Design, Analysis, and Experimentation of Space Deployable Segmented Solar Concentrator

Abstract

To improve the optical concentrator ratio of space solar power stations (SSPSs), this paper proposes a deployable segmented solar concentrator (DSSC) based on an afocal reflective system. First, a novel concept of an afocal reflective concentrator composed of segmented primary and secondary mirrors is introduced, and the deployable mechanism for the segmented primary mirror is described in detail. Subsequently, a model for the comprehensive error of the deployable mechanism with 3D revolute joint clearances and link length errors is established based on the “massless link” equivalent model of the clearance in revolute joints and the homogeneous transfer matrix. Sensitivity analysis evaluates the impact of various geometric errors of the deployable mechanism on the comprehensive error. Finally, a prototype experimental system is built to verify the concentration ratio of the concentrator and the pose error of the deployable mechanism. The experimental results show that the DSSC geometric concentration ratio reaches 5.36 to 6, and the optical concentration ratio reaches 24.7 to 32.2. The repeatability of the deployable mechanism is ±50 µm and ±1.2′, meeting the tolerance requirements of the optical system. The proposed afocal reflective DSSC can be used for solar energy concentration, improving the utilization of solar energy.

1. Introduction

Solar energy is regarded as one of the most plentiful and readily available energy sources. The solar energy density in space reaches up to 1367 W/m², which is approximately five times that observed on the Earth’s surface [1,2]. In comparison to ground-based solar power stations, SSPSs are less susceptible to external factors such as climate and geography, which confers upon them a considerable advantage in terms of solar energy conversion efficiency. The development of SSPSs has been rapid, in line with the advancement of space science and technology, following the initial proposal by Glaser [3]. An SSPS is a device designed to collect solar energy in space, convert it into electrical energy, and transmit that energy wirelessly to Earth [4,5].

SSPSs can be mainly classified into two types: non-concentrating mode and sunlight-concentrating mode. The structures of non-concentrating SSPSs are simple and highly scalable, with examples like the Sail-Tower SSPS [2] and Tethered-SPS [6]. However, non-concentrating SSPSs face issues such as overly large photovoltaic arrays and lower system efficiency. Concentrating SSPSs, which use concentrators, reduce the size of the photovoltaic array while significantly improving the system’s power-to-mass ratio. Arbitrarily large phased array (ALPHA) [7] and orb-shape membrane energy gathering array (OMEGA) [8] are typical examples of concentrating SSPSs. The concentrator of ALPHA is a conical structure composed of numerous thin-film hexagonal reflectors connected together [9,10]. Each planar reflector requires independent control for real-time solar tracking, making attitude control complex. Additionally, there are issues such as reflector occlusion or light leakage. The concentrator of OMEGA is a spherical structure. The spherical concentrator is constructed from the semi-transparent and semi-reflecting film. Solar light passes through the film to reach the interior of the concentrator, while the film also reflects the sunlight onto the photovoltaic array subsystem [8,11]. This design avoids complex attitude control, but it imposes special requirements on the film, and the reflectivity of the film is relatively low. Fan et al. [12] proposed a secondary-reflection OMEGA concentrator, which can achieve better concentration performance and a more uniform energy density distribution of the focal spot. Both SSPS-ALPHA and SSPS-OMEGA employ films as the material for their concentrator components. SSPS-OMEGA has been enhanced through the incorporation of ε-near-zero (ENZ) metamaterial, as detailed in reference [13]. Films as concentrator materials offer a number of advantages, including light weight, low cost, and flexible deployment. However, they also present a number of drawbacks, including low reflectivity, poor durability, and difficulty in maintaining surface shape. Li et al. [14] used a Fresnel lens array as the concentrator for SSPS and transmitted the condensed sunlight to the photovoltaic panel via a bundle of optical fibers. This configuration improves the uniformity of the energy density distribution of the focal spot. However, the transmissive optical systems have higher energy losses, and the integration of Fresnel lenses on a large scale presents certain challenges. Compared to transmissive lenses, metal-coated mirrors as space concentrators offer a number of advantages. These include higher reflectivity, stronger reliability, stable surface shape, a broader temperature range, and a wider working spectrum. As a result, they hold greater promise for use in large-scale energy concentration systems.

Owing to the current limitations of rocket payload enclosures, SSPSs must be assembled in space. The concentrator of SSPS-ALPHA is connected and assembled by “Push-Me/Pull-You” robotic arms [10]. Liu et al. [15] designed a crank-slider mechanism for deploying the thin-film reflective surface of the SSPS-OMEGA concentrator. Technologies such as segmented optical systems and mirror folding and unfolding, have already been successfully applied in the James Webb Space Telescope (JWST) [16,17]. Consequently, the technology for deploying segmented mirrors in orbit has become an essential solution to address the challenges of manufacturing and launching large-diameter mirrors. Furthermore, research on deployable mechanisms with high positioning accuracy and stability represents a crucial step toward advancing this technology.

Due to manufacturing and assembly errors, clearance in revolute joints and link length errors are the most important types of uncertainties in deployable mechanism systems. Therefore, joint clearances and link length errors are also classified as random geometric errors. These uncertainties significantly affect the motion accuracy, reliability, and dynamic performance of the mechanism. The main modeling approaches for rotational joints with clearance include the massless link approach, spring-damper approach, and momentum exchange approach, etc. [18,19]. Erkaya et al. [20,21] treated clearance joints as massless links and investigated their influence on the trajectory and transmission angle of mechanism. Chen et al. [22], using screw theory, equivalently modeled hinge clearance and link length errors as virtual screws and analyzed the accuracy characteristics of the mechanism during its motion. Ren et al. [23] selected continuous contact force models and an improved Coulomb friction model to calculate the normal collision force and tangential friction force under contact conditions. Most of the literature primarily focuses on the radial clearance in rotational joints. However, in practical rotational joints, both radial and axial clearances exist simultaneously, which can cause planar mechanisms to deviate from ideal planar motion, resulting in more complex spatial motion. Brutti et al. [24] introduced a nonlinear contact force model that considers contact elastic deformation, with springs and damping elements placed between the shaft and the hole, and modeled rotational joints with three-dimensional clearances. Yan et al. [25] considered the axial clearance in rotational joints and established a comprehensive three-dimensional model for rotational joints with both radial and axial clearances, using a contact force model to reflect different contact collision situations. Bai et al. [26,27] analyzed the impact of rotational joints with three-dimensional clearance on the dynamic characteristics of satellite antenna systems, based on the established normal and tangential contact force models. Among the three modeling approaches for rotational joints with clearance, the massless link model is notable for its simplicity, ease of computation, and suitability for error and accuracy analysis in the design of low-speed deployable mechanisms. In the work of Li et al. [28], the solution to the dynamic equations of a spatial deployable mechanism with rotational joint errors was presented using the Monte Carlo method. Ding et al. [29] developed a spatial error model for a deployable mechanism that considers rotational joint clearance and link length errors. They then employed discrete sampling points to numerically calculate the error space. Liu et al. [30] employed four sensitivity analysis methods to investigate the sensitivity of the kinematic and dynamic performance of a crank-slider mechanism to clearance and link length errors. The Sobol index is regarded as the most appropriate sensitivity metric for mechanical systems, assisting researchers in identifying the principal influences of each input variable on the output response.

Existing film concentrators face issues such as low reflectivity, poor durability, and difficulty in maintaining surface shape. At the same time, the space deployment of solar concentrators is still in the planning stage. To improve the concentration efficiency and concentration ratio of solar concentrators and to achieve a higher-quality output spot, a new concept of DSSC is proposed, which adopts a segmented mirror deployment design based on an afocal reflective system. The afocal reflective system offers higher reflectivity, greater transmission efficiency, and a wider operational spectrum. Additionally, to facilitate the in-orbit deployment of the large-scale afocal reflective system, a deployment mechanism for the segmented primary mirror has been designed, along with an error analysis. The comprehensive error analysis of the deployment mechanism provides theoretical support for the space deployment of solar concentrators.

This paper is organized as follows. Section 2 introduces the DSSC, which consists of an afocal reflective optical system with segmented primary mirrors and secondary mirror and a deployable mechanism for the segmented primary mirrors. Section 3 develops a probability model of the comprehensive error of the mechanism with joint clearances, together with a sensitivity analysis model based on the Sobol method. Section 4 establishes a prototype model and conducted numerical simulations. Section 5 tests the performance indicators of the prototype.

2. DSSC and Deployable Mechanism Design

2.1. Research Method

The purpose of the DSSC is to concentrate solar energy from an infinite distance and output it at a given energy density. According to Equation (1), the square of the ratio of the collection aperture of the optical system to the spot diameter at the target location should be greater than the ratio of the energy density at the target location to the energy density received by the concentrator.

$${\displaystyle \frac{D_{\mathrm{in}}}{D_{\mathrm{out}}}}\eta =\sqrt{{\displaystyle \frac{S_{\mathrm{in}}}{S_{\mathrm{out}}}}}\eta =\sqrt{{\displaystyle \frac{W_{\mathrm{out}}}{W_{\mathrm{in}}}}}$$

(1)

where D_in is the aperture of the DSSC (m), D_out is the spot diameter at the target location (m), S_in is the collection area of the concentrator (m²), S_out is the spot area at the target location (m²), W_in is the energy density received by the concentrator, which is the solar constant over the entire spectral range, approximately 1367 W/m², W_out is the energy density of the spot (W/m²), and η is the energy concentration efficiency of the optical system.

To better evaluate the ability to enhance the optical energy density per unit area, the concentrator performance is measured using the concentration ratio indicator. The optical concentrator ratio is defined as the ratio of the energy density of the spot W_out to the energy density received by the concentrator W_in. The geometric concentrator ratio is defined as the ratio of the area collected by the concentrator S_in to the spot area at the target location S_out. The optical concentrator ratio and the geometric concentrator ratio can be expressed as [31]:

$$C_{\mathrm{opt}}={\displaystyle \frac{W_{\mathrm{out}}}{W_{\mathrm{in}}}},C_{\mathrm{geo}}={\displaystyle \frac{S_{\mathrm{in}}}{S_{\mathrm{out}}}}$$

(2)

2.2. Concept Design

In comparison to transmissive lenses, reflective optical systems present a number of advantages, including ease of manufacturing, potential for lightweight design, and broad temperature adaptability. Moreover, metallic reflective coatings offer high reflectivity across a broad spectral range. In consideration of the comprehensive factors, including the reflectivity and effective aperture utilization of the optical system, with a certain design margin, the energy collection aperture of the optical system is set at 6 m. The rays converging at the focal point of the parabolic mirror become parallel after reflection. Based on the above principle, an afocal reflective system consisting of a parabolic primary mirror and a secondary mirror is constructed, as shown in Figure 1a. Parallel rays incident on the parabolic primary mirror are reflected to a point that lies on the focal plane of the secondary mirror. The secondary mirror is positioned such that its focal plane coincides with this intermediate image plane. Upon reflection from the secondary mirror, the rays are re-collimated and propagate parallel to the optical axis. This afocal configuration ensures that the output beam remains collimated with a controlled and stable beam diameter. This afocal reflective system can concentrate solar energy and transmit it over a distance of L. In order to satisfy the requisite specifications for launch, it is essential that the afocal reflective system be capable of segmented folding and on-orbit deployment. Based on constraints such as fill factor, volume, and reliability for the segmentation and folding method of the primary mirror, the mirror is divided into six hexagonal segments. The DSSC system is shown in Figure 1b.

The six segmented mirrors fold in both directions along the optical axis. The folded and deployed states of the primary mirror are illustrated in Figure 2. An actively driven crank-linkage mechanism is used as the deployable mechanism. The hinges A, A’ and hinges D, D’ are fixed to the central flange. The segmented mirror support plates 1 and 2 are connected to the central flange through hinges D’ and D’, respectively. The deployable mechanism is actuated by motors installed at hinges A and A’, which provide angular displacement, enabling the unfolding action to be completed under the push of links L₁, L₂ and links L₁’, L₂’.

3. Deployable Mechanism Error Modeling and Sensitivity Analysis

3.1. Error Modeling of Revolute Joints with Clearance

It is inevitable that machining errors will result in a clearance between the bearing and the journal. The deployment action of the mechanism is a low-speed, uniform motion with minimal impact during the process. Ignoring the impact, it is assumed that the bearing and the journal remain in continuous contact. Therefore, an “massless link” equivalent model is used to simulate the revolute pair with a clearance, as shown in Figure 3. The “massless link” represents the line connecting the centers of the bearing and the journal (mm), with its length denoted as r_i (the subscript i represents A, B, C, D).

$$r_i=r_{bi}-r_{ji}$$

(3)

where r_bi and r_ji represent the bearing radius and journal radius of revolute joint i (mm).

As shown in Figure 3, the axial clearance d_i (mm) can be defined as:

$$d_i=(d_b-d_j)/2$$

(4)

where d_b and d_j represent the axial length of the bearing and the axial length of the journal of revolute joint i (mm).

It is generally assumed that machining errors within tolerance limits follow a normal distribution, i.e., r_i ~ N(μ_ri, σ_ri) and d_i ~ N(μ_di, σ_di). According to the 3σ criterion, the mean μ_ri and standard deviation σ_ri of r_i, as well as the mean μ_di and standard deviation σ_di of d_i are calculated.

$${\mu }_{ri}={\displaystyle \frac{1}{2}}\times (r_{i\max }+r_{i\min }), {\sigma }_{ri}={\displaystyle \frac{1}{3}}\times (r_{i\max }-{\mu }_{ri})$$

(5)

$${\mu }_{di}={\displaystyle \frac{1}{2}}\times (d_{i\max }+d_{i\min }), {\sigma }_{di}={\displaystyle \frac{1}{3}}\times (d_{i\max }-{\mu }_{di})$$

(6)

In the formula, r_imax and r_imin represent the maximum and minimum radial clearances between the bearing and the journal (mm), respectively, while d_imax and d_imin represent the maximum and minimum axial clearances between the bearing and the journal (mm), respectively.

In actual mechanisms, the simultaneous presence of radial and axial clearances in the revolute joint allows for the journal to tilt and shift axially relative to the bearing axis. This results in the planar mechanism deviating from ideal planar motion, thereby generating a more complex spatial motion. To describe the spatial positions of the journal and the bearing, it is necessary to regard them as rigid bodies. Local coordinate systems O_ji-x_jiy_jiz_ji and O_bi-x_biy_biz_bi, where z_ji and z_bi are aligned with the axes of the journal and the bearing, respectively, as shown in Figure 4.

The pose of the journal can be represented by a six-dimensional vector [x, y, z, α, β, γ]^T, where the first three components [x, y, z]^T represent the positional displacement of the journal’s coordinate system along the bearing’s coordinate system (mm), and [α, β, γ]^T represent the orientation change of the journal relative to the bearing (deg).

It is generally assumed that the probability of the clearance error vector in the kinematic pair having its endpoint at any point within or on the boundary of the clearance space is equal. This implies that the angular displacements α, β, and γ can be regarded as uniformly distributed random variables. Based on the geometric relationship between the journal and the bearing, the six-dimensional vector [x, y, z, α, β, γ]^T can be expressed as

$$\left\{\begin{array}{l}{x}_i=r_i-{\displaystyle \frac{d_i}{2}}\sin {\beta }_i, y_i=r_i-{\displaystyle \frac{d_i}{2}}\sin {\alpha }_i, z_i={\displaystyle \frac{d_i}{2}}\\ {\alpha }_i\sim U(-{\tan }^{-1}(r_i/{\displaystyle \frac{d_b}{2}}),{\tan }^{-1}(r_i/{\displaystyle \frac{d_b}{2}}))\\ {\beta }_i\sim U(-{\tan }^{-1}(r_i/{\displaystyle \frac{d_b}{2}}),{\tan }^{-1}(r_i/{\displaystyle \frac{d_b}{2}}))\\ {\gamma }_i\sim U(-\pi ,\pi )\end{array}\right.$$

(7)

3.2. Comprehensive Error Modeling of Deployable Mechanism

The transfer matrix ${}_{O_{ij}}{}^{O_{ib}}\mathit{T}$ representing the pose change of the shaft neck coordinate system O_ji-x_jiy_jiz_ji relative to the hole body coordinate system O_bi-x_biy_biz_bi is expressed as:

$${}_{O_{ji}}{}^{O_{bi}}\mathit{T}=\left[\begin{array}{cccc}c\gamma c\beta & c\gamma s\beta s\alpha -s\gamma c\alpha & c\gamma s\beta c\alpha +s\gamma s\alpha & x\\ s\gamma c\beta & s\gamma s\beta s\alpha +c\gamma c\alpha & s\gamma s\beta c\alpha -c\gamma s\alpha & y\\ -s\beta & c\beta s\alpha & c\beta c\alpha & z\\ 0& 0& 0& 1\end{array}\right]$$

(8)

where c is the abbreviation of cos and s is the abbreviation of sin.

The pose errors of the end marker point P of the deployable mechanism is determined by using the transfer matrix to express the kinematic loop. A fixed coordinate system B-x_By_Bz_B and the marker point coordinate system P-x_Py_Pz_P are established, as shown in Figure 5. From the perspective of the kinematic loop, the deployable mechanism includes two transfer paths. The first transfer path goes from the fixed coordinate system B-x_By_Bz_B through the kinematic pair D to the marker point coordinate system P-x_Py_Pz_P. The second transfer path goes from the fixed coordinate system B-x_By_Bz_B through the kinematic pairs A, B, and C to the marker point coordinate system P-x_Py_Pz_P.

In the absence of machining errors, the positions of the axes and holes in each kinematic pair coincide with their ideal positions. The actual position of the marker point coincides with the ideal position. Based on the kinematic loop formed by the two transfer paths, the loop constraint equation can be described as [32,33]:

$$\left({}_{O_D}{}^{B}T{}_P{}^{O_D}T\right)P_{\mathrm{actual}}=\left({}_{O_A}{}^{B}T{}_{O_B}{}^{O_A}T{}_{O_C}{}^{O_B}T{}_P{}^{O_C}T\right)P_{\mathrm{ideal}}$$

(9)

where P_actual = [P_ax,P_ay,P_az,1]^T represents the actual position coordinates of the marker point, and P_ideal = [P_ix,P_iy,P_iz,1]^T represents the ideal position coordinates of the marker point. The pose of the base coordinate system B-x_By_Bz_B is defined as a six-dimensional vector [0,0,0,0,0,0]^T, with $B=I_{4\times 4}$, and the pose matrix of the marker point coordinate system P-x_Py_Pz_P, can be computed.

$$P_{\mathrm{ideal}}={\left({}_{O_A}{}^{B}T{}_{O_B}{}^{O_A}T{}_{O_C}{}^{O_B}T{}_P{}^{O_C}T\right)}^{-1}B$$

(10)

Under ideal conditions, the position coordinates of the marker point can be obtained from Equation (11).

$$P_{\mathrm{actual}}={\left({}_{O_D}{}^{B}T{}_P{}^{O_D}T\right)}^{-1}\left({}_{O_A}{}^{B}T{}_{O_B}{}^{O_A}T{}_{O_C}{}^{O_B}T{}_P{}^{O_C}T\right)P_{\mathrm{ideal}}$$

(11)

After considering the error terms in the clearance kinematic pairs, the constraint equation, including the transfer matrix of the journal and bearing pose errors can be described as:

$$\left({}_{O_{bD}}{}^{B}T{}_{O_{jD}}{}^{O_{bD}}T{}_P{}^{O_{jD}}T\right)P_{\mathrm{actual}}=\left({}_{O_{bA}}{}^{B}T{}_{O_{jA}}{}^{O_{bA}}T{}_{O_{bB}}{}^{O_{jA}}T{}_{O_{jB}}{}^{O_{bB}}T{}_{O_{bC}}{}^{O_{jB}}T{}_{O_{jC}}{}^{O_{bC}}T{}_P{}^{O_{jC}}T\right)P_{\mathrm{ideal}}$$

(12)

The actual pose matrix P_actual of the marker point coordinate system P-x_Py_Pz_P can be described as:

$$P_{\mathrm{actual}}={\left({}_{O_{bD}}{}^{B}T{}_{O_{jD}}{}^{O_{bD}}T{}_P{}^{O_{jD}}T\right)}^{-1}\left({}_{O_{bA}}{}^{B}T{}_{O_{jA}}{}^{O_{bA}}T{}_{O_{bB}}{}^{O_{jA}}T{}_{O_{jB}}{}^{O_{bB}}T{}_{O_{bC}}{}^{O_{jB}}T{}_{O_{jC}}{}^{O_{bC}}T{}_P{}^{O_{jC}}T\right)P_{\mathrm{ideal}}$$

(13)

The pose errors model of the end marker point of the deployable mechanism can be described as:

$$E=P_{\mathrm{actual}}-P_{\mathrm{ideal}}=\left[\begin{array}{cccc}0& -{\mathsf{\epsilon }}_z& {\mathsf{\epsilon }}_y& {\mathsf{\delta }}_x\\ {\mathsf{\epsilon }}_z& 0& -{\mathsf{\epsilon }}_x& {\mathsf{\delta }}_y\\ -{\mathsf{\epsilon }}_y& {\mathsf{\epsilon }}_x& 0& {\mathsf{\delta }}_z\\ 0& 0& 0& 0\end{array}\right]$$

(14)

where δ_x, δ_y, and δ_z are the position error components of the marker point in the x, y, and z directions, respectively, and ε_x, ε_y, and ε_z are the attitude error components of the marker point around the x, y, and z axes, respectively.

3.3. Sensitivity Analysis Based on the Sobol Method

Based on the geometric errors model, sensitivity analysis is used to quantify the contribution of each error component to the comprehensive error, thereby identifying the error components with a dominant influence.

The Sobol method is a widely used sensitivity analysis approach that utilizes variance to describe the uncertainty between input variables and model output responses, while also considering interactions among input variables [34,35]. This paper employs a Monte Carlo-based Sobol sensitivity analysis method to elucidate the influence of significant geometric error components within the geometric error model on the comprehensive error of the deployable mechanism. According to the methodology, the influence of each input variable on the model output is quantified by the ratio of partial variance to total variance, which is referred to as the sensitivity coefficient. Sensitivity coefficients are typically classified into two categories based on disparate evaluation criteria: first order sensitivity coefficient and total sensitivity coefficient. The first-order sensitivity coefficient directly reflects the contribution of each individual input parameter to the model output. In contrast, the total sensitivity coefficient is considered to be the sum of the first-order and higher-order sensitivity coefficients of each input parameter [33]. This accounts for the coupling effect of different input parameters on the model output. In accordance with the definition provided in the literature, the geometric errors model is defined as a function f(x).

$$f(x)=f(X_1,X_2,\dots ,X_{\mathrm{n}})$$

(15)

where X_j (j = 1, 2, …, n, n = 10) represents the geometric error components.

The first-order sensitivity coefficient S_j and the total sensitivity coefficient S_Tj can be expressed as:

$$S_j={\displaystyle \frac{V_j}{V(f)}}={\displaystyle \frac{V_{X_j}\left(E_{X_{~j}}(f\left|X_j\right.)\right)}{V(f)}}$$

(16)

$$S_{Tj}={\displaystyle \frac{E_{X_{~j}}\left(V_{X_j}(f\left|X_{~j}\right.)\right)}{V(f)}}$$

(17)

The Monte Carlo algorithm was used to calculate the partial variance of each geometric error component and the total variance of all geometric error terms. Therefore, the expressions for the expectation and variance in Equations (16) and (17) can be written as

$$V_{X_j}\left(E_{X_{~j}}(f\left|X_j\right.)\right)\approx {\displaystyle \frac{1}{N}}{\displaystyle \sum _{l=1}^{N}f{(B)}_l}(f{(A_B^{(j)})}_l-f{(A)}_l)$$

(18)

$$E_{X_{~j}}\left(V_{X_j}(f\left|X_{~j}\right.)\right)\approx {\displaystyle \frac{1}{2N}}{\displaystyle \sum _{l=1}^N{(f{(A)}_l-f{(A_B^{(j)})}_l)}^2}$$

(19)

$$V(f)\approx {\displaystyle \frac{1}{N}}{\displaystyle \sum _{l=1}^N{(f{(A)}_l)}^2-}{\displaystyle \frac{1}{N}}({\displaystyle \sum _{l=1}^{N}f{(A)}_l{)}^2}$$

(20)

where A, B, and $A_B^{(j)}$ are the sample matrices required for the Monte Carlo algorithm, with matrix sizes of $N\times j$, where N is the number of samples and j is the number of geometric error components. The mixed matrix $A_B^{(j)}$ is formed by replacing the j-th column of matrix A with the j-th column of matrix B, resulting in $N\times (j+2)$ sets of input data.

4. Scaled-Down Model Prototype Simulation Analysis

To further investigate the performance of the DSSC and verify the accuracy of the optical system analysis and deployable mechanism error analysis, it is necessary to conduct a scaled-down design and manufacture a prototype. The prototype selected for investigation was a single segmented mirror. Given that the contour of a single segmented mirror has a relatively minor effect on the energy density of the light spot, a circular mirror, which is more readily fabricable, was selected.

4.1. Parameter Definition

A 12:1 scaled-down design was applied to the DSSC, and the optical system parameters of the prototype listed in

Table 1. Optical system parameters of the prototype.

Mirror	Radius of Curvature (mm)	Conic Coefficient	Half Aperture (mm)	Spacing (mm)
Segmented mirror	−800	−1	75	−333.33
Secondary mirror	−133.33	−1	45

were applied. It should be noted that the x-axis of optical system is defined along the direction of the emitted light.

According to the coordinate definition of the optical system, a negative radius of curvature indicates that the curvature center of the mirror surface is located in the negative direction of the optical axis. Therefore, the primary mirror is a concave mirror, and the secondary mirror is a convex mirror. Parallel light incident on the parabolic primary mirror is reflected to form converging light. After divergence by the convex secondary mirror, the rays become parallel again.

In accordance with the specified shaft-hole fit tolerances of the deployable mechanism, the distribution parameters for each geometric error component, which follows a normal distribution, are presented in

Table 2. Distribution parameters of geometric error components.

Error Component	Range (µm)	Mean (µm)	Standard Deviation (µm)
Radial clearance rA of revolute joint A	[2.5, 37]	19.75	5.75
Radial clearance rB of revolute joint B	[1.5, 32]	16.75	5.08
Radial clearance rC of revolute joint C	[1.5, 32]	16.75	5.08
Radial clearance rD of revolute joint D	[8, 22.5]	15.25	2.42
Axial clearance dA of revolute joint A	[−60, 60]	0	20
Axial clearance dB of revolute joint B	[−60, 60]	0	20
Axial clearance dC of revolute joint C	[−60, 60]	0	20
Axial clearance dD of revolute joint D	[−60, 60]	0	20
Length error ΔL1 of link L1	[−50, 50]	0	16.67
Length error ΔL2 of link L2	[−20, 20]	0	6.67

.

4.2. Afocal Reflective System Simulation

The prototype system simulates and validates the scenario where only a single sub-aperture primary mirror is introduced into the optical path. The optical path diagram and the footprint diagram of the image plane for the afocal reflective system are shown in Figure 6. The diameter of collimated beam from the afocal reflective system achieves a compression ratio of 6:1 relative to the primary mirror, with an area ratio of 36:1 between the two. The simulation effectively demonstrates that a single segmented mirror can converge the parallel incident light, ultimately producing a spot on the image plane that meets the design specifications for spot diameter and exhibits a high energy density, providing a foundation for the construction of the full DSSC.

In this design, an afocal reflective system is employed for the collection of solar energy, which is then emitted as a parallel beam. The quality of the emitted light spot is evaluated based on the parallelism of the outgoing rays, with the angular radius introduced as a defining parameter.

$$\delta =\arctan ({\displaystyle \frac{D_{\mathrm{out}}-D_{\mathrm{in}}/C_{\mathrm{geo}}}{2L}})$$

(21)

The angular radius of the prototype has been established at 1°. The specified tolerance range is shown in

Table 3. Tolerances of the prototype optical system.

Mirror	Translation Error Along the y-Axis (µm)	Translation Error Along the z-Axis (µm)	Tip (′)	Tilt (′)	Piston (mm)
Segmented mirror	±50	±50	±5	±5	±5
Secondary mirror	±50	±50	±5	±5

. In accordance with the aforementioned tolerance range, 200 Monte Carlo iterations were conducted. The adjustment range of compensator is set to ±1 mm for the primary–secondary mirror spacing, ±1 mm for the x and y eccentricities of secondary mirror, and ±1° for the x and y tilts of secondary mirror, with a wavelength of λ = 632.8 nm. In the results, over 90% meet the requirement of an angular radius less than 1°. The diameter of a single light spot at 1 m from the prototype’s emitted beam is 25 mm. For the emitted beam of segmented primary mirror, the radius of the light spot at a distance of 1 m is 37.5 ± 15 mm.

The diameter of the spot at the emission distance L, influenced by the divergence angle of the sunlight, can be expressed as:

$${D_{\mathrm{out}}}^{\prime }=D_{\mathrm{out}}+2L\tan ({\displaystyle \frac{\varpi \kappa }{2}})$$

(22)

where the divergence angle of sunlight $\kappa$ is 32′. As the sunlight passes through the optical system, while the energy converges, and the divergence angle also amplifies, with an amplification factor $\varpi =D_{\mathrm{in}}/D_{\mathrm{out}}$. At the same time, the diameter of the exit spot is also influenced by the optical system’s aberrations and the emission distance. The diameter of the outgoing beam increases as the propagation distance increases. The simulation shows that at a distance of 300 mm, the diameter of the spot is 44.14 mm. Based on the given emission distance L, we can fine-tune the axial position of the secondary mirror to focus the system, appropriately converging the beam and reducing the spot size to the required dimensions.

4.3. Solution of Comprehensive Error and Sensitivity Analysis of the Deployable Mechanism

As shown in Figure 7, an error transmission model was established for the deployable mechanism with clearance in the revolute joints. In accordance with the distribution parameters of the geometric error components as delineated in Table 2, the sample size N for each geometric component is set to 10,000. Following the sampling definitions of the Sobol method, a total of 120,000 sets of input data were generated.

Based on Equation (14), solve for the pose error components of the end marker point in each direction, and record the number of failures n. The optical system allows for an error range as shown in Table 2, with values exceeding the allowed error range being considered in the failure domain F. The failure probability estimate P_f is given by the ratio of the number of sample points n falling within the failure domain F to the total number of sample points N. The motion accuracy reliability P_r of the mechanism can thus be expressed as:

$$P_r=1-P_f=1-n/N$$

(23)

The distribution of comprehensive error and reliability of motion accuracy, as shown in Figure 8. With regard to the position error of the end marker, it can be observed that the average error along the x-direction is the most significant. The standard deviation of the error is the greatest along the z-direction, followed by the x-direction. This indicates that the geometric errors of the deployable mechanism cause the largest pose error in the x-direction, and it also suggests that the motion accuracy reliability in the x-direction is the lowest. On the other hand, the reliability of motion accuracy in the y-direction is the highest. The rotational error of the end marker exhibits similar levels of error in all directions. The smallest orientation error occurs in the z-direction. It is worth noting that the optical system tolerance range of the prototype is relatively large, and the rotational errors are all within the tolerance range. However, the pose errors introduced by the geometric errors should not be ignored.

To identify the influence of various random geometric errors on the pose error of the deployable mechanism, the sensitivity values of these errors to the pose error are calculated and normalized. Figure 9 illustrates the first-order sensitivity coefficients of pose errors along the x, y, and z directions. On the one hand, with regard to end-effector position errors, it can be observed that the influence of ΔL₁ and ΔL₂ is more pronounced in the x and y directions. In contrast, joint clearance, particularly axial clearance, is found to exert a greater influence on z-direction position errors compared to link length errors. On the other hand, the attitude error of the deployable mechanism is influenced by both axial clearance and radial clearance in joints. In the x-direction, radial clearance exerts a greater influence on the end-effector attitude error than axial clearance. In the y-direction, d_A and r_D are the most sensitive error components. In the z-direction, ΔL₁ is the most sensitive error component, while other error components exert a relatively uniform influence. This indicates that random geometric errors have a considerable impact on the attitude error of the deployable mechanism. It is noteworthy that there is a considerable discrepancy between the first-order sensitivity coefficients and the global sensitivity coefficients of the various geometric errors, which serves to illustrate the existence of a strong coupling effect among the geometric errors, as illustrated in Figure 10.

5. Prototype Fabrication and Experimental Investigation

5.1. Experimental System

To further study the performance of the DSSC and verify the design specifications, it is necessary to test the repeatability of the deployable mechanism and the concentration ratio of the optical system. Figure 11 illustrates the prototype system of the DSSC and the coordinate definitions. The deployable mechanism is controlled by the controller and electrical box and unfolds in accordance with a pre-determined action sequence. The deployment process is repeated multiple times, and the pose of the deployed segmented mirrors is determined using a 6-axis absolute arm measurement device and reference points. Subsequently, the secondary mirror is adjusted through a pose adjustment mechanism to correct the deployment errors of the segmented mirrors. Thereafter, the concentration ratio of DSSC is tested on the calibrated system. The experimental procedure is illustrated in Figure 12.

5.2. Positioning Repeatability Evaluation

Repeatability of motion refers to the deployable mechanism’s ability to reach the same position consistently, and it can be evaluated based on the error observed when repeatedly moving to the same position. The repeatability test is performed by moving from the stowed state to the deployed state, then repeating this sequence several times. The repeatability test setup is shown in Figure 13a.

The base coordinate system B-x_By_Bz_B is defined by the identification point of primary mirror and the circular hole centers, with the x-axis set as the optical axis direction. The coordinates of the end marker point of the deployable mechanism are taken as the segmented mirror reference coordinate system. The center of the sub-mirror mounting surface is used as the origin of the segmented mirror reference coordinate system P-x_Py_Pz_P, with the same orientation as the base coordinate B-x_By_Bz_B, as shown in Figure 11. Both coordinate systems, B-x_By_Bz_B and P-x_Py_Pz_P, are established using a 6-axis absolute arm measuring machine. The deployable mechanism performs the deployment sequence 15 times, resulting in pose error data as illustrated in Figure 14. It is worth noting that the repeatability requirements for the optical system are ±50 µm and ±5′. Figure 14 shows that the deployable mechanism achieves a repeatability of ±50 µm and ±1.2′, which is considered excellent performance, meeting the repeatability requirements for the deployable mechanism. These results indicate that the mechanism is suitable for the deployment of segmented mirrors.

5.3. Concentrating Ratio Evaluation

The concentration ratio is a key indicator of the performance of a solar concentrator system; the higher the concentration ratio, the better the system’s energy-focusing capability. As shown in Figure 13b, a solar radiometer and a spot measurement board are used at different distances to measure the energy density and spot diameter of the emitted beam.

The incident light is introduced into the DSSC via a collimator. Three distinct energy densities of incident light were employed, and the diameter and energy density of the emitted beam were evaluated at distances of 0.3 m and 1 m. By comparing the energy densities of the emitted and incident beams, the optical concentration ratio was obtained. By comparing the spot diameters of the emitted and incident beams, the geometric concentration ratio was derived. The experimental results are shown in Figure 15. At distances of 0.3 m and 1 m, the diameter of the emitted beam was 25–28 mm. The geometric concentration ratio reached 5.36–6, and the optical concentration ratio reached 24.7–32.2. The results indicate that the spot diameter aligns with the simulation, and the emitted beam diameter and geometric concentration ratio meet the design requirements. This confirms that the primary and secondary mirror pose errors are within the allowable tolerances. With consideration for a concentration efficiency of 82.12–92.45%, the optical and geometric concentration ratios exhibit a squared relationship, in accordance with the design specifications. It is worth noting that the non-ideal parallel nature of the incident beam in a laboratory setting may have negative effects on the concentration ratio and efficiency measurements.

6. Discussion

A prototype equipped with a single segmented mirror was used as the research subject to verify that deployment mechanism error modeling and sensitivity analysis can significantly aid in ensuring that the DSSC meets the optical system’s tolerance requirements. Furthermore, when light of approximately parallel incidence passes through the deployed primary mirror and the calibrated secondary mirror, the experimentally obtained output spot diameter meets the design specifications. The complete DSSC consists of six identical segmented mirrors, evenly distributed in a circular configuration and operating independently. It can thus be concluded that the complete output of the DSSC is constituted by the aggregate of the individual spot outputs produced by the six segmented mirrors. The experimental results provide a robust validation of the rationality of the prototype design, thereby substantiating the viability of the full DSSC system. It is worth noting that to achieve better concentration performance, the complete DSSC must be equipped with a larger aperture. Concurrently, the optical system will necessitate more stringent tolerance limits. Consequently, during the design and construction of the DSSC, greater precision in the control of the pose and surface accuracy of both the segmented mirrors and the secondary mirror is imperative.

The most advanced current SSPS concepts are SSPS-ALPHA and SSPS-OMEGA. SSPS-OMEGA features a spherical concentrator made of semi-transparent, semi-reflective films, which significantly enhances energy collection efficiency and allows effective operation across various angles and orientations. Nevertheless, the complexity inherent in the design and manufacture of such films, in conjunction with the inevitable occurrence of optical efficiency losses, constitutes a considerable challenge. Compared to fully reflective mirror-based concentrators, this design may suffer from slightly lower efficiency. Moreover, the fabrication and assembly of the spherical concentrator are technically demanding, as large-scale flexible films are difficult to maintain with high optical quality. Additionally, spherical designs may encounter issues related to heat accumulation. SPS-ALPHA, with its conical configuration, offers flexible and efficient solar energy collection along with good structural stability, making it well-suited for large-scale solar power collection and transmission. The system’s modular architecture provides flexibility but also introduces complexity in deployment due to the need for thousands of reflectors and deployment modules. Furthermore, mutual shading among reflectors is a potential drawback. In contrast, the DSSC proposed in this study uses segmented large-aperture reflective mirrors as primary collectors, which offer higher optical reflectivity and thus enhance energy collection performance. These mirrors also maintain high surface shape stability, exhibit strong resistance to degradation and environmental stress, and are capable of long-term stable operation in harsh space conditions. The afocal reflective system further improves the concentration ratio and yields higher-quality output light spots. For large-scale optical systems, in-orbit deployment is technically much less demanding than in-orbit assembly, making DSSC more favorable for space-based implementation. However, quality control remains a significant challenge for DSSC. Future work will focus on optimizing the structural and optical parameters of the DSSC to achieve higher energy density and uniformity of the output light spot.

It is also important to note that the divergence angle of sun light rays is an unavoidable factor that adversely affects the concentration ratio. Due to this effect, the area of the output light spot increases with the propagation distance. Similarly, in experiments, the light produced by point sources is not ideally collimated, and its divergence also contributes to an increase in spot size, affecting the geometric concentration ratio and resulting in reduced and uneven energy density of the output spot. Therefore, future work should focus on evaluating the relationship between working distance and concentration ratio, and on optimizing the optical system parameters to improve the quality of the output light spot.

7. Conclusions

To improve the energy concentration efficiency and optical concentrator ratio of SSPS, a deployable segmented solar concentrator with an afocal reflective design is proposed. The designed deployable mechanism can achieve bidirectional folding and unfolding of the segmented primary mirror, and the repeatability of the deployable mechanism meets the optical system tolerance requirements.

Then, a pose error analysis of the deployable mechanism was conducted, considering the 3D clearance of revolute joints and link length errors. Based on the “massless link” equivalent model, both axial and radial clearances of the kinematic pairs were included in the modeling. Local coordinate systems were established for the axes and holes of each kinematic pair, and a model for the comprehensive error of the mechanism, accounting for the clearance of the kinematic pairs, was developed using homogeneous transformation matrices. Subsequently, the Sobol method was applied for sensitivity analysis of the comprehensive error, identifying the sensitive geometric error components of the deployable mechanism. A simulation analysis of the scaled-down prototype was conducted, and the resulting spot diameter was calculated within the specified tolerance range of the optical system. Monte Carlo algorithms were employed to resolve the probability model of the pose error of deployable mechanism, in addition to the first-order and global sensitivity values.

Based on this, a prototype experimental system was built. The experimental results show that the DSSC geometric concentration ratio reaches 5.36 to 6, and the optical concentration ratio ranges from 24.7 to 32.2. The repeatability of the deployable mechanism is ±50 µm and ±1.2′. The results indicate that the designed afocal reflective segmented deployable solar concentrator is suitable for energy concentration and that the deployable mechanism is capable of performing the required deployment motion of the segmented mirror. The proposed error analysis method is an effective means of simulating the comprehensive error of the deployable mechanism. Furthermore, the sensitivity analysis results provide valuable insights for subsequent tolerance design and processing. This research has a positive impact on improving energy collection efficiency for SSPSs and enables earlier deployment in space.