Dynamics modeling and impact response of a rescue robot with two flexible manipulators

Abstract

Rescue robots have been widely used in rescue work with mobility, adaptability, and flexibility. However, during the implementation of rescue tasks, rescue robots are often affected by external impacts, which will result in a significant decrease in the accuracy of the robot’s end effector, thereby affecting the efficiency of rescue. Developing an accurate dynamic model is the prerequisite to increase the control efficiency. The present paper presents a novel dynamic modeling method based on the transfer matrix method and analyzes the impact response characterizations of a rescue robot with single and dual flexible manipulators. The rescue robot with a single flexible manipulator is analyzed at the beginning. Firstly, the dynamic model of individual components is established using the transfer matrix method. And the overall model is obtained by iterating the transfer matrices and transfer equations of each component. Secondly, the natural frequency and vibration modes of the system can be determined when the boundary conditions are given. Thirdly, the orthogonality of the augmented eigenvectors of the dynamic model is verified, and the impact response of the end effector was obtained. Finally, an experimental setup was constructed to verify the accuracy of using the multi-body system transfer matrix method for dynamic response analysis. Additionally, this novel approach of dynamic modeling and impact response analysis was applied to the rescue robot with two flexible manipulators, and its effectiveness was evaluated. Overall, the modeling method presented in this paper provides a theoretical basis for further improving the accuracy and reducing the vibration of the end effector of rescue robots.

Appendix

1.1 A.1. Spatial rigid body with one end input and one end output transfer matrix

$$ U_{1} = \left[ {\begin{array}{*{20}c} 1 & {b_{2} } & { - b_{1} } & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ {mw^{2} (b_{2} - c_{c2} )} & {u_{42} } & {u_{43} } & 1 & 0 & {b_{2} } \\ { - mw^{2} (b_{1} - c_{c1} )} & {u_{52} } & {u_{53} } & 0 & 1 & { - b_{1} } \\ {mw^{2} } & {mw^{2} c_{c2} } & { - mw^{2} c_{c1} } & 0 & 0 & 1 \\ \end{array} } \right] $$

$$ \begin{gathered} u_{42} = - w^{2} [J_{I,x} - m(b_{2} c_{c2} + b_{3} c_{c3} )] \, , \, u_{43} = - w^{2} (J_{I,xy} + mb_{2} c_{c1} ) \hfill \\ u_{52} = - w^{2} (J_{I,xy} + mb_{1} c_{c2} ){ , }u_{53} = - w^{2} [J_{I,y} - m(b_{3} c_{c3} + b_{1} c_{c1} )] \hfill \\ \end{gathered} $$

1.2 A.2. Planar rigid body with one end input and one end output transfer matrix

$$ U = \left[ {\begin{array}{*{20}c} 1 & 0 & { - b_{2} } & 0 & 0 & 0 \\ 0 & 1 & {b_{1} } & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ { - mw^{2} (b_{2} - c_{{c_{2} }} )} & {mw^{2} (b_{1} - c_{{c_{1} }} )} & { - w^{2} [m(b_{2} c_{{c_{2} }} + b_{1} c_{{c_{1} }} )]} & 1 & { - b_{2} } & {b_{1} } \\ {mw^{2} } & 0 & { - mw^{2} c_{{c_{2} }} } & 0 & 1 & 0 \\ 0 & {mw^{2} } & {mw^{2} c_{{c_{1} }} } & 0 & 0 & 1 \\ \end{array} } \right] $$

1.3 A.3. Transverse vibration beam transfer matrix

$$ U = B\left( l \right)B^{ - 1} \left( 0 \right) = \left[ {\begin{array}{*{20}c} {S(\lambda l)} & {\frac{T(\lambda l)}{\lambda }} & {\frac{U(\lambda l)}{{EI\lambda^{2} }}} & {\frac{V(\lambda l)}{{EI\lambda^{3} }}} \\ {\lambda V(\lambda l)} & {S(\lambda l)} & {\frac{T(\lambda l)}{{EI\lambda }}} & {\frac{U(\lambda l)}{{EI\lambda^{2} }}} \\ {EI\lambda^{2} U(\lambda l)} & {EI\lambda V(\lambda l)} & {S(\lambda l)} & {\frac{T(\lambda l)}{\lambda }} \\ {EI\lambda^{3} T(\lambda l)} & {EI\lambda^{2} U(\lambda l)} & {\lambda V(\lambda l)} & {S(\lambda l)} \\ \end{array} } \right] $$

$$ \lambda = \sqrt[4]{{{{\mathop m\limits^{\_} \omega^{2} } \mathord{\left/ {\vphantom {{\mathop m\limits^{\_} \omega^{2} } {(EI)}}} \right. \kern-0pt} {(EI)}}}} $$

$$ S\left( x \right) = \frac{\cosh x + \cos x}{2},\;\;\;T\left( x \right) = \frac{\sinh x + \sin x}{2} $$

$$ U\left( x \right) = \frac{\cosh x - \cos x}{2},\;\;\;\;V\left( x \right) = \frac{\sinh x - \sin x}{2} $$

1.4 A.4. Planar flexible joint transfer matrix

$$ {\mathbf{U}}_{t}^{\prime } = \left[ {\begin{array}{*{20}c} {I_{3} } & {U_{1,2} } \\ {0_{3 \times 3} } & {I_{3} } \\ \end{array} } \right]_{6 \times 6} \;\;\;\;{\mathbf{U}}_{m}^{\prime } = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ {mw^{2} } & 0 & 1 & 0 \\ 0 & {mw^{2} } & 0 & 1 \\ \end{array} } \right]\;\;\;\;U = {\mathbf{U}}_{t}^{\prime } {\mathbf{U}}_{m}^{\prime } $$

$$ {\mathbf{U}}_{1,2} = \left[ {\begin{array}{*{20}c} 0 & { - 1/K_{{\text{x}}} } & 0 \\ 0 & 0 & { - 1/K_{{\text{y}}} } \\ { - 1/K_{{\text{z}}}^{\prime } } & 0 & 0 \\ \end{array} } \right] $$

where ${\mathbf{U}}^{\prime}_{t}$ is the transfer matrix of the plane elastic hinge. ${\mathbf{U}}^{\prime}_{m}$ is the plane concentration mass transfer matrix. ${\mathbf{U}}$ is the transfer matrix of the plane flexible joint.

1.5 A.5. Spatial longitudinal vibration spring transfer matrix

$$ {\mathbf{U}} = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{3} } & {{\mathbf{\rm O}}_{3 \times 3} } \\ {\mathbf{K}} & {{\mathbf{I}}_{3} } \\ \end{array} } \right]_{6 \times 6} \;\;\;\;{\mathbf{K}} = \left[ {\begin{array}{*{20}c} { - 1/K_{{\text{x}}} } & 0 & 0 \\ 0 & { - 1/K_{{\text{y}}} } & 0 \\ 0 & 0 & { - 1/K_{{\text{z}}} } \\ \end{array} } \right] $$

1.6 A.6. Spatial flexible joints transfer matrix

$$ {\mathbf{U}}_{t} = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{3} } & {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{U}}_{14} } \\ {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{I}}_{3} } & {{\mathbf{U}}_{23} } & {{\mathbf{O}}_{3 \times 3} } \\ {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{I}}_{3} } & {{\mathbf{O}}_{3 \times 3} } \\ {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{O}}_{3 \times 3} } & {{\mathbf{I}}_{3} } \\ \end{array} } \right]_{12 \times 12} \;\;\;\;{\mathbf{U}}_{m} = \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{3} } & {{\mathbf{O}}_{3 \times 3} } \\ {{\mathbf{U}}_{21} } & {{\mathbf{I}}_{3} } \\ \end{array} } \right]_{6 \times 6} \;\;\;\;\;{\mathbf{U}} = {\mathbf{U}}^{\prime}_{t} {\mathbf{U}}^{\prime}_{m} $$

$$ {\mathbf{U}}_{23} = \left[ {\begin{array}{*{20}c} {1/K_{x}{\prime} } & 0 & 0 \\ 0 & {1/K_{y}{\prime} } & 0 \\ 0 & 0 & {1/K_{z}{\prime} } \\ \end{array} } \right]\;\;\;{\mathbf{U}}_{21} = \left[ {\begin{array}{*{20}c} {m\omega^{2} } & 0 & 0 \\ 0 & {m\omega^{2} } & 0 \\ 0 & 0 & {m\omega^{2} } \\ \end{array} } \right]\;\;\;{\mathbf{U}}_{14} = \left[ {\begin{array}{*{20}c} { - 1/K_{{\text{x}}} } & 0 & 0 \\ 0 & { - 1/K_{{\text{y}}} } & 0 \\ 0 & 0 & { - 1/K_{{\text{z}}} } \\ \end{array} } \right] $$

where ${\mathbf{U}}^{\prime}_{t}$ is the transfer matrix of the spatial elastic hinge. ${\mathbf{U}}^{\prime}_{m}$ is the spatial concentration mass transfer matrix. ${\mathbf{U}}$ is the transfer matrix of the spatial flexible joint.

1.7 A.7. Spatial vibrating Euler–Bernoulli beam transfer matrix

${\text{u}} = \left[ {\begin{array}{*{20}l} {u_{{1,1}} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {u_{{1,10}} } & 0 & 0 \\ 0 & {u_{{2,2}} } & 0 & 0 & 0 & {u_{{2,6}} } & 0 & 0 & {u_{{2,9}} } & 0 & {u_{{2,11}} } & 0 \\ 0 & 0 & {u_{{3,3}} } & 0 & {u_{{3,5}} } & 0 & 0 & {u_{{3,8}} } & 0 & 0 & 0 & {u_{{3,12}} } \\ 0 & 0 & 0 & {u_{{4,4}} } & 0 & 0 & {u_{{4,7}} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {u_{{5,3}} } & 0 & {u_{{5,5}} } & 0 & 0 & {u_{{5,8}} } & 0 & 0 & 0 & {u_{{5,12}} } \\ 0 & {u_{{6,2}} } & 0 & 0 & 0 & {u_{{6,6}} } & 0 & 0 & {u_{{6,9}} } & 0 & {u_{{6,11}} } & 0 \\ 0 & 0 & 0 & {u_{{7,4}} } & 0 & 0 & {u_{{7,7}} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {u_{{8,3}} } & 0 & {u_{{8,5}} } & 0 & 0 & {u_{{8,8}} } & 0 & 0 & 0 & {u_{{8,12}} } \\ 0 & {u_{{9,2}} } & 0 & 0 & 0 & {u_{{9,6}} } & 0 & 0 & {u_{{9,9}} } & 0 & {u_{{9,11}} } & 0 \\ {u_{{10,1}} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {u_{{10,10}} } & 0 & 0 \\ 0 & {u_{{11,2}} } & 0 & 0 & 0 & {u_{{11,6}} } & 0 & 0 & {u_{{11,9}} } & 0 & {u_{{11,11}} } & 0 \\ 0 & 0 & {u_{{12,3}} } & 0 & {u_{{12,5}} } & 0 & 0 & {u_{{12,8}} } & 0 & 0 & 0 & {u_{{12,12}} } \\ \end{array} } \right]$$u_{1,1} = u_{10,10} = \cos \beta_{x} x_{i} \;\;\;\;\;u_{1,10} = \frac{{ - \sin \beta_{x} x_{i} }}{{\beta_{x} EA_{i} }}\;\;\;\;u_{10,1} = \beta_{x} EA_{i} \sin \beta_{x} x_{i}$$u_{4,4} = u_{7,7} = \cos \gamma_{{\theta_{x} }} x_{i} \;\;\;\;\;u_{4,7} = \frac{{ - \sin \gamma_{{\theta_{x} }} x_{i} }}{{\gamma_{{\theta_{x} }} \left( {GJ_{p} } \right)_{i} }}\;\;\;\;u_{7,4} = \gamma_{{\theta_{x} }} x_{i} \left( {GJ_{p} } \right)_{i} \sin \gamma_{{\theta_{x} }} x_{i}$$u_{2,2} = u_{6,6} = u_{9,9} = u_{11,11} = S(\lambda_{y} x_{i} )\;\;\;\;u_{3,3} = u_{5,5} = u_{8,8} = u_{12,12} = S(\lambda_{z} x_{i} )$ $u_{2,6} = u_{9,11} = \frac{{T(\lambda_{y} x_{i} )}}{{\lambda_{y} }}\;\;\;\;u_{2,9} = u_{6,11} = \frac{{U(\lambda_{y} x_{i} )}}{{EI_{z,i} \lambda_{y}^{2} }}\;\;\;\;u_{2,11} = \frac{{V(\lambda_{y} x_{i} )}}{{EI_{z,i} \lambda_{y}^{3} }}$ $u_{5,8} = \frac{{T(\lambda_{z} x_{i} )}}{{EI_{z,i} \lambda_{z} }}\;\;\;u_{3,5} = u_{8,12} = \frac{{ - T(\lambda_{z} x_{i} )}}{{\lambda_{z} }}\;\;\;u_{3,8} = u_{5,12} = \frac{{ - U(\lambda_{z} x_{i} )}}{{EI_{y,i} \lambda_{z}^{2} }}$ $u_{3,12} = \frac{{V(\lambda_{z} x_{i} )}}{{EI_{y,i} \lambda_{z}^{3} }}\;\;\;u_{6,9} = \frac{{T(\lambda_{y} x_{i} )}}{{EI_{z,i} \lambda_{y} }}\;\;\;\;u_{5,3} = u_{12,8} = - \lambda_{z} V(\lambda_{z} x_{i} )$ $u_{6,2} = u_{11,9} = - \lambda_{y} V(\lambda_{y} x_{i} )\;\;\;\;u_{8,3} = u_{12,5} = - EI_{y,i} \lambda_{z}^{2} U(\lambda_{z} x_{i} )\;\;\;\;u_{8,5} = EI_{y,i} \lambda_{z} V(\lambda_{z} x_{i} )$ $u_{9,2} = u_{11,6} = EI_{z,i} \lambda_{y}^{2} U(\lambda_{y} x_{i} )\;\;\;u_{9,6} = EI_{z,i} \lambda_{y} V(\lambda_{y} x_{i} )\;\;\;u_{11,2} = EI_{z,i} \lambda_{y}^{3} T(\lambda_{y} x_{i} )$ $u_{12,3} = EI_{y,i} \lambda_{z}^{3} T(\lambda_{z} x_{i} )\;\;\;\lambda_{y} =$$\gamma_{{\theta_{x} }} = \sqrt {{{\left( {\rho J_{p} } \right)_{i} \omega^{2} } \mathord{\left/ {\vphantom {{\left( {\rho J_{p} } \right)_{i} \omega^{2} } {\left( {GJ_{p} } \right)}}} \right. \kern-0pt} {\left( {GJ_{p} } \right)}}} \;\;\;\;0 \le x_{i} \le l_{i}$