CN110806690A - Lossless convex optimization implementation method for unmanned aerial vehicle flight path planning - Google Patents

Lossless convex optimization implementation method for unmanned aerial vehicle flight path planning Download PDF

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CN110806690A
CN110806690A CN201810886186.3A CN201810886186A CN110806690A CN 110806690 A CN110806690 A CN 110806690A CN 201810886186 A CN201810886186 A CN 201810886186A CN 110806690 A CN110806690 A CN 110806690A
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李建勋
张哲�
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Shanghai Jiaotong University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G—PHYSICS
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    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/20—Instruments for performing navigational calculations
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/10—Simultaneous control of position or course in three dimensions
    • G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft

Abstract

A lossless convex optimization implementation method for unmanned aerial vehicle track planning is based on a lossless convex idea and a generalized Banders decomposition algorithm, and utilizes a series of convex optimization and mixed integer linear programming to obtain an optimal solution of a non-convex mixed integer nonlinear programming problem, so that the calculation efficiency is greatly improved on the basis of the minimum cost.

Description

Lossless convex optimization implementation method for unmanned aerial vehicle flight path planning
Technical Field
The invention relates to a technology in the field of unmanned aerial vehicle control, in particular to a lossless convex optimization method for a non-convex unmanned aerial vehicle flight path planning problem with control constraint.
Background
Unmanned aerial vehicle route planning refers to planning one or more safe flight routes based on the factors such as the self performance of the unmanned aerial vehicle, the task time, the energy consumption, the enemy information, the terrain environment, the weather conditions and the like. Due to the fact that the factors to be considered are multiple and complex, a plurality of technical problems need to be solved and perfected in unmanned aerial vehicle track research. In order to describe specific constraints, a 0-1 variable is introduced into a model to express a logical relationship, so that the original linear programming problem becomes a mixed integer nonlinear programming problem and even a non-convex mixed integer nonlinear programming problem.
In most cases, the solution obtained by directly solving the non-convex optimization problem is a feasible solution or a local minimum solution rather than a global optimal solution. The non-convex mixed integer programming problem is characterized in that integer variables are added on the basis of the non-convex problem, so that the difficulty of the problem is further increased, and the problem becomes one of the most difficult problems to solve in the optimization field.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a lossless convex optimization realization method for unmanned aerial vehicle track planning, which is based on a lossless convex idea and a generalized Banders decomposition algorithm and utilizes a series of convex optimization and mixed integer linear programming to obtain an optimal solution of a non-convex mixed integer nonlinear programming problem, thereby greatly improving the calculation efficiency on the basis of the minimum cost.
The invention is realized by the following technical scheme:
the invention relates to a lossless convex optimization implementation method for unmanned aerial vehicle track planning, which comprises the following steps:
the method comprises the steps of firstly, correcting a linear target function of an unmanned aerial vehicle track planning problem into a non-convex function related to energy consumption, correcting a linear state constraint into a convex function state constraint, correcting a control constraint into a non-convex norm range constraint, and then establishing a non-linear optimal control problem model of the unmanned aerial vehicle track planning.
The nonlinear optimal control problem model for unmanned aerial vehicle track planning is as follows:
Figure BDA0001755719100000021
wherein: j is an objective function, x is a state variable, u is a control input, and the current time step is i belongs to [0, T ∈]，
Figure BDA0001755719100000022
For a convex function of the control variable u, l is
Figure BDA0001755719100000023
A, B are constant matrices to describe the system equation, M, p, q, r are constant matrices or constant vectors to describe the constraints of the state variable x, p1,ρ2Two constants are used to describe the upper and lower bounds of the control variable u within a certain range, and δ is the buffer time domain.
Figure BDA0001755719100000024
Is an affine function with respect to x and b, where b is a 0-1 variable used to describe the logical relationship of a particular constraint, C is a constant matrix used to describe the logical variable b, x0And xfRespectively an initial state and a terminal state,
Figure BDA0001755719100000025
is a feasible field of the state variable,for control of the feasible fields of the input, nbThe dimension of the logical variable b.
And step two, relaxing the nonlinear optimal control problem model obtained in the step one to obtain a convex mixed integer linear programming model, specifically, converting the non-convex mixed integer nonlinear programming problem into a convex mixed integer nonlinear programming problem by introducing a relaxation variable gamma, introducing an auxiliary variable β, eliminating a target function of the relaxation problem to obtain a feasibility problem of the relaxation problem, and introducing an auxiliary variable α to obtain a generalized Banders decomposition problem by a relaxation method.
The convex mixed integer nonlinear programming problem is as follows:
Figure BDA0001755719100000027
wherein: Γ is the newly introduced relaxation variable. Setting a new variable z ═ xT,uT,Γ)TObtaining an equivalent simplified model on the right side, wherein f (z) is an objective function of equivalent transformation, g (z, b) is less than or equal to 0 and is a constraint condition of simplification,
Figure BDA0001755719100000028
is the feasible field of the new variable z pair.
The feasibility problem of the convex relaxation problem is that:
Figure BDA0001755719100000031
β is an introduced auxiliary variable used to judge the feasibility of the problem, gj(z, b) is a concrete representation of each line of the function vector g (z, b), nqIs the number of rows. n isbIs the dimension of the logical variable b.
The generalized Banders decomposition problem refers to:
Figure BDA0001755719100000032
wherein α is an introduced auxiliary variable, λ is a Lagrangian variable, and L (z)n,bn,λn) A simplified form of lagrangian function for the convex mixed integer nonlinear programming problem in step two,
Figure BDA0001755719100000033
is the gradient of the Lagrangian function with respect to the variable b, n being the index variable, IkAnd JkAnd mu is a dual variable of z, which is an index set of the kth iteration.
Step three, according to the starting point position x of the unmanned aerial vehicle0And the end position xfMaximum speed V of unmanned aerial vehiclemaxMaximum acceleration AmaxAssigning the barrier information to corresponding variables in the nonlinear optimal control problem model of the unmanned aerial vehicle flight path planning in the step one to obtain an initial planning flight path;
step four, obtaining an optimal solution through an iterative convex mixed integer nonlinear programming problem, a feasibility problem of a relaxation problem and a generalized Banders decomposition problem, and specifically comprising the following steps of:
1) judging whether the convex relaxation problem is feasible, and executing the step 2) when the convex relaxation problem is feasible, or executing the step 3); wherein the convex relaxation problem is the convex mixed integer nonlinear programming problem in the step two.
The judgment means that: and if the feasibility problem of the relaxation problem in the step two is solved, the convex relaxation problem is feasible, otherwise, the convex relaxation problem is not feasible.
2) Solving the convex relaxation problem and obtaining an original dual optimal solution pair (z)k,λk) Then, for index set IkAnd JkMake a correction, Jk＝Jk-1∪{k},Ik＝Ik-1And sets the upper bound Up of the problemk＝min{Upk-1,f(zk)}；
3) Solving the feasibility problem of the convex relaxation problem and obtaining the original dual optimal solution pair (z)k,μk) Then, for index set IkAnd JkMake a correction, Jk＝Jk-1∪{k},Ik＝Ik-1；
4) Solving the generalized Banders decomposition problem and obtaining an original dual optimal solution pair (α)k,
Figure BDA0001755719100000034
) Further setting the lower limit Low of the problemk＝αk；
5) Step 6) is executed when the upper and lower boundaries of the problem are not equal and the error range is not within the error range, otherwise step 7) is executed;
6) by accumulating the metrics for an updated metric set and assigning k to k +1 to the corresponding variable,
Figure BDA0001755719100000041
return to step 2
7) And combining the iteration of the previous k steps to obtain a solution pair of the problem and outputting the optimal solution of the problem.
And step five, according to the obtained optimal solution of the convex relaxation problem in the step four, eliminating the introduced relaxation variable gamma, the auxiliary variable β and the auxiliary variable α to obtain the optimal solution of the nonlinear optimal control problem model of the unmanned aerial vehicle track planning, namely the complete optimal track.
The invention relates to a system for realizing the method, which comprises the following steps: the input module is connected with the modeling module and transmits input information, the modeling module is connected with the resolving module and transmits model information, the resolving module is connected with the output module and transmits resolving information, and the input module is connected with the modeling module and transmits the input information, wherein: and inputting various conditions and constraints of actual problems to the input module, standardizing the actual problems by the input module and then transmitting the standardized actual problems to the modeling module, establishing a nonlinear optimal control problem of the unmanned aerial vehicle flight path planning by the modeling module according to input information and transmitting the nonlinear optimal control problem to the resolving module, transmitting the resolving module to the output module according to the model information and an iterative resolving method in the fourth step, solving the problem result, and outputting the obtained resolving information after the output module standardizes the resolving result.
Technical effects
Compared with the prior art, the linear model of the traditional unmanned aerial vehicle flight path planning is modified according to the actual situation, the non-convex flight path planning problem with control constraint is formed, the non-convex mixed integer non-linear planning problem is solved, the optimal solution of the original problem is obtained by utilizing convex optimization and mixed integer linear planning through the ideas of generalized Banders decomposition and lossless convexity, and the calculation time is greatly reduced on the basis of obtaining the minimum cost.
Drawings
FIG. 1 is a schematic flow diagram of the present invention;
FIG. 2(a) is a three-dimensional track chart of an example in which the size of obstacles is 6;
FIG. 2(b) is a three-dimensional track chart in the case where the obstacle size is 11 in the embodiment;
FIG. 3(a) is a top two-dimensional track chart of the embodiment with 6 obstacle scales;
FIG. 3(b) is a top two-dimensional track chart of the embodiment where the obstacle size is 11;
FIG. 4(a) is a graph showing simulation results in the example in which the obstacle size is 6;
fig. 4(b) is a graph of 11 simulation results of the obstacle size in the example.
Detailed Description
As shown in fig. 1, a non-destructive convex optimization method for a non-convex unmanned aerial vehicle flight path planning problem with control constraints according to this embodiment includes the specific steps of:
step 1, initialization: initial conditions of the input question, arbitrary
Figure BDA0001755719100000042
An error epsilon;
step 2, judging the convex relaxation problem PR (b)k) If it is feasible, execute step 3 when feasibleOtherwise, executing step 4;
step 3, solving the convex relaxation problem PR (b)k) Obtaining the optimal original couple (z)k,λk) Setting Ik＝Ik-1∪{k},Jk＝Jk-1，Upk＝min{Upk-1,f(zk) When Upk＝f(zk) Then (z)*,b*)＝(zk,bk)；
Step 4, solving the convex relaxation problem PRF (b)k) Obtaining the optimal original couple (z)k,λk) Setting Jk＝Jk-1∪{k},Ik＝Ik-1；
Step 5, solving the generalized Banders decomposition problem PR-GBDkObtaining the optimal original pair (α)k,
Figure BDA0001755719100000051
)，Lowk＝αk；
Step 6, when Upk-LowkLess than or equal to epsilon or k is more than kmaxIf not, executing step 8, otherwise, executing step 7;
step 7, updating k to k +1,
Figure BDA0001755719100000052
and executing the step 2;
step 8, when k is less than or equal to kmaxOutput z*The first two items ofOtherwise the output is not feasible.
Real-time data
Firstly, an unmanned aerial vehicle is considered to carry out track planning in urban environment in implementation, building modeling with different sizes is realized by taking a cuboid as an obstacle, two map environment scenes with different sizes and different obstacle numbers are specifically implemented, and specific scene parameters are shown in table 1
Table 1 specific scene parameter settings
Figure BDA0001755719100000054
Second, setting is implemented
The algorithm operating environment is as follows: MATLAB2014a, CPU model Intel Core i7, main frequency 6.4GHz and 8GB memory. The convex optimization problem solver is CVX, and the non-convex optimization problem solver is CPLEX. The specific implementation settings are shown in Table 2
Table 2 implementation environment settings
Third, implement the content
In the implementation, CPLEX is adopted to directly solve the mixed integer programming problem and the lossless convex generalized Banders decomposition algorithm is adopted to carry out comparison, wherein the direct solving method is marked as MINLP, the algorithm of the invention is marked as LC-GBD, the implementation result is shown in table 1, and the path comparison graph generated by the implementation is shown in fig. 2-3.
TABLE 3 implementation results for different scenarios
Figure BDA0001755719100000062
As shown in fig. 2, a 3D flight path display diagram including 6 obstacle scenes and a 3D flight path display diagram including 11 obstacle scenes are shown, as can be seen: the direct solving method is compared with the result track 3d effect of the method, and the track effect of the method is better.
Fig. 3 shows a 2D track top view containing 6 obstacle scenes and a 2D track top view containing 11 obstacle scenes, as can be seen: the direct solving method is compared with the result track 2d effect of the method, and the track effect of the method is better. .
As shown in fig. 4, the histogram and the line chart of the implementation result of scene containing 6 obstacles and the histogram and the line chart of the implementation result of scene 2 containing 11 obstacles are shown as follows: the comparison between the direct solving method and the objective function value and solving time of the method shows that the method has better effect.
From the implementation results of the above tables and graphs, it can be seen that when the lossless saliency method is used, the obtained solution is better than the solution for directly solving the non-convex problem, so that the cost is lower. Therefore, the method provided by the invention can obtain the optimal solution of the original problem, and simultaneously greatly reduces the calculation time.
The foregoing embodiments may be modified in many different ways by those skilled in the art without departing from the spirit and scope of the invention, which is defined by the appended claims and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

Claims (6)

1. A non-destructive convex optimization implementation method for unmanned aerial vehicle flight path planning is characterized by comprising the following steps:
modifying a linear target function of an unmanned aerial vehicle track planning problem into a non-convex function related to energy consumption, modifying a linear state constraint into a convex function state constraint, modifying a control constraint into a non-convex norm range constraint, and then establishing a non-linear optimal control problem model of the unmanned aerial vehicle track planning;
step two, relaxing the nonlinear optimal control problem model obtained in the step one to obtain a convex mixed integer linear programming model, wherein the convex mixed integer linear programming model is obtained by introducing a relaxation variable gamma, converting a non-convex mixed integer nonlinear programming problem into a convex mixed integer nonlinear programming problem, introducing an auxiliary variable β, and eliminating a target function of the relaxation problem to obtain a feasibility problem of the relaxation problem, introducing an auxiliary variable α, and obtaining a generalized Banders decomposition problem by a relaxation method;
step three, according to the starting point position x of the unmanned aerial vehicle0And the end position xfMaximum speed V of unmanned aerial vehiclemaxMaximum acceleration AmaxBarrier and the likeAssigning obstacle information to corresponding variables in the nonlinear optimal control problem model of the unmanned aerial vehicle track planning in the step one to obtain an initial planning track;
step four, obtaining an optimal solution through an iterative convex mixed integer nonlinear programming problem, a feasibility problem of a relaxation problem and a generalized Banders decomposition problem;
step five, according to the obtained optimal solution of the convex relaxation problem in the step four, the introduced relaxation variable gamma, the auxiliary variable β and the auxiliary variable α are omitted, and the optimal solution of the nonlinear optimal control problem model of the unmanned aerial vehicle track planning, namely the complete optimal track, is obtained;
the nonlinear optimal control problem model for unmanned aerial vehicle track planning is as follows:
Figure RE-FDA0001809552850000011
s.t.x(i+1)＝A(i)x(i)+B(i)u(i),i＝0,1,…,T-1
Figure RE-FDA0001809552850000012
Figure RE-FDA0001809552850000013
Figure RE-FDA0001809552850000014
Figure RE-FDA0001809552850000015
wherein: j is an objective function, x is a state variable, u is a control input, and the current time step is i belongs to [0, T ∈]，
Figure RE-FDA0001809552850000016
For a convex function of the control variable u, l is
Figure RE-FDA0001809552850000017
A, B are constant matrices to describe the system equation, M, p, q, r are constant matrices or constant vectors to describe the constraints of the state variable x, p1,ρ2Two constants are used to describe the upper and lower bounds of the control variable u within a certain range, and δ is the buffer time domain.
Figure RE-FDA0001809552850000018
Is an affine function with respect to x and b, where b is a 0-1 variable used to describe the logical relationship of a particular constraint, C is a constant matrix used to describe the logical variable b, x0And xfRespectively an initial state and a terminal state,
Figure RE-FDA0001809552850000021
is a feasible field of the state variable,
Figure RE-FDA0001809552850000022
for control of the feasible fields of the input, nbThe dimension of the logical variable b.
2. The method of claim 1, wherein the convex mixed integer non-linear programming problem is:wherein: Γ is the newly introduced relaxation variable. Setting a new variable z ═ xT,uT,Γ)TObtaining an equivalent simplified model on the right side, wherein f (z) is an objective function of equivalent transformation, g (z, b) is less than or equal to 0 and is a constraint condition of simplification,is the feasible field of the new variable z pair.
3. The method of claim 1, wherein the feasibility problem of the convex relaxation problem is:
Figure RE-FDA0001809552850000025
β is an introduced auxiliary variable used to judge the feasibility of the problem, gj(z, b) is a concrete representation of each line of the function vector g (z, b), nqIs the number of rows. n isbIs the dimension of the logical variable b.
4. The method of claim 1, wherein the generalized Banders decomposition problem is:
Figure RE-FDA0001809552850000026
wherein α is an introduced auxiliary variable, λ is a Lagrangian variable, and L (z)n,bn,λn) A simplified form of lagrangian function for the convex mixed integer nonlinear programming problem in step two,is the gradient of the Lagrangian function with respect to the variable b, n being the index variable, IkAnd JkAnd mu is a dual variable of z, which is an index set of the kth iteration.
5. The method of claim 1, wherein said iterating comprises:
1) judging whether the convex relaxation problem is feasible, and executing the step 2) when the convex relaxation problem is feasible, or executing the step 3); wherein the convex relaxation problem is the convex mixed integer nonlinear programming problem in the step two;
the judgment means that: if the feasibility problem of the relaxation problem in the step two is solved, the convex relaxation problem is feasible, otherwise, the convex relaxation problem is not feasible;
2) solving the convex relaxation problem and obtaining an original dual optimal solution pair (z)k,λk) Then, for index set IkAnd JkMake a correction, Jk＝Jk-1∪{k},Ik＝Ik-1And sets the upper bound Up of the problemk＝min{Upk-1,f(zk)}；
3) Solving the feasibility problem of the convex relaxation problem and obtaining the original dual optimal solution pair (z)k,μk) Then, for index set IkAnd JkMake a correction, Jk＝Jk-1∪{k},Ik＝Ik-1；
4) Solving the generalized Banders decomposition problem and obtaining the original dual optimal solution pair
Figure RE-FDA0001809552850000031
Further setting the lower limit Low of the problemk＝αk；
5) Step 6) is executed when the upper and lower boundaries of the problem are not equal and the error range is not within the error range, otherwise step 7) is executed;
6) by accumulating the metrics for an updated metric set and assigning k to k +1 to the corresponding variable,
Figure RE-FDA0001809552850000032
returning to the step 2;
7) and combining the iteration of the previous k steps to obtain a solution pair of the problem and outputting the optimal solution of the problem.
6. A system for implementing the method of any preceding claim, comprising: the input module is connected with the modeling module and transmits input information, the modeling module is connected with the resolving module and transmits model information, the resolving module is connected with the output module and transmits resolving information, and the input module is connected with the modeling module and transmits the input information, wherein: and inputting various conditions and constraints of actual problems to the input module, standardizing the actual problems by the input module and then transmitting the standardized actual problems to the modeling module, establishing a nonlinear optimal control problem of the unmanned aerial vehicle flight path planning by the modeling module according to input information and transmitting the nonlinear optimal control problem to the resolving module, transmitting the resolving module to the output module according to the model information and an iterative resolving method in the fourth step, solving the problem result, and outputting the obtained resolving information after the output module standardizes the resolving result.
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Application publication date: 20200218