2.1 Inverse kinematics

According to Fig. 1, for i = 1,2,3, each limb loop closure equation can be shown as the following:

(1) \begin{equation} \boldsymbol{{s}}_i= \ell_{i} \hat{\boldsymbol{{s}}}_i=\boldsymbol{{x}}+\boldsymbol{{b}}_i-\boldsymbol{{a}}_i,\end{equation}

in which $\hat{\boldsymbol{{s}}}_i$ is defined as a unit vector along the direction of the limb i. Hence, joint variable, $\ell_i$ , may be derived as $ \ell_i=\sqrt{\left [\boldsymbol{{x}}+\boldsymbol{{b}}_i-\boldsymbol{{a}}_i\right]^T \left [\boldsymbol{{x}}+\boldsymbol{{b}}_i-\boldsymbol{{a}}_i\right]} $ .

2.2 Rotation matrix analysis

In each closed kinematic chain, there are two passive angles, $q_{1_i},q_{2_i}$ , which are related to the revolute joints of each universal joint. According to Fig. 2, these angles can be derived as follows:

(2a) \begin{align} q_{1_i}=atan2 \left( {^{i}\mathrm{\boldsymbol{{s}}}_{y_i}},{-^{i}\mathrm{\boldsymbol{{s}}}_{z_i}} \right),\end{align}

(2b) \begin{align} q_{2_i}=asin \left( {^{i}\mathrm{\boldsymbol{{s}}}_{x_i}}/{\ell_i} \right), \end{align}

in which, ${}^{i}\mathrm{\boldsymbol{{s}}}_{i}$ is defined as ${}^{i}\mathrm{\boldsymbol{{s}}}_{i}={\boldsymbol{{R}}_y}^T (\alpha_i) \; \boldsymbol{{s}}_i $ . Furthermore, the unit vector of each universal joint can be derived as follows:

(3a) \begin{align} \hat{\boldsymbol{{u}}}_{i}=\boldsymbol{{R}}_y (\alpha_i) \left[-1,0,0 \right]^T, \end{align}

(3b) \begin{align} \hat{\boldsymbol{{v}}}_{i}=\boldsymbol{{R}}_y (\alpha_i) \left[0,\cos(q_{1_i}),\sin(q_{1_i})\right]^T, \end{align}

(3c) \begin{align} \hat{\boldsymbol{{w}}}_{i}= \hat{\boldsymbol{{u}}}_{i} \times \hat{\boldsymbol{{v}}}_{i}. \end{align}

Figure 2.

Passive angles for each chain.

Therefore, the rotation matrix of each limb relative to the base coordinate system may be derived as:

(4) \begin{equation} ^{0}\mathrm{\boldsymbol{{R}}}_{i}=\begin{bmatrix} \hat{\boldsymbol{{x}}} \cdot \hat{\boldsymbol{{s}}}_{i} & \hat{\boldsymbol{{x}}} \cdot \hat{\boldsymbol{{v}}}_{i} & \hat{\boldsymbol{{x}}} \cdot \left( \hat{\boldsymbol{{s}}}_{i} \times \hat{\boldsymbol{{v}}}_{i} \right ) \\\hat{\boldsymbol{{y}}} \cdot \hat{\boldsymbol{{s}}}_{i} & \hat{\boldsymbol{{y}}} \cdot \hat{\boldsymbol{{v}}}_{i} & \hat{\boldsymbol{{y}}} \cdot \left( \hat{\boldsymbol{{s}}}_{i} \times \hat{\boldsymbol{{v}}}_{i} \right ) \\\hat{\boldsymbol{{z}}} \cdot \hat{\boldsymbol{{s}}}_{i} & \hat{\boldsymbol{{z}}} \cdot \hat{\boldsymbol{{v}}}_{i} & \hat{\boldsymbol{{z}}} \cdot \left( \hat{\boldsymbol{{s}}}_{i} \times \hat{\boldsymbol{{v}}}_{i} \right )\end{bmatrix},\end{equation}

in which, $\hat{\boldsymbol{{x}}}$ , $\hat{\boldsymbol{{y}}}$ , and $\hat{\boldsymbol{{z}}}$ represent unit Cartesian axes.

2.3 Limb analysis

Accordingto Fig. 3, the position of the center of mass (CM) of the cylinder and the piston of each limb may be written as:

(5a) \begin{align} \boldsymbol{{c}}_{1_i} =\boldsymbol{{a}}_i+ c_{1_i} \hat{\boldsymbol{{s}}}_i, \end{align}

(5b) \begin{align} \boldsymbol{{c}}_{2_i}= \boldsymbol{{a}}_i+ \left( \ell_i - c_{2_i} \right) \hat{\boldsymbol{{s}}}_i. \end{align}

Figure 3.

CM position of each cylinder and piston.

3.1 Angular velocity

In a similar way, by cross-multiplying both sides of Eq. (6) by $\hat{\boldsymbol{{s}}}_i$ , the perpendicular term of angular velocity of each limb will be derived as follows:

(9) \begin{equation}\boldsymbol{\omega}_{{\bot_i}}= \frac{ \hat{\boldsymbol{{s}}}_{{i}_\times}}{\ell_i} \dot{\boldsymbol{{x}}},\end{equation}

in which $\hat{\boldsymbol{{s}}}_{{i}_\times}$ represent $3\times 3$ skew-symmetric matrix generated from vector $\hat{\boldsymbol{{s}}}_i$ . It should be noted that $\boldsymbol{\omega}_{\bot_i} \neq \boldsymbol{\omega}_{i}$ , and this is one of the common mistakes in the literature on the dynamic analysis of parallel robots with universal joints and has been carefully examined in refs. [Reference Carricato and Gosselin32, Reference Vakil, Pendar and Zohoor33]. Consequently, two elements of the angular velocity of each limb may be presented as $\boldsymbol{\omega}_{\bot_i}$ , which is normal to the limb axis, and $\omega_{s_i} $ , that is in the direction of the limb axis, as the following:

(10) \begin{equation} \boldsymbol{\omega}_i=\boldsymbol{\omega}_{\bot_i}+ \omega_{s_i} \hat{\boldsymbol{{s}}}_i= \boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}}.\end{equation}

Now, in order to extract $\omega_{s_i}$ , according to Fig. 2, $\boldsymbol{\omega}_i$ may be decomposed into two components along the unit vectors of $\hat{\boldsymbol{{u}}}_i$ and $\hat{\boldsymbol{{v}}}_i$ of the universal joint as follows:

(11) \begin{equation} \boldsymbol{\omega}_i= \omega_{u_i} \hat{\boldsymbol{{u}}}_i + \omega_{v_i} \hat{\boldsymbol{{v}}}_i.\end{equation}

Since each limb’s angular velocity is placed on the $\hat{\boldsymbol{{u}}}_i-\hat{\boldsymbol{{v}}}_i$ plane, the relationship of $ \hat{\boldsymbol{{w}}}_i \cdot \boldsymbol{\omega}_i =0$ may be established. Hence, by dot multiplying both sides of Eq. (10) by $\hat{\boldsymbol{{w}}}_i$ , the scalar value of $\omega_{s_i}$ would be derived as:

(12) \begin{equation} \omega_{s_i}= \frac{ - \hat{\boldsymbol{{w}}}_i \cdot \boldsymbol{\omega}_{\bot_i} } { \hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i },\end{equation}

where $\boldsymbol{\omega}_{{\bot_i}}$ was previously defined in Eq. (9). Now, by rewriting Eq. (10) as a function of $\dot{\boldsymbol{{x}}}$ , $\boldsymbol{{J}}_{\omega_i}$ may be derived as follows:

(13) \begin{equation} \boldsymbol{{J}}_{\omega_i}= \boldsymbol{{J}}_{\omega_{\bot_i}} + \hat{\boldsymbol{{s}}}_i \; \boldsymbol{\mathcal{J}}_{\omega_{s_i}},\end{equation}

in which $ \;\boldsymbol{{J}}_{\omega_{\bot_i}}$ and $\boldsymbol{\mathcal{J}}_{\omega_{s_i}}$ are defined as:

(14a) \begin{align} \boldsymbol{{J}}_{\omega_{\bot_i}} = \frac{\hat{\boldsymbol{{s}}}_{i_\times}} {\ell_i}\;\;,\;\; \in \mathbb{ R}^{3\times 3} , \end{align}

(14b) \begin{align} \boldsymbol{\mathcal{J}}_{\omega_{s_i}} = \left(\frac{-{\hat{\boldsymbol{{w}}}_i}{}^T} {\hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i }\right) \boldsymbol{{J}}_{\omega_{\bot_i}} \;\;,\;\; \in \mathbb{ R}^{1\times 3}. \end{align}

In order to derive scalar values of $\omega_{u_i}$ and $\omega_{v_i}$ , substitution of (11) into (6) and dot multiplying both sides by $\hat{\boldsymbol{{v}}}_i$ and $\hat{\boldsymbol{{u}}}_i$ , respectively, results in:

(15a) \begin{align} \omega_{u_i} = \frac{\hat{\boldsymbol{{v}}}_i \cdot \dot{\boldsymbol{{x}}}- \hat{\boldsymbol{{v}}}_i \cdot \hat{\boldsymbol{{s}}}_i \; \dot{\ell}_i }{ \ell_i \; \hat{\boldsymbol{{v}}}_i \cdot \left( \hat{\boldsymbol{{u}}}_i \times \hat{\boldsymbol{{s}}}_i \right) }, \end{align}

(15b) \begin{align} \omega_{v_i}= \frac{\hat{\boldsymbol{{u}}}_i \cdot \dot{\boldsymbol{{x}}}- \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \; \dot{\ell}_i }{ \ell_i \; \hat{\boldsymbol{{u}}}_i \cdot \left( \hat{\boldsymbol{{v}}}_i \times \hat{\boldsymbol{{s}}}_i \right) }. \end{align}

Substitution properties $\hat{\boldsymbol{{v}}}_i\cdot \left( \hat{\boldsymbol{{u}}}_i \times \hat{\boldsymbol{{s}}}_i \right)=-\hat{\boldsymbol{{s}}}_i \cdot \left( \hat{\boldsymbol{{u}}}_i \times \hat{\boldsymbol{{v}}}_i \right)$ , $\hat{\boldsymbol{{u}}}_i \cdot \left( \hat{\boldsymbol{{v}}}_i \times \hat{\boldsymbol{{s}}}_i \right)=\hat{\boldsymbol{{s}}}_i \cdot \left( \hat{\boldsymbol{{u}}}_i \times \hat{\boldsymbol{{v}}}_i \right)$ , and $\left( \hat{\boldsymbol{{u}}}_i \times \hat{\boldsymbol{{v}}}_i \right)=\hat{\boldsymbol{{w}}}_i$ in (15) result in:

(16a) \begin{align} \omega_{u_i} = -\left( \frac{\hat{\boldsymbol{{v}}}_i \cdot \dot{\boldsymbol{{x}}}- \hat{\boldsymbol{{v}}}_i \cdot \hat{\boldsymbol{{s}}}_i \; \dot{\ell}_i }{ \ell_i \hat{\boldsymbol{{s}}}_i \cdot \hat{\boldsymbol{{w}}}_i } \right)= \boldsymbol{\mathcal{J}}_{\omega_{u_i}} \dot{\boldsymbol{{x}}} , \end{align}

(16b) \begin{align} \omega_{v_i}= +\left( \frac{\hat{\boldsymbol{{u}}}_i \cdot \dot{\boldsymbol{{x}}}- \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \; \dot{\ell}_i }{ \ell_i \hat{\boldsymbol{{s}}}_i \cdot \hat{\boldsymbol{{w}}}_i } \right)= \boldsymbol{\mathcal{J}}_{\omega_{v_i}} \dot{\boldsymbol{{x}}}. \end{align}

On the other hand, $\boldsymbol{\mathcal{J}}_{\omega_{u_i}}$ and $\boldsymbol{\mathcal{J}}_{\omega_{v_i}}$ may be derived by the substitution of $\dot{\ell}_i= {\hat{\boldsymbol{{s}}}_i}{}^T \dot{\boldsymbol{{x}}} $ into (16) and some manipulations, as follows:

(17a) \begin{align} \boldsymbol{\mathcal{J}}_{\omega_{u_i}} = \frac{{\hat{\boldsymbol{{v}}}_i}{}^T \left(- \boldsymbol{{I}}_{3\times 3} + \hat{\boldsymbol{{s}}}_i {\hat{\boldsymbol{{s}}}_i}{}^T \right) }{ \ell_i \hat{\boldsymbol{{s}}}_i \cdot \hat{\boldsymbol{{w}}}_i }\;\;,\;\; \in \mathbb{ R}^{1\times 3} , \end{align}

(17b) \begin{align} \boldsymbol{\mathcal{J}}_{\omega_{v_i}}= \frac{{\hat{\boldsymbol{{u}}}_i}{}^T \left(+ \boldsymbol{{I}}_{3\times 3} - \hat{\boldsymbol{{s}}}_i {\hat{\boldsymbol{{s}}}_i}{}^T \right) }{ \ell_i \hat{\boldsymbol{{s}}}_i \cdot \hat{\boldsymbol{{w}}}_i }\;\;,\;\; \in \mathbb{ R}^{1\times 3}. \end{align}

in which, $\boldsymbol{{I}}_{3 \times 3}$ denotes $3 \times 3$ identity matrix.

3.2 Linear velocity

Bytime derivative of both sides of Eq. (5), the linear velocity of the CM of the cylinder and the piston of each limb may be derived as:

(18a) \begin{align} \boldsymbol{{v}}_{c_{1_i}}=c_{1_i} \left(\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i\right), \end{align}

(18b) \begin{align} \boldsymbol{{v}}_{c_{2_i}}=(\ell_i-c_{2_i}) \left(\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i\right)+\dot{\ell}_i \hat{\boldsymbol{{s}}}_{i}. \end{align}

Using equations (7) and (10), and anti-commutative property of cross-product, these linear velocities may be written as a function of $\dot{\boldsymbol{{x}}}$ as follow:

(19a) \begin{align} \boldsymbol{{v}}_{c_{1_i}}=-c_{1_i} \hat{\boldsymbol{{s}}}_{i_\times} \boldsymbol{\omega}_i= \left( -c_{1_i} \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i} \right)\dot{\boldsymbol{{x}}}, \end{align}

(19b) \begin{align} \boldsymbol{{v}}_{c_{2_i}}= -\left( \left(\ell_i-c_{2_i}\right) \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i} \right)\dot{\boldsymbol{{x}}}+\left( {\hat{\boldsymbol{{s}}}_i} {\hat{\boldsymbol{{s}}}_i}{}^T \right) \dot{\boldsymbol{{x}}}. \end{align}

Therefore, the linear Jacobians of each limb may be derived as follows:

(20a) \begin{align} \boldsymbol{{J}}_{v_{1_i}}= -c_{1_i} \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i}, \end{align}

(20b) \begin{align} \boldsymbol{{J}}_{v_{2_i}}= - \left(\ell_i-c_{2_i}\right) \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i} + {\hat{\boldsymbol{{s}}}_i} {\hat{\boldsymbol{{s}}}_i}{}^T. \end{align}

Finally, general Jacobian matrix of each limb can be written as $\boldsymbol{{J}}_{1_i}=\left[ {\boldsymbol{{J}}_{v_{1_i}}}^T, {\boldsymbol{{J}}_{\omega_i}}^T \right]^T$ and $\boldsymbol{{J}}_{2_i}=\left[ {\boldsymbol{{J}}_{v_{2_i}}}^T, {\boldsymbol{{J}}_{\omega_i}}^T\right]^T$ .

4.1 Angular acceleration

In order to derive $\dot{\boldsymbol{\omega}}_{\bot_i} $ , one should cross-multiply both sides of the Eq. (22) by $\hat{\boldsymbol{{s}}}_i$ , which results in:

(23) \begin{equation}\dot{\boldsymbol{\omega}}_{\bot_i} = \frac{ \hat{\boldsymbol{{s}}}_i \times \left(\ddot{\boldsymbol{{x}}}- 2\dot{\ell}_i \left(\boldsymbol{\omega}_i \times {\hat{\boldsymbol{{s}}}_i} \right)- \ell_i \left(\boldsymbol{\omega}_i \times \left(\boldsymbol{\omega}_i \times {\hat{\boldsymbol{{s}}}_i} \right) \right) \right)}{\ell_i}= \boldsymbol{{J}}_{\omega_{\bot_i}} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_{\bot_i}} \dot{\boldsymbol{{x}}}.\end{equation}

Using anti-commutative property of cross-product and Eq. (10), $\dot{\boldsymbol{{J}}}_{\omega_{\bot_i}}$ may be derived as follow:

(24) \begin{equation} \dot{\boldsymbol{{J}}}_{\omega_{\bot_i}}=\frac{ 2 \;\dot{\ell}_i\; {\hat{\boldsymbol{{s}}}_{i_\times}}{}^2 \;\boldsymbol{{J}}_{\omega_i} +\ell_i \;\hat{\boldsymbol{{s}}}_{i_\times} \boldsymbol{\omega}_{i_{\times}} \hat{\boldsymbol{{s}}}_{i_\times} \boldsymbol{{J}}_{\omega_i}}{\ell_i}.\end{equation}

To calculate $\dot \omega_{u_i}$ and $\dot \omega_{s_i}$ scalar values, time derivative of equations of (10) and (11) will result in:

(25a) \begin{align} \dot{\boldsymbol{\omega}}_i=\dot{\boldsymbol{\omega}}_{\bot_i}+ \dot \omega_{s_i} \hat{\boldsymbol{{s}}}_i= \boldsymbol{{J}}_{\omega_i} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_i} \dot{\boldsymbol{{x}}}, \end{align}

(25b) \begin{align} \dot{\boldsymbol{\omega}}_i= \dot \omega_{u_i} \hat{\boldsymbol{{u}}}_i + \dot \omega_{v_i} \hat{\boldsymbol{{v}}}_i+ \omega_{u_i} \omega_{v_i} \hat{\boldsymbol{{w}}}_i. \end{align}

Now, noting that $\hat{\boldsymbol{{v}}}_i \cdot \hat{\boldsymbol{{s}}}_i=0$ , $\dot \omega_{u_i}$ and $\dot \omega_{s_i}$ can be calculated by equating (25a) and (25b), and applying dot product with $\hat{\boldsymbol{{u}}}_i$ and $\hat{\boldsymbol{{s}}}_i$ , respectively, as follows:

(26a) \begin{align} \dot \omega_{u_i}= \left(\dot{\boldsymbol{\omega}}_{\bot_i}+ \dot \omega_{s_i} \hat{\boldsymbol{{s}}}_i \right) \cdot \hat{\boldsymbol{{u}}}_i , \end{align}

(26b) \begin{align} \dot{\omega}_{s_i}= \dot \omega_{u_i} \left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) + \omega_{u_i} \omega_{v_i} \left(\hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right). \end{align}

Now, $\dot \omega_{s_i}$ would be derived explicitly by substituting (26a) into (26b), as follows:

(27) \begin{equation}\dot{\omega}_{s_i}= \frac{ \left(\dot{\boldsymbol{\omega}}_{\bot_i} \cdot \hat{\boldsymbol{{u}}}_i\right) \left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right)+ \omega_{u_i} \omega_{v_i} \left(\hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) }{1-\left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right)^2}.\end{equation}

On the other hand, $\dot{\omega}_{s_i}=\boldsymbol{\mathcal{J}}_{\omega_{s_i}} \ddot{\boldsymbol{{x}}}+ \dot{ \boldsymbol{\mathcal{J}}}_{\omega_{s_i}} \dot{\boldsymbol{{x}}}$ is another type of representation of Eq. (27). Now, to derive $\boldsymbol{\mathcal{J}}_{\omega_{s_i}}$ and $\dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}}$ , by the substitution of equations (16) and (23) into Eq. (27) and by some manipulations, $\dot{\omega}_{s_i}$ should be written as follows:

(28) \begin{equation}\dot{\omega}_{s_i}= \frac{ \left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) {\hat{\boldsymbol{{u}}}_i}{}^T \left( \boldsymbol{{J}}_{\omega_{\bot_i}} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_{\bot_i}} \ddot{\boldsymbol{{x}}} \right)+ \left(\hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) \left( \boldsymbol{\mathcal{J}}_{\omega_{u_i}} \dot{\boldsymbol{{x}}} \;\boldsymbol{\mathcal{J}}_{\omega_{v_i}} \dot{\boldsymbol{{x}}} \right) }{1-\left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right)^2}= \boldsymbol{\mathcal{J}}_{\omega_{s_i}} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}} \dot{\boldsymbol{{x}}}.\end{equation}

Therefore, $\boldsymbol{\mathcal{J}}_{\omega_{s_i}}$ and $\dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}}$ may be derived as:

(29a) \begin{align} \boldsymbol{\mathcal{J}}_{\omega_{s_i}}= \frac{ \left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) {\hat{\boldsymbol{{u}}}_i}{}^T }{1-\left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right)^2} \;\boldsymbol{{J}}_{\omega_{\bot_i}}, \end{align}

(29b) \begin{align} \dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}}= \frac{ \left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) {\hat{\boldsymbol{{u}}}_i}{}^T \dot{\boldsymbol{{J}}}_{\omega_{\bot_i}} + \left(\hat{\boldsymbol{{w}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right) \left( \boldsymbol{\mathcal{J}}_{\omega_{u_i}} \dot{\boldsymbol{{x}}} \;\boldsymbol{\mathcal{J}}_{\omega_{v_i}} \right) }{1-\left( \hat{\boldsymbol{{u}}}_i \cdot \hat{\boldsymbol{{s}}}_i \right)^2}. \end{align}

It should be noted that $\boldsymbol{\mathcal{J}}_{\omega_{s_i}}$ can be derived from both (14b) and (29a) equations, and the equality of these two equations is easily proveable. Eventually, by rewriting Eq. (25) as $\dot{\boldsymbol{\omega}}_i= \boldsymbol{{J}}_{\omega_i} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_i} \dot{\boldsymbol{{x}}}$ , in which $\boldsymbol{{J}}_{\omega_i}$ is shown in Eq. (13), $\dot{\boldsymbol{{J}}}_{\omega_i}$ may be derived as:

(30) \begin{equation} \dot{\boldsymbol{{J}}}_{\omega_i}=\dot{\boldsymbol{{J}}}_{\omega_{\bot_i}} + \hat{\boldsymbol{{s}}}_i \; \dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}},\end{equation}

where, Eq. (24) and (29b) define $\dot{\boldsymbol{{J}}}_{\omega_{\bot_i}}$ and $\dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}}$ , respectively.

4.2 Linear acceleration

By differentiation of Eq. (18), the acceleration vectors of the piston and the cylinder CM may bederived as:

(31a) \begin{align} \boldsymbol{{a}}_{c_{1_i}}=c_{1_i}\left(\dot{\boldsymbol{\omega}}_i \times \hat{\boldsymbol{{s}}}_i + \boldsymbol{\omega}_i \times (\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i)\right) = \boldsymbol{{J}}_{v_{1_i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{1_i}} \dot{\boldsymbol{{x}}}, \end{align}

(31b) \begin{align} \boldsymbol{{a}}_{c_{2_i}}=(\ell_i-c_{2_i}) \left(\dot{\boldsymbol{\omega}}_i \times \hat{\boldsymbol{{s}}}_i + \boldsymbol{\omega}_i \times (\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i) \right)+2 \dot{\ell}_i (\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i)+\ddot{\ell}_i \hat{\boldsymbol{{s}}}_i = \boldsymbol{{J}}_{v_{2_i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{2_i}} \dot{\boldsymbol{{x}}}. \end{align}

Using anti-commutative property of cross-product, and equations (10), (21) and (25a), different terms of Eq. (31) can be written as a function of $\dot{\boldsymbol{{x}}}$ and $\ddot{\boldsymbol{{x}}}$ as follows:

(32a) \begin{align} \dot{\boldsymbol{\omega}}_i \times \hat{\boldsymbol{{s}}}_i = - \hat{\boldsymbol{{s}}}_{i_\times} \left(\boldsymbol{{J}}_{\omega_i} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_i} \dot{\boldsymbol{{x}}} \right), \end{align}

(32b) \begin{align} \boldsymbol{\omega}_i \times (\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i)=-\boldsymbol{\omega}_{i_{\times}}\; \hat{\boldsymbol{{s}}}_{i_\times} \; \boldsymbol{\omega}_i= -\boldsymbol{\omega}_{i_{\times}} \;\hat{\boldsymbol{{s}}}_{i_\times}\; \left(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}} \right), \end{align}

(32c) \begin{align} 2 \dot{\ell}_i (\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i)= -2 \dot{\ell}_i \hat{\boldsymbol{{s}}}_{i_\times} \;\left(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}} \right), \end{align}

(32d) \begin{align} \hat{\boldsymbol{{s}}}_i \ddot{\ell}_i = \hat{\boldsymbol{{s}}}_i \left( {\hat{\boldsymbol{{s}}}_i}{}^T \ddot{\boldsymbol{{x}}} + \left(\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i \right)^T \dot{\boldsymbol{{x}}} \right). \end{align}

Therefore, Eq. (31) can be rewritten as follows:

(33) \begin{equation} \boldsymbol{{a}}_{c_{1_i}}=- \left( c_{1_i} \hat{\boldsymbol{{s}}}_{i_\times} \boldsymbol{{J}}_{\omega_i} \right) \ddot{\boldsymbol{{x}}} - c_{1_i} \left( \hat{\boldsymbol{{s}}}_{i_\times} \dot{\boldsymbol{{J}}}_{\omega_i} + \boldsymbol{\omega}_{i_{\times}}\; \hat{\boldsymbol{{s}}}_{i_\times}\; \boldsymbol{{J}}_{\omega_i} \right) \dot{\boldsymbol{{x}}} = \boldsymbol{{J}}_{v_{1_i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{1_i}} \dot{\boldsymbol{{x}}},\end{equation}

(34) \begin{equation}\begin{split} \boldsymbol{{a}}_{c_{2_i}}= & \left( -(\ell_i-c_{2_i}) \hat{\boldsymbol{{s}}}_{i_\times} \; \boldsymbol{{J}}_{\omega_i} + \hat{\boldsymbol{{s}}}_i {\hat{\boldsymbol{{s}}}_i}{}^T \right ) \ddot{\boldsymbol{{x}}} + \\& \left( - (\ell_i-c_{2_i}) \left( \hat{\boldsymbol{{s}}}_{i_\times}\;\dot{\boldsymbol{{J}}}_{\omega_i} +\boldsymbol{\omega}_{i_{\times}}\; \hat{\boldsymbol{{s}}}_{i_\times}\; \boldsymbol{{J}}_{\omega_i} \right) -2 \dot{\ell}_i \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i} + \hat{\boldsymbol{{s}}}_i \left(\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i \right)^T \right) \dot{\boldsymbol{{x}}} = \boldsymbol{{J}}_{v_{2_i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{2_i}} \dot{\boldsymbol{{x}}}.\end{split}\end{equation}

It should be noted that $\boldsymbol{{J}}_{v_{1_i}}$ and $\boldsymbol{{J}}_{v_{2_i}}$ are similar to Eq. (20), and $\dot{\boldsymbol{{J}}}_{v_{1_i}}$ and $\dot{\boldsymbol{{J}}}_{v_{2_i}}$ may be derived as:

(35a) \begin{align} \dot{\boldsymbol{{J}}}_{v_{1_i}}=- c_{1_i} \left( \hat{\boldsymbol{{s}}}_{i_\times}\;\dot{\boldsymbol{{J}}}_{\omega_i} +\boldsymbol{\omega}_{i_{\times}}\; \hat{\boldsymbol{{s}}}_{i_\times}\; \boldsymbol{{J}}_{\omega_i} \right), \end{align}

(35b) \begin{align} \dot{\boldsymbol{{J}}}_{v_{2_i}}= - (\ell_i-c_{2_i}) \left( \hat{\boldsymbol{{s}}}_{i_\times}\;\dot{\boldsymbol{{J}}}_{\omega_i} +\boldsymbol{\omega}_{i_{\times}}\; \hat{\boldsymbol{{s}}}_{i_\times}\; \boldsymbol{{J}}_{\omega_i} \right) -2 \dot{\ell}_i \hat{\boldsymbol{{s}}}_{i_\times} \;\boldsymbol{{J}}_{\omega_i} + \hat{\boldsymbol{{s}}}_i \left(\boldsymbol{\omega}_i \times \hat{\boldsymbol{{s}}}_i \right)^T. \end{align}

5.1 Implicit analysis

It can be proved that the inverse dynamics of the 3-UPU TPM, using the principle of virtual work, can be derived as follows [Reference Tsai22]:

(36) \begin{equation} \boldsymbol{{J}}^{T} \tau + \boldsymbol{{f}}_p+ \sum_{i=1}^{3} \left( {\boldsymbol{{J}}_{{1_i}}}^{T} \boldsymbol{{f}}_{1_i} + {\boldsymbol{{J}}_{{2_i}}}^{T} \boldsymbol{{f}}_{2_i} \right) =0,\end{equation}

in which, $\boldsymbol{{J}}$ is the Jacobian matrix of the whole manipulator, $\boldsymbol{\tau}$ is the actuator generalized forces, and ${\boldsymbol{{J}}_{{1_i}}}$ and ${\boldsymbol{{J}}_{{2_i}}}$ represent the general Jacobian matrix of the cylinder and piston of each limb. Furthermore, $\boldsymbol{{f}}_p$ and $\boldsymbol{{f}}_i$ represent the inertia forces exerted at the CM of the cylinder and piston of each limb. By neglecting the friction terms and considering that in 3-UPU TPM the moving platform has no rotational motion, these inertial forces can be derived as follows [Reference Zhao and Gao34]:

(37a) \begin{align} \boldsymbol{{f}}_p =- m_p\left(\ddot{\boldsymbol{{x}}}-\boldsymbol{{g}}_0 \right) + \boldsymbol{{f}}_{d}, \end{align}

(37b) \begin{align} \boldsymbol{{f}}_{1_i} =-\begin{bmatrix} m_{1_i}\left(\boldsymbol{{a}}_{c_{1_i}}-\boldsymbol{{g}}_0 \right) \\ ^{0}\mathrm{\boldsymbol{{I}}}_{1_i} \; \dot{\boldsymbol{\omega}}_{i} +\boldsymbol{\omega}_{i} \times \left(^{0}\mathrm{\boldsymbol{{I}}}_{1_i}\;\boldsymbol{\omega}_{i} \right ) \end{bmatrix}, \end{align}

(37c) \begin{align} \boldsymbol{{f}}_{2_i} =-\begin{bmatrix} m_{2_i}\left(\boldsymbol{{a}}_{c_{2_i}}-\boldsymbol{{g}}_0 \right) \\ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \; \dot{\boldsymbol{\omega}}_{i} +\boldsymbol{\omega}_{i} \times \left(^{0}\mathrm{\boldsymbol{{I}}}_{2_i}\;\boldsymbol{\omega}_{i} \right ) \end{bmatrix}. \end{align}

in which ${}^{0}\mathrm{\boldsymbol{{I}}}_{1_i}$ and ${}^{0}\mathrm{\boldsymbol{{I}}}_{2_i}$ denote the moment of inertia of the cylinder and piston of each limb, about the CM frame with respect to the base coordinate system. Moreover, $\boldsymbol{{f}}_d$ indicates the external disturbance wrench applied to the moving platform CM. Therefore, the inverse dynamic model of 3-UPU TPM can be formulated as:

(38) \begin{equation} \boldsymbol{{J}}^{T} \tau =\boldsymbol{{f}}= -\left(\boldsymbol{{f}}_p+ \sum_{i=1}^{3} \left({\boldsymbol{{J}}_{{1_i}}}^{T} \boldsymbol{{f}}_{1_i} + {\boldsymbol{{J}}_{{2_i}}}^{T} \boldsymbol{{f}}_{2_i}\right) \right),\end{equation}

where $\boldsymbol{{f}}= \boldsymbol{{J}}^{T} \tau$ represent projection of the actuator forces $\boldsymbol{\tau}$ , on the moving platform. Equation (38) represents the implicit dynamic formulation of the 3-UPU TPM. According to this equation, actuator forces, $\boldsymbol{\tau}$ could be uniquely obtained by $\boldsymbol{\tau}=\boldsymbol{{J}}^{-T}\boldsymbol{{f}}$ at non-singular configurations of robot.

5.2 Explicit dynamics

In order to derive the explicit form of dynamic formulations adopting the principle of virtual work, implicit dynamic equation (38) is reconsidered. In Section 3, it was shown that the velocity and acceleration of the CM of the cylinder and piston of each limb can be extracted as follows:

(39a) \begin{align} \boldsymbol{{v}}_{c_{1_i}} =\boldsymbol{{J}}_{v_{1_i}} \dot{\boldsymbol{{x}}}\;\;,\;\; \boldsymbol{{v}}_{c_{2_i}} =\boldsymbol{{J}}_{v_{2_i}} \dot{\boldsymbol{{x}}} \;\;, \;\; \boldsymbol{{a}}_{c_{1_i}} =\boldsymbol{{J}}_{v_{1_i}} \ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{1_i}} \dot{\boldsymbol{{x}}} \;\;,\;\;\boldsymbol{{a}}_{c_{2_i}}= \boldsymbol{{J}}_{v_{2_i}} \ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{v_{2_i}} \dot{\boldsymbol{{x}}}, \end{align}

(39b) \begin{align} \boldsymbol{\omega}_i= \boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}} \;\;, \;\; \dot \omega_i= \boldsymbol{{J}}_{\omega_i} \ddot{\boldsymbol{{x}}} + \dot{\boldsymbol{{J}}}_{\omega_i} \dot{\boldsymbol{{x}}}. \end{align}

Substitute Eq. (39) in Eq. (37) results in:

(40a) \begin{align} \boldsymbol{{f}}_{p}=- m_p\left(\ddot{\boldsymbol{{x}}}-\boldsymbol{{g}}_0\right)+ \boldsymbol{{f}}_d, \end{align}

(40b) \begin{align} \boldsymbol{{f}}_{1_i}=-\begin{bmatrix} m_{1_i} \left( \boldsymbol{{J}}_{v_{1_i}} \ddot{\boldsymbol{{x}}}+\dot{\boldsymbol{{J}}}_{v_{1_i}} \dot{\boldsymbol{{x}}}-\boldsymbol{{g}}_0 \right) \\ ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}\;\left(\boldsymbol{{J}}_{\omega_{i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{\omega_i}\dot{\boldsymbol{{x}}} \right) +\left(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}}\right)_{\times} \;^{0}\mathrm{\boldsymbol{{I}}}_{1_{i}}\;\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}} \end{bmatrix}, \end{align}

(40c) \begin{align} \boldsymbol{{f}}_{2_i}=-\begin{bmatrix} m_{2_i} \left( \boldsymbol{{J}}_{v_{2_i}} \ddot{\boldsymbol{{x}}}+\dot{\boldsymbol{{J}}}_{v_{2_i}} \dot{\boldsymbol{{x}}}-\boldsymbol{{g}}_0 \right) \\ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i}\;\left(\boldsymbol{{J}}_{\omega_{i}}\ddot{\boldsymbol{{x}}}+ \dot{\boldsymbol{{J}}}_{\omega_i}\dot{\boldsymbol{{x}}} \right) +\left(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}}\right)_{\times} \;^{0}\mathrm{\boldsymbol{{I}}}_{2_i}\;\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}} \end{bmatrix}. \end{align}

Using inertial forces Eq. (40), Eq. (38) may be rewritten as:

(41) \begin{equation}\begin{split} \boldsymbol{{J}}^{T} \tau=\boldsymbol{{f}}+\boldsymbol{{f}}_d =&\sum_{i=1}^{3} m_{1_i} {\boldsymbol{{J}}_{v_{1_i}}}^T \left((\boldsymbol{{J}}_{v_{1_i}} \ddot{\boldsymbol{{x}}}+\dot{\boldsymbol{{J}}}_{v_{1_i}} \dot{\boldsymbol{{x}}})-\boldsymbol{{g}}_0\right)+\sum_{i=1}^{3} m_{2_i} {\boldsymbol{{J}}_{v_{2_i}}}^T \left((\boldsymbol{{J}}_{v_{2_i}} \ddot{\boldsymbol{{x}}}+\dot{\boldsymbol{{J}}}_{v_{2_i}} \dot{\boldsymbol{{x}}})-\boldsymbol{{g}}_0\right)+ \\[5pt] & \sum_{i=1}^{3} {\boldsymbol{{J}}_{\omega_i}}^T \left( \left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \right )\;(\boldsymbol{{J}}_{\omega_i}\ddot{\boldsymbol{{x}}}+\boldsymbol{{J}}_{\omega_i}\dot{\boldsymbol{{x}}}) +(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}})_{\times} \;\left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \right )\;\boldsymbol{{J}}_{\omega_i}\dot{\boldsymbol{{x}}}\right) + \\ &m_p (\ddot{\boldsymbol{{x}}} -\boldsymbol{{g}}_0).\end{split}\end{equation}

Therefore, explicit dynamic formulation of 3-UPU TPM can be derived as:

(42) \begin{equation} \boldsymbol{{J}}^{T} \tau =\boldsymbol{{f}}+\boldsymbol{{f}}_d=\boldsymbol{{M}} (\boldsymbol{{x}}) \ddot{\boldsymbol{{x}}} +\boldsymbol{{C}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})\dot{\boldsymbol{{x}}}+\boldsymbol{{g}}(\boldsymbol{{x}}),\end{equation}

where $\boldsymbol{{M}}(\boldsymbol{{x}})$ , $\boldsymbol{{C}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})$ and $\boldsymbol{{g}}(\boldsymbol{{x}})$ denote the mass matrix, centrifugal, and Coriolis matrix and gravity vector of the robot, respectively, and can be derived as follows:

(43) \begin{equation} \boldsymbol{{M}}(\boldsymbol{{x}})=m_p+\sum_{i=1}^{3} \left({\boldsymbol{{J}}_{v_{1_i}}}^T\; m_{1_i}\;{\boldsymbol{{J}}_{v_{1_i}}}+{\boldsymbol{{J}}_{v_{2_i}}}^T\; m_{2_i}\;{\boldsymbol{{J}}_{v_{2_i}}}+{\boldsymbol{{J}}_{\omega_i}}^T\;\left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \right )\;\boldsymbol{{J}}_{\omega_i}\right),\end{equation}

(44) \begin{equation}\begin{split} \boldsymbol{{C}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})= \sum_{i=1}^{3} & \left( {\boldsymbol{{J}}_{v_{1_i}}}^T\; m_{1_i}\;{\dot{\boldsymbol{{J}}}_{v_{1_i}}}+{\boldsymbol{{J}}_{v_{2_i}}}^T\; m_{2_i}\;{\dot{\boldsymbol{{J}}}_{v_{2_i}}}+{\boldsymbol{{J}}_{\omega_i}}^T\;\left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \right )\;\dot{\boldsymbol{{J}}}_{\omega_i}+ \cdot \cdot \cdot \right .\\[5pt] & \left. {\boldsymbol{{J}}_{\omega_i}}^T\;(\boldsymbol{{J}}_{\omega_i} \dot{\boldsymbol{{x}}})_{\times} \;\left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ ^{0}\mathrm{\boldsymbol{{I}}}_{2_i} \right )\;\boldsymbol{{J}}_{\omega_i} \right) ,\end{split}\end{equation}

(45) \begin{equation} \boldsymbol{{g}}(\boldsymbol{{x}})=-m_p\;\boldsymbol{{g}}_0 - \sum_{i=1}^{3} \left( m_{1_i}\; {\boldsymbol{{J}}_{v_{1_i}}}^T + m_{2_i}\; {\boldsymbol{{J}}_{v_{2_i}}}^T \right) \boldsymbol{{g}}_0.\end{equation}

Equation (42) is a multi-input/multi-output (MIMO) nonlinear equation with some advantageous properties for designing model-based controllers as follows [Reference Taghirad21]:

Property 1. $\boldsymbol{{M}}(\boldsymbol{{x}})$ is positive definite in all configurations.

Property 2. $\left[\dot{\boldsymbol{{M}}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})-2\boldsymbol{{C}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})\right]$ is skew-symmetric.

The first property may be easily proved, considering the quadratic structure in its formulation. However, to prove the second property, first, the time derivative of $\dot{\boldsymbol{{{M}}}}(\boldsymbol{{x}},\dot{\boldsymbol{{x}}})$ may be given as:

(46) \begin{equation}\begin{split}\dot{\boldsymbol{{M}}}= \;& {\dot{\boldsymbol{{J}}}_{v_{1_i}}}{}^T\; m_{1_i}\;{\boldsymbol{{J}}_{v_{1_i}}}+{\boldsymbol{{J}}_{v_{1_i}}}{}^T\; m_{1_i}\;{\dot{\boldsymbol{{J}}}_{v_{1_i}}}+{\dot{\boldsymbol{{J}}}_{v_{2_i}}}{}^T\; m_{2_i}\;{\boldsymbol{{J}}_{v_{2_i}}}+{\boldsymbol{{J}}_{v_{2_i}}}{}^T\; m_{2_i}\;{\dot{\boldsymbol{{J}}}_{v_{2_i}}}+ {\dot{\boldsymbol{{J}}}_{\omega_i}}{}^T \left(^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ {^{0}\mathrm{\boldsymbol{{I}}}_{2_i}} \right )\boldsymbol{{J}}_{\omega_i}+ \\[5pt] &{\boldsymbol{{J}}_{\omega_i}}{}^T \left( \boldsymbol{\omega}_{i_\times} \left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ {^{0}\mathrm{\boldsymbol{{I}}}_{2_i}} \right )+ \left( ^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+{^{0}\mathrm{\boldsymbol{{I}}}_{2_i}} \right ) {\boldsymbol{\omega}_{i_\times}}^T \right )\boldsymbol{{J}}_{\omega_i} +{\boldsymbol{{J}}_{\omega_i}}^T \left(^{0}\mathrm{\boldsymbol{{I}}}_{1_i}+ {^{0}\mathrm{\boldsymbol{{I}}}_{2_i}} \right ) \dot{\boldsymbol{{J}}}_{\omega_i},\end{split} \end{equation}

Hence, it may be verified that the following equation always holds:

(47) \begin{equation}\left( \dot{\boldsymbol{{{M}}}} - 2 \boldsymbol{{C}} \right)^T + \left( \dot{\boldsymbol{{{M}}}} - 2 \boldsymbol{{C}} \right)=0. \end{equation}

6.1 Model validation

Inorder to validate the explicit dynamics given in Eq. (42), CAD model of the 3-UPU TPM is imported into MSC-ADAMS^® software. Then, the output force of the actuators resulted from applying the desired trajectory, $\boldsymbol{{x}} = \boldsymbol{{x}}_d$ , to both the MSC-ADAMS and the robot’s explicit dynamic model is logged. By comparing these two results, the accuracy of the dynamic model is determined. A view of the 3-UPU TPM model in MSC-ADAMS may be seen in Fig. 4. Besides, the parameters of the considered 3-UPU TPM model have also been reported in Table I.

Figure 4.

3-UPU TPM model in MSC-ADAMS^®.

Table I.

Parameters of the considered 3-UPU TPM

To verify the dynamics model of the 3-UPU manipulator, validation trajectory is considered as $\boldsymbol{{x}} =a \left[sin(2 \pi t), sin(2 \pi t), cos(2 \pi t) \right]^T+\boldsymbol{{x}}_0$ , where a and $\boldsymbol{{x}}_0$ is defined as $a = 0.2\;(m)$ and $\boldsymbol{{x}}_0=[0;1.1;0]^T\;(m)$ , respectively. Moreover, the external disturbance $\boldsymbol{{f}}_d$ is considered zero. Therefore, by giving these trajectories to the explicit dynamics model and the MSC-ADAMS^® model, the results of the explicit dynamic verification and its error are reported in Figs. 5 and 6, respectively.

Figure 5.

Explicit dynamics verification of 3-UPU TPM with MSC-ADAMS^®.

Figure 6.

Error of explicit dynamics verification with MSC-ADAMS^®.

As shown in Figs. 5 and 6, respectively, the explicit dynamic formulation of the 3-UPU TPM is verified with the accuracy of $10^{-4}$ , and applying other trajectories such as the third-order polynomial and multi-sine trajectories will not affect the result, and the verification error will always remain in the $10^{-4}$ bound. In addition, to ensure the validity of Eq. (47), which is an important feature in the design of model-base controllers, its value is reported in Fig. 7. As it can be seen in Fig. 7, the Eq. (47) is always valid because the difference between the two sides of the equation is equal to the machine precision in numerical calculations.

Figure 7.

Error of verification for $\dot{\boldsymbol{{M}}}-2\boldsymbol{{C}}$ .

6.2 Comparison of simple and complete dynamic model

Inthis section, by comparing the accuracy of the simple and the achieved complete dynamic mode, the effect of considering the simple model of universal joints on the dynamic model is examined. For this purpose, it is necessary to consider $ \boldsymbol{\omega}_{i}=\boldsymbol{\omega}_{\bot_i} $ and $ \dot{\boldsymbol{\omega}}_{i}=\dot{\boldsymbol{\omega}}_{\bot_i} $ to create a simplified model of the robot [Reference Hraiech, Chebbi, Affi and Romdhane18, Reference Xiangzhou, Yougao and Hongzan20]. Hence, it suffices to set $\boldsymbol{\mathcal{J}}_{\omega_{s_i}}=0$ and $\dot{\boldsymbol{\mathcal{J}}}_{\omega_{s_i}}=0$ in equations (13) and (30), respectively. In addition, assuming no disturbance force is applied to the moving platform, forces of the actuators can be classified into the following parts:

(48) \begin{equation}\tau =\boldsymbol{\tau}_M+ \boldsymbol{\tau}_C+ \boldsymbol{\tau}_g,\end{equation}

in which $\boldsymbol{\tau}_M$ denotes actuator forces generated from mass matrix terms, $\boldsymbol{\tau}_C$ is actuator forces generated from Centrifugal and Coriolis terms, and $\boldsymbol{\tau}_g$ shows actuator forces generated from the gravitational terms. It should be noted that angular velocity and acceleration terms of limbs have no effect on $\boldsymbol{\tau}_g$ , and therefore, $\boldsymbol{\tau}_g$ is not considered in the later comparison. In addition, kinetic energy $K=\frac{1}{2} \; \dot{\boldsymbol{{x}}} ^T \boldsymbol{{M}}(\boldsymbol{{x}}) \dot{\boldsymbol{{x}}}$ is used as another measurement of the robot mass matrix values.

Therefore, using the following normalized root mean square error (NRMSE) [Reference Ljung and Singh35], the comparison will be drawn:

(49) \begin{equation}NRMSE= 100 \left( 1- \frac{|| \boldsymbol{\tau}_F- \boldsymbol{\tau}_S ||}{||\boldsymbol{\tau}_F- mean\left(\boldsymbol{\tau}_F \right)||} \right)\end{equation}

in which $\boldsymbol{\tau}_F$ and $\boldsymbol{\tau}_S$ denote actuator forces generated from the full and the simple dynamic model, respectively. Accordingly, by taking the parameters reported in Table I and the validation trajectory ofSection 6.1 into account, the results of comparing the kinematic and dynamic sections are reported in Tables II and III, respectively.

Table II.

NRMSE of angular velocity and acceleration of 3-UPU TPM limbs

Table III.

NRMSE of different parts of actuator forces

As canbe seen from Table II, the greatest loss of accuracy occurs in the y component of the angular velocity and acceleration of the limbs. Moreover, by examining Table III, it can be deduced that for the values assumed for the robot and trajectory parameters, the simple and complete models are almost the same, and uncertainty only exists in $\boldsymbol{\tau}_C$ . However, since $\boldsymbol{\tau}_c$ is relatively smaller than $\boldsymbol{\tau}_M$ , uncertainty in $\boldsymbol{\tau}$ would be insignificant.

On the other hand, in cases where the amount of $\boldsymbol{\tau}_c$ is high, it can be expected that the values of uncertainties in $\boldsymbol{\tau}$ will increase. Therefore, a trajectory similar to Section 6.1 but with a variable amplitude is considered to investigate the effect of accelerating the trajectory^{Footnote 2} on the uncertainties of $\boldsymbol{\tau}$ . Thus, the results of only the robot dynamics are examined and reported in Table IV.

Table IV.

Effect of faster trajectory on the NRMSE of different parts of actuator forces

As can be seen from Table IV, by accelerating the trajectory, the accuracy of the model decreases, and this is more evident in centrifugal and Coriolis terms, as expected. In this case, there is significant structural uncertainty in the dynamic model, which can reduce the accuracy of the model-based controllers to control the robot in fast trajectories. Consequently, using the complete dynamic model of 3-UPU TPM in fast trajectories is strongly recommended.