1 Introduction
2 Introduction of Atlas Method
Atlas elements | Physical significance | Helical vector type |
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Force constraints | Line vector \( {\$ }^{r} = ({\mathbf{S}} ; { }{\mathbf{S}}_{0} ) \) | |
Couple constraint | Couple vector \( {\$ }^{r} = (0 ; { }{\mathbf{S}}) \) | |
The rotating DOF | Line vector \( {\$} = ({\mathbf{S}} ; { }{\mathbf{S}}_{0} ) \) | |
The moving DOF | Couple vector \( {\$ } = (0 ; { }{\mathbf{S}}) \) |
3 Synthesis Method of Redundant Actuated Parallel Mechanism with Closed Loop Units
4 Type Synthesis of the First 2R1T Closed-Loop Unit Redundant Actuated Parallel Mechanism
4.1 Preliminary Synthesis of 2R1T Redundant Parallel Mechanisms with Closed-Loop Unit
Constraint allocation schemes |
4.2 Instantaneous Verification of the First 2R1T Redundant Actuated Parallel Mechanism with Closed Loop Unit
4.2.1 Verification of DOF Under Initial Configuration
4.2.2 Verification of DOF Under General Configuration
5 Type Synthesis of the Second 2R1T Closed-Loop Unit Redundant Actuated Parallel Mechanism
5.1 Preliminary Synthesis of 2R1T Redundant Parallel Mechanisms with Closed-Loop Unit
The second type PMs with closed loop unit | |
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\( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right] - 2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{\text{UPR}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right] - 2{\text{UPR}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{\text{URP}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{\text{URP}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right] - 2{\text{URP}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} -
\overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right] - 2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R}} (P\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (P\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[
{5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right] - 2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\overline{\text{R}}} \text{PR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{{\text{P}}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\overline{\text{R}}} \text{RP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\overline{\text{R}}} \text{PP})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {5\overline{\text{R}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R} }}} \overline{\text{R}} } \right] - 2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {{\text{P}} \, \overline{\text{R}} \, \overline{\text{U}} \,\overline{\text{R}}\, {\text{P}}} \right]{ - }2{\text{P}} \,\underline{R} \,(\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {\overline{\text{R}} \text{P}\overline{\text{U}} \text{P}\overline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) | \( 2\left[ {(\text{RR}\overline{\text{R}} \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {{\text{P}} \, \overline{\text{R}} \, \overline{\text{U}} \, \overline{\text{R}} \, {\text{P}}} \right]{ - }2\overline{\text{U}} \overline{\text{R}} \, \overline{\text{R}} \) \( 2\left[ {\overline{\text{R}} P\overline{\text{U}} P\overline{\text{R}} } \right]{ - }2\overline{\text{U}} \overline{\text{R}} \overline{\text{R}} \) \( 2\left[ {(\text{RR}\overline{\overline{\text{R}}} \text{RR})_{\text{E}} - \underline{{\overline{{\overline{\text{R}} }} }} \underline{\text{R}} } \right] - 2{\text{UPR}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {{\text{P}} \,\overline{\text{R}}\, \overline{\text{U}} \,\overline{\text{R}}\, {\text{P}}} \right]{ - }2{\text{UPR}} \) \( 2\left[ {\overline{\text{R}} \text{P}\overline{\text{U}} \text{P}\overline{\text{R}} } \right]{ - }2{\text{UPR}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {{\text{P}} \,\overline{\text{R}}\, \overline{\text{U}} \,\overline{\text{R}}\, {\text{P}}} \right]{ - }2{\text{URP}} \) \( 2\left[ {\overline{\text{R}} \text{P}\overline{\text{U}} \text{P}\overline{\text{R}} } \right]{ - }2{\text{URP}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(PP\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {{\text{P}} \,\overline{\text{R}} \, \overline{\text{U}} \overline{\text{R}} \, {\text{P}}} \right]{ - }2\underline{R} \left( {\underline{R} RP} \right)_{E} \) \( 2\left[ {\overline{\text{R}} P\overline{\text{U}} P\overline{\text{R}} } \right]{ - }2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{RP}\overline{\text{R}} \text{PR})_{\text{E}} - \overline{{\underline{\text{R}} }} \overline{{\underline{\text{R}} }} } \right] - 2\underline{\text{R}} \left( {\underline{\text{R}} \text{RP}} \right)_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \, \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \(
2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{{\text{R}} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {{\text{P}}{\overline{\text{R}}} {\overline{\text{U}}} {\overline{\text{R}}} {{\text{P}}}} \right]{ - }2{\underline{\text{R}}} \left( {{\underline{\text{R}}} {{\text{PR}}}} \right)_{{{\text{E}}}} \) \( 2\left[ {\overline{\text{R}} \text{P}\overline{\text{U}} \text{P}\overline{\text{R}} } \right]{ - }2\underline{\text{R} } \left( {\underline{\text{R}} \text{PR}} \right)_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \, \overline{\text{R}} \text{RR})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}}
\) \( 2\left[
{(\text{PR}\overline{\text{R}} \, \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \, \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (P\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {({\text{PP}} \overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2\underline{\text{R}} (\text{P}\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {{\text{P}} {\overline{\text{R}}} {\overline{\text{U}}} {\overline{\text{R}}} {{\text{P}}}} \right]{ - }2 {\underline{\text{R}}} ({{\text{P}}} {\underline{\text{RR}}})_{{{\text{E}}}}\) \( 2\left[ {\overline{\text{R}} \text{P}\overline{\text{U}} \text{P}\overline{\text{R}} } \right]{ - }2\underline{\text{R}} (P\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RR}\overline{\text{R}} \overline{\text{R}} \text{R}R)_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PR}\overline{\text{R}} \overline{\text{R}} \text{RP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{R} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{RRURR})_{\text{E}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) \( 2\left[ {(\text{PP}\overline{\text{R}} \overline{\text{R}} \text{PP})_{\text{E}} - \overline{{\underline{\text{R}} }} \underline{\text{R}} } \right]{ - }2{\text{P}}\underline{\text{R}} (\underline{\text{R}} \text{R})_{\text{E}} \) |
5.2 Instantaneous Verification of the Second 2R1T Redundant Actuated Parallel Mechanism with Closed Loop Unit
5.2.1 Verification of DOF under Initial Configuration
5.2.2 Verification of DOF Under General Configuration
6 Type Synthesis of 2T1R Redundant Parallel Mechanisms with Few Branched Chains and Closed-Loop Units
6.1 Preliminary Synthesis of 2T1R Redundant Parallel Mechanisms with Few Branched Chains and Closed-Loop Units
Constraint allocation schemes |
Branch chain constraint line graph and dual DOF graph with couple | Redundant DOF graph | Corresponding branched chain structure | |
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Chains 1 | \( (RR\overline{\overline{\text{R}}} {\text{RR}})_{E} { - }\overline{\overline{\text{R}}} \) \( (R{\text{P}}\overline{\overline{\text{R}}} {\text{PR}})_{E} { - }\overline{\overline{\text{R}}} \) \( ({\text{P}}R\overline{\overline{\text{R}}} {\text{RP}})_{E} { - }\overline{\overline{\text{R}}} \) \( ({\text{RR}} {\overline{\text{R}}} {\overline{\text{R}}} {{\text{RR}}})_{{\text{E}}} { - } {\overline{\text{R}}}\) \( ({\text{RP}} {\overline{\text{R}}} {\overline{\text{R}}} {\text{PR}})_{{\text{E}}} { - } {\overline{\text{R}}} \) \( ({\text{PR}} {\overline{\text{R}}} {\overline{\text{R}}} {\text{RP}})_{{\text{E}}} { - } {\overline{\text{R}}}\) \( 5\overline{R} - \overline{R} \) \( ({\text{P}}R\overline{R} {\text{RP}})_{E} { - }\overline{\text{R}} \) \( (R{\text{P}}\overline{R} {\text{PR}})_{E} { - }\overline{\text{R}} \) \( ({\text{P}}RR\overline{\overline{\text{R}}} {\text{RR}})_{E} { - }\overline{\overline{\text{R}}} \) \( ({\text{P}}R{\text{P}}\overline{\overline{\text{R}}} {\text{PR}})_{E} { - }\overline{\overline{\text{R}}} \) \( ({\text{PRR}} {\overline{\text{R}}} {\overline{\text{R}}} {\text{RR}})_{{\text{E}}} { - } {\overline{\text{R}}}\) \( ({\text{PRP}} {\overline{\text{R}}} {\overline{\text{R}}} {\text{PR}})_{{\text{E}}} { - } {\overline{\text{R}}}\) \( {\text{P}}5\overline{R} - \overline{R} \) \( ({\text{P}}R{\text{P}}\overline{R} {\text{PR}})_{E} { - }\overline{\text{R}} \) | ||
Chains 2 | |||
Branch chain assembly requirements | The force constraints provided by the two branches need to be parallel in space The force constraints provided by each branch chain are perpendicular to the plane formed by the two couple constraints provided by the branch chain The rank of couple constraint is 2 |
Some 2T1R parallel mechanism representatives | |
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\( 2\left[ {(R{\text{P}}\overline{\overline{\text{R}}} {\text{PR}})_{E} { - }\overline{\overline{\text{R}}} } \right] \) | \( 2\left[ {({\text{P}}RR\overline{\overline{\text{R}}} {\text{RR}})_{E} { - }\overline{\overline{\text{R}}} } \right] \) |
\( 2\left[ {({\text{P}}R\overline{\overline{\text{R}}} {\text{RP}})_{E} { - }\overline{\overline{\text{R}}} } \right] \) | \( 2\left[ {({\text{P}}R{\text{P}}\overline{R} {\text{PR}})_{E} { - }\overline{\text{R}} } \right] \) |
\( 2\left[ {5\overline{R} - \overline{R} } \right] \) | \( 2\left[ {({\text{RR}} {\overline{\text{R}}} {\overline{\text{R}}} {\text{RR}})_{{\text{E}}} { - } {\overline{\text{R}}} } \right] \) |
\( 2\left[ {({\text{P}}R\overline{R} {\text{RP}})_{E} { - }\overline{\text{R}} } \right] \) | \( 2\left[ {(R{\text{P}} \, \overline{\text{R}} \, \overline{\text{R}} \, {\text{PR}})_{E} { - }\overline{\text{R}} } \right] \) |
\( 2\left[ {(R{\text{P}}\overline{R} {\text{PR}})_{E} { - }\overline{\text{R}} } \right] \)\( 2\left[ {({\text{P}} \,R{\text{P}} \,\overline{\text{R}} \, \overline{\text{R}} \, {\text{PR}})_{E} { - }\overline{\text{R}} } \right] \) | \( 2\left[ {({\text{P}} \, R \, \overline{\text{R}} \, \overline{\text{R}} \, {\text{RP}})_{E} { - }\overline{\text{R}} } \right] \) |
\( 2\left[ {(RR\overline{\overline{R}} RR)_{E} - \overline{\overline{R}} } \right] \) | \( 2\left[ {{\text{P}}5\overline{R} - \overline{R} } \right] \) |
\( 2\left[ {({\text{P}} \, RR \,\overline{\text{R}} \, \overline{\text{R}} \, {\text{RR}})_{E} { - }\overline{\text{R}} } \right] \) | \( 2\left[ {({\text{P}}R{\text{P}}\overline{\overline{\text{R}}} {\text{PR}})_{E} { - }\overline{\overline{\text{R}}} } \right] \) |