CN114594785A - Unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization - Google Patents

Unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization Download PDF

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CN114594785A
CN114594785A CN202210049148.9A CN202210049148A CN114594785A CN 114594785 A CN114594785 A CN 114594785A CN 202210049148 A CN202210049148 A CN 202210049148A CN 114594785 A CN114594785 A CN 114594785A
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CN114594785B (en
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刘新福
张国旭
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Beijing Institute of Technology BIT
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Abstract

The invention discloses an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization, and belongs to the technical field of aerospace. The implementation method of the invention comprises the following steps: constructing an optimal control problem in the shortest time, and converting the optimal control problem into a fixed time optimal control problem; introducing dimensionless state quantity theta to replace the tangent value of the course angle, and introducing constraint
Figure DDA0003473178460000011
The non-convex terms in the dynamics and objective function are transferred to the constraint, and the definition variable u transfers the non-convex terms to the inequality constraint | u | is less than or equal to omegamaxδ3V; introducing integer variable etajThe obstacle avoidance constraint of any shape of the two-dimensional plane becomes linear constraint; constrain the generated non-convex inequality to be less than or equal to omegamaxδ3V, carrying out linearization treatment; and (4) solving the mixed integer second-order cone optimization problem in an iteration mode, and realizing the optimal planning of the obstacle avoidance real-time track of the unmanned aerial vehicle. The invention has the following advantages: (1) solving for mixingThe problem of integer second-order cone optimization is solved, and planning is efficient; (2) there is no strict limitation on the shape of the obstacle; (3) time optimization can be realized; (4) the planning reliability is high.

Description

Unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization
Technical Field
The invention relates to an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method, in particular to an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization and suitable for considering a shortest time index, and belongs to the technical field of aerospace.
Background
With the rapid development of autonomous control, communication technology, and high-performance materials, unmanned aerial vehicles have been widely used for various tasks, with more typical fields being military reconnaissance, fire safety, field rescue, and the like. The above tasks put forward higher requirements on the autonomous decision making and flight capability of the unmanned aerial vehicle. Especially to autonomic flight task, unmanned aerial vehicle must possess the ability of keeping away the barrier on line. The obstacle avoidance capability of the unmanned aerial vehicle means that the unmanned aerial vehicle can rapidly and repeatedly plan a feasible and even optimal flight trajectory without collision in the flight process. This requires that the unmanned aerial vehicle can plan a safe flight trajectory with certain optimality for obstacles of different shapes. Generally, optimization methods can be used to solve the trajectory planning problem described above. However, limited to the construction of mathematical models, the existing research and techniques are mostly limited to unmanned aerial vehicle trajectory planning for elliptical or circular obstacles, which makes the application of such techniques limited. In order to solve the problem of obstacle avoidance of two-dimensional arbitrary shapes, a mixed integer-based linear programming Method (MILP) considers the introduction of integer variables and obtains good effect. However, such a method introduces a large number of integer variables, which makes the solution of the whole optimization problem inefficient, and it is often difficult to meet the requirement of computational efficiency in practical application. Therefore, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization not only can be suitable for unmanned aerial vehicle trajectory planning aiming at two-dimensional obstacles in any shapes, but also can effectively reduce the number of integer variables compared with an MILP (mixed integer linear regression) method, and further meets the task requirement of unmanned aerial vehicle on-line obstacle avoidance planning.
In the developed unmanned aerial vehicle trajectory planning method, an unmanned aerial vehicle trajectory planning problem with nonlinear motion constraint and non-convex path constraint is considered in the prior art [1] (see: Wang, Liu, Long, and the like.) of the multi-unmanned aerial vehicle trajectory planning based on penalty function sequence convex planning [ J ]. aviation academic report, 2016,37(010):3149 and 3158.), a mathematical expression of a circular obstacle is established, and a sequence convex optimization algorithm is used for solving the unmanned aerial vehicle trajectory planning problem so as to better weigh optimality and timeliness and improve the unmanned aerial vehicle obstacle avoidance capability, but only the circular obstacle is considered, so that the method has limitation in application.
Prior art [2](see: Maia M H, RKH)
Figure RE-GDA0003605304550000021
On the use of mixed-integer linear programming for predictive control with avoidance constraints[J].International Journal ofRobust&Nonlinear Control,2010, 19(7):822-828.) the method introduces integer variables to construct the unmanned aerial vehicle trajectory planning problem aiming at irregular obstacles (polynomial obstacles), obtains good obstacle avoidance effect and obtains a safe and optimal flight trajectory. However, this method introduces too many integer variables to reduce the problem solving efficiency, and furthermore, the linear unmanned aerial vehicle model is considered, so that this method cannot be applied to the nonlinear unmanned aerial vehicle model.
Therefore, for unmanned aerial vehicle obstacle avoidance real-time trajectory planning, not only the construction of two-dimensional obstacles in any shapes needs to be considered, but also the number of integer variables in the problem can be reduced, the solving efficiency of the optimization problem can be improved, and the application scene of the problem is obviously expanded.
Disclosure of Invention
The invention discloses an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization, which aims to solve the technical problems that: under the constraint of two-dimensional obstacle avoidance in any shape, considering a two-dimensional nonlinear motion model of the unmanned aerial vehicle, and realizing the real-time trajectory planning of the unmanned aerial vehicle obstacle avoidance based on mixed integer second-order cone optimization; has the following advantages: (1) solving the mixed integer second-order cone optimization problem, and realizing efficient planning; (2) there is no strict limitation on the shape of the obstacle; (3) time optimization can be realized; (4) the planning reliability is high.
The purpose of the invention is realized by the following technical scheme:
the invention discloses an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization, which comprises the steps of firstly establishing an inertial coordinate system with the current position of an unmanned aerial vehicle as an origin; establishing a kinematic equation and a constraint condition of the unmanned aerial vehicle under an inertial coordinate system with the current position of the unmanned aerial vehicle as an origin, and establishing the shortest timeThe optimal control problem of (2); converting the shortest time optimal control problem into a fixed time optimal control problem by using the abscissa x as an independent variable; and introducing dimensionless state quantities
Figure RE-GDA0003605304550000022
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics into the last item of the dynamic equation, and introducing constraint
Figure RE-GDA0003605304550000023
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3V; by introducing integer variables ηjMaking the obstacle avoidance constraint of any shape of the two-dimensional plane become a linear constraint containing integer variables; constrain the generated non-convex inequality to be less than or equal to omegamaxδ3Performing linearization treatment on the/V to obtain a mixed integer second-order cone optimization problem; iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the obstacle avoidance real-time track optimal planning of the unmanned aerial vehicle based on the mixed integer second-order cone optimization; when the problem of the mixed integer second-order cone optimization is solved only once, the approximate optimal solution of the original problem can be obtained, and the unmanned aerial vehicle obstacle avoidance real-time trajectory planning is realized.
The invention discloses an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization, which comprises the following steps:
the method comprises the following steps: and establishing an inertial coordinate system with the current position of the unmanned aerial vehicle as an origin.
And establishing an inertial coordinate system by taking the current position of the unmanned aerial vehicle as an origin O, wherein the x axis points to the target point, and the y axis and the x axis form a right-hand rectangular coordinate system, namely establishing the inertial coordinate system by taking the current position of the unmanned aerial vehicle as the origin.
Step two: and D, establishing a kinematic equation and constraint conditions of the unmanned aerial vehicle under the inertial coordinate system established in the step one, and establishing an optimal control problem in the shortest time. The constraint conditions comprise control constraint, terminal constraint and obstacle avoidance constraint of any shape of the two-dimensional plane, and the obstacle avoidance constraint of any shape of the two-dimensional plane is established in a set form, so that the method is suitable for planning the obstacles in any shape of the two-dimensional plane.
Step 2.1: establishing a kinematic equation of the unmanned aerial vehicle under the inertial coordinate system established in the step one; control constraints and terminal constraints are established, and obstacle avoidance constraints of any two-dimensional plane shape are established in a set form, so that the method is suitable for planning the two-dimensional plane obstacle with any shape.
The kinematic equation of the unmanned plane in the inertial coordinate system is expressed as follows:
Figure RE-GDA0003605304550000031
wherein x represents the abscissa position of the unmanned aerial vehicle, y represents the ordinate position of the unmanned aerial vehicle, θ represents the course angle of the unmanned aerial vehicle, V represents the flight speed (constant) of the unmanned aerial vehicle, and ω represents the rate of change of the course angle of the unmanned aerial vehicle, which is a controlled variable.
The control constraints of the drone are as follows:
|ω|≤ωmax (2)
wherein, ω ismaxRepresenting the maximum allowable heading angular rate.
The obstacle avoidance constraint using the form of a set to represent the drone is as follows:
Figure RE-GDA0003605304550000041
wherein X represents the entire two-dimensional planar space, Xo,jRepresenting the planar space occupied by the jth obstacle, and M representing the total number of obstacles.
The terminal constraints of the drone are as follows:
x(tf)＝xf,y(tf)＝yf,θ(tf)＝θf (4)
wherein, tfRepresenting time of flight of the drone, xf、yfθfRespectively representing a given terminal abscissa, ordinate and heading angle.
Step 2.2: and establishing the optimal control problem in the shortest time.
The optimal control problem is obtained by considering the optimal index with the shortest time
Figure RE-GDA0003605304550000042
Wherein, tfThe time of flight is a variable that needs to be solved optimally.
Step three: the x abscissa is used as an independent variable, the shortest time optimal control problem established in the step two is converted into a fixed time optimal control problem, and discretization processing is facilitated; and introducing dimensionless state quantities
Figure RE-GDA0003605304550000043
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics to the last item of the dynamic equation, and introducing the constraint
Figure RE-GDA0003605304550000044
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3And V, six-convexity in subsequent steps is facilitated, and optimization efficiency is improved.
Step 3.1: and (4) converting the shortest time optimal control problem established in the step two into a fixed time optimal control problem by using the abscissa x as an independent variable, so that discretization processing is facilitated.
In the shortest time problem and the inertial frame established in step one, the abscissa x is monotonically increasing. Choosing the abscissa x as the argument, the kinematic equation becomes:
Figure RE-GDA0003605304550000051
where y 'and θ' represent the derivatives of the variables y and θ with respect to the abscissa x. The form of formula (I) is unchanged.
The obstacle avoidance constraint in the formula becomes:
Figure RE-GDA0003605304550000052
wherein the content of the first and second substances,
Figure RE-GDA0003605304550000053
and is provided with
Figure RE-GDA0003605304550000054
A set of representations Xo,jThe boundary value of (2). The formula becomes:
y(xf)＝yf,θ(xf)＝θf (8)
the optimization objective in the equation becomes:
Figure RE-GDA0003605304550000055
step 3.2: introducing dimensionless state quantities
Figure RE-GDA0003605304550000056
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics to the last item of the dynamic equation, and introducing the constraint
Figure RE-GDA0003605304550000057
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3And V, six-convexity in subsequent steps is facilitated, and optimization efficiency is improved.
The dimensionless state quantity is introduced by the formula (10)
Figure RE-GDA0003605304550000058
Replacing the tangent value of the heading angle:
Figure RE-GDA0003605304550000059
using equation (10), the kinematic equation becomes:
Figure RE-GDA00036053045500000510
at this time, the state θ in the original kinematics has been already set
Figure RE-GDA00036053045500000511
Instead, the non-linearity in the dynamics is concentrated to the last term of the kinetic equation, and the objective function in the equation becomes:
Figure RE-GDA00036053045500000512
introducing constraints by equation (13)
Figure RE-GDA0003605304550000061
The non-convex terms in the dynamics and objective functions are passed to the process this constraint:
Figure RE-GDA0003605304550000062
the kinematics in this equation becomes:
Figure RE-GDA0003605304550000063
and the optimization objective in the equation becomes the following linear form:
Figure RE-GDA0003605304550000064
the variable u is defined by the formula as follows:
u:＝δ3ω/V (16)
using formula (xl), the kinetics in formula (xl) become the following linear form:
Figure RE-GDA0003605304550000065
by formula, the non-convex terms in formula are transferred to formula
|u|≤ωmaxδ3/V (18)
And the terminal constraint in the formula becomes
Figure RE-GDA0003605304550000066
Step four: after step three using the abscissa x as the independent variable, by introducing the integer variable ηjAnd D, enabling the obstacle avoidance constraint of any two-dimensional plane shape in the step two to be linear constraint containing integer variables, and further improving the planning and solving efficiency.
For any obstacle j, by introducing an integer variable ηjAnd the obstacle avoidance constraint formula of any shape of the two-dimensional plane is changed into a linear constraint (20), so that the planning and solving efficiency is further improved.
Figure RE-GDA0003605304550000067
Where D is a sufficiently large constant.
Step five: for the process constraint in step three
Figure RE-GDA0003605304550000068
Performing relaxation treatment to obtain constraint
Figure RE-GDA0003605304550000071
And the relaxation form of the optimal control problem in the shortest time is obtained by combining the third step and the fourth step, so that the planning and solving efficiency is improved.
For the process constraint in step three
Figure RE-GDA0003605304550000072
The formula (II) is subjected to a relaxation treatment, and the formula (II) is relaxed into a convex form shown in a formula (22):
Figure RE-GDA0003605304550000073
the formula is convex, i.e. the relaxation problem is as follows:
Figure RE-GDA0003605304550000074
step six: the non-convex inequality generated in the step three is restricted to be less than or equal to omegamaxδ3Performing linearization treatment on the/V, and discretizing to obtain a mixed integer second-order cone optimization problem; iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the obstacle avoidance real-time track optimal planning of the unmanned aerial vehicle based on the mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, if the mixed integer second-order cone optimization problem is solved for once, an approximate optimal solution of the original problem can be obtained, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning is realized, and the planning efficiency can be improved.
The non-convex inequality generated in the step three is constrained to be | u | less than or equal to omegamaxδ3V is linearized at a given initial value delta(0)After that, the equation can be linearized as:
Figure RE-GDA0003605304550000075
the number of discrete points is (N +1), and the optimization variables in the present problem include: y ═ y0y1...yN]T，
Figure RE-GDA0003605304550000076
u＝[u0u1...uN]T，δ＝[δ0δ1...δN]T，η＝[η0η1...ηM]T. We define variables
Figure RE-GDA0003605304550000077
The optimization problem of the mixed integer second-order cone obtained after discretization is as follows:
Figure RE-GDA0003605304550000081
where c, p and b are vectors with appropriate dimensions, Θ and H are matrices with appropriate dimensions,Kis the cartesian product of the second order cone. gmRepresenting constraints in the formula. Problem Problem D (delta)(0)) Is a mixed integer second order cone optimization problem, and an initial section delta is given(0)Iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the unmanned aerial vehicle obstacle avoidance real-time track optimal planning based on the mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, and if the mixed integer second-order cone optimization problem is solved only once, an approximate optimal solution of the original problem can be obtained, so that the real-time trajectory planning of obstacle avoidance of the unmanned aerial vehicle is realized.
Has the advantages that:
1. the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on the mixed integer second-order cone optimization uses the abscissa x as an independent variable, converts the established shortest time optimal control problem into a fixed time optimal control problem, and facilitates discretization processing; and introducing dimensionless state quantities
Figure RE-GDA0003605304550000082
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics into the last item of the dynamic equation, and introducing constraint
Figure RE-GDA0003605304550000083
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3V, convenient bump processing, process constraint
Figure RE-GDA0003605304550000084
Performing relaxation treatment to obtain constraint
Figure RE-GDA0003605304550000085
The solution efficiency is improved by the aid of the convex form.
2. The invention discloses an unmanned aerial vehicle obstacle avoidance real-time track planning method based on mixed integer second-order cone optimization, which is characterized in that after an abscissa x is used as an independent variable, an integer variable eta is introducedjAnd the obstacle avoidance constraint of any shape of the two-dimensional plane becomes a linear constraint containing integer variables, so that the planning and solving efficiency is further improved.
3. The invention discloses an unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization, which is used for establishing obstacle avoidance constraints of any two-dimensional plane shape in a set form in the process of establishing constraint conditions, so that the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method is suitable for planning obstacles of any two-dimensional plane shape.
4. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization is different from a conventional convex optimization method which cannot generate feasible solutions before convergence, if the mixed integer second-order cone optimization problem is solved for once, the approximate optimal solution of the original problem can be obtained, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning is achieved, and the planning efficiency and the reliability can be improved.
Drawings
Fig. 1 is a schematic diagram of the coordinate system establishment of the unmanned aerial vehicle in step 1 of the invention;
FIG. 2 is a flow chart of the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization;
FIG. 3 is the flight trajectory of the UAV when the solution is iterated and solved only once in this embodiment;
fig. 4 is a variation curve of the heading angle of the drone when the solution is iteratively solved and solved only once in this embodiment.
Fig. 5 is a variation curve of the heading angular velocity of the unmanned aerial vehicle when the solution is iteratively solved and solved only once in the embodiment.
Detailed Description
In order to better illustrate the objects and advantages of the present invention, the present invention is explained in detail below by performing simulation analysis on the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization.
Example 1:
as shown in fig. 2, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization disclosed in this embodiment includes the following steps:
the method comprises the following steps: and establishing an inertial coordinate system with the current position of the unmanned aerial vehicle as an origin.
And establishing an inertial coordinate system by taking the current position of the unmanned aerial vehicle as an origin O, wherein the x axis points to the target point, and the y axis and the x axis form a right-hand rectangular coordinate system, namely establishing the inertial coordinate system by taking the current position of the unmanned aerial vehicle as the origin.
Step two: and D, establishing a kinematic equation and constraint conditions of the unmanned aerial vehicle under the inertial coordinate system established in the step one, and establishing an optimal control problem in the shortest time. The constraint conditions comprise control constraint, terminal constraint and obstacle avoidance constraint of any shape of the two-dimensional plane, and the obstacle avoidance constraint of any shape of the two-dimensional plane is established in a set form, so that the method is suitable for planning the obstacles in any shape of the two-dimensional plane.
Step 2.1: establishing a kinematic equation of the unmanned aerial vehicle under the inertial coordinate system established in the step one; control constraints and terminal constraints are established, and obstacle avoidance constraints of any two-dimensional plane shape are established in a set mode, so that the method is suitable for planning obstacles of any two-dimensional plane shape.
The kinematic equation of the unmanned aerial vehicle in the inertial coordinate system is expressed as follows:
Figure RE-GDA0003605304550000101
wherein x represents the abscissa position of the drone, y represents the ordinate position of the drone, θ represents the heading angle of the drone, V represents the flight speed (constant) of the drone, and ω represents the rate of change of the heading angle of the drone, which is the control quantity.
The control constraints of the drone are as follows:
|ω|≤ωmax (26)
wherein, ω ismaxRepresenting the maximum allowable heading angular rate.
The obstacle avoidance constraint using the form of a set to represent the drone is as follows:
Figure RE-GDA0003605304550000102
wherein X represents the entire two-dimensional planar space, Xo,jRepresenting the planar space occupied by the jth obstacle, and M representing the total number of obstacles.
The terminal constraints of the drone are as follows:
x(tf)＝xf,y(tf)＝yf,θ(tf)＝θf (28)
wherein, tfRepresenting time of flight, x, of the dronef、yfθfRespectively representing a given terminal abscissa, ordinate and heading angle.
Step 2.2: and establishing the optimal control problem in the shortest time.
The optimal control problem can be obtained by considering the optimal index with the shortest time
Figure RE-GDA0003605304550000111
Wherein, tfThe time of flight is a variable that needs to be solved optimally.
Step three: using x as the argument, the maximum established in step twoThe short-time optimal control problem is converted into the fixed-time optimal control problem, so that discretization processing is facilitated; and introducing dimensionless state quantities
Figure RE-GDA0003605304550000112
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics to the last item of the dynamic equation, and introducing the constraint
Figure RE-GDA0003605304550000113
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3And V, the six-convexity of the subsequent steps is facilitated, and the solving efficiency is improved.
Step 3.1: and (4) converting the shortest time optimal control problem established in the step two into a fixed time optimal control problem by using the abscissa x as an independent variable, so that discretization processing is facilitated.
In the shortest time problem and the inertial frame established in step one, the abscissa x is monotonically increasing. Choosing the abscissa x as the argument, the kinematic equation becomes:
Figure RE-GDA0003605304550000114
where y 'and θ' represent the derivatives of the variables y and θ with respect to the abscissa x. The form of formula (I) is unchanged. The obstacle avoidance constraint in the formula becomes:
Figure RE-GDA0003605304550000115
wherein the content of the first and second substances,
Figure RE-GDA0003605304550000116
and is
Figure RE-GDA0003605304550000117
A set of representations Xo,jThe boundary value of (1). The formula becomes:
y(xf)＝yf,θ(xf)＝θf (32)
the optimization objective in the equation becomes:
Figure RE-GDA0003605304550000121
step 3.2: introducing dimensionless state quantities
Figure RE-GDA0003605304550000122
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics to the last item of the dynamic equation, and introducing the constraint
Figure RE-GDA0003605304550000123
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3And V, the six-convexity of the subsequent steps is facilitated, and the solving efficiency is improved.
Introducing dimensionless state quantities by formula
Figure RE-GDA0003605304550000124
Replacing the tangent value of the heading angle:
Figure RE-GDA0003605304550000125
using the formula, the kinematic equation in the formula becomes:
Figure RE-GDA0003605304550000126
at this time, the state θ in the original kinematics has been already set
Figure RE-GDA0003605304550000127
Instead, the non-linearities in the dynamics are concentrated into the last term of the equation of the dynamics, whereBecomes:
Figure RE-GDA0003605304550000128
introducing constraints by formula
Figure RE-GDA0003605304550000129
The non-convex terms in the dynamics and objective function are passed to the process constraint:
Figure RE-GDA00036053045500001210
the kinematics in this equation becomes:
Figure RE-GDA00036053045500001211
and the optimization objective in the equation becomes the following linear form:
Figure RE-GDA00036053045500001212
the variable u is defined by the formula as follows:
u:＝δ3ω/V (40)
using formula (xl), the kinetics in formula (xl) become the following linear form:
Figure RE-GDA0003605304550000131
by formula, the non-convex terms in formula are transferred into formula:
|u|≤ωmaxδ3/V (42)
and the terminal constraints in the equation become:
Figure RE-GDA0003605304550000132
step four: after step three using the abscissa x as the independent variable, by introducing the integer variable ηjAnd D, enabling the obstacle avoidance constraint of any two-dimensional plane shape in the step two to be linear constraint containing integer variables, and further improving the planning and solving efficiency.
For any obstacle j, by introducing an integer variable ηjAnd the obstacle avoidance constraint formula of any shape of the two-dimensional plane is changed into linear constraint, so that the planning and solving efficiency is further improved.
Figure RE-GDA0003605304550000133
Where D is a sufficiently large constant.
Step five: for the process constraint in step three
Figure RE-GDA0003605304550000134
Performing relaxation treatment to obtain constraint
Figure RE-GDA0003605304550000135
And the relaxation form of the optimal control problem in the shortest time is obtained by combining the third step and the fourth step, so that the planning and solving efficiency is improved.
For the process constraint in step three
Figure RE-GDA0003605304550000136
The formula is subjected to relaxation treatment, and the formula is relaxed into a convex form shown in the formula:
Figure RE-GDA0003605304550000137
the formula is convex, i.e. the relaxation problem is as follows:
Figure RE-GDA0003605304550000138
step six: the non-convex inequality generated in the step three is restricted to be less than or equal to omegamaxδ3Performing linearization treatment on the/V, and discretizing to obtain a mixed integer second-order cone optimization problem; iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the obstacle avoidance real-time track optimal planning of the unmanned aerial vehicle based on the mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, if the mixed integer second-order cone optimization problem is solved for once, an approximate optimal solution of the original problem can be obtained, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning is realized, and the planning efficiency can be improved.
The non-convex inequality generated in the step three is restricted to be less than or equal to omegamaxδ3V is linearized at a given initial value delta(0)After that, the equation can be linearized as:
Figure RE-GDA0003605304550000141
the number of discrete points is (N +1), and the optimization variables in the present problem include: y ═ y0y1...yN]T，
Figure RE-GDA0003605304550000142
u＝[u0u1...uN]T，δ＝[δ0δ1...δN]T，η＝[η0η1...ηM]T. We define variables
Figure RE-GDA0003605304550000143
The optimization problem of the mixed integer second-order cone obtained after discretization is as follows:
Figure RE-GDA0003605304550000144
where c, p and b are vectors with the appropriate dimensions, Θ and HIs a matrix with the appropriate dimensions and,Kis the cartesian product of the second order cone. gmRepresenting constraints in the formula. Problem Problem D (delta)(0)) Is a mixed integer second order cone optimization problem, and an initial section delta is given(0)Iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the unmanned aerial vehicle obstacle avoidance real-time track optimal planning based on the mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, and if the mixed integer second-order cone optimization problem is solved only once, an approximate optimal solution of the original problem can be obtained, so that the obstacle avoidance real-time trajectory planning of the unmanned aerial vehicle is realized
In order to verify the feasibility of the method, the initial position of the unmanned aerial vehicle is selected to be (0,0) m, and the terminal position is selected to be
Figure RE-GDA0003605304550000145
The limit for the rate of change of the given course angle is 20 °/s, and the specific data is shown in table 1.
Table 1 unmanned aerial vehicle initialization settings
Numerical value
Initial position (x)0,y0) (0,0)m
Initial heading angle θ0 0°
Terminal position (x)f,yf) (110,0)m
Terminal course angle theta f 0°
Limitation omega of course angular velocity max 20°/s
In step six of the present technique, reference is made to iteratively solving the Problem Problem D (δ)(0)) The solution of the original optimal control problem can be obtained, and if the problem is solved once, the approximate optimal solution of the original problem can be obtained. In order to verify the advantages of the method in the aspects of the optimality of iterative solution, the calculation efficiency of solving the solution once and the like, the method is provided with simulation comparative analysis of the two solving modes.
By using the initialization setting of the unmanned aerial vehicle in table 1, the simulation results shown in fig. 3 and fig. 4 are obtained by using iterative solution and only solving once, fig. 3 shows the flight trajectories of the unmanned aerial vehicle in two solution modes, and it can be seen that the unmanned aerial vehicle can effectively complete obstacle avoidance and successfully reach the specified terminal state no matter which solution mode is used, thereby verifying the effectiveness of the invention. In addition, fig. 4 and 5 respectively show the variation curves of the heading angle and the heading angular velocity of the unmanned aerial vehicle, and it can be seen that the solution obtained by solving once is indeed an approximate optimal solution, and only the heading angular velocity has a little difference at the saturation position. This verifies the correctness of the two solutions of the present invention. In addition, the flight time, the calculation time and the iteration times obtained by iterative solution and solution only once are counted in table 2, so that the time consumed by only solving once is the least, and the calculation efficiency is the highest. If a user wants to obtain the optimal solution, the user can use an iterative mode to solve, and if the user wants to improve the solving efficiency, the user can obtain the approximate optimal solution by only solving once.
TABLE 2 time of flight, computation time and number of iterations obtained by iterative solution and solving for only one time
Iterative solution Solved only once
Time of flight tf(s) 23.95 23.98
Calculating the time tc(s) 0.652 0.151
Number of iterations 3 1
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention, and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (7)

1. An unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second-order cone optimization is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
the method comprises the following steps: establishing an inertial coordinate system with the current position of the unmanned aerial vehicle as an origin;
step two: establishing a kinematic equation and constraint conditions of the unmanned aerial vehicle under the inertial coordinate system established in the step one, and establishing an optimal control problem in the shortest time; the constraint conditions comprise control constraints, terminal constraints and obstacle avoidance constraints of any two-dimensional plane shape, and the obstacle avoidance constraints of any two-dimensional plane shape are established in a set form;
step three: the x abscissa is used as an independent variable, the shortest time optimal control problem established in the step two is converted into a fixed time optimal control problem, and discretization processing is facilitated; and introducing a dimensionless state quantity theta to replace the tangent value of the heading angle, concentrating the nonlinearity in the dynamics into the last item of the dynamic equation, and introducing constraint
Figure RE-FDA0003605304540000011
Continuing to transfer the non-convex terms in the dynamics and objective function into the constraint, and defining the variable u to transfer the non-convex terms into the inequality constraint | u | ≦ ωmaxδ3V, the six-convexity of the subsequent steps is facilitated, and the optimization efficiency is improved;
step four: after step three using the abscissa x as the independent variable, by introducing the integer variable ηjThe obstacle avoidance constraint of any shape of the two-dimensional plane in the step two is made to be linear constraint containing integer variables, and the planning and solving efficiency is further improved;
step five: for the process constraint in step three
Figure RE-FDA0003605304540000012
Performing relaxation treatment to obtain constraint
Figure RE-FDA0003605304540000013
The relaxation form of the optimal control problem in the shortest time is obtained by combining the third step and the fourth step, and the planning and solving efficiency is improved;
step six: the non-convex inequality generated in the step three is restricted to be less than or equal to omegamaxδ3Performing linearization treatment on the/V, and discretizing to obtain a mixed integer second-order cone optimization problem; iteratively solving the mixed integerThe second-order cone optimization problem obtains the optimal solution of the optimal control problem in the shortest time, and the optimal planning of the obstacle avoidance real-time track of the unmanned aerial vehicle is realized based on mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, if the mixed integer second-order cone optimization problem is solved for once, an approximate optimal solution of the original problem can be obtained, the unmanned aerial vehicle obstacle avoidance real-time trajectory planning is realized, and the planning efficiency can be improved.
2. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization as claimed in claim 1, wherein: the first implementation method of the method is that,
and establishing an inertial coordinate system by taking the current position of the unmanned aerial vehicle as an origin O, wherein the x axis points to the target point, and the y axis and the x axis form a right-hand rectangular coordinate system, namely establishing the inertial coordinate system by taking the current position of the unmanned aerial vehicle as the origin.
3. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization as claimed in claim 2, wherein: the second step of the method is realized by the following steps,
step 2.1: establishing a kinematic equation of the unmanned aerial vehicle under the inertial coordinate system established in the step one; establishing control constraints and terminal constraints, and establishing obstacle avoidance constraints of any two-dimensional plane shape in a set form, so that the method is suitable for planning obstacles of any two-dimensional plane shape;
the kinematic equation of the unmanned plane in the inertial coordinate system is expressed as follows:
Figure RE-FDA0003605304540000021
wherein x represents the abscissa position of the unmanned aerial vehicle, y represents the ordinate position of the unmanned aerial vehicle, theta represents the course angle of the unmanned aerial vehicle, V represents the flight speed of the unmanned aerial vehicle, and omega represents the change rate of the course angle of the unmanned aerial vehicle, and is a control quantity;
the control constraints of the drone are as follows:
|ω|≤ωmax (2)
wherein, ω ismaxRepresenting the maximum allowable heading angular rate;
the obstacle avoidance constraint using the form of a set to represent the drone is as follows:
Figure RE-FDA0003605304540000022
wherein X represents the entire two-dimensional planar space, Xo,jRepresenting the planar space occupied by the jth obstacle, M representing the total number of obstacles;
the terminal constraints of the drone are as follows:
x(tf)＝xf,y(tf)＝yf,θ(tf)＝θf (4)
wherein, tfRepresenting time of flight, x, of the dronef、yfθfRespectively representing a given terminal abscissa, a given terminal ordinate and a given terminal course angle;
step 2.2: establishing an optimal control problem in the shortest time;
the optimal control problem is obtained by considering the optimal index with the shortest time
Figure RE-FDA0003605304540000031
Wherein, tfThe time of flight is a variable that needs to be solved optimally.
4. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization as claimed in claim 3, wherein: the third step is to realize the method as follows,
step 3.1: the x abscissa is used as an independent variable, the shortest time optimal control problem established in the step two is converted into a fixed time optimal control problem, and discretization processing is facilitated;
in the shortest time problem and the inertial coordinate system established in the first step, the abscissa x is monotonically increased; choosing the abscissa x as the argument, the kinematic equation becomes:
Figure RE-FDA0003605304540000032
wherein y 'and θ' represent the derivatives of the variables y and θ with respect to the abscissa x; the form of formula (I) is unchanged; the obstacle avoidance constraint in the formula becomes:
Figure RE-FDA0003605304540000033
wherein the content of the first and second substances,
Figure RE-FDA0003605304540000034
and is
Figure RE-FDA0003605304540000035
A set of representations Xo,jA boundary value of (d); the formula becomes:
y(xf)＝yf,θ(xf)＝θf (8)
the optimization objective in the equation becomes:
Figure RE-FDA0003605304540000036
step 3.2: introducing dimensionless state quantities
Figure RE-FDA0003605304540000037
Replacing the tangent value of the course angle, concentrating the nonlinearity in the dynamics into the last item of the dynamic equation, and introducing constraint
Figure RE-FDA0003605304540000041
Continuing to transfer non-convex terms in the dynamics and objective function toIn this constraint, and defining the variable u, the non-convex term is transferred to the inequality constraint | u | ≦ ωmaxδ3V, the six-convexity of the subsequent steps is facilitated, and the optimization efficiency is improved;
the dimensionless state quantity is introduced by the formula (10)
Figure RE-FDA0003605304540000042
Replacing the tangent value of the heading angle:
Figure RE-FDA0003605304540000043
using equation (10), the kinematic equation becomes:
Figure RE-FDA0003605304540000044
at this time, the state θ in the original kinematics has been already set
Figure RE-FDA0003605304540000045
Instead, the non-linearity in the dynamics is concentrated to the last term of the kinetic equation, and the objective function in the equation becomes:
Figure RE-FDA0003605304540000046
introducing constraints by equation (13)
Figure RE-FDA0003605304540000047
The non-convex terms in the dynamics and objective function are passed to the process constraint:
Figure RE-FDA0003605304540000048
the kinematics in this equation becomes:
Figure RE-FDA0003605304540000049
and the optimization objective in the equation becomes the following linear form:
Figure RE-FDA00036053045400000410
the variable u is defined by the formula as follows:
u:＝δ3ω/V (16)
using formula (xl), the kinetics in formula (xl) become the following linear form:
Figure RE-FDA00036053045400000411
by formula, the non-convex terms in formula are transferred to formula
|u|≤ωmaxδ3/V (18)
And the terminal constraint in the formula becomes
Figure RE-FDA0003605304540000051
5. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization as claimed in claim 4, wherein: the implementation method of the fourth step is that,
for any obstacle j, by introducing an integer variable ηjThe obstacle avoidance constraint formula of any shape of the two-dimensional plane is changed into a linear constraint (20), so that the planning and solving efficiency is further improved;
Figure RE-FDA0003605304540000052
where D is a sufficiently large constant.
6. The unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization as claimed in claim 5, wherein: the fifth step is to realize that the method is that,
for the process constraint in step three
Figure RE-FDA0003605304540000053
The formula (II) is subjected to a relaxation treatment, and the formula (II) is relaxed into a convex form shown in a formula (22):
Figure RE-FDA0003605304540000054
the formula is convex, i.e. the relaxation problem is as follows:
Figure RE-FDA0003605304540000055
7. the unmanned aerial vehicle obstacle avoidance real-time trajectory planning method based on mixed integer second order cone optimization of claim 6, characterized in that: the sixth realization method comprises the following steps of,
the non-convex inequality generated in the step three is restricted to be less than or equal to omegamaxδ3V is linearized at a given initial value delta(0)After that, the equation can be linearized as:
Figure RE-FDA0003605304540000056
the number of discrete points is (N +1), and the optimization variables in the present problem include: y ═ y0y1...yN]T，
Figure RE-FDA0003605304540000057
u＝[u0u1...uN]T，δ＝[δ0δ1...δN]T，η＝[η0η1...ηM]T(ii) a We define variables
Figure RE-FDA0003605304540000061
The mixed integer second-order cone optimization problem obtained after discretization is as follows:
Figure RE-FDA0003605304540000062
where c, p and b are vectors with the appropriate dimensions, Θ and H are matrices with the appropriate dimensions, and K is the Cartesian product of the second order cones; g is a radical of formulamRepresenting constraints in the formula; problem Problem D (delta)(0)) Is a mixed integer second order cone optimization problem, and an initial section delta is given(0)Iteratively solving the mixed integer second-order cone optimization problem to obtain the optimal solution of the optimal control problem in the shortest time, and realizing the unmanned aerial vehicle obstacle avoidance real-time track optimal planning based on the mixed integer second-order cone optimization; different from the conventional convex optimization method, a feasible solution cannot be generated before convergence, and if the mixed integer second-order cone optimization problem is solved for once, an approximate optimal solution of the original problem can be obtained, so that the real-time trajectory planning of unmanned aerial vehicle obstacle avoidance is realized.
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