CN115291504A - Rocket sublevel recovery landing stage power descent guidance method based on tail end error - Google Patents

Rocket sublevel recovery landing stage power descent guidance method based on tail end error Download PDF

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CN115291504A
CN115291504A CN202210601592.7A CN202210601592A CN115291504A CN 115291504 A CN115291504 A CN 115291504A CN 202210601592 A CN202210601592 A CN 202210601592A CN 115291504 A CN115291504 A CN 115291504A
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socp problem
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CN115291504B (en
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杨广文
陈宇澍
陈海鹏
王禄
甘庆忠
张武生
柳嘉
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National Supercomputing Center In Wuxi
Shanghai Aerospace Control Technology Institute
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Abstract

The invention provides a power descent guidance method for a rocket substage recovery landing stage based on a tail end error, relates to a rocket guidance recovery method, and solves the problem of power descent guidance for a rocket substage vertical recovery landing stage under nonlinear aerodynamic force by converting a PDG problem under the nonlinear aerodynamic force into an SOCP problem. The method has low calculation cost and less calculation resource occupation, can effectively suppress the position error in a specific direction, can make the subproblems feasible under reasonable assumption, and has interpretability; meanwhile, programs running on domestic equipment such as a processor on the rocket and the like are all independently researched and developed, and only the calculation result of open-source software is needed when the general equipment is preprocessed.

Description

Rocket sublevel recovery landing stage power descent guidance method based on tail end error
Technical Field
The invention relates to a rocket guidance recovery method, in particular to a rocket sublevel recovery land section power descent guidance method based on a tail end error.
Background
The rocket substage vertical recovery landing stage needs to start an engine to perform reverse thrust deceleration, and the process is called power descent. In order to realize accurate vertical soft landing, dynamic descent guidance (PDG) is required, namely, trajectory planning is carried out according to the current state observed by the sensor and the position of a preset landing point, and a control signal is generated.
Because the initial conditions (speed and position) of the starting moment (engine ignition) of the mission are large in dispersion deviation, the track is difficult to calculate in advance, and the power descent guidance needs to be carried out on line so as to meet the process constraints of thrust adjustment, residual fuel and the like and the terminal constraints of position, speed, attitude and the like, thereby realizing the established flight mission. Therefore, dynamic descent guidance has high real-time requirements. However, in the rocket-borne computers, the computing power is often relatively insufficient due to the considerations of volume, weight, power consumption, radiation protection and the like, and the contradiction in the aspect of computing speed is very prominent. Particularly, the influence of nonlinear aerodynamic force must be considered in the recovery of rocket launching sub-stages in the earth atmosphere, so that the calculation amount is further greatly increased, and the real-time challenge is more serious. In addition, the power descent guidance also faces the technical difficulties of high requirements on landing position, speed and attitude precision, minimized fuel consumption and the like.
For PDG problem solution without aerodynamic force consideration, there have been some mature research results internationally, and typically a series of papers published by JPL laboratories in the united states and by the university of washington, the cooperative unit, etc. In the work, the PDG problem is systematically described, the motion equation and various constraints of the spacecraft are analyzed, solutions such as lossless convexity and quality approximate convexity are provided for non-convex constraints in the PDG problem, and the PDG problem is finally converted into a second-order cone programming (SOCP) problem to be solved. If the total flight time is fixed, the PDG problem without considering aerodynamic force can be converted into a single SOCP problem solution, but the total flight time needs to be searched through multiple attempts (such as dichotomy, interpolation and the like). If the total flight time is adjusted, the PDG problem without pneumatic action is similar to the PDG problem with pneumatic action, and needs to be converted into a series of SOCP problems through sequence convex action to be solved.
Whereas PDG solutions taking into account non-linear aerodynamic forces are still under investigation in published efforts. The PDG problem considering the nonlinear aerodynamic force is solved by adopting a sequence convex method (hereinafter referred to as an acceleration error method) for relaxation based on acceleration errors at Washington university, the total flight time is solved through local approximate convex, and the acceleration errors are introduced to avoid infeasibility of the problem. The algorithm has practical application prospect, but the calculated amount is greatly increased compared with that without pneumatic power, and the requirement on the performance of the rocket-borne computer is high. In addition, the acceleration error represents the violation degree of the dynamic constraint based on the physical law, and under the condition that any constraint such as the terminal position, the speed, the attitude constraint and the like cannot be met, the problem inadequacy is reflected on the acceleration error, and only a solution with a large acceleration error is obtained, but the solution does not exist in practice and cannot reflect the flight trajectory and the terminal state which can be achieved in practice. SpaceX is not disclosed in the art, is expected to be closely related to JPL and the research work of Washington university, and requires a high performance arrow-mounted computer. The north navigation solves the PDG problem considering the action of air resistance and lift under the two-dimensional condition, and provides an algorithm which does not need to solve the total flight time, but still needs to solve through sequence projection, and the SOCP problem obtained through local linearization is not feasible, thereby causing the failure of the solution.
The SOCP problem is typically solved by a dedicated solver. JPL series thesis uses the SOCP problem efficient solver BSOCP (programs are not disclosed, but algorithm schemes are disclosed in the thesis), under the non-pneumatic condition, the sparsity of the problem structure is good, the calculation efficiency is high, but under the pneumatic condition, the sparsity of the problem structure is slightly poor, the algorithm can lead the number of non-zero elements of a coefficient matrix to be increased rapidly when a linear equation set is solved, and further the calculation efficiency is reduced rapidly. In the currently disclosed solver, for a common configuration of the PDG problem, the efficiency of the ECOS is the highest, and is greatly lower than the BSOCP in the absence of pneumatics, and is predicted to be higher than the BSOCP in the presence of pneumatics, but the computational efficiency still has room for improvement, and does not support hot start, which is not favorable for accelerating convergence by using a solution of a prophase problem in sequence projection.
At present, the main difficulties in solving the problem of power reduction guidance under the action of nonlinear aerodynamic force on line are as follows:
firstly, constraint needs to be effectively relaxed, so that the SOCP subproblem of local approximation is feasible under reasonable assumption, and solving failure caused by infeasibility of factor problem is avoided. Secondly, the problem relaxation and dispersion processes are prevented from introducing too strong nonlinearity, so that convergence is slow, and the number of sub-problems to be solved is increased. Thirdly, the problem that the structure of the subproblem is too complex, so that the calculation cost of a single subproblem is excessively increased is avoided. Fourthly, the SOCP problem is solved efficiently.
Disclosure of Invention
Aiming at the technical problems, the invention provides a rocket sublevel recovery landing segment power descent guidance method based on a tail end error, which aims to solve the problem of rocket sublevel vertical recovery landing segment power descent guidance under nonlinear aerodynamic force.
In order to achieve the purpose, the invention adopts the technical scheme that:
the invention provides a rocket substage recovery landing stage power descent guidance method based on terminal errors, which comprises the following steps
Step 101: inputting an initial total time of flight t f Discrete time step number k f Upper limit of sub-problem number n SC Time step change confidence domain upper limit η Δt Thrust variation confidence range upper limit η T A confidence domain adjustment coefficient beta;
step 102: carrying out initialization;
step 103: if i =1, generating an SOCP problem constraint, and at this time, not using a trust domain constraint, or using a larger trust domain boundary; if i is more than 1, updating SOCP problem constraint;
step 104: solving the SOCP problem; calculating the tail end position and speed error through adding fine grid numerical simulation;
step 105: calculated improvement ratio ξ = (J) L0,i -J L0,i-1 )/(J L1,i -J L1,i-1 ) (ii) a If xi is more than or equal to xi max At that time, if there is a letterUpon activation of a domain boundary, the boundary to be activated corresponds to the radius of trust (η) Δt Or η T ) Multiplying by beta; if xi is less than or equal to xi min Confidence domain radius η Δt ,η T Are all divided by beta; if xi is less than or equal to 0, rejecting SOCP problem and resetting to the solution of last subproblem, i.e. m i [k]＝m i-1 [k],Γ i [k]＝Γ i-1 [k]，
r i [k]＝r i-1 [k],v i [k]＝v i-1 [k],a i [k]＝a i-1 [k],a R,i [k]＝a R,i-1 [k],k∈{0,1,…,k f },Δt i ＝Δt i-1 ；
Step 106: updating the total time of flight t f ；
Step 107: if the end position and the speed error are smaller than the preset limits, executing step 109;
step 108: if i < n SC I = i +1, and steps 103 to 106 are repeatedly executed; if i = n SC Then go to step 109;
step 109: outputting a total time of flight t f Mass m, position r, velocity v, thrust T;
wherein,
Figure BDA0003669543310000041
is the planned objective function value, J L1 ＝-m(k f )+ω κr ||r[k f ]||+ω κv ||v[k f ]| is the simulated objective function value; m, r, v are the mass, position, velocity in the SOCP solution,
Figure BDA0003669543310000042
the quality, position and speed are calculated through numerical simulation, and i is the sequence number of the sub-problem;
the initialization in step 102 specifically includes:
Figure BDA0003669543310000043
the rocket substage recovery landing stage power descent guidance method based on the tail end error provided by the invention is preferably usedThe method can support hot start, and is accelerated by utilizing the correlation among SOCP problems; when the hot start is used, before the step of solving the SOCP problem, the step of outputting an initial value (x) of the current SOCP problem by a hot start initial value correction method of the SOCP problem 0 ,y 0 ,s 0 )；
The SOCP problem warm start initial value correction method comprises the following steps:
step 201: inputting the solution of the existing prior SOCP subproblem
Figure BDA0003669543310000044
Current sub-problem equation constraint coefficient matrix A, right-hand term b, cone constraint dimension (l, S) 1 ,S 2 ,…,S m ) Correction coefficient η 0 (default value 0.001);
step 202: initialization is performed, η = max (1/(| a | | | luminance) ∞ +||b|| ∞ ),η 0 )；
Step 203: initialization, i =1;
Figure BDA0003669543310000045
step 204:
Figure BDA0003669543310000046
Figure BDA0003669543310000047
i+＝S j ，
Figure BDA0003669543310000048
step 205, if i is less than or equal to m, repeatedly executing step 204; if i > m, go to step 206;
step 206: outputting the initial value (x) of the current SOCP problem 0 ,y 0 ,s 0 )。
The rocket substage recovery landing stage power descent guidance method based on the tail end error provided by the invention preferably further comprises the step of outputting the initial value of the current SOCP problem by an initial value hot start correction method of the SOCP problem before the step of solving the SOCP problem(x 0 ,y 0 ,s 0 ) (ii) a Step 105 may be omitted.
The technical scheme has the following advantages or beneficial effects:
the invention provides a rocket substage recovery landing stage power descent guidance method based on terminal errors, which solves the problem of power descent guidance of a rocket substage vertical recovery landing stage under nonlinear aerodynamic force by converting the PDG problem under the nonlinear aerodynamic force into the SOCP problem. The invention can realize low calculation cost and small occupation of calculation resources, can effectively suppress the position error in a specific direction, can make the subproblems feasible under reasonable assumption, and has interpretability; meanwhile, programs running on domestic equipment such as a processor on a rocket are all independently researched and developed, and only the calculation result of open-source software is needed when the general equipment is preprocessed.
Detailed Description
The present invention will be further described with reference to specific examples, but the present invention is not limited thereto.
Example 1:
the PDG problem of nonlinear aerodynamic force is considered, that is, under a specific initial value (including initial position, velocity and fuel mass of the rocket, and using a predetermined landing point as an origin of a coordinate system), trajectory planning is performed by adjusting the magnitude and direction of thrust, so that the rocket can land accurately at the predetermined landing point, the direction is vertical to the ground and upward during landing, the velocity is zero, and the fuel consumption is minimized. In the aspect of rocket attitude, namely approximately considering that the tail of the rocket points to the thrust direction, the influence of a sideslip angle is not considered.
The problem can be described as
An objective function:
Figure BDA0003669543310000051
initial and edge values:
Figure BDA0003669543310000052
power restraint:
Figure BDA0003669543310000061
and (3) state constraint:
Figure BDA0003669543310000062
and (4) controlling and constraining:
Figure BDA0003669543310000063
wherein r is position, v is velocity, a is acceleration, T is thrust, and m is mass; theta gs Is the minimum angle (anti-collision) between the connecting line of the current position of the rocket and the landing point and the ground T,max The maximum included angle between the thrust direction and the speed direction; without special mention, | x | | | all represent two norms of the vector x. t is t f As a result of the total time of flight,
Figure BDA0003669543310000064
lower and upper bounds of the thrust rate of change, ρ 0 For a high gas density at the predetermined landing point, c ρ Is the atmospheric density attenuation coefficient, m dry Is the arrow-shaped structural mass (without fuel), alpha is the ratio of specific fuel consumption to thrust, C D Is the coefficient of air resistance, S ref Is a rocket reference area.
According to the thought of the literature, the control constraint is subjected to lossless convex transformation and converted into the control constraint
||T(t)||≤Γ(t)
0≤T min ≤Γ(t)≤T max
Figure BDA0003669543310000065
The original problem is now transformed into a series of sub-problems of local linear approximation for solution. To efficiently control the amount of computation and make the local approximation sub-problem generally feasible, the present invention introduces a relaxation scheme based on the end error. The scheme relaxes the constraints of the position and the speed of the tail end, namely, the tail end position and the speed can have any errors, and the position and speed errors are added into an objective function for punishment.
The discretized PDG local linear approximation subproblem (numbered i) can be described as:
an objective function:
Figure BDA0003669543310000066
initial and edge values:
Figure BDA0003669543310000067
power restraint:
Figure BDA0003669543310000071
wherein, D [ k ] is not used as a solving variable and is substituted into a [ k ] during actual solving;
and (3) state constraint:
Figure BDA0003669543310000072
and (4) controlling and constraining:
Figure BDA0003669543310000073
and (3) end error constraint:
Figure BDA0003669543310000074
trust domain constraints:
Figure BDA0003669543310000075
wherein D is air resistance, Δ t is discrete time step, κ r ,κ v For the penalty factor, a larger value should be taken to suppress the end position and velocity error, η Δt ,η T The confidence domain upper bound of the time step and the thrust variation between adjacent subproblems.
When the ith sub-problem (hereinafter referred to as sub-problem i) is solved, the first i-1 sub-problems are solved and the variable values are determined. Therefore, the power constraint in the discretized PDG local linear approximation subproblem solution has no nonlinearity, and non-convexity cannot be caused; while the other constraints are convex constraints. Therefore, for each index i, the local linear approximation, the discretized PDG local linear approximation sub-problem is a convex optimization problem. More specifically, the discretized PDG local linear approximation sub-problem is a second-order cone optimization problem (SOCP), i.e. the PDG problem under nonlinear aerodynamic forces can be converted into a SOCP problem solution.
The standard form of the SOCP problem is:
Figure BDA0003669543310000081
wherein,
Figure BDA0003669543310000082
representing a linear cone, and l is the dimension of the linear cone; k S Representing a second-order cone, and m is the number of the second-order cones;
linear cone
Figure BDA0003669543310000083
And a second order cone K S The definitions are respectively:
Figure BDA0003669543310000084
in this embodiment, the discrete PDG local linear approximation problem can be converted into the standard form of the SOCP problem by:
the unconstrained variable omega amount conversion form is: ω = ω + -ω - Wherein
Figure BDA0003669543310000085
Linear constraint of h T ω < d translates into: h is T (ω + -ω - )+ω s = d, wherein ω + ,ω - ,ω s ∈K L ；
The secondary constraint | | omega | | < d is transformed into:
Figure BDA0003669543310000086
therefore, in this embodiment, the PDG problem under non-linear aerodynamic forces can be solved by transforming the sequence convex into a series of local linear approximation sub-problems. The rocket substage recovery landing stage power descent guidance method based on the tail end error (hereinafter referred to as tail end error method) provided by the embodiment 1 of the invention comprises the following steps:
step 101: inputting an initial total time of flight t f Discrete time step number k f Upper limit of sub-problem number n SC And a time step change confidence domain upper limit eta Δt Thrust variation confidence region upper limit eta T A confidence domain adjustment coefficient beta;
step 102: carrying out initialization;
step 103: if i =1, generating an SOCP problem constraint, and in this case, not using a trust domain constraint or using a larger trust domain boundary; if i is more than 1, updating SOCP problem constraint;
step 104: solving the SOCP problem; calculating the tail end position and speed error through adding fine grid numerical simulation;
step 105: calculated improvement ratio ξ = (J) L0,i -J L0,i-1 )/(J L1,i -J L1,i-1 ) (ii) a If xi is more than or equal to xi max If the confidence domain boundary is activated, the boundary to be activated corresponds to the confidence domain radius (eta) Δt Or η T ) Multiplying by beta; if xi is less than or equal to xi min Radius of confidence field η Δt ,η T Are all divided by beta; if xi is less than or equal to 0, rejecting the SOCP problem and resetting the SOCP problem to be the solution of the last subproblem, namely m i [k]＝m i-1 [k],Γ i [k]＝Γ i-1 [k]，
r i [k]＝r i-1 [k],v i [k]＝v i-1 [k],a i [k]＝a i-1 [k],a R,i [k]＝a R,i-1 [k],k∈{0,1,…,k f },Δt i ＝Δt i-1 ；
Step 106: updating the total time of flight t f ；
Step 107: if the end position and the speed error are smaller than the preset limits, executing step 109;
step 108: if i < n SC I = i +1, and steps 103 to 106 are repeatedly executed; if i = n SC Then go to step 109;
step 109: outputting a total time of flight t f Mass m, position r, velocity v, thrust T;
wherein,
Figure BDA0003669543310000091
is the planned objective function value, J L1 ＝-m(k f )+ω κr ||r[k f ]||+ω κv ||v[k f ]| is the simulated objective function value; m, r, v are the mass, position, velocity in the SOCP solution,
Figure BDA0003669543310000092
the quality, position and speed are calculated through numerical simulation, and i is the sequence number of the sub-problem;
the step 102 initialization specifically includes:
Figure BDA0003669543310000093
in order to further improve the calculation efficiency, hot start can be adopted for solving the SOCP problem. The hot start means that when a plurality of convex optimization sub-problems need to be solved in sequence convexity and the like, and the parameter change of the current sub-problem is smaller than that of a solved sub-problem, an initial value is generated through solution calculation of the sub-problem in the early stage, so that the initial value error is reduced, and the solution is accelerated. The interior point method adds punishment through barrier functions, and limits a search area in a constraint space. The hot start of the interior point method is relatively difficult because the barrier function tends to have large gradients, curvatures and changes rapidly near the constraint boundary. If the previous solution is close to the constraint boundary, it is used as the initial value, which often results in poor validity of updating the direction, and the allowed step size is small, and the convergence of the subsequent iteration step is slow. To avoid this, the initial value is adjusted to deviate from the boundary. The specific adjustment mode is to add an offset constructed based on a cold start initial value to an item close to a constraint boundary in the existing solution.
The SOCP problem is solved by adopting an interior point method, and generally, the original problem and the dual problem are not only considered. The dual problem form of the SOCP standard problem is:
Figure BDA0003669543310000101
correspondingly, in the hot start, not only the original variable x but also the dual variable s satisfying the same cone constraint are deviated from the boundary. Therefore, in the present embodiment, before "solving the SOCP problem", outputting the initial value (x) of the current SOCP problem by the SOCP problem hot start initial value correction method 0 ,y 0 ,s 0 )；
The method for correcting the initial hot start value of the SOCP problem specifically comprises the following steps:
step 201: inputting the solution of the existing early SOCP subproblem
Figure BDA0003669543310000102
Current sub-problem equation constraint coefficient matrix A, right-hand term b, cone constraint dimension (l, S) 1 ,S 2 ,…,S m ) Correction coefficient η 0 (default value 0.001);
step 202: initialization is performed, η = max (1/(| a | | | luminance) ∞ +||b|| ∞ ),η 0 )；
Step 203: initialization, i =1;
Figure BDA0003669543310000103
step 204:
Figure BDA0003669543310000104
Figure BDA0003669543310000105
step 205, if i is less than or equal to m, repeatedly executing step 204; if i > m, go to step 206;
step 206: outputting the initial value (x) of the current SOCP problem 0 ,y 0 ,s 0 )。
If the number of steps for solving the single SOCP problem is small, the optimal solution may not be approached, but only a certain improvement is provided, so that the confidence domain cannot be adjusted according to the target function improvement proportion, and the confidence domain should be kept constant. In this case, step 105 may be omitted; the omitted rocket sublevel recovery landing section power descent guidance method based on the tail end error specifically comprises the following steps:
step 301: inputting an initial total time of flight t f Discrete time step number k f Upper limit of sub-problem number n SC And a time step change confidence domain upper limit eta Δt Thrust variation confidence range upper limit η T A confidence domain adjustment coefficient beta;
step 302: the initialization is carried out such that,
Figure BDA0003669543310000111
step 303: if i =1, generating an SOCP problem constraint, and in this case, not using a trust domain constraint or using a larger trust domain boundary; if i is more than 1, updating SOCP problem constraint;
step 304: outputting the initial value (x) of the current SOCP problem by the SOCP problem hot start initial value correction method 0 ,y 0 ,s 0 )；
Step 305: solving the SOCP problem; calculating the tail end position and speed error through adding fine grid numerical simulation;
step 306: updating the total time of flight t f ；
Step 307: if the end position and the speed error are smaller than the preset bounds, go to step 309;
step 308: if i < n SC I = i +1, and repeatedly executing steps 303 to 306; if i = n SC Then go to step 309;
step 309: outputting a total time of flight t f Mass m, position r, velocity v, thrust T.
The following beneficial effects are provided for the embodiment 1 of the invention:
1. the calculation cost is low. The improvement of the calculated amount mainly comes from three aspects: one is the single SOCP problem solving computation. For single SOAnd (3) CP problem solving, namely directly discretizing and solving the original problem which is not relaxed, wherein when the tail end error method is converted into the SOCP standard problem, a variable representing a tail end speed mode exists, and only a tail end position mode needs to be added. The mode needs to be implemented only by a four-dimensional second order cone constraint. The variables and the calculated amount are increased less. In the acceleration error method, N is needed to increase a three-dimensional acceleration error at each time grid point k A four-dimensional second order cone constraint, or a 3N k A second order cone constraint of +1 dimension. Traditional SOCP solver (based on [12 ]]And similar algorithms) when a single iterative step is used to solve the system of linear equations, the number of non-zero elements in the coefficient matrix due to a second order cone constraint is proportional to the square of the dimension, and thus N is used k The four-dimensional second-order cone constraint calculation amount is lower, but the use of the high-dimensional second-order cone constraint is beneficial to reasonably distributing acceleration errors to each time grid point, and no matter which mode is adopted, huge calculation amount increase is caused. The ECOS and the FSOCP solver perform sparsification processing on the second-order cone constraint, the number of nonzero elements of a coefficient matrix caused by the second-order cone constraint is in direct proportion to the order, the calculation speed of processing the high-dimensional second-order cone is obviously higher than that of the traditional solver, and even if the solvers are used, the increment of the calculation amount of the acceleration error method is still larger than that of the terminal error method. The second is the number of SOCP problems to solve. The number of the SOCP problem solving is difficult to analyze theoretically, but the problem solved by the acceleration error method is more complex and strong in nonlinearity; in addition, the method reduces the end error by minimizing the acceleration error in the objective function, although the end error is 0 (ignoring numerical discrete error) when the acceleration error is 0, the acceleration error and the end error do not have direct correspondence when the acceleration error is not 0, which may cause the end error to be reduced slowly. And the end error method directly adds the end error into the objective function for minimization, thereby being capable of quickly reducing the end error. Numerical experiments show that when cold start is adopted, the number of SOCP problems needing to be solved by the terminal error method is obviously lower than that of the terminal error method. 3. Is a warm start. By hot start, the initial error of the SOCP problem can be reduced, thereby speeding up the solution. As the solution changes during the SOCP solution, the constraint actually changes, and the constraint is assumed during the single SOCP solution in the sequence convexThe beam is not changed, and the improvement on the original problem is limited when the solving precision is higher. Through hot start, the upper bound of the solving steps of the single SOCP problem can be limited, namely, the SOCP problem is not solved and completed each time, but the constraint is updated in time, so that the total calculation cost is reduced.
2. The subproblems can be made feasible under reasonable assumptions, and the solution is interpretable. The acceleration error method allows the dynamic constraint to have any error, namely the rocket is allowed to have no consideration on the dynamic constraint and move, the SOCP subproblem is feasible as long as other constraints are not in conflict with each other, but because the existence of the acceleration error actually violates the physical law, the obtained solution has no practical significance when the acceleration error is nonzero (actually, the acceleration error cannot be reduced to a very low level), and the actual motion condition and the reachable terminal state cannot be reflected. The end error method does not break the power constraints, and the control constraints (thrust magnitude, angle, rate of change) can be directly satisfied by the settings, so its sub-problem feasibility depends on whether there is a feasible solution to the state constraints (including collision avoidance, maximum speed and fuel quality). The fuel mass constraint, i.e., fuel mass is not negative, can be satisfied as long as the total flight time (not greater than the time to consume all fuel at minimum thrust) is reasonably set. When the maximum speed constraint meets the requirement of initial speed, if the maximum thrust is greater than gravity (generally all meet), when the speed approaches the maximum speed, the included angle between the resultant force of the thrust, the resistance and the gravity and the speed is greater than pi/2, so that the speed is not increased, and a feasible solution is provided under the general condition. The collision avoidance constraint is | | [ r x (t),r z (t)] T ||tanθ gs ≤r y (t), i.e., where the initial height value is large and the downward velocity component is not too large, it may not be feasible in extreme cases. In general, the terminal error method allows the position and the speed of the rocket at the last time point to be any values allowed by other constraints, eliminates the infeasibility caused by the incapability of meeting the terminal position and speed constraints, and enables the local SOCP approximation subproblem to be feasible under a general reasonable condition. After hot start is introduced, the single SOCP problem is only solved for a certain number of steps, and even if the subproblems are not feasible (the power constraint error can not be reduced to be close to 0), the power constraint error can be enabled to have a certain valueThe amplitude is reduced, the power constraint can be adjusted after the hot start, and the error of resistance approximation is reduced, so that the infeasibility caused by overlarge initial resistance value error is reduced. When the numerical discrete error and the local linear approximation error are not considered, the track planned by the terminal error method is the actual motion track, the actual intermediate state and the actual terminal state can be reflected, and the interpretability is sufficient.
3. The position error in the specific direction can be effectively suppressed. For example, when a rocket lands, the position error in the vertical direction affects the safety, and the landing site in the horizontal direction usually has a certain range, so that the influence of the smaller position error is small. If all the directional errors are required to be small, time grids are required to be thinned, iteration steps are increased, and the calculation cost is high. By applying the tail end error method, the error of the direction can be suppressed only by adding a larger weight to a specific direction when the tail end position error is calculated, and reasonable errors in other directions are allowed.
4. And the autonomous controllability is strong. By using a self-developed FSOCP solver (pre-processing in a ground-based generic device to generate operand address files for different discrete time steps), all libraries (except the C language standard library) running on a domestic embedded device can be developed autonomously.
The guidance problem is solved by adopting the rocket substage recovery landing stage power descent guidance method based on the terminal error provided by the embodiment, and the specific flight task parameters are set as the following table 1:
Figure BDA0003669543310000131
Figure BDA0003669543310000141
TABLE 1 flight mission parameters
Note: air density in ρ = ρ 0 exp(-c ρ r y ) And (4) calculating. Coefficient of variation of density c ρ The height is 1000m per liter, and the air density is 0.0001The degree is reduced to 90.48 percent of the original degree.
The aerodynamic coefficients are as follows:
mach number 0.2 0.3 0.5 1.0
Coefficient of resistance C a 0.32 0.32 0.51 0.53
TABLE 2 aerodynamic coefficients
The computing environment is a notebook computer, the CPU is R7 5800H (basic frequency is 3.2GHz, maximum acceleration frequency is 4.4 GHz), the memory is 16GB, and the programs run in series. Since the preprocessing can be completed in advance, rather than acquiring initial conditions to start, the calculation time of FSOCP and ECOS only counts the solution time, and does not count the preparation time. When the sequence is embossed, the termination condition is that the error of the terminal position and the speed is less than 1m and 0.1m/s, respectively. The number of discrete time segments is fixed at 30 (corresponding to a number of time grid points of 31). The maximum number of solved subproblems is 30, and if the number of subproblems reaches the upper limit and still does not meet the convergence condition, the number is counted as the failure of the solution.
Using different initial values of total time of flight, the number of sub-problems solved when each method reaches the convergence condition, the calculated time and remaining fuel ratio are shown in Table 3. During cold start, the number of iteration steps of ECOS single SOCP solving is automatically set, the upper limit of FSOCP iteration is 40 steps (if the iteration steps meet the convergence condition before reaching the upper limit, the iteration steps are terminated, otherwise, the iteration steps reach the upper limit, the iteration steps are terminated). Comparing the results of the acceleration error method and the end error method, the end error method greatly reduces the number of SOCP subproblems to be solved during cold start, and the average subproblem number in the numerical result is only 60% of the latter. FSOCP is more computationally efficient than ECOS. Due to the reduction of the number of subproblems and the improvement of the solving efficiency of a single subproblem, the average solving time of the end error method (FSOCP) is only 13.2% of that of the acceleration error method (ECOS). When FSOCP is used in each case, the average calculation time of the end error method is 54.6% of that of the acceleration error method. Compared with cold start, the hot start (supported by FSOCP only) further greatly reduces the total calculation cost, although the number of sub-problems is increased due to the fact that the number of iteration steps of a single sub-problem is small, the total iteration steps are greatly reduced, the calculation efficiency is improved by more than 5 times, and the average calculation time is only 2.1% of the acceleration error method (ECOS) and 8.8% of the acceleration error method (FSOCP). In terms of residual fuel quality, since the PDG problem under nonlinear aerodynamic forces is a non-convex problem, each method may converge to a different locally optimal solution, and based on the results in table 3, the residual fuel of the end error method (FSOCP, hot start) is greatly superior to the other methods, and the end error method (FSOCP, cold start) is slightly less than the acceleration error method (ECOS), but with a smaller difference.
Figure BDA0003669543310000151
TABLE 3 number of subproblems solved at convergence of each method, calculation time and remaining fuel comparison
In order to verify the capability of the tail end error method for suppressing the position error in the specific direction, the components in the vertical direction in the tail end position error are weighted according to different weights, and the obtained planning result is shown in a table 4. In the solution, a terminal error method (FSOCP, hot start, the upper limit of the iteration step number of each SOCP subproblem is 5. Therefore, the position error in the vertical direction can be effectively suppressed under the condition that the time grid number is not increased and the total iteration step number is not increased greatly by giving a larger weight to the vertical direction in the position error.
Figure BDA0003669543310000152
Figure BDA0003669543310000161
TABLE 4 planning results of weighting the vertical component of the end position error by different weights
In order to compare the solution success rate and the calculation efficiency of the terminal error method and the acceleration error method for one time, monte Carlo simulation is carried out. And taking the original initial position, speed and mass as expectations, adding normal random disturbance to the initial position, speed and mass, wherein the disturbance standard deviation of the position in each direction is 300m, the disturbance standard deviation of the speed in each direction is 30m/s, and the disturbance standard deviation of the fuel mass is 300kg. And considering various delays in the actual situation, the position, the speed and the quality after 2.2s of numerical simulation are taken as initial values of the planning. Each algorithm uses the FSOCP solver. The number of sub-problems at cold start is 30; when the sub-problem iteration step upper limit is 1 and 5 respectively during the hot start, the sub-problem number upper limit is 120 and 60. When the terminal position and the speed error are respectively less than 1m and 0.1m/s, the operation is ended, and the planning is recorded as successful; when the upper limit of the sub-problem number is reached, the end error is still not lower than the threshold value, and the failure of planning is marked. And recording the planning success rate and the average calculation time when the planning is successful.
The monte carlo simulation planning success rate and the calculation time are shown in table 5. Because the initial value fluctuation range is very large, part of random initial values may be really infeasible, and because the planning considering the nonlinear aerodynamic force is a non-convex problem, the difficulty of planning part of the initial values is very large, and the solution meeting the precision requirement is difficult to converge under the practical iteration limitation in some algorithms. As can be seen from the results, the end error method is more computationally efficient than the acceleration error method (FSOCP) at cold start, but the success rate of planning is slightly lower. After the hot start is used, the planning success rate and the calculation efficiency of the terminal error method are both higher than those of the acceleration error method, and when each subproblem is solved for 1 step, the calculation cost is only 10.7% of that of the acceleration error method (FSOCP). In addition, the calculation efficiency and the solution success rate of the acceleration error method using the FSOCP are higher than those of the ECOS.
Figure BDA0003669543310000162
TABLE 5 Monte Carlo simulation planning success ratio and calculation time comparison
The results verify that the terminal error method has higher calculation efficiency than the acceleration error method, has the capability of repeatedly pressing the terminal position error in the specific direction, and can obtain a higher solution success rate than the acceleration error method through hot start.
Those skilled in the art will appreciate that variations may be implemented by those skilled in the art in combination with the prior art and the above-described embodiments, and will not be described in detail herein. Such variations do not affect the essence of the present invention, and are not described herein.
The above description is of the preferred embodiment of the invention. It is to be understood that the invention is not limited to the particular embodiments described above, in that devices and structures not described in detail are understood to be implemented in a manner common in the art; it will be understood by those skilled in the art that various changes and modifications may be made, or equivalents may be modified, without departing from the spirit of the invention. Therefore, any simple modification, equivalent change and modification made to the above embodiments according to the technical spirit of the present invention will still fall within the protection scope of the technical scheme of the present invention.

Claims (3)

1. A rocket substage recovery landing stage power descent guidance method based on terminal errors is characterized by comprising
Step 101: inputting an initial total time of flight t f Discrete time step number k f Sub-problems in numberLimit of n SC Time step change confidence domain upper limit η Δt Thrust variation confidence region upper limit eta T A confidence domain adjustment coefficient beta;
step 102: carrying out initialization;
step 103: if i =1, generating an SOCP problem constraint, and at this time, not using a trust domain constraint, or using a larger trust domain boundary; if i is more than 1, updating SOCP problem constraint;
step 104: solving the SOCP problem; calculating the tail end position and speed error through adding fine grid numerical simulation;
step 105: calculated improvement ratio ξ = (J) L0,i -J L0,i-1 )/(J L1,i -J L1,i-1 ) (ii) a If xi is more than or equal to xi max If the confidence domain boundary is activated, the activated boundary corresponds to the confidence domain radius (eta) Δt Or η T ) Multiplying by beta; if xi is less than or equal to xi min Confidence domain radius η Δt ,η T Are all divided by beta; if xi is less than or equal to 0, rejecting the SOCP problem and resetting the SOCP problem to be the solution of the last subproblem, namely m i [k]＝m i-1 [k],Γ i [k]＝Γ i-1 [k]，r i [k]＝r i-1 [k],v i [k]＝v i-1 [k],a i [k]＝a i-1 [k],a R,i [k]＝a R,i-1 [k],k∈{0,1,…,k f },Δt i ＝Δt i-1 ；
Step 106: updating the total time of flight t f ；
Step 107: if the end position and the speed error are smaller than the preset limits, executing step 109;
step 108: if i < n SC I = i +1, and steps 103 to 106 are repeatedly executed; if i = n SC Then go to step 109;
step 109: outputting a total time of flight t f Mass m, position r, velocity v, thrust T;
wherein,
Figure FDA0003669543300000011
is the planned objective function value; j. the design is a square L1 ＝-m(k f )+ω κr ||r[k f ]||+ω κv ||v[k f ]| is the simulated objective function value;
Figure FDA0003669543300000012
mass, position and speed calculated through numerical simulation, wherein m, r and v are mass, position and speed in the SOCP solving result; i is the sub-question number;
the initialization in step 102 specifically includes:
Figure FDA0003669543300000021
2. a rocket substage recovery landing stage power descent guidance method for tip error based on claim 1 wherein hot start can be supported, acceleration is performed using correlations between SOCP problems; when the hot start is used, the step of solving the SOCP problem also comprises the step of outputting an initial value (x) of the current SOCP problem by a hot start initial value correction method of the SOCP problem 0 ,y 0 ,s 0 )；
The method for correcting the initial hot start value of the SOCP problem comprises the following steps:
step 201: inputting the solution of the existing prior SOCP subproblem
Figure FDA0003669543300000022
The current subproblem equality constraint coefficient matrix A, the right-hand term b, the cone constraint dimension (l, S) 1 ,S 2 ,…,S m ) Correction coefficient η 0 (default value 0.001);
step 202: initialization is performed with η = max (1/(| | a | | |) yellow ∞ +||b|| ∞ ),η 0 )；
Step 203: initialization, i =1;
Figure FDA0003669543300000023
step 204:
Figure FDA0003669543300000024
Figure FDA0003669543300000025
i+＝S j ，
Figure FDA0003669543300000026
step 205, if i is less than or equal to m, repeatedly executing step 204; if i > m, go to step 206;
step 206: outputting the initial value (x) of the current SOCP problem 0 ,y 0 ,s 0 )。
3. A rocket substage recovery landing gear power down guidance method based on end error as claimed in claim 2, characterized in that before "solving SOCP problem" it also includes outputting initial value (x) of current SOCP problem by SOCP problem hot start initial value correction method 0 ,y 0 ,s 0 ) (ii) a Step 105 may be omitted.
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