Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator

Abstract

With the development of space technology, lighter and larger space manipulators will be born, of which flexible characteristics are more obvious. The manipulator vibration caused by the flexibility not only reduces the efficiency of the manipulator but also affects the accuracy of the operation. The flexibility of space manipulator mainly comes from structural flexibility of links and transmission flexibility of harmonic gear reducer in joints. The vibrations generated by these two kinds of flexibility are coupled and transformed mutually, making the dynamics characteristics of space manipulator system complicated. Therefore it is difficult to assess respective effects of these flexibilities on vibrations of the manipulator tip. And the characteristics of integrated vibration of manipulator tip with different link and joint stiffnesses are not very clear. In this paper, the dynamic equations of multi-link multi-DOF flexible manipulator are established. Then, vibration responses of the tip under different elastic modulus, damping and joint stiffness were studied, and vibration characteristics of the tip with both link and joint were also analyzed. Moreover, the effects of motion planning on the vibration of the tip were analyzed. Finally, the vibration characteristics of the manipulator with flexible joints and links are verified by a two-degree-of-freedom manipulator experimental system. Dynamics analysis results presented some useful rules for the path planning and control to suppress the vibration of the flexible space manipulator.

Appendix: Expressions of matrixes \(\boldsymbol{G}_{jk}\), \(\boldsymbol{\beta} _{jk}\), \(\boldsymbol{M} _{i}\) and \(\boldsymbol{f} _{i}\)

The kinematic relationship between two adjacent rigid bodies B_i and B_j and the kinematic relationship between a rigid body (B_i) and a flexible body (B_j) are shown in Figs. 29 and 30, respectively. Matrixes $\boldsymbol{G}$ and $\boldsymbol{g}$ have the recursive forms as follows:

$$\begin{aligned} \boldsymbol{G}_{j,k} =& \left \{ \textstyle\begin{array}{l@{\quad}l} \boldsymbol{\varGamma}_{ij}\boldsymbol{G}_{i,k} & \mbox{if } \mathrm{B}_{k} < \mathrm{B}_{j} \\ \boldsymbol{U}_{ij} & \mbox{if } \mathrm{B}_{k} = \mathrm{B}_{j} \\ 0 & \mbox{otherwise} \end{array}\displaystyle \right .\quad \bigl( j,k = 1, \ldots,n; i = i^{ +} ( j ) \bigr) \end{aligned}$$

(21)

$$\begin{aligned} \boldsymbol{g}_{j,k} =& \left \{ \textstyle\begin{array}{l@{\quad}l} \boldsymbol{\varGamma}_{ij}\boldsymbol{g}_{i,k} & \mbox{if } \mathrm{B}_{k} < \mathrm{B}_{j} \\ \boldsymbol{\beta}_{ij} & \mbox{if } \mathrm{B}_{k} = \mathrm{B}_{j} \\ 0 & \mbox{otherwise} \end{array}\displaystyle \right . \quad \bigl( j,k = 1, \ldots,n; i = i^{ +} ( j ) \bigr) \end{aligned}$$

(22)

where

$$\begin{aligned} \boldsymbol{\varGamma}_{ij} =& \left \{ \textstyle\begin{array}{l@{\quad}l} = \left [ \textstyle\begin{array}{c@{\quad}c} \boldsymbol{E}_{3 \times3} & - \tilde{\boldsymbol{r}}_{ij} \\ \boldsymbol{O}_{3 \times3} & \boldsymbol{E}_{3 \times3} \end{array}\displaystyle \right ] \in\mathrm{R}^{6 \times6}, & \bigl(\mbox{if } \mathrm{B}_{j} \mbox{ is rigid body}, i = i^{ +} ( j ) \bigr) \\ = \left [ \textstyle\begin{array}{c@{\quad}c} \boldsymbol{E}_{3 \times3} & - \tilde{\boldsymbol{r}}_{ij} \\ \boldsymbol{O}_{3 \times3} & \boldsymbol{E}_{3 \times3} \\ \boldsymbol{O}_{s \times3} & \boldsymbol{O}_{s \times3} \end{array}\displaystyle \right ] \in\mathrm{R}^{ ( 6 + s ) \times6}, & \bigl(\mbox{if } \mathrm{B}_{j} \mbox{ is flexible body}; i = i^{ +} ( j )\bigr) \end{array}\displaystyle \right . \end{aligned}$$

(23)

$$\begin{aligned} \boldsymbol{\varGamma}_{ij} =& \left \{ \textstyle\begin{array}{l@{\quad}l} \left [ \textstyle\begin{array}{c} \tilde{\boldsymbol{s}}_{j\mathrm{Q}}^{\mathrm{H}_{j}}\boldsymbol{h}_{j} \\ \boldsymbol{h}_{j} \end{array}\displaystyle \right ] \in\mathrm{R}^{6 \times1}, & \bigl(\mbox{if } \mathrm{B}_{j} \mbox{ is rigid body}; i = i^{ +} ( j ) \bigr) \\ \left [ \textstyle\begin{array}{c@{\quad}c} \tilde{\stackrel{\frown}{\boldsymbol{s}}}_{j\mathrm{Q}}^{\mathrm{H}_{j}}\boldsymbol{h}_{j} & - \boldsymbol{\varLambda}_{ji} \\ \boldsymbol{h}_{j} & - \boldsymbol{A}_{0j}\boldsymbol{\varphi}_{ \mathrm{r}}^{\mathrm{Q}} \\ \boldsymbol{O}_{s \times1} & \boldsymbol{E}_{s \times s} \end{array}\displaystyle \right ] \in\mathrm{R}^{ ( 6 + s ) \times ( 1 + s ) }, & \bigl(\mbox{if } \mathrm{B}_{j} \mbox{ is flexible body}; i = i^{ +} ( j ) \bigr) \end{array}\displaystyle \right . \end{aligned}$$

(24)

$$\begin{aligned} \boldsymbol{\beta}_{ij} =& \dot{\boldsymbol{\varGamma}}_{ij} \boldsymbol{V}_{i} + \dot{\boldsymbol{U}}_{ij}\dot{\boldsymbol{y}} _{j}, \quad \bigl( i = i^{ +} ( j ) \bigr) . \end{aligned}$$

(25)

$\boldsymbol{E}_{m\times n}$ and $\boldsymbol{O}_{m\times n}$ represent the $m\times n$ unit matrix and the $m\times n$ zero matrix, respectively. $\boldsymbol{r} _{ij}$ represents the relative position between the origin of frame $O _{i} x _{i} y _{i} z _{i}$ and the origin of frame $O _{j} x _{j} y _{j} z _{j}$. $\boldsymbol{s}_{j\mathrm{Q}}^{ \mathrm{H}_{j}}$ is the position vector from the origin of frame $O _{j} x _{j} y _{j} z _{j}$ to point $\mathrm{P}_{\mathrm{H}j}$. $\boldsymbol{h}_{j}$ is the vector of the rotation axis of hinge H_j. $\boldsymbol{\stackrel{\frown}{s}}_{j\mathrm{Q}}^{\mathrm{H}_{j}}$ is the position vector from point $O _{j}$ to point $\stackrel{\frown}{ \mathrm{Q}}_{\mathrm{H}j}$. $\boldsymbol{A}_{0j}$ represents the orientation of B_j with respect to the inertial frame. The columns of modal matrix $\boldsymbol{\varphi}_{\mathrm{r}}^{\mathrm{Q}} \in\mathrm{R}^{3 \times s}$ are composed of the deformation modes associated with small rotation variables of hinge definition point $\mathrm{Q}_{\mathrm{H}j}$. $\boldsymbol{\varLambda} _{ji}$ is a matrix related to the deformation of a flexible body.

Fig. 29

Kinematic relationship between two adjacent rigid bodies

Fig. 30

Kinematic relationship between a rigid body and a flexible body

The expressions of generalized mass matrix $\boldsymbol{M} _{i}$ and force vector $\boldsymbol{f} _{i}$ are as follows:

$$\begin{aligned} \boldsymbol{M}_{i} =& \left \{ \textstyle\begin{array}{l} \left [ \textstyle\begin{array}{c@{\quad}c} m_{i}\boldsymbol{E}_{3 \times3} & \boldsymbol{O}_{3 \times3} \\ \boldsymbol{O}_{3 \times3} & \boldsymbol{I}_{i} \end{array}\displaystyle \right ] \in\mathrm{R}^{6 \times6}, \quad (\mbox{if }\mathrm{B}_{i} \mbox{ is rigid body} ) \\ \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} ( \sum_{k = 1}^{N} m_{k} ) \boldsymbol{E}_{3 \times3} & - \sum_{k = 1}^{N} m_{k}\tilde{\boldsymbol{s}}_{k} & \boldsymbol{A}_{0i} \sum_{k = 1}^{N} m_{k}\boldsymbol{\varphi}_{\mathrm{t}}^{k} \\ & - \sum_{k = 1}^{N} m_{k}\tilde{\boldsymbol{s}}_{k} \tilde{\boldsymbol{s}}_{k} & \boldsymbol{A}_{0i}\sum_{k = 1}^{N} m_{k}\tilde{\boldsymbol{s}}_{k}\boldsymbol{\varphi}_{\mathrm{t}}^{k} \\ \mathrm{Symmetric} & & \sum_{k = 1}^{N} m_{k} ( \boldsymbol{\varphi}_{\mathrm{t}}^{k} ) ^{\mathrm{T}} \boldsymbol{\varphi}_{\mathrm{t}}^{k} \end{array}\displaystyle \right ] \in\mathrm{R}^{ ( 6 + s ) \times ( 6 + s ) }, \\ \quad (\mbox{if } \mathrm{B}_{i} \mbox{ is flexible body}) \end{array}\displaystyle \right . \end{aligned}$$

(26)

$$\begin{aligned} \boldsymbol{f}_{i} =& \left \{ \textstyle\begin{array}{l@{\quad}l} -\boldsymbol{w}_{i} + \boldsymbol{f}_{i}^{\circ} \in\mathrm{R}^{6 \times1}, & (\mbox{if } \mathrm{B}_{i} \mbox{ is rigid body}) \\ -\boldsymbol{w}_{i} + \boldsymbol{f}_{i}^{\circ} - \boldsymbol{f}_{i}^{\mathrm{u}}, & (\mbox{if } \mathrm{B}_{i} \mbox{ is flexible body}) \end{array}\displaystyle \right . \end{aligned}$$

(27)

where $m _{i}$ is the mass for rigid body B_i, $\boldsymbol{I} _{i}$ is the $3\times 3$ inertia matrix of rigid body B_i. For flexible body B_i, $N$ is the number of finite element nodes, $m_{k}$ is the lumped mass at node $k$, and $\mathbf{s}_{k}$ is the position vector from the origin of the floating reference frame to node $k$. The columns of modal matrix $\boldsymbol{\varphi}_{\mathrm{t}}^{\mathrm{Q}} \in\mathrm{R}^{3 \times s}$ are composed of deformation modes associated with translational displacement coordinates of hinge definition point $\mathrm{Q}_{\mathrm{H}j}$. $\boldsymbol{w} _{i}$ is the quadratic velocity term defining the Coriolis force and centrifugal forces, $\boldsymbol{f}_{i}^{\mathrm{o}}$ is a $6\times 1$ vector composed of the external forces and torques exerted on B_i. $\boldsymbol{f} _{i}^{\mathrm{u}}$ is the generalized deformation force of flexible body B_i.