1 Introduction
2 Basic Concepts on the Used Graph Theory
3 Graph Framework
3.1 Kinematic Function Unit
Elements | Meanings | Coded values |
---|---|---|
x1 | Rotational motion | R |
Translational motion | T | |
Swinging motion | S | |
x2 | Continuous motion | 1 |
Intermittent motion | −1 | |
x3 | Reciprocating motion | 1 |
Non-reciprocating motion | −1 |
3.2 Kinematic Link Graph
Mechanisms | KFUs |
---|---|
Crank-rocker | \(\{ CR_{1}^{RS} ,(R,1, - 1),(S,1,1)\}\) |
\(\{ CR_{1}^{SR} ,(S,1,1),(R,1, - 1)\}\) | |
Spur gear | \(\{ SG_{1}^{RR} ,(R,1, - 1),(R,1, - 1)\}\) |
Geneva wheel | \(\{ GW_{1}^{RR} ,(R,1, - 1),(R, - 1, - 1)\}\) |
Cam-follower | \(\{ CF_{1}^{RT} ,(R,1, - 1),(T,1,1)\}\) |
Pawl-Ratchet wheel | \(\{ PRW_{1}^{SR} ,(S,1,1),(R, - 1, - 1)\}\) |
System input | \(\{ SI_{1}^{R} ,(0,0,0),(R,1, - 1)\}\) |
System output | \(\{ SO_{1}^{R} ,(R, - 1, - 1),(0,0,0)\}\) |
3.3 Graph Representation of Design Candidate
3.3.1 Walk Representation
3.3.2 Path Representation
3.4 Weight Matrix Theorem
4 Polynomial Operations
4.1 Polynomial-Walk Operation
4.2 Edge Sequence Operation
Execution steps | Results for each step |
k = 1, k + 1 = 2 < (8/2 = 4) | \(e_{in} = e_{2}\),\(e_{out} = e_{16}\), thus \(S_{1 - set}^{in} = \{ \{ e_{2} \} \}\),\(S_{1 - set}^{out} = \{ \{ e_{16} \} \}\), then \(S_{2 - set}^{in} = \{ \{ e_{2} ,e_{6} \} ,\{ e_{2} ,e_{7} \} \}\), \(S_{2 - set}^{out} = \{ \{ e_{7} ,e_{16} \} \}\) |
k = 2, (k + 1 = 3) < (8/2 = 4) | \(S_{3 - set}^{in} = \{ \{ e_{2} ,e_{6} ,e_{9} \} ,\{ e_{2} ,e_{6} ,e_{10} \} \}\), \(S_{3 - set}^{out} = \{ \{ e_{9} ,e_{7} ,e_{16} \} ,\{ e_{13} ,e_{7} ,e_{16} \} \}\) |
k = 3, (k + 1 = 4) = (8/2 = 4) | \(S_{4 - set}^{in} = \{ \{ e_{2} ,e_{6} ,e_{9} ,e_{6} \} ,\{ e_{2} ,e_{6} ,e_{10} ,e_{13} \} ,\) \(\{ e_{2} ,e_{6} ,e_{9} ,e_{7} \} \}\), \(S_{4 - set}^{out} = \{ \{ e_{6} ,e_{9} ,e_{7} ,e_{16} \} ,\{ e_{10} ,e_{13} ,e_{7} ,e_{16} \} \} .\) Since \(Ter(\{ e_{2} ,e_{6} ,e_{9} ,e_{6} \} ) = Init(\{ e_{10} ,e_{13} ,e_{7} ,e_{16} \} )\) = \(CR_{1}^{SR}\), and \(Ter(\{ e_{2} ,e_{6} ,e_{10} ,e_{13} \} ) = Init(\{ e_{6} ,e_{9} ,e_{7} ,e_{16} \} )\) = \(CR_{1}^{RS}\), \(ES_{1} (M_{1}^{Kw} )\) = \(\{ e_{2} ,e_{6} ,e_{9} ,e_{6} \} \cup \{ e_{10} ,e_{13} ,e_{7} ,e_{16} \}\) = \(\{ e_{2} ,e_{6} ,e_{9} ,e_{6} ,e_{10} ,e_{13} ,e_{7} ,e_{16} \},\) \(ES_{2} (M_{1}^{Kw} )\) = \(\{ e_{2} ,e_{6} ,e_{10} ,e_{13} \} \cup \{ e_{6} ,e_{9} ,e_{7} ,e_{16} \}\) = \(\{ e_{2} ,e_{6} ,e_{10} ,e_{13} ,e_{6} ,e_{9} ,e_{7} ,e_{16} \}\) |
4.3 Vertex Sequence Operation
5 Computational Flowchart of Synthesis Approach
- Step 1: Set design requirements and extract KFUs from the chosen mechanisms.
- Step 2: Construct KLG based on the two connection rules, and generate its vertices set V(KLG) and weighted matrix \(A_{\omega } (KLG)\) so that the KLG can be recognized and used in computer.
- Step 3: Run polynomial-walk operation to compute polynomial-walk \(P_{W}^{{N_{\text{max} } }} (KLG)\), and then separate and save all its monomial-walks \(M_{i}^{Kw}\)in set \(S_{{M^{Kw} }}\).
- Step 4: Run edge sequence operation for each \(M_{i}^{Kw}\) to figure out all its edge sequences \(ES_{\ell } (M_{{^{i} }}^{Kw} )\), and then save them in set SES.
- Step 5: Run vertex sequence operation for each \(ES_{\ell } (M_{{^{i} }}^{Kw} )\) to figure out its vertex sequence \(VS_{\ell } (M_{{^{i} }}^{Kw} )\). Then, a walk representation \(W_{\ell }\) of \(M_{i}^{Kw}\) is formulated.
- Step 6: Generated and save the vertex sequence \(VS^{P} (W_{\ell } )\) and adjacency matrix AP(\(W_{\ell }\)) to the path representation of \(W_{\ell }\).
6 Design Example
6.1 Extract and Formulate the KFUs
Mechanisms | KFUs |
---|---|
Slider-crank | \(\{ SC_{1}^{RT} ,(R,1, - 1),(T,1,1)\}\) |
\(\{ SC_{1}^{TR} ,(T,1,1),(R,1, - 1)\}\) | |
Spur gear | \(\{ SG_{1}^{RR} ,(R,1, - 1),(R,1, - 1)\}\) |
Worm-gear | \(\{ WG_{1}^{RR} ,(R,1, - 1),(R,1, - 1)\}\) |
Cam-follower | \(\{ CF_{1}^{RT} ,(R,1, - 1),(T,1,1)\}\) |
System input | \(\{ SI_{1}^{R} ,(0,0,0),(R,1, - 1)\}\) |
System output | \(\{ SO_{1}^{R} ,(T,1,1),(0,0,0)\}\) |
6.2 Construct Kinematic Link Graph KLG
6.3 Compute Polynomial-Walk
6.4 Compute the Walk Representations and Path Representations to the Design Candidates
ID | Monomial-walks | Walk representations | Path representations | Design candidates |
---|---|---|---|---|
\(M_{1}^{Kw}\) | \(e_{2} e_{10}\) | \(W_{1} (M_{1}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{1}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{1}^{Kw} ) = \{ e_{2} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{1}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{1}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{2}^{Kw}\) | \(e_{4} e_{20}\) | \(W_{1} (M_{2}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{2}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{2}^{Kw} ) = \{ e_{4} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{2}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{2}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{3}^{Kw}\) | \(e_{1} e_{6} e_{10}\) | \(W_{1} (M_{3}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{3}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{3}^{Kw} ) = \{ e_{1} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{3}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{3}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{4}^{Kw}\) | \(e_{1} e_{8} e_{20}\) | \(W_{1} (M_{4}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{4}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{4}^{Kw} ) = \{ e_{1} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{4}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{4}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{5}^{Kw}\) | \(e_{3} e_{10} e_{16}\) | \(W_{1} (M_{5}^{Kw} ) =\left\{ {\begin{array}{*{20}l} {VS_{1} (M_{5}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{5}^{Kw} ) = \{ e_{3} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{5}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{5}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{6}^{Kw}\) | \(e_{3} e_{18} e_{20}\) | \(W_{1} (M_{6}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{6}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{6}^{Kw} ) = \{ e_{3} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{6}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{6}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{7}^{Kw}\) | \(e_{1} e_{5} e_{6} e_{10}\) | \(W_{1} (M_{7}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{7}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{7}^{Kw} ) = \{ e_{1} ,e_{5} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{7}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{7}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{8}^{Kw}\) | \(e_{2} e_{9} e_{10} e_{12}\) | \(W_{1} (M_{8}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{8}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{8}^{Kw} ) = \{ e_{2} ,e_{9} ,e_{12} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{8}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,SC_{2}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{8}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{9}^{Kw}\) | \(e_{1} e_{5} e_{8} e_{20}\) | \(W_{1} (M_{9}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{9}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{1}^{RR} ,{\kern 1pt} {\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{9}^{Kw} ) = \{ e_{1} ,e_{5} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{9}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,WG_{2}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{9}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{10}^{Kw}\) | \(e_{1} e_{7} e_{10} e_{16}\) | \(W_{1} (M_{10}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{10}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{10}^{Kw} ) = \{ e_{1} ,e_{7} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{10}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{10}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{11}^{Kw}\) | \(e_{3} e_{6} e_{10} e_{15}\) | \(W_{1} (M_{11}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{11}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{11}^{Kw} ) = \{ e_{3} ,e_{15} ,e_{6} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{11}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{11}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{12}^{Kw}\) | \(e_{2} e_{9} e_{14} e_{20}\) | \(W_{1} (M_{12}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{12}^{Kw} ) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{12}^{Kw} ) = \{ e_{2} ,e_{9} ,e_{14} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{12}^{Kw} )) = \{ SI_{1}^{R} ,SC_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{12}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{13}^{Kw}\) | \(e_{4} e_{10} e_{12} e_{19}\) | \(W_{1} (M_{13}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{13}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{12}^{Kw} ) = \{ e_{4} ,e_{19} ,e_{12} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{13}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{13}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{14}^{Kw}\) | \(e_{1} e_{7} e_{18} e_{20}\) | \(W_{1} (M_{14}^{Kw} ) =\left\{ {\begin{array}{*{20}l} {VS_{1} (M_{14}^{Kw} ) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,{\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{14}^{Kw} ) = \{ e_{1} ,e_{7} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{14}^{Kw} )) = \{ SI_{1}^{R} ,WG_{1}^{RR} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{14}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{15}^{Kw}\) | \(e_{3} e_{8} e_{15} e_{20}\) | \(W_{1} (M_{15}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{15}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{15}^{Kw} ) = \{ e_{3} ,e_{15} ,e_{8} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{15}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,WG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{15}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{16}^{Kw}\) | \(e_{3} e_{10} e_{16} e_{17}\) | \(W_{1} (M_{16}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{16}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{1}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{16}^{Kw} ) = \{ e_{3} ,e_{17} ,e_{16} ,e_{10} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{16}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{2}^{RR} ,SC_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{16}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{17}^{Kw}\) | \(e_{4} e_{14} e_{19} e_{20}\) | \(W_{1} (M_{17}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{17}^{Kw} ) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,{\kern 1pt} CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{17}^{Kw} ) = \{ e_{4} ,e_{19} ,e_{14} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{17}^{Kw} )) = \{ SI_{1}^{R} ,CF_{1}^{RT} ,SC_{1}^{TR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{17}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) | |
\(M_{18}^{Kw}\) | \(e_{3} e_{17} e_{18} e_{20}\) | \(W_{1} (M_{18}^{Kw} ) = \left\{ {\begin{array}{*{20}l} {VS_{1} (M_{18}^{Kw} ) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{1}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \} } \hfill \\ {ES_{1} (M_{18}^{Kw} ) = \{ e_{3} ,e_{17} ,e_{18} ,e_{20} \} } \hfill \\ \end{array} } \right.\) | \(VS^{P} (W_{1} (M_{18}^{Kw} )) = \{ SI_{1}^{R} ,SG_{1}^{RR} ,SG_{2}^{RR} ,CF_{1}^{RT} ,SO_{1}^{T} \}\) \(A^{P} (W_{1} (M_{18}^{Kw} )) = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]\) |