CN112287560A - Solver design method for rocket online trajectory planning - Google Patents

Solver design method for rocket online trajectory planning Download PDF

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CN112287560A
CN112287560A CN202011262143.1A CN202011262143A CN112287560A CN 112287560 A CN112287560 A CN 112287560A CN 202011262143 A CN202011262143 A CN 202011262143A CN 112287560 A CN112287560 A CN 112287560A
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rocket
following
trajectory planning
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CN112287560B (en
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程晓明
王晋麟
张惠平
徐帆
魏小丹
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Beijing Aerospace Automatic Control Research Institute
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Abstract

The invention relates to a solver design method for rocket online trajectory planning, which is a solver design method based on rocket trajectory planning problems and belongs to the technical field of aerospace guidance control. The invention designs a convex optimization solving method aiming at rocket trajectory planning, further improves the solving speed, and can meet the requirement of the rocket online trajectory planning problem on solving instantaneity. The explicit coding technology designed by the invention carries out off-line explicit coding on some complex mathematical calculation processes in the convex optimization solving process, and then carries the off-line explicit coding on the embedded platform again, so that the solving speed of the rocket trajectory planning problem can be further improved, and the requirement of on-line planning on the real-time performance is met.

Description

Solver design method for rocket online trajectory planning
Technical Field
The invention relates to a solver design method for rocket online trajectory planning, which is a solver design method based on rocket trajectory planning problems and belongs to the technical field of aerospace guidance control.
Background
At present, regarding the convex optimization problem of rocket trajectory planning, only foreign design solvers such as ECOS, CVXGEN and the like can be relied on, and the solvers have two problems: firstly, the internal code is in a 'black box' state; secondly, the method belongs to a universal solver, and a problem of multiple versatility is considered during design, so that the solving speed is low, and the real-time requirement of a large-scale planning problem of multi-stage rocket trajectory planning cannot be met.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method can be used for solving the convex optimization problem of rocket trajectory planning on line. The reliability and the stability of the rocket during the on-line trajectory planning can be ensured. The real-time requirement of the rocket for online trajectory planning under fault conditions or other conditions is met. Firstly, aiming at the convex optimization problem of rocket trajectory planning, carrying out corresponding real-time interior point algorithm design, and then carrying out customized explicit coding of the algorithm; and finally, carrying out simulation verification on the algorithm by using the real-time interior point algorithm through a simulation experiment.
The technical solution of the invention is as follows:
a solver design method for rocket online trajectory planning, the method comprises the following steps:
the method comprises the following steps of firstly, constructing a rocket track planning problem into a standard convex planning problem, and specifically comprises the following steps:
taking the fuel province as an optimization index, firstly describing the rocket path planning problem as follows:
Figure BDA0002774966360000021
wherein r ═ x, y, z]TIs a position vector, v ═ vx,vx,vx]TIs the velocity vector, m is the aircraft mass, g ═ gx,gx,gx]TIs a gravity acceleration vector, T ═ Tx,Tx,Tx]TRepresenting the aircraft thrust vector. I isspIs the specific impulse of the aircraft, g0Is HaipingThe magnitude of the gravitational acceleration of the surface.
Constructing the trajectory planning problem into a standard convex planning problem form by an equal-interval discrete and state transfer method:
Figure BDA0002774966360000022
wherein x represents a state vector (including position, speed and mass), and A, G matrix, b, s and h represent a matrix and a vector obtained by converting the formula (1) to represent dynamic constraint and state constraint.
And step two, solving the standard convex programming problem constructed in the step one, wherein the concrete steps are as follows:
(1) initializing variables;
the iteration variables x and s are initialized by the following equation.
Figure BDA0002774966360000023
Figure BDA0002774966360000024
s＝r+αe (5)
The iteration variables y and z are initialized by the following equation.
Figure BDA0002774966360000031
Figure BDA0002774966360000032
z＝r+αe (8)
(2) Updating a Nesterov-Todd scaling matrix, wherein the specific method comprises the following steps:
the Nesterov-Todd scaling matrix W is calculated by variables z and s, and satisfies the following conditions:
W-Ts＝Wz (16)
w is a block diagonal matrix whose corner blocks can be computed from z and s for each constraint. For non-negative image limit, the corner blocks are:
Figure BDA0002774966360000033
for a second-order cone, the angle block calculation method is as follows:
Figure BDA0002774966360000034
wherein
Figure BDA0002774966360000035
After W is calculated, it is defined
λ＝W-Ts＝Wz (20)
(3) Determining the affine search direction, specifically comprising the following steps:
affine search direction (Δ x)a,Δya,Δza,Δsa,Δτa,Δκa) Obtained by solving the following system of equations.
Figure BDA0002774966360000041
λ·(WΔza+W-1Δsa)＝-λ·λ (21)
κΔτa+τΔκa＝-τκ
(4) Carrying out ray-imitating search by using the affine search direction obtained in the step (3), wherein the specific method comprises the following steps: finding the maximum aa∈[0,1]Make a pair
Figure BDA0002774966360000042
All have:
Figure BDA0002774966360000043
(5) determining a combined search direction, wherein the specific method comprises the following steps:
the combined search direction (Δ x, Δ y, Δ z, Δ s, Δ τ, Δ κ) is obtained by solving the following system of equations.
Figure BDA0002774966360000044
λ·(WΔz+W-1Δs)＝σμe-λ·λ-W-1Δsa·WΔza (24)
κΔτ+τΔκ＝σμ-τκ-ΔτaΔκa
Wherein σ ═ 1- αa)3；
(6) And (5) carrying out combined line search by using the combined search direction obtained in the step (5), wherein the specific method comprises the following steps: finding the maximum alpha E [0,1 ∈]Make a pair
Figure BDA0002774966360000045
All have:
Figure BDA0002774966360000046
(7) updating the initialized variables in the step (1);
Figure BDA0002774966360000051
(8) calculating dual gaps and residual errors;
the residual is calculated as:
Figure BDA0002774966360000052
the main residuals are:
Figure BDA0002774966360000053
the dual residuals are:
Figure BDA0002774966360000054
the main non-solution criterion is as follows:
Figure BDA0002774966360000055
the dual non-solution criterion is as follows:
Figure BDA0002774966360000056
(9) determining a termination condition of the variable update in the step (7);
definition of
ρ＝max{-cTx,-bTy-hTz} (14)
When in use
Figure BDA0002774966360000057
When the updating is finished, the updating is stopped;
thirdly, obtaining a control variable of each discrete point;
extracting a control variable T ═ T at each discrete pointx,Tx,Tx]TObtaining the pitch angle instruction of the rocket at each discrete point
Figure BDA0002774966360000061
With yaw angle command psicI.e. pitch angle command
Figure BDA0002774966360000062
With yaw angle command psicThe result of rocket online trajectory planning is obtained;
Figure BDA0002774966360000063
Figure BDA0002774966360000064
and fourthly, using the result of the online track planning obtained in the third step for the guidance and control of the rocket.
Advantageous effects
The invention designs a convex optimization solving method aiming at rocket trajectory planning, further improves the solving speed, and can meet the requirement of the rocket online trajectory planning problem on solving instantaneity.
The explicit coding technology designed by the invention carries out off-line explicit coding on some complex mathematical calculation processes in the convex optimization solving process, and then carries the off-line explicit coding on the embedded platform again, so that the solving speed of the rocket trajectory planning problem can be further improved, and the requirement of on-line planning on the real-time performance is met.
Detailed Description
The present invention will be further described with reference to the following examples.
A solver design method for rocket online trajectory planning, the method comprises the following steps:
the method comprises the following steps of firstly, constructing a rocket track planning problem into a standard convex planning problem, and specifically comprises the following steps:
taking the fuel province as an optimization index, firstly describing the rocket path planning problem as follows:
Figure BDA0002774966360000065
wherein r ═ x, y, z]TIs a position vector, v ═ vx,vx,vx]TIs the velocity vector, m is the aircraft mass, g ═ gx,gx,gx]TIs a gravity acceleration vector, T ═ Tx,Tx,Tx]TRepresenting the aircraft thrust vector. I isspIs the specific impulse of the aircraft, g0The magnitude of the gravitational acceleration at sea level.
Constructing the trajectory planning problem into a standard convex planning problem form by an equal-interval discrete and state transfer method:
Figure BDA0002774966360000071
wherein x represents a state vector (including position, speed and mass), and A, G matrix, b, s and h represent a matrix and a vector obtained by converting the formula (1) to represent dynamic constraint and state constraint.
And step two, solving the standard convex programming problem constructed in the step one, wherein the concrete steps are as follows:
(1) initializing variables;
the iteration variables x and s are initialized by the following equation.
Figure BDA0002774966360000072
Figure BDA0002774966360000073
s＝r+αe (5)
The iteration variables y and z are initialized by the following equation.
Figure BDA0002774966360000074
Figure BDA0002774966360000075
z＝r+αe (8)
(2) Updating a Nesterov-Todd scaling matrix, wherein the specific method comprises the following steps:
the Nesterov-Todd scaling matrix W is calculated by variables z and s, and satisfies the following conditions:
W-Ts＝Wz (16)
w is a block diagonal matrix whose corner blocks can be computed from z and s for each constraint. For non-negative image limit, the corner blocks are:
Figure BDA0002774966360000081
for a second-order cone, the angle block calculation method is as follows:
Figure BDA0002774966360000082
wherein
Figure BDA0002774966360000083
After W is calculated, it is defined
λ＝W-Ts＝Wz (20)
(3) Determining the affine search direction, specifically comprising the following steps:
affine search direction (Δ x)a,Δya,Δza,Δsa,Δτa,Δκa) Obtained by solving the following system of equations.
Figure BDA0002774966360000084
λ·(WΔza+W-1Δsa)＝-λ·λ (21)
κΔτa+τΔκa＝-τκ
(4) Carrying out ray-imitating search by using the affine search direction obtained in the step (3), wherein the specific method comprises the following steps:
finding the maximum aa∈[0,1]Make a pair
Figure BDA0002774966360000085
All have:
Figure BDA0002774966360000086
(5) determining a combined search direction, wherein the specific method comprises the following steps:
the combined search direction (Δ x, Δ y, Δ z, Δ s, Δ τ, Δ κ) is obtained by solving the following system of equations.
Figure BDA0002774966360000091
λ·(WΔz+W-1Δs)＝σμe-λ·λ-W-1Δsa·WΔza (24)
κΔτ+τΔκ＝σμ-τκ-ΔτaΔκa
Wherein σ ═ 1- αa)3；
(6) And (5) carrying out combined line search by using the combined search direction obtained in the step (5), wherein the specific method comprises the following steps: finding the maximum alpha E [0,1 ∈]Make a pair
Figure BDA0002774966360000092
All have:
Figure BDA0002774966360000093
(7) updating the initialized variables in the step (1);
Figure BDA0002774966360000094
(8) calculating dual gaps and residual errors;
the residual is calculated as:
Figure BDA0002774966360000095
the main residuals are:
Figure BDA0002774966360000096
the dual residuals are:
Figure BDA0002774966360000101
the main non-solution criterion is as follows:
Figure BDA0002774966360000102
the dual non-solution criterion is as follows:
Figure BDA0002774966360000103
(9) determining a termination condition of the variable update in the step (7);
definition of
ρ＝max{-cTx,-bTy-hTz} (14)
When in use
Figure BDA0002774966360000104
When the updating is finished, the updating is stopped;
thirdly, obtaining a control variable of each discrete point;
extracting a control variable T ═ T at each discrete pointx,Tx,Tx]TObtaining the pitch angle instruction of the rocket at each discrete point
Figure BDA0002774966360000105
With yaw angle command psicI.e. pitch angle command
Figure BDA0002774966360000106
With yaw angle command psicThe result of rocket online trajectory planning is obtained;
Figure BDA0002774966360000107
Figure BDA0002774966360000108
and fourthly, using the result of the online track planning obtained in the third step for the guidance and control of the rocket.
Examples
By taking a certain rocket as an object, the explicit coding solver provided by the invention is utilized to solve the trajectory planning problem on line to obtain a simulation result, including the orbit elements entering the orbit finally, the flight trajectory and attitude angle from the fault moment to the moment of entering the circular orbit.
The circular orbit emergency planning requirements are as follows:
planning a target: at the end of the flight phase, the rocket enters a circular mooring path of maximum radius.
Planning conditions are as follows: the thrust failure time was set to 125s, and the thrust remained unchanged after the drop, and no fuel leakage occurred.
When the main engine flies to 125s, the thrust is set to be reduced by 30%, under the condition that the available fuel is not changed, the flying time is prolonged, the rail entering at the final fuel exhaustion moment is a circular rail with the height of 150km, and the planning time is less than 1 s.
In the simulation process, the computational efficiency of the convex optimization solver is tested, the running environment is ARM Raspberry Pi 3B (Raspberry Pi Model 3B), the quad-core 64-bit ARM Cortex-A53 processor, the single-core main frequency is 1.2GHz, the RAM is 1GB, the double-precision floating point operational capability is 150.231M Flops, and the on-chip RAM is 1M. The results of calculation efficiency and calculation accuracy are shown in the following table.
TABLE 2 results of the calculation efficiency and calculation accuracy of the currently tested examples
Figure BDA0002774966360000111
Through the analysis of the customized solver and the tests performed at present, the explicit coding solver designed by the invention can complete the rocket online trajectory planning task meeting the orbit entering precision within 1 s.

Claims (4)

1. A solver design method for rocket online trajectory planning is characterized by comprising the following steps:
firstly, constructing a rocket path planning problem into a standard convex planning problem
Secondly, solving the standard convex programming problem constructed in the first step;
thirdly, acquiring control variables of each discrete point to obtain a result of online track planning;
and fourthly, using the result of the online track planning obtained in the third step for the guidance and control of the rocket.
2. A solver design method for rocket online trajectory planning according to claim 1, characterized in that:
the specific method for constructing the rocket path planning problem into the standard convex planning problem in the first step is as follows:
taking the fuel province as an optimization index, firstly describing the rocket path planning problem as follows:
Figure FDA0002774966350000011
wherein r ═ x, y, z]TIs a position vector, v ═ vx,vx,vx]TIs the velocity vector, m is the aircraft mass, g ═ gx,gx,gx]TIs a gravity acceleration vector, T ═ Tx,Tx,Tx]TRepresenting an aircraft thrust vector; i isspIs the specific impulse of the aircraft, g0The gravity acceleration at sea level;
the trajectory planning problem is constructed into a standard convex planning problem form:
Figure FDA0002774966350000012
wherein x represents a state vector comprising position, velocity and mass, and A, G matrix, b, s, h represents a matrix and a vector obtained by conversion of the formula (1), and represents kinetic constraint and state constraint.
3. A solver design method for rocket online trajectory planning according to claim 1, characterized in that:
the concrete steps of solving the standard convex programming problem constructed in the first step in the second step are as follows:
(1) initializing variables;
the iteration variables x and s are initialized by the following formula;
Figure FDA0002774966350000021
Figure FDA0002774966350000022
s＝r+αe (5)
the iteration variables y and z are initialized by the following formula;
Figure FDA0002774966350000023
Figure FDA0002774966350000024
z＝r+αe (8)
(2) updating a Nesterov-Todd scaling matrix, wherein the specific method comprises the following steps:
the Nesterov-Todd scaling matrix W is calculated by variables z and s, and satisfies the following conditions:
W-Ts＝Wz (16)
w is a block diagonal matrix whose corner blocks can be computed from z and s for each constraint; for non-negative image limit, the corner blocks are:
Figure FDA0002774966350000025
for a second-order cone, the angle block calculation method is as follows:
Figure FDA0002774966350000026
wherein
Figure FDA0002774966350000031
After W is calculated, it is defined
λ＝W-Ts＝Wz (20)
(3) Determining the affine search direction, specifically comprising the following steps:
affine search direction (Δ x)a,Δya,Δza,Δsa,Δτa,Δκa) The method is obtained by solving the following equation system;
Figure FDA0002774966350000032
(4) carrying out ray-imitating search by using the affine search direction obtained in the step (3), wherein the specific method comprises the following steps:
finding the maximum aa∈[0,1]Make a pair
Figure FDA0002774966350000035
All have:
Figure FDA0002774966350000033
(5) determining a combined search direction, wherein the specific method comprises the following steps:
the combined search direction (Δ x, Δ y, Δ z, Δ s, Δ τ, Δ κ) is obtained by solving the following equation system;
Figure FDA0002774966350000034
wherein σ ═ 1- αa)3；
(6) And (5) carrying out combined line search by using the combined search direction obtained in the step (5), wherein the specific method comprises the following steps: finding the maximum alpha E [0,1 ∈]Make a pair
Figure FDA0002774966350000047
All have:
Figure FDA0002774966350000041
(7) updating the initialized variables in the step (1);
Figure FDA0002774966350000042
(8) calculating dual gaps and residual errors;
the residual is calculated as:
Figure FDA0002774966350000043
the main residuals are:
Figure FDA0002774966350000044
the dual residuals are:
Figure FDA0002774966350000045
the main non-solution criterion is as follows:
Figure FDA0002774966350000046
the dual non-solution criterion is as follows:
Figure FDA0002774966350000051
(9) determining a termination condition of the variable update in the step (7);
definition of
ρ＝max{-cTx,-bTy-hTz} (14)
When in use
Figure FDA0002774966350000052
When the update is stopped, the update is stopped.
4. A solver design method for rocket online trajectory planning according to claim 1, characterized in that: the method for acquiring the control variable of each discrete point in the third step comprises the following steps:
extracting a control variable T ═ T at each discrete pointx,Tx,Tx]TObtaining the pitch angle instruction of the rocket at each discrete point
Figure FDA0002774966350000055
With yaw angle command psicI.e. pitch angle command
Figure FDA0002774966350000056
With yaw angle command psicThe result of rocket online trajectory planning is obtained;
Figure FDA0002774966350000053
Figure FDA0002774966350000054
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