CN116520822A - Smooth curvature parameterized representation path generation method - Google Patents
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- 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
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
- G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
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- G—PHYSICS
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- G06Q10/00—Administration; Management
- G06Q10/04—Forecasting or optimisation specially adapted for administrative or management purposes, e.g. linear programming or "cutting stock problem"
- G06Q10/047—Optimisation of routes or paths, e.g. travelling salesman problem
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- G—PHYSICS
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- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T11/00—2D [Two Dimensional] image generation
- G06T11/20—Drawing from basic elements, e.g. lines or circles
- G06T11/203—Drawing of straight lines or curves
Abstract
The invention discloses a path generation method of smooth curvature parameterization representation, which comprises the following steps: step S1: obtaining rough curvature through adjacent sampling points; step S2: obtaining a piecewise linear curvature curve by using a convex optimization method; step S3: the curvature of each point is adjusted to spatially fit the original path. The invention has the advantages of simple principle, wide application range, small hardware cost, capability of improving path planning precision and the like.
Description
Technical Field
The invention mainly relates to the technical field of unmanned aerial vehicles, in particular to a smooth path generation method with parameterized curvature.
Background
In the unmanned field, unmanned paths need to be planned, and generation of paths is guaranteed by adopting a path tracking control mode. In the prior art, it is the most common technical approach to generally employ a series of discrete cartesian point-described paths. The most basic road model is a linear interpolation model; other path generation methods, such as Du Binsi curves, bezier curves, polynomials, spline functions, etc., are all spatially fitting these discrete points to represent the path.
In the unmanned field, continuous curvature is critical in path generation, as continuous curvature means a smooth steering angle. When the curvature is discontinuous, control instability and off-course conditions tend to occur at this point. There are still some disadvantages in existing various path representation or generation methods, such as:
a. discrete waypoints often need to be added with links for calculating to obtain continuous curvature in a control algorithm;
b. the path curvature discontinuity represented by the Du Binsi curve;
c. paths represented by bezier curves or splines may have a high degree of accuracy, but the curvature tends to oscillate, meaning that the vehicle steering wheel will turn back and forth.
Although many practitioners continue to study and optimize the above technology in recent years, the route generation method based on optimization is also widely studied and applied, but the difficulty of realizing the technology in real time is still faced with large calculation amount, and sometimes the optimization may not have a proper solution, which brings a hindrance to further optimization promotion in the control technology in the unmanned area.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at the technical problems existing in the prior art, the invention provides the path generation method for smooth curvature parameterization representation, which has the advantages of simple principle, wide application range, small hardware cost and capability of improving path planning precision.
In order to solve the technical problems, the invention adopts the following technical scheme:
a method of path generation for a smooth curvature parameterized representation, comprising:
step S1: obtaining rough curvature through adjacent sampling points; obtaining a corresponding curvature for each path sampling point, and assuming the curvature between every two adjacent path sampling points to change linearly to obtain a curve of curvature-along path length;
step S2: obtaining a piecewise linear curvature curve by using a convex optimization method; the piecewise linear fitting of the curvature curve is solved by converting the piecewise linear problem into an unconstrained convex optimization problem;
step S3: adjusting the curvature of each point to fit the original path in space; and (2) adjusting the curvature parameter at each sampling point on the basis of the piecewise linearized curvature curve in the step (S2) so that the deviation between the road represented by the whole and the road represented by the original discrete point meets the precision.
As a further improvement of the process of the invention: in the step S1, a corresponding quadratic parameter equation is solved by each cartesian coordinate sampling point and several adjacent points around, and a rough curvature at each sampling point is obtained by a calculation formula in the form of a curvature scalar in a planar coordinate system.
As a further improvement of the process of the invention: the flow of the step S1 includes:
step S101: the length along the path of each sampling point is calculated according to the following formula:
wherein the subscript is the index of the sampling point;respectively represent +.>Cartesian abscissa of each path sampling point in a space coordinate system, +.>Respectively represent +.>Cartesian ordinate of each path sampling point in a space coordinate system, < >>Represents the total number of sampling points in the path, +.>Respectively represent +.>Arc length parameters of each path sampling point, representing distance along the path from the start point, +.>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Represents the total number of sampling points, starting sampling point +.>；
Step S102: converting the sampling points from Cartesian coordinates to a curvature domain: and constructing a secondary parameter equation by adopting a plurality of adjacent points to interpolate, solving the coefficient of the secondary parameter equation, and obtaining the curvature of the point through derivation.
As a further improvement of the process of the invention: the step S102 includes:
constructing a quadratic parameter equation by the above method, whereinFor passing through adjacent 3 routesCoefficients of quadratic parameter equation formed by spatial fitting of radial sampling points, +.>Respectively a Cartesian abscissa and an ordinate of the path sampling point in a space coordinate system; adopts->Point and->Length along the path between points> and />As parameter->And->：
The coefficients of the quadratic parameter equation are obtained by the following formula:
the curvature is obtained by a calculation formula of the curvature under a plane coordinate system:
wherein ,for curvature (S)>Is the first and second derivatives of the quadratic parameter equation.
As a further improvement of the process of the invention: in the step S2, a convex optimization method is used to perform linear trend filtering on the curve of curvature-along path length obtained in the step S1 to obtain a piecewise linear curvature curve.
As a further improvement of the process of the invention: in the step S2, piecewise linear fitting is performed on the curvature curve by converting the piecewise linear problem into the unconstrained convex optimization problem, which is described as follows:
wherein ,for the curvature of each sampling point obtained in step S1, and (2)>The optimized curvatures of the i, i+1 and i+2 path sampling points are the curvatures of the sampling points after piecewise linearization, and the objective function minimizes the curvature norm and considers the linear trend; />Is a variable with adjustable parameters.
As a further improvement of the process of the invention: in step S3, the curvature is adjusted by: and sampling the curvature slope, and selecting the curvature slope corresponding to the smallest deviation as the curvature adjustment slope, thereby obtaining the curvature after adjustment.
As a further improvement of the process of the invention: in step S3, starting from the first path sampling point, the curve at each sampling point is adjusted point by pointRate ofUntil the last path sampling point; wherein (1)>For the +.sup.th obtained in step S1>Curvature of the sampling points.
As a further improvement of the process of the invention: the flow of the step S3 includes:
step S301: an initial guess of the curvature slope is calculated: for each sampling point with curvature to be adjusted, selecting the curvature of the previous point and the curvature of the five subsequent points, wherein the curvatures of the seven sampling points are obtained by piecewise linearization obtained in the step S2, and obtaining an initial guess value of curvature slope through a least square method;
step S302: sampling curvature slope within a certain range around the curvature slope, selecting the curvature slope that minimizes the evaluation function, and obtaining the curvature of the corresponding second point as the adjusted curvature.
Compared with the prior art, the invention has the advantages that:
the smooth curvature parameterized representation path generation method has the advantages of simple principle, wide application range, small hardware cost and capability of improving path planning precision, can meet the precision of road representation while obtaining a road with continuous and smooth curvature, is convenient for generating the fastest curve in subsequent track planning by a curvature curve, can calculate and obtain a feedforward steering angle through the curvature curve in the design of a path tracking feedback feedforward controller, has simple algorithm steps, is easy to understand in principle and has small calculated amount.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
Fig. 2 is a schematic representation of a road represented by discrete waypoints employed in a specific embodiment of the invention.
Fig. 3 is a schematic view showing the effect of obtaining the rough curvature through step S1 in the embodiment of the present invention.
FIG. 4 is a schematic diagram showing the effect of piecewise linearization obtained in step S2 in an embodiment of the invention.
Fig. 5 is a schematic view showing the effect of sampling curvature slope in step S3 according to an embodiment of the present invention.
Fig. 6 is a schematic diagram showing the effects of the curvature curve obtained in step S3 and the piecewise curvature curve in step S2 according to the embodiment of the present invention.
Detailed Description
The invention will be described in further detail with reference to the drawings and the specific examples.
The smooth curvature parameterized representation path generation method is mainly applied to the technical field of unmanned control and is used for realizing unmanned full-intelligent and high-precision control, and the core principle is as follows: the invention adopts the curve that the curvature changes along the length of the path to parameterize the path, and the curvature can be conveniently continuous on the curvature curve; the invention further evaluates the smoothness of the curvature through an evaluation function, so that the smoothness of the curvature curve can be continuously optimized, and the deviation between the whole path obtained by Fresnel integration and the path represented by the original discrete road point can meet a certain precision by slightly adjusting the curvature corresponding to each sampling point.
As shown in fig. 1-6, the method for generating a path of a smooth curvature parameterized representation of the present invention comprises the steps of:
step S1: obtaining rough curvature through adjacent sampling points; obtaining a corresponding curvature for each path sampling point, and assuming the curvature between every two adjacent path sampling points to change linearly to obtain a curve of curvature-along path length;
namely: solving a corresponding secondary parameter equation through each Cartesian coordinate sampling point and a plurality of surrounding adjacent points, and obtaining rough curvature at each sampling point through a curvature scalar form calculation formula under a plane coordinate system;
step S2: obtaining a piecewise linear curvature curve by using a convex optimization method; the piecewise linear fitting of the curvature curve is solved by converting the piecewise linear problem into an unconstrained convex optimization problem;
namely: performing linear trend filtering on the curve of the curvature-along path length obtained in the step S1 by using a convex optimization method to obtain a piecewise linear curvature curve;
step S3: adjusting the curvature of each point to fit the original path in space; and (2) adjusting the curvature parameter at each sampling point on the basis of the piecewise linearized curvature curve in the step (S2) so that the deviation between the road represented by the whole and the road represented by the original discrete point meets the precision.
Namely: performing tiny adjustment on curvature parameters at each sampling point on the basis of the piecewise linearization curvature curve in the step S2, so that the deviation between the road represented by the whole and the road represented by the original discrete point meets the accuracy;
in a specific application example, the flow of step S1 may include:
step S101: the length along the path of each sampling point is calculated according to the following formula:
wherein the subscript (e.g) Index for sampling point; />Respectively represent +.>Cartesian abscissa of each path sampling point in a space coordinate system, +.>Respectively represent +.>Cartesian ordinate of each path sampling point in a space coordinate system, < >>Represents the total number of sampling points in the path, +.>Respectively represent +.>Arc length parameters of each path sampling point, representing distance along the path from the start point, +.>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Representing the total number of sampling points, the initial sampling point。
Step S102: converting the sampling points from Cartesian coordinates to a curvature domain: because no expression is used for deriving to obtain a scalar form of curvature, interpolation is needed to be carried out by constructing a secondary parameter equation by adopting a plurality of adjacent points, coefficients of the secondary parameter equation are solved, and the curvature at the point is obtained by deriving, and the specific steps are as follows:
from the above, a quadratic parameter equation can be constructed, in whichFor coefficients of the quadratic parameter equation formed by spatial fitting of adjacent 3 path sampling points +.>Respectively the cartesian abscissa and the ordinate of the path sampling point in the space coordinate system. For->In point, there is also a need for +.>And solving the secondary parameter equation at three points. Adopts->Point and->Length along the path between points> and />As parameter->And->：
The coefficients of the quadratic parameter equation can be obtained by:
then, the curvature can be obtained by a calculation formula of the curvature under the plane coordinate system (the curvature of the instantaneous steering center is positive when left and negative when right):
wherein ,for curvature (S)>Is the first and second derivatives of the quadratic parameter equation.
In a specific application example, the flow of step S2 may include:
the piecewise linear fitting of curvature curves is solved by converting the piecewise linear problem into an unconstrained convex optimization problem, which is described as follows:
wherein ,for the curvature of each sampling point obtained in step S1, and (2)>The optimized curvatures of the i, i+1 and i+2 path sampling points are the curvatures of the sampling points after piecewise linearization, and the objective function minimizes the curvature norm and considers the linear trend; />For a parameter-adjustable variable, +.>The greater the degree of linearization of the result, the higher, +.>The smaller the result is, the smaller the curvature difference between the result and the original sampling point is, the result in the figure is +.>。
In a specific application example, in step S3, the curvature is preferably adjusted in the following manner: and sampling the curvature slope, and selecting the curvature slope corresponding to the smallest deviation as the curvature adjustment slope, thereby obtaining the curvature after adjustment.
In a specific application example, the flow of step S3 may include: starting from the first path sampling point, adjusting the curvature at each sampling point by pointUntil the last path sampling point; wherein (1)>For the +.sup.th obtained in step S1>Curvature of the sampling points.
As a preferred example, the flow of step S3 specifically includes:
step S301: an initial guess of the curvature slope is calculated: for each sampling point with curvature to be adjusted, the curvature of the previous point and the curvature of the five subsequent points are selected, the curvatures of seven sampling points are obtained by piecewise linearization obtained in the step S2, and the initial guess value of the curvature slope is obtained by a least square method, and the formula is as follows:
wherein Is an initial guess of the curvature slope obtained by the least square method.
Step S302: sampling curvature slope within a certain range around the curvature slope, selecting the curvature slope that minimizes the evaluation function, and obtaining the curvature of the corresponding second point as the adjusted curvature.
The evaluation function minimizes the square of the error of six points, including the point where the curvature needs to be adjusted, from the original cartesian point, as follows:
wherein The corresponding Cartesian coordinates obtained by integrating the curvature are the following Fresnel integral respectively:
in the formula For course angle of path, initial point +.>The course angle of the path at can be determined by the point +.>The heading angle of the connected line segment is approximate. In a computer, the integration can be approximated by a numerical integration method, for example, the integration of course angles, a trapezoidal integration method is adopted, and in the integration of the obtained position, the accuracy can be satisfied by adopting a simpson formula.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above examples, and all technical solutions belonging to the concept of the present invention belong to the protection scope of the present invention. It should be noted that modifications and adaptations to the invention without departing from the principles thereof are intended to be within the scope of the invention as set forth in the following claims.
Claims (9)
1. A method of generating a path for a smooth parameterized representation of curvature, comprising:
step S1: obtaining rough curvature through adjacent sampling points; obtaining a corresponding curvature for each path sampling point, and assuming the curvature between every two adjacent path sampling points to change linearly to obtain a curve of curvature-along path length;
step S2: obtaining a piecewise linear curvature curve by using a convex optimization method; the piecewise linear fitting of the curvature curve is solved by converting the piecewise linear problem into an unconstrained convex optimization problem;
step S3: adjusting the curvature of each point to fit the original path in space; and (2) adjusting the curvature parameter at each sampling point on the basis of the piecewise linearized curvature curve in the step (S2) so that the deviation between the road represented by the whole and the road represented by the original discrete point meets the precision.
2. The method for generating a path of a smooth curvature parameterized representation according to claim 1, wherein in the step S1, the corresponding quadratic parameter equation is solved by each cartesian coordinate sampling point and several neighboring points around, and the rough curvature at each sampling point is obtained by the calculation formula of the scalar form of curvature in the plane coordinate system.
3. The method for generating a path for a smooth curvature parameterized representation according to claim 2, wherein the flow of step S1 comprises:
step S101: the length along the path of each sampling point is calculated according to the following formula:
wherein the subscript is the index of the sampling point;respectively represent +.>Cartesian abscissa of each path sampling point in a space coordinate system, +.>Respectively represent +.>Cartesian ordinate of each path sampling point in a space coordinate system, < >>Represents the total number of sampling points in the path, +.>Respectively represent +.>Arc length parameters of each path sampling point, representing distance along the path from the start point, +.>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Indicate->Cartesian coordinates of the individual sampling points +.>Value of axis>Represents the total number of sampling points, starting sampling point +.>；
Step S102: converting the sampling points from Cartesian coordinates to a curvature domain: and constructing a secondary parameter equation by adopting a plurality of adjacent points to interpolate, solving the coefficient of the secondary parameter equation, and obtaining the curvature of the point through derivation.
4. A method of generating a path for a smooth curvature parameterized representation according to claim 3, wherein step S102 comprises:
constructing a quadratic parameter equation by the above method, whereinTo sample point nulls by adjacent 3 pathsCoefficients of quadratic parameter equation formed by inter-fitting, +.>Respectively a Cartesian abscissa and an ordinate of the path sampling point in a space coordinate system; adopts->Point and->Length along the path between points> and />As a parameterAnd->：
The coefficients of the quadratic parameter equation are obtained by the following formula:
the curvature is obtained by a calculation formula of the curvature under a plane coordinate system:
wherein ,for curvature (S)>Is the first and second derivatives of the quadratic parameter equation.
5. The method for generating a path represented by a smooth curvature parameterization according to any of claims 1-4, wherein in step S2, the curvature-along-path-length curve obtained in step S1 is subjected to linear trend filtering by using a convex optimization method to obtain a piecewise linearized curvature curve.
6. The method for generating a path for a smooth parametric representation of curvature according to claim 5, wherein in step S2, piecewise linear fitting of the curvature curve is performed by converting piecewise linear problem into unconstrained convex optimization problem solution, as follows:
wherein ,for the curvature of each sampling point obtained in step S1, and (2)>The optimized curvatures of the i, i+1 and i+2 path sampling points are the curvatures of the sampling points after piecewise linearization, and the objective function minimizes the curvature norm and considers the linear trend; />Is a variable with adjustable parameters.
7. The method of generating a path for a smooth curvature parameterized representation according to claim 1, wherein in step S3, the curvature is adjusted in the manner of: and sampling the curvature slope, and selecting the curvature slope corresponding to the smallest deviation as the curvature adjustment slope, thereby obtaining the curvature after adjustment.
8. The method of generating a path for a smooth curvature parameterized representation according to claim 1, wherein in step S3, starting from a first path sampling point, the curvature at each sampling point is adjusted point by pointUntil the last path sampling point; wherein (1)>For the +.sup.th obtained in step S1>Curvature of the sampling points.
9. The method for generating a path for a smooth curvature parameterized representation according to claim 8, wherein the flow of step S3 comprises:
step S301: an initial guess of the curvature slope is calculated: for each sampling point with curvature to be adjusted, selecting the curvature of the previous point and the curvature of the five subsequent points, wherein the curvatures of the seven sampling points are obtained by piecewise linearization obtained in the step S2, and obtaining an initial guess value of curvature slope through a least square method;
step S302: sampling curvature slope within a certain range around the curvature slope, selecting the curvature slope that minimizes the evaluation function, and obtaining the curvature of the corresponding second point as the adjusted curvature.
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