Robust tracking control method for flexible attached trajectory of small celestial body

Abstract

This invention discloses a robust tracking control method for flexible attachment trajectories on small celestial bodies, belonging to the field of deep space probe control technology. The implementation method is as follows: While retaining the flexibility characteristics, the structure of the flexible probe is simplified. A "spring-damping-torsion spring" model of the flexible probe is established in both the flexible node reference coordinate system and the node fixed coordinate system, enabling it to possess both flexibility and, as far as possible, effectively estimate and control the state of its attachment process. A nominal trajectory error tracking control model is established based on the dynamic model. A performance index function is designed to transform the flexible probe attachment trajectory tracking control problem into an optimal problem. An HJB equation is established based on the performance index function, and the expression for the optimal control force is calculated. Through strategy iteration, the control law and performance index function are formed into a closed-loop equation. A single-evaluation neural network is used to approximate the optimal control law and optimal performance index function, achieving robust tracking of the nominal trajectory while reducing probe fuel consumption.

Description

Technical Field

[0001] This invention relates to a robust tracking control method for flexible attachment trajectories on small celestial bodies, belonging to the field of deep space probe control technology. Background Technology

[0002] There are two main attachment methods used in past asteroid landing missions: one is the "rigid + buffer" method. Due to the weak gravity of asteroids, this method makes the probe prone to bounce and overturning, greatly increasing the risk of mission failure. The other is the "contact and go" method, used by the OSIRIS-REx mission, Hayabusa, and Hayabusa2 to successfully land on asteroid surfaces. However, the contact time is too short to obtain sufficient samples. In small body exploration missions, flexible multi-node probes can reduce the risk of overturning and bounce when the probe contacts the asteroid surface. The lander uses flexible deformable materials and has a disc-shaped shape. Compared to rigid landers, flexible landers increase the contact area with the asteroid surface, reducing the risk of overturning and tumbling when the probe's attitude deviates. Furthermore, the unique flexible material can dissipate residual kinetic energy during terminal velocity deviations, preventing bounce and escape.

[0003] The flexible lander considers a mass cluster with specific functions and rigid characteristics as the smallest unit, and the control actuators are installed on the mass cluster. Flexible connections exist between the mass clusters, and during attachment, the lander is subjected to flexible forces generated by relative tension, compression, bending, and torsion. Compared to rigid-structured probes, flexible probes have more complex dynamic characteristics, posing significant challenges to controller design. Furthermore, during flexible attachment, it is difficult to accurately model the dynamics of the flexible probe, and the interaction forces between nodes have strong uncertainties, greatly increasing the difficulty of controlling the safe and stable attachment of the flexible probe. For the trajectory tracking control problem of the flexible probe landing on the surface of a small celestial body, the influence of uncertain forces between nodes needs to be considered. Based on this, the controller output needs to accurately track the desired commands to ensure the probe's safe and stable attachment to the asteroid surface. Summary of the Invention

[0004] The purpose of this invention is to provide a robust tracking control method for flexible attachment trajectories on small celestial bodies. A "spring-damping-torsion spring" model of the flexible detector is established in both the flexible node reference coordinate system and the node fixed coordinate system, enabling it to possess both flexibility and, as far as possible, effective estimation and control of the attachment process's state. Based on the dynamic model, a nominal trajectory error tracking control model is established. A performance index function is designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. The HJB equation is established based on the performance index function, and the expression for the optimal control force is calculated. Through strategy iteration, the control law and performance index function are formed into a closed-loop equation. A single-evaluation neural network is used to approximate the optimal control law and optimal performance index function, achieving robust tracking of the nominal trajectory while reducing detector fuel consumption.

[0005] The objective of this invention is achieved through the following technical solution.

[0006] This invention discloses a robust tracking control method for flexible attachment trajectories on small celestial bodies. While preserving the flexibility characteristics, it simplifies the structure of the flexible detector. A "spring-damping-torsion spring" model is established for the flexible detector within both the flexible node reference coordinate system and the node fixed coordinate system, ensuring both flexibility and effective estimation and control of the attachment process. Addressing the uncertainty of the interaction forces between nodes, a nominal trajectory error tracking control model is established based on a dynamic model. A performance index function is designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. An HJB equation is established based on the performance index function, and the expression for the optimal control force is calculated. Through strategy iteration, the control law and performance index function are formed into a closed-loop equation. A single-evaluation neural network is used to approximate the optimal control law and optimal performance index function, achieving robust tracking of the nominal trajectory while effectively reducing detector fuel consumption.

[0007] The robust tracking control method for flexible attachment trajectories of small celestial bodies disclosed in this invention includes the following steps:

[0008] Step 1: Take the mass cluster area with specific functions and rigid characteristics on the flexible lander as the smallest constituent unit, establish an equivalent dynamic model of the flexible body and a generalized spring-damping-torsion spring model to approximate the flexible force, so that the generalized spring-damping-torsion spring model has both flexible characteristics and can effectively estimate and control the state of its attachment process as much as possible.

[0009] The minimum constituent unit is considered as a mass cluster with specific functions and rigid characteristics on a flexible lander. Control actuators are installed on these mass clusters. Flexible connections exist between these mass clusters, and during attachment, they are subjected to flexible forces generated by relative tension, compression, bending, and torsion of the flexible bodies. For a flexible lander with a planar shape, the three-mass cluster model is the simplest configuration for equivalently simulating the motion of the flexible lander.

[0010] To facilitate the description of the motion of flexible detector nodes, the following coordinate system is defined:

[0011] (1) Nodal reference coordinate system Origin O is Located at the centroid of the node, x is axis and y is The axis lies on the plane containing the three nodes, z s The axis is perpendicular to the plane containing the three nodes.

[0012] (2) Node-fixed coordinate system Origin O ia Located at the centroid of the node, z ia Coinciding with the principal axis of minimum inertia of node i, x ia y coincides with the principal axis of maximum inertia of node i. ia The axis satisfies the right-hand rule.

[0013] Considering the flexible forces and flexible moments acting on node i, the translational dynamic equation of node i in the asteroid fixed coordinate system is:

[0014]

[0015] Where r i v i and T i These represent the position, velocity, and thrust vector of node i in the asteroid's fixed coordinate system, respectively; ω is the spin angular velocity vector of the small celestial body; and m... i For the mass of node i, F if Let be the flexible force acting on node i. The dynamic models of nodes j and k have the same expression. The calculation steps for the flexible force are given below.

[0016] Flexible torque T on node i if Approximated by a torsion spring

[0017]

[0018] Where θ i and β i These represent the rotation angles of the fixed coordinate system at node i relative to the x-axis and y-axis of the reference coordinate system, respectively, with the rotation sequence being ZXY. The transformation matrix K1, K2, C1, and C2 are constant coefficients to be determined for transforming node i from the reference coordinate system to the asteroid fixed coordinate system.

[0019] Flexible force F if From tension and compression internal forces and bending and torsional internal forces constitute:

[0020]

[0021] The bending and torsional internal forces are generated by the torsion spring torque to ensure the balance of internal forces and internal moments of the flexible lander. The flexible torque acting at node i is T. if Based on the equilibrium of internal forces and internal moments, the bending and torsional internal forces satisfy...

[0022]

[0023] and The bending and torsional internal forces acting on nodes j and k are respectively, r ij With r ik Let i and j represent the position vectors between nodes i and j, and i and k, respectively. The bending and torsional internal forces have zero components in the x and y directions of the node reference coordinate system. At this time, equation (2) has a unique solution. Solve for the reaction forces generated by the flexible moments acting on the three nodes in sequence to obtain the bending and torsional internal forces of the nodes.

[0024] The tensile and compressive internal forces consist of conservative forces and dissipative forces. These are approximated using generalized spring terms and generalized damping terms, respectively, resulting in a generalized spring-damped model for flexible attachment. The tensile and compressive internal forces acting on node i are... Represented as

[0025]

[0026] in and These represent the tensile and compressive internal forces between nodes i and j, and between nodes i and k, respectively. Meanwhile, Represented as

[0027]

[0028] Where v ij =v j -v i ,g(r ij ) represents the generalized elasticity term, h(v) ij ) represents the generalized damping term, ε ij Represents the unmodeled term, g(r) ij ) and h(v ij Further writing

[0029]

[0030] Where l0 is the initial length between the two nodes of the flexible lander, and g(||r ij ||-l0) and h(v) ij Let g(||r) be a scalar function, and let g(||r) be a scalar function. ij ||-l0)>0 and h(v)ij ) > 0. According to the definition of equation (5), the elastic term and the damping term satisfy

[0031]

[0032] Linearize the elastic and damping terms near the equilibrium point.

[0033]

[0034] in and Let be the coefficients of the spring and damping terms. To account for modeling errors, the tensile and compressive internal forces are expressed as follows:

[0035]

[0036] The spring and damping coefficients, r, represent the tension and compression internal forces between node i and node k, respectively. ik , Let i be the relative position and velocity vector between node i and node k. To compensate for the modeling error of the tensile and compressive internal forces between node i and node k, the above approximate models of bending and torsional internal forces, tensile and compressive internal forces, and flexible moment are substituted into formulas (1) and (2) to obtain the dynamic models of each node of the flexible lander.

[0037] Step 2: Establish a nominal trajectory error tracking control model based on the dynamic model, and design a performance index function to transform the flexible detector attachment trajectory tracking control problem into an optimal problem, so that the control force can take into account the fuel consumption and robustness of the small celestial body landing trajectory tracking system.

[0038] The flexible adhesion dynamics model of equation (1) is transformed into an error tracking control model.

[0039]

[0040] in: U x U y U z Let f(x) represent the partial derivatives of the gravitational potential of the nodes in the three axes of the asteroid fixed coordinate system, f(x) represent the disturbance caused by the uncertainty of the interaction forces between the nodes due to the modeling error of the flexible force, and x represent the state variables of each node of the flexible detector. The following performance index V(x) is designed, and the control law u is calculated and solved to minimize the performance index:

[0041]

[0042] Where Q and R are positive definite constant weight matrices, and the performance index includes three terms, f 2max Used to handle the impact of uncertainty on the system, x T Qx represents the system convergence rate, u T Ru represents the energy state of the system, and t is the time parameter. The uncertain disturbance f(x) is bounded, meaning there always exists a non-negative function. Make

[0043]

[0044] The robust control problem of an uncertain system can be equivalent to an energy-optimal problem with a performance index function of formula (12).

[0045] Step 3: Based on the performance index function of the small celestial body flexible attachment trajectory tracking system designed in Step 2, establish the HJB equation and construct the expression for the optimal control force.

[0046] According to the optimal control criterion, the performance index shown in equation (12) satisfies the following HJB equation.

[0047]

[0048] in To minimize the performance index function, the following equation is obtained.

[0049]

[0050] Optimal control solution u * for

[0051]

[0052] Step 4: Form a closed-loop equation by iterating the strategy, and use a single-evaluation neural network to approximate the optimal control law and the optimal performance index function. Based on the optimal control law and the optimal performance index function, robust tracking of the nominal trajectory can be achieved while effectively reducing the detector's fuel consumption.

[0053] A single evaluation network is used to fit and minimize the performance index function:

[0054] V * (x)=W T σ(x)+ v (x) (17)

[0055] Among them W T To evaluate the weights of a neural network, σ(x) is the activation function of the neuron, and ε v (x) represents the approximation error of the neural network, and the partial derivative of the performance index is:

[0056]

[0057] Substituting the above equation into the Hamiltonian equation, we get:

[0058]

[0059] Use W c Let the ideal weights of the neural network be represented, then the evaluation network can be written as:

[0060]

[0061] Substituting the above equation into the Hamiltonian equation, we get:

[0062]

[0063] e c To measure the error in fitting the Hamiltonian function and to train an ideal neural network with optimal weights, gradient descent is used to update the evaluation network weights.

[0064]

[0065] Where α c To evaluate the learning efficiency of the network, the formula is rewritten as follows:

[0066]

[0067] in The difference between equation (21) and equation (23) represents the fitting error of the neural network to the Hamiltonian.

[0068]

[0069] make use To represent the estimation bias of the neural network, the above equation can be rewritten as:

[0070]

[0071] The derivative of the weight bias in a neural network is:

[0072]

[0073] When an evaluation network is used instead of the performance index function, the optimal control law obtained is:

[0074]

[0075] Thus, based on the landing dynamics model of Xiaotian, a nominal trajectory error tracking control model was established. The performance index function was designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. The HJB equation was established based on the performance index function, and the expression of the optimal control force was calculated. Through strategy iteration, the control law and the performance index function were formed into a closed-loop equation. A single evaluation neural network was used to approximate the optimal control law and the optimal performance index function, so as to achieve robust tracking of the nominal trajectory while reducing the detector's fuel consumption.

[0076] Beneficial effects:

[0077] 1. The robust tracking control method for flexible attachment trajectories of small celestial bodies disclosed in this invention simplifies the structure of the flexible detector while retaining its flexible characteristics. A "spring-damped" model of the flexible detector is established in both the flexible node reference coordinate system and the node fixed coordinate system. The mass clusters on the flexible lander, possessing specific functions and rigid characteristics, are considered as the smallest constituent units. The flexible lander installs control actuators on these mass clusters. Flexible connections exist between the mass clusters, and during attachment, the lander is subjected to flexible forces generated by relative tension, compression, bending, and torsion of the flexible body. This ensures that the lander possesses both flexible characteristics and allows for effective estimation and control of its attachment process state.

[0078] 2. The robust tracking control method for flexible attachment trajectories of small celestial bodies disclosed in this invention addresses the disturbance problem caused by the uncertainty of the interaction forces between nodes of the flexible detector. Based on a dynamic model, a nominal trajectory error tracking control model is established. A performance index function is designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. An HJB equation is established based on the performance index function, and the expression for the optimal control force is calculated. Through strategy iteration, the control law and performance index function are formed into a closed-loop equation. A single-evaluation neural network is used to approximate the optimal control law and optimal performance index function, achieving robust tracking of the nominal trajectory while effectively reducing the detector's fuel consumption. Attached Figure Description

[0079] Figure 1 This is a flowchart of the robust optimal trajectory tracking control method for flexible attachment to small celestial bodies disclosed in this invention;

[0080] Figure 2 It is an approximate model of the three-mass clustering region of a flexible lander;

[0081] Figure 3 (a) and Figure 3 (b) are schematic diagrams of the flexible lander's node reference coordinate system and the fixed coordinate system, respectively;

[0082] Figure 4 This describes the centroid motion trajectory of a flexible detector using the robust optimal trajectory tracking control method for small celestial bodies disclosed in this invention.

[0083] Figure 5 This is a diagram showing the trajectory tracking error of a three-node flexible detector using the robust optimal trajectory tracking control method for small celestial bodies disclosed in this invention. Figure 5 (a) represents the trajectory tracking error (x1, y1, z1) of node 1; Figure 5 (b) represents the trajectory tracking error (x2, y2, z2) at node 2; Figure 5 (c) represents the trajectory tracking error (x3, y3, z3) for node 3;

[0084] Figure 6 This is the result of online training of neural network weights in the small celestial body flexible attachment robust optimal trajectory tracking control method disclosed in this invention; Detailed Implementation

[0085] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0086] Numerical simulations were performed using 433Eros as the target celestial body for verification. In the fixed coordinate system of the celestial body, the initial position r0 of the probe's center of mass is [10177, 6956, 8256] m, the initial velocity v0 is [-2, 1.2, -1.7] m / s, and the target position r... f The value is [676,5121,449]m, and the transition time is t. f The time was set to 720s. Nominal trajectories were designed for the three nodes of the flexible detector using ZEM / ZEV guidance laws, and trajectory tracking was performed using the designed robust optimal control law. The mass ratio of the flexible node to the flexible material mass was 13.2:1, the distance between the nodes was 0.6m, and the three nodes were arranged in an equilateral triangle. The spring constant k was... ij r With damping coefficient k iv r The values ​​are 640 and 0.5 respectively; the number of hidden layers in the evaluation network is selected as l. n =9, the initial weights of the neural network are 0, and the evaluation network learning rate is 0.1.

[0087] like Figure 1 As shown in the example, the robust optimal trajectory tracking control method for flexible attachment to small celestial bodies disclosed in this example has the following specific implementation steps:

[0088] Step 1: Take the mass cluster area with specific functions and rigid characteristics on the flexible lander as the smallest constituent unit, establish an equivalent dynamic model of the flexible body and a generalized spring-damping-torsion spring model to approximate the flexible force, so that the generalized spring-damping-torsion spring model has both flexible characteristics and can effectively estimate and control the state of its attachment process as much as possible.

[0089] The minimum constituent unit is considered as a mass cluster with specific functions and rigid characteristics on a flexible lander. Control actuators are installed on these mass clusters. Flexible connections exist between these mass clusters, and during attachment, they are subjected to flexible forces generated by relative tension, compression, bending, and torsion of the flexible bodies. For a flexible lander with a planar shape, the three-mass cluster model is the simplest configuration for equivalently simulating the motion of the flexible lander.

[0090] To facilitate the description of the motion of flexible detector nodes, the following coordinate system is defined:

[0091] (1) Nodal reference coordinate system Origin O is Located at the centroid of the node, x is axis and y is The axis lies on the plane containing the three nodes, z s The axis is perpendicular to the plane containing the three nodes.

[0092] (2) Node-fixed coordinate system Origin O ia Located at the centroid of the node, z ia Coinciding with the principal axis of minimum inertia of node i, x ia y coincides with the principal axis of maximum inertia of node i. ia The axis satisfies the right-hand rule.

[0093] Considering the flexible forces and flexible moments acting on node i, the translational dynamic equation of node i in the asteroid fixed coordinate system is:

[0094]

[0095] Where r i v i and T i These represent the position, velocity, and thrust vector of node i in the asteroid's fixed coordinate system, respectively; ω is the spin angular velocity vector of the small celestial body; and m... i For the mass of node i, F if Let be the flexible force acting on node i. The dynamic models of nodes j and k have the same expression. The calculation steps for the flexible force are given below.

[0096] Flexible torque T on node i if Approximated by a torsion spring

[0097]

[0098] Where θ i and β i These represent the rotation angles of the fixed coordinate system at node i relative to the x-axis and y-axis of the reference coordinate system, respectively, with the rotation sequence being ZXY. The transformation matrix K1, K2, C1, and C2 are constant coefficients to be determined for transforming node i from the reference coordinate system to the asteroid fixed coordinate system.

[0099] Flexible force F if From tension and compression internal forces and bending and torsional internal forces constitute:

[0100]

[0101] The bending and torsional internal forces are generated by the torsion spring torque to ensure the balance of internal forces and internal moments of the flexible lander. The flexible torque acting at node i is T. if Based on the equilibrium of internal forces and internal moments, the bending and torsional internal forces satisfy...

[0102]

[0103] and The bending and torsional internal forces acting on nodes j and k are respectively, r ij With r ik Let i and j represent the position vectors between nodes i and j, and i and k, respectively. The bending and torsional internal forces have zero components in the x and y directions of the node reference coordinate system. At this time, the system of equations (29) has a unique solution. The reaction forces generated by the flexible moments acting on the three nodes are solved in sequence to obtain the bending and torsional internal forces of the nodes.

[0104] The tensile and compressive internal forces consist of conservative forces and dissipative forces. These are approximated using generalized spring terms and generalized damping terms, respectively, resulting in a generalized spring-damped model for flexible attachment. The tensile and compressive internal forces acting on node i are... Represented as

[0105]

[0106] in and These represent the tensile and compressive internal forces between nodes i and j, and between nodes i and k, respectively. Meanwhile, Represented as

[0107]

[0108] Where v ij =v j -v i ,g(r ij ) represents the generalized elasticity term, h(v) ij ) represents the generalized damping term, ε ij Represents the unmodeled term, g(r) ij ) and h(v ij Further writing

[0109]

[0110] Where l0 is the initial length between the two nodes of the flexible lander, and g(||r ij ||-l0) and h(v) ij Let g(||r) be a scalar function, and let g(||r) be a scalar function. ij ||-l0)>0 and h(v) ij ) > 0. According to the definition of equation (5), the elastic term and the damping term satisfy

[0111]

[0112] Linearize the elastic and damping terms near the equilibrium point.

[0113]

[0114] in and Let be the coefficients of the spring and damping terms. To account for modeling errors, the tensile and compressive internal forces are expressed as follows:

[0115]

[0116] The spring and damping coefficients, r, represent the tension and compression internal forces between node i and node k, respectively. ik , Let i be the relative position and velocity vector between node i and node k. To compensate for the modeling error of the tensile and compressive internal forces between node i and node k, the above approximate models of bending and torsional internal forces, tensile and compressive internal forces, and flexible moment are substituted into formulas (28) and (29) to obtain the dynamic models of each node of the flexible lander.

[0117] Step 2: Establish a nominal trajectory error tracking control model based on the dynamic model, and design a performance index function to transform the flexible detector attachment trajectory tracking control problem into an optimal problem, so that the control force can take into account the fuel consumption and robustness of the small celestial body landing trajectory tracking system.

[0118] The flexible adhesion dynamics model of equation (28) is transformed into an error tracking control model.

[0119]

[0120] in: U x U y U zLet f(x) represent the partial derivatives of the gravitational potential of the nodes in the three axes of the asteroid fixed coordinate system, f(x) represent the disturbance caused by the uncertainty of the interaction forces between the nodes due to the modeling error of the flexible force, and x represent the state variables of each node of the flexible detector. The following performance index V(x) is designed, and the control law u is calculated and solved to minimize the performance index:

[0121]

[0122] Where Q and R are positive definite constant weight matrices, and the performance index includes three terms, f 2 max Used to handle the impact of uncertainty on the system, x T Qx represents the system convergence rate, u T Ru represents the energy state of the system, and t is the time parameter. The uncertain disturbance f(x) is bounded, meaning there always exists a non-negative function. Make

[0123]

[0124] According to the following theorem, the robust control problem of an uncertain system can be equivalently transformed into an energy-optimal problem with a performance index function of formula (39).

[0125] Theorem: The optimal control solution for a small body landing control system with the above performance indices exists and satisfies the following conditions.

[0126] f 2 max (x)>f T (x)Rf(x) (41)

[0127] The optimal solution for this system is the robust solution of the control system, meaning the system converges in an energy-optimal manner with a steady-state error approaching zero. This control system can simultaneously consider both fuel consumption and robustness.

[0128] Step 3: Based on the performance index function of the small celestial body flexible attachment trajectory tracking system designed in Step 2, establish the HJB equation and calculate the expression of the optimal control force.

[0129] According to the optimal control criterion, the performance index shown in equation (39) satisfies the following HJB equation.

[0130]

[0131] in V * Let (x) be the performance index function to be minimized, resulting in the following equation:

[0132]

[0133] Optimal control solution u * for

[0134]

[0135] Step 4: Through strategy iteration, the control law and performance index function are formed into a closed-loop equation. A single-evaluation neural network is used to approximate the optimal control law and optimal performance index function, which can effectively reduce the fuel consumption of the detector while achieving robust tracking of the nominal trajectory.

[0136] A single evaluation network is used to fit and minimize the performance index function:

[0137] V * (x)=W T σ(x)+ε v (x) (45)

[0138] Among them W T To evaluate the weights of a neural network, σ(x) is the activation function of the neuron, and ε v (x) represents the approximation error of the neural network, and the partial derivative of the performance index is:

[0139]

[0140] Substituting the above equation into the Hamiltonian equation, we get:

[0141]

[0142] Use W c Let the ideal weights of the neural network be represented, then the evaluation network can be written as:

[0143]

[0144] Substituting the above equation into the Hamiltonian equation, we get:

[0145]

[0146] e c To measure the error in fitting the Hamiltonian function and to train an ideal neural network with optimal weights, gradient descent is used to update the evaluation network weights.

[0147]

[0148] Where α c To evaluate the learning efficiency of the network, the formula is rewritten as follows:

[0149]

[0150] in The difference between equation (49) and equation (51) represents the fitting error of the neural network to the Hamiltonian.

[0151]

[0152] make use To represent the estimation bias of the neural network, the above equation can be rewritten as:

[0153]

[0154] The derivative of the weight bias in a neural network is:

[0155]

[0156] When an evaluation network is used instead of the performance index function, the optimal control law obtained is:

[0157]

[0158] Thus, based on the landing dynamics model of Xiaotian, a nominal trajectory error tracking control model was established. The performance index function was designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. The HJB equation was established based on the performance index function, and the expression of the optimal control force was calculated. Through strategy iteration, the control law and the performance index function were formed into a closed-loop equation. A single evaluation neural network was used to approximate the optimal control law and the optimal performance index function, so as to achieve robust tracking of the nominal trajectory while reducing the detector's fuel consumption.

[0159] like Figure 5 , Figure 6 As shown in the embodiment, the trajectory tracking error of the three nodes in the three-axis direction gradually converges to zero and remains stable after 10 seconds, and the neural network weights converge. The embodiment shows that the robust tracking control method for flexible attachment trajectory of small celestial bodies proposed in this invention can effectively handle the disturbance caused by the uncertain interaction force between nodes, and the optimal control force can enable the detector to accurately track the nominal trajectory.

[0160] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. 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 within the scope of protection of the present invention.

Claims

1. A method for robust tracking control of flexible attached trajectories of small celestial bodies, characterized in that: Includes the following steps, Step 1: Take the mass cluster area with specific functions and rigid characteristics on the flexible lander as the smallest constituent unit, establish the equivalent dynamic model of the flexible body and the generalized spring-damping-torsion spring model to approximate the flexible force, so that the generalized spring-damping-torsion spring model has both flexible characteristics and can effectively estimate and control the state of its attachment process as much as possible. Step 2: Establish a nominal trajectory error tracking control model based on the dynamic model, and design a performance index function to transform the flexible detector attachment trajectory tracking control problem into an optimal problem, so that the control force can take into account the fuel consumption and robustness of the small celestial body landing trajectory tracking system. Step 3: Based on the performance index function of the small celestial body flexible attachment trajectory tracking system designed in Step 2, establish the HJB equation and construct the expression for the optimal control force; Step 3 is implemented as follows: According to the optimal control criterion, the performance index shown in equation (12) satisfies the following HJB equation. (14) in , Q and R are positive definite constant weight matrices. The performance index includes three terms: x is the state variable of each node of the flexible detector, u is the control law, and f is the weight matrix. 2 max Used to handle the impact of uncertainty on the system, x T Qx represents the system convergence rate, u T Ru represents the energy status of the system. , To minimize the performance index function, we obtain the following equation: (15) Optimal control solution for (16) Step 4: Form a closed-loop equation by iterating the strategy, and use a single evaluation neural network to approximate the optimal control law and the optimal performance index function. Based on the optimal control law and the optimal performance index function, robust tracking of the nominal trajectory can be achieved while effectively reducing the detector's fuel consumption. Step 4 is implemented as follows: A single evaluation network is used to fit and minimize the performance index function: (17) Among them W T To evaluate the weights of a neural network, It is the neuron activation function. For the approximation error of the neural network, the partial derivative of the performance index is: (18) Substituting the above equation into the Hamiltonian equation, we get: (19) W c represents the ideal weight of the neural network, then the evaluation network can be written as: (20) Substituting the above equation into the Hamiltonian equation, we get: (21) e c To fit the error of Hamiltonian function, in order to train the ideal neural network weight, the gradient descent method is used to update the evaluation network weight (22) in To evaluate the learning efficiency of the network; Rewrite the expression as follows (23) in The difference between equation (21) and equation (23) represents the fitting error of the neural network to the Hamiltonian. (24) make ,use To represent the estimation bias of the neural network, the above equation can be rewritten as: (25) The derivative of the weight bias in a neural network is: (26) When an evaluation network is used instead of the performance index function, the optimal control law obtained is: (27) Based on the landing dynamics model of Xiaotian, a nominal trajectory error tracking control model is established. A performance index function is designed to transform the flexible detector attachment trajectory tracking control problem into an optimal problem. The HJB equation is established based on the performance index function, and the expression of the optimal control force is calculated. Through strategy iteration, the control law and the performance index function are formed into a closed-loop equation. A single evaluation neural network is used to approximate the optimal control law and the optimal performance index function, so as to achieve robust tracking of the nominal trajectory while reducing the detector's fuel consumption.

2. The robust tracking control method for flexible attachment trajectory of small celestial bodies as described in claim 1, characterized in that: Step 1 is implemented as follows: The mass clusters with specific functions and rigid characteristics on the flexible lander are regarded as the smallest constituent units. The flexible lander installs control actuators on the mass clusters. There are flexible connections between the mass clusters. During the attachment process, the flexible lander is subjected to flexible forces generated by the relative tension, compression, bending and torsion of the flexible body. For a flexible lander with a planar shape, the three-mass cluster model is the simplest configuration for equivalently simulating the motion of the flexible lander. To facilitate the description of the motion of flexible detector nodes, the following coordinate system is defined: (1) Nodal reference coordinate system :origin Located at the centroid of the node, shaft and The axis lies on the plane containing the three nodes. The axis is perpendicular to the plane containing the three nodes; (2) Node-fixed coordinate system :origin Located at the centroid of the node, With nodes Minimum moment of inertia principal axes coincide. With nodes Maximum moment of inertia coincides with principal axis. The axis satisfies the right-hand rule; Considering that the node is subjected to flexible forces and flexible moments, the node The translational dynamic equations in the fixed coordinate system of the asteroid are as follows: (1) in , and They are nodes Position, velocity, and thrust vector in the asteroid's fixed coordinate system The spin angular velocity vector of the small celestial body. For nodes quality Let be the flexible force acting on node i. The dynamic models of nodes j and k have the same expression. The calculation steps for the flexible force are given below. Flexible torque on node i Approximated by a torsion spring (2) in and They are nodes Fixed coordinate system relative to reference coordinate system shaft and The angle of rotation of the shaft, the sequence of rotation is as follows , For nodes The transformation matrices K1, K2, C1, and C2 for transforming from the reference coordinate system to the asteroid fixed coordinate system are constant coefficients to be determined. Flexible force From tension and compression internal forces and bending and torsional internal forces constitute: (3) The bending and torsional internal forces are generated by the torsion spring torque to ensure the balance of internal forces and internal moments of the flexible lander; nodes The flexible torque acting on the upper part is Based on the equilibrium of internal forces and internal moments, the bending and torsional internal forces satisfy... (4) and Let be the bending and torsional internal forces acting on nodes j and k, respectively. and These represent the position vectors between nodes i and j, and between node i and k, respectively, and the bending and torsional internal forces in the nodal reference coordinate system. shaft and When the axial component is 0, the equation set (2) has a unique solution; solve the reaction forces generated by the flexible torques acting on the three nodes in sequence to obtain the bending and torsional internal forces of the nodes; The tensile and compressive internal forces consist of conservative forces and dissipative forces. These are approximated using generalized spring terms and generalized damping terms, respectively, to obtain a generalized spring-damped model for flexible attachment; nodes The tensile and compressive internal forces it is subjected to Represented as (5) in and Representing nodes respectively With nodes The tensile and compressive internal forces between nodes and between nodes i and k; simultaneously Represented as (6) in , Represents the generalized elasticity term. Represents the generalized damping term. Represents unmodeled items. and Further writing (7) in This is the initial length between the two nodes of the flexible lander. and It is a scalar function, and has and According to the definition of equation (5), the elastic term and the damping term satisfy... (8) Linearize the elastic and damping terms near the equilibrium point. (9) in and Let be the coefficients of the spring and damping terms; To account for modeling errors, the tensile and compressive internal forces are expressed as follows: (10) , The spring and damping coefficients, respectively, represent the tension and compression internal forces between node i and node k. , Let i be the relative position and velocity vector between node i and node k. To compensate for the modeling error of the tensile and compressive internal forces between node i and node k, the above approximate models of bending and torsional internal forces, tensile and compressive internal forces, and flexible moment are substituted into formulas (1) and (2) to obtain the dynamic models of each node of the flexible lander.

3. The robust tracking control method for flexible attachment trajectory of small celestial bodies as described in claim 2, characterized in that: Step 2 is implemented as follows: The flexible adhesion dynamics model of equation (1) is transformed into an error tracking control model. (11) in: , , , , point Let f(x) represent the partial derivatives of the gravitational potential of the nodes in the three axes of the asteroid fixed coordinate system, f(x) represent the perturbation caused by the uncertainty of the interaction forces between the nodes due to the modeling error of the flexible force, and x represent the state variables of each node of the flexible detector. The following performance indicators are designed. Calculate and solve the control law u to minimize the performance index: (12) Where Q and R are positive definite constant weight matrices, and the performance index includes three terms, f 2 max Used to handle the impact of uncertainty on the system, x T Qx represents the system convergence rate, u T Ru represents the energy state of the system, and t is a time parameter; the uncertain disturbance f(x) is bounded, that is, there always exists a non-negative function. , making (13) The robust control problem of an uncertain system can be equivalent to an energy-optimal problem with a performance index function of formula (12).

Images