In the field of system control, consensus algorithms have attracted widespread attention in the research of multi-agent formation control [1-2], and related applications such as clustering, rendezvous and formation control have emerged. The main idea of the consensus algorithm is to use the communication information between individual agents to control the multi-agent system, so that the state of each agent can reach a common value.
In this paper, under the constraint that only part of the following spacecraft can obtain the status information of the pilot spacecraft, this paper studies the multi-spacecraft attitude-orbit coupling distributed collaborative tracking control considering input saturation in view of the problems of unmodeled dynamics and external environment interference of the system, so that the pilot spacecraft that follows the spacecraft tracking dynamics is studied.
Spacecraft model
In this paper, MRPs are used to describe spacecraft attitudes, and the attitude kinematics and dynamic equations of the i-th spacecraft are
where σi = eitan/4 = [σi1 σi2 σi3]T represents the i-th spacecraft attitude, and ωi = [ωi1 ωi2 ωi3]T is the projection of the angular velocity of the i-th spacecraft ontology coordinate system relative to the reference frame in this system.
Ji is a positively definite symmetric spacecraft rotational inertia matrix, S(a) is defined as any 3 × 3 3 oblique symmetry matrix of any 3 × 1 vector a, τi = [τi1 τi2 τi3]T is the control input acting on the i-th spacecraft, τdi = [τdi1 τdi2 τdi3]T is the interference acting on the i-th spacecraft, in equation (1).
thereinto
Represents the 2 - norm of a vector.
Equations (1) and (2) are subjected to a series of transformations to obtain the equations of motion of spacecraft in the form Euler-Lagrange
thereinto
Note that Mσi (σi) is a symmetric positive definite matrix.
Spacecraft relative orbital dynamics model
In the LVLH reference frame, the equation of relative motion between the ith spacecraft and the reference point is described as the form Euler-Lagrange as follows
Here it is defined that ρi = [xi yi zi]T is the position vector of the ith spacecraft relative to the reference point, and Fi = [fxifyi fzi]T is the control force exerted on the i-th spacecraft.
Fdi = [fdxi fdyi fdzi]T is the environmental disturbance force acting on the i-th spacecraft, mi is the mass of the i-th spacecraft, and μ, Rc and θ are the gravitational constants of the Earth, the distance from the reference point to the center of the Earth, and the true periscopic angle of the reference orbit.
Chebyshev neural network
The neural network structure in this paper adopts a single-layer Chebyshev neural network based on Chebyshev polynomials, and its functional connection network consists of a set of orthogonal Chebyshev polynomials. Based on the approximation characteristics of CNNs, any continuous nonlinear function fN(X) can be approximated by CNNs, that is
where W* is the optimal weight matrix of the CNN and εf is the approximation error of the CNN.
For system (7), make the following assumptions globally:
Hypothesis 1. The communication topology between the following spacecraft is undirected, and the pilot spacecraft cannot obtain information to track the spacecraft.
Hypothesis 2. The position, velocity, attitude, and angular velocity of the pilot spacecraft are bounded, and its first and second derivatives are also bounded, that is, p0, p0, ̈p0 are bounded.
Design of attitude-rail coupling distributed collaborative control law considering input saturation
A description of the problem
Spacecraft operating in orbit, with the consumption of fuel, its mass and moment of inertia and other parameters change, that is, some parameters of the system cannot be accurately obtained.
In the actual system, because the actuator of the control system can only provide limited driving capacity, there is always a phenomenon of control input saturation in the system.
The saturation characteristic seriously affects the performance of adaptive controllers, so it is necessary to consider the problem of control input saturation when designing controllers based on adaptive learning.
In this paper, for the multi-spacecraft system with various external interference and uncertainty parameters, considering the influence of control input coupling between spacecraft attitude and orbital dynamics and the output saturation characteristics of the actuator, this paper studies the multi-spacecraft attitude-orbit coupling distributed collaborative control when only part of the following spacecraft can obtain the pilot spacecraft state.
Consider introducing a pilot spacecraft into a formation and treating it as the i+1st spacecraft, noting that the pilot spacecraft is 0. The communication topology between the following spacecraft and with the pilot spacecraft is denoted as Figure G. Here the pilot spacecraft cannot obtain information about the status of the following spacecraft, ie
Therefore, a0i = 0. It is assumed that only part of the following spacecraft can obtain the status of the pilot spacecraft. If following the spacecraft can obtain information about the pilot spacecraft, then ai0 = 1, otherwise ai0 ≠ 1. Notice that the Laplacian matrix of Figure G is
where H = L + diag(a10,...,an0) symmetric positive definite.
Attitude and rail coupling distributed collaborative control
In this section, considering the situation that the pilot spacecraft has a time-varying angular velocity, three distributed finite time sliding mode estimators are designed for the partial tracking spacecraft that cannot obtain the pilot spacecraft, and the position and attitude, velocity and angular velocity and acceleration and angular acceleration of the pilot spacecraft are estimated respectively
thereinto
is a normal number, satisfied
Time T exists, when t > T,
Analysis of finite time convergence of estimators is omitted here.
The following auxiliary variables are defined
When t > T, the estimated value in the auxiliary variable is replaced with the true value, and then equation (11) is reduced to
Combined with equation (7), both sides of the above equation are multiplied by Mi(·) to obtain the error kinetic equation shown in the following equation
Due to the good function approximation ability of neural networks, they are often used to compensate for uncertain models in systems. In this paper, neural network functions are used to compensate for the unknown nonlinear term fNi in the system (13).
Note that ̈pri contains the first derivative of the neighbor following the state of the spacecraft, that is, the nonlinear function contains the acceleration following the spacecraft as well as the angular acceleration information.
When approximating this nonlinear function, this information needs to be used as input to the neural network function, but acceleration and angular acceleration cannot be measured. Although this information can be estimated by designing higher-order sliding mode observers, it is possible to approximate nonlinear functions by using only partial information PI as input to neural network functions.
Further, since the relative velocity and relative angular velocity of the neighbor following the spacecraft are difficult to measure, the input of the neural network function should be avoided as much as possible to contain the velocity and angular velocity information of the neighbor spacecraft. Consider that each following spacecraft will eventually converge to the state of the pilot spacecraft.
Simulation analysis
This section simulates the proposed distributed attitude-rail coupling control law (16) considering the saturation of control inputs. Take, for example, the formation flight of six spacecraft. The communication topological relationship between the following spacecraft and the pilot spacecraft is shown in Figure 2, where 0 represents the pilot spacecraft, 1 ~ 6 represents the following spacecraft, and the path between spacecraft represents the information transfer relationship between them.
Considering the pilot spacecraft with time-varying state, a distributed sliding mode estimator (10) is designed for the following spacecraft that cannot obtain the pilot spacecraft information to estimate the state of the pilot spacecraft. The initial position and velocity of each following spacecraft relative to the reference orbit are zero, and the initial attitude parameters are shown in Table 1. The control parameters are shown in Table 2. The simulation results are shown in Figure 3 ~ Figure 7.
Figure 3 shows the time-dependent curve of the 3D trajectory of each following spacecraft, and the darkest line in the figure represents the 3D curve of the pilot spacecraft. From the figure, it can be seen that each following spacecraft converges to the position of the pilot spacecraft.
Figure 4 shows the change curve of the control force required by each following spacecraft, and it can be seen from the simulation curve that the control force acting on the following spacecraft has saturation characteristics and can ensure the tracking of the dynamic pilot spacecraft.
Figure 5 shows the position change curve of each following spacecraft in the LVLH coordinate system, and the three lines pointed to by the desired position arrow in Figure 5 are the curves of the position of the pilot spacecraft at LVLH coordinates. From the above simulation curve, it can be seen that under the action of the distributed control law designed in this paper, the tracking of the position and speed of the pilot spacecraft is realized by following the spacecraft.
Figure 6 shows the attitude change curve of each spacecraft. The three lines pointed by the desired position arrows in Figure 6 represent the change curve in the attitude of the pilot spacecraft. Similarly, the distributed collaborative control law designed in this paper can ensure that each following spacecraft tracks the pilot spacecraft.
Figure 7 shows the control torque change curve required by each spacecraft to track the pilot spacecraft, and it can be seen from Figure 7 that the control torque change law reflects the saturation characteristics of the designed control law.
Based on the above simulation studies, it can be seen that the distributed attitude-orbit coupling cooperative control law designed in this section considering input saturation is feasible and effective.
summary
In this paper, the problem of relative orbit and attitude coupling cooperative control of multi-spacecraft systems with control input coupling is studied. Under the constraint that only part of the pilot spacecraft can be followed, considering the unmodeled dynamics and external environmental interference of the multi-spacecraft system, a distributed collaborative adaptive control law with input saturation is proposed, so that the pilot spacecraft that follows the spacecraft tracking dynamics is proposed.
Finally, the simulation analysis is carried out by taking the formation flight of six spacecraft as an example, and the results show that the cooperative control law designed in this paper is effective and feasible. The next step is to consider the problem of collision avoidance and information packet loss in the actual formation flight of the multi-spacecraft system.
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