CN108388135A - A kind of Mars landing track optimized controlling method based on convex optimization - Google Patents

Abstract

The present invention relates to a Mars landing trajectory optimization control method based on convex optimization. The method comprises the following steps: (1) establishing a Mars landing dynamics model and performing convex optimization; (2) performing a single off-line optimization with the goal of fuel optimization The nominal trajectory is obtained; (3) The model predictive control in the rolling time domain is used to track the nominal trajectory to complete the landing control. Compared with the prior art, the present invention effectively combines convex optimization and model predictive control in the rolling time domain, effectively reduces modeling errors and cumulative errors caused by disturbances, and improves landing accuracy.

Description

An Optimal Control Method for Mars Landing Trajectory Based on Convex Optimization

technical field

The invention relates to a Mars landing trajectory optimization control method, in particular to a Mars landing trajectory optimization control method based on convex optimization.

Background technique

In recent years, with the continuous improvement of science and technology, people's demand for space resources has ushered in a peak period, and the interest in Mars exploration has been rekindled. In this context, issues such as precise landing of manned aircraft have become more and more important. more important. In the field of aviation, the problem that an aerospace lander accurately lands on a given target location with an error of less than a few hundred meters under guidance is called a precision landing problem, and trajectory optimization is a key technology in precision landing.

In order to ensure a safe landing on Mars, it is necessary to comprehensively consider the scientific research value of each site and the overall terrain conditions (such as slope, roughness). For missions of natural exploration nature, the specific landing point within the predetermined error range is not particularly important. For example, in NASA's Mars Exploration Rovers mission, the landing error can be as large as 35km. On the other hand, in the requirements of some scientific research tasks, it may be necessary to accurately land on dangerous terrain, or to land on a designated ground location such as a fuel supply point. For example, with the continuous deepening of my country's Chang'e project and manned spaceflight project, future lunar exploration will be an important research direction for our country. In this context, it can be foreseen that detectors with trajectory optimization and high-precision fixed-point soft landing capabilities are necessary. indispensable

But at the same time, the difficulty of the Mars landing mission lies in its many constraints and a large number of nonlinear time-varying items, so it is difficult to meet the system requirements with general optimization methods. For general three-dimensional problems with additional state and control constraints, it is often difficult to obtain the analytical solution of the optimal thrust, so it is particularly important to be able to adapt to the airborne fast trajectory optimization algorithm.

In related research at home and abroad, the commonly used methods are the direct configuration method, the direct multiple shooting method and the numerical solution method using the Legendre pseudospectral method, as well as the traditional polynomial-based guidance method, in which the trajectory based on the quartic polynomial in time Used in the Apollo missions. However, when the convergence of related algorithms is unclear, the real-time airborne nonlinear programming solution obtained by general iterative algorithms may not be the desired result. Since the precise landing on Mars requires real-time calculation of the optimal trajectory onboard, an algorithm with definite convergence and guaranteed convergence to the global optimum should be designed according to the structure of the problem. Convex optimization has the property of global optimality, so the trajectory optimization based on convex optimization will be the research direction.

Contents of the invention

The object of the present invention is to provide a Mars landing trajectory optimization control method based on convex optimization in order to overcome the above-mentioned defects in the prior art.

The purpose of the present invention can be achieved through the following technical solutions:

A Mars landing trajectory optimization control method based on convex optimization, characterized in that the method comprises the following steps:

(1) Establish a Mars landing dynamics model and perform convex optimization;

(2) Perform a single offline optimization with the goal of fuel optimization to obtain the nominal trajectory;

(3) The model predictive control in the rolling time domain is used to track the nominal trajectory to complete the landing control.

Step (1) The Mars landing dynamics model is specifically:

_Tc = (nTcosφ)e,

Among them, r∈IR ³ , v∈IR ³ , T _c ∈IR ³ , r is the position vector of the spacecraft relative to the target point, v is the velocity vector of the spacecraft in the ground fixed coordinate system, T _c is the thrust vector, IR ³ represents the xyz three-axis coordinate system, is the component of the derivative of the position vector of the spacecraft relative to the target point in the direction of the i axis, v _i is the component of the velocity vector of the spacecraft in the fixed coordinate system on the ground in the direction of the i axis, i=x, y, z, m is the spacecraft mass, μ is the gravitational constant of Mars, α is the constant of the fuel consumption rate, g∈IR ³ , g is the gravitational acceleration constant of Mars, I _sp is the specific impulse of the thruster, e∈IR ³ , and e is the instantaneous Thrust direction vector, g _e is the earth's gravity constant, φ is the angle between the propeller and the instantaneous thrust direction vector e, n is the number of sub-propellers.

Step (1) Convex optimization is specifically:

(11) Establish thrust amplitude constraints:

ρ ₁ = nT ₁ cos φ,

ρ ₂ = nT ₂ cos φ,

Among them, ρ ₁ and ρ ₂ are the lower and upper bounds of the thrust amplitude, t _f is the termination time of the power descent landing process, n is the number of sub-thrusters, T ₁ is the non-zero minimum thrust of the sub-thrusts, T ₂ is The maximum thrust of the sub-propeller;

Displacement non-negativity constraints:

r _x > 0,

Among them, r _x is the component of the position vector of the spacecraft relative to the target point in the x-axis direction;

Altitude constraint:

in, is a given safety angle, θ _alt (t) is the altitude angle from the spacecraft to the target point;

(12) Introduce the slack variable Γ(t) and the additional constraint ‖T _c (t)‖≤Γ(t) for variable substitution Then the thrust constraints after convex optimization are obtained:

ρ ₁ and ρ ₂ are the lower and upper bounds of the thrust amplitude, z and z ₀ represent the mass variable and its initial value after variable substitution, and σ represents the new variable after the slack variable transformation.

Step (2) The optimization model after a single offline optimization with the goal of fuel optimization is:

0<ρ ₁ ≤‖T _c (t)‖≤ρ ₂ ,

‖SX‖+c ^T X ≤ 0,

m(0)=m _wet ,

r(0)=r ₀ r(t _f )=0,

Among them, S and c are coefficient matrices:

m _wet represents the initial mass of the lander when it is fully loaded with fuel, Denotes the final mass of the lander when it lands, and X represents the state vector containing the position and velocity information of the spacecraft.

Step (2) obtaining the nominal trajectory is specifically: using the CVX toolbox with Mosek as the solver to solve the optimization model, and obtain the state vector X representing the position and velocity information of the spacecraft.

Step (3) is specifically:

(31) Build a predictive model:

X(k+i|k)＝A(k+i-1|k)X(k+i-1|k)+B(k+i-1|k)U(k+i-1|k) ,

Among them, X(k+i|k) represents the predicted value of state information at time k+i at time k, and U(k+i-1|k) represents k+i-1 obtained from the optimization result at time k The input amount at time, A(k+i-1|k) represents the predicted value of the state matrix A at time k+i-1 at time k, and B(k+i-1|k) represents the value of k at time k The predicted value of the input matrix B at +i-1 time, the state information includes spacecraft position and velocity information, and the input quantity is thrust;

(32) Establish optimization objective function:

Among them, X _opt (k+i) represents the target value of trajectory tracking at time k+i, Q is a symmetric positive definite parameter matrix, p is the step size of rolling optimization, and X(N|k) represents the final time at time k. State information prediction value, X _opt (N) represents the target value of trajectory tracking at the final moment, and R is a symmetrical positive definite parameter matrix;

(33) Take the prediction model established in step (31) as the constraint condition, and use the objective function in (32) as the optimization index, perform a finite time-domain online rolling optimization with a step size of p at any time k, and at time k Get optimized p optimal control sequences U(k+i|k);

(34) At each current moment, only the control quantity and position information at the current moment in the optimization results are brought into the Mars landing dynamics model to obtain the real position information at the next moment, and it is used as the optimized initial information at the next moment. Roll the optimization in the time domain until the end, and use the optimized control amount at the current moment as input each time to complete the control work of the lander.

Compared with prior art, the present invention has following advantage:

(1) Through the optimization algorithm based on convex optimization, the present invention can ensure that the result obtained from the solution is the global optimum, avoiding the deficiency that the traditional method may converge to the local optimum, so the obtained fuel optimal trajectory is the result of the global optimum ;

(2) The nominal trajectory is tracked by the model predictive control in the rolling time domain, which can effectively avoid the cumulative error caused by the modeling error or the actual disturbance, and reduce the drift between the real trajectory and the optimized trajectory. The control method of sub-offline optimization is more accurate;

(3) The convex optimization and model predictive control in the rolling time domain are effectively combined to effectively reduce the cumulative error caused by modeling errors and disturbances, and improve the landing accuracy.

Description of drawings

Fig. 1 is the block flow diagram of the Mars landing trajectory optimization control method based on convex optimization in the present invention;

Figure 2 is a schematic diagram of the altitude angle constraint cone during the landing process;

Figure 3 is a flow chart of the model predictive control algorithm.

Detailed ways

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments. Note that the description of the following embodiments is merely an illustration in nature, and the present invention is not intended to limit the applicable objects or uses thereof, and the present invention is not limited to the following embodiments.

Example

As shown in Figure 1, a Mars landing trajectory optimization control method based on convex optimization, the method includes the following steps:

(1) Establish a Mars landing dynamics model and perform convex optimization;

(2) Perform a single offline optimization with the goal of fuel optimization to obtain the nominal trajectory;

(3) The model predictive control in the rolling time domain is used to track the nominal trajectory to complete the landing control.

Step (1) The Mars landing dynamics model is specifically:

_Tc = (nTcosφ)e,

Among them, r∈IR ³ , v∈IR ³ , T _c ∈IR ³ , r is the position vector of the spacecraft relative to the target point, v is the velocity vector of the spacecraft in the ground fixed coordinate system, T _c is the thrust vector, IR ³ represents the xyz three-axis coordinate system, is the component of the derivative of the position vector of the spacecraft relative to the target point in the direction of the i axis, v _i is the component of the velocity vector of the spacecraft in the fixed coordinate system on the ground in the direction of the i axis, i=x, y, z, m is the spacecraft mass, μ is the gravitational constant of Mars, α is the constant of the fuel consumption rate, g∈IR ³ , g is the gravitational acceleration constant of Mars, I _sp is the specific impulse of the thruster, e∈IR ³ , and e is the instantaneous Thrust direction vector, g _e is the earth's gravity constant, φ is the angle between the propeller and the instantaneous thrust direction vector e, n is the number of sub-propellers.

Step (1) Convex optimization is specifically:

(11) Establish thrust amplitude constraints:

ρ ₁ = nT ₁ cos φ,

ρ ₂ = nT ₂ cos φ,

Among them, ρ ₁ and ρ ₂ are the lower and upper bounds of the thrust amplitude, t _f is the termination time of the power descent landing process, n is the number of sub-thrusters, T ₁ is the non-zero minimum thrust of the sub-thrusts, T ₂ is The maximum thrust of the sub-propeller;

Displacement non-negativity constraints:

r _x > 0,

Among them, r _x is the component of the position vector of the spacecraft relative to the target point in the x-axis direction;

Altitude constraint:

in, is a given safety angle, θ _alt (t) is the altitude angle from the spacecraft to the target point, and the altitude angle constraint cone diagram is shown in Figure 2;

(12) Introduce the slack variable Γ(t) and the additional constraint ‖T _c (t)‖≤Γ(t) for variable substitution Then the thrust constraints after convex optimization are obtained:

ρ ₁ and ρ ₂ are the lower and upper bounds of the thrust amplitude, z and z ₀ represent the mass variable and its initial value after variable substitution, and σ represents the new variable after the slack variable transformation.

Step (2) The optimization model after a single offline optimization with the goal of fuel optimization is:

0<ρ ₁ ≤‖T _c (t)‖≤ρ ₂ ,

‖SX‖+c ^T X ≤ 0,

m(0)=m _wet ,

r(0)=r ₀ r(t _f )=0,

Among them, S and c are coefficient matrices:

m _wet represents the initial mass of the lander when it is fully loaded with fuel, Denotes the final mass of the lander when it lands, and X represents the state vector containing the position and velocity information of the spacecraft.

Step (2) obtaining the nominal trajectory is specifically: using the CVX toolbox with Mosek as the solver to solve the optimization model, and obtain the state vector X representing the position and velocity information of the spacecraft.

Step (3) is specifically:

(31) Build a predictive model:

X(k+i|k)＝A(k+i-1|k)X(k+i-1|k)+B(k+i-1|k)U(k+i-1|k) ,

Among them, X(k+i|k) represents the predicted value of state information at time k+i at time k, and U(k+i-1|k) represents k+i-1 obtained from the optimization result at time k The input amount at time, A(k+i-1|k) represents the predicted value of the state matrix A at time k+i-1 at time k, and B(k+i-1|k) represents the value of k at time k The predicted value of the input matrix B at +i-1 time, the state information includes spacecraft position and velocity information, and the input quantity is thrust;

(32) Establish optimization objective function:

Among them, X _opt (k+i) represents the target value of trajectory tracking at time k+i, Q is a symmetric positive definite parameter matrix, p is the step size of rolling optimization, and X(N|k) represents the final time at time k. State information prediction value, X _opt (N) represents the target value of trajectory tracking at the final moment, and R is a symmetrical positive definite parameter matrix;

(33) Take the prediction model established in step (31) as the constraint condition, and use the objective function in (32) as the optimization index, perform a finite time-domain online rolling optimization with a step size of p at any time k, and at time k Get optimized p optimal control sequences U(k+i|k);

(34) At each current moment, only the control quantity and position information at the current moment in the optimization results are brought into the Mars landing dynamics model to obtain the real position information at the next moment, and it is used as the optimized initial information at the next moment. Roll the optimization in the time domain until the end, and use the optimized control amount at the current moment as input each time to complete the control work of the lander.

In actual implementation, it mainly consists of three parts, namely sensor module, guidance and control module and actuator module.

The sensor module includes a radar detector placed on the bottom of the Mars lander and an IMU inertial measurement unit that measures the real-time flight status of the lander. The guidance control module is a calculation and control unit with an embedded processor as the core, which has strong computing power. The actuator is composed of n sub-thrusts arranged symmetrically at the bottom of the lander, and the thrust is adjusted by the opening and closing of the throttle valve.

Furthermore, under the framework of the above system, the core module where the guidance and control is located can be designed according to the actual task requirements.

The specific implementation steps of the above-mentioned Mars landing trajectory optimization and control scheme based on convex optimization are as follows:

The first step: the design of the embedded controller. Under the condition that the lander is only affected by gravity and anti-thrust, ignoring other forces caused by wind and the assumption of a uniform gravity field, a translational motion model of the Mars lander during landing is established.

After completing the establishment of the motion model for the research object, set a safe altitude angle according to physical factors, engineering constraints, and actual mission requirements, and analyze and convexize the constraints of the lander in consideration of the constraints of the altitude angle. The optimization and control model after convexization is obtained, and it is written into the embedded controller of the calculation control module before each task execution.

The second step: a single offline optimization. According to the optimization problem established in the first step, through the built-in CVX toolbox in the embedded system, the corresponding CPU operation is called for a single offline optimization, and the optimal thrust and optimal trajectory with the best fuel consumption are obtained as the nominal Trajectories and reference inputs are stored in the microcomputer unit.

The third step: state measurement and input. According to the IMU unit in the sensor module, the speed and acceleration of the lander during the landing process are measured in real time, and the current relative position information between the lander and the target point is measured through the radar detector, and the measured value of the state information at this moment is used as an input to connect to the guidance Embedded with the control module.

Step 4: Model prediction. At each moment, the current state is used as the initial information to establish a good motion control model to predict the flight trajectory in the next period of time, and to solve the optimal control problem in this period of time through optimization. For the accuracy of the model, at each moment, only the control information at the current moment among the many control quantities in the optimization results is used, and the state at the next moment is obtained according to the current state information and control quantities as the basis for the next rolling time-domain optimization. initial information. Each optimization only optimizes a small section on the time domain axis, and the optimization is repeated continuously, which is equivalent to rolling forward in the time domain until the complete optimization of the entire process is achieved. The specific algorithm flow is shown in Figure 3.

The specific model is:

X(k+i|k)＝A(k+i-1|k)X(k+i-1|k)+B(k+i-1|k)U(k+i-1|k) ,

In the optimization of the rolling time domain, the main consideration for selecting the performance index is the tracking of the nominal trajectory, and the corresponding optimization objective function is:

Step 5: Scroll optimization and real-time control.

(1) Transform the problem into the form of model prediction in the rolling time domain. Determine the rolling time domain value P, and adjust the optimization time to k=0 given the initial state information.

(2) Determine the value of Q in the objective function, use the CVX toolbox to solve the optimal control problem in the rolling time domain, and obtain a series of state information and control information in the rolling time domain.

(3) At the current moment, only the current control amount and current position information in the optimization result are used, and brought into the aircraft motion model to obtain the real position information at the next moment, and use it as the initial information for the next moment optimization.

(4) Repeat the above steps to optimize iteratively, stop the optimization when k=N, and obtain a complete optimization result, and the real-time control process ends.

The above-mentioned embodiments are merely examples, and do not limit the scope of the present invention. These embodiments can also be implemented in various other forms, and various omissions, substitutions, and changes can be made without departing from the scope of the technical idea of the present invention.

Claims (6)

1. A Mars landing trajectory optimization control method based on convex optimization is characterized by comprising the following steps:

(1) establishing a Mars landing dynamic model and performing convex optimization;

(2) carrying out single off-line optimization by taking the fuel optimization as a target to obtain a nominal track;

(3) and tracking the nominal track by adopting model prediction control of a rolling time domain to finish landing control.

2. The Mars landing trajectory optimization control method based on convex optimization according to claim 1, wherein the Mars landing dynamics model in the step (1) is specifically as follows:

T_c＝(nTcosφ)e，

wherein r ∈ IR³，v∈IR³，T_c∈IR³R is the position vector of the spacecraft relative to the target point, v is the velocity vector of the spacecraft in the ground fixed coordinate system, T_cThrust vector, IR, to which the spacecraft is subjected³Representing an xyz three-axis coordinate system,is the component of the derivative of the spacecraft with respect to the target point position vector in the direction of the i-axis, v_iIs the component of the velocity vector of the spacecraft in the fixed coordinate system on the ground in the direction of the i axis, i is x, y, z, m is the mass of the spacecraft, mu is the gravitational constant of mars, α is the constant of the fuel consumption rate, g belongs to IR³G is the gravitational acceleration constant of Mars, I_spFor the specific impulse of the propeller, e ∈ IR³E is the instantaneous thrust direction vector, g_eIs the earth gravity constant, phi is the included angle between the propeller and the instantaneous thrust direction vector e, and n is the number of the sub-propellers.

3. The Mars landing trajectory optimization control method based on convex optimization according to claim 2, characterized in that the convex optimization in the step (1) is specifically:

(11) establishing thrust amplitude constraint:

ρ₁＝nT₁cosφ，

ρ₂＝nT₂cosφ，

where ρ is₁And ρ₂Lower and upper limits of thrust amplitude, t_fThe final time of the power descent landing process, n is the number of the sub-propellers, T₁Is the non-zero minimum thrust, T, of the sub-thruster₂The maximum thrust of the sub-thruster is obtained;

displacement non-negative constraint:

r_x＞0，

wherein r is_xIs the component of the spacecraft in the direction of the x axis relative to the position vector of the target point;

height angle constraint:

wherein,for a given safety angle, θ_alt(t) is the altitude of the spacecraft to the target point;

(12) introducing a relaxation variable Γ (T) and an additional constraint | T_c(t) | < Γ (t), variable substitutionAnd further obtaining thrust constraint conditions after convex optimization:

ρ₁and ρ₂Lower and upper bounds for thrust amplitude, z and z₀Representing the quality variable after the variable replacement and the initial value thereof, and sigma representing the new variable after the relaxation variable transformation.

4. The Mars landing trajectory optimization control method based on convex optimization according to claim 3, wherein the optimization model of the step (2) after single off-line optimization with the fuel optimization as the target is as follows:

0＜ρ₁≤‖T_c(t)‖≤ρ₂，

‖SX‖+c^TX≤0，

m(0)＝m_wet，

r(0)＝r₀r(t_f)＝0，

wherein S and c are both coefficient matrices:

m_wetindicating the initial mass at full lander fuel,representing the final mass of the lander when landed and X representing a state vector containing spacecraft position and velocity information.

5. The Mars landing trajectory optimization control method based on convex optimization according to claim 4, wherein the step (2) of obtaining the nominal trajectory specifically comprises: and solving the optimization model by adopting a CVX tool box with Mosek as a solver to obtain a state vector X representing the position and speed information of the spacecraft.

6. The Mars landing trajectory optimization control method based on convex optimization according to claim 2, wherein the step (3) is specifically as follows:

(31) establishing a prediction model:

X(k+i|k)＝A(k+i-1|k)X(k+i-1|k)+B(k+i-1|k)U(k+i-1|k)，

wherein X (k + i | k) represents a predicted value of state information at the time k to the time k + i, U (k + i-1| k) represents an input quantity at the time k-1 obtained in an optimization result at the time k, A (k + i-1| k) represents a predicted value of a state matrix A at the time k to the time k + i-1, B (k + i-1| k) represents a predicted value of an input matrix B at the time k to the time k + i-1, the state information comprises spacecraft position and speed information, and the input quantity is thrust;

(32) establishing an optimization objective function:

wherein, X_opt(k + i) represents the target value of the trajectory tracking at the time k + i, Q is a symmetric positively determined parameter matrix, p is the step size of the rolling optimization, X (N | k) represents the predicted value of the state information at the time k to the final time, and X_opt(N) represents the final time trace trackingR is a symmetric positive definite parameter matrix;

(33) taking the prediction model established in the step (31) as a constraint condition, taking the objective function in the step (32) as an optimization index, performing finite time domain online rolling optimization with the step length of p at any time k, and obtaining p optimized control sequences U (k + i | k) at the time k;

(34) and at each current moment, substituting the control quantity and the position information of the current moment in the optimization result into the Mars landing dynamics model to obtain the real position information of the next moment, taking the real position information as the optimization initial information of the next moment, performing rolling optimization on the whole time domain until the optimization is finished, and finishing the control work of the lander by taking the optimization control quantity of the current moment as input each time.