2. OMPR

Based on the grinding process requirements, the OMPR requires at least five DOFs. The robot used in this study was a five-DOF OMPR developed using a three-DOF parallel (3UPS + UP: U - hook joint, P - moving pair, S - ball joint) manipulator combined with a two-DOF serial manipulator. The five-DOF OMPR consisted of a fixed platform, drive branch chain, constrained branch chain, moving platform, and two-DOF serial manipulator. The topological structure and prototype of the OMPR are shown in Fig. 1(a) and (b); there are three UPS drive branches and one UP constraint branch link between the movable and fixed platforms. The world coordinate system, O–XYZ, is established at the center point, O, of the mirror to be processed, the Z axis is downward, the Y axis points to the center point of the robot fixed platform, and the X axis is determined based on the right-hand rule. The fixed platform is installed at an angle of 30° with the Z axis of the world coordinate system. A fixed coordinate system, O ₁–X ₁ Y ₁ Z ₁, is established on the center, O ₁, of the fixed platform, the Y ₁ axis is along the $\overline{\overline{{O_{1}U_{1}}}}$ direction, the Z ₁ axis is perpendicular to the fixed platform, and the X ₁ axis is determined by the right-hand rule. A moving coordinate system, O ₂–x ₂ y ₂ z ₂, is established on the central point, O _2, of the moving platform, the y ₂ axis is along the $\overline{\overline{{O_{2}u_{1}}}}$ direction, and the z ₂ axis is perpendicular to the moving platform. Moreover, the x ₂ axis is determined by the right-hand rule. The coordinate system, O ₃–x ₃ y ₃ z ₃, is established on the junction point, O _3, of the two series rotating heads. The x ₃ axis is collinear with the axis of the secondary rotating head, and the y ₃ axis is determined by the right-hand rule. The secondary rotating head coordinate system, O ₄–x ₄ y ₄ z ₄, is set up on tool spindle O ₄ point, the z ₄ axis is down the tool spindle, the y ₄ axis is along the direction, and the x ₄ axis is determined by the right-hand rule. The tool coordinate system, S–uvw, is established on the end tool execution point, S, and the w axis is the normal direction of the mirror to be processed. The u axis is the tangent direction of the tool path, and the v axis is determined based on the right-hand rule. The end grinding tool uses computer controlled optical surfacing (CCOS) grinding system, and its structure is shown in Fig. 1(c). The CCOS grinding system consists of a rotating motor and a revolving motor, which uses a smaller grinding head for grinding.

Figure 1. OMPR.

When the OMPR performs a processing task, a transformation relationship needs to be established to convert the end execution point in the world coordinate system to the fixed coordinate system.

(1) \begin{equation} {_{}^{O}}{\textbf{P}}{}=\textbf{q}_{1}+{_{}^{O}}{\textbf{R}}{_{O_{1}}^{}}{_{}^{O_{1}}}{\textbf{P}}{}, \end{equation}

where ^O P is the position vector of any point in the world coordinate system, q ₁ is the position vector of the center point of the fixed platform under O–XYZ, ${_{}^{O_{1}}}{\textbf{P}}{}$ is the position vector of point P under O ₁ –X ₁ Y ₁ Z ₁, and ${_{}^{O}}{\textbf{R}}{_{O_{1}}^{}}$ is the transformation matrix from O–XYZ to O ₁ –X ₁ Y ₁ Z ₁.

(1) Inverse solution of series module

In the world coordinate system, the end point of the grinding tool on the curved surface is $\textbf{S}=\left[\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right]^{\mathrm{T}}$ , and a smooth trajectory curve through the point S is given as follows:

(2) \begin{align} \boldsymbol{\Gamma }\colon \begin{cases} x=x(t)\\[3pt] y=y(t)\\[3pt] z=z(t). \end{cases} \end{align}

Subsequently, the tangent vector, u , of $\boldsymbol{\Gamma }$ at point S and the normal vector, w , of the surface are

(3) \begin{equation} \begin{cases} \textbf{u}=\dfrac{\left[\begin{array}{l@{\quad}l@{\quad}l} \dot{x}_{0}\left(t_{0}\right) & \dot{y}_{0}(t) & \dot{z}_{0}(t) \end{array}\right]^{\mathrm{T}}}{\sqrt{\dot{x}_{0}\left(t_{0}\right)^{2}+\dot{y}_{0}(t)^{2}+\dot{z}_{0}(t)^{2}}}\\[12pt] \textbf{w}=\dfrac{\left[\begin{array}{l@{\quad}l@{\quad}l} \mathrm{\ell }_{x}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right) & \mathrm{\ell }_{y}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right) & \mathrm{\ell }_{z}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right) \end{array}\right]^{\mathrm{T}}}{\sqrt{\mathrm{\ell }_{x}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right)^{2}+\mathrm{\ell }_{y}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right)^{2}+\mathrm{\ell }_{z}\!\left(\begin{array}{l@{\quad}l@{\quad}l} x_{0} & y_{0} & z_{0} \end{array}\right)^{2}}}, \end{cases} \end{equation}

where $\mathrm{\ell }_{x}$ is the partial derivative of the surface equation to x, and t ₀ is the value at a certain time.

Subsequently, the posture matrix of the end grinding tool coordinate system in the O–XYZ coordinate system is obtained as

(4) \begin{equation} \textbf{R}_{d}=[\begin{array}{l@{\quad}l@{\quad}l} \textbf{u} & \textbf{v} & \textbf{w} \end{array}]_{3\times 3}. \end{equation}

where $\textbf{v}=\textbf{u}\times \textbf{w}$ .

Thus, the vector of point O ₄ in the O–XYZ coordinate system is

(5) \begin{equation} {_{}^{O}}{\textbf{O}}{_{4}^{}}=\textbf{S}-\textbf{R}_{d}\cdot \textbf{q}_{2} \end{equation}

where $\textbf{q}_{2}$ is the position vector of point S in the O ₄–x ₄ y ₄ z ₄.

Figure 2. Schematic of numerical search method.

In Figure 2(a), the relative position vector of O ₃ and O ₂ is typically unchanged in the moving platform coordinate system, O ₂–x ₂ y ₂ z ₂. Therefore,

(6) \begin{equation} \textbf{r}_{2}=\textbf{r}_{3}-d_{1}\textbf{n}, \end{equation}

where r ₂ is the position vector of O ₂ under O ₁ –X ₁ Y ₁ Z ₁, r ₃ is the position vector of O ₃ under O ₁ –X ₁ Y ₁ Z ₁, n is the direction vector of $\overline{\overline{{O_{3}O_{1}}}}$ , and d ₁ is the distance between O ₂ and O ₃.

Based on the structure of the two tandem rotating heads and cutter tools, the O ₃ point is continuously distributed on the plane circle with the O ₄ point as the center and d ₂ as the radius. According to the characteristics of the vector cross-product, two vertical vectors in the plane circle are found,

(7) \begin{align} \begin{cases} \boldsymbol{\lambda }_{\mathrm{1}}=\dfrac{^{O}\textbf{O}_{4}\times \textbf{E}^{\mathrm{1}}}{\left| ^{O}\textbf{O}_{4}\times \textbf{E}^{\mathrm{1}}\right| }\\[12pt] \boldsymbol{\lambda }_{2}=\dfrac{\boldsymbol{\lambda }_{\mathrm{1}}\times ^{O}\textbf{O}_{4}}{\left| \boldsymbol{\lambda }_{\mathrm{1}}\times ^{O}\textbf{O}_{4}\right| }, \end{cases} \end{align}

where E ¹ is a unit vector, λ ₁ is the unit direction vector perpendicular to ^O O ₄, and λ ₂ is the unit direction vector perpendicular to λ ₁ and ^O O ₄.

Using the numerical search method, on the circle having O ₄ as the center, w as the central axis, and d ₂ as the radius, vectors λ ₁ and λ ₂ are searched clockwise in a certain step $\varsigma$ from 0 to 2 π, and a series of scattered points are obtained. The positions of the scattered points in the world coordinate system can be expressed as

(8) \begin{align} \begin{cases} x_{s}=^{O}O_{4x}+d_{2}\lambda _{1x}\cos \left(\varsigma \right)+d_{2}\lambda _{2x}\sin \left(\varsigma \right)\\[5pt] y_{s}=^{O}O_{4y}+d_{2}\lambda _{1y}\cos \left(\varsigma \right)+d_{2}\lambda _{2y}\sin \left(\varsigma \right)\\[5pt] z_{s}=^{O}O_{4z}+d_{2}\lambda _{1z}\cos \left(\varsigma \right)+d_{2}\lambda _{2z}\sin \left(\varsigma \right), \end{cases} \end{align}

where ^O O _4x , ^O O _4y , and ^O O _4z are the components of ^O O ₄ along the coordinate axis of the world coordinate system; λ _1x , λ _1y , and λ _1z are the components of λ ₁ along each coordinate axis of the world coordinate system; and λ _2x , λ _2y , and λ _2z are the components of λ ₂ along each coordinate axis of the world coordinate system.

As shown in Fig. 2(b), the motion of point O ₃ in the parallel module essentially adopts the fixed platform center point, O ₁, as the fulcrum to swing around the X ₁ and Y ₁ axes. Therefore, the minimum distance to the O ₁ point can be used to constrain the scattered points, and the position vector of the O ₃ point in the world coordinate system O–XYZ can be obtained.

During the machining process, angles $\psi$ and $\varphi$ of the two tandem rotating heads are fine-tuned. Figure 3 shows the posture of the OMPR.

Figure 3. Attitude diagram of series module.

At present, the positions of points S, O ₄, O ₃, and O ₁ in the world coordinate system are known, and the rotation angle of the first rotating head and secondary rotating head is obtained as

(9) \begin{equation} \varphi =\arccos \left(\frac{\overline{\overline{{O_{3}O_{4}}}}\cdot \overline{\overline{{O_{3}O_{1}}}}}{\left| \overline{\overline{{O_{3}O_{4}}}}\right| \left| \overline{\overline{{O_{3}O_{1}}}}\right| }\right). \end{equation}

(10) \begin{equation} \psi =\arccos \left(\textbf{x}_{3}\cdot \textbf{x}_{2}'\right) \end{equation}

where x ₃ and $\textbf{x}_{2}\mathit{'}$ are the unit vector expressions of the x ₃ and $x_{2}\mathit{'}$ axes.

(2) Kinematics of parallel module

Based on Fig. 1, O ₁, O ₂, U _i , and u _i can form closed-loop vectors (i = 1, 2, 3); therefore,

(11) \begin{align} \begin{cases} \textbf{r}_{Ui}+l_{i}\textbf{e}_{i}=\textbf{r}_{ui}+\textbf{r}_{2}\\[5pt] \textbf{r}_{2}=L_{2}\textbf{e}_{L}, \end{cases} \end{align}

where L ₂ is the length of the UP branch chain with O ₂ as the end, l _i and e _i are the length and unit direction vector of branch chain i, e _L = n is the unit direction vector of the UP branch chain under O ₁ –X ₁ Y ₁ Z ₁, r ₂ is the position vector of point O ₂ under O ₁ –X ₁ Y ₁ Z ₁, r _Ui is the position vector of U _i in O ₁ –X ₁ Y ₁ Z ₁, and r _ui is the position vector of u _i in O ₁ —X ₁ Y ₁ Z ₁.

The lengths of the UPS and UP chains change as follows:

(12) \begin{align}l_{i}=\sqrt{\left(\textbf{r}_{2}+\textbf{r}_{ui}-\textbf{r}_{Ui}\right)^{\mathrm{T}}\left(\textbf{r}_{2}+\textbf{r}_{ui}-\textbf{r}_{Ui}\right)}\end{align}

and

(13) \begin{equation} L_{2}={\textbf{e}_{L}}^{\mathrm{T}}\textbf{r}_{2}. \end{equation}

Furthermore, the velocity of the UP constrained branch chain and the angular velocity of the moving platform are obtained as follows:

(14) \begin{align} \dot{L}_{2}={\textbf{e}_{L}}^{\mathrm{T}}\dot{\textbf{r}}_{2}\quad \textrm{and}\end{align}

(15) \begin{equation} \boldsymbol{\omega }=\frac{\textbf{e}_{L}\times \dot{\textbf{r}}_{2}}{L_{2}}=\frac{\textbf{e}_{L\times }}{L_{2}}\dot{\textbf{r}}_{2}, \end{equation}

where $\textbf{e}_{L\times }=\left[\begin{array}{l@{\quad}l@{\quad}l} 0 & -e_{Lz} & e_{Ly}\\[5pt] e_{Lz} & 0 & -e_{Lx}\\[5pt] -e_{Ly} & e_{Lx} & 0 \end{array}\right]$ is the third-order antisymmetric matrix of $\textbf{e}_{L}$ and $\dot{\textbf{r}}_{2}$ is the speed of the moving platform.

The expansion acceleration of the UP branch chain and the angular acceleration of the moving platform are

(16) \begin{equation} \ddot{L}_{2}={\textbf{e}_{L}}^{\mathrm{T}}\ddot{\textbf{r}}_{2}+\left(\boldsymbol{\omega }\times \textbf{e}_{L}\right)^{\mathrm{T}}\dot{\textbf{r}}_{2}=\textbf{J}_{L}\ddot{\textbf{r}}_{2}+\dot{\textbf{J}}_{L}\dot{\textbf{r}}_{2}, \end{equation}

(17) \begin{equation} \dot{\boldsymbol{\omega }}=\frac{\textbf{e}_{L\times }}{L_{2}}\ddot{\textbf{r}}_{2}+\left[\frac{\left(\textbf{e}_{L}\times \boldsymbol{\omega }_{d}\right)_{\times }}{L_{2}}-\frac{\dot{L}_{2}}{{L_{2}}^{2}}\textbf{e}_{L\times }\right]\dot{\textbf{r}}_{2}=\textbf{J}_{d\omega }\ddot{\textbf{r}}_{2}+\dot{\textbf{J}}_{d\omega }\dot{\textbf{r}}_{2}. \end{equation}

The speed and acceleration of the UPS branch chain are

(18) \begin{align} \begin{cases} \dot{\textbf{L}}=\textbf{J}\dot{\textbf{r}}_{2}\\[5pt] \ddot{\textbf{L}}=\textbf{J}\ddot{\textbf{r}}_{2}+\dot{\textbf{J}}\dot{\textbf{r}} \end{cases}\!\!\!\!\!\!, \end{align}

where $\textbf{J}={\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{e}_{1}-\frac{{\textbf{e}_{1}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u1}}{L_{2}} & \textbf{e}_{2}-\frac{{\textbf{e}_{2}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u2}}{L_{2}} & \textbf{e}_{3}-\frac{{\textbf{e}_{3}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u3}}{L_{2}} \end{array}\right]^{\mathrm{T}}}_{3\times 3}$ ,

$\begin{array}{l} \dot{\textbf{J}}=\left[\begin{array}{l@{\quad}l@{\quad}l} \boldsymbol{\omega }_{1}\times \textbf{e}_{1} & \boldsymbol{\omega }_{2}\times \textbf{e}_{2} & \boldsymbol{\omega }_{3}\times \textbf{e}_{3} \end{array}\right]^{\mathrm{T}}+\frac{\dot{L}_{2}}{{L_{2}}^{2}}\left[\begin{array}{l@{\quad}l@{\quad}l} {\textbf{e}_{1}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u1} & {\textbf{e}_{2}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u2} & {\textbf{e}_{3}}^{\mathrm{T}}\textbf{e}_{L}\textbf{r}_{u3} \end{array}\right]^{\mathrm{T}}\\[4pt] -\frac{1}{L_{2}}\left[\begin{array}{l} \left(\boldsymbol{\omega }_{1}\times \textbf{e}_{1}\right)^{\mathrm{T}}\textbf{e}_{L}{\textbf{r}_{u1}}^{\mathrm{T}}+{\textbf{e}_{1}}^{\mathrm{T}}\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right){\textbf{r}_{u1}}^{\mathrm{T}}+{\textbf{e}_{1}}^{\mathrm{T}}\textbf{e}_{L}\left(\boldsymbol{\omega }_{d}\times \textbf{r}_{u1}\right)^{\mathrm{T}}\\[5pt] \left(\boldsymbol{\omega }_{2}\times \textbf{e}_{2}\right)^{\mathrm{T}}\textbf{e}_{L}{\textbf{r}_{u2}}^{\mathrm{T}}+{\textbf{e}_{2}}^{\mathrm{T}}\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right){\textbf{r}_{u2}}^{\mathrm{T}}+{\textbf{e}_{2}}^{\mathrm{T}}\textbf{e}_{L}\left(\boldsymbol{\omega }_{d}\times \textbf{r}_{u2}\right)^{\mathrm{T}}\\[5pt] \left(\boldsymbol{\omega }_{3}\times \textbf{e}_{3}\right)^{\mathrm{T}}\textbf{e}_{L}{\textbf{r}_{u3}}^{\mathrm{T}}+{\textbf{e}_{3}}^{\mathrm{T}}\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right){\textbf{r}_{u3}}^{\mathrm{T}}+{\textbf{e}_{3}}^{\mathrm{T}}\textbf{e}_{L}\left(\boldsymbol{\omega }_{d}\times \textbf{r}_{u3}\right)^{\mathrm{T}} \end{array}\right] \end{array}$ ,

$\dot{\textbf{L}}=\left[\begin{array}{l@{\quad}l@{\quad}l} \dot{l}_{1} & \dot{l}_{2} & \dot{l}_{3} \end{array}\right]^{\mathrm{T}}, \ddot{\textbf{L}}=\left[\begin{array}{l@{\quad}l@{\quad}l} \ddot{l}_{1} & \ddot{l}_{2} & \ddot{l}_{3} \end{array}\right]^{\mathrm{T}}$ .

The angular velocity and angular acceleration of the UPS branch chain are

(19) \begin{equation} \boldsymbol{\omega }_{i}=\frac{1}{l_{i}}\left[\textbf{e}_{i\times }+\left({\textbf{e}_{i}}^{\mathrm{T}}\textbf{r}_{ui}\textbf{E}-\textbf{r}_{ui}{\textbf{e}_{i}}^{\mathrm{T}}\right)\frac{\textbf{e}_{L\times }}{L_{2}}\right]\dot{\textbf{r}}_{2}=\textbf{J}_{\omega i}\dot{\textbf{r}}_{2}, \end{equation}

(20) \begin{equation} \dot{\boldsymbol{\omega }}_{i}=\textbf{J}_{\omega i}\ddot{\textbf{r}}_{2}+\dot{\textbf{J}}_{\omega i}\dot{\textbf{r}}_{2}, \end{equation}

where $\begin{array}{l} \dot{\textbf{J}}_{\omega i}=-\dfrac{\dot{l}_{i}}{{l_{i}}^{2}}\left[\textbf{e}_{i\times }+\left({\textbf{e}_{i}}^{\mathrm{T}}\textbf{r}_{ui}\textbf{E}-\textbf{r}_{ui}{\textbf{e}_{i}}^{\mathrm{T}}\right)\textbf{J}_{d\omega }\right]+\dfrac{1}{l_{i}}\left(\boldsymbol{\omega }_{i}\times \textbf{e}_{i}\right)_{\times }+\left({\textbf{e}_{i}}^{\mathrm{T}}\textbf{r}_{ui}\textbf{E}-\textbf{r}_{ui}{\textbf{e}_{i}}^{\mathrm{T}}\right)\dot{\textbf{J}}_{d\omega }\\[5pt] +[(\boldsymbol{\omega }_{i}\times \textbf{e}_{i})^{\mathrm{T}}\textbf{r}_{ui}\textbf{E}+{\textbf{e}_{i}}^{\mathrm{T}}(\boldsymbol{\omega }_{d}\times \textbf{r}_{ui})\textbf{E}-(\boldsymbol{\omega }_{d}\times \textbf{r}_{ui}){\textbf{e}_{i}}^{\mathrm{T}}-\textbf{r}_{ui}(\boldsymbol{\omega }_{i}\times \textbf{e}_{i})^{\mathrm{T}}]\textbf{J}_{d\omega }\} \end{array}$ and E is the third-order unit matrix.

The center of mass velocity of the UPS and UP branch chains is

(21) \begin{equation} \textbf{v}_{ci}=\textbf{e}_{i}\textbf{J}_{i}\dot{\textbf{r}}_{2}-\left(l_{i}-l_{bi}\right)\textbf{e}_{i\times }\textbf{J}_{\omega i}\dot{\textbf{r}}_{2}, \end{equation}

(22) \begin{equation} \textbf{v}_{Ly}=\textbf{e}_{L}{\textbf{e}_{L}}^{\mathrm{T}}\dot{\textbf{r}}_{2}-\left(L_{2}-L_{y}\right)\frac{1}{L_{2}}\left(\textbf{E}-\textbf{e}_{L}{\textbf{e}_{L}}^{\mathrm{T}}\right)\dot{\textbf{r}}_{2}. \end{equation}

The accelerations of the center of masses of the UPS and UP branch chains are

(23) \begin{equation} \begin{array}{l} \dot{\textbf{v}}_{ci}=\left\{\left(\boldsymbol{\omega }_{i}\times \textbf{e}_{i}\right)\textbf{J}_{i}+\textbf{e}_{i}\dot{\textbf{J}}_{i}+\dot{l}_{i}\textbf{e}_{i\times }\textbf{J}_{\omega i}-\left(l_{i}-l_{bi}\right)\left[\left(\boldsymbol{\omega }_{i}\times \textbf{e}_{i}\right)_{\times }\right]\textbf{J}_{\omega i}-\left(l_{i}-l_{bi}\right)\textbf{e}_{i\times }\dot{\textbf{J}}_{\omega i}\right\}\dot{\textbf{r}}_{2}\\[5pt] \;\qquad +\;\left[\textbf{e}_{i}\textbf{J}_{i}-\left(l_{i}-l_{bi}\right)\textbf{e}_{i\times }\textbf{J}_{\omega i}\right]\ddot{\textbf{r}}_{2} \end{array}, \end{equation}

(24) \begin{equation} \begin{array}{l} \dot{\textbf{v}}_{Ly}=\left[\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right)\textbf{J}_{y}+\textbf{e}_{L}\dot{\textbf{J}}_{y}-\left(L_{2}-L_{y}\right)\dfrac{1}{L_{2}}\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right)\left(\boldsymbol{\omega }_{d}\times \textbf{e}_{L}\right)^{\mathrm{T}}+\dfrac{L_{y}\dot{L}_{2}}{{L_{2}}^{2}}\left(\textbf{E}-\textbf{e}_{L}{\textbf{e}_{L}}^{\mathrm{T}}\right)\right]\dot{\textbf{r}}_{2}\\[5pt] \;\qquad +\;\left[\textbf{e}_{L}{\textbf{e}_{L}}^{\mathrm{T}}-\left(L_{2}-L_{y}\right)\dfrac{1}{L_{2}}\left(\textbf{E}-\textbf{e}_{L}{\textbf{e}_{L}}^{\mathrm{T}}\right)\right]\ddot{\textbf{r}}_{2} \end{array}, \end{equation}

where L _y is the distance from the center of mass of driving chain 2 to O ₂, and J _y is the component of J in the y-direction.

Then, the dynamic model of the OMPR can be expressed as:

(25) \begin{equation} \boldsymbol{\tau }=\textbf{M}_{q}\cdot \ddot{\textbf{L}}\boldsymbol{+}\textbf{C}_{q}\cdot \dot{\textbf{L}}+\textbf{G}_{q} \end{equation}

where $\boldsymbol\tau$ is the control input torque, M _q is the inertia matrix, $\textbf{M}_{q}=\left(\textbf{J}^{-1}\right)^{\mathrm{T}}\textbf{M}\textbf{J}^{-1}$ , C _q is Coriolis force, $\textbf{C}_{q}=\left(\textbf{J}^{-1}\right)^{\mathrm{T}}\textbf{M}\dot{\textbf{J}}^{-1}+\left(\textbf{J}^{-1}\right)^{\mathrm{T}}\textbf{C}\textbf{J}^{-1}$ , and G _q is gravity, $\textbf{G}_{q}=\left(\textbf{J}^{-1}\right)^{\mathrm{T}}\textbf{G}$ . Due to the coupling force in the motion process of the parallel manipulator, the diagonal parts of M _q and C _q represent the force on the current joint, namely inertial force, Coriolis force, and centrifugal force, while the nondiagonal part is the force generated by one joint on other joints, that is, the coupling force. The following relationship can be obtained:

(26) \begin{align} \begin{cases} \textbf{M}_{qf}=\textbf{M}_{qc}\cdot \ddot{\textbf{L}}\\[5pt] \textbf{C}_{qf}=\textbf{C}_{qc}\cdot \dot{\textbf{L}}\\[5pt] \textbf{F}_{o}=\textbf{M}_{qo}\cdot \ddot{\textbf{L}}+\textbf{C}_{qo}\cdot \dot{\textbf{L}} \end{cases} \end{align}

where M _qf , C _qf , F _o are inertia force, Coriolis force and coupling force, respectively. And,

$\textbf{M}_{qc}=\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{M}_{q}\left(1,1\right) & & \\[5pt] & \textbf{M}_{q}\left(2,2\right) & \\[5pt] & & \textbf{M}_{q}\left(3,3\right) \end{array}\right], \textbf{M}_{qo}=\left[\begin{array}{l@{\quad}l@{\quad}l} & \textbf{M}_{q}\left(1,2\right) & \textbf{M}_{q}\left(1,3\right)\\[5pt] \textbf{M}_{q}\left(2,1\right) & & \textbf{M}_{q}\left(2,3\right)\\[5pt] \textbf{M}_{q}\left(3,1\right) & \textbf{M}_{q}\left(3,2\right) & \end{array}\right]$ ,

$\textbf{C}_{qc}=\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{C}_{q}\left(1,1\right) & & \\[5pt] & \textbf{C}_{q}\left(2,2\right) & \\[5pt] & & \textbf{C}_{q}\left(3,3\right) \end{array}\right], \textbf{C}_{qo}=\left[\begin{array}{l@{\quad}l@{\quad}l} & \textbf{C}_{q}\left(1,2\right) & \textbf{C}_{q}\left(1,3\right)\\[5pt] \textbf{C}_{q}\left(2,1\right) & & \textbf{C}_{q}\left(2,3\right)\\[5pt] \textbf{C}_{q}\left(3,1\right) & \textbf{C}_{q}\left(3,2\right) & \end{array}\right]$ .

Then, the dynamic model can be further expressed as:

(27) \begin{equation} \boldsymbol{\tau }=\textbf{M}_{qf}\cdot \ddot{\textbf{L}}\boldsymbol{+}\textbf{C}_{qf}\cdot \dot{\textbf{L}}+\textbf{G}_{q}+\textbf{F}_{o} \end{equation}

Since the active branch chain of the OMPR belongs to the single input single output model, it can be expressed as a linear constant continuous system:

(28) \begin{align} \begin{cases} \dot{x}(t)=Ax(t)+Bu\\[5pt] y=Cx(t) \end{cases} \end{align}

where $x(t)\in \textbf{R}^{n}, u\in \textbf{R}^{m}, A\in \textbf{R}^{n\times n}$ is a nonsingular matrix, $B\in \textbf{R}^{n\times m}, C\in \textbf{R}^{r\times n}$ .

3. Contour error analysis

In the process of optical mirror surface processing, the grinding tool accurately traverses each removal point on the mirror surface to be processed at a certain speed. As shown in Fig. 4, the commonly used tool paths at present are the grid, concentric circle, and spiral types.

Figure 4. Tool paths.

For the OMPR to meet the processing requirements of different types of optical mirrors, the robot must be driven by the control system ensuring that the grinding tool can complete the above three types of contour movements.

The motion trajectory of the end grinding tool of the OMPR is composed of a series of interpolation points. The positions of these points can be described as vectors along the x, y, and z directions in the reference coordinate system, O–XYZ.

(29) \begin{equation} \textbf{s}(t)=\left[\begin{array}{l@{\quad}l@{\quad}l} s_{x}(t) & s_{y}(t) & s_{z}(t) \end{array}\right]^{\mathrm{T}}\!\textbf{,}\;\, t\in R, \end{equation}

where s (t) represents the trajectory parameter of the grinding tool, which is a second-order derivable continuous smooth ideal contour.

Subsequently, the unit tangent vector of the motion trajectory can be expressed as

(30) \begin{equation} \textbf{t}(t)=\frac{\dot{\textbf{s}}(t)}{\left| \dot{\textbf{s}}(t)\right| }, \end{equation}

where $\dot{\textbf{s}}(t)$ is the velocity vector.

Thus, the principal normal vector at any point of the ideal trajectory of the grinding tool is defined as follows:

(31) \begin{equation} \textbf{n}(t)=\frac{\rho \cdot \dot{\textbf{t}}(t)}{\left| \rho \cdot \dot{\textbf{t}}(t)\right| }, \end{equation}

where $\rho$ is the radius of curvature.

The tangent vector and the principal normal vector define a two-dimensional plane, which is called the osculating plane. In the three-dimensional space, there is a secondary normal vector at each point on the trajectory curve, which is perpendicular to the tangent vector and the principal normal vector. It is the second normal vector of the trajectory curve and is expressed as

(32) \begin{equation} \textbf{b}(t)=\textbf{t}(t)\times \textbf{n}(t). \end{equation}

t (t), n (t), and b (t) form the Frenet coordinate system in the ideal position. As shown in Figure 5, D and A are the vectors of points D and A in O–XYZ, respectively, and the tracking error of the two points in O–XYZ is E _d . A _F is the position vector of point A in the Frenet coordinate system. Thus, the transformation relationship between the reference coordinate system, O–XYZ, and the Frenet coordinate system of point A is expressed as follows:

Figure 5. Frenet coordinate system transformation.

(33) \begin{equation} \begin{array}{l} T_{F}\colon \textbf{A}=\textbf{R}_{F}\textbf{A}_{F}+\boldsymbol{D}\\[5pt] T_{W}\colon \textbf{A}_{F}={\textbf{R}_{F}}^{-1}\textbf{E}_{d} \end{array}, \end{equation}

where $\textbf{R}_{F}=\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}$ is the coordinate transformation matrix, and ${\textbf{R}_{F}}^{-1}={\textbf{R}_{F}}^{\mathrm{T}}$ . $T_{F}$ represents the transformation from the Frenet coordinate system to the reference coordinate system, and $T_{W}$ represents the transformation from the reference coordinate system to the Frenet coordinate system.

In the three-dimensional space, based on the Frenet coordinate system, the curvature and radius of curvature at a certain point can be obtained from the ideal trajectory of the grinding tool, and an osculating circle can be drawn. The contour error is solved using the shortest distance from the actual movement point of the grinding tool to the osculating circle.

The contour error of the spiral trajectory of the grinding tool in the three-dimensional space is shown in Fig. 6(a). Point Q is the projection point of point A on the osculating plane. Point E, which is closest to point A on the osculating circle, can be obtained. Thus, the contour error vector, ε , can be expressed as

Figure 6. Three-dimensional contour error based on osculating circle.

(34) \begin{equation} \boldsymbol{\varepsilon }=\left[\overline{\overline{{O_{o}A}}}\cdot \textbf{b}(t)\right]\cdot \textbf{b}(t)+\left[\rho -\left(\overline{\overline{{O_{o}A}}}\cdot \textbf{b}_{e}\right)\right]\cdot \textbf{b}_{e}, \end{equation}

where $\textbf{b}_{e}=\frac{\overline{\overline{{O_{o}Q}}}}{\left| \overline{\overline{{O_{o}Q}}}\right| }$ , and it is the unit orthogonal vector of b in the osculating plane.

The front view of a point projected onto the osculating plane is shown in Fig. 6(b), and the osculating circle is located in the osculating plane of the Frenet coordinate system. Among them, points E, Q, and O _o are on the same straight line. The contour error based on the osculating circle can be obtained as

(35) \begin{equation} \boldsymbol{\varepsilon }=\left[\begin{array}{l} 0\\[5pt] 0\\[5pt] -E_{b} \end{array}\right]+\left(\frac{\rho }{\sqrt{{E_{t}}^{2}+\left(E_{n}-\rho \right)^{2}}}-1\right)\cdot \left[\begin{array}{l} E_{t}\\[5pt] E_{n}-\rho \\[5pt] 0 \end{array}\right]. \end{equation}

where E _t , E _n, and E _b are the components of the actual point of the end track along each axis of the Frenet coordinate system.

4. Feedforward combined multi-axis cross-coupling contour control

Compared to the parallel module, the series module has a lower speed, smaller range of motion, and smaller error. The series module can achieve higher motion accuracy by adopting a three-loop feedback control and speed feedforward. In this study, a cross-coupling control system is used to compensate the contour error at the end of the parallel module. Moreover, the parallel module single-joint servo control comprises speed and dynamic feedforward. The contour control strategy of the OMPR is shown in Fig. 7. Transfer functions G ₁ ^P (s), G ₂ ^P (s), G ₃ ^P (s), G ₄ ^P (s), and G ₅ ^P (s) are used to replace the relationship between the input and the output. q _a1, q _a2, q _a3, q _a4, and q _a5 are the actual output joint displacements of the control system. Tracking errors E _q1, E _q2, and E _q3 of the parallel module single-branch chain control system are input into the contour error model, and the contour error, ε, in the task space is calculated. The contour error control quantity, ε _c , is obtained using the cross-coupling controller, and a mathematical model of the compensation channel is established. The control variable, ε _c , is converted into the joint space and assigned to position loop input terminals u _q1, u _q2, and u _q3 of the three drive branch chain servo control systems to obtain actual kinematic servo system input values q ₁, q ₂, and q ₃ after compensation. C _x , C _y , and C _z are the forward channel gain coefficients; C _q1, C _q2, and C _q3 are the compensation channel gain coefficients, and G _c is the cross-coupling controller. G _v4(s) and G _v5(s) are the transfer functions of the speed loop, G _i4(s) and G _i5(s) are the transfer functions of the current loop, and K _sv is the feedforward function of the speed loop.

Figure 7. Contour control strategy of OMPR.

The closed-loop control of UPS branch chain includes position loop, speed loop, current loop, and servo motor. And the structure of single-branch chain servo control system is shown in Fig. 8.

Figure 8. Servo control system of the UPS branch chain.

In the figure, K _p is the proportional coefficient of the position loop, K _pv is the proportional coefficient of the speed loop, $\tau\!$ _v is the time constant of the velocity loop, K _pI is the proportional coefficient of the current loop, $\tau\!$ _I is the current loop time constant, K _w is the proportional coefficient of PWM inverter, T _w is the switching frequency of PWM inverter, $T_{h}=0.3\sim 0.5T_{w}, K_{h}=1$ , R is the armature resistance, L is the armature inductance, K _T is the torque coefficient, K _e is the back EMF coefficient, J is the load moment of inertia of the motor, and B ₁ is the viscous resistance.

To simplify the calculation, make: $K_{pv}\left(1{+}\frac{1}{\tau _{v}s}\right)=G_{1}, K_{pI}\left(1{+}\frac{1}{\tau _{I}s}\right)=G_{2}, \frac{K_{w}}{T_{w}{+}1}=G_{3}, \frac{1}{Ls{+}R}=G_{4}, \frac{K_{T}}{Js{+}B_{1}} = G_{5}, \frac{K_{he}}{T_{he}{+}1} = G_{6}$ . Then, the transfer function of UPS branch chain is

(36) \begin{equation} {G_{i}}^{P}\left(s\right)=\frac{G_{1}G_{2}G_{3}G_{4}G_{5}K_{p}}{\left(1\textbf{--}G_{2}G_{3}G_{4}G_{6}\right)\left(1\textbf{--}G_{4}G_{5}K_{e}\right)s\textbf{--}G_{1}G_{2}G_{3}G_{4}G_{5}s\textbf{--}G_{1}G_{2}G_{3}G_{4}G_{5}K_{p}}i=1,2,3 \end{equation}

(1) Forward channel cross-coupling gain

To establish the transfer function for system analysis, the inverse of the error Jacobian matrix, J _e ^-1, is used to map the tracking error of the parallel module from the joint space to the task space, and the tracking error of the end moving platform of the parallel module is obtained as follows:

(37) \begin{equation} \textbf{E}_{d}={\left[\left(\begin{array}{l@{\quad}l@{\quad}l} \textbf{e}_{1} & \textbf{e}_{2} & \textbf{e}_{3} \end{array}\right)^{\mathrm{T}}-\frac{1}{L_{2}}\left(\begin{array}{l@{\quad}l@{\quad}l} \textbf{r}_{u1}{\textbf{e}_{1}}^{\mathrm{T}}\textbf{e}_{L} & \textbf{r}_{u2}{\textbf{e}_{2}}^{\mathrm{T}}\textbf{e}_{L} & \textbf{r}_{u3}{\textbf{e}_{3}}^{\mathrm{T}}\textbf{e}_{L} \end{array}\right)^{\mathrm{T}}\right]_{3\times 3}}^{-1}\cdot \left[\begin{array}{l} E_{q1}\\[5pt] E_{q2}\\[5pt] E_{q3} \end{array}\right]_{3\times 1}. \end{equation}

The tracking errors in any direction of X ₁, Y ₁, and Z ₁ at the end of the parallel module are caused by the coupling of the tracking errors of the three UPS branches. So the end tracking error, ^F E _d , of the parallel module in the Frenet coordinate system can be calculated using the single-axis tracking error of the driving branch chain as follows:

(38) \begin{equation} {_{}^{F}}{\textbf{E}}{_{d}^{}}={\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}}^{-1}\cdot {\textbf{J}_{e}}^{-1}\cdot \textbf{E}_{q}. \end{equation}

The contour error in the task space of the parallel module can be calculated utilizing ^F E _d based on the osculating circle approximation methods.

(39) \begin{equation} \boldsymbol{\varepsilon }=\textbf{K}\cdot {_{}^{F}}{\boldsymbol{E }}{_{d}^{}}+\boldsymbol{\delta }, \end{equation}

where K is the contour error transfer matrix, $\textbf{K}=\left[\begin{array}{l@{\quad}l@{\quad}l} H & 0 & 0\\[5pt] 0 & H & 0\\[5pt] 0 & 0 & -1 \end{array}\right], \boldsymbol{\delta }=\left[\begin{array}{l} 0\\[5pt] -H\cdot \rho \\[5pt] 0 \end{array}\right]$ , and $H=\frac{\rho }{\sqrt{\left[\left({\textbf{J}_{e}}^{-1}\cdot \textbf{E}_{q}\right)^{\mathrm{T}}\textbf{t}\right]^{2}+\left[\left({\textbf{J}_{e}}^{-1}\cdot \textbf{E}_{q}\right)^{\mathrm{T}}\textbf{n}-\rho \right]^{2}}}-1$ .

Combining Fig. 7 and Eqs. (37) and (38), the forward channel gain of the cross-coupling control system is as follows:

(40) \begin{equation} \begin{cases} \textbf{C}_{x}=\textbf{K}\cdot {\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}}^{-1}\cdot {\textbf{J}_{e}}^{-1}\left(\colon \mathrm{,}1\right)\\[5pt] \textbf{C}_{y}=\textbf{K}\cdot {\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}}^{-1}\cdot {\textbf{J}_{e}}^{-1}\left(\colon \mathrm{,}2\right)\\[5pt] \textbf{C}_{x}=\textbf{K}\cdot {\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}}^{-1}\cdot {\textbf{J}_{e}}^{-1}\left(\colon \mathrm{,}3\right) \end{cases}, \end{equation}

The forward channel gain, $\boldsymbol\varepsilon$ , is obtained as follows:

(41) \begin{equation} \boldsymbol{\varepsilon }=\textbf{C}_{x}\cdot E_{q1}+\textbf{C}_{y}\cdot E_{q2}+\textbf{C}_{z}\cdot E_{q3}+\boldsymbol{\delta }. \end{equation}

(2) Compensation channel cross-coupling gain

After the contour error, $\boldsymbol{\varepsilon }$ , is calculated using the cross-coupling controller, the contour error compensation amount, $\boldsymbol{\varepsilon }_{c}$ , is obtained. It is converted into the task space of the parallel module, and the contour error, $\boldsymbol{\varepsilon }_{r}$ , is obtained in the fixed platform coordinate system using Eq. (33) as follows:

(42) \begin{equation} \boldsymbol{\varepsilon }_{r}=\left[\begin{array}{l@{\quad}l@{\quad}l} \varepsilon _{rx} & \varepsilon _{ry} & \varepsilon _{rz} \end{array}\right]^{\mathrm{T}}=\left[\begin{array}{l@{\quad}l@{\quad}l} \textbf{t} & \textbf{n} & \textbf{b} \end{array}\right]_{3\times 3}\cdot \boldsymbol{\varepsilon }_{c}, \end{equation}

Thus, the position loop compensations, u _q1, u _q2, and u _q3, of the three driving branches in the joint space corresponding to the task space compensation are as follows:

(43) \begin{equation} \textbf{u}_{q}=\textbf{J}_{e}\boldsymbol{\varepsilon }_{r}, \end{equation}

where $\textbf{u}_{q}=\left[\begin{array}{l@{\quad}l@{\quad}l} u_{q1} & u_{q2} & u_{q3} \end{array}\right]^{\mathrm{T}}$ .

Combining Fig. 7 and Eqs. (42) and (43), the gain coefficient matrix of the compensation channel is obtained as follows:

(44) \begin{equation} \left\{\begin{array}{l@{\quad}l@{\quad}l} \textbf{C}_{q1}=\textbf{J}_{e}(1,\boldsymbol{:})\cdot [\textbf{t} \quad \textbf{n} \quad \textbf{b}]_{3\times 3}\\[5pt] \textbf{C}_{q2}=\textbf{J}_{e}(2,\boldsymbol{:})\cdot [\textbf{t} \quad \textbf{n} \quad \textbf{b} ]_{3\times 3}\\[5pt] \textbf{C}_{q3}=\textbf{J}_{e}(3,\textbf{:})\cdot [\textbf{t} \quad \textbf{n} \quad \textbf{b} ]_{3\times 3} \end{array}\right. ,\end{equation}

(3) Lyapunov stability analysis

According to Eq. (28), the observer of linear system can be defined as:

(45) \begin{align} \begin{cases} \dot{\hat{x}}=A\hat{x}+Bu+\textbf{O}_{g}\left(y-\hat{y}\right)\\[5pt] \hat{y}=C^{\mathrm{T}}\hat{x} \end{cases} \end{align}

where O _g is the observation gain matrix. Then the dynamic error can be expressed as:

(46) \begin{equation} E=\left(A-O_{g}C^{\mathrm{T}}\right)\tilde{x} \end{equation}

where $\tilde{x}$ is the state estimation error matrix. From Fig. 7, the tracking error $\tilde{x}$ = [E _q1, E _q2, E _q3] of the single-axis UPS branch chain of the control system can be expressed as:

(47) \begin{equation} \begin{array}{l} E_{q1}=l_{1}\dfrac{\left(1-{G_{1}}^{P}\right)\left(1+\textbf{C}_{q1}\textbf{C}_{x}G_{c}{G_{1}}^{P}\right)}{1+C_{l}G_{c}}+l_{2}\dfrac{\left(1-{G_{2}}^{P}\right)\textbf{C}_{q1}\textbf{C}_{y}G_{c}{G_{1}}^{P}}{1+C_{l}G_{c}}\\[9pt] \;\;\qquad +\;l_{3}\dfrac{\left(1-{G_{3}}^{P}\right)\textbf{C}_{q1}\textbf{C}_{z}G_{c}{G_{1}}^{P}}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q1}\boldsymbol{\delta }{G_{1}}^{P}}{1+C_{l}G_{c}} \end{array} \end{equation}

(48) \begin{equation} \begin{array}{l} E_{q2}=l_{1}\dfrac{\left(1-{G_{1}}^{P}\right)\textbf{C}_{q2}\textbf{C}_{x}G_{c}{G_{2}}^{P}}{1+C_{l}G_{c}}+l_{2}\dfrac{\left(1-{G_{2}}^{P}\right)\left(1+\textbf{C}_{q2}\textbf{C}_{y}G_{c}{G_{2}}^{P}\right)}{1+C_{l}G_{c}}\\[9pt] \;\;\qquad+\;l_{3}\dfrac{\left(1-{G_{3}}^{P}\right)\textbf{C}_{q2}\textbf{C}_{z}G_{c}{G_{2}}^{P}}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q2}\boldsymbol{\delta }{G_{2}}^{P}}{1+C_{l}G_{c}} \end{array} \end{equation}

(49) \begin{equation} \begin{array}{l} E_{q3}=l_{1}\dfrac{\left(1-{G_{1}}^{P}\right)\textbf{C}_{q3}\textbf{C}_{x}G_{c}{G_{3}}^{P}}{1+C_{l}G_{c}}+l_{2}\dfrac{\left(1-{G_{2}}^{P}\right)\textbf{C}_{q3}\textbf{C}_{y}G_{c}{G_{3}}^{P}}{1+C_{l}G_{c}}\\[9pt] \;\;\qquad+\;l_{3}\dfrac{\left(1-{G_{3}}^{P}\right)\left(1+\textbf{C}_{q3}\textbf{C}_{z}G_{c}{G_{3}}^{P}\right)}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q3}\boldsymbol{\delta }{G_{3}}^{P}}{1+C_{l}G_{c}} \end{array} \end{equation}

where $C_{l}={G^{p}}_{1}\textbf{C}_{q1}\textbf{C}_{x}+{G^{p}}_{2}\textbf{C}_{q2}\textbf{C}_{y}+{G^{p}}_{3}\textbf{C}_{q3}\textbf{C}_{z}$ .

Similarly, from Fig. 7, the system input x = [q ₁, q ₂, q ₃] adjusted by the multi-axis cross-coupling contour control system is deduced, and the following is obtained:

(50) \begin{equation} \begin{array}{l} q_{1}=l_{1}\dfrac{1+\left(\textbf{C}_{q1}\textbf{C}_{x}+\textbf{C}_{q1}\textbf{C}_{x}{G_{1}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+l_{2}\dfrac{\textbf{C}_{q1}\textbf{C}_{y}\left(1+{G_{2}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}\\[9pt] \qquad +\;l_{3}\dfrac{\textbf{C}_{q1}\textbf{C}_{z}\left(1+{G_{3}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q1}\boldsymbol{\delta }}{1+C_{l}G_{c}} \end{array} \end{equation}

(51) \begin{equation} \begin{array}{l} q_{2}=l_{1}\dfrac{\textbf{C}_{q2}\textbf{C}_{x}\left(1+{G_{1}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+l_{2}\dfrac{1+\left(\textbf{C}_{q2}\textbf{C}_{y}+\textbf{C}_{q2}\textbf{C}_{y}{G_{2}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}\\[9pt] \qquad +\;l_{3}\dfrac{\textbf{C}_{q2}\textbf{C}_{z}\left(1+{G_{3}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q2}\boldsymbol{\delta }}{1+C_{l}G_{c}} \end{array} \end{equation}

(52) \begin{equation} \begin{array}{l} q_{3}=l_{1}\dfrac{\textbf{C}_{q3}\textbf{C}_{x}\left(1+{G_{1}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+l_{2}\dfrac{\textbf{C}_{q3}\textbf{C}_{y}\left(1+{G_{2}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}\\[9pt] \qquad +\;l_{3}\dfrac{1+\left(\textbf{C}_{q3}\textbf{C}_{z}+\textbf{C}_{q3}\textbf{C}_{z}{G_{3}}^{P}\right)\!G_{c}}{1+C_{l}G_{c}}+\dfrac{G_{c}\textbf{C}_{q3}\boldsymbol{\delta }}{1+C_{l}G_{c}} \end{array} \end{equation}

Based on Eqs. (50)–(52), system inputs q ₁, q ₂, and q ₃ regulated by the cross-coupling controller are cross-coupling terms. The cross-coupling control system outputs a certain correction signal based on the contour error generated by the OMPR during the contour movement process and compensates it to the position loop input end of the single-joint control system, completing the closed-loop contour control.

Based on this, the Lyapunov closed-loop equation can be constructed as follows:

(53) \begin{equation} V(t)=x^{T}Px+\frac{1}{2}E^{T}P_{1}E \end{equation}

where P = P ^T > 0 and P ₁ = P ₁^T > 0 $\cdot$

The necessary and sufficient conditions for the stability of the control system are as follows: for the scalar function V with continuous first-order partial derivative, if V is positive definite and $\dot{V}$ is negative definite. Given Q = Q ^T > 0, $Q_{0}={Q_{0}}^{T}\gt 0$ , and

(54) \begin{align} \begin{cases} -Q=-PA-A^{T}P \\[5pt] -Q_{1}=\left(A-O_{g}C^{\mathrm{T}}\right)^{\mathrm{T}}P_{1}+P_{1}\left(A-O_{g}C^{\mathrm{T}}\right) \end{cases} \end{align}

Find the full derivative of V,

(55) \begin{equation} \dot{V}=-x^{T}Qx-E^{T}Q_{1}E. \end{equation}

Order, $Q_{0}=\left[\begin{array}{l@{\quad}l} Q & 0\\[5pt] 0 & \frac{1}{2}Q_{1} \end{array}\right]$ . Then,

(56) \begin{equation} \dot{V}=\left[\begin{array}{l@{\quad}l} x^{\mathrm{T}} & E \end{array}^{\mathrm{T}}\right]Q_{0}\left[\begin{array}{l} x\\[5pt] E \end{array}\right]\leq -\lambda _{\min }\left(Q_{0}\right)\left|\left|\left[\begin{array}{l} x\\[5pt] E \end{array}\right]\right|\right|^{2} \end{equation}

where $\lambda _{\min }\left(Q_{0}\right)$ is the minimum eigenvalue of matrix Q ₀. Because of $\lambda _{\min }\left(Q_{0}\right)\gt 0$ , it can prove that $\dot{V}$ is negative definite, and it is proved that the closed-loop system is asymptotically stable.

(4) Simulation analysis

According to the feedforward combined multi-axis cross-coupling contour control system established, the control system is simulated and analyzed by Matlab/Simulink. Three trajectories of line, arc, and spiral in the world coordinate system are analyzed respectively. The trajectory equations are

(57) \begin{equation} \begin{array}{l} \text{Line}\colon \begin{cases} X=70t-R_{g}\\[5pt] Y=0\times t_{1}\\[5pt] Z=0\times t_{1}+1097 \end{cases} t_{1}=0\colon 10\\ \\ \mathrm{Arc}\colon \begin{cases} X=200\cos \left(0.2\pi t_{2}+\dfrac{3}{2}\pi \right)\\[5pt] Y=200\sin \left(0.2\pi t_{2}+\dfrac{3}{2}\pi \right)\\[5pt] Z=0\times t_{2}+1097 \end{cases} t_{2}=0\colon 10\\ \\ \text{Spiral}\colon \begin{cases} X=20t_{3}\cos \left(\dfrac{\pi }{2}t_{3}+\dfrac{\pi }{2}\right)\\[5pt] Y=20t_{3}\sin \left(\dfrac{\pi }{2}t_{3}+\dfrac{\pi }{2}\right)\\[5pt] Z=0\times t_{3}+1097 \end{cases} t_{3}=0\colon 10 \end{array} \end{equation}

As can be seen from Fig. 9(a), when only cross-coupling contour control is used for compensation, the contour error is significantly reduced. When the feedforward combined multi-axis cross-coupling contour control is used, the contour error is reduced from 0.015 to 0.004 mm, and the overall variation law of contour error is smoother. It can be seen from Fig. 9(b) that the contour error of the arc trajectory presents a regular fluctuating trend, which is caused by the error accumulation of the motion branch. When the feedforward combined multi-axis cross-coupling contour control is used, the error is reduced from the peak value of 0.193 to 0.018 mm. As can be seen from Fig. 9(c), with the increase of spiral contour, the contour error also increases, and the maximum contour error is obtained at the maximum curvature. Using feedforward combined multi-axis cross-coupling contour control, the maximum contour error can be reduced from 0.218 to 0.022 mm, and the variation trend of contour error is smoother. Therefore, the designed feedforward combined multi-axis cross-coupling contour control can effectively reduce the contour error, and the effect is better than the cross-coupling contour compensation control.

Figure 9. Contour error simulation.

5. Experimental analysis

To verify the effectiveness of the established contour control model, a control system is built for the OMPR. Contour control experiments are conducted for three types of common trajectories in optical mirror processing. The experimental site and the topology of CCOS are shown in Fig. 10. The multi-axis motion controller is a programmable multi-axes controller (PMAC). On a PC, the path contour of the grinding tool in the world coordinate system is planned. Considering that the numerical search method for the inverse solution of the series module reduces the operation speed of the controller, the PC is used for the offline inverse solution of the series module. Moreover, the angle change of the two series swivel heads and the position of the parallel module moving platform are input to the PMAC to complete the control task. The displacement and velocity of five axes of the machining robot are collected, and the position and velocity of the end execution point are calculated through the forward kinematics model. Because the CCOS has pneumatic active position compensation function in the Z axis direction, only the motion parameters in the X and Y directions have research significance. During the experiment, the revolution speed of CCOS grinding system was set to 30 r/min and the rotation speed was set to 180 r/min. The linear contour, arc contour, and spiral contour are tested respectively. The trajectory equation is the same as the simulation trajectory, as shown in Eq. (57).

Figure 10. Experimental site.

5.1 Linear contour control compensation experiment

In the experiment, the trajectory of the grinding tool was a linear contour along the X axis of the world coordinate system, the length is 700 mm, the movement period was 20 s, and the sampling period was 20 ms. The experiment was conducted on an aluminum plate instead of an optical mirror. Based on the established contour error model, the contour error of the OMPR can be obtained by mapping the tracking error of each joint of the parallel module to the workspace. During the experiment, no compensation control, traditional cross-coupling compensation, and feedforward combined multi-axis cross-coupling compensation contour control experiments were conducted. The torque control mode is adopted for the linear contour experiment, and the driving force of the three driving branches in the experiment of feedforward combined multi-axis cross-coupling contour control compensation is shown in Fig. 11. Due to the inclined arrangement of the OMPR, the gravity is different in the three driving components. In order to counteract gravity, the UPS1 needs to output the force pointing to the moving platform along the branch chain, which is specified as the positive direction here. The UPS2 and UPS3 need to output the force pointing to the fixed platform along the branch chain, which is specified as the negative direction here. The UPS1 bears most of the gravity component, so the output force of the UPS1 is the largest.

Figure 11. Driving force output signal in linear contour linear contour.

The experimental results are shown in Fig. 12. When using the traditional PID control, the contour error generated by the parallel modules is approximately 0.027 mm. When using the traditional cross-coupling compensation control strategy, the contour error is reduced to 0.018 mm. When the feedforward combined multi-axis cross-coupling control strategy is used for compensation, the contour error is further reduced to 0.006 mm. The variation trend of the experimental value and the theoretical value is the same, and the difference is only 0.002 mm, which is caused by the random error of the robot. It can be seen that the effect of the feedforward combined multi-axis cross-coupling control strategy is better than that of the traditional multi-axis cross-coupling control system.

Figure 12. Experiment of linear contour error control compensation.

The velocity stability has a direct impact on the surface accuracy of the optical mirror. In the experiment, the velocity errors in X and Y directions in the processing of the optical mirror were tested respectively. The experimental results are shown in Fig. 13. When the compensation control mode is not adopted, the velocity errors in the X and Y directions of the linear contour are 0.025 and 0.021 mm/s. Through traditional cross-coupling compensation, the velocity errors in X and Y directions are reduced to 0.013 and 0.019 mm/s. Through feedforward combined multi-axis cross-coupling contour control, the velocity errors in X and Y directions are reduced to 0.005 and 0.006 mm/s. Therefore, the designed control compensation strategy not only reduces the velocity errors of the OMPR but also makes its motion more stable.

Figure 13. Linear contour velocity error.

5.2 Arc contour control compensation experiment

Taking the machined mirror as the center point, given the space arc contour trajectory of the grinding tool radius of 200 mm, the movement period is 20 s, and the sampling period is 20 ms. The experimental steps are the same as those of the linear contour control experiment. The torque control mode is adopted for the arc contour experiment, and the driving force of the three driving branches in the experiment of feedforward combined multi-axis cross-coupling contour control compensation is shown in Fig. 14. During the movement along the arc contour, the three UPS branch chains have the process of elongation and shortening. When the branch chain moves to the longest position, the heavy torque is the largest, and the driving force output value also reaches the maximum.

Figure 14. Driving force output signal in linear contour arc contour.

The experimental results are shown in Fig. 15. When the control compensation is not conducted, the contour error of the OMPR is approximately sinusoidal, and the error peak is 0.223 mm. When the traditional cross-coupling compensation control strategy is adopted, the contour error is reduced to 0.069 mm. When the feedforward combined multi-axis cross-coupling control strategy is used for compensation, the contour error is further reduced to approximately 0.021 mm. The variation trend of the experimental value and the theoretical value is the same, and the difference is only 0.003 mm. It is verified that the feedforward combined multi-axis cross-coupling control strategy still exhibits good compensation performance in a circular motion trajectory.

Figure 15. Experiment of arc contour error control compensation.

At the same time, the velocity errors in X and Y directions during processing of the optical mirror under arc contour are collected. The experimental results are shown in Fig. 16. When the compensation control mode is not adopted, the velocity errors in the X and Y directions of the arc contour are 0.055 and 0.051 mm/s. Through traditional cross-coupling compensation, the velocity errors in X and Y directions are reduced to 0.031 and 0.028 mm/s. Through feedforward combined multi-axis cross-coupling contour control, the velocity errors in X and Y directions are reduced to 0.008 and 0.010 mm/s. It can be seen that the designed control strategy also has a good effect on the velocity error compensation of arc contour.

Figure 16. Arc contour velocity error.

5.3 Spiral contour control compensation experiment

The given end grinding tool is a spiral contour with the starting point at the origin of the world coordinate system, the movement time is 40 s, and the sampling period is 40 ms. The experimental steps are the same as before. The torque control mode is adopted for the spiral contour experiment, and the driving force of the three driving branches in the experiment of feedforward combined multi-axis cross-coupling contour control compensation is shown in Fig. 17. When the OMPR moves along the spiral contour, the amplitude of elongation and shortening of the three UPS branches increases with the gradual increase of the curvature radius of the spiral.

Figure 17. Driving force output signal in linear contour spiral contour.

The experimental results are shown in Fig. 18. For the spiral contour, the contour error at the end of the parallel module in the control system increases as the curvature of the spiral increases. When using the traditional PID control, the maximum contour error is 0.213 mm. When the traditional cross-coupling compensation control strategy is adopted, the contour error is reduced to approximately 0.121 mm. When the feedforward combined multi-axis cross-coupling control strategy is used for compensation, the contour error is further reduced to approximately 0.030 mm. The variation trend of the experimental value and the theoretical value is the same, and the difference is only 0.008 mm. It can be seen that the feedforward combined multi-axis cross-coupling control strategy also exhibits good compensation performance in the spiral trajectory.

Figure 18. Experiment of spiral contour error control compensation.

Similarly, the velocity errors of the OMPR in the X and Y directions under the spiral contour are measured. The experimental results are shown in Fig. 19. With the increase of the curvature of the spiral contour, the motion velocity errors of the machining robot in the X and Y directions also fluctuate greatly with the increase of the curvature. When the compensation control mode is not adopted, the velocity errors in the X and Y directions of the arc contour are 0.083 and 0.062 mm/s. Through traditional cross-coupling compensation, the velocity errors in X and Y directions are reduced to 0.037 and 0.025 mm/s. Through feedforward combined multi-axis cross-coupling contour control, the velocity errors in X and Y directions are reduced to 0.011 and 0.010 mm/s. It can be seen that the designed control strategy also has a good effect on the trajectory of large curvature.

Figure 19. Spiral contour velocity error.