Flexible attachment trajectory convex programming method for small celestial bodies

Abstract

The application discloses a small celestial body flexible attachment trajectory convex programming method and belongs to the technical field of deep space exploration.The application constructs a small celestial body flexible attachment trajectory programming problem according to initial state information of a flexible lander, terminal state information and constraint conditions needed to be met in an attachment process.The flexible attachment trajectory programming problem has a strong nonlinear flexible attachment dynamics model and a non-convex flexible constraint.A homotopy parameter is introduced into the flexible attachment dynamics model to construct an auxiliary problem for small celestial body flexible attachment trajectory programming.The auxiliary problem is connected to the rigid attachment trajectory programming problem and the flexible attachment trajectory programming problem through the homotopy parameter, and a sequence of auxiliary sub-problems constructed by iteratively solving the linearized and discretized auxiliary problem is used to avoid directly solving the complex flexible attachment problem, so that the online programming of the small celestial body flexible attachment fuel consumption optimal trajectory is realized under the condition of limited computing resources, and the convergence and robustness of the small celestial body flexible attachment trajectory convex programming are improved.

Description

Technical Field

[0001] The present invention relates to a trajectory planning method, in particular to a convex planning method for a small celestial body flexible attachment trajectory, and belongs to the field of deep space exploration technology. Background Art

[0002] The surface morphology of small celestial bodies is complex and their gravity is weak. Traditional rigid landers are prone to overturning and rebounding during the attachment process. Compared with rigid landers, the surface shape of flexible landers can increase the contact area with the surface of small celestial bodies and reduce the risk of overturning and rolling after the lander is attached. At the same time, the flexible lander produces flexible deformation when attached to the surface, and uses internal damping to consume the residual kinetic energy generated by surface collision, thereby reducing the risk of rebound and improving the reliability of small celestial body attachment missions. For this reason, this patent focuses on the flexible attachment mission of small celestial bodies. In the flexible attachment mission of small celestial bodies, trajectory planning is a necessary prerequisite for achieving precise landing. The flexible attachment trajectory planning problem has a strongly nonlinear dynamic model, and the flexible internal force changes at high frequency during the attachment process, resulting in poor convergence using traditional sequential convex optimization methods.

[0003] Considering the above factors, it is necessary to quickly plan a landing trajectory that satisfies various state and control constraints for the flexible attachment trajectory of small celestial bodies. This patent uses fuel consumption during landing as a performance indicator and proposes a convex planning method for the flexible attachment trajectory of small celestial bodies. Summary of the Invention

[0004] The main purpose of the present invention is to provide a convex planning method for the flexible attachment trajectory of a small celestial body. This method constructs the flexible attachment trajectory planning problem for a small celestial body based on the initial and terminal state information of the flexible lander and the constraints that need to be met during the attachment process. Compared to the traditional rigid attachment trajectory planning problem, the flexible attachment trajectory planning problem has a strongly nonlinear flexible attachment dynamics model and non-convex flexible constraints. Using traditional sequential convex optimization methods to directly solve the flexible attachment trajectory planning problem has poor convergence. To this end, an auxiliary problem for the flexible attachment trajectory planning of a small celestial body is constructed by introducing homotopy parameters into the flexible attachment dynamics model. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through homotopy parameters. By iteratively solving a sequence of auxiliary subproblems constructed from the linearized and discretized auxiliary problem, online planning of the optimal trajectory for the flexible attachment fuel consumption of a small celestial body is achieved, thereby improving the convergence and robustness of the convex planning of the flexible attachment trajectory of a small celestial body.

[0005] The object of the present invention is achieved through the following technical solutions.

[0006] The present invention discloses a convex planning method for the flexible attachment trajectory of a small celestial body. This method constructs a flexible attachment trajectory planning problem for a small celestial body based on the initial and terminal state information of a flexible lander and the constraints that must be met during the attachment process. By introducing homotopy parameters into the flexible attachment dynamics model, an auxiliary problem for flexible attachment trajectory planning is constructed. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem via the homotopy parameters. The auxiliary problem is then convexified and discretized, transforming it into a small celestial body attachment trajectory planning problem that can be solved using convex optimization. When the homotopy parameter is 0, the auxiliary problem degenerates into a rigid attachment trajectory planning problem. First, the auxiliary problem is solved when the homotopy parameter is 0 to provide initial values ​​for the flexible attachment trajectory planning problem. The homotopy parameter is then incremented, and the solution to the rigid attachment trajectory planning problem is used as the initial parameter to iteratively solve subsequent auxiliary problems until the auxiliary problem is solved when the homotopy parameter is 1. This results in the optimal fuel consumption trajectory for the flexible attachment trajectory planning problem to be planned, thereby achieving convex planning for the flexible attachment trajectory of a small celestial body. The present invention has the advantages of high convergence and strong robustness.

[0007] The present invention discloses a convex planning method for a small celestial body flexible attachment trajectory, comprising the following steps:

[0008] Step 1: Taking optimal fuel consumption as the performance indicator, construct a small celestial body flexible attachment trajectory planning problem based on the flexible lander's initial state, terminal state information, and constraints that need to be met during the attachment process in the fixed coordinate system at the center of the small celestial body. The constraints include thrust amplitude constraint, inter-node distance constraint, equivalent surface inclination constraint, glide angle constraint, minimum mass constraint, and boundary constraint. Among them, the inter-node distance constraint and equivalent surface inclination constraint are constraints unique to the flexible attachment trajectory planning task. The flexible lander has a planar shape and is composed of nodes with rigid characteristics and flexible structures. Autonomous navigation and guidance control are achieved by installing navigation sensors and actuators on the nodes.

[0009] Define the equivalent surface as the plane where the three nodes are located, and define the center position r of the equivalent surface o , center velocity v o and normal vector n s for

[0010]

[0011] Among them, r i and v i (i=1,2,3) represent the position vector of the node in the fixed coordinate system of the small celestial body center, r ij =r j -r i .

[0012] Define the equivalent surface reference coordinate system The center of the equivalent surface is the coordinate origin O s , z s Axis points to the equivalent surface normal, x s Axis and the center of the equivalent surface point to node 1, y s Axis and x s Axis, z s The flexible lander attitude changes from the small celestial body fixed coordinate system to the equivalent surface reference coordinate system. The transformation matrix A s Approximate representation.

[0013] Define the equivalent surface inclination angle θ s is the landing plane normal vector and the equivalent surface normal vector n s Angle

[0014]

[0015] The plane normal vector Pointing outside the small celestial body.

[0016] Since the three nodes have similar properties, in order to simplify the expression, subscript i represents the number of any node of the flexible lander, and subscript j and subscript k represent the numbers of the two nodes adjacent to the node.

[0017] During the flexible attachment process, the dynamic equation of node i in the fixed coordinate system at the center of the small celestial body is:

[0018]

[0019] Where ω=[ω x ,ω y ,ω z ] T is the rotational angular velocity vector of the small celestial body, g(r i )=[g ix ,g iy ,g iz ] T is the local gravitational acceleration vector, T i =[T ix ,T iy ,T iz ] T is the thrust vector, is the thrust amplitude, F if =[F ifx ,F ify ,F ifz ] T is the flexible internal force vector, m i is the quality of node i.

[0020] The flexible internal force consists of conservative force terms and dissipative force terms. The conservative force terms and dissipative force terms are approximated by three-dimensional spring terms and damping terms respectively. At this time, the flexible internal force on the node is expressed as

[0021]

[0022] in, and are the coefficient matrices of the spring term and the damping term respectively. During the attachment process, the mass change equation of the node is expressed as

[0023]

[0024] Among them, I sp is the thruster specific impulse, g E is the acceleration due to gravity at sea level.

[0025] During the attachment process, the flexible lander needs to meet the following state and control constraints:

[0026] (a) Thrust amplitude constraint

[0027] Thrust amplitude output by the flexible lander node thruster||T i ||Bounded

[0028] 0<T min ≤||T i ≤||T max (6)

[0029] Among them, T min is the lower limit of thrust amplitude, T max is the upper bound of the thrust amplitude.

[0030] (b) Inter-node distance constraint

[0031] During the attachment process, excessive compression or stretching of the flexible lander will lead to structural damage. Therefore, the distance between any two nodes i and j of the flexible lander needs to satisfy

[0032] l min ≤||r j -r i ||≤l max (7)

[0033] Among them, l min is the minimum distance between nodes, l max The maximum distance between nodes, l0 is the original distance between nodes.

[0034] (c) Equivalent surface inclination constraint

[0035] In order to prevent the flexible lander from tilting excessively during the attachment process, the equivalent surface inclination angle must be less than the maximum inclination angle.

[0036]

[0037] Among them, θ max Indicates the maximum tilt angle allowed during the attachment process.

[0038] (d) Glide angle constraint

[0039] During the attachment process, in order to ensure that the landing point is always within the observation range of the optical camera and to meet the terminal obstacle avoidance requirements, the glide angle constraint is used to constrain the flexible lander to a cone centered on the landing point. The center position of the equivalent surface is used to approximate the overall position of the flexible lander. In this case, the glide angle constraint is expressed as

[0040]

[0041] in, represents the position vector of the center of the equivalent surface at the landing point, δ max Indicates the maximum glide angle.

[0042] (e) Minimum mass constraint

[0043] During the attachment process, the mass of each node needs to be greater than the dry weight m dry

[0044] m i (t)≥m dry (10)

[0045] In addition to the above constraints, the flexible attachment task also needs to satisfy conventional boundary constraints

[0046]

[0047] Among them, m wet represents the initial mass of the flexible lander node, Represents the position and velocity of node i at the initial moment and the position and velocity at the terminal moment.

[0048] Introducing optimal fuel consumption performance indicators

[0049]

[0050] Based on the constructed dynamic equation (3) and mass change equation (5), the trajectory optimization problem is constructed with the optimal flexible attachment fuel consumption as shown in formula (12) as the optimization goal and the control constraints (6), state constraints (7 to (10) and boundary conditions (11) as constraints.

[0051] Step 2: An auxiliary problem for small body flexible attachment trajectory planning is constructed by introducing homotopy parameters into the flexible attachment dynamics model. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through the homotopy parameters.

[0052] Introducing the homotopy parameter ε∈[0,1] into the flexible adhesion dynamics equation (3), the dynamics equation of the auxiliary problem is obtained

[0053]

[0054] The auxiliary problem is composed of the performance index (12), the dynamic equation (13), the mass change equation (5), the control constraint (6), the state constraints (7 to (10), and the boundary condition (11). The auxiliary problem smoothly connects the rigid adhesion trajectory planning problem and the flexible adhesion trajectory planning problem through the dynamic equation (13) containing homotopy parameters.

[0055] Step 3: Introduce slack variables, convexify the thrust amplitude constraint in the auxiliary problem, and convexify the dynamic equations and other constraints in the auxiliary problem, thereby convexifying the non-convex auxiliary problem and discretizing the auxiliary problem, and then transforming the auxiliary problem into a small celestial body attachment trajectory planning problem that can be solved using convex optimization.

[0056] The auxiliary problem has nonlinear flexible adhesion dynamics, nonconvex control constraints, nonconvex node distance constraints and equivalent surface inclination constraints. The slack variable Γ is introduced. i ,make

[0057] ||T i ||≤Г i (14)

[0058] Use slack variable Г i Substituting the performance index (12), thrust amplitude constraint (6) and thrust amplitude in the mass change equation (5) for the auxiliary problem, we obtain

[0059]

[0060]

[0061]

[0062] In order to ensure the equivalence of variable substitution, the thrust relaxation constraint is added to the relaxed problem

[0063] ||T i ||=Г i (18)

[0064] The thrust relaxation constraint (18) and the minimum distance constraint between nodes (7) are convexified using the first-order Taylor expansion to obtain

[0065]

[0066]

[0067] in, represents the control quantity of node i in the nominal trajectory, represents the relative position vector from node i to node j in the nominal trajectory.

[0068] After convexifying the equivalent surface inclination constraint, we get

[0069]

[0070]

[0071] in, represents the equivalent surface normal vector in the nominal trajectory.

[0072] Before convexifying the dynamic equation (13) of the auxiliary problem, define the state variables and control variables in

[0073]

[0074] The dynamic equation (13) and the mass change equation (5) are equivalent to the nonlinear vector function f of the state vector and the control vector defined by equation (23):

[0075]

[0076] Each time the auxiliary problem is solved, the homology parameter ε is a constant.

[0077] Define the nominal trajectory, and its state and control quantities are and Performing a first-order Taylor expansion on the right side of equation (24) near the nominal trajectory yields the convexified flexible adhesion dynamics:

[0078]

[0079] in

[0080]

[0081] After convexifying the auxiliary problem, the auxiliary problem is further discretized. In the actual attachment mission, the control frequency dt of the flexible lander is determined, and the trajectory can be discretized into N evenly distributed discrete points, where

[0082]

[0083] For the convenience of expression, define two sets

[0084]

[0085] At the same time, the time at the discrete point q is defined as

[0086]

[0087] In order to improve the feasibility of the optimization problem, the control sequence is first-order held at each time interval. Therefore, in the interval t∈[t q ,t q+1 ], let u(t) be u q and u q+1 Function

[0088]

[0089] in

[0090]

[0091] Use Φ A (t q+1 ,t q ) means zero input, from state Transfer to state The state transfer matrix Φ A (t q+1 ,t q ) is determined by the differential equation shown in formula (32).

[0092]

[0093] Combining equations (30) and (32), the state x q To state x q+1 The discrete-time dynamic equation is expressed as

[0094]

[0095] in

[0096]

[0097] In order to make the performance indicators, state constraints, control constraints and boundary conditions in the auxiliary problem q , Therefore, the performance indicators and constraints must satisfy equations (35) to (42).

[0098] Performance indicators

[0099]

[0100] Dynamic Constraints

[0101]

[0102] Thrust amplitude constraint

[0103]

[0104] Flexible constraints

[0105]

[0106]

[0107] Glide angle constraint

[0108]

[0109] Minimum mass constraint

[0110] m i,q ≥m dry (41)

[0111] Boundary conditions

[0112]

[0113] Step 4. Solve the auxiliary problem when the homology parameter is 0 to provide the initial value for the flexible attachment trajectory planning problem of the small celestial body. Increase the homology parameter and use the solution of the rigid attachment trajectory planning problem as the initial parameter to iteratively solve the subsequent auxiliary problems until the auxiliary problem with the homology parameter being 1 is solved. The optimal fuel consumption trajectory of the flexible attachment trajectory planning problem is obtained, that is, the convex planning of the flexible attachment trajectory of the small celestial body is realized.

[0114] Formula (43) is used as the auxiliary problem when the nominal trajectory is convexified and discretized ε = 0. At this time, the auxiliary problem degenerates into the rigid attachment trajectory planning problem, that is, the first auxiliary problem in the auxiliary sub-problem sequence is obtained and solved.

[0115]

[0116] The relative deviation between state and control is defined as

[0117]

[0118] Among them, p(p∈{1,…,P}) represents the p-th iteration, and P represents the total number of iterations.

[0119] The convergence condition for determining the convergence of the auxiliary problem sub-problem sequence is shown in formula (45):

[0120]

[0121] Among them, Δ tol is the upper bound of the allowed error.

[0122] In the pth round of iteration, first use the nominal trajectory of this round The auxiliary problem is convexified and discretized, and then solved. If the optimal solution does not meet the convergence condition shown in formula (45), the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration At the same time, if ε p <1, then let ε p+1 =ε p +Δ ε , where Δ ε is the homotopy parameter increment, and enters the p+1th round of iteration; if the optimal solution meets the convergence conditions shown in formula (45), but ε p <1, then let ε p+1 =1, and the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration Enter the p+1th iteration; If the optimal solution satisfies the convergence condition shown in formula (45), and ε p =1, then the optimal trajectory of flexible attachment burnup of small celestial body is outputted finally.

[0123] The method also includes step 5, performing guidance control based on the fuel-optimal trajectory of the flexible attachment trajectory planning problem obtained in step 4, so that the flexible lander can achieve efficient and precise attachment with minimal fuel consumption while satisfying various constraints.

[0124] Beneficial effects:

[0125] 1. The convex planning method for the flexible attachment trajectory of a small celestial body disclosed in the present invention is aimed at a flexible lander with a planar shape and composed of nodes with rigid characteristics and flexible structures. It takes optimal fuel consumption as the performance indicator. According to the initial state and terminal state information of the flexible lander in the fixed coordinate system at the center of the small celestial body and the constraints that need to be met during the attachment process, the small celestial body flexible attachment trajectory planning problem is constructed for the first time. Convex planning is introduced in the flexible attachment to improve the efficiency of the flexible attachment planning solution. It is more suitable for the trajectory planning needs of the flexible attachment of small celestial bodies under the conditions of limited computing resources.

[0126] 2. Compared with the traditional rigid attachment trajectory planning problem, the flexible attachment trajectory planning problem has a strongly nonlinear flexible attachment dynamics model and non-convex flexible constraints. The convergence of the flexible attachment trajectory planning problem directly solved using the traditional sequential convex optimization method is poor. The convex planning method for the flexible attachment trajectory of small celestial bodies disclosed in the present invention introduces homotopy parameters into the flexible attachment dynamics model to construct an auxiliary problem for the flexible attachment trajectory planning of small celestial bodies. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through the homotopy parameters. The auxiliary problem is convexified and discretized to transform the auxiliary problem into an auxiliary problem for the small celestial body attachment trajectory planning problem that can be solved using convex optimization. The sequence of auxiliary sub-problems composed of the convexified and discretized auxiliary problems is iteratively solved to avoid directly solving the complex flexible attachment problem, realize the rapid generation of the optimal trajectory for the flexible attachment fuel consumption of small celestial bodies, and thus improve the convergence of the flexible attachment trajectory planning of small celestial bodies. BRIEF DESCRIPTION OF THE DRAWINGS

[0127] Figure 1 This is a flow chart of the convex planning method for the flexible attachment trajectory of small celestial bodies;

[0128] Figure 2 Schematic diagram of the flexible lander normal vector and the equivalent surface reference coordinate system;

[0129] Figure 3 This is the relationship between the three-axis position and velocity of the flexible lander node 1's optimal fuel consumption trajectory and time;

[0130] Figure 4 This is the optimal thrust curve for the fuel consumption of node 1 of the flexible lander.

[0131] Figure 5 is the curve of the flexible internal force between the nodes of the flexible lander changing with time, where: Figure 5 (a) is the relationship between the flexible internal force of node 1 and time. Figure 5 (b) is the relationship between the flexible internal force of node 2 and time. Figure 5 (c) is a graph showing the relationship between the flexible internal force acting on node 3 and time. DETAILED DESCRIPTION

[0132] In order to better illustrate the purpose and advantages of the present invention, the invention is further described below with reference to an example and corresponding drawings.

[0133] like Figure 1 As shown, the convex planning method for the flexible attachment trajectory of a small celestial body disclosed in this embodiment is specifically implemented in the following steps:

[0134] Step 1: Taking optimal fuel consumption as the performance indicator, construct a small celestial body flexible attachment trajectory planning problem based on the flexible lander's initial state, terminal state information, and constraints that need to be met during the attachment process in the fixed coordinate system at the center of the small celestial body. The constraints include thrust amplitude constraint, inter-node distance constraint, equivalent surface inclination constraint, glide angle constraint, minimum mass constraint, and boundary constraint. Among them, the inter-node distance constraint and equivalent surface inclination constraint are constraints unique to the flexible attachment trajectory planning task. The flexible lander has a planar shape and is composed of nodes with rigid characteristics and flexible structures. Autonomous navigation and guidance control are achieved by installing navigation sensors and actuators on the nodes.

[0135] Define the equivalent surface as the plane where the three nodes are located, and define the center position r of the equivalent surface o , center velocity v o and normal vector n s for

[0136]

[0137] Among them, r i and v i (i=1,2,3) represent the position vector of the node in the fixed coordinate system of the small celestial body center, r ij =r j -r i .

[0138] Define the equivalent surface reference coordinate system The center of the equivalent surface is the coordinate origin O s , z s Axis points to the equivalent surface normal, x s Axis and the center of the equivalent surface point to node 1, y s Axis and x s Axis, z s The axes form a right-handed coordinate system. Figure 2 The equivalent surface normal vector and equivalent surface reference coordinate system during the attachment process are shown. The attitude of the flexible lander changes from the fixed coordinate system of the small celestial body to the equivalent surface reference coordinate system. The transformation matrix A s Approximate representation.

[0139] Define the equivalent surface inclination angle θ s is the landing plane normal vector and the equivalent surface normal vector n s Angle

[0140]

[0141] The plane normal vector Pointing outside the small celestial body.

[0142] Since the three nodes have similar properties, in order to simplify the expression, subscript i represents the number of any node of the flexible lander, and subscript j and subscript k represent the numbers of the two nodes adjacent to the node.

[0143] During the flexible attachment process, the dynamic equation of node i in the fixed coordinate system at the center of the small celestial body is:

[0144]

[0145] Where ω=[ω x ,ω y ,ω z ] T is the rotational angular velocity vector of the small celestial body, g(r i )=[g ix ,g iy ,g iz ] T is the local gravitational acceleration vector, T i =[T ix ,T iy ,T iz ] T is the thrust vector, is the thrust amplitude, F if =[F ifx ,F ify ,F ifz ] T is the flexible internal force vector, m i is the quality of node i.

[0146] The flexible internal force consists of conservative force terms and dissipative force terms. The conservative force terms and dissipative force terms are approximated by three-dimensional spring terms and damping terms respectively. At this time, the flexible internal force on the node is expressed as

[0147]

[0148] in, and are the coefficient matrices of the spring term and the damping term respectively. During the attachment process, the mass change equation of the node is expressed as

[0149]

[0150] Among them, I sp =150s is the thruster specific impulse, g E =9.81m / s 2 is the acceleration due to gravity at sea level.

[0151] During the attachment process, the flexible lander needs to meet the following state and control constraints:

[0152] (a) Thrust amplitude constraint

[0153] Thrust amplitude output by the flexible lander node thruster||T i ||Bounded

[0154] 0<T min ≤||T i ||≤T max (51)

[0155] Among them, T min =2N is the lower limit of thrust amplitude, T max =20N is the upper limit of thrust amplitude.

[0156] (b) Inter-node distance constraint

[0157] During the attachment process, excessive compression or stretching of the flexible lander will lead to structural damage. Therefore, the distance between any two nodes i and j of the flexible lander needs to satisfy

[0158] l min ≤||r j -r i ||≤l max (52)

[0159] Among them, l min =0.8l0 is the minimum distance between nodes, l max =1.2l0 Maximum distance between nodes, is the original distance between nodes.

[0160] (c) Equivalent surface inclination constraint

[0161] In order to prevent the flexible lander from tilting excessively during the attachment process, the equivalent surface inclination angle must be less than the maximum inclination angle.

[0162]

[0163] Among them, θ max =30deg indicates the maximum tilt angle allowed during the attachment process.

[0164] (d) Glide angle constraint

[0165] During the attachment process, in order to ensure that the landing point is always within the observation range of the optical camera and to meet the terminal obstacle avoidance requirements, the glide angle constraint is used to constrain the flexible lander to a cone centered on the landing point. The center position of the equivalent surface is used to approximate the overall position of the flexible lander. In this case, the glide angle constraint is expressed as

[0166]

[0167] in, represents the position vector of the center of the equivalent surface at the landing point, δmax =30deg indicates the maximum glide angle.

[0168] (e) Minimum mass constraint

[0169] During the attachment process, the mass of each node needs to be greater than the dry weight m dry =100kg

[0170] m i (t)≥m dry (55)

[0171] In addition to the above constraints, the flexible attachment task also needs to satisfy conventional boundary constraints

[0172]

[0173] Among them, m wet =300kg represents the initial mass of the flexible lander node, Represents the position and velocity of node i at the initial moment and the position and velocity at the terminal moment.

[0174] The initial position, initial velocity and initial posture of the equivalent surface center are r o (0) = [7254.79, -6031.66, -8496.68] m and v o (0) = [-1.23, 1.44, -0.42] m / s and q(t0) = [0.1759, 0.7309, 0.6007, -0.2720]; the end position, end velocity and end attitude of the equivalent surface center are r o (t f )=[6843.68,-4678.87,-4545.95]m and v o (t f )=[0,0,0]m / s and q(t f )=[0.1759,0.7309,0.6007,-0.2720].

[0175] In this example, the distance between the flexible lander nodes at the initial and final moments is set to the original length, and the velocity of each node is equal to the velocity of the center of the equivalent surface. At this time, Equation (57) is used to inversely solve the position vector and velocity vector of each node at the initial and final moments.

[0176]

[0177] Introducing optimal fuel consumption performance indicators

[0178]

[0179] Based on the constructed dynamic equation (48) and mass change equation (50), the trajectory optimization problem is constructed with the optimal flexible attachment fuel consumption as shown in formula (58) as the optimization goal, and the control constraints (51), state constraints (52 to (55) and boundary conditions (56) as constraints.

[0180] Step 2: An auxiliary problem for small body flexible attachment trajectory planning is constructed by introducing homotopy parameters into the flexible attachment dynamics model. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through the homotopy parameters.

[0181] Introducing the homotopy parameter ε∈[0,1] into the flexible adhesion dynamics equation (48), the dynamics equation of the auxiliary problem is obtained

[0182]

[0183] The auxiliary problem is composed of a performance index (58), a dynamic equation (59), a mass change equation (50), a control constraint (51), state constraints (52 to (55), and a boundary condition (56). The auxiliary problem smoothly connects the rigid adhesion trajectory planning problem and the flexible adhesion trajectory planning problem through the dynamic equation (59) containing homotopy parameters.

[0184] Step 3: Introduce slack variables, convexify the thrust amplitude constraint in the auxiliary problem, and convexify the dynamic equations and other constraints in the auxiliary problem, thereby convexifying the non-convex auxiliary problem and discretizing the auxiliary problem, and then transforming the auxiliary problem into a small celestial body attachment trajectory planning problem that can be solved using convex optimization.

[0185] The auxiliary problem has nonlinear flexible adhesion dynamics, nonconvex control constraints, nonconvex node distance constraints and equivalent surface inclination constraints. The slack variable Γ is introduced. i ,make

[0186] ||T i ||=Γ i (60)

[0187] Use slack variable Γ i Substituting the performance index (58), thrust amplitude constraint (51) and thrust amplitude in the mass change equation (50) for the auxiliary problem, we obtain

[0188]

[0189]

[0190]

[0191] In order to ensure the equivalence of variable substitution, the thrust relaxation constraint is added to the relaxed problem

[0192] ||T i ||=Γ i (64)

[0193] The thrust relaxation constraint (64) and the minimum distance constraint between nodes (52) are convexified using the first-order Taylor expansion to obtain

[0194]

[0195]

[0196] in, represents the control quantity of node i in the nominal trajectory, represents the relative position vector from node i to node j in the nominal trajectory.

[0197] After convexifying the equivalent surface inclination constraint, we get

[0198]

[0199]

[0200] in, represents the equivalent surface normal vector in the nominal trajectory.

[0201] Before convexifying the dynamic equations (59) of the auxiliary problem, define the state variables and control variables in

[0202]

[0203] The dynamic equation (59) and the mass change equation (50) are equivalent to the nonlinear vector function f of the state vector and the control vector defined by equation (70):

[0204]

[0205] Each time the auxiliary problem is solved, the homology parameter ε is a constant.

[0206] Define the nominal trajectory, and its state and control quantities are and Performing a first-order Taylor expansion on the right side of Equation (70) near the nominal trajectory yields the convexified flexible adhesion dynamics:

[0207]

[0208] in

[0209]

[0210] After convexifying the auxiliary problem, the auxiliary problem is further discretized. In the actual attachment mission, the control frequency dt of the flexible lander is determined, and the trajectory can be discretized into N evenly distributed discrete points, where

[0211]

[0212] For the convenience of expression, define two sets

[0213]

[0214] At the same time, the time at the discrete point q is defined as

[0215]

[0216] In order to improve the feasibility of the optimization problem, the control sequence is first-order held at each time interval. Therefore, in the interval t∈[t q ,t q+1 ], let u(t) be u q and u q+1 Function

[0217]

[0218] in

[0219]

[0220] Use Φ A (t q+1 ,t q ) means that under zero input, the state Transfer to state The state transfer matrix Φ A (t q+1 ,t q ) is determined by the differential equation shown in formula (32).

[0221]

[0222] Combining equations (76) and (78), the state x q To state x q+1 The discrete-time dynamic equation is expressed as

[0223]

[0224] in

[0225]

[0226] In order to make the performance indicators, state constraints, control constraints and boundary conditions in the auxiliary problem q , Therefore, the performance indicators and constraints must satisfy equations (81) to (88).

[0227] Performance indicators

[0228]

[0229] Dynamic Constraints

[0230]

[0231] Thrust amplitude constraint

[0232]

[0233] Flexible constraints

[0234]

[0235]

[0236] Glide angle constraint

[0237]

[0238] Minimum mass constraint

[0239] m i,q ≥m dry (87)

[0240] Boundary conditions

[0241]

[0242] Step 4. Solve the auxiliary problem when the homology parameter is 0 to provide the initial value for the flexible attachment trajectory planning problem of the small celestial body. Increase the homology parameter and use the solution of the rigid attachment trajectory planning problem as the initial parameter to iteratively solve the subsequent auxiliary problems until the auxiliary problem with the homology parameter being 1 is solved. The optimal fuel consumption trajectory of the flexible attachment trajectory planning problem is obtained, that is, the convex planning of the flexible attachment trajectory of the small celestial body is realized.

[0243] Formula (89) is used as the auxiliary problem when the nominal trajectory is convexified and discretized ε = 0. At this time, the auxiliary problem degenerates into the rigid attachment trajectory planning problem, that is, the first auxiliary problem in the auxiliary sub-problem sequence is obtained and solved.

[0244]

[0245] The relative deviation between state and control is defined as

[0246]

[0247] Among them, p(p∈{1,…,P-1}) represents the p-th iteration, and P represents the total number of iterations.

[0248] The convergence condition for determining the convergence of the auxiliary problem sub-problem sequence is shown in formula (45):

[0249]

[0250] Among them, Δ tol is the upper bound of the allowed error.

[0251] In the pth round of iteration, first use the nominal trajectory of this round The auxiliary problem is convexified and discretized, and then solved. If the optimal solution does not meet the convergence condition shown in formula (91), the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration At the same time, if ε p <1, then let ε p+1 =ε p +Δ ε , where Δ ε =0.2 is the homotopy parameter increment, and enters the p+1th round of iteration; if the optimal solution meets the convergence conditions shown in formula (91), but ε p <1, the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration And let ε p+1 =1, enter the p+1th round of iteration; if the optimal solution meets the convergence conditions shown in formula (91), and ε p =1, then the optimal trajectory of flexible attachment burnup of small celestial body is outputted finally.

[0252] The method also includes step 5, performing guidance control based on the fuel-optimal trajectory of the flexible attachment trajectory planning problem obtained in step 4, so that the flexible lander can achieve efficient and precise attachment with minimal fuel consumption while satisfying various constraints.

[0253] Figure 3 This is a diagram showing the relationship between the three-axis position and velocity of the flexible lander node 1's optimal fuel consumption trajectory and time. Figure 4 This is the optimal thrust curve for the fuel consumption of node 1 of the flexible lander. Figure 5 is the curve of the flexible internal force between each node of the flexible lander changing with time.

[0254] The above specific description further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A convex planning method for the flexible attachment trajectory of a small celestial body, characterized by: The following steps are included: Step 1: Taking optimal fuel consumption as the performance indicator, construct a small celestial body flexible attachment trajectory planning problem based on the flexible lander's initial state and terminal state information in the fixed coordinate system at the center of the small celestial body and the constraints that need to be met during the attachment process; the constraints include thrust amplitude constraint, inter-node distance constraint, equivalent surface inclination constraint, glide angle constraint, minimum mass constraint, and boundary constraint, among which the inter-node distance constraint and equivalent surface inclination constraint are constraints unique to the flexible attachment trajectory planning task; the flexible lander has a planar shape and is composed of nodes with rigid characteristics and flexible structures. Autonomous navigation and guidance control are achieved by installing navigation sensors and actuators on the nodes; Step 1 is implemented as follows: Define the equivalent surface as the plane where the three nodes are located, and define the center position r of the equivalent surface o , center velocity v o and normal vector n s for Among them, r i and v i Respectively represent the position vector of the node in the fixed coordinate system of the small celestial body center, i=1,2,3, r ij =r j -r i ; Define the equivalent surface reference coordinate system The center of the equivalent surface is the coordinate origin O s , z s Axis points to the equivalent surface normal, x s Axis and the center of the equivalent surface point to node 1, y s Axis and x s Axis, z s The axes form a right-handed coordinate system; the attitude of the flexible lander changes from the small celestial body fixed coordinate system to the equivalent surface reference coordinate system The transformation matrix A s Approximate representation; Define the equivalent surface inclination angle θ s is the landing plane normal vector and the equivalent surface normal vector n s Angle The plane normal vector Pointing to the outside of the small celestial body; Since the three nodes have similar properties, in order to simplify the description, subscript i represents the number of any node of the flexible lander, and subscript j and subscript k represent the numbers of the two nodes adjacent to the node; During the flexible attachment process, the dynamic equation of node i in the fixed coordinate system at the center of the small celestial body is: Where ω=[ω x ,ω y ,ω z ] T is the rotational angular velocity vector of the small celestial body, g(r i )=[g ix ,g iy ,g iz ] T is the local gravitational acceleration vector, T i =[T ix ,T iy ,T iz ] T is the thrust vector, is the thrust amplitude, F if =[F ifx ,F ify ,F ifz ] T is the flexible internal force vector, m i is the mass of node i; The flexible internal force consists of conservative force terms and dissipative force terms. The conservative force terms and dissipative force terms are approximated by three-dimensional spring terms and damping terms respectively. At this time, the flexible internal force on the node is expressed as in, and are the coefficient matrices of the spring term and the damping term respectively; during the attachment process, the mass change equation of the node is expressed as Among them, I sp is the thruster specific impulse, g E is the acceleration of gravity at sea level; During the attachment process, the flexible lander needs to meet the following state and control constraints: (a) Thrust amplitude constraint Thrust amplitude output by the flexible lander node thruster||T i ||Bounded 0<T min ≤||T i ||≤T max (6) Among them, T min is the lower limit of thrust amplitude, T max is the upper bound of the thrust amplitude; (b) Inter-node distance constraint During the attachment process, excessive compression or stretching of the flexible lander will lead to structural damage; therefore, the distance between any two nodes i and j of the flexible lander needs to satisfy l min ≤||r j -r i ||≤l max (7) Among them, l min is the minimum distance between nodes, l max The maximum distance between nodes, l0 is the original distance between nodes; (c) Equivalent surface inclination constraint In order to prevent the flexible lander from tilting excessively during the attachment process, the equivalent surface inclination angle must be less than the maximum inclination angle. Among them, θ max Indicates the maximum tilt angle allowed during the attachment process; (d) Glide angle constraint During the attachment process, in order to ensure that the landing point is always within the observation range of the optical camera and to meet the terminal obstacle avoidance requirements, the glide angle constraint is used to constrain the flexible lander to a cone centered on the landing point. The center position of the equivalent surface is used to approximate the overall position of the flexible lander. At this time, the glide angle constraint is expressed as in, represents the position vector of the center of the equivalent surface at the landing point, δ max represents the maximum glide angle; (e) Minimum mass constraint During the attachment process, the mass of each node needs to be greater than the dry weight m dry m i (t)≥m dry (10) In addition to the above constraints, the flexible attachment task also needs to satisfy conventional boundary constraints Among them, m wet represents the initial mass of the flexible lander node, Represents the position and velocity of node i at the initial moment and the position and velocity at the terminal moment; Introducing optimal fuel consumption performance indicators Based on the constructed dynamic equation (3) and mass change equation (5), the trajectory optimization problem is constructed with the flexible attachment fuel consumption optimization as shown in formula (12) as the optimization goal and the control constraints (6), state constraints (7 to (10) and boundary conditions (11) as constraints. Step 2: By introducing homotopy parameters into the flexible attachment dynamics model, an auxiliary problem for small celestial body flexible attachment trajectory planning is constructed. The auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through the homotopy parameters. Step 2 is implemented as follows: Introducing the homotopy parameter ε∈[0,1] into the flexible adhesion dynamics equation (3), the dynamics equation of the auxiliary problem is obtained The auxiliary problem is composed of a performance index (12), a dynamic equation (13), a mass change equation (5), a control constraint (6), state constraints (7 to (10) and a boundary condition (11); the auxiliary problem smoothly connects the rigid attachment trajectory planning problem and the flexible attachment trajectory planning problem through the dynamic equation (13) containing homotopy parameters; Step 3: Introduce slack variables to convexify the thrust amplitude constraint in the auxiliary problem, and convexify the dynamic equations and other constraints in the auxiliary problem, thereby convexifying the non-convex auxiliary problem and discretizing the auxiliary problem, thereby transforming the auxiliary problem into a small celestial body attachment trajectory planning problem that can be solved using convex optimization; Step 4. Solve the auxiliary problem when the homology parameter is 0 to provide the initial value for the flexible attachment trajectory planning problem of the small celestial body. Increase the homology parameter and use the solution of the rigid attachment trajectory planning problem as the initial parameter to iteratively solve the subsequent auxiliary problems until the auxiliary problem with the homology parameter being 1 is solved. The optimal fuel consumption trajectory of the flexible attachment trajectory planning problem is obtained, that is, the convex planning of the flexible attachment trajectory of the small celestial body is realized.

2. The convex planning method for the flexible attachment trajectory of a small celestial body according to claim 1, characterized in that: The method also includes step 5, performing guidance control based on the fuel-optimal trajectory of the flexible attachment trajectory planning problem obtained in step 4, so that the flexible lander can achieve efficient and precise attachment with minimal fuel consumption while satisfying various constraints.

3. The convex planning method for the flexible attachment trajectory of a small celestial body according to claim 1, characterized in that: Step 3 is implemented as follows: The auxiliary problem has nonlinear flexible adhesion dynamics, nonconvex control constraints, nonconvex node distance constraints and equivalent surface inclination constraints; the slack variable Γ is introduced i ,make ||T i ||≤Γ i (14) Use slack variable Γ i Substituting the performance index (12), thrust amplitude constraint (6) and thrust amplitude in the mass change equation (5) for the auxiliary problem, we obtain In order to ensure the equivalence of variable substitution, the thrust relaxation constraint is added to the relaxed problem ||T i ||=C i (18) The thrust relaxation constraint (18) and the minimum distance constraint between nodes are convexified using the first-order Taylor expansion to obtain in, represents the control quantity of node i in the nominal trajectory, represents the relative position vector from node i to node j in the nominal trajectory; After convexifying the equivalent surface inclination constraint, we get in, represents the equivalent surface normal vector in the nominal trajectory; Before convexifying the dynamic equation (13) of the auxiliary problem, define the state variables and control variables in The dynamic equation (13) and the mass change equation (5) are equivalent to the nonlinear vector function f of the state vector and the control vector defined by equation (23): Each time the auxiliary problem is solved, the homotopy parameter ε is a constant; Define the nominal trajectory, and its state and control quantities are and Performing a first-order Taylor expansion on the right side of equation (24) near the nominal trajectory yields the convexified flexible adhesion dynamics: in After convexifying the auxiliary problem, the auxiliary problem is further discretized. In the actual attachment mission, the control frequency dt of the flexible lander is determined, and the trajectory can be discretized into N evenly distributed discrete points, where For the convenience of expression, define two sets At the same time, the time at the discrete point q is defined as In order to improve the feasibility of the optimization problem, the control sequence is first-order held at each time interval; therefore, in the interval t∈[t q ,t q+1 ], let u(t) be u q and u q+1 Function in Use Φ A (t q+1 ,t q ) means that under zero input, the state Transfer to state The state transfer matrix of A (t q+1 ,t q ) is determined by the differential equation shown in formula (32); Combining equations (30) and (32), the state x q To state x q+1 The discrete-time dynamic equation is expressed as in In order to make the performance indicators, state constraints, control constraints and boundary conditions in the auxiliary problem q , Therefore, the performance indicators and constraints must satisfy equations (35) to (42); Performance indicators Dynamic Constraints Thrust amplitude constraint Flexible constraints Glide angle constraint Minimum mass constraint m i,q ≥m dry (41) Boundary conditions 。 4. The convex planning method for the flexible attachment trajectory of a small celestial body according to claim 3, characterized in that: Step 4 is implemented as follows: Formula (43) is used as the auxiliary problem when the nominal trajectory is convexified and discretized ε = 0. At this time, the auxiliary problem degenerates into the rigid attachment trajectory planning problem, that is, the first auxiliary problem in the auxiliary sub-problem sequence is obtained and solved; The relative deviation between state and control is defined as Where p represents the pth iteration, p∈{1,…,P} represents the total number of iterations; The convergence condition for determining the convergence of the auxiliary problem sub-problem sequence is shown in formula (45): Among them, Δ tol is the upper bound of the allowed error; In the pth round of iteration, first use the nominal trajectory of this round The auxiliary problem is convexified and discretized, and then solved. If the optimal solution does not satisfy the convergence condition shown in formula (45), the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration. At the same time, if ε p <1, then let ε p+1 =ε p +Δ ε , where Δ ε is the homotopy parameter increment, and enters the p+1th round of iteration; if the optimal solution meets the convergence conditions shown in formula (45), but ε p <1, then let ε p+1 =1, and the optimal solution of this round is used as the nominal trajectory of the p+1th round of iteration Enter the p+1th iteration; if the optimal solution satisfies the convergence condition shown in formula (45), and ε p =1, then the optimal trajectory of flexible attachment burnup of small celestial body is outputted finally.

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