Inverse dynamic modelling of a three-legged six-degree-of-freedom parallel mechanism

Abstract

The Monash Epicyclic Parallel Manipulator (MEPaM) is a novel six-degree-of-freedom (dof) parallel mechanism with base mounted actuators. Closed form equations of the inverse dynamic model of MEPaM are obtained through two different methods, with simulation showing these models to be equivalent. The base inertial parameters for the dynamic model of MEPaM are determined, reducing the number of inertial parameters from 100 to 28. This significantly simplifies the dynamic calibration model and thus the number of computations required.

Appendices

Appendix A: Recursive energy relations

Letting the transformation matrix ${^{B}\mathbf{T}_{j}} = \bigl[{\scriptsize\begin{matrix}{}\mathbf{i} & \mathbf{j} & \mathbf{k} & {^{B}\mathbf{p}_{j}} \cr 0 & 0 & 0 & 1\end{matrix}}\bigr]$ with $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ being $3\times1$ vectors which describe the orientation of link $j$ with respect to the base frame, the angular velocity of link $j$ be $\boldsymbol{\omega}_{j} = [\omega_{xj} \ \omega_{yj} \ \omega_{zj}] ^{T}$ and the linear velocity of link $j$ be $\mathbf{v}_{j} = [v_{xj} \ v_{yj} \ v_{zj}]^{T}$, the recursive energy relations associated with the link’s inertial parameter vector $\boldsymbol{\chi}_{j}$ are [19, 20]

$$\begin{aligned} \textstyle\begin{array}{rcl@{\qquad}rcl} e_{XX_{j}} &=& \frac{1}{2}\omega_{xj}^{2}, &e_{XY_{j}} &=& \omega_{xj}\omega _{yj}, \\ e_{XZ_{j}} &=& \omega_{xj}\omega_{zj}, & e_{YY_{j}} &=& \frac{1}{2}\omega _{yj}^{2}, \\ e_{YZ_{j}} &=& \omega_{yj}\omega_{zj}, & e_{ZZ_{j}} &=& \frac{1}{2}\omega _{zj}^{2}, \\ e_{MX_{j}} &=& \omega_{xj}v_{yj}- \omega_{yj}v_{zj}-{^{B}\mathbf{g}^{T}} \mathbf {i}, & e_{MY_{j}} &=& \omega_{xj}v_{zj}- \omega_{zj}v_{xj}-{^{B}\mathbf {g}^{T}} \mathbf{j}, \\ e_{MZ_{j}} &=& \omega_{zj}v_{xj}- \omega_{xj}v_{zj}-{^{B}\mathbf{g}^{T}} \mathbf {k}, &e_{m_{j}} &=& \frac{1}{2}{\mathbf{v}_{j}^{T}} \mathbf{v}_{j} - {^{B}\mathbf {g}^{T}} {^{B}\mathbf{p}_{j}}. \end{array}\displaystyle \end{aligned}$$

(49)

Appendix B: Grouping relations for serial legs

A complete derivation and overview of the grouping relations for serial and tree robots can be found in [13, 18, 19]. The rules and expressions used to determine the base inertial parameters of the serial legs of MEPaM are considered below. Using the rules developed in [13, 18, 19], where possible, the parameters of link $3i$ were grouped with the parameters of link $2i$ which were subsequently grouped with the parameters of link $1i$. With reference to Table 1, $\gamma_{j} = 0$ and $b_{j} = 0$ for $j = 2$ and 3, hence the rules which pertain to serial robots suffice. These are (in terms of the Khalil–Kleinfinger parameters):

• If joint $j$ is revolute, the parameters $YY_{j}$, $MZ_{j}$ and $m_{j}$ can be grouped with the parameters of link $j$ and link $j-1$. The resulting parameters are:

  $$\begin{aligned} XX_{Rj} &= XX_{j} - YY_{j}, \\ XX_{Rj-1} &= XX_{j-1} + YY_{j} + 2d_{j}MZ_{j} + {d_{j}}^{2}m_{j}, \\ XY_{Rj-1} &= XY_{j-1} + a_{j}s_{\alpha_{j}}MZ_{j}+a_{j}d_{j}s_{\alpha_{j}}m_{j}, \\ XZ_{Rj-1} &= XZ_{j-1} - a_{j}c_{\alpha_{j}}MZ_{j}-a_{j}d_{j}c_{\alpha_{j}}m_{j}, \\ YY_{Rj-1} &= YY_{j-1} + c_{\alpha_{j}}^{2}YY_{j} + 2d_{j}c_{\alpha_{j}}^{2}MZ_{j} + \bigl(a_{j}^{2}+d_{j}^{2}{c_{\alpha_{j}}}^{2} \bigr)m_{j}, \\ YZ_{Rj-1} &= YZ_{j-1} + c_{\alpha_{j}}s_{\alpha_{j}}YY_{j} + 2d_{j}c_{\alpha _{j}}s_{\alpha_{j}}MZ_{j} + {d_{j}}^{2}c_{\alpha_{j}}s_{\alpha_{j}}m_{j}, \\ ZZ_{Rj-1} &= ZZ_{j-1} + s_{\alpha_{j}}^{2}YY_{j} + 2d_{j}s_{\alpha_{j}}^{2}MZ_{j} + \bigl(a_{j}^{2}+d_{j}^{2}{s_{\alpha_{j}}}^{2} \bigr)m_{j}, \\ MX_{Rj-1} &= MX_{j-1} + a_{j}M_{j}, \\ MY_{Rj-1} &= MY_{j-1} - s_{\alpha_{j}}MZ_{j} - d_{j}s_{\alpha_{j}}m_{j}, \\ MZ_{Rj-1} &= MZ_{j-1} + c_{\alpha_{j}}MZ_{j} + d_{j}c_{\alpha_{j}}m_{j}. \\ m_{Rj-1} &= m_{j-1} + m_{j}. \end{aligned}$$

• If joint $j$ is prismatic, the parameters of the inertia tensor of link $j$ ($XX_{j}, XY_{j}, XZ_{j}, YY_{j}, YZ_{j}, ZZ_{j}$) can be grouped with those of link $j-1$. The resulting parameters are:

  $$\begin{aligned} XX_{Rj-1} &= XX_{j-1} + {c_{\theta_{j}}}^{2}XX_{j} - 2c_{\theta_{j}}s_{\theta _{j}}XY_{j} + {s_{\theta_{j}}}^{2}YY_{j}, \\ XY_{Rj-1} &= XY_{j-1} + c_{\theta_{j}}s_{\theta_{j}}c_{\alpha_{j}}XX_{j} + \bigl(c_{\theta_{j}}^{2}-s_{\theta_{j}}^{2} \bigr)c_{\alpha_{j}}XY_{j}-c_{\theta_{j}}s_{\alpha _{j}}XZ_{j} \\ &\quad{}-c_{\theta_{j}}s_{\theta_{j}}c_{\alpha_{j}}YY_{j}+s_{\theta _{j}}s_{\alpha_{j}}YZ_{j}, \\ XZ_{Rj-1} &= XZ_{j-1} + c_{\theta_{j}}s_{\theta_{j}}s_{\alpha_{j}}XX_{j} + \bigl(c_{\theta_{j}}^{2}-s_{\theta_{j}}^{2} \bigr)s_{\alpha_{j}}XY_{j}+c_{\theta_{j}}c_{\alpha _{j}}XZ_{j} \\ &\quad{}-c_{\theta_{j}}s_{\theta_{j}}s_{\alpha_{j}}YY_{j}-s_{\theta _{j}}c_{\alpha_{j}}YZ_{j}, \\ YY_{Rj-1} &= YY_{j-1} + s_{\theta_{j}}^{2}c_{\alpha_{j}}^{2}XX_{j} + 2c_{\theta _{j}}s_{\theta_{j}}c_{\alpha_{j}}^{2}XY_{j}-2s_{\theta_{j}}c_{\alpha_{j}}s_{\alpha _{j}}XZ_{j}+c_{\theta_{j}}^{2}c_{\alpha_{j}}^{2}YY_{j} \\ &\quad{}-2c_{\theta_{j}}c_{\alpha _{j}}s_{\alpha_{j}}YZ_{j} + s_{\alpha_{j}}^{2}ZZ_{j}, \\ YZ_{Rj-1}& = YZ_{j-1} + s_{\theta_{j}}^{2}c_{\alpha_{j}}s_{\alpha_{j}}XX_{j} + 2c_{\theta_{j}}s_{\theta_{j}}c_{\alpha_{j}}s_{\alpha_{j}}XY_{j}+s_{\theta _{j}} \bigl(c_{\alpha_{j}}^{2}-s_{\alpha_{j}}^{2} \bigr)XZ_{j} \\ &\quad{}+c_{\theta_{j}}^{2}c_{\alpha _{j}}s_{\alpha_{j}}YY_{j}+2c_{\theta_{j}} \bigl(c_{\alpha_{j}}^{2}-s_{\alpha_{j}}^{2} \bigr)YZ_{j} - c_{\alpha_{j}}s_{\alpha_{j}}ZZ_{j}, \\ ZZ_{Rj-1} &= ZZ_{j-1} + s_{\theta_{j}}^{2}s_{\alpha_{j}}^{2}XX_{j} + 2c_{\theta _{j}}s_{\theta_{j}}s_{\alpha_{j}}^{2}XY_{j}+2s_{\theta_{j}}c_{\alpha_{j}}s_{\alpha _{j}}XZ_{j}+c_{\theta_{j}}^{2}s_{\alpha_{j}}^{2}YY_{j} \\ &\quad{}+2c_{\theta_{j}}c_{\alpha _{j}}s_{\alpha_{j}}YZ_{j} + c_{\alpha_{j}}^{2}ZZ_{j}. \end{aligned}$$

• If the axis of the prismatic joint $j$ is parallel to the nearest revolute joint axis $i$ ($i$ is not necessarily $j-1$), then $MZ_{j}$ has no effect on the dynamic model and the parameters $MX_{j}$ and $MY_{j}$ can be grouped as follows:

  $$\begin{aligned} MX_{Rj-1}& = MX_{j-1} + c_{\theta_{j}}MX_{j} - s_{\theta_{j}}MY_{j}, \\ MY_{Rj-1} &= MY_{j-1} + s_{\theta_{j}}c_{\alpha_{j}}MX_{j} + c_{\theta _{j}}c_{\alpha_{j}}MY_{j}, \\ MZ_{Rj-1}& = MZ_{j-1} + s_{\theta_{j}}s_{\alpha_{j}}MX_{j} + c_{\theta _{j}}s_{\alpha_{j}}MY_{j}, \\ ZZ_{Ri} &= ZZ_{i} + 2a_{j}c_{\theta_{j}}MX_{j} - 2a_{j}s_{\theta_{j}}MY_{j}. \end{aligned}$$

• If joint $j$ is revolute and articulated on the base, the parameters $XX_{j}$, $XY_{j}$, $XZ_{j}$, $YY_{j}$, $YZ_{j}$, $MZ_{j}$ and $m_{j}$ have no effect on the dynamic model and can be eliminated.

Appendix C: Grouping leg parameters with platform

For the parameters $m_{3i}$ to be grouped with the parameters of the platform, the energy function $e_{m_{3i}}$ needs to be in a form that is a linear combination of the platform energy functions, i.e.

$$ e_{m_{3i}} = \kappa_{1}e_{XX_{P}} + \kappa_{2}e_{XY_{P}} + \cdots+ \kappa_{10}e_{m_{P}} . $$

(50)

This is possible by use of the velocity of the connection point of the legs to the platform, i.e.

$$ {^{P}\mathbf{v}_{3i}} = {^{P} \mathbf{v}_{P}} + {^{P}\boldsymbol{\omega }_{P}} \times{^{P}\boldsymbol{\rho}_{i}} $$

(51)

where ${^{P}\mathbf{v}_{P}} = [^{P}v_{xP} \ ^{P}v_{yP} \ ^{P}v_{zP}]^{T}$, ${^{P}\boldsymbol{\omega}_{P}} = [^{P}\omega_{xP} \ ^{P}\omega_{yP} \ ^{P}\omega_{zP}]^{T}$ and ${^{P}\boldsymbol{\rho}_{i}} = [^{P}b_{xi} \ ^{P}b_{yi} \ ^{P}b_{zi}]^{T}$.

Let the transformation of the platform frame to the base frame be

$$ {^{B}\mathbf{T}_{P}} = \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} & {^{B}\mathbf{p}_{P}} \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

(52)

where $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are $3\times1$ vectors which describe the orientation of the platform with respect to the base frame and ${^{B}\mathbf{p}_{P}}$ is the translation between the origin of the two frames. Then the co-ordinates of the connection points in the base frame are

$$\begin{aligned} {^{B}\mathbf{b}_{i}} &= \begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} & {^{B}\mathbf{p}_{P}} \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} {^{P}\mathbf{b}_{i}} \\ 1 \end{bmatrix} \\ &= {^{B}\mathbf{p}_{P}} + {^{P}b_{xi}} \mathbf{i} + {^{P}b_{yi}}\mathbf{j} + {^{P}b_{iz}} \mathbf{k}. \end{aligned}$$

(53)

Therefore, the energy function for the masses $m_{3i}$ can be determined using Eq. (49) for $e_{m_{j}}$, Eq. (51) for the velocity of link $3i$ and Eq. (53) for the connection point expressed in $\mathcal{F}_{B}$, i.e.

$$\begin{aligned} e_{m_{3i}} &= \frac{1}{2}{^{P} \mathbf{v}_{3i}^{T}} {^{P}\mathbf{v}_{3i}} - {^{B}\mathbf{g}^{T}} {^{B}\mathbf{b}_{i}} \\ &= \frac{1}{2} \bigl({^{P}b_{yi}}^{2} + {^{P}b_{zi}}^{2} \bigr){^{P}\omega _{xP}}^{2} - {^{P}b_{xi}} {^{P}b_{yi}} {^{P}\omega_{xP}} {^{P}\omega_{yP}} - {^{P}b_{xi}} {^{P}b_{zi}} {^{P}\omega_{xP}} {^{P}\omega_{zP}} \\ &\quad{}+ \frac {1}{2} \bigl({^{P}b_{xi}}^{2} + {^{P}b_{zi}}^{2} \bigr){^{P} \omega_{yP}}^{2} - {^{P}b_{yi}} {^{P}b_{zi}} {^{P}\omega_{yP}} {^{P}\omega_{zP}} + \frac{1}{2} \bigl({^{P}b_{xi}}^{2} + {^{P}b_{yi}}^{2} \bigr){^{P} \omega_{zP}}^{2} \\ &\quad{}+{^{P}b_{xi}} \bigl({^{P}\omega_{zP}} {^{P}v_{yP}}-{^{P}\omega _{yP}} {^{P}v_{zP}}-{^{B}\mathbf{g}^{T}} \mathbf{i} \bigr) +{^{P}b_{yi}} \bigl({^{P} \omega_{xP}} {^{P}v_{zP}}-{^{P} \omega_{zP}} {^{P}v_{xP}} -{^{B}\mathbf{g}^{T}}\mathbf{j} \bigr) \\ &\quad{} +{^{P}b_{zi}} \bigl({^{P}\omega _{yP}} {^{P}v_{xP}}-{^{P}\omega_{xP}} {^{P}v_{yP}}-{^{B}\mathbf{g}^{T}}\mathbf {k} \bigr) + \frac{1}{2}{^{P}\mathbf{v}_{P}^{T}} {^{P}\mathbf{v}_{P}} - {^{B}\mathbf {g}^{T}} {^{B}\mathbf{p}_{P}}. \end{aligned}$$

(54)

With aid of Eq. (49), Eq. (54) is rearranged to

$$\begin{aligned} e_{m_{3i}} &= \bigl({^{P}b_{yi}}^{2} + {^{P}b_{zi}}^{2} \bigr)e_{XX_{P}} - {^{P}b_{xi}} {^{P}b_{yi}}e_{XY_{P}} - {^{P}b_{xi}} {^{P}b_{zi}}e_{XZ_{P}} + \bigl({^{P}b_{xi}}^{2} + {^{P}b_{zi}}^{2} \bigr)e_{YY_{P}} \\ &\quad{} - {^{P}b_{yi}} {^{P}b_{zi}}e_{YZ_{P}} + \bigl({^{P}b_{xi}}^{2} + {^{P}b_{yi}}^{2} \bigr)e_{ZZ_{P}} +{^{P}b_{xi}}e_{MX_{P}} \\ &\quad{}+{^{P}b_{yi}}e_{MY_{P}} +{^{P}b_{zi}}e_{MZ_{P}} + e_{m_{P}}. \end{aligned}$$

(55)

As a result of Eq. (55), it is possible to group the $m_{3i}$ parameters with the inertial parameters of the platform. Due to the assignment of the platform frame, $\mathcal{F}_{P}$, the co-ordinates of the platform vertices in $\mathcal{F}_{P}$ are:

$$\begin{aligned} &{^{P}\mathbf{b}_{1}} = \begin{bmatrix}d_{31}\\0\\0 \end{bmatrix} , \qquad {^{P}\mathbf{b}_{2}} = \begin{bmatrix}-d_{32}s_{2}\\d_{32}c_{2}\\0 \end{bmatrix} ,\qquad {^{P}\mathbf{b}_{3}} = \begin{bmatrix}-d_{33}s_{3}\\-d_{33}c_{3}\\0 \end{bmatrix} . \end{aligned}$$

Hence, the grouping relations are

$$\begin{aligned} XX_{RP} &= XX_{P} + \sum_{i=1}^{3} \bigl({^{P}b_{yi}}^{2} + {^{P}b_{zi}}^{2} \bigr)m_{3i} =XX_{P} + d_{32}^{2}c_{2}^{2}m_{32} + d_{33}^{2}c_{3}^{2}m_{33}, \\ XY_{RP} &= XY_{P} - \sum_{i=1}^{3}{^{P}b_{xi}} {^{P}b_{yi}}m_{3i} = XY_{P} + d_{32}^{2}s_{2}c_{2}m_{32} - d_{33}^{2}s_{3}c_{3}m_{33}, \\ XZ_{RP}& = XZ_{P} - \sum_{i=1}^{3}{^{P}b_{xi}} {^{P}b_{zi}}m_{3i} = XZ_{P}, \\ YY_{RP}& = YY_{P} + \sum_{i=1}^{3} \bigl({^{P}b_{xi}}^{2} + {^{P}b_{zi}}^{2} \bigr)m_{3i}=YY_{P} + d_{31}^{2}m_{31} + d_{32}^{2}s_{2}^{2}m_{32} + d_{33}^{2}s_{3}^{2}m_{33}, \\ YZ_{RP} &= YZ_{P} - \sum_{i=1}^{3}{^{P}b_{yi}} {^{P}b_{zi}}m_{3i} = YZ_{P}, \\ ZZ_{RP} &= ZZ_{P} + \sum_{i=1}^{3} \bigl({^{P}b_{xi}}^{2} + {^{P}b_{yi}}^{2} \bigr)m_{3i} =ZZ_{P} + d_{31}^{2}m_{31} + d_{32}^{2}m_{32} + d_{33}^{2}m_{33}, \\ MX_{RP}& = MX_{P} + \sum_{i=1}^{3}{^{P}b_{xi}}m_{3i} = MX_{P} + d_{31}m_{31} - d_{32}s_{2}m_{32} - d_{33}s_{3}m_{33}, \\ MY_{RP}& = MY_{P} + \sum_{i=1}^{3}{^{P}b_{yi}}m_{3i} = MY_{P} + d_{32}c_{2}m_{32} - d_{33}c_{3}m_{33}, \\ MZ_{RP}& = MZ_{P} + \sum_{i=1}^{3}{^{P}b_{zi}}m_{3i} = MZ_{P}, \\ m_{RP} &= m_{P} + \sum_{i=1}^{3}m_{3i} = m_{P} + m_{31} + m_{32} + m_{33}. \end{aligned}$$