Time-optimal three-axis reorientation method and apparatus for inertia-symmetric rigid body spacecraft
By introducing a new inertial axis coordinate system and Radau pseudospectral discretization technique into an inertial symmetric rigid spacecraft, the problem of low computational efficiency for time-optimal retargeting of inertial symmetric rigid spacecraft is solved, and fast and accurate trajectory generation of inertial rigid body spacecraft is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING XINGXU ZHIYUAN AEROSPACE TECHNOLOGY CO LTD
- Filing Date
- 2026-03-07
- Publication Date
- 2026-06-09
Smart Images
Figure CN122166334A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of time-optimal redirection technology, and relates to a time-optimal three-axis redirection method and device for an inertial symmetric rigid spacecraft. Background Technology
[0002] The time-optimal redirection problem for rigid-body spacecraft has fundamental research value from both engineering practice and mathematical theory perspectives. In space missions, operations such as pointing, turning, or multi-target maneuvers often require spacecraft to transition from one attitude to another in the shortest possible time. Maneuvers based on the Euler principal axis (characteristic axis) are often considered the natural choice for achieving rapid rotational maneuvers because they represent the shortest angular path between two orientations, and have become an industry standard applied to the control systems of various current spacecraft. In reality, whether the characteristic axis maneuver is optimal depends on the specific definition of the allowable set of control torques. For the time-optimal redirection problem of inertial symmetric rigid spacecraft (ISRS), Bai and Junkins proved analytically that when the total amplitude of the control torques is constrained but the direction is freely selectable, the characteristic axis maneuver is indeed time-optimal. Furthermore, Bilimoria and Wie first pointed out that under three-axis control conditions, the optimal redirection time is actually less than the time required for the characteristic axis maneuver, because the nutation component can provide additional control torque for the redirection axis. Summary of the Invention
[0003] To address the problems existing in the above-mentioned traditional methods, this invention proposes a time-optimal three-axis retargeting method and device for inertial symmetric rigid spacecraft.
[0004] To achieve the above objectives, the embodiments of the present invention adopt the following technical solutions: On the one hand, a time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft is provided, the method comprising the following steps: Step 1: Based on the inertial symmetry of the spacecraft and the a priori time-optimal control switch structure, determine the analytical expression of the angular velocity of the spacecraft during the maneuver; the analytical expression of the angular velocity is completely determined by two optimizable parameters.
[0005] Step 2: A new inertia axis coordinate system is introduced. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft redirection trajectory described by quaternions are derived based on the analytical expression of angular velocity.
[0006] Step 3: The Radau pseudospectral method is used to discretize the quaternion dynamic equations, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem.
[0007] Step 4: Using the given two parameter values, the discretized quaternion dynamic equations are transformed into a system of linear equations. Solving the system of linear equations yields the corresponding discrete quaternion numerical sequence.
[0008] Step 5: Based on the discrete quaternion numerical sequence, optimize the two parameters using the gradient method or Newton's method, and determine the time-optimal retargeting trajectory based on the optimization results.
[0009] On the other hand, a time-optimal three-axis redirection device for an inertial-symmetric rigid spacecraft is also provided, the device comprising: The angular velocity analytical expression determination module is used to determine the analytical expression of the spacecraft's angular velocity during the maneuver, based on the spacecraft's inertia symmetry and the a priori time-optimal control switch structure; the analytical expression of angular velocity is completely determined by two optimizable parameters.
[0010] A new inertia axis coordinate system construction module is used to introduce a new inertia axis coordinate system. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft reorientation trajectory described by quaternions are derived based on the analytical expression of angular velocity.
[0011] The Radau pseudospectral method discretization module is used to discretize the quaternion dynamics equations using the Radau pseudospectral method, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem.
[0012] The Discrete Quaternion Numerical Sequence Determination Module is used to transform the discretized quaternion dynamic equations into a system of linear equations using two given parameter values, solve the system of linear equations, and obtain the corresponding discrete quaternion numerical sequence.
[0013] The time-optimal redirection trajectory determination module is used to optimize two parameters based on a discrete quaternion numerical sequence using either the gradient method or Newton's method, and determine the time-optimal redirection trajectory based on the optimization results.
[0014] One of the above technical solutions has the following advantages and beneficial effects: The aforementioned time-optimal three-axis retargeting method and apparatus for inertial symmetric rigid spacecraft includes the following steps: determining an analytical expression for angular velocity based on the spacecraft's inertial symmetry and a priori time-optimal control switch structure; explicitly deriving the symmetry characteristics of the retargeting trajectory by introducing a new fixed principal axis coordinate system; discretizing the quaternion dynamic equations using the Radau pseudospectral discretization method, thereby transforming the time-optimal retargeting problem into a nonlinear programming problem; converting the discretized quaternion dynamic equations into a linear system of equations and solving them using two given parameter values to obtain the corresponding discrete quaternion numerical sequence; optimizing the two parameters using the gradient method or Newton's method, and determining the time-optimal retargeting trajectory based on the optimization results. This method eliminates the need for numerical integration or solving large-scale nonlinear programming problems, significantly improving computational efficiency and enabling rapid and accurate trajectory generation for inertial rigid spacecraft. Attached Figure Description
[0015] To more clearly illustrate the technical solutions in the embodiments of this application or the conventional technology, the drawings used in the description of the embodiments or the conventional technology will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0016] Figure 1 This is a flowchart illustrating the time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft in one embodiment. Figure 2 In one embodiment, there are two solid principal axis coordinate systems. I and N A schematic diagram; Figure 3 This is a schematic diagram of the control structure for the optimal solution of the 5-switching time in one embodiment, wherein... Figure 3 (a) to (d) represent the control structures for the optimal 5-switching time solution, respectively. A schematic diagram; Figure 4 This is a schematic diagram of the control structure for the optimal solution of 6-switching time in one embodiment, wherein... Figure 4 (a) to (d) represent the control structures for the optimal 6-switching time solution, respectively. A schematic diagram; Figure 5 In one embodiment, during the optimal reorientation process of 5 switching times... and A schematic diagram of the curve changing over time, where Figure 5 (a) Figure 5 (c) and Figure 5 (e) represents the five optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time. Figure 5 (b) Figure 5 (d) and Figure 5 (f) represents the five optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time; Figure 6 In one embodiment, during the optimal reorientation process of 5 switching times... A schematic diagram of the curve changing over time, where Figure 6 (a) to (d) represent the five optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 7 In one embodiment, during the optimal reorientation process of 5 switching times... A schematic diagram of the curve changing over time, where Figure 7 (a) to (d) represent the five optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 8 In one embodiment, during the optimal reorientation process of 6 switching times and A schematic diagram of the curve changing over time, where Figure 8 (a) Figure 8 (c) and Figure 8 (e) represents the six optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time. Figure 8 (b) Figure 8 (d) and Figure 8 (f) represents the six optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time; Figure 9 In one embodiment, during the optimal reorientation process of 6 switching times A schematic diagram of the curve changing over time, where Figure 9 (a) to (d) represent the six optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 10 In one embodiment, during the optimal reorientation process of 6 switching times A schematic diagram of the curve changing over time, where Figure 10 (a) to (d) represent the six optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time. Detailed Implementation
[0017] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.
[0018] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in this specification is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.
[0019] It should be noted that, in this document, the reference to "embodiment" means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of the invention. The presentation of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. Those skilled in the art will understand that the embodiments described herein can be combined with other embodiments. The term "and / or" as used herein refers to any combination of one or more of the associated listed items, and all possible combinations, including such combinations.
[0020] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
[0021] The dimensionless Euler rotational motion of an ISRS (Inertial Symmetric Rigid Spacecraft) in vector form, in an inertial axisymmetric coordinate system with the ISRS center of mass as the origin, can be expressed as follows: (1) in, It is a dimensionless angular velocity vector. This is a dimensionless control torque vector. The quaternion-represented dynamic equation is: (2) in, and These are the scalar and vector parts of the quaternion, respectively. These are Euler vectors (characteristic axes). Let be the rotation angle. The differential equation for the quaternion is: (3) in, , .
[0022] Command subscript I and T Let these represent the quantities related to the initial and final values, respectively. Without loss of generality, the initial time is always fixed at zero, i.e. The technical problem solved by this application is to find the rotational trajectory of a rigid body under given constraints (the trajectory of the quaternion changing with time), and to make the end time... Minimize. Therefore, the time-optimal control problem can be formulated as minimizing the following performance metrics: (4) in, J This is the optimal control quantity.
[0023] Simultaneously, the ISRS is redirected from the initial state to the final state according to formulas (1) and (3). Let... It is the Euclidean norm of quaternions. and The constraints are: (5) (6) in, For optimal control torque, .
[0024] Under the performance index of formula (4), the Hamiltonian function can be defined as: (7) in, and They are respectively and The corresponding conjugate state vector. Based on the Hamiltonian function, the conjugate differential equation can be derived as follows: , (8) Since the triaxial reversal dynamics equations are implicit functions of time, the Hamiltonian function associated with the time-optimal solution is always −1, that is: (9) According to the Pontryagin extreme value principle, the optimal control torque under constraint formula (6) is... for: (10) Among them, when the switching function exist When zero occurs within a finite time interval, s This represents the corresponding singular control law. This application only considers the switching function. exist The case where the isolated point on the arc is zero, therefore there is no corresponding singular arc.
[0025] In one embodiment, such as Figure 1 As shown, a time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft is provided, which may include the following processing steps 1 to 5: Step 1: Based on the inertial symmetry of the spacecraft and the a priori time-optimal control switch structure, determine the analytical expression of the angular velocity of the spacecraft during the maneuver; the analytical expression of the angular velocity is completely determined by two optimizable parameters.
[0026] First, we introduce two inertia axis body coordinate systems. I and N .make I and N Each consists of two sets of right-handed orthogonal vectors. and definition, and Relationship such as Figure 2 As shown, Figure 2 In the middle, "O" represents the centroid of the ISRS. The rotation angle is... and equal, , , and On the same plane, by Figure 2 visible, and This is the Euler vector in formula (2). .
[0027] exist Figure 2 coordinate system in I Next, depending on the direction. rotation angle The value of , the time-optimal triaxial redirection solution of ISRS may contain 5 or 6 switches. The corresponding control structures of the 5-switch and 6-switch time-optimal solutions are as follows: Figure 3 and Figure 4 As shown in the figure. The dark and light bars represent control torque values of +1 and -1, respectively.
[0028] The switching function of the time-optimal solution is completely determined by two optimizable parameters. According to... Figure 3 and Figure 4 It can be seen that the time interval This allows for the division into a grid consisting of 6 segments. Furthermore, given the control structure, the angular velocity vector has an analytical solution.
[0029] Step 2: A new inertia axis coordinate system is introduced. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft redirection trajectory described by quaternions are derived based on the analytical expression of angular velocity.
[0030] Specifically, I The boundary conditions in the coordinate system are: (11) , (12) in, This represents the zero vector of size 3×1. N In coordinate system, remain unchanged. and The value is: , (13) Suppose that an inertia axis is applied The dimensionless control torque is . The constraints are shown in formula (6). According to... Figure 2 It can be seen that for the coordinate system I and N Redirection below, inertia axis They have the same initial and final positions. The control structures are identical in both coordinate systems. Therefore, the time-optimal triaxial relocation solutions in both coordinate systems have the same relocation and switching times.
[0031] Introducing a coordinate system N The optimization of the redirection problem can be simplified because of the quaternions in this coordinate system. q It has symmetry.
[0032] It should be noted that and . N In the coordinate system, the boundary conditions in formula (13) satisfy , Then quaternion q The symmetry is as follows: Lemma 1: For control structures ( )Down and The optimal value, N Quaternions in coordinate system q satisfy: (14) Proof: For control structures ( )Down and The optimal value of ω corresponds to an analytical solution for the angular velocity. The following conclusion can be drawn: (15) in, See formula (3). Based on the derivation of the dynamic function shown in formula (3), it can be seen that... satisfy: (16) Substituting formula (15) into (16) yields: (17) According to formula (3) It can also be expressed as: (18) It should be noted that Therefore, view As a whole, according to formulas (17) and (18), we know Q.E.D.
[0033] definition for The quaternion corresponding to time step, i.e. Formula (14) requires Must meet: (19) Formula (19) is the optimal formula for numerical optimization using the symmetry in Lemma 1. and The conditions that need to be met.
[0034] Step 3: The Radau pseudospectral method is used to discretize the quaternion dynamic equations, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem.
[0035] Step 4: Using the given two parameter values, the discretized quaternion dynamic equations are transformed into a system of linear equations. Solving the system of linear equations yields the corresponding discrete quaternion numerical sequence.
[0036] Step 5: Based on the discrete quaternion numerical sequence, optimize the two parameters using the gradient method or Newton's method, and determine the time-optimal retargeting trajectory based on the optimization results.
[0037] The aforementioned time-optimal three-axis retargeting method for inertial symmetric rigid spacecraft includes: determining an analytical expression for angular velocity based on the spacecraft's inertial symmetry and a priori time-optimal control switch structure; explicitly deriving the symmetry characteristics of the retargeting trajectory by introducing a new fixed principal axis coordinate system; discretizing the quaternion dynamics equations using the Radau pseudospectral discretization method, thereby transforming the time-optimal retargeting problem into a nonlinear programming problem; converting the discretized quaternion dynamics equations into a linear system of equations and solving them using two given parameter values to obtain the corresponding discrete quaternion numerical sequence; optimizing the two parameters using the gradient method or Newton's method, and determining the time-optimal retargeting trajectory based on the optimization results. This method eliminates the need for numerical integration or solving large-scale nonlinear programming problems, significantly improving computational efficiency and enabling fast and accurate trajectory generation for inertial rigid spacecraft.
[0038] In one embodiment, the time-optimal control switch structure set a priori in step 1 is the switching time of the 5-switching-time-optimal solution; wherein the switching time of the 5-switching-time-optimal solution is: (20) in, , , , , For five switching times, This is the first optimizable parameter. , The second optimizable parameter is the total redirection time.
[0039] In one embodiment, the time-optimal control switch structure set a priori in step 1 is the switching time of the 6-switching-time-optimal solution; wherein, the switching time of the 6-switching-time-optimal solution is: (twenty one) in, , , , , For five switching times, This is the first optimizable parameter. , The second optimizable parameter is the total redirection time.
[0040] In one embodiment, step 3 includes: based on the quaternion symmetry derived in step 2, only the time interval needs to be analyzed. First, Divide into a grid containing three segments, and define independent variables. , ; will the first k Each segment Using independent variables Representing this, we obtain the result using independent variables. The time indicated t for: (twenty two) in, , , , The optimal switching time is... .
[0041] Based on the quaternion differential equations of inertial symmetric rigid spacecraft and using independent variables The time indicated t The expression is obtained in the new inertia axis body coordinate system with independent variables. The first k The quaternion dynamic equations for each segment are: (twenty three) (twenty four) in, For the new inertia axis body coordinate system, quaternion. W Angular velocity, superscript Indicates the first k Relevant quantities for segmentation; This is an intermediate quantity.
[0042] The quaternion dynamics equations are discretized using the Radau pseudospectral method, resulting in the discretized quaternion dynamics equations as follows: (25) (26) Boundary constraints: The continuity condition of quaternions between segments is satisfied by the equality constraints: ,in ; in, For the basis of Lagrange polynomials, , , , ; for At Legendre-Gauss-Radau The value; ; , They are respectively and Legendre polynomial of order 1; For the first k The segmented Radau pseudospectral differential matrix ( Size is ) i Line number j The elements of the column, that is .
[0043] In one embodiment, define variables , , , , , for: , (27) , , (28) The discretized quaternion dynamic equations are transformed into matrix form as follows: (29) , (30) The quaternion constraint is: (31) in, This represents the Kronecker tensor product of two matrices. For size Radau pseudospectral differential synthesis matrix of order 1 , .
[0044] In one embodiment, step 4 includes: setting boundary conditions, initial values and ranges of variation for two optimizable parameters; isolating variables. The discretized quaternion dynamic equations can be transformed into collocation conditions in matrix form and rewritten as follows: (32) , (33) (34) , (35) , (36) in, for The zero matrix, for The identity matrix.
[0045] If the coefficient matrix Always reversible, and ,but ;make ,but ;set up ,definition ,but ; Optimize two optimizable parameters to find the equation that satisfies of and make Take the minimum value, where It is a matrix that embodies the symmetry of quaternions; combined with the corresponding Hamiltonian function and switching function, we obtain an overdetermined linear equation system about discrete conjugate quantities, and solve it by Moore-Penrose pseudo-inverse to obtain the corresponding sequence of discrete conjugate quantities to verify the optimality of the solution.
[0046] In one embodiment, when using the 5-switch time optimal solution, the corresponding matrix reflecting quaternion symmetry is: , (37) When using the 6-switching time optimal solution, the corresponding matrix reflecting quaternion symmetry is: , (38) in, , , , These are four matrices that demonstrate quaternion symmetry.
[0047] In one embodiment, step 5 includes: optimizing two optimizable parameters using a gradient-based or Newton-Raphson-Raphson iterative method to obtain optimal values for the two optimizable parameters; during the optimization process, finite differences are used to provide... Approximate values of the partial derivatives of the two optimizable parameters; determine the time-optimal retargeting trajectory based on the optimal values of the two optimizable parameters.
[0048] Specifically, the numerical optimization method includes the following processes: exist N Seeking in coordinate system and When the optimal value is found, Lemma 1 can be used to reduce the number of mesh segments. First, let... Divided into a grid containing three segments , ,in, and Let be the switching time in formulas (20) and (21). Let As a new independent variable, Available It is represented as shown in formula (22).
[0049] Let the superscript "( k )” indicates the first The correlation of each segment is then treated as an independent variable. The first The piecewise quaternion differential equations are shown in equations (23) and (24).
[0050] Discretization using Radau pseudospectral, It can be approximated as: (39) (40) in, For the basis of Lagrange polynomials, Not the value of the point allocation. and They are exist And Legendre-Gauss-Radau (LGR) point An approximation of . Let . and They are respectively and Legendre polynomials of order LGR. The LGR point is the polynomial of order LGR. The roots are: (41) In formula (39) about The differential is: (42) Let the differential approximation on the right side of formula (42) be equal to the equation on the right side of formula (23). The collocation condition is shown in formulas (25) and (26).
[0051] The boundary constraints are: (43) The continuity condition of quaternions between segments is satisfied by the following equation: (44) When implementing the algorithm, it can be used replace This removes redundant variables and the equality constraints of formula (44). To simplify the expression, the variables are defined as shown in formulas (27) and (28). , , , , , .
[0052] The collocation condition in formula (25) can be written in a compact form as shown in formula (29). The quaternion constraint in formula (5) can be expressed as shown in formula (31).
[0053] At this point, the time-optimal relocation problem has been transformed into a nonlinear programming problem, with the corresponding variables being: and The constraints are given in equations (29) and (31). This nonlinear programming problem can be optimized using Snopt. In fact, the ISRS time-optimal redirection can be optimized in a more direct way without solving the corresponding nonlinear programming problem.
[0054] First, isolate variables The collocation condition in formula (29) can be rewritten as shown in formulas (32) to (36). It should be noted that for a given... and angular velocity There is an analytical solution. Therefore, and Known.
[0055] Secondly, suppose that the coefficient matrix in formula (32) Always reversible and ,but Let formula (32) contain... ,but for: (45) make ,definition . for: (46) Therefore, it can be seen that in formula (46) Directly from the given and Confirmed. It should be noted that... Therefore, for the optimal value and , Because it satisfies the constraints in formula (19).
[0056] At this point, only optimization is needed. and To find the ones that satisfy formula (19) And make The minimum value is taken. Numerical optimization methods are shown in Table 1. Clearly, this method does not require numerical integration or solving large nonlinear programming problems.
[0057] Table 1 Numerical Optimization Algorithm
[0058] In step 2, optimization can be achieved using gradient-based or Newton-based iteration methods. However, about and The gradient has no explicit expression. During the optimization process, finite differences are used to provide the corresponding approximation. It should be noted that the numerical optimization algorithm shown in Table 1 does not impose the constraint on the discrete quaternion modulus in formula (31). The specific reasons are analyzed as follows: Define conjugate quaternions It can be verified that it satisfies the following equation: (47) in," " represents the product of two quaternions. Because Automatic satisfaction, see formula (13) The value. Therefore, according to formula (47), we can theoretically derive... (That is, the quaternion modulus satisfies the constraints). For a given... and The constraint accuracy of the quaternion model depends on the inverse matrix operation in formula (45). Furthermore, since the integral and differential forms of the Radau pseudospectral method are equal, this also guarantees the coefficient matrix of formula (32). It is always an invertible matrix. Therefore, the discrete quaternion calculated by formula (45) is equal to the value obtained by Gaussian integration. For each segment, the quaternion dynamic equation has high-order continuity, so the calculation accuracy increases with the number of collocation points. It increases with the increase of.
[0059] In step 3, for the optimized... and , and This is subsequently determined. Radau pseudospectral discretization can be used to discretize the conjugate quantities on each segment; for simplicity, this will not be discussed further in this paper. Combining the corresponding Hamiltonian function and switching function, an overdetermined linear equation system about the discrete conjugate quantities can be obtained and solved using the Moore-Penrose pseudoinverse. For specific methods, please refer to reference 1 (Patterson, MA, and Rao, AV, “Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems,” Journal of Spacecraft and Rockets, Vol. 49, No. 2, 2012, pp. 363-377. https: / / doi.org / 10.2514 / 1.A32071), only requiring the quantity describing the attitude to be replaced from the modified Rodriguez parameter (MRP) with the quaternion used in this paper.
[0060] Define the optimal solution corresponding to 5-switching time. for: (48) Define the optimal solution corresponding to 6-switching time. for: (49) make Indicates by and The augmented conjugate state vector is formed. According to formula (8), we know... and The expression is: (50) for Suppose and satisfy: (51) (52) in, Given by formulas (37) and (38), It is given by formula (48) and formula (49). The symmetry properties are as follows: Lemma 2: Given the optimal value and , According to the control structure in the coordinate system Optimized satisfy: (53) Proof: Given the optimal value and , and That is, it is certain and satisfies: (53) Combining formula (50), The expression is: (54) Substituting formula (53) into formula (54) yields: (55) Combining formula (50), It can also be expressed as: (56) Thus, based on the assumptions of formula (51) and considering... For the whole, it can be derived from formulas (55) and (56). Q.E.D.
[0061] In summary, given the optimal value and , In the interval There is an analytical solution. In the interval The discrete solution is known. In the interval The discrete solution can be determined according to reference 1. Therefore, and In the interval It can be approximated by Lagrange polynomials. In the interval This can be obtained from Lemmas 1 and 2.
[0062] In one specific embodiment, the numerical examples are all within the control structure. The LGR points for each segment are obtained as follows: For any , The range of variation is or Give, The range of variation is: (57) in, This indicates the time of the corresponding redirection along the inertia axis.
[0063] and The initial value is: 5 - Switch mode: , 6-Switch Mode: , .
[0064] Given the initial values and range of the variables, the optimization problem discussed in this application is transformed into a simple minimum optimization under nonlinear constraints, which can be solved using GNU Octave. For example, the optimized result and for: 5. When the optimal switching time is found: , ; 6-Optimal solution for switching time: , ; N In coordinate system, the estimated and The conjugate state estimates at time t are shown in Table 2. The curves showing the changes of the state and conjugate state over time are as follows: Figures 5 to 10 As shown.
[0065] Table 2. Conjugate State Estimation Results
[0066] The optimality of the simulation example can be verified by numerically testing the optimality conditions. The key indicator is the consistency between the control torque and the switching function (indirectly verified by angular velocity), see [link to relevant documentation]. Figure 5 and Figure 8 ,in Figure 5 (a) Figure 5 (c) and Figure 5 (e) represents the five optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time. Figure 5 (b) Figure 5 (d) and Figure 5 (f) represents the five optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time; where Figure 8 (a) Figure 8 (c) and Figure 8(e) represents the six optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time. Figure 8 (b) Figure 8 (d) and Figure 8 (f) represents the six optimal reorientation processes during switching time. , and A schematic diagram of the curve changing over time. and The symmetry can be obtained from Figure 6 , Figure 7 , Figure 9 and Figure 10 verify, Figure 6 (a) to (d) represent the five optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 7 (a) to (d) represent the five optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 9 (a) to (d) represent the six optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time; Figure 10 (a) to (d) represent the six optimal reorientation processes during switching time, respectively. , , and A schematic diagram of the curve changing over time. The estimated conjugate quantities in Table 2 also satisfy the condition... hour, Meanwhile, the hypothesis used to verify Lemma 2... This has also been verified.
[0067] It should be understood that, although the above Figure 1 The steps are shown sequentially as indicated by the arrows, but these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise explicitly stated in this document, there is no strict order in which these steps are executed; they can be performed in other orders. Furthermore, the above... Figure 1At least some of the steps may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed in turn or alternately with other steps or at least some of the sub-steps or stages of other steps.
[0068] In one embodiment, a time-optimal three-axis redirection device for an inertial-symmetric rigid spacecraft is also provided, the device comprising: The angular velocity analytical expression determination module is used to determine the analytical expression of the spacecraft's angular velocity during the maneuver, based on the spacecraft's inertia symmetry and the a priori time-optimal control switch structure; the analytical expression of angular velocity is completely determined by two optimizable parameters.
[0069] A new inertia axis coordinate system construction module is used to introduce a new inertia axis coordinate system. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft reorientation trajectory described by quaternions are derived based on the analytical expression of angular velocity.
[0070] The Radau pseudospectral method discretization module is used to discretize the quaternion dynamics equations using the Radau pseudospectral method, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem.
[0071] The Discrete Quaternion Numerical Sequence Determination Module is used to transform the discretized quaternion dynamic equations into a system of linear equations using two given parameter values, solve the system of linear equations, and obtain the corresponding discrete quaternion numerical sequence.
[0072] The time-optimal redirection trajectory determination module is used to optimize two parameters based on a discrete quaternion numerical sequence using either the gradient method or Newton's method, and determine the time-optimal redirection trajectory based on the optimization results.
[0073] In one embodiment, the time-optimal control switch structure set a priori in the angular velocity analytical expression determination module is the switching time of the 5-switching time optimal solution; wherein the switching time of the 5-switching time optimal solution is as shown in formula (20).
[0074] In one embodiment, the time-optimal control switch structure set a priori in the angular velocity analytical expression determination module is the switching time of the 6-switching time optimal solution; wherein, the switching time of the 6-switching time optimal solution is as shown in formula (21).
[0075] In one embodiment, the Radau pseudospectral discretization module is also used to construct the quaternion symmetry derived in the new inertia axis coordinate system module, requiring only the analysis of the time interval. First, Divide into a grid containing three segments, and define independent variables. , ; will the first k Each segment Using independent variables Representing this, we obtain the result using independent variables. The time indicated t Its expression is shown in formula (22).
[0076] Based on the quaternion differential equations of inertial symmetric rigid spacecraft and using independent variables The time indicated t The expression is obtained in the new inertia axis body coordinate system with independent variables. The first k The piecewise quaternion dynamic equations are shown in Equations (23) and (24).
[0077] The quaternion dynamics equations are discretized using the Radau pseudospectral method, resulting in the discretized quaternion dynamics equations shown in formulas (25) and (26).
[0078] In one embodiment, the variables in the Radau pseudospectral discrete module are defined as shown in equations (27) and (28). , , , , , ; The discretized quaternion dynamic equations are transformed into matrix form as shown in equations (28) and (29); the quaternion constraints are: .
[0079] In one embodiment, the discrete quaternion numerical sequence determination module is also used to set boundary conditions, initial values and ranges of variation of two optimizable parameters; and isolation variables. The discretized quaternion dynamic equations can be transformed into collocation conditions in matrix form and rewritten as shown in equations (32) to (36); if the coefficient matrix Always reversible, and ,but ;make ,but ;set up ,definition ,but ; Optimize two optimizable parameters to find the equation that satisfies of and make Take the minimum value, where It is a matrix that embodies the symmetry of quaternions; combined with the corresponding Hamiltonian function and switching function, we obtain an overdetermined linear equation system about discrete conjugate quantities, and solve it by Moore-Penrose pseudo-inverse to obtain the corresponding sequence of discrete conjugate quantities to verify the optimality of the solution.
[0080] In one embodiment, the discrete quaternion numerical sequence determination module is also used to determine the matrix that embodies the quaternion symmetry when the 5-switching time optimal solution is adopted, as shown in formula (37).
[0081] When the optimal solution for 6-switching time is adopted, the corresponding matrix that reflects the quaternion symmetry is shown in formula (38).
[0082] In one embodiment, the time-optimal relocation trajectory determination module is further used to optimize two optimizable parameters based on gradient or Newton's method, obtaining the optimal values of the two optimizable parameters; during the optimization process, finite differences are used to provide... Approximate values of the partial derivatives of the two optimizable parameters; determine the time-optimal retargeting trajectory based on the optimal values of the two optimizable parameters.
[0083] It is understood that for a detailed explanation of the time-optimal three-axis redirection device for inertial symmetric rigid spacecraft, please refer to the corresponding explanations of the various embodiments of the time-optimal three-axis redirection method for inertial symmetric rigid spacecraft above, which will not be repeated here. Each module in the aforementioned time-optimal three-axis redirection device for inertial symmetric rigid spacecraft can be implemented entirely or partially through software, hardware, or a combination thereof. Each module can be embedded in hardware or independently of a device with data processing capabilities, or stored in software in the memory of the aforementioned device, so that the processor can call and execute the operations corresponding to each module. The aforementioned device can be, but is not limited to, various types of data processing computer devices already existing in the art.
[0084] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0085] The above embodiments are merely illustrative of several implementation methods of this application, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and all such modifications and improvements fall within the scope of protection of this application.
Claims
1. A time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft, characterized in that, Including the following steps: Step 1: Based on the inertial symmetry of the spacecraft and the a priori time-optimal control switch structure, determine the analytical expression of the angular velocity of the spacecraft during the maneuver; wherein the analytical expression of the angular velocity is completely determined by two optimizable parameters; Step 2: A new inertia axis coordinate system is introduced. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft redirection trajectory described by quaternions are derived based on the analytical expression of angular velocity. Step 3: The Radau pseudospectral method is used to discretize the quaternion dynamic equations, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem; Step 4: Using the given two parameter values, the discretized quaternion dynamic equations are transformed into a system of linear equations. The system of linear equations is solved to obtain the corresponding discrete quaternion numerical sequence. Step 5: Based on the discrete quaternion numerical sequence, optimize the two parameters using the gradient method or Newton's method, and determine the time-optimal retargeting trajectory based on the optimization results.
2. The time-optimal three-axis retargeting method for inertial symmetric rigid spacecraft according to claim 1, characterized in that, The time-optimal control switch structure set a priori in step 1 is the switching time of the 5-switching-time-optimal solution; where the switching time of the 5-switching-time-optimal solution is: ； in, , , , , For five switching times, This is the first optimizable parameter. , The second optimizable parameter is , where is the total redirection time.
3. The time-optimal three-axis redirection method for inertial symmetric rigid spacecraft according to claim 1, characterized in that, In step 1, the a priori optimal control switch structure is the switching time of the 6-switching-time optimal solution; where the switching time of the 6-switching-time optimal solution is: ； in, , , , , For five switching times, This is the first optimizable parameter. , The second optimizable parameter is , where is the total redirection time.
4. The time-optimal three-axis redirection method for inertial symmetric rigid spacecraft according to claim 1, characterized in that, Step 3 includes: Based on the quaternion symmetry derived in step 2, only the time interval is analyzed. First, Divide into a grid containing three segments, and define independent variables. , ; will the first k Each segment Using independent variables Representing this, we obtain the result using independent variables. The time indicated t for: ； in, , , , The optimal switching time is... ; Based on the quaternion differential equations of inertial symmetric rigid spacecraft and using independent variables The time indicated t The expression is obtained in the new inertia axis body coordinate system with independent variables. The first k The quaternion dynamic equations for each segment are: ； ； in, For the new inertia axis body coordinate system, quaternion. W Angular velocity, superscript Indicates the first k Relevant quantities for segmentation; As an intermediate quantity; The quaternion dynamics equations are discretized using the Radau pseudospectral method, resulting in the discretized quaternion dynamics equations as follows: ； ； Boundary constraints: The continuity condition of quaternions between segments is satisfied by the equality constraints: ,in ; in, For the basis of Lagrange polynomials, , , , ; for At Legendre-Gauss-Radau The value; ; , They are respectively and Legendre polynomial of order 1; For the first k The segmented Radau pseudospectral differential matrix The i Line number j The elements of the column.
5. The time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft according to claim 4, characterized in that, Define variables , , , , , for: ， ； ， ， ； The discretized quaternion dynamic equations are transformed into matrix form as follows: ； ， ； The quaternion constraint is: ;in, This represents the Kronecker tensor product of two matrices. For size Radau pseudospectral differential synthesis matrix of order 1 .
6. The time-optimal three-axis redirection method for inertial symmetric rigid spacecraft according to claim 1, characterized in that, Step 4 includes: Set boundary conditions, initial values and range of variation for the two optimizable parameters; Isolation variables The discretized quaternion dynamics equations are transformed into collocation conditions in matrix form and rewritten as follows: ； ， ； ； ， ； ， ； in, for The zero matrix, for The identity matrix; If the coefficient matrix Always reversible, and ,but ;make ,but ; set up ,definition ,but ; Optimize two optimizable parameters to find a solution that satisfies the equation of and make Take the minimum value, where It is a matrix that embodies the symmetry of quaternions; By combining the corresponding Hamiltonian function and switching function, an overdetermined linear system of equations about discrete conjugate quantities is obtained. The Moore-Penrose pseudoinverse is then used to solve the system and obtain the corresponding sequence of discrete conjugate quantities to verify the optimality of the solution.
7. The time-optimal three-axis redirection method for inertial symmetric rigid spacecraft according to claim 6, characterized in that, When using the 5-switch time optimal solution, the corresponding matrix reflecting quaternion symmetry is: ， ； When using the 6-switching time optimal solution, the corresponding matrix reflecting quaternion symmetry is: ， ； in, , , , These are four matrices that demonstrate quaternion symmetry.
8. The time-optimal three-axis retargeting method for an inertial symmetric rigid spacecraft according to claim 1, characterized in that, Based on the discrete quaternion numerical sequence, the two parameters are optimized using the gradient method or Newton's method. The time-optimal retargeting trajectory is determined based on the optimization results, including: Gradient-based or Newton-Raphson-Raphson iterative methods optimize two optimizable parameters to obtain their optimal values. During the optimization process, finite differences are used to provide... Approximate values of the partial derivatives of the two optimizable parameters; The optimal time-based redirection trajectory is determined based on the optimal values of the two optimizable parameters.
9. A time-optimal three-axis redirection device for an inertial symmetric rigid spacecraft, characterized in that, include: The angular velocity analytical expression determination module is used to determine the analytical expression of the spacecraft's angular velocity during the maneuver, based on the spacecraft's inertial symmetry and the a priori time-optimal control switch structure. The analytical expression for the angular velocity is entirely determined by two optimizable parameters; A new inertia axis coordinate system construction module is used to introduce a new inertia axis coordinate system. Under the new inertia axis coordinate system, the symmetry characteristics of the spacecraft redirection trajectory described by quaternions are derived based on the analytical expression of angular velocity. The Radau pseudospectral method discretization module is used to discretize the quaternion dynamic equations using the Radau pseudospectral method, transforming the continuous-time domain time-optimal redirection control problem into a parameterized nonlinear programming problem. The discrete quaternion numerical sequence determination module is used to transform the discretized quaternion dynamic equations into a system of linear equations using two given parameter values, solve the system of linear equations, and obtain the corresponding discrete quaternion numerical sequence. The time-optimal redirection trajectory determination module is used to optimize the two parameters based on the discrete quaternary numerical sequence using the gradient method or Newton's method, and determine the time-optimal redirection trajectory based on the optimization results.