Kinematic and dynamic modeling of a planar parallel manipulator served as CNC tool holder

Abstract

This paper introduces a novel planar parallel manipulator used as the tool holder in a 4-axis CNC machine. The manipulator has two translational and one rotational degree-of-freedom (DOF) while one of the translational DOFs is decoupled from the other two DOFs. The inverse and direct position kinematics of the manipulator are solved in closed form. Velocity, acceleration and singularity analyses are implemented using Jacobian matrices and it is shown that the proposed manipulator can be easily designed to have a singularity-free workspace. An analytical method is presented to determine workspace of the manipulator. A closed form solution is also presented for the inverse and direct dynamics of the manipulator by Newton–Euler method. Moreover, a kinematic conditioning index and a dynamic conditioning index are evaluated on the workspace revealing that the manipulator has a good dexterity especially in the center of the workspace.

Appendices

Appendix 1

(a) Calculation of $\dot{\theta }^{2}$ as a function of ${\varvec{\uprho }}$ and ${\dot{\varvec{\uprho }}}$

Considering the velocity equation (24), the term $\dot{\theta }$ can be obtained as

$$\begin{aligned} \dot{\theta }=\mathbf{k}^{T}{\dot{\varvec{\uprho }}} \end{aligned}$$

(71)

where k is a $3\times 1$ vector, and is defined as

$$\begin{aligned} \mathbf{k}=[{\begin{array}{lll} {k_1 }&{} {k_2 }&{} {k_3 } \\ \end{array} }]^{T}=\mathbf{J}^{T}{} \mathbf{l}_3 \end{aligned}$$

(72)

and

$$\begin{aligned} \mathbf{l}_3 =[{\begin{array}{lll} 0&{} 0&{} 1 \\ \end{array} }]^{T} \end{aligned}$$

(73)

As a consequence, the term $\dot{\theta }^{2}$ will be obtained as

$$\begin{aligned} \dot{\theta }^{2}={\dot{\varvec{\uprho }}}^{T}{} \mathbf{G}_1 {\dot{\varvec{\uprho }}} \end{aligned}$$

(74)

where

$$\begin{aligned} \mathbf{G}_1 =\left[ {{\begin{array}{lll} {k_1^2 }&{}\quad {k_1 k_3 }&{}\quad {k_1 k_3 } \\ {k_1 k_3 }&{} \quad {k_3^2 }&{} \quad {k_3^2 } \\ {k_1 k_3 }&{}\quad {k_3^2 }&{}\quad {k_3^2 } \\ \end{array} }} \right] _{3\times 3} \end{aligned}$$

(75)

(b) Calculation of $\dot{\alpha }_i$ and $\dot{\alpha }_i^2$ as a function of ${\varvec{\uprho }}$ and ${\dot{\varvec{\uprho }}}$.

Parameter $\dot{\alpha }_i$ can be obtained through multiplying both sides of Eq. (15) by Ed $_{i}$ as follows

$$\begin{aligned} \dot{\alpha }_i =\frac{(\mathbf{Ed}_i )^{T}{\dot{\mathbf{h}}}+(\mathbf{Ed}_i )^{T}{} \mathbf{Eb}_i \dot{\theta }}{d_i^2 }, \quad i = 1, 2 \end{aligned}$$

(76)

or

$$\begin{aligned} \dot{\alpha }_i =\mathbf{t}_i^T {\dot{\varvec{\uppsi }}}, \quad i = 1, 2 \end{aligned}$$

(77)

where t $_{i}$ is a $3\times 1$ vector, and is defined as

$$\begin{aligned} \mathbf{t}_i =d_i^{-2} [{\begin{array}{lll} 0&{} {d_{i,x} }&{} {\mathbf{d}_i^T \mathbf{b}_i } \\ \end{array} }]^{T} \end{aligned}$$

(78)

Substituting the value of ${\dot{\varvec{\uppsi }}}$ from Eq. (24) into Eq. (77), we have

$$\begin{aligned} \dot{\alpha }_i =\mathbf{u}_i^T {\dot{\varvec{\uprho }}} \end{aligned}$$

(79)

where u $_{i}$ is a $3\times 1$ vector, as follows

$$\begin{aligned} \mathbf{u}_i =[{\begin{array}{lll} {u_{1,i} }&{} {u_{2,i} }&{} {u_{3,i} } \\ \end{array} }]^{T}=(\mathbf{M}^{-1}{} \mathbf{K})^{T}{} \mathbf{t}_i \end{aligned}$$

(80)

As a result, the variable $\dot{\alpha }_i^2$ will be

$$\begin{aligned} \dot{\alpha }_i^2 ={\dot{\varvec{\uprho }}}^{T}{} \mathbf{G}_{2,i} {\dot{\varvec{\uprho }}}, \quad i = 1, 2 \end{aligned}$$

(81)

where

$$\begin{aligned} \mathbf{G}_{2,i} =\left[ {{\begin{array}{lll} {u_{1,i}^2 }&{} \quad {u_{1,i} u_{3,i} }&{}\quad {u_{1,i} u_{3,i} } \\ {u_{1,i} u_{3,i} }&{}\quad {u_{3,i}^2 }&{}\quad {u_{3,i}^2 } \\ {u_{1,i} u_{3,i} }&{}\quad {u_{3,i}^2 }&{}\quad {u_{3,i}^2 } \\ \end{array} }} \right] _{3\times 3} \end{aligned}$$

(82)

(c) Calculation of $\dot{d}_i \dot{\alpha }_i$ and $\ddot{\alpha }_i$ as a function of ${\varvec{\uprho }}$ and ${\dot{\varvec{\uprho }}}$.

Multiplying both sides of Eq. (25a) by Ed $_{i}$, yields

$$\begin{aligned}&d_i (d_i \ddot{\alpha }_i +\dot{d}_i \dot{\alpha }_i )=(\mathbf{Ed}_i )^{T}{\ddot{\mathbf{h}}}\nonumber \\&\quad -(\mathbf{Ed}_i )^{T}{} \mathbf{b}_i \dot{\theta }^{2}+(\mathbf{Ed}_i )^{T}{} \mathbf{Eb}_i \ddot{\theta }, \quad i = 1, 2 \end{aligned}$$

(83)

Eqs. (83) can be rewritten as

$$\begin{aligned} d_i (d_i \ddot{\alpha }_i +\dot{d}_i \dot{\alpha }_i )=-(\mathbf{Ed}_i )^{T}{} \mathbf{b}_i \dot{\theta }^{2}+d_i^2 \mathbf{t}_i^T {\ddot{\varvec{\uppsi }}}, \quad i = 1, 2 \end{aligned}$$

(84)

Taking into account Eq. (79), the term $\dot{d}_i \dot{\alpha }_i$ is obtained as follows

$$\begin{aligned} \dot{d}_i \dot{\alpha }_i ={\dot{\varvec{\uprho }}}^{T}{} \mathbf{G}_{3,i} {\dot{\varvec{\uprho }}}, \quad i = 1, 2 \end{aligned}$$

(85)

where

$$\begin{aligned}&{} \mathbf{G}_{3,1} =\left[ {{\begin{array}{lll} {u_{1,1} }&{} \quad {u_{2,1} }&{}\quad {u_{3,1} } \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{} \quad 0 \\ \end{array} }} \right] _{3\times 3} ,\nonumber \\&\quad \mathbf{G}_{3,2} =\left[ {{\begin{array}{lll} 0&{}\quad 0&{}\quad 0 \\ {u_{1,2} }&{}\quad {u_{2,2} }&{}\quad {u_{3,2} } \\ 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right] _{3\times 3} \end{aligned}$$

(86)

Now, introducing Eqs. (74), (85) and (33) into (84), and solving the resultant equation for $\ddot{\alpha }_i$, gives

$$\begin{aligned} \ddot{\alpha }_i =\mathbf{v}_i^T {\ddot{\varvec{\uprho }}}+{\dot{\varvec{\uprho }}}^{T}{} \mathbf{G}_{4,i} {\dot{\varvec{\uprho }}}, \quad i = 1, 2 \end{aligned}$$

(87)

where

$$\begin{aligned}&{} \mathbf{v}_i =d_i^{-2} \mathbf{J}^{T}{} \mathbf{t}_i \end{aligned}$$

(88)

$$\begin{aligned}&{} \mathbf{G}_{4,i} =d_i^{-2} (-2d_i \mathbf{G}_{3,i} -(\mathbf{Ed}_i )^{T}\mathbf{b}_i \mathbf{G}_1 \nonumber \\&\quad +\,(\mathbf{t}_i^T \mathbf{M}^{-1}{} \mathbf{l}_1 )\mathbf{N}_1 +(\mathbf{t}_i^T \mathbf{M}^{-1}{} \mathbf{l}_2 )\mathbf{N}_2 ) \end{aligned}$$

(89)

and

$$\begin{aligned} \mathbf{l}_1 =[{\begin{array}{lll} 1&{} 0&{} 0 \\ \end{array} }]^{T}, \quad \mathbf{l}_2 =[{\begin{array}{lll} 0&{} 1&{} 0 \\ \end{array} }]^{T} \end{aligned}$$

(90)

Appendix 2

$$\begin{aligned}&{\tilde{\mathbf{A}}}_{1,i} ={\tilde{\mathbf{C}}}+\mathbf{Es}_u \mathbf{v}_i^T , \quad {\tilde{\mathbf{A}}}_{2,i} =({\tilde{\mathbf{F}}}\nonumber \\&\quad +\,\mathbf{n}_i \mathbf{l}_i^T +d_i \mathbf{En}_i \mathbf{v}_i^T +\mathbf{Es}_l \mathbf{v}_i^T ), \quad i = 1, 2 \end{aligned}$$

(91)

where

$$\begin{aligned} {\tilde{\mathbf{F}}}=\left[ {{\begin{array}{lll} 0&{}\quad 0&{}\quad 1 \\ 0&{} \quad 0&{}\quad 0 \\ \end{array} }} \right] _{2\times 3} \end{aligned}$$

(92)

Moreover, ${\tilde{\mathbf{B}}}_{1,i}$ and ${\tilde{\mathbf{B}}}_{2,i}$ constitute of $3\times 3$ submatrices $\mathbf{G}_{5,i}$, $\mathbf{G}_{6,i}$, $\mathbf{G}_{7,i}$ and $\mathbf{G}_{8,i}$, as follows

$$\begin{aligned} {\tilde{\mathbf{B}}}_{1,i} =\left[ {{\begin{array}{l} {\mathbf{G}_{5,i} } \\ {\mathbf{G}_{6,i} } \\ \end{array} }} \right] _{6\times 3} , \quad {\tilde{\mathbf{B}}}_{2,i} =\left[ {{\begin{array}{l} {\mathbf{G}_{7,i} } \\ {\mathbf{G}_{8,i} } \\ \end{array} }} \right] _{6\times 3} \end{aligned}$$

(93)

where

$$\begin{aligned} \begin{array}{lll} &{}&{}{} \mathbf{G}_{5,i} =(\mathbf{Es}_u )^{T}{\hat{\mathbf{i}}}{} \mathbf G _{4,i} -\mathbf{s}_u^T {\hat{\mathbf{i}}}{} \mathbf G _{2,i} \\ &{}&{}{} \mathbf{G}_{6,i} =(\mathbf{Es}_u )^{T}{\hat{\mathbf{j}}}{} \mathbf G _{4,i} -\mathbf{s}_u^T {\hat{\mathbf{j}}}{} \mathbf G _{2,i} \\ &{}&{}{} \mathbf{G}_{7,i} =-(d_i \mathbf{n}_i +\mathbf{s}_l )^{T}{\hat{\mathbf{i}}}{} \mathbf G _{2,i} +(d_i \mathbf{En}_i +\mathbf{Es}_l )^{T}{\hat{\mathbf{i}}}{} \mathbf G _{4,i}\\ &{}&{}\qquad \qquad +\,2(\mathbf{En}_i )^{T}{\hat{\mathbf{i}}}{} \mathbf G _{3,i} \\ &{}&{}{} \mathbf{G}_{8,i} =-(d_i \mathbf{n}_i +\mathbf{s}_l )^{T}{\hat{\mathbf{j}}}{} \mathbf G _{2,i} +(d_i \mathbf{En}_i +\mathbf{Es}_l )^{T}{\hat{\mathbf{j}}}{} \mathbf G _{4,i}\\ &{}&{}\qquad \qquad +\,2(\mathbf{En}_i )^{T}{\hat{\mathbf{j}}}{} \mathbf G _{3,i} \\ \end{array} \end{aligned}$$

(94)

Appendix 3

$$\begin{aligned} {\tilde{\mathbf{a}}}_{1,i}= & {} \mathbf{z}_1^T {\tilde{\mathbf{A}}}_{1,i} +\mathbf{z}_2^T {\tilde{\mathbf{A}}}_{2,i} +d_i^{-1} (I_u +I_l +m_l d_i^2 )\mathbf{v}_i^T , \nonumber \\ {\tilde{\mathbf{b}}}_{1,i}= & {} -(\mathbf{z}_1 +\mathbf{z}_2 )^{T} \end{aligned}$$

(95)

$$\begin{aligned} {\tilde{\mathbf{a}}}_{2,i}= & {} -m_l \mathbf{n}_i^T {\tilde{\mathbf{A}}}_{2,i} , \quad {\tilde{\mathbf{b}}}_{2,i} =m_l \mathbf{n}_i^T \end{aligned}$$

(96)

$$\begin{aligned} \mathbf{G}_{9,i}= & {} (\mathbf{z}_{1,x} \mathbf{G}_{5,i} +\,\mathbf{z}_{1,y} \mathbf{G}_{6,i} +\,\mathbf{z}_{2,x} \mathbf{G}_{7,i} \nonumber \\&\quad +\,\mathbf{z}_{2,y} \mathbf{G}_{8,i} +d_i^{-1} (I_u +I_l +m_l d_i^2 )\mathbf{G}_{4,i} ) \end{aligned}$$

(97)

$$\begin{aligned} \mathbf{G}_{10,i}= & {} -(m_l \mathbf{n}_{i,x} \mathbf{G}_{7,i} +m_l \mathbf{n}_{i,y} \mathbf{G}_{8,i} ) \end{aligned}$$

(98)

with

$$\begin{aligned} \mathbf{z}_1= & {} m_u d_i^{-1} \left( \left| {\mathbf{s}_u^T \mathbf{En}_i } \right| \mathbf{n}_i -\left| {\mathbf{s}_u^T \mathbf{n}_i } \right| \mathbf{En}_i\right) \end{aligned}$$

(99)

$$\begin{aligned} \mathbf{z}_2= & {} m_l d_i^{-1} \left( \left| {(\mathbf{d}_i +\mathbf{s}_l )^{T}{} \mathbf{En}_i } \right| \mathbf{n}_i -\left| {(\mathbf{d}_i +\mathbf{s}_l )^{T}{} \mathbf{n}_i } \right| \mathbf{En}_i\right) \nonumber \\ \end{aligned}$$

(100)