In the field of solar energy harvesting, the efficiency of photovoltaic panels is significantly influenced by their ability to track the sun’s trajectory. Fixed-mount solar panels, while cost-effective, suffer from substantial energy loss due to the relative motion between the sun and the earth over daily and seasonal cycles. To address this limitation, we propose a novel 1T2R (one translation and two rotations) parallel mechanism specifically designed for adjusting the attitude and position of solar panels. This mechanism offers a larger orientation range, simpler structure, and lower control complexity compared to existing solutions. In this paper, we detail the topological design based on the Position and Orientation Characteristic (POC) set theory, kinematic modeling, forward and inverse position solutions, and motion simulation. Our results demonstrate that the mechanism can achieve continuous orientation changes over a wide angular range, making it an ideal candidate for advanced solar tracking systems.
Introduction
Sustainable energy development has placed solar power at the forefront of renewable energy sources. Solar panels, as the primary conversion devices, are deployed globally. However, conventional fixed-angle installations cannot adapt to the changing solar altitude and azimuth angles throughout the day and across seasons. To maximize annual energy yield, active solar tracking mechanisms are required. Among various tracking architectures, parallel mechanisms offer high stiffness, low inertia, and precise positioning. In our research, we focus on designing a parallel mechanism with three degrees of freedom (DOF) — one translational movement along the vertical axis and two rotational movements about the horizontal axes — to optimize the orientation of solar panels toward the sun. The mechanism is driven by three linear actuators, providing a compact and robust solution for practical solar panel applications.
Compared to existing designs, our proposed mechanism exhibits superior symmetry, reduced coupling, and straightforward control. The following sections describe the systematic synthesis, kinematic analysis, and validation of this mechanism.
Topological Synthesis of Branches
We employed the POC set methodology to generate viable branch chains that can realize 1T2R output motions. The POC set of the moving platform, denoted as $M_{Pa}$, is the intersection of the POC sets of all branches. For a 1T2R mechanism, the required $M_{Pa}$ is:
$$ M_{Pa} = \begin{pmatrix} t^1(\parallel P_{21}) \\ r^2(\parallel \Diamond(R_{22},R_{23})) \end{pmatrix} $$
Based on the theory, several single-open-chain (SOC) and hybrid open-chain (HSOC) types were synthesized. Table 1 summarizes the candidate branch types meeting the 1T2R requirement, where $t$ denotes translational degrees and $r$ denotes rotational degrees.
| Branch ID | Joint Sequence | POC Set | Remarks |
|---|---|---|---|
| B1 | $R_{11}\perp R_{12}(\perp P_{13})\perp R_{14}\perp R_{15}$ | $\begin{pmatrix} t^3 \\ r^2(\parallel \Diamond(R_{11},R_{12})) \end{pmatrix}$ | SOC |
| B2 | $P_{21}\perp R_{22}\perp R_{23}$ | $\begin{pmatrix} t^1(\parallel P_a) \\ r^2(\parallel \Diamond(R_{22},R_{23})) \end{pmatrix}$ | SOC |
| B3 | $R_{31}\perp R_{32}(\perp P_{33})\parallel R_{34}\perp R_{35}$ | $\begin{pmatrix} t^3 \\ r^2(\parallel \Diamond(R_{34},R_{35})) \end{pmatrix}$ | SOC |
| B4 | $R_{41}\perp P_{42}\perp R_{43}\perp R_{44}$ | $\begin{pmatrix} t^2 \\ r^2 \end{pmatrix}$ | HSOC |
| B5 | $P_{51}\perp R_{52}\perp R_{53}\perp R_{54}$ | $\begin{pmatrix} t^2 \\ r^2 \end{pmatrix}$ | HSOC |
From these candidates, we selected three branches that provide the best trade-off between structural simplicity and functional performance. The selected branches are B1, B2, and B3. Their assembly yields a parallel mechanism composed of a fixed base, a moving platform, and three limbs. The mechanism is shown schematically in the attached figure (the real structure incorporates the following link).
Topological Characteristics of the Mechanism
The POC set of each branch is computed using the serial combination rule $M_{bi} = \bigcup_{i=1}^m M_{Ji}$. For branch B1 (SOC1), the 5 joints give:
$$ M_{b1} = \begin{pmatrix} t^3 \\ r^2(\parallel \Diamond(R_{11},R_{12})) \end{pmatrix} $$
For branch B2 (SOC2):
$$ M_{b2} = \begin{pmatrix} t^1(\parallel P_a) \\ r^2(\parallel \Diamond(R_{22},R_{23})) \end{pmatrix} $$
For branch B3 (SOC3):
$$ M_{b3} = \begin{pmatrix} t^3 \\ r^2(\parallel \Diamond(R_{34},R_{35})) \end{pmatrix} $$
The POC set of the moving platform is obtained by intersecting the POC sets of the three branches:
$$ M_{Pa} = \bigcap_{j=1}^3 M_{bj} = \begin{pmatrix} t^1(\parallel P_{21}) \\ r^2(\parallel \Diamond(R_{22},R_{23})) \end{pmatrix} $$
This confirms that the mechanism provides exactly one translation and two rotations. To verify the degree of freedom, we used the general DOF formula for spatial parallel mechanisms:
$$ F = \sum_{i=1}^m f_i – \sum_{j=1}^v \xi_{Lj} $$
where $v = m – n + 1$ is the number of independent loops, $f_i$ is the DOF of joint $i$, and $\xi_{Lj}$ is the number of independent displacement equations of loop $j$. For our mechanism, there are two independent loops. The first loop (branches B1 and B2) yields $\xi_{L1}=5$, and the second loop (first loop plus branch B3) yields $\xi_{L2}=5$. The total joint DOF sum is $f_1+f_2+…+f_{13}=13$ (counting all revolute and prismatic joints). Therefore:
$$ F = 13 – (5+5) = 3 $$
Thus, the mechanism has three degrees of freedom, matching the desired 1T2R motion. The coupling degree $\kappa$ was also evaluated using the SOC-based method. The constrained degree of each loop is defined as:
$$ \Delta_j = \sum_{i=1}^{m_j} f_i – I_j – \xi_{Lj} $$
For loop 1: $\Delta_1 = 5 – 1 – 5 = -1$ (assuming one actuator in the loop). For loop 2: $\Delta_2 = 8 – 2 – 5 = 1$. The coupling degree of the whole mechanism is:
$$ \kappa = \frac{1}{2} \sum_{j=1}^v |\Delta_j| = \frac{1}{2}(1+1)=1 $$
A coupling degree of 1 indicates that the position solutions of the two loops are partially coupled, requiring simultaneous solution of the loop equations.
Kinematic Position Analysis
We established a coordinate system on the fixed base (global frame $Oxyz$) and on the moving platform (local frame $O’x’y’z’$). Both platforms are equilateral triangles with side length $2a_1 = 752.1\ \text{mm}$. The centers are at the midpoints of the base edges. The three limbs are labeled as limb I, II, and III, with corresponding actuator lengths $l_1, l_2, l_3$. The geometric constraints are shown in the schematic model (see figure in the article).
Forward Position Solution
Given the actuator lengths $l_1, l_2, l_3$, we need to compute the position $(x_{O’}, y_{O’}, z_{O’})$ and orientation angles $\alpha, \beta$ of the moving platform. From the first loop (limbs I and II), the projection onto the $xOz$ plane yields:
$$ l_1^2 – \left[2a_1 – 2a_1\cos\alpha\right]^2 = \left[l_2 – 2a_1\sin\alpha\right]^2 $$
Solving for $\alpha$ gives $\alpha = f_1(l_1, l_2)$. Then the position of point $O’$ along the $x$ and $z$ axes are:
$$ x_{O’} = a_1 – a_1\cos\alpha $$
$$ y_{O’} = 0 $$
$$ z_{O’} = l_2 – a_1\sin\alpha $$
From the second loop (including limb III), we consider the projection onto the $yOz$ plane. The constraint is:
$$ l_3^2 – \left[\sqrt{3}a_1 – \sqrt{3}a_1\cos\beta\right]^2 = \left[z_{O’} + \sqrt{3}a_1\sin\beta\right]^2 $$
Solving yields $\beta = f_2(l_1, l_2, l_3)$. Table 2 shows the forward solution result for a sample input set.
| Case | $x_{O’}$ (mm) | $y_{O’}$ (mm) | $z_{O’}$ (mm) | $\alpha$ (°) | $\beta$ (°) |
|---|---|---|---|---|---|
| 1 | 60.60 | 0 | 742.07 | 32.98 | 18.97 |
| 2 | 60.60 | 0 | 332.65 | 32.98 | 18.97 |
| 3 | 467.48 | 0 | 1266.90 | -75.93 | 4.61 |
| 4 | 467.48 | 0 | 902.13 | -75.93 | 4.61 |
These four cases represent different configurations satisfying the same actuator lengths, indicating the existence of multiple assembly modes.
Inverse Position Solution
Given the desired pose $(x_{O’}, y_{O’}, z_{O’}, \alpha, \beta)$, we compute the required actuator lengths. Using the geometry of the first loop:
$$ l_1 = \sqrt{(x_{O’} – a_1\cos\alpha + a_1)^2 + (z_{O’} – a_1\sin\alpha)^2} $$
$$ l_2 = \sqrt{(z_{O’} + a_1\sin\alpha)^2} $$
And from the second loop:
$$ l_3 = \sqrt{(y_{O’} + \sqrt{3}a_1\cos\beta – \sqrt{3}a_1)^2 + (z_{O’} + \sqrt{3}a_1\sin\beta)^2} $$
For verification, we used the forward solution Case 1 from Table 2 as input to the inverse equations. Table 3 lists the eight possible inverse solutions due to the quadratic nature of the equations.
| Solution | $l_1$ (mm) | $l_2$ (mm) | $l_3$ (mm) |
|---|---|---|---|
| 1 | 954.50 | 537.37 | 954.51 |
| 2 | 550.87 | 537.37 | 954.51 |
| 3 | 954.50 | 946.77 | 954.51 |
| 4 | 954.50 | 537.37 | 531.47 |
| 5 | 550.87 | 946.77 | 954.51 |
| 6 | 550.87 | 537.37 | 531.47 |
| 7 | 954.50 | 946.77 | 531.47 |
| 8 | 550.87 | 946.77 | 531.47 |
The first solution exactly matches the original actuator lengths used in the forward calculation, confirming the correctness of both models. The other seven solutions correspond to different assembly modes of the mechanism, which must be considered in path planning for solar panel tracking.
Motion Simulation and Workspace Analysis
To evaluate the dynamic performance and orientation capability of the mechanism, we built a virtual prototype in ADAMS. The joints were defined according to the topological relationships, and three prismatic actuators were driven with prescribed sinusoidal functions:
$$ l_1 = 30\sin\left(\frac{2\pi}{30}t\right) \quad \text{(mm)} $$
$$ l_2 = 20\sin\left(\frac{2\pi}{1.5}t\right) \quad \text{(mm)} $$
$$ l_3 = 15\sin\left(\frac{2\pi}{3}t\right) \quad \text{(mm)} $$
The simulation ran for 30 seconds, and the resulting orientation angles $\alpha$ and $\beta$ were recorded. The maximum and minimum values are summarized in Table 4.
| Angle | Minimum (°) | Maximum (°) | Range (°) |
|---|---|---|---|
| $\alpha$ (rotation about y-axis) | -45.16 | 45.29 | 90.45 |
| $\beta$ (rotation about x-axis) | -45.16 | 45.29 | 90.45 |
The mechanism achieved a symmetric angular range of approximately 90° in both rotational directions. The translational motion along the z-axis also varied smoothly without singularities during the entire simulation. These results indicate that the proposed parallel mechanism can effectively orient a solar panel over a wide solid angle, covering the necessary daily and seasonal sun positions.
Furthermore, the continuous and smooth nature of the angle-time curves (not shown here) confirms that the mechanism is free from sudden jumps or kinematic singularities within the tested range. This smoothness is critical for practical control systems to avoid mechanical shocks and ensure stable operation of the solar panel.
Conclusion
We have designed, analyzed, and simulated a novel 1T2R parallel mechanism specifically for solar panel attitude and position adjustment. The mechanism features three identical branches composed of revolute and prismatic joints, resulting in a symmetric structure with a coupling degree of 1. The forward and inverse position solutions were derived analytically and verified through numerical examples. ADAMS simulation demonstrated that the mechanism can achieve a continuous orientation range of about 90° in two perpendicular rotational axes, which is sufficient for efficient solar tracking in most geographical locations. The simplicity of the kinematic equations and the modular structure make the mechanism easy to manufacture and control. Future work will focus on building a physical prototype and integrating a closed-loop control system for real-time solar tracking.