Nonlinear dynamic analysis of the extended telescopic joints manipulator with flexible links

Abstract

In this paper, the open-chain arm containing flexible links connected through telescopic joints is investigated so as to replace robotic chains with revolute joints. In this manipulator, telescopic joints are considered by a simultaneous motion combination of the prismatic and revolute joints. In comparison with revolute–prismatic joints in which prismatic joint hubs are utilized, telescopic joints lack the hub, and the linear motion is generated via a gear. Accordingly, the flexible link is not placed inside the hub. In fact, the link is merely divided into two sections each at a time step. The front section which is considered in the motion equations is placed at the back of the hinge’s point. This part solely creates vibration, which probably affects the motion of other sections. This type of manipulator’s joints due to the structure benefits consists of the optimum weight, simple mechanism, extended workspace acceptable for robotic tools, industrial machinery, and domestic systems. To derive motion equations of N-link flexible manipulator with telescopic joints system, the Euler–Lagrange formulation is employed. The obtained equations are in the time-varying form, and the manipulator link length changes during motion. The derived equations are simulated for a two-link system with different elasticity values, and the results are then analyzed to verify the outcomes. The bifurcation analysis has also been used to prevent unstable vibrations and chaotic responses. It should be noticed that this type of manipulator can be used in a fast and precise mechatronics' system for utilization in space exploration, space station, and spacecraft as well.

Appendices

Appendix 1

Here, the terms of Eq. (7) are completely shown:

$$ \begin{aligned} & C_{2ij} = \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\mathbf{x}}_{i}^{T} {\mathbf{r}}_{ij} d\eta_{i} } \quad {\tilde{\mathbf{C}}}_{1ij} = \int_{0}^{{l_{i} }} {\mu_{i} {\tilde{\mathbf{r}}}_{ij} d\eta_{i} } ;\quad {\mathbf{C}}_{1ij} = \int_{0}^{{l_{i} }} {\mu_{i} {\mathbf{r}}_{ij} d\eta_{i} } ; \\ & C_{5ij} = \int_{0}^{{l_{i} }} {\mu_{i} \eta_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {\tilde{\mathbf{r}}}_{ij} d\eta_{i} } \quad C_{4i} = \int_{{ - \left( {L_{i} - l_{i} } \right)}}^{{l_{i} }} {\mu_{i} \eta_{i}^{2} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {}^{i}{\tilde{\mathbf{x}}}_{i} d\eta_{i} } ;\quad {\mathbf{C}}_{3ijk} = \int_{0}^{{l_{i} }} {\mu_{i} {\mathbf{r}}_{ij}^{T} {\mathbf{r}}_{ik} d\eta_{i} } ; \\ & {\mathbf{C}}_{8ij} = \int_{0}^{{l_{i} }} {\mu_{i} \eta_{i} {}^{i}{\tilde{\mathbf{x}}}^{T}_{i} {\mathbf{r}}_{ij} d\eta_{i} } \quad {\mathbf{C}}_{7ij} = \int_{0}^{{l_{i} }} {\mu_{i} {\tilde{\mathbf{x}}}^{T}_{ij} {\mathbf{r}}_{ik} d\eta_{i} } ;\quad C_{6ijk} = \int_{0}^{{l_{i} }} {\mu_{i} {\tilde{\mathbf{r}}}^{T}_{ij} {\tilde{\mathbf{r}}}_{ik} d\eta_{i} } ; \\ & {{\varvec{\upalpha}}}_{ij} = C_{5ij} + \sum\limits_{k = 1}^{{m_{i} }} {\delta_{ik} \left( t \right)C_{6ijk} } \quad {{\varvec{\upbeta}}}_{ij} = {\mathbf{C}}_{8ij} + \sum\limits_{k = 1}^{{m_{i} }} {\delta_{ik} \left( t \right){\mathbf{C}}_{9ijk} } ;\quad {\mathbf{C}}_{9ijk} = \int_{0}^{{l_{i} }} {\mu_{i} {\tilde{\mathbf{r}}}_{ij} {\mathbf{r}}_{ik} d\eta_{i} } ; \\ & C^{\prime}_{5ij} = \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {\tilde{\mathbf{r}}}_{ij} d\eta_{i} } \quad C^{\prime}_{4i} = 2\int_{{ - \left( {L_{i} - l_{i} } \right)}}^{{l_{i} }} {\mu_{i} \eta_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {}^{i}{\tilde{\mathbf{x}}}_{i} d\eta_{i} } ;\quad \dot{C}_{4i} = 2\dot{\eta }_{i} \int_{{ - \left( {L_{i} - l_{i} } \right)}}^{{l_{i} }} {\mu_{i} \eta_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {}^{i}{\tilde{\mathbf{x}}}_{i} d\eta_{i} } ; \\ & {\dot{\mathbf{\alpha }}}_{ij} = \dot{C}_{5ij} + \sum\limits_{k = 1}^{{m_{i} }} {\dot{\delta }_{ik} \left( t \right)C_{6ijk} } \quad {\mathbf{\alpha^{\prime}}}_{ij} = \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {\tilde{\mathbf{r}}}_{ij} d\eta_{i} } ;\quad \dot{C}_{5ij} = \dot{\eta }_{i} \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\tilde{\mathbf{x}}}_{i}^{T} {\tilde{\mathbf{r}}}_{ij} d\eta_{i} } ; \\ & {\mathbf{C^{\prime}}}_{8ij} = {\mathbf{C}}_{7ij} = \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\tilde{\mathbf{x}}}^{T}_{i} {\mathbf{r}}_{ij} d\eta_{i} } \quad {\mathbf{C^{\prime}}}_{8ij} = {\mathbf{C}}_{7ij} = \int_{0}^{{l_{i} }} {\mu_{i} {}^{i}{\tilde{\mathbf{x}}}^{T}_{i} {\mathbf{r}}_{ij} d\eta_{i} } ;\quad {\dot{\mathbf{\beta }}}_{ij} = {\dot{\mathbf{C}}}_{8ij} + \sum\limits_{k = 1}^{{m_{i} }} {\dot{\delta }_{ik} \left( t \right){\mathbf{C}}_{9ijk} } ; \\ \end{aligned} $$

(47)

where ${}^{i}{\tilde{\mathbf{x}}}_{i}$ and ${\tilde{\mathbf{r}}}_{ij}$ are, respectively, the skew-symmetric matrices associated with the vectors ${}^{i}{\mathbf{x}}_{i}$ and ${\mathbf{r}}_{ij}$.

Appendix 2: The derivative of the kinetic energy of elastic robotic chain

The derivative of kinetic energy is obtained with respect to $\dot{q}$ and $q$, and then the velocities $\dot{q}$ are differentiated with respect to time:

$$ \begin{aligned} \frac{d}{dt}\left( {\frac{\partial T}{{\partial \dot{q}_{j} }}} \right) & = \sum\limits_{i = 1}^{n} {\frac{{\partial {\mathbf{\ddot{r}}}^{T}_{{o_{i} }} }}{{\partial \dot{q}_{j} }}\left( {B_{0i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{1i} - B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \\ & \quad + \sum\limits_{i = 1}^{n} {\frac{{\partial {\dot{\mathbf{r}}}^{T}_{{o_{i} }} }}{{\partial \dot{q}_{j} }}\left( {B_{0i} {\mathbf{\ddot{r}}}_{{o_{i} }} + {}^{i}{\dot{\mathbf{B}}}_{1i} - 2\dot{B}_{2i} {}^{i}{{\varvec{\upomega}}}_{i} - B_{2i} {}^{i}{\dot{\mathbf{\omega }}}_{i} - {}^{i}{\tilde{\mathbf{\omega }}}_{i} B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \\ & \quad + \sum\limits_{i = 1}^{n} {\frac{{\partial {}^{i}{\dot{\mathbf{\omega }}}^{T}_{i} }}{{\partial \dot{q}_{j} }}\left( {B_{2i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} + {}^{i}{\mathbf{B}}_{5i} } \right)} \\ & \quad + \sum\limits_{i = 1}^{n} {\frac{{\partial {}^{i}{\vec{\mathbf{\omega }}}^{T}_{i} }}{{\partial \dot{q}_{j} }}\left( {B_{2i} {\mathbf{\ddot{r}}}_{{o_{i} }} + {}^{i}{\dot{\mathbf{B}}}_{5i} + {}^{i}{\dot{\mathbf{B}}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} + {}^{i}{\mathbf{B}}_{4i} {}^{i}{\dot{\mathbf{\omega }}}_{i} + {}^{i}{\tilde{\mathbf{\omega }}}_{i} {\mathbf{B}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \\ \end{aligned} $$

(48)

$$ \frac{\partial T}{{\partial q_{j} }} = 0 $$

The derivative of kinetic energy with respect to $\dot{\delta }_{jf}$ and $\delta_{jf}$ encompasses differentiation with respect to ${{\varvec{\upalpha}}}_{ij} ,{{\varvec{\upbeta}}}_{ij} ,{\mathbf{B}}_{1i} ,B_{2i} ,B_{3i} ,{\mathbf{B}}_{4i} ,{\mathbf{B}}_{5i}$ in addition to ${\dot{\mathbf{r}}}_{{o_{i} }}$ and ${}^{i}{{\varvec{\upomega}}}_{i}$.

$$ \begin{gathered} \frac{d}{dt}\left( {\frac{\partial T}{{\partial \dot{\delta }_{if} }}} \right) = \sum\limits_{i = j + 1}^{n} {\frac{{\partial {\mathbf{\ddot{r}}}^{T}_{{o_{i} }} }}{{\partial \dot{\delta }_{if} }}\left( {B_{0i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{1i} - B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum\limits_{i = j + 1}^{n} {\frac{{\partial {\dot{\mathbf{r}}}^{T}_{{o_{i} }} }}{{\partial \dot{\delta }_{if} }}\left( {B_{0i} {\mathbf{\ddot{r}}}_{{o_{i} }} + {}^{i}{\dot{\mathbf{B}}}_{1i} - 2\dot{B}_{2i} {}^{i}{{\varvec{\upomega}}}_{i} - B_{2i} {}^{i}{\dot{\mathbf{\omega }}}_{i} - {}^{i}{\tilde{\mathbf{\omega }}}_{i} B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum\limits_{i = j + 1}^{n} {\frac{{\partial {}^{i}{\dot{\mathbf{\omega }}}^{T}_{i} }}{{\partial \dot{\delta }_{if} }}\left( {B_{2i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum\limits_{i = j + 1}^{n} {\frac{{\partial {}^{i}{{\varvec{\upomega}}}_{i}^{T} }}{{\partial \dot{\delta }_{if} }}\left( {B_{2i} .{\mathbf{\ddot{r}}}_{{o_{i} }} + {}^{i}{\dot{\mathbf{B}}}_{5i} + {}^{i}{\dot{\mathbf{B}}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} + {}^{i}{\mathbf{B}}_{4i} {}^{i}{\dot{\mathbf{\omega }}}_{i} + {}^{i}{\tilde{\mathbf{\omega }}}_{i} {\mathbf{B}}_{4i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\mathbf{\ddot{r}}}^{T}_{{o_{i} }} {\mathbf{C}}_{1if} + \sum\limits_{k = 1}^{{m_{i} }} {\ddot{\delta }_{ik} {\mathbf{C}}_{3ijk} } + \ddot{\eta }_{j} {\mathbf{C}}_{2ij} + {}^{i}{\dot{\mathbf{\omega }}}^{T}_{i} {{\varvec{\upbeta}}}_{ij} + {}^{i}{\vec{\mathbf{\omega }}}^{T}_{i} {\dot{\mathbf{\beta }}}_{ij} \hfill \\ \end{gathered} $$

(49)

$$ \begin{gathered} - \frac{\partial T}{{\partial \delta_{if} }} = - \sum\limits_{i = j + 1}^{n} {\frac{{\partial {\dot{\mathbf{r}}}^{T}_{{o_{i} }} }}{{\partial \delta_{if} }}\left( {B_{0i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{1i} - B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} + {}^{i}{{\varvec{\upomega}}}_{i}^{T} \sum\limits_{j = 1}^{{m_{i} }} {\dot{\delta }_{ij} {\mathbf{C}}_{9ijk} } \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {}^{i}{{\varvec{\upomega}}}_{i}^{T} \dot{\eta }_{i} {\mathbf{C}}_{7ij} - \frac{1}{2}{}^{i}{{\varvec{\upomega}}}_{i}^{T} (C_{5ij}^{T} ){}^{i}{{\varvec{\upomega}}}_{i} - \frac{1}{2}{}^{i}{{\varvec{\upomega}}}_{i}^{T} (\sum\limits_{k = 1}^{{m_{i} }} {\delta_{ik} {\mathbf{C}}_{6ijk} } ){}^{i}{{\varvec{\upomega}}}_{i} \hfill \\ \end{gathered} $$

Finally, the differentiation of kinetic energy with respect to $\dot{\eta },\eta$ is presented:

$$\begin{aligned} \frac{d}{{dt}}\left( {\frac{{\partial T}}{{\partial \dot{\eta }_{j} }}} \right) & = \sum\limits_{{i = j + 1}}^{n} {\frac{{\partial {\mathbf{\ddot{r}}}^{T} _{{o_{i} }} }}{{\partial \dot{\eta }_{j} }}\left( {B_{{0i}} {\mathbf{\dot{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{{1i}} - B_{{2i}} {}^{i}{\mathbf{\omega }}_{i} } \right)} \\ & \quad + \sum\limits_{{i = j + 1}}^{n} {\frac{{\partial {\mathbf{\dot{r}}}^{T} _{{o_{i} }} }}{{\partial \dot{\eta }_{j} }}\left( {B_{{0i}} {\mathbf{\ddot{r}}}_{{o_{i} }} + {}^{i}{\mathbf{\dot{B}}}_{{1i}} - 2\dot{B}_{{2i}} {}^{i}{\mathbf{\omega }}_{i} - B_{{2i}} {}^{i}{\mathbf{\dot{\omega }}}_{i} - {}^{i}{\mathbf{\tilde{\omega }}}_{i} B_{{2i}} {}^{i}{\mathbf{\omega }}_{i} } \right)} \\ & \quad + {\mathbf{\ddot{r}}}^{T} _{{o_{i} }} B_{{0i}} {}^{i}{\mathbf{x}}_{i} + B_{{0i}} \ddot{\eta }_{i} + \sum\limits_{{k = 1}}^{{m_{i} }} {\ddot{\delta }_{{ik}} {\mathbf{C}}_{{2ij}} } - {}^{i}{\mathbf{\dot{\omega }}}_{j}^{T} \sum\limits_{{j = 1}}^{{m_{i} }} {\delta _{{ij}} {\mathbf{C}}_{{7ij}} } - {}^{i}{\mathbf{\omega }}_{j}^{T} \sum\limits_{{j = 1}}^{{m_{i} }} {\dot{\delta }_{{ij}} {\mathbf{C}}_{{7ij}} } \\ - \frac{\partial T}{{\partial \eta }} &= - \sum\limits_{i = j + 1}^{n} {\frac{{\partial {\dot{\mathbf{r}}}^{T}_{{o_{i} }} }}{\partial \eta }\left( {B_{0i} {\dot{\mathbf{r}}}_{{o_{i} }} + {}^{i}{\mathbf{B}}_{1i} - B_{2i} {}^{i}{{\varvec{\upomega}}}_{i} } \right)} \hfill \\ &\quad - \frac{1}{2}{}^{i}{{\varvec{\upomega}}}_{i}^{T} \left( {C^{\prime}_{4i} + \sum\limits_{j = 1}^{{m_{i} }} {\delta_{ij} (C_{5ij}^{\prime T} + } {\mathbf{\alpha^{\prime}}}_{ij} )} \right){}^{i}{{\varvec{\upomega}}}_{i} - {}^{i}{\dot{\mathbf{\omega }}}_{j}^{T} \sum\limits_{j = 1}^{{m_{i} }} {\dot{\delta }_{ij} {\mathbf{C^{\prime}}}_{8ij} } \hfill \\\end{aligned}$$

(50)

Appendix 3: Derivatives of system potential energy and Rayleigh dissipation function

Taking the partial derivative of strain potential energy with respect to state variables results in

$$ \begin{gathered} \frac{{\partial V_{e} }}{{\partial \delta_{if} }} = \sum\limits_{k = 1}^{{m_{i} }} {\delta_{ik} \left( t \right){\mathbf{K}}_{jkf} } - {\mathbf{g}}^{T} \frac{{\partial {}^{i}{\mathbf{R}}_{j} }}{{\partial \delta_{jf} }}\left( {\int_{0}^{{l_{i} }} {\mu \left\{ {\begin{array}{*{20}c} {u_{i} } & {v_{i} } & {w_{i} } \\ \end{array} } \right\}d\eta + \mu l\eta {}^{i}{\mathbf{x}}_{i} } } \right) - {\mathbf{g}}^{T} {}^{i}{\mathbf{R}}_{j} \int_{0}^{{l_{i} }} {\mu {\mathbf{r}}_{ij} d\eta } \hfill \\ \frac{{\partial V_{e} }}{{\partial q_{j} }} = - {\mathbf{g}}^{T} \frac{{\partial {}^{i}{\mathbf{R}}_{j} }}{{\partial q_{j} }}\left( {\int_{0}^{{l_{i} }} {\mu \left\{ {\begin{array}{*{20}c} {u_{i} } & {v_{i} } & {w_{i} } \\ \end{array} } \right\}d\eta + \mu l\eta {}^{i}{\mathbf{x}}_{i} } } \right);\,\,\,\,\,\,\,\,\frac{{\partial V_{e} }}{{\partial \eta_{j} }} = - {\mathbf{g}}^{T} {}^{i}{\mathbf{R}}_{j} \left( {\mu l{}^{i}{\mathbf{x}}_{i} } \right) \hfill \\ \end{gathered} $$

(51)

In addition, the generalized forces related to nonconservative forces are found by differentiation using the Rayleigh dissipation function with respect to generalized velocities. Taking the derivative of Rayleigh dissipation function with respect to ${\delta }_{ij}$ results in

$$ \frac{\partial D}{{\partial \dot{\delta }_{jf} }} = \sum\limits_{k = 1}^{{m_{j} }} {\dot{\delta }_{jk} \left( t \right){\mathbf{D}}_{jkf} } $$

(52)

Similar to ${K}_{jkf}$, ${D}_{jkf}$ for a manipulator with known dimensions can be calculated.