Solar energy is one of the most promising renewable energy sources due to its sustainability, environmental friendliness, and wide availability. However, the efficiency of solar panels is significantly influenced by their orientation relative to the sun. Fixed-angle installations, which are commonly used, cannot adapt to the daily and seasonal variations in the sun’s position, resulting in suboptimal energy capture. To address this limitation, we propose a novel parallel mechanism capable of adjusting the attitude and position of solar panels, enabling a larger range of solar radiation reception. This paper presents the systematic design, kinematic analysis, and simulation of a 1T2R (one translation and two rotations) parallel mechanism tailored for solar tracking applications.
The increasing global installed capacity of photovoltaic (PV) systems—reaching 890 GW in China by the end of 2024—highlights the urgency of improving the energy yield per panel. Traditional solar trackers often adopt serial mechanisms or complex multi-axis systems, which suffer from high cost, low stiffness, or limited workspace. Parallel mechanisms, on the other hand, offer advantages such as high rigidity, high precision, and compact structure. Among them, mechanisms with one translational and two rotational degrees of freedom (1T2R) are particularly suitable for solar panels because they can adjust the panel’s tilt angle (elevation) and azimuth angle while also providing vertical displacement to avoid ground obstacles or optimize spacing.
In this work, we employ the Position and Orientation Characteristic (POC) set theory for topological synthesis of the parallel mechanism. This method is intuitive and avoids complex algebraic computations, making it ideal for conceptual design. We first analyze the required motion pattern for solar panels, then synthesize feasible limb configurations, and finally select an optimal architecture with symmetry, simplicity, and ease of control. The inverse and forward kinematics are derived analytically, and a numerical example validates the correctness of the equations. The workspace is evaluated using MATLAB, and dynamic simulation in ADAMS confirms the smooth motion and large rotational range of the mechanism. The results demonstrate that the proposed mechanism can achieve an elevation angle range of approximately -45° to +45° and an azimuth angle range of similar magnitude, which is sufficient for most terrestrial solar tracking applications.
1. Topological Synthesis of the 1T2R Parallel Mechanism
Based on the POC set theory, the motion output of the moving platform is the intersection of the POC sets of all limbs. Therefore, each limb must provide at least the required motion capabilities. For a 1T2R mechanism, the limb POC set can be expressed as one of the following types:
$$ M_{bj} = \begin{bmatrix} t^3 \\ r^3 \end{bmatrix}, \begin{bmatrix} t^3 \\ r^2 \end{bmatrix}, \begin{bmatrix} t^2 \\ r^3 \end{bmatrix}, \begin{bmatrix} t^2 \\ r^2 \end{bmatrix}, \begin{bmatrix} t^1 \\ r^3 \end{bmatrix}, \begin{bmatrix} t^1 \\ r^2 \end{bmatrix} $$
Here, \( t^i \) denotes the number of independent translational degrees of freedom, and \( r^j \) denotes the number of independent rotational degrees of freedom. Since solar panels require two rotations (elevation and azimuth) and one translation (vertical lift), we focus on limbs with POC sets containing at least \( t^1 \) and \( r^2 \). Additionally, to simplify control and manufacturing, we prefer limbs that include a prismatic (P) joint as the actuator, because linear actuators (e.g., electric cylinders) provide large stroke and are easy to control.
Figure 1 shows several candidate limb structures that satisfy the requirements. Among them, we select three limbs with the following topologies:
- Limb I: SOC1 { –R11⊥R12 (⊥P13) ⊥R14⊥R15– } – a 5-DOF chain with two universal joints and one prismatic joint.
- Limb II: SOC2 { –P21⊥R22⊥R23– } – a 3-DOF chain with one prismatic and two revolute joints forming a universal joint.
- Limb III: SOC3 { –R31⊥R32 (⊥P33) || R34⊥R35– } – a 5-DOF chain similar to Limb I but with a different arrangement of revolute axes.
These three limbs are assembled between a fixed base and a moving platform. Both the base and the platform are equilateral triangles with side length \( 2a_1 \). The assembly constraints are set so that the axes of certain revolute joints are parallel or collinear, as detailed in the original text. The resulting 1T2R parallel mechanism is shown in Figure 2 (3D model).
To verify the mobility, we compute the POC set of the moving platform using the intersection rule:
$$ M_{Pa} = \bigcap_{j=1}^{3} M_{bj} = \begin{bmatrix} t^1(||P_{21}) \\ r^2(||\Diamond(R_{22},R_{23})) \end{bmatrix} $$
This confirms that the platform has one translational degree of freedom along the direction of the prismatic joint in Limb II, and two rotational degrees of freedom about axes perpendicular to that direction. The degree of freedom can also be calculated using the constrained screw theory; the result is \( F = 3 \), consistent with the POC analysis.
The coupling degree \( \kappa \) of the mechanism is determined by the method of single-open-chain (SOC) decomposition. For the two independent loops (Loop1: Limb I + Limb II; Loop2: Loop1 + Limb III), we obtain:
$$ \xi_{L1} = 5, \quad \xi_{L2} = 5 $$
$$ \Delta_1 = \sum f_i – I_1 – \xi_{L1} = 1 $$
$$ \Delta_2 = \sum f_i – I_2 – \xi_{L2} = -1 $$
$$ \kappa = \frac{1}{2}\sum \Delta_j = 1 $$
A coupling degree of 1 indicates that the position analysis requires solving a single-loop constraint equation, which is manageable analytically. This property simplifies the control algorithm for real-time tracking.
2. Forward and Inverse Kinematics
Kinematic modeling is essential for understanding the relationship between actuator strokes and platform pose. We define a fixed coordinate system \( Oxyz \) attached to the base, and a moving coordinate system \( O’x’y’z’ \) attached to the platform. The origin \( O \) is at the midpoint of base vertices \( A_1 \) and \( A_2 \), and the origin \( O’ \) is at the midpoint of platform vertices \( B_1 \) and \( B_2 \). The geometric parameters are set as: side length of base/platform = \( 2a_1 = 752.1 \) mm.
2.1 Forward Kinematics
Given the actuator lengths \( l_1, l_2, l_3 \) (corresponding to the three prismatic joints), we want to find the platform position \( [x_{O’}, y_{O’}, z_{O’}]^T \) and orientation angles \( \alpha \) (rotation about y-axis) and \( \beta \) (rotation about x-axis). The forward kinematics is solved by decoupling the mechanism into two loops.
Loop 1: This loop involves Limb I and Limb II, which constrain the platform’s translation in the x-direction and the rotation \( \alpha \). From the geometric projection shown in Figure 3 (conceptual), we derive:
$$ l_1^2 – \left[ 2a_1 – (2a_1) \cos\alpha \right]^2 = \left[ l_2 – (2a_1) \sin\alpha \right]^2 $$
Solving for \( \alpha \) yields an explicit function: \( \alpha = f_1(l_1,l_2) \). Then the platform coordinates can be expressed as:
$$ x_{O’} = a_1 – a_1 \cos\alpha $$
$$ y_{O’} = 0 $$
$$ z_{O’} = l_2 – a_1 \sin\alpha $$
Note that \( y_{O’} = 0 \) because of the symmetry of the loop constraint; the platform’s center lies in the plane defined by the base’s x-z plane for this loop.
Loop 2: This loop involves the second circuit (including Limb III) and determines the rotation \( \beta \) and the remaining translation. Projecting onto the y-z plane gives:
$$ l_3^2 – \left[ \sqrt{3}a_1 – (\sqrt{3}a_1) \cos\beta \right]^2 = \left[ z_{O’} + \sqrt{3}a_1 \sin\beta \right]^2 $$
Using the previously computed \( z_{O’} \), we obtain \( \beta = f_2(l_1,l_2,l_3) \). Thus, the forward kinematic solution is explicit and does not require iterative numerical methods.
2.2 Inverse Kinematics
Given the platform pose \( [x_{O’}, y_{O’}, z_{O’}, \alpha, \beta] \), we compute the required actuator lengths. Using vector geometry:
$$ l_1 = \sqrt{ \left( x_{O’} – a_1 \cos\alpha + a_1 \right)^2 + \left( z_{O’} – a_1 \sin\alpha \right)^2 } $$
$$ l_2 = \sqrt{ \left( z_{O’} + a_1 \sin\alpha \right)^2 } \quad \text{(absolute value)} $$
$$ l_3 = \sqrt{ \left( y_{O’} + \sqrt{3}a_1 \cos\beta – \sqrt{3}a_1 \right)^2 + \left( z_{O’} + \sqrt{3}a_1 \sin\beta \right)^2 } $$
Note that \( y_{O’} \) is zero from the forward kinematics equation, but in the inverse problem, it can be specified arbitrarily; however, due to mechanism constraint, \( y_{O’} \) must remain zero for a consistent solution. This indicates that the mechanism has a constrained motion: the platform’s center moves only in the x-z plane, while the two rotations are about the y and x axes respectively. This characteristic is acceptable for solar panels because the vertical translation and two tilt angles are sufficient for tracking.
2.3 Numerical Verification
To validate the kinematic equations, we select a test case. Set \( l_1 = 954.51 \) mm, \( l_2 = 537.36 \) mm, \( l_3 = 954.51 \) mm. The forward kinematics yields the platform pose given in Table 1.
| Case | xO’ (mm) | yO’ (mm) | zO’ (mm) | α (°) | β (°) |
|---|---|---|---|---|---|
| 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 |
For case 1, we compute the inverse kinematics, obtaining eight possible combinations of \( l_1, l_2, l_3 \), as shown in Table 2. The first row matches the original inputs exactly, confirming the consistency of forward and inverse solutions.
| No. | l1 (mm) | l2 (mm) | l3 (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 existence of multiple inverse solutions (eight) is typical for parallel mechanisms due to assembly modes. In practical control, the solution that lies within the actuator stroke limits and avoids singularities is selected.
3. Workspace and Motion Simulation
To evaluate the angular range of solar panels achievable by the mechanism, we perform a Monte Carlo workspace analysis using MATLAB. The actuator stroke limits are set as: minimum 300 mm, maximum 1200 mm for all three legs. The resulting workspace in terms of orientation angles is shown in Figure 4 (conceptual). The mechanism can achieve an elevation angle \( \alpha \) from -45.2° to 45.3° and an azimuth angle \( \beta \) from -45.1° to 45.3°. This range is sufficient for most low-latitude to mid-latitude regions where the sun’s altitude varies within ±45°.
We then import the 3D CAD model into ADAMS software to perform dynamic simulation. Three actuator inputs are defined as harmonic functions to represent typical tracking motion:
$$ l_1 = 30 \sin\left(\frac{2\pi}{30} t\right) $$
$$ l_2 = 20 \sin\left(\frac{2\pi}{1.5} t\right) $$
$$ l_3 = 15 \sin\left(\frac{2\pi}{3} t\right) $$
The simulation results (Figure 5) show that both \( \alpha \) and \( \beta \) change smoothly without abrupt jumps, and the vertical displacement \( z_{O’} \) also varies continuously. This confirms that the mechanism can operate stably under typical solar tracking scenarios. The maximum angular velocities and accelerations are within acceptable limits for electric linear actuators.
4. Discussion and Conclusions
We have presented a systematic design methodology for a 1T2R parallel mechanism intended for solar panels attitude adjustment. The key contributions include:
- Topological synthesis using POC set theory to generate a novel architecture with three limbs, each containing a prismatic actuator.
- Explicit forward and inverse kinematic solutions that enable real-time control without heavy computational burden.
- Verification of correctness through numerical example and ADAMS simulation.
- Workspace evaluation showing a rotational range of ±45° in both elevation and azimuth, which is compatible with dual-axis solar tracking requirements.
Compared to existing solar tracking mechanisms, our design offers several advantages: (1) symmetric structure simplifies manufacturing and assembly; (2) coupling degree \( \kappa = 1 \) ensures that the kinematics can be solved in closed form; (3) the use of three linear actuators (preferably electric cylinders) allows easy integration with PV systems that already generate electricity. The mechanism is particularly suitable for arid or semi-arid regions where dust and maintenance are concerns, as the simple structure minimizes wear points.
Future work will focus on experimental validation using a physical prototype, as well as optimization of link lengths to further increase the workspace or improve stiffness. Additionally, a closed-loop control algorithm based on sun position sensors will be implemented to achieve automatic tracking with high accuracy.
In conclusion, the proposed 1T2R parallel mechanism provides an effective solution for enhancing the energy yield of solar panels through active attitude adjustment. Its straightforward kinematics, large angular range, and structural simplicity make it a strong candidate for next-generation solar trackers.