3.1. The coordinate systems for the novel dish solar tracking platform

The schematic diagram of the coordinate systems of the novel dish solar tracking platform is shown in Fig. 3. Points P and M are the centroids of the 3-RPS fixed base and moving platform, respectively. Point O is the centroid of the dish concentrator and its projection on the ground is point H. During sun tracking, the coordinates of points P, O and H are constant and the position of point M changes with the movement of the three telescopic rods of 3-RPS. Take points O, H, M and P as coordinate origins, then establish the dish coordinate system O-xyz, the horizontal coordinate system H-XYZ, the moving coordinate system M-X_mY_mZ_m and the fixed coordinate system P-X_pY_pZ_p, which will be abbreviated as {O}, {H}, {M}, {P} in this paper. The coordinate axes X, Y and Z point to the east, north and zenith, respectively. The projection of point M on the O-xy plane is M_t. The vertical distances from point P to H and from point M_t to O are equal, which is d. The radii of the fixed platform and moving platform of the 3-RPS are r _b and r _m , respectively. $\phi$ is the angle between the vector ${\vec{\mathit{O}\mathit{M}_{\mathit{t}}}}$ and the x axis of the coordinate system {O} and is also the angle between the vector $\vec {HP}$ and the X axis of the coordinate system {H}. $\alpha$ _s and $\gamma$ _s are altitude angle and azimuth angle, respectively. Point G and angle $\eta$ are the intersection point and the included angle of the normal line passing through point M and the normal line passing through point O, respectively.

Figure 3. Schematic diagram of the coordinate system for the novel dish solar tracking platform.

3.2. Coordinate transformation matrix

3.2.1. Rotation transformation matrix from {M} to {O}

Let the coordinates of the origin M of coordinate system {M} in coordinate system {O} be (x_{m_O} , y_{m_O} , z_{m_O} ). Since the moving platform of the 3-RPS is on the tangent plane of the rotating paraboloid and point M is the tangent point of the two planes, point M satisfies the rotating paraboloid equation shown in Eq. (1):

(1) \begin{equation} F\!\left(x_{m\_ o},y_{m\_ o},z_{m\_ o}\right)={x_{m\_ o}}^{2}+{y_{m\_ o}}^{2}-4fz_{m\_ o}=0 \end{equation}

Then the coordinates of point M in the coordinate system {O} can be written as:

(2) \begin{equation} M\_ _{O}=\left(\begin{array}{c@{\quad}c@{\quad}c} d\!\cos \phi, & d\!\sin \phi, & \left(d^{2}\cos \phi ^{2}+d^{2}\sin \phi ^{2}\right)/4f \end{array}\right)^{T} \end{equation}

where f is the focal length of the rotating paraboloid dish concentrator, $f=\dfrac{D\!\left(1+\cos \!\left(\theta_{\textrm{rim}}\right)\right)}{4\sin \!\left(\theta_{\textrm{rim}}\right)}$ , D and $\theta_{\textrm{rim}}$ are the diameter and edge angle, respectively.

The equation for the normal line passing through point M can be expressed as:

(3) \begin{equation} \frac{\mathit{x}-y_{m\_ o}}{\partial F/x_{m\_ o}}=\frac{x-y_{m\_ o}}{\partial F/y_{m\_ o}}=\frac{z-z_{m\_ o}}{\partial F/z_{m\_ o}} \end{equation}

where $\partial F/x_{m\_ o}, \partial F/y_{m\_ o}$ and $\partial F/z_{m\_ o}$ can be obtained from the partial derivative of Eq. (1) as follows:

(4) \begin{equation} \left\{\frac{\partial F}{x_{m\_ o}}=2x_{m\_o}, \frac{\partial F}{y_{m\_o}} = 2y_{m\_o}, \frac{\partial F}{z_{m\_o}} = -4f \right \} \end{equation}

Let the coordinates of point G in coordinate system {O} be (0, 0, z _{G_O} )^T, then substitute the coordinates of point G into Eq. (4), and the included angle $\eta$ of the normal line passing through point M and the normal line passing through point O can be obtained by combining Eqs. (4)–(7).

(5) \begin{equation} \left| {\overrightarrow{GO}}\right| =\sqrt{\left(-z_{G\mathit{\_ }\mathrm{o}}\right)^{2}} \end{equation}

(6) \begin{equation} \left| \overrightarrow{{{GM_{o}}}}\right| =\sqrt{\left(x_{m\mathit{\_ }o}\right)^{2}+\left(y_{m\mathit{\_ }o}\right)^{2}+\left(z_{m\mathit{\_ }o}-z_{G\mathit{\_ }o}\right)^{2}} \end{equation}

(7) \begin{equation} \eta =\mathit{\arccos }\left(\frac{\overrightarrow{{{GO}}}\cdot \overrightarrow{{{GM}}}}{\left| \overrightarrow{{{GO}}}\right| \cdot \left| \overrightarrow{{{GM}}}\right| }\right)\begin{array}{c@{\quad}c} , & 0\leq \eta \leq 90 \end{array} \end{equation}

It can be seen from Fig. 3 that the coordinate system {O} can be obtained by rotating {M} clockwise $\eta$ around Y_m-axis and clockwise $\phi$ around Z_m $^{\backprime}$ -axis. The homogeneous transformation matrix of frame {M} to frame{O} can be expressed as in Eq. (8):

(8) \begin{equation} {}^{O}{T}{_{M}^{}}=\left[\begin{array}{c@{\quad}c} {}^{O}{R}{_{M}^{}} & M_{\_ O}\\[5pt] 0 & 1 \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\cos \eta \cos \phi & \sin \phi & -\cos \phi \,\sin \eta & d\!\cos \phi \\[5pt] -\cos \eta \cos \phi & -\cos \phi & -\sin \eta \,\sin \phi & d\!\cos \phi \\[5pt] -\sin \eta & 0 & \cos \eta & \dfrac{d^{2}\cos \phi ^{2}+d^{2}\sin \phi ^{2}}{4f}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

3.2.2. Rotation transformation matrix from {P} to {O}

It can also be seen from Fig. 3 that the coordinate system {P} rotates $(\phi -180)^{\mathit{^{\circ}}}$ around the Z axis to obtain coordinate system {H}, and the coordinates of point P in the coordinate system {H} are $(\mathit{d}\cos \phi,\mathit{d}\sin \phi,0)^{T}$ . Then, the rotation transformation matrix of coordinate system {P} with respect to coordinate system {H} can be expressed as:

(9) \begin{equation} ^{H}T_{P}=\left[\begin{array}{c@{\quad}c} ^{H}R_{{}^{P}{}{}} & P_{\_ H}\\[5pt] 0 & 1 \end{array}\right] = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -\cos \phi & \sin \phi & 0 & d\!\cos \phi \\[5pt] -\sin \phi & -\cos \phi & 0 & d\!\sin \phi \\[5pt] 0 & 0 & 1 & 0\\[5pt] 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

3.2.3. The transformation matrix of the dish concentrator with respect to {H}

For solar tracking, the attitude of the dish concentrator is known, and only the length of the three telescopic rods of the 3-RPS needs to be determined. To obtain the analytical relationship between the length of the telescopic rod, the solar altitude angle and the solar azimuth angle, the attitude of the dish parabolic concentrator can be described as follows:

(10) \begin{equation} \Phi _{\textrm{dish}} = \left(x_{O\_H},y_{O\_H},z_{O\_H},\alpha,\beta,\gamma \right) \end{equation}

where x _{O_H} , y _{O_H} and z _{O_H} are the coordinates of the mass center of the parabolic dish concentrator in the coordinate system {H}. $\alpha$ , $\beta$ and $\gamma$ are the Euler angles in the form of ZYZ, which are used to describe the attitude of the dish concentrator with respect to coordinate system {H}, and the corresponding homogeneous transformation matrix is

(11) \begin{equation} {}^{H}{T}{_{O}^{}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \mathit{\cos }\mathit{\alpha }\cos \beta \cos \gamma -\sin \alpha \sin \gamma & -\mathit{\cos }\mathit{\alpha }\cos \beta \sin \gamma -\sin \alpha \cos \gamma & \cos \alpha \sin \beta & x_{O\_ H}\\[5pt] \mathit{\sin }\mathit{\alpha }\cos \beta \cos \gamma +\cos \alpha \sin \gamma & -\mathit{\sin }\mathit{\alpha }\cos \beta \sin \gamma +\cos \alpha \cos \gamma & \sin \alpha \sin \beta & y_{O\_ H}\\[5pt] -\sin \beta \cos \gamma & \sin \beta \sin \gamma & \cos \beta & z_{O\_ H}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

Then the attitude transformation matrix of the 3-RPS moving platform with respect to the horizon coordinate system {H} is

(12) \begin{equation} {}^{H}{T}{_{M}^{}}=\left[\begin{array}{c@{\quad}c} {}^{H}{R}{_{M}^{}} & M\_ H\\[5pt] 0 & 1 \end{array}\right]=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \mathit{n}_{1} & o_{1} & a_{1} & x_{m\_ H}\\[5pt] n_{2} & o_{2} & a_{2} & \mathit{y}_{m\_ H}\\[5pt] n_{3} & o_{3} & a_{3} & z_{m\_ H}\\[5pt] 0 & 0 & 0 & 1 \end{array}\right] \end{equation}

where M_H = (x _{m_H} , y _{m_H} , z _{m_H} ) ^T describes the position of the origin of the coordinate system {M} and orientation vectors n = (n ₁ , n ₂ , n ₃ ) ^T , o = (o ₁ , o ₂ , o ₃ ) ^T , a = (a ₁ , a ₂ , a ₃ ) ^T are the directional cosines of the axes X_m , Y_m , Z_m with respect to the coordinate system {H}. From the algorithm of continuous transformation of the rigid body relative to the coordinate system, the following equation can be obtained.

(13) \begin{equation} {}^{H}{T}{_{M}^{}}={}^{H}{T}{_{O}^{}}{}^{O}{T}{_{M}^{}} \end{equation}

Then combine Eq. (8) and Eqs. (11)–(13) to obtain the elements of the ${}^{H}{T}{_{M}^{}}$ matrix as follows:

\begin{align*} n_{1}&=\cos \eta \cos \phi \,\left(\sin \alpha \sin \gamma -\cos \alpha \,\cos \beta \cos \gamma \right)+\cos \eta \,\sin \phi \,\left(\cos \gamma \,\sin \alpha +\cos \alpha \,\cos \beta \,\sin \gamma \right)\\[3pt] &\quad -\cos \alpha \,\sin \beta \sin \eta\\[3pt] n_{2}&= -\cos \eta \cos \phi \,\left(\cos \alpha \,\sin \gamma +\cos \beta \,\cos \gamma \,\sin \alpha \right)-\cos \eta \,\sin \phi \,\left(\cos \alpha \,\cos \gamma -\cos \beta \,\sin \alpha \,\sin \gamma \right)\\[3pt] &\quad -\sin \alpha \,\sin \beta \,\sin \eta \\[3pt] n_{3}&=\sin \beta \,\cos \eta \,\cos \gamma \,\cos \phi -\cos \beta \,\sin \eta -\sin \beta \,\cos \eta \,\sin \gamma \,\sin \phi \\[3pt] o_{1}&=\cos \phi \,\left(\cos \gamma \,\sin \alpha +\cos \alpha \,\cos \beta \,\sin \gamma \right)-\sin \phi \,\left(\sin \alpha \,\sin \gamma -\cos \alpha \,\cos \beta \,\cos \gamma \right) \\[3pt] o_{2}&=\sin \phi \,\left(\cos \alpha \,\sin \gamma +\cos \beta \,\cos \gamma \,\sin \alpha \right)-\cos \phi \,\left(\cos \alpha \,\cos \gamma -\cos \beta \,\sin \alpha \,\sin \gamma \right) \\[3pt] o_{3}&=-\sin \eta \,\sin \phi \,\left(\cos \alpha \,\cos \gamma -\cos \beta \,\sin \alpha \,\sin \gamma \right)-\cos \phi \,\sin \eta \,\left(\cos \alpha \,\sin \gamma +\cos \beta \,\cos \gamma \,\sin \alpha \right)\\[3pt] &\quad +\cos \eta \,\sin \alpha \,\sin \beta \\[3pt] a_{1}&=\cos \phi \,\sin \eta \,\left(\sin \alpha \,\sin \gamma -\cos \alpha \,\cos \beta \,\cos \gamma \right)+\sin \eta \,\sin \phi \,\left(\cos \gamma \,\sin \alpha +\cos \alpha \,\cos \beta \,\sin \gamma \right) \\[3pt] &\quad +\cos \alpha \cos \eta \,\sin \beta \\[3pt] a_{2} &= -\sin \eta \,\sin \phi \,\left(\cos \alpha \,\cos \gamma -\cos \beta \,\sin \alpha \,\sin \gamma \right)-\cos \phi \,\sin \eta \,\left(\cos \alpha \sin \gamma +\cos \beta \,\cos \gamma \,\sin \alpha \right)\\[3pt] &\quad +\cos \eta \,\sin \alpha \sin \beta \\[3pt] a_{3}&=\cos \left(\beta \right)\,\cos \left(\eta \right)+\cos \left(\gamma \right)\,\cos \left(\phi \right)\,\sin \left(\beta \right)\,\sin \left(\eta \right)-\sin \left(\beta \right)\,\sin \left(\eta \right)\,\sin \left(\gamma \right)\,\sin \left(\phi \right) \\[3pt] x_{m-H}&=\cos \alpha \,\sin \beta \,\left(d^{2}\cos \phi ^{2}+d^{2}\sin \phi ^{2}\right)/\left(4f\right)-d\!\sin \phi \,\left(\cos \gamma \sin \alpha +\cos \alpha \,\cos \beta \,\sin \gamma \right)\\[3pt] & \quad -d\!\cos \phi \,\left(\sin \alpha \,\sin \gamma -\cos \alpha \,\cos \beta \,\cos \gamma \right) \\[3pt] y_{m-H}&=\sin \alpha \,\sin \beta \,\left(d^{2}\cos \phi ^{2}+d^{2}\sin \phi ^{2}\right)/\left(4f\right)+d\!\cos \phi \,\left(\cos \alpha \,\sin \gamma +\cos \beta \,\cos \gamma \sin \alpha \right)\\[3pt] &\quad +d\!\sin \phi \,\left(\cos \alpha \,\cos \gamma -\cos \beta \,\sin \alpha \,\sin \gamma \right) \\[3pt] z_{m-H} &= z_{o-H}+\cos \beta \,\left(d^{2}\cos \phi ^{2}+d^{2}\sin \phi ^{2}\right)/\left(4f\right)-d\!\cos \gamma \,\cos \phi \,\sin \beta +d\!\sin \beta \,\sin \gamma \,\sin \phi \end{align*}

The movement of the 3-RPS moving platform is constrained by spherical joints, so its three legs can only move in the planes of $y_{{S_{1\_ H}}} = 0, y_{{S_{2}}\_ H} = -\sqrt{3}x$ and $y_{{S_{3\_ H}}} = \sqrt{3}x_{{S_{3}}\_ H}$ . Then the constraint relations of $x_{m-H}$ and $y_{m-H}$ are obtained as follows:

(14) \begin{equation} x_{m-H}=-0.5r_{m}\left(\begin{array}{l} \cos \eta \,\cos \phi \left(\cos \alpha ^{2}\cos \beta -\cos \alpha ^{2}+1\right)+0.5\sin 2\alpha \,\cos \eta \,\sin \phi \left(\cos \beta -1\right)\\[5pt] -\cos \phi \left(\cos \alpha ^{2}+\cos \beta \,\sin \alpha ^{2}\right)+0.5\sin 2\alpha \,\sin \phi \left(\cos \beta -1\right)+\cos \alpha \,\sin \beta \,\sin \eta \end{array}\right) \end{equation}

(15) \begin{equation} y_{m-H}=r_{m}\!\left(\cos \eta \sin \phi\!\left(\cos \alpha ^{2}+\cos \beta \sin \alpha ^{2}\right)+\cos \alpha \cos \eta \cos \phi \sin \alpha \!\left(\cos \beta -1\right)+\sin \alpha \sin \beta \sin \eta \right) \end{equation}

As shown in Fig. 3, since the parabolic dish concentrator cannot rotate around the Z axis and the path of the sun in a day is basically symmetrical with respect to the ZHX plane, the relationship between the Euler angle, altitude angle $\alpha _{S}$ and azimuth angle $\gamma _{S}$ is obtained

(16) \begin{equation} \left\{\begin{array}{l} \alpha = 90-\gamma _{s}\\[5pt] \beta = 90-\alpha _{s}\\[5pt] \gamma =-\alpha \end{array}\right. \end{equation}

4.1. Length of the 3-RPS telescopic rod

According to the geometric shape of the 3-RPS, the coordinates of the rotary joints in coordinate system {P} are given by $\overrightarrow{\mathit{P}\mathit{B}_{\mathit{1}}}=(\mathit{r}_{\mathit{b}},0, 0)^{\mathrm{T}}$ , $\overrightarrow{\mathit{P}\mathit{B}_{\mathit{2}}}=\left(-\frac{1}{2}\mathit{r}_{\mathit{b}},\frac{\sqrt{3}}{2}\mathit{r}_{\mathit{b}},0\right)^{\mathrm{T}}$ and $\overrightarrow{\mathit{P}\mathit{B}_{\mathit{3}}}=\left(-\frac{1}{2}\mathit{r}_{\mathit{b}},-\frac{\sqrt{3}}{2}\mathit{r}_{\mathit{b}},0\right)^{\mathrm{T}}$ . The coordinates of the spherical joints in coordinate system {M} are given by $\overrightarrow{\mathit{M}\mathit{S}_{\mathit{1}}}=(\mathit{r}_{\mathit{m}},0,0)^{\mathrm{T}}$ , $\overrightarrow{\mathit{M}\mathit{S}_{\mathit{2}}}=\left(-\frac{1}{2}\mathit{r}_{\mathit{m}},\frac{\sqrt{3}}{2}\mathit{r}_{\mathit{m}},0\right)^{\mathrm{T}}$ and $\overrightarrow{\mathit{M}\mathit{S}_{\mathit{3}}}=\left(-\frac{1}{2}\mathit{r}_{\mathit{m}},-\frac{\sqrt{3}}{2}\mathit{r}_{\mathit{m}},0\right)^{\mathrm{T}}$ . It can be seen from Eq. (9) and Eqs. (12)–(15), ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{P}}$ is determined by parameters $\mathit{\phi }$ and d, and ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{M}}$ is determined by parameters $\mathit{z}_{\mathit{o}\mathit{-}\mathit{H}}\mathit{,}\mathit{\alpha }_{\mathit{s}}\mathit{,}\mathit{\gamma }_{\mathit{s}}\mathit{,}\mathit{\eta }\mathit{,}\mathit{\phi }\mathit{,}\mathit{r}_{\mathit{m}}$ and d. In order to express the relationship more clearly, ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{P}}$ and ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{M}}$ can be written as ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{P}}(\mathit{\phi }\mathit{,}\mathit{d})$ and ${}^{\mathit{H}}{\mathit{T}}{}_{\mathit{M}}(\mathit{z}_{\mathit{o}\mathit{-}\mathit{H}},\mathit{\alpha }_{\mathit{s}}, \mathit{\gamma }_{\mathit{s}}, \mathit{\eta }, \mathit{\phi },\mathit{r}_{\mathit{m}},\mathit{d})$ . Then the position vector of the spherical joints S _i (i = 1, 2, 3) with respect to the coordinate system {H} is given as

(17) \begin{equation} \left[\begin{array}{c} \overrightarrow{{{B_{i}S_{i}}}}\\[5pt] 1 \end{array}\right]={}^{H}{T}{_{M}^{}}\left(z_{o-H},\alpha _{\mathrm{s}},\gamma _{s},\eta,\phi,r_{m},d\right)\left[\begin{array}{c} \overrightarrow{{{MS_{i}}}}\\[5pt] 1 \end{array}\right]-{}^{H}{T}{_{P}^{}}\left(\phi,d\right)\left[\begin{array}{c} \overrightarrow{{{PB_{i}}}}\\[5pt] 1 \end{array}\right] \end{equation}

Then the length l _i (i = 1, 2, 3) of each telescopic rod of the 3-RPS can be written as:

(18) \begin{equation} l_{i}=\left\| \overrightarrow{{{B_{i}S_{i}}}}\right\| \end{equation}

According to the comprehensive analysis of Eqs. (17)–(18), the telescopic rod lengths of the 3-RPS in this novel dish concentrator platform are mainly determined by the parameters z _{O_H} , $\alpha$ _s , $\beta$ _s , $\phi$ , $\eta$ , r _m , r _b and d. But the parameters $\phi$ , $\eta$ , r _m , r _b and d are related to the structure sizes and installation position of the 3-RPS and the parabolic dish concentrator. Therefore, only the three parameters of $\alpha$ _s , $\beta$ _s and z _{O_H} are needed after the sizes and installation position are known, and solar tracking can be realized. The position and orientation of the 3-RPS moving platform can be written as:

(19) \begin{equation} W_{\textrm{3-RPS}}=\Phi \!\left(\alpha _{S},\gamma _{s},z_{o\_ H}\right) \end{equation}

4.2. The velocity of the 3-RPS manipulator

${}^{H}{{\boldsymbol{{V}}}}{_{M}^{}}$ is the linear velocity of the center point of the moving platform; ${}^{H}{\boldsymbol{\omega }}{_{M}^{}}$ is the angular velocity of the moving platform; ${}^{H}{{\boldsymbol{{r}}}}{_{Si}^{}}$ is the radius vector from the spherical joint $S_{{\boldsymbol{{i}}}}$ to point M; ${}^{H}{{\boldsymbol{{V}}}}{_{Si}^{}}$ is the velocity of the spherical joint $S_{i}$ ; L _i is the length vector of the telescopic rod and ${}^{H}{{\boldsymbol{{e}}}}{_{i}^{}}$ is the unit vector of L _i where ${}^{H}{{\boldsymbol{{e}}}}{_{i}^{}}={\boldsymbol{{L}}}_{i}/l_{i}$ , and $\dot{l}_{i}$ is the drive velocity of L _i .

The velocity vector of spherical joint $S_{i}$ can be written as:

(20) \begin{equation} {}^{H}{{\boldsymbol{{V}}}}{_{S_i}^{}}={}^{H}{{\boldsymbol{{V}}}}{_{M}^{}}+{}^{H}{\boldsymbol{\omega }}{_{M}^{}}\times {}^{H}{{\boldsymbol{{r}}}}{_{S_i}^{}} \end{equation}

The driving velocity $\dot{l}_{i}$ can be expressed as the projection of ${}^{H}{{\boldsymbol{{V}}}}{_{S_i}}$ on L _i .

(21) \begin{equation} \dot{\!l}_{i}={}^{H}{{\boldsymbol{{V}}}}{_{S_i}^{}}\cdot {}^{H}{{\boldsymbol{{e}}}}{_{i}^{}}=\left[\begin{array}{c} {}^{H}{{\boldsymbol{{e}}}}{_{i}^{}}T \!\left({}^{H}{{\boldsymbol{{r}}}}{_{si}^{}}\times {}^{H}{{\boldsymbol{{e}}}}{_{i}^{}}\right)^{T} \end{array}\right]\left[\begin{array}{c} {}^{H}{{\boldsymbol{{V}}}}{_{M}^{}}\\[5pt] {}^{H}{\boldsymbol{\omega }}{_{M}^{}} \end{array}\right] \end{equation}

The length vector of the three telescopic rods of the 3-RPS is

(22) \begin{equation} \dot{{\boldsymbol{{L}}}}_{i}=J_{A}\left[\begin{array}{c} {}^{H}{{\boldsymbol{{V}}}}{_{M}^{}}\\[5pt] {}^{H}{\boldsymbol{\omega }}{_{M}^{}} \end{array}\right] \end{equation}

where $\dot{{\boldsymbol{{L}}}}_{i} = [\begin{array}{c@{\quad}c@{\quad}c} \dot{l}_{1}\mathit{,} & \dot{l}_{2}\mathit{,} & \dot{l}_{3} \end{array}]^{T}, J_{A}=\left[\begin{array}{c@{\quad}c} {}^{H}{{\boldsymbol{{e}}}}_{{1}}T & ({}^{H}{r}_{{S1}}\times {}^{H}{{\boldsymbol{{e}}}}_{{1}})^{T}\\[5pt] {}^{H}{{\boldsymbol{{e}}}}_{{2}}T & ({}^{H}{r}_{{S2}}\times {}^{H}{{\boldsymbol{{e}}}}_{{2}})^{T}\\[5pt] {}^{H}{{\boldsymbol{{e}}}}_{{3}}T & ({}^{H}{r}_{{S3}}\times {}^{H}{{\boldsymbol{{e}}}}_{{3}})^{T} \end{array}\right]_{3\times 6}$

Derive from Eqs. (14)–(15) to obtain the linear velocity of the 3-RPS moving platform as follows:

(23) \begin{equation} ^{H}V_{M}=Jv\left[\begin{array}{c} \dot{\alpha }\\[5pt] \dot{\beta }\\[5pt] \dot{Z}_{-H} \end{array}\right]=\left[Jv1,Jv2,Jv3\right]\left[\begin{array}{c} \dot{\alpha }\\[5pt] \dot{\beta }\\[5pt] \dot{Z}_{-H} \end{array}\right] \end{equation}

where

\begin{equation*} J_{v1}=\left[\begin{array}{c} 0.5r_{m}\,\left(\begin{array}{c} \left(\cos \beta -1\right)\cos \phi \sin 2\,\alpha +\sin \eta \sin \alpha \,\sin \beta \,\\[5pt] -\left(\cos \beta -1\right)\left(k_{2}+\sin \phi \right)\cos 2\,\alpha +k_{1}\left(\cos \beta -1\right)\sin 2\,\alpha \, \end{array}\right)\\[5pt] r_{m}\,\left(\begin{array}{c} k_{1}\left(\cos \beta -1\right)\cos \alpha ^{2}+\sin \eta \,\sin \beta \,\cos \alpha \\[5pt] +\;2k_{2}\left(\cos \beta -1\right)\cos \alpha \,\sin \alpha -k_{1}\left(\cos \beta -1\right)\sin \alpha ^{2} \end{array}\right)\\[5pt] 0 \end{array}\right] \end{equation*}

\begin{equation*} J_{v2}=\left[\begin{array}{c} 0.5r_{m}\,\left(\sin \beta \,\cos \alpha ^{2}k_{4}+k_{1}\sin 2\,\alpha \,\,\sin \beta -\sin \eta \,\cos \alpha \cos \beta -\cos \phi \sin \beta \right)\\[5pt] -r_{m}\,\sin \alpha \!\left(k_{1}\cos \alpha \sin \beta -\sin \eta \cos \beta +k_{2}\sin \alpha \,\sin \beta \,\right)\\[5pt] 0 \end{array}\right] \end{equation*}

\begin{equation*} Jv_{3}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 1 \end{array}\right]^{T} \end{equation*}

$k_{1}=\cos \eta \,\cos \phi, k_{2}=\sin \phi \cos \eta, k_{3}=0.5\sin \phi (\cos \eta +1), k_{4}=k_{1}+\cos \phi$ .

The relationship between the angular velocity of the 3-RPS moving platform and the rotation matrix is as follows:

(24) \begin{equation} {}^{H}{\omega }{}\times =^{H}{\dot{R}_{M}}^{H}{R_{M}}^{-1}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & {}^{-H}{\omega }{_{z}^{}} & {}^{H}{\omega }{_{y}^{}}\\[5pt] {}^{H}{\omega }{_{z}^{}} & 0 & -{}^{H}{\omega }{_{x}^{}}\\[5pt] -{}^{H}{\omega }{_{y}^{}} & {}^{H}{\omega }{_{x}^{}} & 0 \end{array}\right] \end{equation}

The angular velocity of the 3-RPS moving platform is obtained by combining Eq. (12) and Eq. (24) as follows:

(25) \begin{equation} ^{H}\omega _{M}=\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & -\sin \alpha \, & \sin \beta \,\cos \alpha \\[5pt] 0 & \cos \alpha & \sin \beta \,\sin \alpha \\[5pt] 1 & 0 & \cos \beta \end{array}\right]\left[\begin{array}{c} \dot{\alpha }\\[5pt] \dot{\beta }\\[5pt] -\dot{\alpha } \end{array}\right] \end{equation}

Combining Eq. (23) and Eq. (25) to obtain the velocity of the center of mass of the 3-RPS moving platform.

(26) \begin{equation} \left[\begin{array}{c} {}^{H}{{\boldsymbol{{V}}}}{_{M}^{}}\\[5pt] {}^{H}{\boldsymbol{\omega }}{_{M}^{}} \end{array}\right]=J_{B}\left[\begin{array}{c} \dot{\alpha }\\[5pt] \dot{\beta }\\[5pt] \dot{Z}_{O-H} \end{array}\right] \end{equation}

By combining Eq. (22) and Eq. (26), the driving speed $\dot{{L}}_i$ of the rod can be obtained.

(27) \begin{equation} \dot{{\boldsymbol{{L}}}}_{i}=J_{A}J_{B}\left[\begin{array}{c} \dot{\alpha }\\[5pt] \dot{\beta }\\[5pt] z_{O-H} \end{array}\right] \end{equation}

5.1. Sun position calculation

In order to track the sun in real time, it is necessary to know the solar altitude angle $\alpha$ _s and the solar azimuth angle $\gamma$ _s , which can be obtained by combining Eq. (28) and Eq. (29) [Reference Qin, Yang, Hou and Zhou14, Reference Liu, Liao, Zhang and Mo15]:

(28) \begin{equation} \alpha _{s}=\arcsin \!\left(\sin L_{at}\sin \delta +\cos L_{at}\cos \delta \cos \omega _{s}\right) \end{equation}

(29) \begin{equation} \gamma _{s}=\text{sgn}\omega _{s}|\arccos \left(\frac{\sin \alpha _{s}\sin L_{\textrm{at}}-\sin \delta }{\cos \alpha _{s}\cos L_{\textrm{at}}}\right) \end{equation}

where L _at is the local latitude, $\delta$ is the sun declination angle, $\omega$ _s is the solar hour angle ( $\omega$ _s < 0 in the morning, $\omega$ _s = 0 at noon and $\omega$ _s > 0 in the afternoon), sgn $\omega$ _s is a sign function (when $\omega$ _s < 0, sgn $\omega$ _s = −1; when $\omega$ _s = 0, sgn $\omega$ _s = 0; when $\omega$ _s > 0, sgn $\omega$ _s = 1), $\delta$ and $\omega$ _s can be determined by Eq. (30) and Eq. (31) [Reference Soulayman16] [Reference John and William17]:

(30) \begin{equation} \delta =\frac{180}{\pi }\left(\begin{array}{c} 0.00148\sin 3\lambda -0.399912\cos \lambda \\[5pt] +0.070257\sin \lambda -0.006758\cos 2\lambda \\[5pt] +0.000907\sin 2\lambda -0.002679\cos 3\lambda \\[5pt] +0.006918 \end{array}\right) \end{equation}

(31) \begin{equation} \omega _{s}=15\times \left(S_{t}-12\right) \end{equation}

where $\lambda =\dfrac{360\left(n-1\right)}{365}$ , n is the product day in the order of days, that is, on January 1, n = 1, on January 2, n = 2, …, until December 31, n = 365 (in the common year) or 366 (in the leap year).

(32) \begin{equation} S_{t} = S_{\mathit{m}}+E_{t} \end{equation}

(33) \begin{equation} S_{m}=\left(S_{l}+F_{1}/60\right)-\frac{4}{60}\left(120-\left(L_{o}+\frac{F_{2}}{60}\right)\right) \end{equation}

(34) \begin{equation} E_{t}=\frac{180}{\pi }\cdot 4\left(\begin{array}{c} +0.001868\cos \lambda -0.032077\sin \lambda \\[5pt] -0.014615\cos 2\lambda -0.0409\sin 2\lambda \\[5pt] +0.000075 \end{array}\right) \end{equation}

where S _l is the hourly value at the observation time; F ₁ is the minute value of time; L _o is the degree value of the longitude of the observation location, which is positive in the east longitude and negative in the west longitude; and F ₂ is the minute value of longitude.

The extreme values of the inclination angles of the Earth’s rotation axis relative to the sun’s incident light occur at the spring equinox, summer solstice, autumn equinox and winter solstice. Taking Hunan University of Science and Technology (27.9° N, 112.9° E) as the observation point, the trajectories of $\alpha$ _s and $\gamma$ _s from 9 am to 16 pm on these four special days are shown in Fig. 4, which shows that the change law of $\alpha$ _s in a day increases gradually in the morning, reaches its peak at noon and decreases in the afternoon, and $\alpha$ _s reaches the maximum and minimum values on the summer solstice and the winter solstice, respectively, and $\alpha$ _s at the spring equinox and autumn equinox is nearly equal. The variation of $\gamma$ _s in a day presents a monotonic decreasing trend and reaches the maximum and minimum values on the summer solstice and the winter solstice, respectively, while the values at the spring and autumn equinoxes are almost equal. In this paper, the trajectories of $\alpha$ _s and $\gamma$ _s on these four special days will be taken as the inputs of the 3-RPS prismatic joints.

Figure 4. The curve of the solar altitude and azimuth angles. (a) Solar altitude angle. (b) Solar zenith angle.

5.2. Trajectory planning of the 3-RPS parallel manipulator

The structural parameters of the proposed novel dish solar tracking platform are shown in Table 1. The fixed base centroids of the four 3-RPSs are located in the axis +X, +Y, −X and −Y directions of the horizontal coordinate system {H}, namely $\phi$ = 90(N − 1), N = 1, 2, 3, 4.

Table 1. Structural parameters of the dish solar tracking platform.

According to the inverse solution formula of 3-RPS, the height of the center of mass (z _O-H ) of the dish concentrator will affect the length of the telescopic rod of 3-RPS. When the value of z _O-H is not taken correctly, the number of telescopic rods will reach three or more, making the telescopic rod structure more complex and difficult to control. Therefore, z _O-H needs to be optimized to meet the requirements of two telescopic rods. By transforming the 3-RPS inverse solution formula, the optimized objective function and constraints are obtained as shown in formula (35):

(35) \begin{equation} \mathit{z}_{O-H}=\min \sum _{i=1}^{3}\lambda _{i,N}f\left(\mathit{l}_{i,N},t\right) \end{equation}

\begin{equation*} \begin{array}{c@{\quad}c} s_{\cdot }t_{\cdot } & 2000\leq l_{i,N}\leq 4000\\[5pt] & 9\leq t\leq 16 \end{array} \end{equation*}

where f(l _i,N , t) is a function containing two variables l _i,N and t, which can be converted from Eq. (17), l _i,N represents the length of the ith telescopic rod of the Nth 3-RPS; $\lambda$ _i,N is the influence weight of the ith link length of the Nth 3-RPS, and since the telescopic rods of the 3-RPS are independent of each other, the values of $\lambda$ _i,N can be equal.

Take the parameters in Table 1 into formula (35), and optimize z _O-H by using the GA optimization toolbox of MATLAB software to obtain the results shown in Fig. 5.

Figure 5. Results of z _O-H optimization by GA.

It can be seen from Fig. 5 that when the number of iterations is 75, the best fitness value of z _O-H is 2960.23 mm. Taking this result into the 3-RPS inverse solution formula, calculate the length of the 3-RPS telescopic rod in the spring equinox, summer solstice, autumn equinox and winter solstice, and obtain the motion trajectory as shown in Fig. 6. The minimum and maximum lengths of the telescopic rod are 2105 mm and 3864 mm, respectively. Then a 2-section telescopic rod with a length of both the upper rod and the lower rod of 2000 mm can meet the requirements for the movement of the 3-RPS.

Figure 6. Length of the telescopic rods of the 3-RPS manipulators. (a) March equinox. (b) Summer solstice. (c) September equinox. (d) Winter solstice.

5.3. Attitude simulation of 3-RPS parallel manipulators

In order to analyze the movement posture of four 3-RPSs more intuitively, the kinematics attitude of the 3-RPS is simulated by using MATLAB software. Figures 7–10 show the attitudes of the four 3-RPSs at 10 am, 12 noon and 14 pm on the summer solstice.

Figure 7. Attitude of the 3-RPS moving platform on the +X axis at the summer solstice. (a) 10:00 am. (b) 12:00 noon. (c) 14:00 pm.

Figure 8. Attitude of the 3-RPS moving platform on the +Y axis at the summer solstice. (a) 10:00 am. (b) 12:00 noon. (c) 14:00 pm.

Figure 9. Attitude of the 3-RPS moving platform on the −X axis at the summer solstice. (a) 10:00 am. (b) 12:00 noon. (c) 14:00 pm.

Figure 10. Attitude of the 3-RPS moving platform on the −Y axis at the summer solstice. (a) 10:00 am. (b) 12:00 noon. (c) 14:00 pm.

6.1. Error evaluation index

To ensure that the 3-RPS can stably and accurately track the sun’s trajectory, the time-weighted integral absolute error (ITAE) is used as the performance index of controller design. The expression of ITAE index is as follows [Reference Tang18]:

(36) \begin{equation} J_{\textrm{ITAE}}=\int _{0}^{t}t\left| e\!\left(t\right)\right| dt \end{equation}

The ITAE index includes two factors: time and error. It considers the dynamic and steady-state performance of the control system, which can ensure the response speed and steady-state accuracy of the system. The controller of the 3-RPS will be designed with the goal of minimum ITAE.

6.2. PID controller

The mathematical expression of the PID controller is

(37) \begin{equation} u\!\left(t\right)=K_{P}e\!\left(t\right)+K_{I}\int _{0}^{t}e\!\left(\tau \right)d\tau +K_{D}\frac{de\!\left(t\right)}{dt} \end{equation}

where K _P , K _I and K _D represent the three coefficients of proportion, integral and differential, respectively. The schematic diagram of the PID controller for a single 3-RPS is shown in Fig. 12, where act_ pos, des_pos, des_el and des_vel are the ideal input position, actual output position, ideal input velocity and actual output velocity for the 3-RPS, respectively, and force represents the input driving torque. Since each 3-RPS is in parallel, the PID controller diagrams of the other 3-RPSs are the same as in Fig. 11.

Figure 11. Schematic diagram of the single 3-RPS PID controller.

Figure 12. Flow chart of the PID controller optimized by PSO.

6.3. PSO-PID controller

PSO is an evolutionary algorithm proposed by J. Kennedy and R. C. Eberhart in 1995. It originates from the migration and clustering behavior of birds in the process of foraging [Reference Kennedy and Eberhart19]. Its basic idea is to find the optimal solution through cooperation and information sharing among individuals in the group. The update rule of PSO is to initialize a group of random solutions and then find the optimal solution through iteration. In each iteration, the particle updates itself by tracking the extreme individual value P _best and the extreme global value G _best. After these two optimal values are found, the particle updates its speed and new position through the following formula [Reference Feng, Ma, Yin and Cao20]:

(38) \begin{equation} v_{t+1}=\omega v_{t}+c_{1}\text{rand}\left(\right)\left(p_{\textit{t}\text{best}}-x_{t}\right)+c_{2}\text{rand}\left(\right)\left(G_{\textit{t}\text{best}}-x_{t}\right) \end{equation}

(39) \begin{equation} x_{t+1}=x_{t}+v_{t+1} \end{equation}

where $\omega$ is the inertial weight, t = 1, 2, 3, …, T, and T is the total running time; x _t is the position of the tth particle; v _t is the velocity of the tth particle; p _tbest is the optimal position searched by the tth particle; p _gbest is the optimal position searched by the whole particle swarm; c ₁ and c ₂ are learning factors and rand() is a random number between (0,1). The flow chart of the PID controller optimized by PSO is shown in Fig. 12.

6.4. Trajectory tracking control simulation

A visual simulation model of a rotating parabolic solar power generation system has been established in the Simscape Multibody of MATLAB software, in which driving elements and detection modules were added, and component parameters were set. The control system structure of the visualization platform is shown in Fig. 13, mainly composed of the 3-RPS inverse kinematics models, PSO-PID controller, a multibody physical model of the dish solar tracking platform and the solar trajectory model.

Figure 13. Control schematic diagram of the visual simulation platform.

When the solar altitude angle and the azimuth angle are both 90°, the attitude of the novel dish solar tracking platform is in the initial position. The 3-RPS needs to move from the initial position to the starting point for tracking before sun tracking in the morning and then move to the initial position after sun tracking in the afternoon. The attitude of the initial position is shown in Fig. 14.

Figure 14. Attitude of the concentrator in the initial position of the novel dish platform.

Based on this visualization platform, the simulation experiment of motion control is carried out by taking the sun tracking on the summer solstice. The PID parameters obtained by PSO are shown in Table 2.

Table 2. Parameters of PSO-PID controller.

The PSO-PID controller is designed by using the PID parameters in Table 2. Experiments are carried out on the visual simulation platform, and the error response of each telescopic rod is shown in Fig. 13 when four 3-RPSs of the dish system track the sun trajectory. Table 3 shows the stability error. It can be seen from Fig. 15 and Table 3 that the PSO-PID controllers on the four 3-RPSs can make the telescopic rod error converge rapidly, which can achieve stable and accurate tracking of the system. In the steady state, the error of the first rod of all 3-RPSs is smaller than that of the other two telescopic rods, mainly because the displacement change of the telescopic rod near the centroid of the concentrator is smaller than the outer layer during the whole tracking work. Figure 16 shows the three attitudes of the concentrator at 10:00 am, 12:00 noon and 14:00 pm in the visual simulation platform of the novel dish solar power generation system.

Table 3. Steady-state error.

Figure 15. Error response curve of 3-RPS parallel manipulators. (a) Error response curve of 3-RPS on the +X axis. (b) Error response curve of 3-RPS on the +Y axis. (c) Error response curve of 3-RPS on −X axis. (d) Error response curve of 3-RPS on −Y axis.

Figure 16. Attitude of the novel dish solar tracking platform on summer solstice. (a) 10:00 am. (b) 12:00 noon. (c) 14:00 pm.