \end{array} = 3 \), there must be \( m_{R} \ge 1 \) and \( m_{P} \le 3 \) according to Eqs. (5, 6 and 7). So, there are 3 pair combination schemes according to Eq. (1) and Table 2: 3R1P, 2R2P and 1R3P.

  • Step 3. Determine sub-SOCs contained in the SOC branch

    • Case 1: 3R1P

    According to Eq. (1) and Table 2, this branch contains \( sub - SOC\{ - R\parallel R\parallel R - \} \) and a P pair. The SOC branch shall be \( SOC\{ - R\parallel R\parallel R - P - \} \).

    • Case 2: 2R2P

    According to Eq. (1) and Table 2, this branch contains \( sub - SOC\{ - R\parallel R - P - \} \) and a P pair. The SOC branch shall be \( SOC\{ - R\parallel R - P - P - \} \).

    • Case 3: 1R3P

    According to Eq. (1) and Table 2, this branch contains \( sub - SOC\{ - R - P - P - \} \) and a P pair. The SOC branch shall be \( SOC\{ - R - P - P - P - \} \).

  • Step 4. Check POC set of the obtained SOC branch

3.2 Method for Topological Structure Synthesis of Hybrid Branch

The branch containing one or more loop(s) is called hybrid branch. It is also referred to as HSOC branch.

If two kinematic chains have same POC set, they are considered as topologically equivalent. Similarly, if a sub-PM (PM) and a sub-SOC have same POC set, the sub-PM and the sub-SOC are topologically equivalent. If a SOC branch and a HSOC have the same POC set, the HSOC is topologically equivalent to the SOC branch.

Two-branch sub-PM, i.e. single-loop mechanism, is usually used in HSOC branch. These sub-PMs, together with their topologically equivalent sub-SOCs and there POC sets are listed in Table 3.

Table 3. Two-branch sub-PMs and their topologically equivalent sub-SOCs
Full size table

Generally, a HSOC branch can be obtained by replacing a sub-SOC in a SOC branch with a topologically equivalent sub-PM. According to Eq. (1), this HSOC branch and the original SOC branch have the same POC set.

For example, the SOC branch \( \{ - R\parallel R\parallel C - \} \) in Fig. 1a can be re-expressed as \( \{ - R\parallel R\parallel R\parallel P - \} \). POC set of its sub-SOC \( \{ - R\parallel R\parallel P - \} \) is \( \left[ \begin{aligned} t^{1} (\parallel P) \cup t^{1} ( \bot R) \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} r^{1} (\parallel R) \hfill \\ \end{aligned} \right] \). So this sub-SOC is topologically equivalent to No. 3 sub-PM in Table 3.

Fig. 1.
figure 1

Two topologically equivalent (3T1R) branches

Full size image

Replace sub-SOC \( \{ - R\parallel R\parallel P - \} \) with this No. 3 sub-PM and obtain the HSOC branch in Fig. 1b. The branch in Fig. 1a and the branch in Fig. 1b have the same POC set.

4 Geometrical Conditions for Assembling Branches

Since POC set of each branch contains more elements than (or at least the same number of elements as) POC set of the PM, these branches shall be so assembled between the two platforms that some elements of their POC sets are eliminated to obtain the desired POC set of the PM. The basic equations used to determine geometrical conditions for assembling branches will be discussed in the following two different cases.

Case 1.

After the first k branches are assembled, a sub-PM will be obtained. If POC set of this sub-PM still contains any element that is not included in POC set of the PM, the (k + 1)th branch shall be so assembled between the two platforms that this unwanted element shall be eliminated during intersection operation between POC sets.

  1. A.

    If a rotational element is to be eliminated, the following three intersection operation equations shall be used.

    $$ \left[ {r^{1} (\parallel R_{i} )} \right]_{bi} \cap \left[ {r^{1} (\parallel R_{j} )} \right]_{b\,j} = \left[ {r^{0} } \right]_{\;Pa} ,\quad if\,R_{i} \nparallel R_{j} $$
    (8a)
    $$ \left[ {r^{1} (\parallel R_{i} )} \right]_{bi} \cap \left[ {r^{2} (\parallel \diamondsuit (R_{j1} ,R_{j2} ))} \right]_{{b{\text{j}}}} = \left[ {r^{0} } \right]_{Pa} ,\;\;\;if\;R_{i} \nparallel (\diamondsuit (R_{j1} ,R_{j2} )) $$
    (8b)
    $$ \begin{aligned} & \left[ {r^{2} (\parallel \diamondsuit (R_{1} ,R_{2} ))} \right]_{bi} \cap \left[ {r^{2} (\parallel \diamondsuit (R_{j1} ,R_{j2} ))} \right]_{bj} \\ & = \left[ {r^{1} (\parallel (\diamondsuit (R_{i1} ,R_{i2} ) \cap (\diamondsuit (R_{j1} ,R_{j2} )))} \right]_{Pa} ,if\;(\diamondsuit (R_{i1} ,R_{i2} ))\nparallel (\diamondsuit (R_{j1} ,R_{j2} )) \\ \end{aligned} $$
    (8c)
  1. B.

    If a translational element is to be eliminated, the following three intersection operation equations shall be used.

    $$ \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{bi} \cap \left[ {t^{1} (\parallel P_{j}^{ * } )} \right]_{b\,j} = \left[ {t^{0} } \right]_{Pa} ,if\;P_{i}^{ * } \nparallel P_{j}^{ * } $$
    (9a)
    $$ \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{bi} \cap \left[ {t^{1} (\parallel P_{j}^{ * } )} \right]_{b\,j} = \left[ {t^{0} } \right]_{Pa} ,if\;P_{i}^{ * } \nparallel P_{j}^{ * } $$
    (9b)
    $$ \begin{aligned} & \left[ {t^{2} (\parallel \diamondsuit (P_{1}^{ * } ,P_{2}^{ * } ))} \right]_{bi} \cap \left[ {t^{2} (\parallel \diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } ))} \right]_{bj} \\ & = \left[ {t^{1} (\parallel (\diamondsuit (P_{i1}^{ * } ,P_{i2}^{ * } ) \cap (\diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } )))} \right]_{Pa} ,if\;(\diamondsuit (P_{i1}^{ * } ,P_{i2}^{ * } ))\nparallel (\diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } )) \\ \end{aligned} $$
    (9c)

    Note: P* – translation of P pair or derivative translation of R pair or H pair.

Case 2.

After the first k branches are assembled, a sub-PM will be obtained. If POC set of this sub-PM is identical to POC set of the PM, the (k + 1)th branch shall be so assembled between the two platforms that no element shall be eliminated during intersection operation between POC sets.

  1. C.

    The following intersection operation equations shall not eliminate any rotational element.

    $$ \left[ {r^{1} (\parallel R_{i} )} \right]_{bi} \cap \left[ {r^{1} (\parallel R_{j} )} \right]_{bj\,} = \left[ {r^{1} (\parallel R_{i} )} \right]_{Pa} ,if\;R_{i} \parallel R_{j} $$
    (10a)
    $$ \left[ {r^{1} (\parallel R_{i} )} \right]_{bi} \cap \left[ {r^{2} \diamondsuit \left( {\left( {R_{j1} ,R_{j2} } \right)} \right)} \right] = \left[ {r^{1} (\parallel R_{i} )} \right]_{Pa} ,if\;R_{i} \parallel (\diamondsuit (R_{j1} ,R_{j2} )) $$
    (10b)
    $$ \begin{aligned} & \left[ {r^{2} (\parallel \diamondsuit (R_{1} ,R_{2} ))} \right]_{bi} \cap \left[ {r^{2} (\parallel \diamondsuit (R_{j1} ,R_{j2} ))} \right] \\ & = \left[ {r^{2} (\parallel \diamondsuit R_{i1} ,R_{i2} )} \right]_{Pa} ,if\;(\diamondsuit (R_{i1} ,R_{i2} ))\parallel (\diamondsuit (R_{j1} ,R_{j2} )) \\ \end{aligned} $$
    (10c)
    $$ \left[ {r^{1} (\parallel R_{i} )} \right]_{bi} \cap \left[ {r^{3} } \right]_{bj} = \left[ {r^{1} (\parallel R_{i} )} \right]_{Pa} $$
    (10d)
    $$ \left[ {r^{2} (\parallel \diamondsuit (R_{i1} ,R_{i2} ))} \right]_{bi} \cap \left[ {r^{3} } \right]_{bj} = \left[ {r^{2} (\parallel \diamondsuit (R_{i1} ,R_{i2} ))} \right]_{Pa} $$
    (10e)
    $$ \left[ {r^{3} } \right]_{bi} \cap \left[ {r^{3} } \right]_{bj} = \left[ {r^{3} } \right]_{Pa} $$
    (10f)
  1. D.

    The following intersection operation equations shall no eliminate any translational element.

$$ \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{bi} \cap \left[ {t^{1} (\parallel P_{j}^{ * } )} \right]_{b\,j} = \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{Pa} ,if\;P_{i}^{ * } \parallel P_{j}^{ * } $$
(11a)
$$ \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{bi} \cap \left[ {t^{2} (\parallel \diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } ))} \right] = \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{Pa} ,if\;P_{i}^{ * } \parallel (\diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } )) $$
(11b)
$$ \begin{aligned} & \left[ {t^{2} (\parallel \diamondsuit (P_{1}^{ * } ,P_{2}^{ * } ))} \right]_{bi} \cap \left[ {t^{2} (\parallel \diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } ))} \right] \\ & \ = \left[ {t^{2} (\parallel \diamondsuit P_{i1}^{ * } ,P_{i2}^{ * } )} \right]_{Pa} ,if\;(\diamondsuit (P_{i1}^{ * } ,P_{i2}^{ * } ))\parallel (\diamondsuit (P_{j1}^{ * } ,P_{j2}^{ * } )) \\ \end{aligned} $$
(11c)
$$ \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{bi} \cap \left[ {t^{3} } \right]_{bj} = \left[ {t^{1} (\parallel P_{i}^{ * } )} \right]_{Pa} $$
(11d)
$$ \left[ {t^{2} (\parallel \diamondsuit (P_{i1}^{ * } ,P_{i2}^{ * } ))} \right]_{bi} \cap \left[ {t^{3} } \right]_{bj} = \left[ {t^{2} (\parallel \diamondsuit (P_{i1}^{ * } ,P_{i2}^{ * } ))} \right]_{Pa} $$
(11e)
$$ \left[ {t^{3} } \right]_{bi} \cap \left[ {t^{3} } \right]_{bj} = \left[ {t^{3} } \right]_{Pa} $$
(11f)

By the way, if POC set of the sub-PM obtained after the first k branches are assembled is already identical to POC set of the PM, the (k + 1)th branch can be replaced by any simple branch whose POC set contains at least all elements of POC set of the PM, e.g. the \( SOC\{ - S - S - R - \} \) branch.

5 Topological Structure Synthesis of 3T1R PM

Step 1 Design requirements.

\( M_{Pa} = \left[ {\begin{array}{*{20}c} {t^{ 3} } \\ {r^{ 1} } \\ \end{array} } \right] \); DOF = 4; No P pair or only driving P pair shall be contained; Each branch contains only one driving pair and all driving pairs must be allocated on the same platform.

Step 2 Determine POC set of SOC branch.

According to Eq. (3), POC set of the SOC branch shall be \( M_{{b_{j} }} = \left[ \begin{aligned} t^{3} \hfill \\ r^{1} \hfill \\ \end{aligned} \right] \), \( \left[ \begin{aligned} t^{3} \hfill \\ r^{2} \hfill \\ \end{aligned} \right] \) or \( \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right] \)

Step 3 Topological structure synthesis of SOC branch.

  1. (1)

    Topological structure synthesis of SOC branch with POC set \( \left[ \begin{aligned} t^{3} \hfill \\ r^{1} \hfill \\ \end{aligned} \right] \)

    As discussed in Example 4.1, three candidate SOC branches are obtained (see Table 4).

    Table 4. Feasible SOC branches and corresponding HSOC branches
    Full size table
  2. (2)

    Topological structure synthesis of SOC branch with POC set \( \left[ \begin{aligned} t^{3} \hfill \\ r^{2} \hfill \\ \end{aligned} \right] \)

    Four candidate SOC branches are obtained based on a synthesis process similar to that described in Example 4.1 (see Table 4).

  3. (3)

    Topological structure synthesis of (3T3R) SOC branch with POC set \( \left[ \begin{aligned} t^{3} \hfill \\ r^{3} \hfill \\ \end{aligned} \right] \)

    As we all know, the simplest such SOC branch is \( SOC\{ - S - S - R - \} . \)

Step 4 Topological structure synthesis of HSOC branch.

Sub-SOCs containing P pairs can be replaced by the following sub-PMs.

  1. (1)

    \( sub - SOC\{ - P - \} = sub - PM:\{ - P^{{(4{\text{R}})}} - \} . \)

  2. (2)

    \( sub - SOC\{ - R - P - \} = sub - PM:\{ - R^{(2R2S)} - P^{(2R2S)} - \} . \)

  3. (3)

    \( \,sub - SOC\{ - R - P - P - \} = sub - PM:\{ - R^{(4S)} - P^{(4S)} - P^{(4S)} - \} . \)

  4. (4)

    \( \,sub - SOC\{ - P - P - P - \} = sub - PM:\{ - 2\begin{array}{*{20}c} {SOC} \\ \end{array} \left\{ { - R\parallel R^{(4S)} - P^{(4S)} - P^{(4S)} - } \right\} - \} . \)

  5. (5)

    \( sub - SOC\{ - P - P - \} = sub - PM:\{ - \diamondsuit (P^{(5R1C)} ,P^{(5R1C)} ) - \} . \)

  6. (6)

    \( sub - SOC\{ - P - P - P - \} = sub - PM:\{ - Delta\begin{array}{*{20}c} {PM} \\ \end{array} - \} . \)

After topologically equivalent replacement, 6 types of 3T1R branches and 5 types of 3T2R branches which contain no P pairs are obtained (see Table 4), as shown respectively in Figs. 2 and 3.

Fig. 2.
figure 2

Six types of 3T1R branches without P pairs

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Fig. 3.
figure 3

Five types of 3T2R branches without P pairs

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Step 5 Branch combination scheme.

Based on the 2 SOC branches (No. 4 and No. 8 in Table 4) and 10 HSOC branches (refer to Table 4), different branch combination schemes which can generate the desired POC set and DOF of the PM can be obtained. During branch combination, we shall ensure that the branch shall have simple structure and certain symmetry. Each branch shall have only one driving pair and all driving pairs shall be allocated on the same platform. Table 5 shows some of these branch combination schemes.

Table 5. Branch combination scheme of 3T1R PM
Full size table

Step 6 Determine geometrical conditions for assembling branches.

We will use scheme 1 in Table 5 as an example to explain the process to determine geometrical conditions for assembling branches between two platforms, as shown in Fig. 4a.

  1. (1)

    Select 4 identical SOC branches

    $$ SOC\{ - R_{{{\text{j}}1}} \parallel R_{j2} \parallel R_{j3} - R_{j4} \parallel R_{j5} - \} ,{\text{j}} = 1, 2, 3, 4 $$
  2. (2)

    Select an arbitrary point on moving platform as the base point o’.

  3. (3)

    Determine POC set of SOC branch

    $$ M_{bj} = \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{j3} ,R_{j4} )} \\ \end{array} } \right],{\text{j}} = 1, 2, 3, 4, $$
  4. (4)

    Establish POC equation for PM

    Substitute the desired POC set of the PM and POC set of each branch into Eq. (2) and obtain

    $$ \left[ \begin{aligned} t^{3} \hfill \\ r^{1} \hfill \\ \end{aligned} \right] \Leftarrow \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{13} ,R_{14} )} \\ \end{array} } \right] \cap \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{23} ,R_{24} )} \\ \end{array} } \right] \cap \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{33} ,R_{34} )} \\ \end{array} } \right] \cap \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{43} ,R_{44} )} \\ \end{array} } \right] $$

    Where, “<=” means the POC set on the left side of the equation is to be obtained by intersection operation of all the POC sets on the right side of the equation

  5. (5)

    Determine geometrical conditions for assembling the first two branches.

    In order for the moving platform to obtain the desired POC set, intersection of POC sets of the first two branches must eliminate one rotational element. According to Eq. (8c), when \( R_{13} \nparallel R_{23} \) and \( R_{15} \parallel R_{25} \), the moving platform has only one rotation (||R15) and 3 translations. So the above POC equation can be rewritten as

    $$ \left[ \begin{aligned} t^{3} \hfill \\ r^{1} \hfill \\ \end{aligned} \right] \Leftarrow \left[ \begin{aligned} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} t^{3} \hfill \\ r^{1} (\parallel R_{15} ) \hfill \\ \end{aligned} \right] \cap \left[ \begin{aligned} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} & {} \\ \end{array} t^{3} \hfill \\ r^{2} (\parallel \diamondsuit (R_{33} ,R_{34} ) \hfill \\ \end{aligned} \right] \cap \left[ \begin{aligned} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} & {} \\ \end{array} t^{3} \hfill \\ r^{2} (\parallel \diamondsuit (R_{43} ,R_{44} ) \hfill \\ \end{aligned} \right] $$

    Where, the first POC set on right side of the equation is POC set of the sub-PM form obtained after the first two branches are assembled.

  6. (6)

    Determine geometrical conditions for assembling the other two branches

    Since POC set of the sub-PM formed by the first two branches is already identical to the desired POC set of the PM, assembly of the other two branches shall not change POC set of the PM. According to Eq. (10b), geometrical conditions for assembling these two branches are: \( R_{15} \parallel R_{25} \parallel R_{35} \parallel R_{45} \), \( R_{11} \parallel R_{31} \) and \( R_{21} \parallel R_{41} \).

  7. (7)

    Determine allocation scheme of the 4 branches on platforms

    According to geometrical conditions for assembling branches, there may be several different allocation schemes of the 4 branches on platforms. One feasible allocation scheme is:

    $$ SOC\{ - R_{{{\text{j}}1}} \parallel R_{j2} \parallel \overbrace {{R_{j3} \bot R_{j4} }}^{{}}\parallel R_{j5} - \} ,R_{15} \parallel R_{25} \parallel R_{35} \parallel R_{45} ,R_{11} \parallel R_{31} ,R_{21} \parallel R_{41} ,R_{21} \parallel R_{41} , $$

    (Axes of R11, R21, R31 and R41 lie in the same plane and form a square)

    The PM assembled based on this allocation scheme is shown in Fig. 4a. The PMs shown in Fig. 4b and c also satisfy the geometrical assembling conditions.

  8. (8)

    Discussion

    Since POC set of the sub-PM formed by the first two branches and the two platforms is already identical to the desired POC set of the PM, the other two branches can be replaced by two simpler \( SOC\{ - S - S - R - \} \) branches, see Fig. 4d.

    For each branch combination scheme in Table 5, corresponding PM(s) can be obtained similarly, as shown in Figs. 4, 5, 6, 7, 8, 9, 10, 11 and 12.

Fig. 4.
figure 4

Four PMs with No.6 SOC in Table 4

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Step 7 Check DOF of the PM.

Check the above obtained 18 PMs for their DOFs. Take the PM in Fig. 4a as example, Eqs. (4a, b) are used to calculate its DOF.

  1. (1)

    Check POC set of branch: POC set of each branch is

    $$ M_{bj} = \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{j3} ,R_{j4} )} \\ \end{array} } \right],{\text{j}} = 1, 2, 3, 4. $$
  2. (2)

    Calculate DOF of the PM: Substitute \( M_{bj} \) into Eq. (4b) and obtain \( \xi_{L1} = 6,\;\xi_{L2} = 5,\;\xi_{L3} = 5 \). Then substitute \( \xi_{{L_{j} }} \) into Eq. (4a) and obtain DOF of the PM

    $$ F = \sum\limits_{i = 1}^{m} {f_{i} } - \sum\limits_{j = 1}^{\nu } {\xi_{{L_{j} }} } = 20 - (6 + 5 + 5) = 4 $$

All the other 17 PMs can be checked in the same way.

Step 8 Select driving pairs.

According to criteria for driving pair selection (refer to Sect. 2.3), check whether the 4 R pairs on (Fig. 5) the fixed platform (Fig. 4a) can all be selected as driving pairs.

  1. (1)

    Suppose the 4 R pairs \( (R_{11} ,R_{21} ,R_{31} ,R_{41} ) \) to be driving pairs and lock them, a new PM can be obtained. Topological structure of each branch is \( SOC\{ - R_{j2} \parallel R_{j3} - R_{j4} \parallel R_{j5} - \} ,{\text{j}} = 1, 2, 3, 4 \).

  2. (2)

    According to Eq. (1), POC set of each branch is

    $$ M_{bj} = \left[ {\begin{array}{*{20}c} {t^{3} } \\ {r^{2} (\parallel \diamondsuit (R_{j3} ,R_{j4} )} \\ \end{array} } \right],{\text{j}} = 1, 2, 3, 4. $$
  3. (3)

    Determine DOF of the new PM

    Substitute \( M_{bj} \) into Eq. (4b) and obtain \( \xi_{L1} = 6,\;\xi_{L2} = 5,\;\xi_{L3} = 5 \). Then there is

    $$ F^{ * } = \sum\limits_{i = 1}^{m} {f_{i} } - \sum\limits_{j = 1}^{\nu } {\xi_{{L_{j} }} } = 1 6- (6 + 5 + 5) = 0 $$
  4. (4)

    Since DOF of the obtained new PM is \( F^{ * } = 0 \), the 4 R pairs (Fig. 6) on the fixed platform can be selected as (Fig. 8) driving pairs simultaneously according to Sect. 2.3.

    Fig. 5.
    figure 5

    Two PMs with No.7 HSOC in Table 4

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    Fig. 6.
    figure 6

    Two PMs with No.8 HSOC in Table 4

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All the other 18 PMs can be checked in the same (Fig. 11) way and find that the 4 R pairs on the fixed (Fig. 12) platform of each PM can be selected as driving pairs simultaneously.

6 Conclusions

  1. (1)

    18 types of 3T1R PMs with no P pairs are obtained using the topological structure synthesis of PMs based on POC equations, as shown in Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12. Among these 18 types of PMs, 3 types of 3T1R PMs shown in Figs. 7a, 9a and 10 respectively have been used in SCARA robot design. The other 15 types of 3T1R PMs also feature simple structure and may have the possibility to be used in new SCARA robot design.

    Fig. 7.
    figure 7

    Two PMs with No.9 HSOC in Table 4

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    Fig. 8.
    figure 8

    Two PMs with No.1 HSOC in Table 4

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    Fig. 9.
    figure 9

    Two PMs with No.5 HSOC in Table 4

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    Fig. 10.
    figure 10

    PM with No.6 HSOC in Table 4

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    Fig. 11.
    figure 11

    PM with No.3 HSOC in Table 4

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    Fig. 12.
    figure 12

    Two PMs with No.4 HSOC in Table 4

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  2. (2)

    The topological structure synthesis method based on POC equations discussed in this paper is the further development of our early work on topological structure synthesis of PMs. New achievements of this paper include: (a) general method for topological structure synthesis of simple branches based on POC equation for serial mechanisms; (b) general method for topological structure synthesis of complex branches based on replacement a sub-SOC in a SOC branch with a topologically equivalent sub-PM; (c) general method for determination of geometrical conditions for assembling branches between platforms (for the same geometrical condition, there may be several different branch assembling schemes); (d) general criteria for selection of driving pairs.

  3. (3)

    The three basic equations (Eqs. (1, 2 and 4a, b)) unveil the internal relations among topological structure, POC set and DOF. They are the theoretical basis for mechanism structure analysis and synthesis.