CN110597088A - Vehicle dynamics simulation method under polar coordinate system - Google Patents
Vehicle dynamics simulation method under polar coordinate system Download PDFInfo
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Abstract
The invention discloses a vehicle dynamics simulation method under a polar coordinate system, which is used for describing the motion position and the attitude of a vehicle in space by adopting a spatial polar coordinate system in combination with a Newton's law, a centrifugal force and a Coriolis force calculation method aiming at the problem that the vehicle rotation motion in the vehicle dynamics system has singular points. By utilizing the principle that each instantaneous rotating shaft of the vehicle can change in the rotating process and spatial rotation has inheritance, three translational degrees of freedom and three rotational degrees of freedom in the traditional dynamic model are converted into one translational degree of freedom and five rotational degrees of freedom, the motion of the vehicle in the space can be well described, and meanwhile, the problem of spatial rotation singular points of components such as a vehicle body and the like in a vehicle dynamic system can be effectively solved.
Description
Technical Field
The invention belongs to the technical field of vehicle dynamics simulation, and particularly relates to a vehicle dynamics simulation method under a polar coordinate system.
Background
The core problem for vehicle dynamics is that of dynamics modeling and equation of motion solution. In vehicle system dynamics theory, a common coordinate system is a cartesian coordinate system, which defines three mutually perpendicular vectors as three coordinate directions to describe the linear motion of an object in space. In a cartesian coordinate system, newton establishes a newton equation in 1787 to solve the problems of kinematics and dynamics of a fulcrum. On the basis, the dynamic equation of a rigid body system can be written into a plurality of forms, wherein a simple and widely practical equation is a Newton-Euler equation proposed by Euler in 1750, the Newton-Euler equation defines the coordinate system of the motion of an object in the mass center under a Cartesian coordinate system, the translation and rotation actions of the object are described as independent motion degrees of freedom, the concept of a rotation vector is adopted, and the Newton-Euler equation has a concise expression form by utilizing an even number form, so that the Newton-Euler equation is widely applied to the kinematics and dynamic analysis of open-chain and closed-chain space mechanisms. The newton-euler equation is a quantitative description of euler's law of motion and is an extension of newton's law of motion.
The rotational motion of an object is generally described by using a Kaldo angle, an Euler angle or an Euler quaternion, the Kaldo angle and the Euler angle respectively rotate around three mutually perpendicular rotating shafts in sequence, only the sequence of the rotation is different, but all the rotating singular points have the problem that when any shaft rotates 90 degrees, other shafts of the shaft barrel are overlapped, the Euler rotation can only generate the rotation of one shaft by the overlapped two shafts, and in order to solve the problem of the rotating singular points, the Euler proposes an Euler quaternion method which derives four elements related to the rotational motion of the moment from the instantaneous rotating shaft and the rotating angle of the object according to the finite rotation theory. However, the four elements are not independent of each other, and need to satisfy a certain constraint relationship. Shabanan ect analyzes expressions of Euler angles and Euler parameters, and is applied to a modeling process of wheel system dynamics, although Euler four elements can well describe the rotation motion of a space object and a rotation singular point problem does not exist, a constraint equation introduced among four elements changes a dynamic differential equation into a differential algebraic equation, the complexity of solving the dynamic equation is increased, and the equation solving efficiency is reduced. The complete vehicle-track coupling dynamics theory is built by domestic scholars under a Cartesian coordinate system by adopting the Kaldo angle, the problem of singular point rotation is effectively avoided due to the fact that the rotation motion adopts the small-angle hypothesis, and the problem of large-angle rotation of a space object is limited to a certain extent.
Jal Lou n and Bayo propose a complete Cartesian coordinate system method in 1994, which belongs to an absolute coordinate system modeling method. The characteristic of this method is to avoid using Euler angle and Euler parameter in the ordinary Cartesian method, but utilize the Cartesian coordinates of several reference points and reference vectors which are fixed with the rigid body to describe the space position and attitude of the rigid body. The reference point is selected to be at the center of the hinge, and the reference vector is along the rotating shaft or the translation direction of the hinge, so that the position variable can be reduced by sharing the position among a plurality of rigid bodies. However, the kinetic equation formed by complete cartesian coordinates is essentially the same as that of a general cartesian method, and the jacobian matrix is unified under an absolute coordinate system, so that the solution and calculation are convenient.
The motion of the vehicle in the space can be better described by selecting a proper coordinate system, and the solving precision can be improved. The existing vehicle dynamics models are basically built under a three-dimensional Cartesian coordinate system, and the vehicle dynamics modeling under a three-dimensional space polar coordinate is difficult to directly derive and directly apply to simulation calculation. Jal Yuan n gives the motion equation of the object in the two-dimensional polar coordinate system, but the derivation process is complicated, and the motion equation of the object in the three-dimensional polar coordinate system is not given.
Disclosure of Invention
Aiming at the defects in the prior art, the vehicle dynamics simulation method under the polar coordinate system provided by the invention aims at the problem that singular points exist in the vehicle rotation motion process in the existing vehicle dynamics system.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a vehicle dynamics simulation method under a polar coordinate system comprises the following steps:
s1, setting the total simulation duration T and the simulation step length dt of the vehicle dynamics;
s2, determining parameters of the vehicle to be simulated, and building a vehicle dynamic equation under a polar coordinate system according to the parameters;
s3, setting initial simulation time t0Is 0, the corresponding initial integration step number is 1;
s4, solving the constructed vehicle dynamics equation according to the current simulation time to obtain vehicle freedom degree data under a corresponding polar coordinate system;
s5, increasing the simulation time by a simulation step length dt, increasing the corresponding integration step number by 1, and judging whether the current simulation time T after the simulation step length dt is increased is greater than the total simulation time length T;
if yes, go to step S6;
if not, returning to the step S4;
and S6, outputting vehicle freedom degree data corresponding to the vehicle dynamics equation at present under the polar coordinate system as a vehicle dynamics simulation result.
Further, in step S2, the parameters of the vehicle to be simulated include a vehicle mass m and a moment of inertia J ═ J in a cartesian coordinate system [ J ═ Jx,Jy,Jz]TStress point of vehicle under Cartesian coordinate systemThe force appliedMoment of forceSpatial initial position of vehicleAnd an initial attitude R of the vehicle0＝[α0,β0,γ0]T；
Wherein i is the number of each stress point, and i is 1,2, 3.. M, M is the total number of stress points;
α0,β0,γ0initial angles of rotation around x, y and z axes in a Cartesian coordinate system respectively, and the rotation sequence is that gamma is firstly rotated around the z axis0Rotated by beta about the y-axis0Finally rotate alpha around the x-axis0；
The polar coordinate system in the step S2 comprises a three-dimensional space polar coordinate system and a vehicle mass center polar coordinate system; the three-dimensional space polar coordinate system is used for describing the position of the vehicle mass center in the three-dimensional space, and the vehicle mass center polar coordinate system is used for describing the motion posture of the vehicle in the three-dimensional space.
Further, the step S2 is specifically:
s21, converting the initial position of the vehicle in the Cartesian coordinate system into a three-dimensional polar coordinate system, and calculating the vehicle position rotation matrix in the three-dimensional polar coordinate system according to the initial position
S22, converting the initial attitude of the vehicle into a vehicle mass center polar coordinate system, and calculating a vehicle attitude rotation matrix under the vehicle mass center polar coordinate system according to the initial attitude of the vehicle
S23, rotating matrix according to vehicle positionCalculating resultant force F borne by vehicle in three-dimensional polar coordinate system0；
S24, rotating matrix according to vehicle postureAnd resultant force F experienced by the vehicle0Calculating resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0；
S25, according to the resultant force F0Sum and resultant moment M0And building a vehicle dynamic equation.
Further, in step S21, the conversion formula for converting the initial position of the vehicle in the three-dimensional space coordinate system to the three-dimensional space polar coordinate system is as follows:
in the formula, L is in a three-dimensional polar coordinate system0Is the initial position P of the vehicle0The translation distance along the radius direction of the earth;
θ0is the initial position P of the vehicle0The rotation angle in the direction of the warp angle;
is the initial position P of the vehicle0Rotation angle around latitude direction;
andrespectively an x-axis coordinate, a y-axis coordinate and a z-axis coordinate of the initial position of the vehicle in a Cartesian coordinate system;
vehicle position rotation matrix in three-dimensional space polar coordinate systemComprises the following steps:
in step S22, the conversion formula for converting the initial posture of the vehicle into the polar coordinate system of the center of mass of the vehicle is:
wherein, in a three-dimensional space and a coordinate system,the rotation angle of the vehicle around the longitude direction; psi0The rotation angle of the vehicle around the latitude direction;
φ0for the rotation angle of the vehicle around the radius direction of the earth；
Vehicle attitude rotation matrix under vehicle mass center polar coordinate system
In step S23, a resultant force F experienced by the vehicle in the three-dimensional polar coordinate system0Comprises the following steps:
in the formula (I), the compound is shown in the specification,a vehicle position rotation matrix under a three-dimensional space polar coordinate system;
for the point of force applied to the vehicle in a Cartesian coordinate systemThe force exerted;
in step S24, the resultant moment M borne by the vehicle in the polar coordinate system of the center of mass of the vehicle0：
In the formula (I), the compound is shown in the specification,for the point of force applied to the vehicle in a Cartesian coordinate systemThe moment of (2).
Further, in step S25, the vehicle dynamics equation is constructed as follows:
wherein m is the vehicle mass;
the second derivative of the moving freedom L in a three-dimensional space polar coordinate system;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in the direction of the degree of freedom of movement L;
FQthe rotation theta around the longitude direction and the rotation around the latitude direction of the vehicle in a three-dimensional space polar coordinate systemThe resultant of the centrifugal forces generated;
the second derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in a first rotational degree of freedom θ direction;
FCθthe Coriolis force is generated by the rotation theta of the vehicle around the longitude direction in the three-dimensional space polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe second derivative of (a);
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0Along a second degree of rotational freedomA component of direction;
for the rotation of the vehicle around the latitude direction in a three-dimensional polar coordinate systemThe resulting coriolis force;
for the third rotational degree of freedom of the vehicle in the polar coordinate system of the vehicle mass centerThe moment of inertia of (a);
for the third rotational degree of freedom in the polar coordinate system of the vehicle mass centerThe second derivative of (a);
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0Along a third degree of rotational freedomA component of direction;
Jψthe moment of inertia around the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system;
the second derivative of the fourth rotational degree of freedom psi in the vehicle centroid polar coordinate system is obtained;
resultant moment M borne by vehicle in polar coordinate system of vehicle mass center0A component in the direction of the fourth rotational degree of freedom ψ;
Jφthe moment of inertia around the fifth rotational degree of freedom phi in the vehicle mass center polar coordinate system;
the second derivative of the fifth rotational degree of freedom phi in the vehicle mass center polar coordinate system;
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0A component in the direction of the fifth rotational degree of freedom ψ;
the first derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe first derivative of (a);
the first derivative of the moving freedom degree L in the three-dimensional space polar coordinate system;
the value is the value of an integral step number on the first rotational degree of freedom theta in a three-dimensional polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe value of the last integration step.
Further, in the step S4, according to the current simulation time, the vehicle dynamics equation is solved by a fourth-order longge stoke method;
the vehicle freedom degree data comprises vehicle movement and rotation freedom degree data under a three-dimensional polar coordinate system and vehicle rotation freedom degree data under a vehicle mass center polar coordinate system under an integral step number corresponding to the current simulation time t.
Further, the data of the degree of freedom of movement and rotation of the vehicle under the three-dimensional polar coordinate system comprises Lj,θj, The vehicle dynamics equation established by the fourth-order Rungestota method is subjected to iterative solution to obtain the vehicle dynamics equation;
j is the number of the integration step number in the vehicle dynamics simulation process, and J is 1,2, 3.
LjThe value of the degree of freedom L of movement in a three-dimensional polar coordinate system under the j integral step number;
θjthe value of a first rotational degree of freedom theta in a three-dimensional space polar coordinate system under the jth integral step number;
is a three-dimensional polar coordinate system under the j integral step numberLower second degree of rotational freedomA value of (d);
the first derivative value of the moving freedom degree L under the j integral step number;
the value of the first rotational degree of freedom theta under the j integral step number;
is the second degree of freedom of rotation at the j integral step numberA first derivative value of;
the second derivative value of the motion freedom L under the j integral step number;
the second derivative value of the first rotational degree of freedom theta under the j integral step number;
is the second degree of freedom of rotation at the j integral step numberA second derivative value of;
the vehicle rotational freedom data under the vehicle mass center polar coordinate system comprisesψj,φj、 Carrying out iterative solution on the established vehicle dynamic equation by a fourth-order Runge Kutta method to obtain the vehicle dynamic equation;
wherein the content of the first and second substances,is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA value of (d);
ψjthe value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is the j integral step number;
φjthe value of a fifth rotational degree of freedom phi under a vehicle mass center polar coordinate system under the jth integral step number;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA first derivative value of;
the first derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
the first derivative value of a fifth rotational degree of freedom phi under the jth integral step number and the vehicle mass center polar coordinate system;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA second derivative value of;
the second derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
and the second derivative value of the fifth rotational degree of freedom phi under the j integral step number and the vehicle mass center polar coordinate system.
Further, the method for obtaining the vehicle movement and rotation freedom data in the three-dimensional polar coordinate system of the vehicle specifically comprises the following steps:
a1, determining a vehicle position rotation matrix under the current integral step number j according to the current simulation timeIs as follows;
in the formula (I), the compound is shown in the specification,the first rotational degree of freedom theta of the vehicle at the j integral step numberjAnd a second degree of rotational freedomThe amount of change in the rotation matrix resulting from the change;
in three dimensions for the j-1 th integration stepA rotation matrix of the vehicle position in a space polar coordinate system;
the above-mentionedExpressed as:
a2, rotating matrix according to vehicle position in three-dimensional space polar coordinate systemDetermining resultant force F borne by vehicle under current integral step numberjComprises the following steps:
wherein, when j is 1,
a3, calculating the resultant force F at each integration stepjSubstituting the vehicle dynamics equation, solving the vehicle dynamics equation through a four-order library tower solver, and outputting vehicle movement and rotation freedom degree data under a three-dimensional space polar coordinate system.
Further, the method for obtaining the vehicle rotational freedom data of the vehicle in the vehicle center of mass polar coordinate system specifically comprises the following steps:
b1, determining the vehicle attitude rotation matrix under the current step length according to the current simulation time
In the formula (I), the compound is shown in the specification,when the number is j integral steps, the angle freedom degree generated by the rotation of the vehicle changes to generate a rotation matrix transformation quantity;
the vehicle attitude rotation matrix is the vehicle attitude rotation matrix in the j-1 integral step number;
the above-mentionedExpressed as:
in the formula (I), the compound is shown in the specification,
b2 matrix according to vehicle position and postureDetermining resultant moment M borne by the vehicle under the vehicle mass center polar coordinate system under the current integral step numberjIs as follows;
in the formula (I), the compound is shown in the specification,a vehicle attitude rotation matrix under a vehicle mass center polar coordinate system is obtained;
the stress points of the vehicle under a Cartesian coordinate system are defined;
Fjthe resultant force borne by the vehicle at the j integral step number;
b3, calculating the resultant moment M under each integral stepjSubstituting into the vehicle dynamics equation, and solving the vehicle dynamics equation through a fourth-order Runge Kutta solverAnd outputting the vehicle rotational freedom data under the vehicle mass center polar coordinate system.
Further, in step S6, when it is required to output the current integration step number j, each force point of the vehicleIn the case of absolute coordinate positions in a cartesian coordinate system, this is calculated by the following formula:
in the formula (I), the compound is shown in the specification,in a Cartesian coordinate system, the force pointA position matrix of (a);
for the stress point of the vehicle in a Cartesian coordinate systemWhen converting to the vehicle mass center polar coordinate system, the corresponding rotation matrix;
transposing a matrix for the vehicle attitude in the vehicle mass center polar coordinate system in the jth integration step;
is the ith stress pointThe initial direction vector of the direction of the,wherein the content of the first and second substances,is the ith stress pointIn a moving freedom value under a three-dimensional space polar coordinate system, a superscript T is a transposition operator;
a vehicle position rotation matrix in a three-dimensional space polar coordinate system is obtained in the jth integral step;
Ojis the initial vector direction of the vehicle center point O at the j integral step number, and Oj＝[Lj,0,0]，LjThe value of the moving freedom L in a three-dimensional space polar coordinate system under the j integral step number;
wherein, the stress point of the vehicle under the Cartesian coordinate systemThe conversion formula when converting into the vehicle mass center polar coordinate system is as follows:
in the formula, in a vehicle mass center polar coordinate system,is a stress pointThe translation distance along the radius direction of the earth;
is a stress pointAngle of rotation around warp directionDegree;
is a stress pointAn angle of rotation about the latitudinal direction;
andrespectively the stress point in the Cartesian coordinate system of the vehicle mass centerCoordinates on the x-axis, y-axis, and z-axis;
stress point under vehicle mass center polar coordinate systemOf the rotation matrixComprises the following steps:
the invention has the beneficial effects that:
the invention provides a vehicle dynamics simulation method under a polar coordinate system, which aims at the problem that the vehicle rotation motion in the vehicle dynamics system has singular points, adopts a spatial polar coordinate system in combination with a Newton's law, a centrifugal force and a Coriolis force calculation method, and provides a six-degree-of-freedom dynamics simulation method of a vehicle under the polar coordinate system, and is used for describing the motion position and the posture of the vehicle in the space. By utilizing the principle that each instantaneous rotating shaft of the vehicle can change in the rotating process and spatial rotation has inheritance, three translational degrees of freedom and three rotational degrees of freedom in the traditional dynamic model are converted into one translational degree of freedom and five rotational degrees of freedom, the motion of the vehicle in the space can be well described, and meanwhile, the problem of spatial rotation singular points of components such as a vehicle body and the like in a vehicle dynamic system can be effectively solved.
Drawings
FIG. 1 is a flowchart of a vehicle dynamics simulation method under a polar coordinate system according to the present invention.
FIG. 2 is a flow chart of a method for constructing a vehicle dynamics equation in accordance with the present invention.
FIG. 3 is a diagram illustrating stress points in a Cartesian coordinate system according to an embodiment of the present inventionSchematic displacement diagram of (a).
Fig. 4 is a schematic view of a rotation angle in a cartesian coordinate system according to an embodiment of the present invention.
FIG. 5 is a diagram illustrating stress points in a polar coordinate system according to an embodiment of the present inventionSchematic displacement diagram of (a).
Fig. 6 is a schematic view of a rotation angle under a polar coordinate system according to an embodiment of the invention.
FIG. 7 is a diagram illustrating stress points in a Cartesian coordinate system according to an embodiment of the present inventionSchematic diagram of the trajectory of (1).
FIG. 8 is a view illustrating a stress point in a polar coordinate system according to an embodiment of the present inventionSchematic diagram of the trajectory of (1).
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 1, a vehicle dynamics simulation method in a polar coordinate system includes the following steps:
s1, setting the total simulation duration T and the simulation step length dt of the vehicle dynamics;
s2, determining parameters of the vehicle to be simulated, and building a vehicle dynamic equation under a polar coordinate system according to the parameters;
s3, setting initial simulation time t0Is 0, the corresponding initial integration step number is 1;
s4, solving the constructed vehicle dynamics equation according to the current simulation time to obtain vehicle freedom degree data under a corresponding polar coordinate system;
s5, increasing the simulation time by a simulation step length dt, increasing the corresponding integration step number by 1, and judging whether the current simulation time T after the simulation step length dt is increased is greater than the total simulation time length T;
if yes, go to step S6;
if not, returning to the step S4;
and S6, outputting vehicle freedom degree data corresponding to the vehicle dynamics equation at present under the polar coordinate system as a vehicle dynamics simulation result.
In step S2, the parameters of the vehicle to be simulated include vehicle mass m and moment of inertia J ═ J in a cartesian coordinate system [ J ═ mx,Jy,Jz]TStress point of vehicle under Cartesian coordinate systemThe force appliedMoment of forceSpatial initial position of vehicleAnd an initial attitude R of the vehicle0＝[α0,β0,γ0]T；
Wherein i is the number of each stress point, and i is 1,2, 3.. M, M is the total number of stress points;
α0,β0,γ0initial angles of rotation around x, y and z axes in a Cartesian coordinate system respectively, and the rotation sequence is that gamma is firstly rotated around the z axis0Rotated by beta about the y-axis0Finally rotate alpha around the x-axis0；
The polar coordinate system in the step S2 comprises a three-dimensional space polar coordinate system and a vehicle mass center polar coordinate system; the three-dimensional space polar coordinate system is used for describing the position of the vehicle mass center in the three-dimensional space, and the vehicle mass center polar coordinate system is used for describing the motion posture of the vehicle in the three-dimensional space. Taking the earth as an example, the motion of a single particle in a spatial polar coordinate system comprises translation along the radius direction of the earth and rotation along the longitude and latitude directions.
As shown in fig. 2, step S2 specifically includes:
s21, converting the initial position of the vehicle in the Cartesian coordinate system into a three-dimensional polar coordinate system, and calculating the vehicle position rotation matrix in the three-dimensional polar coordinate system according to the initial position
The conversion formula for converting the initial position of the vehicle in the three-dimensional space coordinate system to the three-dimensional space polar coordinate system is as follows:
in the formula, L is in a three-dimensional polar coordinate system0Is the initial position P of the vehicle0The translation distance along the radius direction of the earth;
θ0is the initial position P of the vehicle0The rotation angle in the direction of the warp angle;
is the initial position P of the vehicle0Rotation angle around latitude direction;
andrespectively an x-axis coordinate, a y-axis coordinate and a z-axis coordinate of the initial position of the vehicle in a Cartesian coordinate system;
vehicle position rotation matrix under three-dimensional space polar coordinate systemComprises the following steps:
s22, converting the initial attitude of the vehicle into a vehicle mass center polar coordinate system, and calculating a vehicle attitude rotation matrix under the vehicle mass center polar coordinate system according to the initial attitude of the vehicle
The initial angle of rotation around the x, y, z axes defined in the cartesian coordinate system coincides with the initial angle defined in the polar coordinate system; therefore, the conversion formula for converting the initial attitude of the vehicle into the polar coordinate system of the center of mass of the vehicle is as follows:
wherein, in a three-dimensional space and a coordinate system,the rotation angle of the vehicle around the longitude direction;
ψ0the rotation angle of the vehicle around the latitude direction;
φ0the rotation angle of the vehicle around the radius direction of the earth;
vehicle attitude rotation matrix under vehicle mass center polar coordinate system
At the moment, the moment of inertia of the vehicle is under the polar coordinate system of the center of mass of the vehicleThe calculation formula is as follows:
wherein J vehicle has a moment of inertia in Cartesian coordinates, and J ═ Jx,Jy,Jz]T；
S23, rotating matrix according to vehicle positionCalculating resultant force F borne by vehicle in three-dimensional polar coordinate system0；
Wherein, the resultant force F borne by the vehicle under the three-dimensional polar coordinate system0Comprises the following steps:
in the formula (I), the compound is shown in the specification,a vehicle position rotation matrix under a three-dimensional space polar coordinate system;
for the point of force applied to the vehicle in a Cartesian coordinate systemThe force exerted;
s24, rotating matrix according to vehicle postureAnd resultant force F experienced by the vehicle0Calculating resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0；
Wherein, the resultant moment M borne by the vehicle under the polar coordinate system of the vehicle mass center0：
In the formula (I), the compound is shown in the specification,for the point of force applied to the vehicle in a Cartesian coordinate systemThe moment of (2).
S25, according to the resultant force F0Sum and resultant moment M0And building a vehicle dynamic equation.
Combined resultant force F according to Newton's law of motion0Sum and resultant moment M0The established vehicle dynamics equation is obtained as follows:
wherein m is the vehicle mass;
the second derivative of the moving freedom L in a three-dimensional space polar coordinate system;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in the direction of the degree of freedom of movement L;
FQis a three-dimensional space poleRotation theta of vehicle around longitude direction and rotation around latitude direction in coordinate systemThe resultant of the centrifugal forces generated;
the second derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in a first rotational degree of freedom θ direction;
FCθthe Coriolis force is generated by the rotation theta of the vehicle around the longitude direction in the three-dimensional space polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe second derivative of (a);
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0Along a second degree of rotational freedomA component of direction;
for the rotation of the vehicle around the latitude direction in a three-dimensional polar coordinate systemThe resulting coriolis force;
for the third rotational degree of freedom of the vehicle in the polar coordinate system of the vehicle mass centerThe moment of inertia of (a);
for the third rotational degree of freedom in the polar coordinate system of the vehicle mass centerThe second derivative of (a);
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0Along a third degree of rotational freedomA component of direction;
Jψthe moment of inertia around the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system;
the second derivative of the fourth rotational degree of freedom psi in the vehicle centroid polar coordinate system is obtained;
resultant moment M borne by vehicle in polar coordinate system of vehicle mass center0A component in the direction of the fourth rotational degree of freedom ψ;
Jφthe moment of inertia around the fifth rotational degree of freedom phi in the vehicle mass center polar coordinate system;
is the quality of the vehicleA second derivative of a fifth rotational degree of freedom phi in the polar cardiac coordinate system;
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0A component in the direction of the fifth rotational degree of freedom ψ;
the first derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe first derivative of (a);
the first derivative of the moving freedom degree L in the three-dimensional space polar coordinate system;
the value is the value of an integral step number on the first rotational degree of freedom theta in a three-dimensional polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe value of the last integration step.
The step S4 is specifically: solving a vehicle dynamic equation by a fourth-order Runge Kutta method according to the current simulation time;
the vehicle freedom degree data comprises vehicle movement and rotation freedom degree data under a three-dimensional polar coordinate system and vehicle rotation freedom degree data under a vehicle mass center polar coordinate system under an integral step number corresponding to the current simulation time t;
the vehicle moving and rotating freedom data under the three-dimensional space polar coordinate system comprises Lj,θj, The vehicle dynamics equation established by the fourth-order Rungestota method is subjected to iterative solution to obtain the vehicle dynamics equation;
j is the number of the integration step number in the vehicle dynamics simulation process, and J is 1,2, 3.
LjThe value of the degree of freedom L of movement in a three-dimensional polar coordinate system under the j integral step number;
θjthe value of a first rotational degree of freedom theta in a three-dimensional space polar coordinate system under the jth integral step number;
is the second rotational degree of freedom in the polar coordinate system of the three-dimensional space under the j integral step numberA value of (d);
the first derivative value of the moving freedom degree L under the j integral step number;
the value of the first rotational degree of freedom theta under the j integral step number;
is the j integral step numberSecond degree of rotational freedomA first derivative value of;
the second derivative value of the motion freedom L under the j integral step number;
the second derivative value of the first rotational degree of freedom theta under the j integral step number;
is the second degree of freedom of rotation at the j integral step numberA second derivative value of;
the vehicle rotational freedom data under the vehicle mass center polar coordinate system comprisesψj,φj、Carrying out iterative solution on the established vehicle dynamic equation by a fourth-order Runge Kutta method to obtain the vehicle dynamic equation;
wherein the content of the first and second substances,is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA value of (d);
ψjthe value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is the j integral step number;
φjthe value of a fifth rotational degree of freedom phi under a vehicle mass center polar coordinate system under the jth integral step number;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA first derivative value of;
the first derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
the first derivative value of a fifth rotational degree of freedom phi under the jth integral step number and the vehicle mass center polar coordinate system;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA second derivative value of;
the second derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
and the second derivative value of the fifth rotational degree of freedom phi under the j integral step number and the vehicle mass center polar coordinate system.
Specifically, the method for obtaining the vehicle movement and rotation freedom data in the three-dimensional polar coordinate system includes:
a1, determining a vehicle position rotation matrix under the current integral step number j according to the current simulation timeIs as follows;
in the formula (I), the compound is shown in the specification,the first rotational degree of freedom theta of the vehicle at the j integral step numberjAnd a second degree of rotational freedomThe amount of change in the rotation matrix resulting from the change;
the rotation matrix of the vehicle position in the three-dimensional space polar coordinate system is the j-1 integral step number;
the formula reflects the inheritance of the rotation matrix in the solving process of the vehicle freedom data;
expressed as:
in the formula (I), the compound is shown in the specification,
a2, rotating matrix according to vehicle position in three-dimensional space polar coordinate systemDetermining vehicle station under current integration step numberApplied force FjComprises the following steps:
wherein, when j is 1,
a3, calculating the resultant force F at each integration stepjSubstituting the vehicle dynamics equation, solving the vehicle dynamics equation through a four-order library tower solver, and outputting vehicle movement and rotation freedom degree data under a three-dimensional space polar coordinate system.
Specifically, the method for obtaining the vehicle rotational freedom data of the vehicle in the vehicle centroid polar coordinate system specifically comprises the following steps:
b1, determining the vehicle attitude rotation matrix under the current step length according to the current simulation time
In the formula (I), the compound is shown in the specification,when the number is j integral steps, the angle freedom degree generated by the rotation of the vehicle changes to generate a rotation matrix transformation quantity;
the vehicle attitude rotation matrix is the vehicle attitude rotation matrix in the j-1 integral step number;
the calculation formula reflects the inheritance of the rotation matrix;
expressed as:
in the formula (I), the compound is shown in the specification,
b2 matrix according to vehicle position and postureDetermining resultant moment M borne by the vehicle under the vehicle mass center polar coordinate system under the current integral step numberjIs as follows;
in the formula (I), the compound is shown in the specification,a vehicle attitude rotation matrix under a vehicle mass center polar coordinate system is obtained;
the stress points of the vehicle under a Cartesian coordinate system are defined;
Fjthe resultant force borne by the vehicle at the j integral step number;
b3, calculating the resultant moment M under each integral stepjSubstituting the vehicle dynamics equation, solving the vehicle dynamics equation through a fourth-order Runge Kutta solver, and outputting vehicle rotational freedom data under a vehicle mass center polar coordinate system.
When the vehicle rotation and movement freedom degree data in the three-dimensional space polar coordinate system and the vehicle rotation freedom degree data in the vehicle mass center polar coordinate system are calculated, the resultant force F borne by the vehicle is obtained by the j integral step numberjAnd the resultant moment MjAnd substituting the obtained data into a vehicle dynamics equation, and obtaining each degree of freedom of movement and rotation of the vehicle and first-order and second-order derivative values thereof under the next integral step number j +1 through the integration of a fourth-order Runge Kutta solver. And the resultant force F borne by the vehicle obtained by updating the integral step number of j +1 againj+1And receiveResultant moment Mj+1And repeating the iterative operation. And finally, stopping iterative operation when the J-th integral step number is reached, and outputting final freedom degree data as a simulation result of vehicle dynamics.
In the above calculation process, it should be noted that, unlike any other method, the rotation matrix of the current integration step number in the present invention is not only related to the variation of each rotational degree of freedom in the current step but also related to the rotation matrix of the previous step. Namely, on the basis of the rotation matrix of the previous step, the rotation matrix of the current variation of each rotational degree of freedom is superposed, and the problem of the rotational singular point can be avoided as long as the current variation of each rotational degree of freedom is smaller than 90 degrees by controlling the number of integration steps.
In one embodiment of the invention, each stress point of the vehicle needs to output the current integral step number jIn the case of absolute coordinate positions in a cartesian coordinate system, this is calculated by the following formula:
in the formula (I), the compound is shown in the specification,in a Cartesian coordinate system, the force pointA position matrix of (a);
for the stress point of the vehicle in a Cartesian coordinate systemWhen converting to the vehicle mass center polar coordinate system, the corresponding rotation matrix;
transposing a matrix for the vehicle attitude in the vehicle mass center polar coordinate system in the jth integration step;
is the ith stress pointThe initial direction vector of the direction of the,wherein the content of the first and second substances,is the ith stress pointIn a moving freedom value under a three-dimensional space polar coordinate system, a superscript T is a transposition operator;
a vehicle position rotation matrix in a three-dimensional space polar coordinate system is obtained in the jth integral step;
Ojis a vector direction matrix of the vehicle center point at the j integral step number, and Oj＝[Oj,0,0]，LjThe value of the moving freedom L in a three-dimensional space polar coordinate system under the j integral step number;
wherein, the stress point of the vehicle under the Cartesian coordinate systemThe conversion formula when converting into the vehicle mass center polar coordinate system is as follows:
in the formula, in a vehicle mass center polar coordinate system,is a stress pointThe translation distance along the radius direction of the earth;
is a stress pointThe angle of rotation around the warp direction;
is a stress pointAn angle of rotation about the latitudinal direction;
andrespectively the stress point in the Cartesian coordinate system of the vehicle mass centerCoordinates on the x-axis, y-axis, and z-axis;
stress point under vehicle mass center polar coordinate systemOf the rotation matrixComprises the following steps:
in one embodiment of the present invention, an example for verifying the correctness of the singular point-free six-degree-of-freedom vehicle dynamics simulation method of the polar coordinate system of the present invention is provided:
let the mass of the vehicle be 100kg as the mass m of the vehicle and the moment of inertia J in the cartesian coordinate systemx＝Jy＝Jz＝120kg·m2Is subjected to forces in two directions, force pointsCoordinates are (3m, 4m, 3m), force Coordinates of (0m, 1m, 1m), forceThe initial position and the attitude of the vehicle are both 0, and the freedom degree of movement of the vehicle is restrainedThe simulation time is 40s, the movement of the vehicle is respectively calculated by adopting the Runge Kutta integral algorithm, and the result is compared with the result calculated by SIMPACK software, and the result is shown in figures 3-8.
As can be seen from fig. 3 to 8, the singular point-free six-degree-of-freedom vehicle dynamics simulation method of the present invention can well describe the motion of the vehicle in the space, and can accurately solve the motion trajectory of the vehicle in the space. The result of the solution is completely consistent with the vehicle dynamics model established in the traditional Cartesian coordinate system.
But three rotational degrees of freedom in polar coordinate systemThe three rotational degrees of freedom result quite differently from a cartesian coordinate system. The angular values of the rotational degrees of freedom (α, β, γ) in a cartesian coordinate system are all large, with the risk of the rotation axes of the multiple degrees of freedom rotating through 90 ° simultaneously, which causesThe singular problem of rotation. In the polar coordinate system, the rotation matrix of the current product step is only related to the rotation angle variation of the current degree of freedom and the rotation matrix of the previous step. As long as the rotation angle variation satisfying the current degree of freedom is less than 90 degrees, the problem of singular rotation is avoided.
The invention has the beneficial effects that:
the invention provides a vehicle dynamics simulation method under a polar coordinate system, which aims at the problem that the vehicle rotation motion in the vehicle dynamics system has singular points, adopts a spatial polar coordinate system in combination with a Newton's law, a centrifugal force and a Coriolis force calculation method, and provides a six-degree-of-freedom dynamics simulation method of a vehicle under the polar coordinate system, and is used for describing the motion position and the posture of the vehicle in the space. By utilizing the principle that each instantaneous rotating shaft of the vehicle can change in the rotating process and spatial rotation has inheritance, three translational degrees of freedom and three rotational degrees of freedom in the traditional dynamic model are converted into one translational degree of freedom and five rotational degrees of freedom, the motion of the vehicle in the space can be well described, and meanwhile, the problem of spatial rotation singular points of components such as a vehicle body and the like in a vehicle dynamic system can be effectively solved.
Claims (10)
1. A vehicle dynamics simulation method under a polar coordinate system is characterized by comprising the following steps:
s1, setting the total simulation duration T and the simulation step length dt of the vehicle dynamics;
s2, determining parameters of the vehicle to be simulated, and building a vehicle dynamic equation under a polar coordinate system according to the parameters;
s3, setting initial simulation time t0Is 0, the corresponding initial integration step number is 1;
s4, solving the constructed vehicle dynamics equation according to the current simulation time to obtain vehicle freedom degree data under a corresponding polar coordinate system;
s5, increasing the simulation time by a simulation step length dt, increasing the corresponding integration step number by 1, and judging whether the current simulation time T after the simulation step length dt is increased is greater than the total simulation time length T;
if yes, go to step S6;
if not, returning to the step S4;
and S6, outputting vehicle freedom degree data corresponding to the vehicle dynamics equation at present under the polar coordinate system as a vehicle dynamics simulation result.
2. The method according to claim 1, wherein in step S2, the parameters of the vehicle to be simulated include vehicle mass m and moment of inertia J ═ J in cartesian coordinatesx,Jy,Jz]TStress point of vehicle under Cartesian coordinate systemThe force appliedMoment of forceSpatial initial position of vehicleAnd an initial attitude R of the vehicle0＝[α0,β0,γ0]T；
Wherein i is the number of each stress point, and i is 1,2, 3.. M, M is the total number of stress points;
α0,β0,γ0initial angles of rotation around x, y and z axes in a Cartesian coordinate system respectively, and the rotation sequence is that gamma is firstly rotated around the z axis0Rotated by beta about the y-axis0Finally rotate alpha around the x-axis0；
The polar coordinate system in the step S2 comprises a three-dimensional space polar coordinate system and a vehicle mass center polar coordinate system; the three-dimensional space polar coordinate system is used for describing the position of the vehicle mass center in the three-dimensional space, and the vehicle mass center polar coordinate system is used for describing the motion posture of the vehicle in the three-dimensional space.
3. The method for simulating vehicle dynamics in a polar coordinate system according to claim 2, wherein the step S2 is specifically as follows:
s21, converting the initial position of the vehicle in the Cartesian coordinate system into a three-dimensional polar coordinate system, and calculating the vehicle position rotation matrix in the three-dimensional polar coordinate system according to the initial position
S22, converting the initial attitude of the vehicle into a vehicle mass center polar coordinate system, and calculating a vehicle attitude rotation matrix under the vehicle mass center polar coordinate system according to the initial attitude of the vehicle
S23, rotating matrix according to vehicle positionCalculating resultant force F borne by vehicle in three-dimensional polar coordinate system0；
S24, rotating matrix according to vehicle postureAnd resultant force F experienced by the vehicle0Calculating resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0；
S25, according to the resultant force F0Sum and resultant moment M0And building a vehicle dynamic equation.
4. The method for simulating vehicle dynamics according to claim 3, wherein in step S21, the transformation formula for transforming the initial position of the vehicle in the three-dimensional space coordinate system to the three-dimensional space polar coordinate system is:
in the formulaIn a three-dimensional polar coordinate system, L0Is the initial position P of the vehicle0The translation distance along the radius direction of the earth;
θ0is the initial position P of the vehicle0The rotation angle in the direction of the warp angle;
is the initial position P of the vehicle0Rotation angle around latitude direction;
andrespectively an x-axis coordinate, a y-axis coordinate and a z-axis coordinate of the initial position of the vehicle in a Cartesian coordinate system;
vehicle position rotation matrix in three-dimensional space polar coordinate systemComprises the following steps:
in step S22, the conversion formula for converting the initial posture of the vehicle into the polar coordinate system of the center of mass of the vehicle is:
wherein, in a three-dimensional space and a coordinate system,the rotation angle of the vehicle around the longitude direction;
ψ0the rotation angle of the vehicle around the latitude direction;
φ0for vehicles around the earth's radiusRotating the angle;
vehicle attitude rotation matrix under vehicle mass center polar coordinate system
In step S23, a resultant force F experienced by the vehicle in the three-dimensional polar coordinate system0Comprises the following steps:
in the formula (I), the compound is shown in the specification,a vehicle position rotation matrix under a three-dimensional space polar coordinate system;
for the point of force applied to the vehicle in a Cartesian coordinate systemThe force exerted;
in step S24, the resultant moment M borne by the vehicle in the polar coordinate system of the center of mass of the vehicle0：
In the formula (I), the compound is shown in the specification,for the point of force applied to the vehicle in a Cartesian coordinate systemThe moment of (2).
5. The method for simulating vehicle dynamics under the polar coordinate system according to claim 4, wherein in the step S25, the vehicle dynamics equation is constructed by:
wherein m is the vehicle mass;
the second derivative of the moving freedom L in a three-dimensional space polar coordinate system;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in the direction of the degree of freedom of movement L;
FQthe rotation theta around the longitude direction and the rotation around the latitude direction of the vehicle in a three-dimensional space polar coordinate systemThe resultant of the centrifugal forces generated;
the second derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0A component in a first rotational degree of freedom θ direction;
FCθthe Coriolis force is generated by the rotation theta of the vehicle around the longitude direction in the three-dimensional space polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe second derivative of (a);
is the resultant force F borne by the vehicle in a three-dimensional polar coordinate system0Along a second degree of rotational freedomA component of direction;
for the rotation of the vehicle around the latitude direction in a three-dimensional polar coordinate systemThe resulting coriolis force;
for the third rotational degree of freedom of the vehicle in the polar coordinate system of the vehicle mass centerThe moment of inertia of (a);
for the third rotational degree of freedom in the polar coordinate system of the vehicle mass centerThe second derivative of (a);
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0Along a third degree of rotational freedomA component of direction;
Jψthe moment of inertia around the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system;
the second derivative of the fourth rotational degree of freedom psi in the vehicle centroid polar coordinate system is obtained;
resultant moment M borne by vehicle in polar coordinate system of vehicle mass center0A component in the direction of the fourth rotational degree of freedom ψ;
Jφthe moment of inertia around the fifth rotational degree of freedom phi in the vehicle mass center polar coordinate system;
the second derivative of the fifth rotational degree of freedom phi in the vehicle mass center polar coordinate system;
for resultant moment M borne by the vehicle in a polar coordinate system of the mass center of the vehicle0A component in the direction of the fifth rotational degree of freedom ψ;
the first derivative of the first rotational degree of freedom theta in the three-dimensional space polar coordinate system is obtained;
in a three-dimensional polar coordinate systemSecond degree of rotational freedomThe first derivative of (a);
the first derivative of the moving freedom degree L in the three-dimensional space polar coordinate system;
the value is the value of an integral step number on the first rotational degree of freedom theta in a three-dimensional polar coordinate system;
for the second rotational degree of freedom in a three-dimensional polar coordinate systemThe value of the last integration step.
6. The method according to claim 5, wherein in step S4, the vehicle dynamics equation is solved by a fourth-order Runge Kutta method according to the current simulation time;
the vehicle freedom degree data comprises vehicle movement and rotation freedom degree data under a three-dimensional polar coordinate system and vehicle rotation freedom degree data under a vehicle mass center polar coordinate system under an integral step number corresponding to the current simulation time t.
7. The method of claim 6, wherein the data of degree of freedom of movement and rotation of the vehicle in the polar coordinate system comprises Lj,θj, The vehicle dynamics equation established by the fourth-order Rungestota method is subjected to iterative solution to obtain the vehicle dynamics equation;
j is the number of the integration step number in the vehicle dynamics simulation process, and J is 1,2, 3.
LjThe value of the degree of freedom L of movement in a three-dimensional polar coordinate system under the j integral step number;
θjthe value of a first rotational degree of freedom theta in a three-dimensional space polar coordinate system under the jth integral step number;
is the second rotational degree of freedom in the polar coordinate system of the three-dimensional space under the j integral step numberA value of (d);
the first derivative value of the moving freedom degree L under the j integral step number;
the value of the first rotational degree of freedom theta under the j integral step number;
is the second degree of freedom of rotation at the j integral step numberA first derivative value of;
the second derivative value of the motion freedom L under the j integral step number;
the second derivative value of the first rotational degree of freedom theta under the j integral step number;
is the second degree of freedom of rotation at the j integral step numberA second derivative value of;
the vehicle rotational freedom data under the vehicle mass center polar coordinate system comprisesψj,φj、 Carrying out iterative solution on the established vehicle dynamic equation by a fourth-order Runge Kutta method to obtain the vehicle dynamic equation;
wherein the content of the first and second substances,is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA value of (d);
ψjthe value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is the j integral step number;
φjis the fifth rotational freedom in the vehicle centroid polar coordinate system in the jth integral step numberA value of degree φ;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA first derivative value of;
the first derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
the first derivative value of a fifth rotational degree of freedom phi under the jth integral step number and the vehicle mass center polar coordinate system;
is the third rotational degree of freedom under the polar coordinate system of the vehicle mass center under the j integral step numberA second derivative value of;
the second derivative value of the fourth rotational degree of freedom psi in the vehicle mass center polar coordinate system is obtained under the j integral step number;
and the second derivative value of the fifth rotational degree of freedom phi under the j integral step number and the vehicle mass center polar coordinate system.
8. The method for simulating vehicle dynamics in a polar coordinate system according to claim 7, wherein the method for obtaining the vehicle motion and rotation degree of freedom data in the polar coordinate system in the three-dimensional space is specifically as follows:
a1, determining a vehicle position rotation matrix under the current integral step number j according to the current simulation timeIs as follows;
in the formula (I), the compound is shown in the specification,the first rotational degree of freedom theta of the vehicle at the j integral step numberjAnd a second degree of rotational freedomThe amount of change in the rotation matrix resulting from the change;
the rotation matrix of the vehicle position in the three-dimensional space polar coordinate system is the j-1 integral step number;
the above-mentionedExpressed as:
a2, rotating matrix according to vehicle position in three-dimensional space polar coordinate systemDetermining resultant force F borne by vehicle under current integral step numberjComprises the following steps:
wherein, when j is 1,
a3, calculating the resultant force F at each integration stepjSubstituting the vehicle dynamics equation, solving the vehicle dynamics equation through a four-order library tower solver, and outputting vehicle movement and rotation freedom degree data under a three-dimensional space polar coordinate system.
9. The method for simulating vehicle dynamics in a polar coordinate system according to claim 7, wherein the method for obtaining the vehicle rotational freedom data of the vehicle in the vehicle center of mass polar coordinate system specifically comprises:
b1, determining the vehicle attitude rotation matrix under the current step length according to the current simulation time
In the formula (I), the compound is shown in the specification,when the number is j integral steps, the angle freedom degree generated by the rotation of the vehicle changes to generate a rotation matrix transformation quantity;
the vehicle attitude rotation matrix is the vehicle attitude rotation matrix in the j-1 integral step number;
the above-mentionedExpressed as:
in the formula (I), the compound is shown in the specification,
b2 matrix according to vehicle position and postureDetermining resultant moment M borne by the vehicle under the vehicle mass center polar coordinate system under the current integral step numberjIs as follows;
in the formula (I), the compound is shown in the specification,a vehicle attitude rotation matrix under a vehicle mass center polar coordinate system is obtained;
the stress points of the vehicle under a Cartesian coordinate system are defined;
Fjthe resultant force borne by the vehicle at the j integral step number;
b3, calculating the resultant moment M under each integral stepjSubstituting the vehicle dynamics equation, solving the vehicle dynamics equation through a fourth-order Runge Kutta solver, and outputting vehicle rotational freedom data under a vehicle mass center polar coordinate system.
10. The method for simulating vehicle dynamics in polar coordinate system according to claim 7, wherein in step S6, when it is required to output the current integration step j, each force point of the vehicleIn the case of absolute coordinate positions in a cartesian coordinate system, this is calculated by the following formula:
in the formula (I), the compound is shown in the specification,in a Cartesian coordinate system, the force pointA position matrix of (a);
for the stress point of the vehicle in a Cartesian coordinate systemWhen converting to the vehicle mass center polar coordinate system, the corresponding rotation matrix;
transposing a matrix for the vehicle attitude in the vehicle mass center polar coordinate system in the jth integration step;
is the ith stress pointThe initial direction vector of the direction of the,wherein the content of the first and second substances,is the ith stress pointIn a moving freedom value under a three-dimensional space polar coordinate system, a superscript T is a transposition operator;
a vehicle position rotation matrix in a three-dimensional space polar coordinate system is obtained in the jth integral step;
Ojis the initial vector direction of the vehicle center point O at the j integral step number, and Oj＝[Lj,0,0]，LjThe value of the moving freedom L in a three-dimensional space polar coordinate system under the j integral step number;
wherein, the stress point of the vehicle under the Cartesian coordinate systemThe conversion formula when converting into the vehicle mass center polar coordinate system is as follows:
in the formula, in a vehicle mass center polar coordinate system,is a stress pointThe translation distance along the radius direction of the earth;
is a stress pointThe angle of rotation around the warp direction;
is a stress pointAn angle of rotation about the latitudinal direction;
andrespectively the stress point in the Cartesian coordinate system of the vehicle mass centerCoordinates on the x-axis, y-axis, and z-axis;
stress point under vehicle mass center polar coordinate systemOf the rotation matrixComprises the following steps:
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