Kinematic and Dynamic Modelling for a Class of Hybrid Robots Composed of m Local ClosedLoop Linkages Appended to an nLink Serial Manipulator
Abstract
:Featured Application
Abstract
1. Introduction
2. Kinematics of a Generalized Mechanism for The Hybrid Robots Class III
3. Dynamic Modelling of the Hybrid Robots Class III
4. Example: Kinematic and Dynamic Modelling of a Real Robot
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
$${r}_{C1}=\left[\begin{array}{c}0\\ \frac{{b}_{1}}{2}\\ 0\end{array}\right]$$

$${r}_{\mathrm{C}7}=\left[\begin{array}{c}\left({L}_{21}+{q}_{22}\frac{{L}_{22}}{2}\right)C{q}_{1}C\left({q}_{2}+{q}_{21}\right)+{c}_{2}C{q}_{1}C{q}_{2}\\ \left({L}_{21}+{q}_{22}\frac{{L}_{22}}{2}\right)S{q}_{1}C\left({q}_{2}+{q}_{21}\right)+{c}_{2}S{q}_{1}C{q}_{2}\\ \left({L}_{21}+{q}_{22}\frac{{L}_{22}}{2}\right)S\left({q}_{2}+{q}_{21}\right)+{c}_{2}S{q}_{2}+{b}_{1}\end{array}\right]$$

$${r}_{C2}=\left[\begin{array}{c}\frac{{L}_{2}}{2}C{q}_{1}C{q}_{2}\\ \frac{{L}_{2}}{2}S{q}_{1}S{q}_{2}\\ \frac{{L}_{2}}{2}S{q}_{2}+{b}_{1}\end{array}\right]$$

$${r}_{C8}=\left[\begin{array}{c}C{q}_{1}\left(\frac{{L}_{42}}{2}+\left({L}_{3}{b}_{3}\right)C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ \mathrm{S}{q}_{1}\left(\frac{{L}_{42}}{2}+\left({L}_{3}{b}_{3}\right)C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ \left({L}_{3}{b}_{3}\right)\mathrm{S}\left({q}_{2}+{q}_{3}\right)+{L}_{2}\mathrm{S}{q}_{2}+{b}_{1}\end{array}\right]$$

$${r}_{C3}=\left[\begin{array}{c}\frac{1}{2}{L}_{11}C{q}_{1}C{q}_{11}{L}_{1}C{q}_{1}\\ \frac{1}{2}{L}_{11}S{q}_{1}C{q}_{11}{L}_{1}S{q}_{1}\\ \frac{1}{2}{L}_{11}S{q}_{11}+{b}_{1}\end{array}\right]$$

$${r}_{C9}=\left[\begin{array}{c}C{q}_{1}\left(\frac{{L}_{4}}{2}+{L}_{42}+\left({L}_{3}{b}_{3}\right)C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ S{q}_{1}\left(\frac{{L}_{4}}{2}+{L}_{42}+\left({L}_{3}{b}_{3}\right)C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ \left({L}_{3}{b}_{3}\right)S\left({q}_{2}+{q}_{3}\right)+{L}_{2}S{q}_{2}+{b}_{1}\end{array}\right]$$

$${r}_{\mathrm{C}4}=\left[\begin{array}{c}\left({L}_{11}+{q}_{12}\frac{{L}_{4}}{2}\right)C{q}_{1}C{q}_{11}{L}_{1}C{q}_{1}\\ \left({L}_{11}+{q}_{12}\frac{{L}_{4}}{2}\right)S{q}_{1}C{q}_{11}{L}_{1}S{q}_{1}\\ \left({L}_{11}+{q}_{12}\frac{{L}_{4}}{2}\right)S{q}_{11}+{b}_{1}\end{array}\right]$$

$${r}_{C10}=\left[\begin{array}{c}C{q}_{1}\left(\frac{{L}_{2}}{2}C{q}_{2}{c}_{1}\right)\\ S{q}_{1}\left(\frac{{L}_{2}}{2}C{q}_{2}{c}_{1}\right)\\ \frac{{L}_{2}}{2}S{q}_{2}+{b}_{1}\end{array}\right]$$

$$\begin{array}{l}{r}_{C5}=\left[\begin{array}{c}C{q}_{1}\left(\frac{\left({L}_{3}2{b}_{3}\right)}{2}C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ S{q}_{1}\left(\frac{\left({L}_{3}2{b}_{3}\right)}{2}C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ \frac{\left({L}_{3}2{b}_{3}\right)}{2}\mathrm{S}\left({q}_{2}+{q}_{3}\right)+{L}_{2}S{q}_{2}+{b}_{1}\end{array}\right]\\ \text{}\end{array}$$

$${r}_{C11}=\left[\begin{array}{c}C{q}_{1}\left({L}_{2}C{q}_{2}\frac{{c}_{2}}{2}\right)\\ S{q}_{1}\left({L}_{2}C{q}_{2}\frac{{c}_{2}}{2}\right)\\ {L}_{2}S{q}_{2}+{b}_{1}\end{array}\right]$$

$$\begin{array}{l}{r}_{C6}=\left[\begin{array}{c}C{q}_{1}\left(\frac{{L}_{21}}{2}C\left({q}_{2}+{q}_{21}\right)+{c}_{2}C{q}_{2}\right)\\ S{q}_{1}\left(\frac{{L}_{21}}{2}C\left({q}_{2}+{q}_{21}\right)+{c}_{2}C{q}_{2}\right)\\ \frac{{L}_{21}}{2}\mathrm{S}\left({q}_{2}+{q}_{21}\right)+{c}_{2}S{q}_{2}+{b}_{1}\end{array}\right]\\ \text{}\end{array}$$

$$\begin{array}{l}{r}_{C12}=\left[\begin{array}{c}C{q}_{1}\left(\frac{\left({L}_{3}{b}_{3}\right)}{2}C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ S{q}_{1}\left(\frac{\left({L}_{3}{b}_{3}\right)}{2}C\left({q}_{2}+{q}_{3}\right)+{L}_{2}C{q}_{2}\right)\\ \frac{\left({L}_{3}{b}_{3}\right)}{2}\mathrm{S}\left({q}_{2}+{q}_{3}\right)+{L}_{2}S{q}_{2}+{c}_{1}+{b}_{1}\end{array}\right]\\ \text{}\end{array}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{1}=\left[\begin{array}{l}0\\ 0\\ {\dot{q}}_{1}\end{array}\right]$$

$$\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{2}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{2}\mathrm{S}{q}_{1}\text{}\\ \text{}{\dot{q}}_{2}\mathrm{C}{q}_{1}\text{}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{1}\end{array}\right]$$

$${\mathsf{\omega}}_{3}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{11}\mathrm{S}{q}_{1}\\ \text{}{\dot{q}}_{11}\mathrm{C}{q}_{1}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{1}\end{array}\right]$$

$${\mathsf{\omega}}_{4}={\omega}_{3}$$

$$\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{5}=\left[\begin{array}{l}\hspace{0.17em}S{q}_{1}\left({\dot{q}}_{2}+{\dot{q}}_{3}\right)\text{}\\ C{q}_{1}\left({\dot{q}}_{2}+{\dot{q}}_{3}\right)\text{}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{1}\end{array}\right]$$

$$\hspace{0.17em}{\mathsf{\omega}}_{6}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}S{q}_{1}\left({\dot{q}}_{2}+{\dot{q}}_{21}\right)\text{}\\ C{q}_{1}\left({\dot{q}}_{2}+{\dot{q}}_{21}\right)\text{}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{1}\end{array}\right]$$

$${\mathsf{\omega}}_{7}={\mathsf{\omega}}_{6}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{8}=\hspace{0.17em}{\mathsf{\omega}}_{1}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{9}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{4}\mathrm{C}{q}_{1}\text{}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{4}\mathrm{S}{q}_{1}\text{}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\dot{q}}_{1}\end{array}\right]$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{10}=\hspace{0.17em}{\mathsf{\omega}}_{2}$$

$$\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{11}=\hspace{0.17em}{\mathsf{\omega}}_{1}$$

$$\hspace{0.17em}\hspace{0.17em}{\mathsf{\omega}}_{12}=\hspace{0.17em}{\mathsf{\omega}}_{5}$$

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$\mathbf{Loop}\text{}{\mathbf{\Theta}}_{1}$  $\mathbf{Loop}\text{}{\mathbf{\Theta}}_{2}$  $\mathbf{Loop}\text{}{\mathbf{\Theta}}_{3}$  $\mathbf{Loop}\text{}{\mathbf{\Theta}}_{4}$  

Link  ${l}_{2}$  ${l}_{11}$  ${l}_{12}$  ${l}_{3}$  ${l}_{21}$  ${l}_{22}$  ${l}_{2}$  ${l}_{31}$  ${l}_{32}$  ${l}_{3}$  ${l}_{41}$  ${l}_{42}$ 
$\theta $  ${q}_{2}$  ${q}_{11}$  0  ${q}_{3}$  ${q}_{21}$  0  ${q}_{2}$  ${q}_{31}$  ${q}_{32}$  ${q}_{3}^{\u2019}$  ${q}_{41}$  ${q}_{42}$ 
$d$  0  0  ${q}_{12}$  0  0  ${q}_{22}$  0  0  0  0  0  0 
$a$  ${L}_{2}$  ${L}_{1}\mathrm{cos}{q}_{11}$  0  ${L}_{3}{b}_{3}$  ${L}_{2}\mathrm{cos}{q}_{21}$  0  ${L}_{2}$  ${L}_{2}$  0  ${L}_{3}{b}_{3}$  ${L}_{3}{b}_{3}$  0 
$\alpha $  0  $\frac{\pi}{2}$  $\frac{\pi}{2}$  0  $\frac{\pi}{2}$  $\frac{\pi}{2}$  0  0  0  0  0  $\frac{\pi}{2}$ 
Index $\mathit{j}$  Link  Mass Centre (m)  Mass (kg)  

${\mathbf{x}}_{\mathbf{C}}$  ${\mathbf{y}}_{\mathbf{C}}$  ${\mathbf{z}}_{\mathbf{C}}$  
1  ${l}_{1}$  0  ${b}_{1}/2$  0  ${m}_{1}=81.3$ 
2  ${l}_{2}$  ${L}_{2}/2$  0  0  ${m}_{2}=50.6$ 
3  ${l}_{11}$  0  0  ${L}_{11}/2$  ${m}_{3}=25.6$ 
4  ${l}_{12}$  0  ${L}_{12}/2$  0  ${m}_{4}=5.70$ 
5  ${l}_{3}$  ${L}_{3}/2$  0  0  ${m}_{5}=58.4$ 
6  ${l}_{21}$  0  0  ${L}_{21}/2$  ${m}_{6}=25.6$ 
7  ${l}_{22}$  0  ${L}_{22}/2$  0  ${m}_{7}=5.70$ 
8  ${l}_{42}$  0  0  ${L}_{42}/2$  ${m}_{8}=37.7$ 
9  ${l}_{4}$  0  0  ${L}_{4}/2$  ${m}_{9}=73.1$ 
10  ${l}_{31}$  ${L}_{2}/2$  0  0  ${m}_{10}=7.20$ 
11  ${l}_{32}$  ${c}_{2}/2$  0  0  ${m}_{11}=12.9$ 
12  ${l}_{41}$  ${L}_{3}/2$  0  0  ${m}_{12}=6.50$ 
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Chu, A.M.; Nguyen, C.D.; Vu, M.H.; Duong, X.B.; Nguyen, T.A.; Le, C.H. Kinematic and Dynamic Modelling for a Class of Hybrid Robots Composed of m Local ClosedLoop Linkages Appended to an nLink Serial Manipulator. Appl. Sci. 2020, 10, 2567. https://doi.org/10.3390/app10072567
Chu AM, Nguyen CD, Vu MH, Duong XB, Nguyen TA, Le CH. Kinematic and Dynamic Modelling for a Class of Hybrid Robots Composed of m Local ClosedLoop Linkages Appended to an nLink Serial Manipulator. Applied Sciences. 2020; 10(7):2567. https://doi.org/10.3390/app10072567
Chicago/Turabian StyleChu, Anh My, Cong Dinh Nguyen, Minh Hoan Vu, Xuan Bien Duong, Tien Anh Nguyen, and Chi Hieu Le. 2020. "Kinematic and Dynamic Modelling for a Class of Hybrid Robots Composed of m Local ClosedLoop Linkages Appended to an nLink Serial Manipulator" Applied Sciences 10, no. 7: 2567. https://doi.org/10.3390/app10072567