A multi-objective optimization method for dish concentrator based on 3-rps tracking mechanism

Abstract

The present application relates to the technical field of solar power generation, and more particularly to a multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism, which comprises the following steps: constructing a unified kinematics model to realize the mapping of the pose workspace and the solar tracking time window; taking the circumscribed radius of the moving platform, the circumscribed radius of the fixed platform, the height of the concentrator center point, and the horizontal installation spacing as design variables, and constructing a four-objective optimization function including the workspace coverage target, the motion risk target, the motion transmission performance target, and the structural compactness target; using an adaptive PSO-NSGA-III hybrid optimization algorithm to solve the four-objective optimization function to obtain an optimal solution set; and selecting design variable parameters from the optimal solution set as the optimization design parameters of the dish concentrator. The present application can solve the problem that multi-objective optimization cannot be achieved in the prior art.

Description

Technical Field

[0001] This invention relates to the field of solar power generation technology, and in particular to a multi-objective optimization method for dish concentrators based on a 3-RPS tracking mechanism. Background Technology

[0002] Disk concentrators are core devices for efficient solar thermal utilization. Their tracking mechanisms need to have a large attitude adjustment range, excellent load-bearing capacity, and stable motion transmission performance to adapt to the sun's all-day trajectory and ensure concentrating efficiency. A dish concentrator includes an absorber and multiple reflectors. The reflectors reflect light back to the absorber, and each reflector is connected to its own tracking mechanism. The tracking mechanism adjusts the reflector's attitude in real time to improve energy harvesting efficiency. The 3-RPS parallel mechanism, with its simple topology, compact layout, and two rotational and one translational degrees of freedom, has become the preferred solution in the field of solar tracking. For example, Chinese invention patent ZL2025113711851 discloses a tracking mechanism that uses an I3-RPS tracking mechanism, which is also a 3-RPS parallel mechanism and has a large working space. The existing invention patent mainly addresses the control problem of the I3-RPS tracking mechanism. However, in practical applications, the dimensions of the dish concentrator in various aspects affect its performance in a coupled manner. Performance considerations, besides workspace, also require comprehensive consideration of other factors such as motion risk, motion transmission performance, and structural compactness. Only a dish concentrator manufactured after comprehensively considering multiple factors can simultaneously meet multi-dimensional usage requirements. Currently, the optimization design methods for dish concentrators only consider workspace factors and cannot achieve multi-objective optimization, thus failing to meet the requirements of long-term, high-reliability, and high-performance solar tracking. Therefore, there is an urgent need to develop a supporting systematic design and optimization method. Summary of the Invention

[0003] The purpose of this invention is to provide a multi-objective optimization method for a disc condenser based on a 3-RPS tracking mechanism, which aims to solve the problem that multi-objective optimization cannot be achieved in the prior art.

[0004] To achieve the above objectives, this invention provides a multi-objective optimization method for a dish condenser based on a 3-RPS tracking mechanism. The dish condenser includes multiple 3-RPS tracking mechanisms, each of which includes a fixed platform, a moving platform, three branches, and a reflector fixed on the moving platform. Three sets of triangularly distributed revolute joints are installed on the fixed platform, and three sets of triangularly distributed composite spherical joints are installed on the moving platform. Each branch includes a pair of corresponding revolute joints and a pair of composite spherical joints, as well as a telescopic rod connecting the pair of revolute joints and the pair of composite spherical joints. The multi-objective optimization method includes the following steps: S1, constructing a unified kinematic model. Model: Establish a multi-coordinate system, and using the transformation between coordinate systems and the solar direction vector calculation model, derive the inverse kinematic equations and motion constraint equations of the 3-RPS tracking mechanism under arbitrary triangular moving and fixed platforms. This achieves the mapping between the 3-RPS tracking mechanism's pose workspace and the solar tracking time window, and further obtains the velocity Jacobian matrix and acceleration Jacobian matrix. S2: Construct a four-objective optimization model: Using the outer radius of the moving platform, the outer radius of the fixed platform, the height of the concentrator center point, and the horizontal installation spacing as design variables, construct a model that includes workspace coverage objectives, motion risk objectives, and motion transmission objectives. The four-objective optimization function for performance and structural compactness is defined, and the feasible domain and motion constraints for each design variable are set. The horizontal installation spacing refers to the straight-line distance between the vertical projection point of the center point of the disc concentrator in the global coordinate system and the vertical projection point of the outer center of the fixed platform in the global coordinate system. The workspace coverage objective is to quantify the proportion of effective, non-interference solar attitudes to the total number of sampled attitudes based on the solar trajectories of typical days during the spring equinox, summer solstice, and winter solstice. The motion risk objective is to quantify the constraint utilization rate of the telescopic rod stroke, the rotation angle of the composite spherical sub-pair, and the rotation angle of the rotary sub-pair, minimizing the motion limit risk. The motion transmission performance objective is based on the speed Jacobian... The reciprocal of the condition number of the matrix is ​​used to maximize the isotropic motion; the structural compactness objective is to minimize the material consumption and volume occupied by the mechanism based on the weighted sum of the normalized design variables; S3, Hybrid algorithm solution: The adaptive PSO-NSGA-III hybrid optimization algorithm is used to solve the four-objective optimization function to obtain the Pareto optimal solution set; This hybrid optimization algorithm uses NSGA-III as a framework and introduces the PSO velocity-position update rule to improve the search efficiency; S4, Output optimized design parameters: Design variable parameters are selected from the Pareto optimal solution set as the optimized design parameters of the disc concentrator.

[0005] Further, in step S1, the method for deriving the inverse kinematic equations of the 3-RPS tracking mechanism is as follows: A multi-coordinate system is established, including a global coordinate system {G}, a fixed coordinate system {A}, a moving coordinate system {B}, a tangent plane coordinate system {D}, a concentrator coordinate system {C}, and three branched coordinate systems {Ei}; the transformation matrix from the branched coordinate system {Ei} to the global coordinate system {G} is obtained using the transformation relationships from the branched coordinate system {Ei} to the fixed coordinate system {A} and from the fixed coordinate system {A} to the global coordinate system {G}; the transformation matrix from the moving coordinate system {B} to the global coordinate system {G} is obtained using the transformation relationships from the moving coordinate system {B} to the tangent plane coordinate system {D}, from the tangent plane coordinate system {D} to the concentrator coordinate system {C}, and from the concentrator coordinate system {C} to the global coordinate system {G}; based on the solar direction vector calculation model and the coordinate system... By transforming the coordinate system, the coordinates of the circumcenters of the fixed and moving platforms in the global coordinate system {G} are obtained. Based on these coordinates, and combined with the geometric parameters of the fixed and moving platforms, the coordinates of the revolute joints and composite spherical joints in the global coordinate system {G} are obtained. Based on the coordinates of each branch's revolute joints and composite spherical joints in the global coordinate system {G}, the vectors, lengths, and unit vectors of each branch are obtained. The method for deriving the motion constraint equations of the 3-RPS tracking mechanism is as follows: based on the physical constraint that the unit direction vector of the revolute joint is perpendicular to the radial vector, the equation of the unit direction vector of the revolute joint is obtained. Based on the physical constraint that the unit direction vector of the revolute joint is perpendicular to the vector of the branch, the orthogonal condition equation is obtained. Solving this orthogonal condition equation yields the motion constraint equations of the 3-RPS tracking mechanism.

[0006] Further, in step S1, the method for obtaining the velocity Jacobian matrix is ​​as follows: the time derivative of the inverse kinematics equation is performed to obtain the velocity mapping relationship between the pose of the moving platform and the branch length, and the velocity Jacobian matrix is ​​obtained based on the velocity mapping relationship; the method for obtaining the acceleration Jacobian matrix is ​​as follows: the time derivative of the velocity mapping relationship is performed to obtain the acceleration mapping relationship between the pose of the moving platform and the branch length, and the acceleration Jacobian matrix is ​​obtained based on the acceleration mapping relationship.

[0007] Furthermore, workspace coverage targets The expression is:

[0008] ,

[0009] In the formula, N T =1,…,T, where T=3 represents three typical dates selected for workspace coverage assessment; This represents the number of effective, interference-free solar attitudes that the 3-RPS tracking mechanism can achieve within the Tth typical day, taking the spring equinox, summer solstice, and winter solstice as three typical days; This represents the total number of sampled solar positions within the T-th typical day; To track time; The solar altitude angle; This is the solar azimuth angle; For design variables.

[0010] Furthermore, sports risk targets The expression is:

[0011] ,

[0012] In the formula, N represents the total number of 3-RPS tracking mechanisms; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N; Let be the actual rotation angle of the composite spherical joint of the i-th branch; This is the maximum permissible rotation angle for the composite ball joint; Let be the actual rotation angle of the revolute joint of the i-th branch; This is the maximum permissible rotation angle of the revolute joint; Let be the actual length of the telescopic rod of the i-th branch; The average length of the telescopic pole. and These are the upper and lower limits of the telescopic pole's length, respectively. This refers to the length adjustment range of the telescopic rod; These are the weighting coefficients for the travel constraint of the telescopic rod, the rotation angle constraint of the composite spherical joint, and the rotation angle constraint of the revolute joint, respectively. To track time; The solar altitude angle; This is the solar azimuth angle.

[0013] Furthermore, motion transmission performance targets The expression is:

[0014] ,

[0015] In the formula, N represents the total number of 3-RPS tracking mechanisms; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N; Let be the vector Jacobian matrix of the nth 3-RPS tracking mechanism in the kth feasible pose; and These are the minimum and maximum singular values ​​of the matrix, respectively; The number of feasible sampling points for the nth 3-RPS tracking mechanism; the reciprocal of the condition number. Characterizes the degree of isotropy of motion.

[0016] Furthermore, the goal of structural compactness The expression is:

[0017] ,

[0018] In the formula, The outer radius of the moving platform; To determine the circumscribed radius of the platform; For horizontal installation spacing; The height of the center point of the concentrator; , , and Parameters , , and The initial design dimensions; , , and Parameters , , and The weighting coefficients are calculated using the analytic hierarchy process (AHP) based on pairwise comparisons of the engineering importance of each structural dimension to the overall material consumption and volume occupied by the machine.

[0019] Further, in step S2, the feasible regions of each design variable are as follows: when the aperture radius of the disc concentrator is R, the feasible region of the outer radius of the moving platform is [0.125R, 0.25R], the feasible region of the outer radius of the fixed platform is [0.09R, 0.18R], the feasible region of the height of the concentrator center point is [1.56R, 2.18R], and the feasible region of the horizontal installation spacing is [0.18R, 0.68R]. The motion constraints are the extension rod stroke constraint, the composite spherical joint rotation angle constraint, and the revolute joint rotation angle constraint.

[0020] Furthermore, in the adaptive PSO-NSGA-III hybrid optimization algorithm, a linearly decreasing inertia weight strategy is adopted to balance the global exploration and local exploitation capabilities, with the inertia weight value being [0.4, 0.9]. A nonlinear time-varying strategy is adopted to adjust the cognitive and social factors, with the adjustment range of both cognitive and social factors being [1, 2]. An adaptive adjustment strategy is adopted for the mutation probability, with the mutation probability value ranging from [0.1, 0.3].

[0021] Furthermore, in the adaptive PSO-NSGA-III hybrid optimization algorithm, the algorithm performance is evaluated by hypervolume, population diversity, and generation distance indicators.

[0022] This invention provides a multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism. It constructs a unified kinematic model adaptable to both arbitrary triangular moving and fixed platforms, achieving precise mapping between the 3-RPS tracking mechanism's pose workspace and the solar tracking time window. The proposed four-objective optimization function, encompassing workspace coverage, motion risk, motion transfer performance, and structural compactness, overcomes the shortcomings of existing methods that only consider workspace. An adaptive PSO-NSGA-III hybrid optimization algorithm is employed to solve the four-objective optimization function, balancing the distribution and convergence of the solution set, resulting in significantly higher search efficiency than standard algorithms. After optimization using this method, the tracking time of the dish concentrator can be effectively improved, motion risk reduced, and structural compactness enhanced, achieving a synergistic improvement in multi-objective performance. Attached Figure Description

[0023] Figure 1 This is a schematic diagram of the multi-coordinate system established in this invention;

[0024] Figure 2 This is a schematic diagram of the geometric structure of the moving platform;

[0025] Figure 3 This is a schematic diagram of the geometric structure of the fixed platform;

[0026] Figure 4 It is the curve of the evaluation index changing with the number of iterations τ;

[0027] Figure 5 These are curves showing how the adaptive parameters and mutation probability parameters of PSO change with the number of iterations;

[0028] Figure 6 It is a Pareto front plot;

[0029] Figure 7 It is a parallel coordinate graph of the distribution of the objective function value and the distribution of the design variables. Detailed Implementation

[0030] The embodiments of the present invention will be described in detail below.

[0031] This embodiment provides a multi-objective optimization method for a dish condenser based on a 3-RPS tracking mechanism. The dish condenser includes multiple 3-RPS tracking mechanisms. Each 3-RPS tracking mechanism includes a fixed platform, a moving platform, three branches, and a mirror fixed on the moving platform. Three sets of triangularly distributed revolute joints are installed on the fixed platform, and three sets of triangularly distributed composite spherical joints are installed on the moving platform. Each branch includes a pair of corresponding revolute joints and composite spherical joints, and a telescopic rod connecting the pair of revolute joints and composite spherical joints. The basic structure of the dish condenser and the 3-RPS tracking mechanism is described in Chinese Invention Patent ZL2025113711851. Since this is prior art, these basic structures are not detailed in this embodiment. The difference between this embodiment and the prior art is that in Chinese Invention Patent ZL2025113711851, the moving platform and the fixed platform are equilateral triangles, while in this embodiment, the moving platform and the fixed platform are arbitrary triangles. In this embodiment, the disc condenser is equipped with four 3-RPS tracking mechanisms, hereinafter N=4.

[0032] The multi-objective optimization method includes the following steps.

[0033] S1. Construct a unified kinematic model: Establish a multi-coordinate system, and with the help of the transformation between the coordinate systems and the solar direction vector calculation model, derive the inverse kinematic equations and motion constraint equations of the 3-RPS tracking mechanism under the arbitrary triangular moving platform and the arbitrary triangular fixed platform. Realize the mapping between the pose workspace of the 3-RPS tracking mechanism (referring to the set of all legal positions and angles that the moving platform can achieve) and the solar tracking time window, and further obtain the velocity Jacobian matrix and the acceleration Jacobian matrix.

[0034] In this step, this embodiment establishes a series of coordinate systems based on the geometric topological features of the dish concentrator, such as... Figure 1 As shown, the established coordinate system includes the global coordinate system {G} (GX). G Y G Z G The coordinate system is fixed to the ground at installation point G and coincides with the horizontal coordinate system, serving as the reference coordinate system for sun position tracking. The coordinate system {A} (AX) is defined. A Y A Z A ) is fixed to the outer center A of the fixed platform, and its Z A The axis is perpendicular to the plane of the fixed platform, X A The axis points to the center E1 of the revolute joint. Moving coordinate system {B} (BX) B Y B Z B Z is fixed to the outer center B of the moving platform. B The axis is perpendicular to the plane of the moving platform, X BThe axis points to the center S1 of the composite sphere sub-center. In Z... B Establish a tangent plane coordinate system {D} (DX) at the orthogonal projection point D of the axis and the condenser surface. D Y D Z D The coordinate axes of the condenser coordinate system {C} are parallel to the moving coordinate system {B}. C Y C Z C With the center point C of the condenser disc as the origin, Z C The axis coincides with the normal to the condenser surface. For the i-th branch, at the center E of the rotational joint... i Establish a branched coordinate system {Ei} (E i -X Ei Y Ei Z Ei The coordinate system is used to perform kinematic analysis of each branch. The definitions of each coordinate system are summarized in Table 1.

[0035] Table 1. Coordinate System Definition

[0036]

[0037] In this embodiment, the coordinate system transformation is transmitted along two independent kinematic chains, ultimately unified to the global coordinate system {G}: one is a branch kinematic chain ({Ei}→{A}→{G}), used to describe the attitude of the rotating joint and the position of the fixed platform relative to the global coordinate system {G}; the other is a moving platform kinematic chain ({B}→{D}→{C}→{G}), used to describe the attitude of the moving platform and the disc condenser of the 3-RPS tracking mechanism relative to the global coordinate system {G}.

[0038] Branched kinematic chain: To establish a kinematic chain in coordinate system {E} i The attitude transformation relationship from coordinate system {A} to coordinate system {E} is first determined by transforming coordinate system {E} into coordinate system {A}. i} around its Y Ei Rotate the axis clockwise Then circle around its Z Ei Rotate the axis counterclockwise ζ Ei Therefore, the rotation matrix can be obtained:

[0039] (1)

[0040] In the formula, For Z Ei Axis and Z A The angle between the axes; ζ Ei For Y A axis and Y Ei The included angle of the axis.

[0041] Rotate coordinate system {A} around Z G Rotate the axis counterclockwise We can obtain:

[0042] (2)

[0043] In the formula, For vectors With X G The included angle between axes; N represents the total number of 3-RPS tracking mechanisms, in this embodiment N=4; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N.

[0044] Combining equations (1) and (2), we obtain the coordinate system {E} i Rotation matrix from coordinate system {G} to coordinate system {G}:

[0045] (3)

[0046] Kinematic chain of moving platform: Since coordinate system {M} is parallel to coordinate system {D}, the rotation matrix from coordinate system {M} to {D} is the identity matrix:

[0047] (4)

[0048] Coordinate system {D} can be obtained by rotating coordinate system {C} twice. First, rotate coordinate system {D} around the Y-axis. D Rotate the axis counterclockwise by η N Then circle around Z D Rotate the axis counterclockwise ψ B Based on this, the rotation matrix from {D} to {C} is derived as follows:

[0049] (5)

[0050] in, (6)

[0051] In the formula, η N The normal vector of the concentrator The normal vector of the circumcenter B of the 3-RPS moving platform The included angle; Let be the unit normal vector at point B; is the unit normal vector of the condenser surface at its center point C; N represents the total number of 3-RPS tracking mechanisms, in this embodiment N=4; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N.

[0052] Because the disc concentrator cannot rotate around Z... G The physical constraints of axis rotation, combined with the sun's east-to-west motion, allow the disc concentrator to track the sun's trajectory almost entirely based on Z-axis rotation. G Y GIt is planar symmetric. Using ZYZ Euler angles to describe its attitude relative to coordinate system {G}, the homogeneous transformation matrix from coordinate system {C} to coordinate system {G} is:

[0053] (7)

[0054] In the formula, θ z1 θ y θ z2 For Euler angles.

[0055] By analyzing the position of the sun in coordinate system {G}, the Euler angles and the solar altitude angle can be derived. Azimuth The relationship between them is as follows:

[0056] (8)

[0057] in,

[0058] (9)

[0059] In the formula, L at The latitude is δ. s ω is the solar declination angle; s Solar hour angle (morning ω) s <0, noon ω s =0, afternoon ω s >0); sgn(ω) s ) is the corresponding sign function, and its value takes the following rules: ω s When <0, sgn(ω) s )=-1,ω s When ω = 0, sgn(ω) s )=0, ω s When sgn(ω) > 0, s =1.

[0060] Substituting equation (8) into equation (7), we get:

[0061] (10)

[0062] The rotation matrix from coordinate system {B} to coordinate system {G} is expressed as:

[0063] (11)

[0064] in,

[0065]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072]

[0073] .

[0074] Depend on Figure 1 It can be seen that the coordinates of the fixed platform circumcenter A in coordinate system {G}, the coordinates of the moving platform circumcenter B in coordinate system {D}, and the coordinates of the condenser center point C in coordinate system {G} are as follows:

[0075] (12)

[0076] (13)

[0077] (14)

[0078] In the formula, It is the horizontal distance between the origin of coordinate system {G} and the origin of coordinate system {A}, and also the straight-line distance between the vertical projection point of the center point of the disc condenser in the global coordinate system and the vertical projection point of the outer center of the fixed platform in the global coordinate system. This is the perpendicular distance between the origin of coordinate system {B} and the origin of coordinate system {D}. This is the perpendicular distance between the origin of coordinate system {G} and the origin of coordinate system {C}.

[0079] Point D lies on the tangent plane of the parabola of revolution, let its coordinates be... Then its coordinates satisfy the following equation of the paraboloid of revolution:

[0080] (15)

[0081] The coordinates of point D in coordinate system {C} can be expressed as:

[0082] (16)

[0083] In the formula, The focal length of the condenser; From point C to point D p The distance.

[0084] The coordinates of point B in coordinate system {G} can be expressed as:

[0085] (17)

[0086] The coordinates of point B can then be determined as follows:

[0087] (18)

[0088] (19)

[0089] (20)

[0090] in,

[0091] .

[0092] A schematic diagram of the geometric parameters of the moving platform is shown below. Figure 2 As shown, the geometric parameters of the fixed platform are illustrated in the diagram. Figure 3 In the picture, , , The moving platform triangles are respectively The lengths of the three sides, which are the vertices. , , The opposite side; , , These are fixed platform triangles. The lengths of the three sides, which are the vertices. , , The opposite side. , , They are respectively vertex , , The interior angle at that location. , , They are respectively vertex , , The interior angle at that location. For vectors Relative to X B The azimuth angle of the axis ranges from (0 to 360°). For vectors Relative to X A The azimuth angle of the axis ranges from (0, 360°); ζ Si For YB axis and Y Si The angle between the axes; ζ Ei For Y A axis and Y Ei The included angle of the axis.

[0093] triangle The geometric features of are fully described by the following set of equations, which define its circumradius ( ). ), side length ( , , ) and interior angles ( , , The relationship between )

[0094] (twenty one)

[0095] (twenty two)

[0096] circumference of the moving platform with interior angles , , The main design parameters are pre-defined, and the corresponding side lengths are... , , The result is obtained through the above equation. For triangles... Its geometric parameters (circumscribed radius) Side length , , and interior angles , , The parameters are defined in a similar manner and satisfy the same triangular relationship. Therefore, by substituting the corresponding fixed platform parameters into equations (21) and (22), the parameters of the fixed platform can be obtained through the same process.

[0097] The moving platform's outer center B points to Vectors of each vertex Relative to X B Azimuth angles of the axis The direction is determined as follows. Similarly, the circumcenter A of the platform is determined to point to... Vectors of each vertex Relative to X A Azimuth angles of the axis It is given by the following formula:

[0098] (twenty three)

[0099] (twenty four)

[0100] The coordinates of the composite spherical pair in coordinate system {B}:

[0101] (25)

[0102] The coordinates of the revolute joint in coordinate system {A}:

[0103] (26)

[0104] The coordinates of the composite spherical joint and the revolute joint in coordinate system {G} are as follows:

[0105] (27)

[0106] (28)

[0107] The length of the i-th branch is:

[0108] (29)

[0109] The unit vector of the i-th branch is:

[0110] (30)

[0111] Based on the above steps, the vector, length, and unit vector of each branch are obtained.

[0112] Since the axis of the revolute joint lies in the plane of the fixed platform and is parallel to the radial vector... Perpendicular, therefore the unit direction vector of the revolute joint. This can be determined by the orthogonality condition, namely:

[0113] (31)

[0114] In the formula, ,in This indicates that in coordinate system {G}, the direction from point A to its i-th rotational joint center E... i The vector.

[0115] Therefore, we can conclude that:

[0116] (32)

[0117] Since the i-th branch can only revolve around the axis Rotation, therefore branch vector Must with Maintaining perpendicularity, the following constraint equations can be obtained:

[0118] (33)

[0119] Solving equation (33), we obtain the following motion constraint equations for the 3-RPS tracking mechanism:

[0120] (34)

[0121] (35)

[0122] (36)

[0123] For the different triangular configurations of the moving platform and the fixed platform in the 3-RPS tracking mechanism, the corresponding constraint equations can be derived by substituting their respective interior angles into equations (34) to (36).

[0124] Based on the above steps, the motion constraint equations of the 3-RPS tracking mechanism are obtained.

[0125] The position Jacobian matrix of each 3-RPS mechanism can be expressed as:

[0126] (37)

[0127] In the formula, Let the distance from point B to point S be represented in coordinate system {G}. i The position vector; the superscript T in the upper right corner represents the transpose operation, which is used to interchange rows and columns of vectors and matrices. This symbol in all formulas below follows this definition.

[0128] Differentiate equation (29) and combine it with By taking the dot product, the velocity of the telescopic rod can be derived, as shown below:

[0129] (38)

[0130] In the formula, and These are the linear velocity and angular velocity of the moving platform, respectively.

[0131] The generalized pose vector of the moving platform is ,but and It can be calculated using the following formula:

[0132] (39)

[0133] (40)

[0134] In the formula, ; To obtain the derivative of the generalized coordinates Calculate the global linear velocity of the moving platform The Jacobian matrix; To obtain the derivative of the generalized coordinates Calculate the global angular velocity of the moving platform The Jacobian matrix.

[0135]

[0136]

[0137]

[0138]

[0139]

[0140] .

[0141] Dynamic platform speed and The relationship between them can be represented as:

[0142] (41)

[0143] Equation (38) can be expressed in the following form:

[0144] (42)

[0145] In the formula, J is the velocity Jacobian matrix.

[0146] Differentiating both sides of equation (42), we can obtain the following relationship between the branch length and the acceleration of the moving platform:

[0147] (43)

[0148] (44)

[0149] (45)

[0150] (46)

[0151] In the formula,

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158]

[0159]

[0160]

[0161]

[0162] .

[0163] Based on the above steps, the velocity Jacobian matrix and the acceleration Jacobian matrix are obtained.

[0164] Based on the derivation of the inverse kinematics equations above, the constraints of the 3-RPS tracking mechanism can be determined as follows:

[0165] (47)

[0166] (48)

[0167] in, and These are the upper and lower limits of the telescopic pole's length, respectively. and The allowable rotation angles of the composite sphere pair are respectively The maximum and minimum values; and These are the allowable rotation angles of the revolute joint. The maximum and minimum values; Let the coordinate system be {S} i Z of} Si The unit vector of the axis in coordinate system {G} satisfies ,and ; Let E be the rotational subcenter of the i-th branch in the global coordinate system {G}. i The unit direction vector to the fixed coordinate origin B.

[0168] Based on the above steps, the travel constraints of the telescopic rod, the rotation angle constraints of the composite ball joint, and the rotation angle constraints of the revolute joint are obtained.

[0169] S2. Construct a four-objective optimization model: Using the circumscribed radius of the moving platform, the circumscribed radius of the fixed platform, the height of the concentrator center point, and the horizontal installation spacing as design variables, construct a four-objective optimization function that includes workspace coverage objectives, motion risk objectives, motion transmission performance objectives, and structural compactness objectives, and set the feasible domain and motion constraints for each design variable; where, the horizontal installation spacing refers to the straight-line distance between the vertical projection point of the disc concentrator center point in the global coordinate system and the vertical projection point of the fixed platform circumscribed center point in the global coordinate system; the workspace coverage objective is to quantify the proportion of effective non-interference solar attitudes to the total number of sampled attitudes based on the solar trajectories of typical days of the vernal equinox, summer solstice, and winter solstice; the motion risk objective is to quantify the constraint utilization rate of the telescopic rod stroke, the composite spherical sub-angle, and the rotary sub-angle, and minimize the motion limit risk; the motion transmission performance objective is to maximize the motion isotropy based on the reciprocal of the condition number of the velocity Jacobian matrix; the structural compactness objective is to minimize the material consumption and volume occupied by the mechanism based on the weighted sum of the normalized design variables.

[0170] Research and analysis show that the branch length, platform radius ratio, concentrator center point height, and maximum allowable rotation angle of the composite spherical pair are the dominant factors affecting the mechanism's workspace. This embodiment fixes the platform configuration, revolute joint layout, and composite spherical pair arrangement; simultaneously, it selects the platform's circumscribed radius... Determine the external radius of the platform Concentrator center point height and horizontal installation spacing This serves as a design variable for subsequent multi-objective optimization. The horizontal installation spacing refers to the straight-line distance between the vertical projection point of the center point of the disc concentrator in the global coordinate system and the vertical projection point of the outer center of the fixed platform in the global coordinate system.

[0171] The design variables in this embodiment are:

[0172] (52)

[0173] The feasible domain of the design variables is determined as follows:

[0174] (1) Ratio of the radius of the moving platform to that of the fixed platform Λ r Analysis shows that Λ r The effect on the pose workspace is non-monotonic. To cover the optimal performance range and ensure sufficient search degrees of freedom, Λ is set. r The range of values ​​for is Λ r ∈[0.7,2.3].

[0175] (2) External radius of the moving platform Each 3-RPS tracking mechanism supports one-quarter of the concentrator surface area. Given the concentrator aperture radius... =1600mm, to ensure that each 3-RPS tracking mechanism has sufficient movement space. It should not exceed At the same time, if If the value is too small, it will reduce the stiffness of the moving platform and increase the difficulty of arranging the composite spherical pair. Therefore, setting... mm.

[0176] (3) Determine the outer radius of the platform To ensure that each 3-RPS tracking mechanism has good force transmission characteristics and avoids singular configurations, the radius ratio of the moving platform to the stationary platform must meet the constraint condition. Combining The range of values ​​for can be obtained. mm.

[0177] (4) Concentrator center point height This parameter is strongly coupled with the branch chain stroke. Given the telescopic cylinder length constraint... mm, and in order to adapt to large tilt angle motion conditions, set mm.

[0178] (5) Horizontal installation spacing : The value of must ensure that point A is located within the projection area of ​​the condenser, while avoiding interference between the four fixed platforms; furthermore... It should be at least greater than Therefore, setting mm.

[0179] For the workspace coverage target, the pose workspace of the 3-RPS tracking mechanism needs to ensure uninterrupted, interference-free, and blind-spot-free solar tracking within a preset time period on key representative days at a specific installation location. Therefore, the optimization process uses the solar trajectory coverage performance on three typical dates—the spring equinox, summer solstice, and winter solstice—as the core basis (these three dates cover both extreme and average solar trajectory conditions throughout the year), which forms the basis for constructing the workspace coverage objective function. The foundation. The mathematical expression for the workspace coverage target is as follows:

[0180] (53)

[0181] In the formula, N T =1,…,T, where T=3 represents three typical dates selected for workspace coverage assessment; N val,T N represents the number of effective, interference-free solar attitudes achievable by the 3-RPS tracking mechanism within the Tth typical day; tol,T t represents the total number of sampled solar positions within the T-th typical day; s To track time; The solar altitude angle; This is the solar azimuth angle; These are the design variables. The objective function is minimized during the optimization process to maximize the mechanism's effective solar tracking coverage.

[0182] To ensure the long-term reliable operation of the 3-RPS tracking mechanism, a motion risk objective function is defined for the motion risk objective. This is used to quantify the maximum utilization of the composite ball joint rotation angle, the revolute joint rotation angle, and the telescopic rod stroke under the most unfavorable working conditions. Its expression is as follows:

[0183] (54)

[0184] In the formula, N represents the total number of 3-RPS tracking mechanisms, which is N=4 in this embodiment; n represents the nth 3-RPS tracking mechanism, where n=1,2,3,…N; Let be the actual rotation angle of the composite spherical joint of the i-th branch; This is the maximum permissible rotation angle for the composite ball joint; Let be the actual rotation angle of the revolute joint of the i-th branch; This is the maximum permissible rotation angle of the revolute joint; Let be the actual length of the telescopic rod of the i-th branch; This is the average length (nominal length) of the telescopic pole. and These are the upper and lower limits of the telescopic pole's length, respectively. This refers to the length adjustment range of the telescopic rod; These are the weighting coefficients for the travel constraint of the telescopic rod, the rotation angle constraint of the composite spherical joint, and the rotation angle constraint of the revolute joint, respectively; t s To track time; The solar altitude angle; This is the solar azimuth angle. The smaller the value, the farther the working point of the mechanism is from the constraint limit, the greater the safety margin, and the lower the motion risk.

[0185] For the motion transmission performance objective, to evaluate the inherent kinematic performance of the mechanism, the motion transmission performance is defined based on the condition number of the Jacobian matrix, and the objective function is: The expression is as follows:

[0186] (55)

[0187] In the formula, N represents the total number of 3-RPS tracking mechanisms, which is N=4 in this embodiment; n represents the nth 3-RPS tracking mechanism, where n=1,2,3,…N; Let be the vector Jacobian matrix of the nth 3-RPS tracking mechanism in the kth feasible pose; and These are the minimum and maximum singular values ​​of the matrix, respectively; The number of feasible sampling points for the nth 3-RPS tracking mechanism. The reciprocal of the condition number. The objective function is characterized by its isotropic nature. During optimization, this objective function is minimized to improve the motion transmission performance and resistance to singular configurations of the 3-RPS tracking mechanism.

[0188] For the goal of structural compactness, achieving a compact structure while meeting functional requirements plays a crucial role in reducing manufacturing costs, material consumption, and installation space. Therefore, the comprehensive compactness and economic indicators are defined as follows:

[0189] (56)

[0190] In the formula: , , and Parameters , , and The initial design dimensions; , , and Parameters , , and The weighting coefficients are calculated using the analytic hierarchy process (AHP). The weighting coefficients are obtained by pairwise comparisons based on the engineering importance of each structural dimension to the overall material consumption and volume occupied by the machine. The calculation process and results of the AHP are as follows: The circumscribed radius of the moving platform is selected. Determine the external radius of the platform Horizontal installation spacing and the height of the concentrator center point Four structural dimensions were used as evaluation indicators. Considering the engineering impact of the dish concentrator's overall material consumption and space occupation, a 1–9 scale method (scale 1 represents equal importance, scales 3, 5, 7, and 9 represent slightly, significantly, strongly, and extremely important, respectively, with even-numbered scales representing a trade-off between adjacent levels) was employed to compare the pairwise importance of the indicators. The values ​​of each scale within the matrix were determined based on the equipment's structural characteristics: the outer radius of the moving platform... The biggest impact is on the overall machine footprint and consumables, and the fixed platform external radius. The second most significant factor is the height of the concentrator's center point. Spacing with horizontal installation The constraint on the overall machine volume is the weakest, and the influence of both is comparable. Based on this, a 4th-order judgment matrix is ​​constructed. The rows and columns of the matrix correspond to the outer radius of the moving platform, the outer radius of the fixed platform, the horizontal installation spacing, and the height of the concentrator center point, respectively. The matrix form is as follows:

[0191]

[0192] The matrix is ​​normalized column-by-column using the summation method, and the average value is taken row-by-row to obtain the initial weight vector. Then, the largest eigenvalue of the matrix is ​​calculated to complete the consistency check. The calculated consistency ratio CR < 0.1 proves that the comparison logic is consistent and the matrix is ​​valid. The qualified initial weights are simplified to two decimal places to obtain the weight values ​​of each component: The results are as follows. =0.45、 =0.35、 =0.1、 =0.1, the sum of the four weights is 1, satisfying the normalization requirement of the weighted summation of the objective function. This objective function is used to quantitatively evaluate the overall structural compactness and material economy of the mechanism, transforming the implicit cost and volume constraints into explicit optimization objectives, ensuring that the optimal design scheme meets performance requirements while possessing better engineering economy and a more compact layout. During the optimization process, this objective function is minimized to obtain a mechanism configuration with lower material consumption and a more compact structure.

[0193] Based on the kinematic analysis, constraints, and sub-objective functions established above, the complete formulation of this multi-objective optimization problem can be constructed as follows:

[0194] (57)

[0195] In the formula, To track time; and These are the minimum and maximum tracking times, respectively.

[0196] S3. Hybrid Algorithm Solution: The four objective optimization functions are solved using an adaptive PSO-NSGA-III hybrid optimization algorithm to obtain the Pareto optimal solution set. This hybrid optimization algorithm uses NSGA-III as a framework and introduces the PSO velocity-position update rule to improve search efficiency.

[0197] To address the issues of high dimensionality, objective coupling, and constraint complexity in multi-objective parameter optimization, this embodiment proposes an adaptive PSO-NSGA-III hybrid optimization algorithm. This algorithm is based on the NSGA-III framework to ensure the effectiveness of the multi-objective solution set; it introduces the velocity-position update rule of PSO to improve population search efficiency; and it employs an adaptive strategy to dynamically balance the global exploration and local exploitation capabilities of the algorithm. The PSO algorithm is characterized by fast convergence and simple implementation, making it suitable for optimization design problems of parallel mechanisms. However, it suffers from poor population diversity and premature convergence in multi-objective optimization scenarios. To overcome these shortcomings, this paper combines PSO with the NSGA-III algorithm to construct a hybrid optimization framework, utilizing the global search capability of PSO to improve the solution efficiency of the NSGA-III solution set.

[0198] For a population size of N pso For a particle swarm with a decision space dimension of K, the standard PSO algorithm iteratively updates the velocity and position of the particles based on an inertial weight strategy.

[0199] (58)

[0200] (59)

[0201] In the formula: This represents the number of iterations. For particle indexing; In the first Inertia weights in the next iteration; As a cognitive learning factor, it represents the strength of a particle's learning towards its own historical best position; In the first The social learning factor in the next iteration represents the strength of a particle's learning toward the global optimal position of the population; Indicates the first The particle in the first The optimal position of an individual in the next iteration; Indicates the particle swarm in the th The global optimal position in the next iteration; and is an independent random number that follows a uniform distribution within the interval (0,1); and The first The first particle , The search speed of each iteration; and The first The first particle , The particle position vector of the next iteration.

[0202] To prevent excessively high particle velocities from causing search instability, boundary constraints must be applied to the velocities:

[0203] (60)

[0204] In the formula, and These represent the minimum and maximum speeds, respectively.

[0205] To ensure that particles always remain within the feasible decision space, a reflection strategy is used to correct out-of-bounds positions:

[0206] (61)

[0207] In the formula, and The first The lower and upper limits of the design variables, k=1,2,…,k; For the first The first particle Dimensional design variables in the th In the next iteration, if the original position exceeds the variable's value range, the out-of-bounds correction is performed using the reflection strategy of this formula, outputting a feasible position that satisfies the constraints; on the right side... The original new position is obtained from preliminary calculations after velocity update without boundary correction; the left side. This represents the feasible new position after being constrained and corrected by the reflection strategy.

[0208] When a position is reflected, the velocity in the corresponding dimension simultaneously reverses and attenuates:

[0209] (62)

[0210] In the formula, This is the velocity decay coefficient, used to balance the stability of the search process; Representing the The first particle Dimensional variable The velocity of the next iteration is corrected by the reverse decay when the position goes out of bounds using this formula.

[0211] NSGA-III is employed as the environment selection mechanism to improve the distribution quality of the Pareto solution set. After non-dominated sorting, a set of uniformly distributed reference points is generated in the normalized objective space based on the Das-Dennis method. Each candidate solution is associated with the reference point with the smallest vertical distance, and solutions located in less crowded regions are preferentially retained. This strategy allows the algorithm to maintain both the convergence and diversity of non-dominated solutions during the optimization process.

[0212] To dynamically balance global exploration and local development capabilities, a linearly decreasing inertia weighting strategy is adopted:

[0213] (64)

[0214] In the formula, , These are the maximum and minimum values ​​of the inertia weight, respectively. This represents the maximum number of iterations.

[0215] Using a nonlinear time-varying strategy to study cognitive factors With social learning factors Adaptive adjustments are made to coordinate the learning process between individual optimality and global optimality for particles:

[0216] (65)

[0217] In the formula, and These are the initial values ​​for the cognitive factor and the social factor, respectively. and It is divided into cognitive factors and social factors, with minimum boundary values.

[0218] To further enhance the global search capability of the population, an adaptive mutation operation is introduced after particle position out-of-bounds correction. To balance population diversity and solution stability, the mutation probability... Adaptive adjustment strategy adopted:

[0219] (66)

[0220] In the formula, For the first The adaptive mutation probability of particles in the next iteration; and These represent the maximum and minimum values ​​of the mutation probability, respectively.

[0221] To comprehensively evaluate the algorithm's performance, three performance metrics widely used in multi-objective optimization are employed: Hypervolume (HV), Population Diversity (PD), and Generation Distance (GD), used to evaluate the convergence, diversity, and uniformity of the Pareto solution set, respectively. The HV metric measures the volume enclosed by the approximate Pareto front and the reference point in the objective space, reflecting both convergence and diversity. A higher HV value indicates faster convergence, stronger stability, and better overall performance. PD is defined as the average Euclidean distance between all non-dominated solution pairs; a higher PD value indicates better solution distribution and diversity. The GD metric measures the average distance from the approximate solution set to the true Pareto front, quantifying the algorithm's convergence performance; a lower GD value indicates better convergence of the solution set.

[0222] Finally, the main parameter configurations of the adaptive PSO-NSGA-III hybrid algorithm in this embodiment are as follows: population size N pso =36, maximum number of iterations =120; the number of reference points is set to 4. In the adaptive PSO strategy, the inertia weight... The value range is [0.4, 0.9], cognitive factor With social learning factors The adjustment range is [1,2]; the mutation probability The value range is [0.1, 0.3].

[0223] Figure 4 The curves showing the variation of three evaluation metrics of the adaptive PSO-NSGA-III algorithm and the standard NSGA-III algorithm with the number of iterations are presented. As shown in (a), the HV value of the adaptive PSO-NSGA-III algorithm gradually increases in the early stage of iteration, stabilizes at a high level after about 20 iterations, and remains high throughout subsequent iterations. In contrast, the standard NSGA-III algorithm fluctuates drastically in the early stage of iteration and fails to converge to a high level, with the final stable HV value being significantly lower than that of the adaptive PSO-NSGA-III algorithm. As shown in (b), the PD value of the adaptive PSO-NSGA-III algorithm remains stable at a high level throughout the iteration process, with only minor fluctuations. In contrast, the PD value of the standard NSGA-III algorithm drops sharply in the early stage of iteration and continues to decrease slowly, approaching 0.02 after the 68th iteration. As shown in (c), the GD value of the adaptive PSO-NSGA-III algorithm drops rapidly in the early stage of iteration, stabilizes at a very low level after about 60 iterations, and remains unchanged. In contrast, the GD value of the standard NSGA-III algorithm remains at a high level throughout all iterations, without a significant downward trend. The above results demonstrate that, for the multi-objective optimization problem in this embodiment, the adaptive PSO-NSGA-III algorithm proposed in this embodiment has significant advantages over the standard NSGA-III algorithm.

[0224] Figure 5 The curves showing the changes in adaptive PSO parameters and mutation probability during the optimization process are presented respectively. Among them, the inertia weight... With the number of iterations Linear decrease; cognitive factor Social learning factors By dynamically adjusting through reverse segmentation, a dynamic balance between global exploration and local development capabilities is achieved. Simultaneously, the mutation probability... The value was smoothly reduced from 0.3 to 0.1, which not only ensured the diversity of the population in the early stage of algorithm iteration, but also avoided the disturbance of high-quality solutions in the later stage, thus ensuring the stability and efficiency of the multi-objective optimization process.

[0225] Figure 6 The Pareto front distribution obtained by the adaptive PSO-NSGA-III algorithm is shown, including two-dimensional scatter plots for each target and a three-dimensional visualization of the target space. The two-dimensional scatter plots (a)–(f) show that the obtained solutions are closely distributed near the Pareto front, exhibiting good continuity and uniformity, indicating that the algorithm has high convergence accuracy and satisfactory coverage. Simultaneously, it can be observed that… and There is a clear negative correlation between them, reflecting the reasonable trade-off between different objectives. The three-dimensional visualization results of (g) further demonstrate... , , A smooth, continuous Pareto front in space, represented by color mapping in the figure. The value of ...

[0226] S4. Output optimized design parameters: Select design variable parameters from the Pareto optimal solution set as the optimized design parameters for the disc concentrator.

[0227] Figure 7 Parallel coordinate plots of the distribution of objective function values ​​and the distribution of design variables are presented. As shown in Table 2, the range of values ​​for each objective corresponding to the obtained Pareto front are: ∈[0.00,0.78]、 ∈[0.48,0.60]、 ∈[0.74,0.88] and ∈[0.40,0.58]. Based on the optimized Pareto solution set, this embodiment finally gives the recommended value range for key structural design parameters: circumscribed radius of the moving platform. ∈[225,392]mm, fixed platform circumscribed radius ∈[174,286]mm, Concentrator center point height ∈[2510,3471]mm, horizontal installation spacing ∈[340,1058]mm.

[0228] Table 2 Comparison of the objective function at the initial design stage and the objective function after the Pareto solution

[0229]

[0230] Table 2 shows that, compared to the initial design, all Pareto optimization solutions significantly extended the effective solar tracking period. Simultaneously, the overall structural compactness of the optimized mechanism was improved, and the compactness objective function... The value decreased from an initial 0.73 to an optimized range of 0.40–0.58. To extend the tracking time, the mechanism needs to move within a larger pose workspace, ensuring that the optimal solution prioritizing tracking coverage corresponds to… , The target value has increased. It is important to emphasize that all Pareto optimal solutions satisfy the allowable constraints of the composite ball joint and actuator stroke. For engineering applications that prioritize long-term operational reliability rather than extreme tracking time, a solution can be selected from the Pareto solution set. As low as 0.48 Design schemes as low as 0.74.

[0231] In summary, the multi-objective optimization method for disc concentrators based on a 3-RPS tracking mechanism of the present invention can solve the problem that multi-objective optimization cannot be achieved in the prior art.

[0232] Finally, it should be noted that the above embodiments are only used to illustrate the preferred technical solutions of the present invention, and are not intended to limit the scope of protection of the present invention. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the essence and scope of the present invention.

Claims

1. A multi-objective optimization method for a dish condenser based on a 3-RPS tracking mechanism, the dish condenser comprising multiple 3-RPS tracking mechanisms, each 3-RPS tracking mechanism comprising a fixed platform, a moving platform, three branches, and a reflector fixed on the moving platform; three sets of triangularly distributed revolute joints are mounted on the fixed platform, and three sets of triangularly distributed composite spherical joints are mounted on the moving platform; each branch comprises a pair of corresponding revolute joints and composite spherical joints, and a telescopic rod connecting the pair of revolute joints and composite spherical joints, characterized in that... This multi-objective optimization method includes the following steps: S1. Construct a unified kinematic model: Establish a multi-coordinate system, and with the help of the transformation between the coordinate systems and the solar direction vector calculation model, derive the inverse kinematic equations and motion constraint equations of the 3-RPS tracking mechanism under the moving platform and the fixed platform of any triangle, realize the mapping between the pose workspace of the 3-RPS tracking mechanism and the solar tracking time window, and further obtain the velocity Jacobian matrix and the acceleration Jacobian matrix. S2. Construct a four-objective optimization model: Using the circumscribed radius of the moving platform, the circumscribed radius of the fixed platform, the height of the concentrator center point, and the horizontal installation spacing as design variables, construct a four-objective optimization function that includes workspace coverage objectives, motion risk objectives, motion transmission performance objectives, and structural compactness objectives, and set the feasible domain and motion constraints for each design variable; where, the horizontal installation spacing refers to the straight-line distance between the vertical projection point of the disc concentrator center point in the global coordinate system and the vertical projection point of the fixed platform circumscribed center point in the global coordinate system; the workspace coverage objective is to quantify the proportion of effective non-interference solar attitudes to the total number of sampled attitudes based on the solar trajectories of typical days of the spring equinox, summer solstice, and winter solstice; the motion risk objective is to quantify the constraint utilization rate of the telescopic rod stroke, the composite spherical sub-angle, and the rotary sub-angle, and minimize the motion limit risk; the motion transmission performance objective is to maximize the motion isotropy based on the reciprocal of the condition number of the velocity Jacobian matrix; the structural compactness objective is to minimize the material consumption and volume occupied by the mechanism based on the weighted sum of the normalized design variables; S3. Hybrid Algorithm Solution: The four-objective optimization function is solved using an adaptive PSO-NSGA-III hybrid optimization algorithm to obtain the Pareto optimal solution set. This hybrid optimization algorithm uses NSGA-III as a framework and introduces the PSO velocity-position update rule to improve search efficiency. S4. Output optimized design parameters: Select design variable parameters from the Pareto optimal solution set as the optimized design parameters for the disc concentrator.

2. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 1, characterized in that, In step S1, the method for deriving the inverse kinematics equations of the 3-RPS tracking mechanism is as follows: a multi-coordinate system is established, including a global coordinate system {G}, a fixed coordinate system {A}, a moving coordinate system {B}, a tangent plane coordinate system {D}, a condenser coordinate system {C}, and three branched coordinate systems {Ei}; by using the transformation relationship from the branched coordinate system {Ei} to the fixed coordinate system {A} and from the fixed coordinate system {A} to the global coordinate system {G}, the transformation matrix from the branched coordinate system {Ei} to the global coordinate system {G} is obtained. Using the transformation relationships from the moving coordinate system {B} to the tangent plane coordinate system {D}, from the tangent plane coordinate system {D} to the concentrator coordinate system {C}, and from the concentrator coordinate system {C} to the global coordinate system {G}, the transformation matrix from the moving coordinate system {B} to the global coordinate system {G} is obtained. Based on the solar direction vector calculation model and coordinate system transformation relationships, the coordinates of the circumcenters of the fixed platform and the moving platform in the global coordinate system {G} are obtained. Based on the coordinates of the circumcenters of the fixed platform and the moving platform in the global coordinate system {G}, combined with the geometric structural parameters of the moving platform and the fixed platform, the coordinates of the revolute joints and composite spherical joints in the global coordinate system {G} are obtained. Based on the coordinates of the revolute joints and composite spherical joints of each branch in the global coordinate system {G}, the vector, length, and unit vector of each branch are obtained. The method for deriving the motion constraint equations of the 3-RPS tracking mechanism is as follows: Based on the physical constraint that the unit direction vector of the revolute joint is perpendicular to the radial vector, the equation of the unit direction vector of the revolute joint is obtained; based on the physical constraint that the unit direction vector of the revolute joint is perpendicular to the vector of the branch, the orthogonal condition equation is obtained; solving the orthogonal condition equation yields the motion constraint equations of the 3-RPS tracking mechanism.

3. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 2, characterized in that, In step S1, the method for obtaining the velocity Jacobian matrix is ​​as follows: the time derivative of the inverse kinematics equation is performed to obtain the velocity mapping relationship between the pose of the moving platform and the branch length, and the velocity Jacobian matrix is ​​obtained based on the velocity mapping relationship. The method for obtaining the acceleration Jacobian matrix is ​​as follows: take the time derivative of the velocity mapping relationship to obtain the acceleration mapping relationship between the pose of the moving platform and the branch length, and obtain the acceleration Jacobian matrix based on the acceleration mapping relationship.

4. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 3, characterized in that, Workspace coverage target The expression is: ， In the formula, N T =1,…,T, where T=3 represents three typical dates selected for workspace coverage assessment; This represents the number of effective, interference-free solar attitudes that the 3-RPS tracking mechanism can achieve within the Tth typical day, taking the spring equinox, summer solstice, and winter solstice as three typical days; This represents the total number of sampled solar positions within the T-th typical day; To track time; The solar altitude angle; This is the solar azimuth angle; For design variables.

5. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 3, characterized in that, Sports risk goals The expression is: ， In the formula, N represents the total number of 3-RPS tracking mechanisms; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N; Let be the actual rotation angle of the composite spherical joint of the i-th branch; This is the maximum permissible rotation angle for the composite ball joint; Let be the actual rotation angle of the revolute joint of the i-th branch; This is the maximum permissible rotation angle of the revolute joint; Let be the actual length of the telescopic rod of the i-th branch; The average length of the telescopic pole. and These are the upper and lower limits of the telescopic pole's length, respectively. This refers to the length adjustment range of the telescopic rod; These are the weighting coefficients for the travel constraint of the telescopic rod, the rotation angle constraint of the composite spherical joint, and the rotation angle constraint of the revolute joint, respectively. To track time; The solar altitude angle; This is the solar azimuth angle.

6. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 3, characterized in that, Motion transmission performance target The expression is: ， In the formula, N represents the total number of 3-RPS tracking mechanisms; n represents the nth 3-RPS tracking mechanism, n=1,2,3,…N; Let be the vector Jacobian matrix of the nth 3-RPS tracking mechanism in the kth feasible pose; and These are the minimum and maximum singular values ​​of the matrix, respectively; The number of feasible sampling points for the nth 3-RPS tracking mechanism; the reciprocal of the condition number. Characterizes the degree of isotropy of motion.

7. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 3, characterized in that, Structural compactness target The expression is: ， In the formula, The outer radius of the moving platform; To determine the circumscribed radius of the platform; For horizontal installation spacing; The height of the center point of the concentrator; , , and Parameters , , and The initial design dimensions; , , and Parameters , , and The weighting coefficients are calculated using the analytic hierarchy process (AHP) based on pairwise comparisons of the engineering importance of each structural dimension to the overall material consumption and volume occupied by the machine.

8. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 3, characterized in that, In step S2, the feasible regions of each design variable are as follows: when the aperture radius of the disc concentrator is R, the feasible region of the outer radius of the moving platform is [0.125R, 0.25R], the feasible region of the outer radius of the fixed platform is [0.09R, 0.18R], the feasible region of the height of the concentrator center point is [1.56R, 2.18R], and the feasible region of the horizontal installation spacing is [0.18R, 0.68R]. The motion constraints are the extension rod stroke constraint, the composite spherical joint rotation angle constraint, and the revolute joint rotation angle constraint.

9. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 1, characterized in that: In the adaptive PSO-NSGA-III hybrid optimization algorithm, a linear decreasing inertia weight strategy is adopted to balance the global exploration and local exploitation capabilities, with the inertia weight value being [0.4, 0.9]. A nonlinear time-varying strategy is adopted to adjust the cognitive and social factors, with the adjustment range of both cognitive and social factors being [1, 2]. An adaptive adjustment strategy is adopted for the mutation probability, with the mutation probability value ranging from [0.1, 0.3].

10. The multi-objective optimization method for a dish concentrator based on a 3-RPS tracking mechanism according to claim 9, characterized in that: In the adaptive PSO-NSGA-III hybrid optimization algorithm, the algorithm performance is evaluated by hypervolume, population diversity, and generation distance indicators.

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