CN110597088A - A Vehicle Dynamics Simulation Method in Polar Coordinate System - Google Patents
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Abstract
本发明公开了一种极坐标系下的车辆动力学仿真方法，针对车辆动力学系统中车辆旋转运动存在奇异点的问题，采用空间极坐标系结合牛顿定律、离心力和科氏力计算方法，给出了车辆在极坐标系下的六自由度动力学仿真方法，用于描述车辆在空间中的运动位置和姿态。利用车辆在转动过程中各瞬时转轴会产生变化、空间旋转具有继承性的原理，将传统动力学模型中的三个平动和三个转动自由度转换为一个平动五个转动自由度，能够很好地描述车辆在空间中的运动，同时能够有效解决车辆动力学系统中车体等部件空间旋转奇异点的问题。
The invention discloses a vehicle dynamics simulation method in a polar coordinate system. Aiming at the problem that there is a singular point in the vehicle rotational motion in the vehicle dynamics system, the spatial polar coordinate system is combined with Newton's law, centrifugal force and Coriolis force calculation methods to give A six-degree-of-freedom dynamics simulation method of the vehicle in the polar coordinate system is proposed, which is used to describe the moving position and attitude of the vehicle in space. Utilizing the principle that each instantaneous rotation axis will change during the rotation of the vehicle and the spatial rotation is inherited, the three translation and three rotation degrees of freedom in the traditional dynamic model are converted into one translation and five rotation degrees of freedom, which can It can well describe the movement of the vehicle in space, and at the same time, it can effectively solve the problem of the singular point of the space rotation of the vehicle body and other components in the vehicle dynamics system.
Description
技术领域technical field
本发明属于车辆动力学仿真技术领域，具体涉及一种极坐标系下的车辆动力学仿真方法。The invention belongs to the technical field of vehicle dynamics simulation, and in particular relates to a vehicle dynamics simulation method in a polar coordinate system.
背景技术Background technique
针对车辆动力学的核心问题是动力学建模和运动方程求解的问题。在车辆系统动力学理论中，常用的坐标系为笛卡尔坐标系，其定义三个相互垂直的向量为三个坐标方向，来描述空间物体的直线运动。在笛卡尔坐标系下，牛顿在1787年建立了牛顿方程解决了支点的运动学和动力学问题。在此基础上，刚体系统的动力学方程可以写成很多种形式，其中一种简单的被广泛实用的方程是1750年欧拉提出的牛顿-欧拉方程，其在笛卡尔坐标系下定义物体运动的坐标系在质心，将物体的平动和转动作作为独立的运动自由度来描述，采用旋转矢量的概念，利用对偶数形式，使牛顿-欧拉方程具有简明的表达形式，在开链和闭链空间机构的运动学和动力分析中得到广泛应用。牛顿-欧拉方程是欧拉运动定律的定量描述，是牛顿运动定律的延伸。The core issues for vehicle dynamics are dynamic modeling and solution of motion equations. In vehicle system dynamics theory, the commonly used coordinate system is the Cartesian coordinate system, which defines three mutually perpendicular vectors as three coordinate directions to describe the linear motion of space objects. Under the Cartesian coordinate system, Newton established Newton's equation in 1787 to solve the kinematics and dynamics problems of the fulcrum. On this basis, the dynamic equation of a rigid body system can be written in many forms, one of which is a simple and widely used equation is the Newton-Euler equation proposed by Euler in 1750, which defines the motion of an object in the Cartesian coordinate system The coordinate system is at the center of mass, and the translation and rotation of the object are described as independent motion degrees of freedom. The concept of rotation vector is adopted, and the even number form is used to make the Newton-Euler equation have a concise expression. In the open chain and It is widely used in the kinematics and dynamic analysis of closed-chain space mechanisms. The Newton-Euler equation is a quantitative description of Euler's law of motion and an extension of Newton's law of motion.
对于物体自身的旋转运动通常采用卡尔丹角、欧拉角或欧拉四元素来描述、卡尔丹角和欧拉角分别绕着三个相互垂直的旋转轴依次旋转，只是旋转的顺序不一样，但都存在旋转奇异点问题，即当任意一轴旋转90度的时候会导致该轴桶其他轴重合，欧拉旋转被重合的两根轴只能产生一个轴的旋转，为了解决旋转奇异点问题，欧拉提出欧拉四元素方法，其根据有限旋转理论，由物体瞬时旋转轴和旋转角度推导得到与此刻旋转运动相关的四个元素。但这四个元素不相互独立，需要满足一定的约束关系。Shabanan ect.分析了欧拉角和欧拉参数的表达式，并应用于车轮系统动力学的建模过程中，虽然欧拉四元素能很好地描述空间物体的旋转运动且不存在旋转奇异点问题，但引入的四元素间的约束方程将动力学微分方程变成微分代数方程，增加动力学方程求解的复杂性，降低了方程求解效率。国内有学者在笛卡尔坐标系下采用卡尔丹角建立了完整的车辆-轨道耦合动力学理论，由于旋转运动采用了小角度假设，有效避免了旋转奇异点问题，但对于空间物体大角度旋转问题具有一定的局限性。The rotational motion of the object itself is usually described by the Cardan angle, the Euler angle or the Euler four elements. The Cardan angle and the Euler angle rotate around three mutually perpendicular rotation axes respectively, but the order of rotation is different. However, there is a problem of rotation singularity, that is, when any axis rotates 90 degrees, it will cause the other axes of the axis barrel to overlap, and the two axes that are overlapped by Euler rotation can only produce a rotation of one axis. In order to solve the problem of rotation singularity , Euler proposed Euler's four-element method, according to the finite rotation theory, the four elements related to the current rotation motion are derived from the object's instantaneous rotation axis and rotation angle. However, these four elements are not independent of each other and need to satisfy certain constraints. Shabanan ect. analyzed the expressions of Euler angles and Euler parameters, and applied them in the modeling process of wheel system dynamics, although the Euler four elements can well describe the rotational motion of space objects and there is no rotational singularity However, the introduced constraint equation between four elements turns the dynamic differential equation into a differential algebraic equation, which increases the complexity of solving the dynamic equation and reduces the efficiency of solving the equation. Some domestic scholars have established a complete vehicle-track coupling dynamics theory using the Cardan angle in the Cartesian coordinate system. Since the rotation motion adopts the small-angle assumption, the problem of rotation singularity is effectively avoided, but for the problem of large-angle rotation of space objects has certain limitations.
Jalón和Bayo在1994年提出完全笛卡尔坐标系方法，属于绝对坐标系建模方法。这种方法的特点是避免使用一般笛卡尔方法中的欧拉角和欧拉参数，而是利用与刚体固结的若干参考点和参考矢量的笛卡尔坐标描述刚体的空间位置与姿态。参考点选择在铰的中心，参考矢量沿铰的转轴或平动方向，可通过多个刚体间共享位置来减少位置变量。但完全笛卡尔坐标所形成的动力学方程与一般笛卡尔方法本质相同，只是其雅克比矩阵统一在一个绝对坐标系下，便于求解计算。Jalón and Bayo proposed the complete Cartesian coordinate system method in 1994, which belongs to the absolute coordinate system modeling method. The characteristic of this method is to avoid the use of Euler angles and Euler parameters in the general Cartesian method, but use the Cartesian coordinates of several reference points and reference vectors fixed with the rigid body to describe the spatial position and attitude of the rigid body. The reference point is selected at the center of the hinge, and the reference vector is along the axis of rotation or translation direction of the hinge. The position variable can be reduced by sharing the position among multiple rigid bodies. However, the kinetic equation formed by complete Cartesian coordinates is essentially the same as the general Cartesian method, except that the Jacobian matrix is unified in an absolute coordinate system, which is convenient for solution calculation.
选择合适的坐标系能够更好地描述车辆在空间中的运动，更能提高求解精度。现有的车辆动力学模型基本上都是建立在三维笛卡尔坐标系下，三维空间极坐标下的车辆动力学建模很难直接推导并直接应用于仿真计算中。Jalón给出了二维平面极坐标系下物体的运动方程，但推导过程比较复杂，且并未给出三维空间极坐标系下物体的运动方程。Choosing an appropriate coordinate system can better describe the motion of the vehicle in space and improve the accuracy of the solution. The existing vehicle dynamics models are basically established in the three-dimensional Cartesian coordinate system, and the vehicle dynamics modeling in the three-dimensional space polar coordinates is difficult to be directly derived and directly applied to the simulation calculation. Jalón gave the equation of motion of the object in the polar coordinate system of the two-dimensional plane, but the derivation process is more complicated, and did not give the equation of motion of the object in the polar coordinate system of the three-dimensional space.
发明内容Contents of the invention
针对现有技术中的上述不足，本发明提供的极坐标系下的车辆动力学仿真方法针对现有的车辆动力学系统中车辆旋转运动过程中存在奇异点的问题。In view of the above-mentioned shortcomings in the prior art, the vehicle dynamics simulation method in the polar coordinate system provided by the present invention aims at the problem of singular points in the vehicle rotational motion process in the existing vehicle dynamics system.
为了达到上述发明目的，本发明采用的技术方案为：一种极坐标系下的车辆动力学仿真方法，包括以下步骤：In order to achieve the above-mentioned purpose of the invention, the technical solution adopted in the present invention is: a vehicle dynamics simulation method under a polar coordinate system, comprising the following steps:
S1、设置车辆动力学的仿真总时长T和仿真步长dt；S1. Set the total simulation time T and simulation step dt of vehicle dynamics;
S2、确定待仿真车辆的参数，并根据其搭建极坐标系下的车辆动力学方程；S2. Determine the parameters of the vehicle to be simulated, and build the vehicle dynamics equation in the polar coordinate system according to them;
S3、设置初始仿真时间t0为0，对应的初始积分步数为1；S3. Set the initial simulation time t0 to be 0, and the corresponding initial integration step number to be 1;
S4、根据当前仿真时间，求解搭建的车辆动力学方程，获得对应的极坐标系下的车辆自由度数据；S4. According to the current simulation time, solve the built vehicle dynamics equation, and obtain the vehicle degree of freedom data in the corresponding polar coordinate system;
S5、使仿真时间增加仿真步长dt，对应的积分步数增加1，并判断增加仿真步长dt后的当前仿真时间t是否大于仿真总时长T；S5. Increase the simulation time by the simulation step size dt, increase the corresponding integration step by 1, and judge whether the current simulation time t after increasing the simulation step size dt is greater than the total simulation time T;
若是，则进入步骤S6；If so, proceed to step S6;
若否，则返回步骤S4；If not, return to step S4;
S6、输出极坐标系下车辆动力学方程当前对应的车辆自由度数据，作为车辆动力学仿真结果。S6. Outputting the current vehicle degree of freedom data corresponding to the vehicle dynamics equation in the polar coordinate system as a vehicle dynamics simulation result.
进一步地，所述步骤S2中，待仿真车辆的参数包括车辆质量m、笛卡尔坐标系下的转动惯量J＝[Jx,Jy,Jz]T、车辆在笛卡尔坐标系下受力点所受的力力矩车辆的空间初始位置和车辆的初始姿态R0＝[α0,β0,γ0]T；Further, in the step S2, the parameters of the vehicle to be simulated include the vehicle mass m, the moment of inertia J=[J x , J y , J z ] T in the Cartesian coordinate system, the force on the vehicle in the Cartesian coordinate system point Force torque The initial position of the vehicle in space and the initial pose of the vehicle R 0 =[α 0 ,β 0 ,γ 0 ] T ;
其中，i为各受力点的编号，且i＝1,2,3,...M，M为受力点的总数；Among them, i is the serial number of each stress point, and i=1,2,3,...M, M is the total number of stress points;
α0,β0,γ0分别笛卡尔坐标系下绕x,y,z轴旋转时的初始角度，且旋转顺序为先绕z轴旋转γ0，再绕y轴旋转β0，最后绕x轴旋转α0；α 0 , β 0 , and γ 0 are the initial angles when rotating around the x, y, and z axes in the Cartesian coordinate system, and the rotation sequence is to first rotate γ 0 around the z axis, then rotate β 0 around the y axis, and finally rotate around x axis rotation α 0 ;
所述步骤S2中的极坐标系包括三维空间极坐标系和车辆质心极坐标系；所述三维空间极坐标系用于描述车辆质心在三维空间中的位置，所述车辆质心极坐标系用于描述车辆在三维空间中的运动姿态。The polar coordinate system in the step S2 includes a three-dimensional space polar coordinate system and a vehicle center-of-mass polar coordinate system; the three-dimensional space polar coordinate system is used to describe the position of the vehicle center of mass in three-dimensional space, and the vehicle center-of-mass polar coordinate system is used for Describe the motion posture of the vehicle in three-dimensional space.
进一步地，所述步骤S2具体为：Further, the step S2 is specifically:
S21、将车辆在笛卡尔坐标系下的初始位置转换至三维空间极坐标系下，并根据其计算三维空间极坐标系下车辆位置旋转矩阵 S21. Convert the initial position of the vehicle in the Cartesian coordinate system to the three-dimensional space polar coordinate system, and calculate the vehicle position rotation matrix in the three-dimensional space polar coordinate system according to it
S22、将车辆的初始姿态转换至车辆质心极坐标系中，并根据其计算车辆质心极坐标系下车辆姿态旋转矩阵 S22. Transform the initial attitude of the vehicle into the vehicle center-of-mass polar coordinate system, and calculate the vehicle attitude rotation matrix in the vehicle center-of-mass polar coordinate system based on it
S23、根据车辆位置旋转矩阵计算三维空间极坐标系下车辆所受合力F0；S23. Rotate the matrix according to the vehicle position Calculate the resultant force F 0 on the vehicle in the three-dimensional space polar coordinate system;
S24、根据车辆姿态旋转矩阵和车辆所受合力F0，计算车辆质心极坐标系下车辆所受合力矩M0；S24. Rotate the matrix according to the vehicle attitude and the resultant force F 0 on the vehicle, calculate the resultant moment M 0 on the vehicle in the polar coordinate system of the center of mass of the vehicle;
S25、根据合力F0和合力矩M0，搭建车辆动力学方程。S25. Establish a vehicle dynamics equation according to the resultant force F 0 and the resultant moment M 0 .
进一步地，所述步骤S21中，将车辆在三维空间坐标系下的初始位置转换至三维空间极坐标系下的转换公式为：Further, in the 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:
式中，在三维空间极坐标系下，L0为车辆初始位置P0沿地球半径方向的平动距离；In the formula, in the three-dimensional space polar coordinate system, L 0 is the translational distance of the initial position P 0 of the vehicle along the radius of the earth;
θ0为车辆初始位置P0绕经度方向的转动角度；θ 0 is the rotation angle of the initial position P 0 of the vehicle around the longitude direction;
为车辆初始位置P0绕纬度方向的转动角度； is the rotation angle of the initial position P0 of the vehicle around the latitude direction ;
和分别为笛卡尔坐标系中车辆初始位置的x轴、y轴和z轴坐标； and are the x-axis, y-axis and z-axis coordinates of the initial position of the vehicle in the Cartesian coordinate system;
所述三维空间极坐标系下车辆位置旋转矩阵为：The vehicle position rotation matrix in the three-dimensional space polar coordinate system for:
所述步骤S22中，将车辆的初始姿态转换至车辆质心极坐标系中的转换公式为：In the step S22, the conversion formula for converting the initial posture of the vehicle into the polar coordinate system of the vehicle center of mass is:
式中，在三维空间及坐标系中，为车辆绕经度方向的转动角度；ψ0为车辆绕纬度方向的转动角度；In the formula, in the three-dimensional space and the coordinate system, is the rotation angle of the vehicle around the longitude direction; ψ 0 is the rotation angle of the vehicle around the latitude direction;
φ0为车辆绕地球半径方向的转动角度；φ 0 is the rotation angle of the vehicle around the radius of the earth;
所述车辆质心极坐标系下车辆姿态旋转矩阵 The vehicle attitude rotation matrix in the polar coordinate system of the vehicle center of mass
所述步骤S23中，三维空间极坐标系下车辆所受合力F0为：In the step S23, the resultant force F0 on the vehicle in the polar coordinate system of the three-dimensional space is :
式中，为三维空间极坐标系下车辆位置旋转矩阵；In the formula, is the vehicle position rotation matrix in the three-dimensional space polar coordinate system;
为车辆在笛卡尔坐标系下受力点所受的力； is the force point of the vehicle in the Cartesian coordinate system the force received;
所述步骤S24中，车辆质心极坐标系下车辆所受合力矩M0：In the step S24, the resultant moment M 0 suffered by the vehicle in the polar coordinate system of the vehicle center of mass:
式中，为车辆在笛卡尔坐标系下受力点的力矩。In the formula, is the force point of the vehicle in the Cartesian coordinate system moment.
进一步地，所述步骤S25中，搭建的车辆动力学方程为：Further, in the step S25, the vehicle dynamics equation built is:
式中，m为车辆质量；In the formula, m is the mass of the vehicle;
为三维空间极坐标系中移动自由度L的二阶导数； is the second derivative of the degree of freedom L of movement in the three-dimensional space polar coordinate system;
为三维空间极坐标系中车辆所受合力F0沿移动自由度L方向的分量； is the component of the resultant force F 0 on the vehicle along the direction of the moving degree of freedom L in the polar coordinate system of the three-dimensional space;
FQ为三维空间极坐标系中车辆绕经度方向的转动θ和绕纬度方向的转动产生的离心力的合力；F Q is the rotation θ of the vehicle around the longitude direction and the rotation around the latitude direction in the three-dimensional space polar coordinate system the resulting centrifugal force;
为三维空间极坐标系中第一旋转自由度θ的二阶导数； is the second derivative of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中车辆所受合力F0沿第一旋转自由度θ方向的分量； is the component of the resultant force F 0 on the vehicle along the direction of the first rotational degree of freedom θ in the polar coordinate system of the three-dimensional space;
FCθ为三维空间极坐标系中车辆绕经度方向的转动θ产生的科氏力；F Cθ is the Coriolis force generated by the rotation θ of the vehicle around the longitude direction in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度的二阶导数； is the second rotational degree of freedom in the three-dimensional space polar coordinate system The second derivative of ;
为三维空间极坐标系中车辆所受合力F0沿第二旋转自由度方向的分量； is the resultant force F 0 on the vehicle in the three-dimensional space polar coordinate system along the second rotational degree of freedom component of direction;
为三维空间极坐标系中车辆绕纬度方向的转动产生的科氏力； is the rotation of the vehicle around the latitude direction in the three-dimensional space polar coordinate system The resulting Coriolis force;
为车辆质心极坐标系中车辆绕第三旋转自由度的转动惯量； is the third rotational degree of freedom of the vehicle in the polar coordinate system of the vehicle center of mass moment of inertia;
为车辆质心极坐标系中第三旋转自由度的二阶导数； is the third rotational degree of freedom in the polar coordinate system of the vehicle center of mass The second derivative of ;
为车辆质心极坐标系中车辆所受合力矩M0沿第三旋转自由度方向的分量； is the resultant moment M 0 of the vehicle in the polar coordinate system of the vehicle center of mass along the third rotational degree of freedom component of direction;
Jψ为车辆质心极坐标系中绕第四旋转自由度ψ的转动惯量；J ψ is the moment of inertia around the fourth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中第四旋转自由度ψ的二阶导数； is the second derivative of the fourth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
车辆质心极坐标系车辆所受合力矩M0沿第四旋转自由度ψ方向的分量； The component of the resultant moment M0 of the vehicle in the polar coordinate system of the vehicle's center of mass along the direction of the fourth rotational degree of freedom ψ;
Jφ为车辆质心极坐标系中绕第五旋转自由度φ的转动惯量；J φ is the moment of inertia around the fifth rotational degree of freedom φ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中第五旋转自由度φ的二阶导数； is the second derivative of the fifth rotational degree of freedom φ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中车辆所受合力矩M0沿第五旋转自由度ψ方向的分量； is the component of the resultant moment M 0 on the vehicle along the direction of the fifth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
为三维空间极坐标系中第一旋转自由度θ的一阶导数； is the first derivative of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度的一阶导数； is the second rotational degree of freedom in the three-dimensional space polar coordinate system The first derivative of ;
为三维空间极坐标系中移动自由度L的一阶导数； is the first derivative of the degree of freedom L of movement in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第一旋转自由度θ上一积分步数时的值； is the value of one integration step on the first rotation degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度上一积分步数时的值。 is the second rotational degree of freedom in the three-dimensional space polar coordinate system The value at the last number of integration steps.
进一步地，所述步骤S4中，根据当前仿真时间，通过四阶龙格库塔法对车辆动力学方程进行求解；Further, in the step S4, according to the current simulation time, the vehicle dynamics equation is solved by the fourth-order Runge-Kutta method;
所述车辆自由度数据包括当前仿真时间t对应的积分步数下，三维空间极坐标系下车辆移动及转动自由度数据及车辆质心极坐标系下车辆转动自由度数据。The vehicle degree of freedom data includes vehicle movement and rotation degree of freedom data in the three-dimensional space polar coordinate system and vehicle rotation degree of freedom data in the vehicle centroid polar coordinate system under the integration steps corresponding to the current simulation time t.
进一步地，所述三维空间极坐标系下车辆移动及转动自由度数据包括Lj,θj, 由四阶龙格库塔法所建立的车辆动力学方程进行迭代求解得到；Further, the vehicle movement and rotation degrees of freedom data in the three-dimensional polar coordinate system include L j , θ j , It is obtained by iteratively solving the vehicle dynamics equation established by the fourth-order Runge-Kutta method;
其中，j为车辆动力学仿真过程中，积分步数的编号，且j＝1,2,3,...J，J为积分步数总数；Wherein, j is the number of integration steps in the process of vehicle dynamics simulation, and j=1,2,3,...J, J is the total number of integration steps;
Lj为第j积分步数下，三维空间极坐标系下移动自由度L的值；L j is the value of the degree of freedom L of movement in the three-dimensional space polar coordinate system under the jth integration step;
θj为第j积分步数下，三维空间极坐标系下第一旋转自由度θ的值；θ j is the value of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system under the jth integration step;
为第j积分步数下，三维空间极坐标系下第二旋转自由度的值； is the second rotational degree of freedom in the polar coordinate system of the three-dimensional space under the jth integration step value;
为第j积分步数下，移动自由度L的一阶导数值； is the first derivative value of the moving degree of freedom L under the jth integration step;
为第j积分步数下，第一旋转自由度θ的值； is the value of the first rotational degree of freedom θ under the jth integration step;
为第j积分步数下，第二旋转自由度的一阶导数值； is the second rotational degree of freedom under the jth integration step The value of the first order derivative;
为第j积分步数下，移动自由度L的二阶导数值； is the second-order derivative value of the moving degree of freedom L under the jth integration step;
为第j积分步数下，第一旋转自由度θ的二阶导数值； is the second derivative value of the first rotational degree of freedom θ at the jth integration step;
为第j积分步数下，第二旋转自由度的二阶导数值； is the second rotational degree of freedom under the jth integration step The second derivative value of ;
所述车辆质心极坐标系下车辆转动自由度数据包括ψj,φj、 由四阶龙格库塔法对所建立的车辆动力学方程进行迭代求解得到；The vehicle rotation degree of freedom data in the polar coordinate system of the vehicle center of mass includes ψ j ,φ j , It is obtained by iteratively solving the established vehicle dynamics equations by the fourth-order Runge-Kutta method;
其中，为第j积分步数下，车辆质心极坐标系下第三旋转自由度的值；in, is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step value;
ψj为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的值；ψ j is the value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
φj为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的值；φ j is the value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第三旋转自由度的一阶导数值； is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step The value of the first order derivative;
为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的一阶导数值； is the first derivative value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的一阶导数值； is the first derivative value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第三旋转自由度的二阶导数值； is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step The second derivative value of ;
为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的二阶导数值； is the second derivative value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的二阶导数值。 is the second derivative value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step.
进一步地，获得所述车辆在三维空间极坐标系下，车辆移动及转动自由度数据的方法具体为：Further, the method for obtaining the vehicle movement and rotation degree of freedom data of the vehicle in the three-dimensional space polar coordinate system is as follows:
A1、根据当前仿真时间，确定当前积分步数j下的车辆位置旋转矩阵为；A1. According to the current simulation time, determine the vehicle position rotation matrix under the current integration step j for;
式中，为第j积分步数时，车辆的第一旋转自由度θj和第二旋转自由度变化产生的旋转矩阵变化量；In the formula, When is the jth integration step, the first rotational degree of freedom θ j and the second rotational degree of freedom of the vehicle The amount of change in the rotation matrix generated by the change;
为第j-1积分步数时在三维空间极坐标系下车辆位置的旋转矩阵； is the rotation matrix of the vehicle position in the three-dimensional space polar coordinate system at the j-1th integration step;
所述表示为：said Expressed as:
A2、根据三维空间极坐标系中车辆位置旋转矩阵确定当前积分步数下车辆所受合力Fj为：A2. According to the vehicle position rotation matrix in the three-dimensional space polar coordinate system Determine the resultant force F j on the vehicle under the current integration steps as:
式中，当j＝1时， In the formula, when j=1,
A3、将每一积分步数下的合力Fj代入车辆动力学方程，并通过四阶库塔求解器对车辆动力学方程进行求解，输出三维空间极坐标系下车辆移动及转动自由度数据。A3. Substitute the resultant force F j under each integration step into the vehicle dynamics equation, and solve the vehicle dynamics equation through the fourth-order Kutta solver, and output the vehicle movement and rotation degree of freedom data in the three-dimensional space polar coordinate system.
进一步地，获得所述车辆在车辆质心极坐标系下的车辆转动自由度数据的方法具体为：Further, the method for obtaining the vehicle rotational degree of freedom data of the vehicle in the polar coordinate system of the vehicle center of mass is specifically as follows:
B1、根据当前仿真时间，确定当前步长下的车辆姿态旋转矩阵 B1. According to the current simulation time, determine the vehicle attitude rotation matrix under the current step size
式中，为第j积分步数时，车辆转动产生的角度自由度变化产生的旋转矩阵变换量；In the formula, When is the j-th integration step, the transformation amount of the rotation matrix generated by the change of the angular degree of freedom caused by the vehicle rotation;
为第j-1积分步数时的车辆姿态旋转矩阵； is the vehicle attitude rotation matrix at the j-1th integration step;
所述表示为：said Expressed as:
式中， In the formula,
B2、根据车辆位置姿态矩阵确定当前积分步数下车辆质心极坐标系下车辆所受合力矩Mj为；B2. According to the vehicle position attitude matrix Determine the resultant moment M j of the vehicle under the polar coordinate system of the center of mass of the vehicle under the current integration steps;
式中，为车辆质心极坐标系下车辆姿态旋转矩阵；In the formula, is the vehicle attitude rotation matrix in the polar coordinate system of the vehicle center of mass;
为车辆在笛卡尔坐标系下的受力点； is the stress point of the vehicle in the Cartesian coordinate system;
Fj为第j积分步数时车辆所受合力；F j is the resultant force on the vehicle at the jth integration step;
B3、将每一积分步数下的合力矩Mj代入车辆动力学方程，并通过四阶龙格库塔求解器求解车辆动力学方程，输出车辆质心极坐标系下的车辆转动自由度数据。B3. Substitute the resultant moment M j under each integration step into the vehicle dynamics equation, and solve the vehicle dynamics equation through the fourth-order Runge-Kutta solver, and output the vehicle rotation degree of freedom data in the polar coordinate system of the vehicle center of mass.
进一步地，所述步骤S6中，当需要输出当前积分步数j下，车辆各受力点在笛卡尔坐标系下的绝对坐标位置时，通过以下公式计算：Further, in the step S6, when it is necessary to output the current integration step number j, each force point of the vehicle In the absolute coordinate position under the Cartesian coordinate system, it is calculated by the following formula:
式中，为在笛卡尔坐标系中，受力点的位置矩阵；In the formula, In the Cartesian coordinate system, the force point The position matrix;
为车辆在笛卡尔坐标系下的受力点转换到车辆质心极坐标系中时，对应的旋转矩阵； is the force point of the vehicle in the Cartesian coordinate system The corresponding rotation matrix when converting to the polar coordinate system of the vehicle center of mass;
为第j积分步下，车辆质心极坐标系中车辆姿态转置矩阵； is the vehicle attitude transposition matrix in the polar coordinate system of the vehicle center of mass at the jth integration step;
为第i个受力点的初始方向矢量，其中，为第i个受力点在三维空间极坐标系下的移动自由度值，上标T为转置运算符； is the i-th stress point The initial direction vector of , in, is the i-th stress point The value of the degree of freedom of movement in the three-dimensional space polar coordinate system, the superscript T is the transpose operator;
为第j积分步下，三维空间极坐标系中车辆位置旋转矩阵； is the vehicle position rotation matrix in the three-dimensional space polar coordinate system under the jth integration step;
Oj为第j积分步数下，车辆中心点O的初始矢量方向，且Oj＝[Lj,0,0]，Lj为第j积分步数下，三维空间极坐标系下的移动自由度L的值；O j is the initial vector direction of the center point O of the vehicle at the jth integration step, and O j = [L j ,0,0], L j is the movement in the three-dimensional space polar coordinate system at the jth integration step The value of the degree of freedom L;
其中，车辆在笛卡尔坐标系下的受力点转换到车辆质心极坐标系中时的转换公式为：Among them, the force point of the vehicle in the Cartesian coordinate system The conversion formula when converting to the polar coordinate system of the vehicle center of mass is:
式中，在车辆质心极坐标系中，为受力点沿地球半径方向的平动距离；In the formula, in the polar coordinate system of the vehicle center of mass, force point Translational distance along the radius of the earth;
为受力点绕经度方向转动的角度； force point The angle of rotation around the longitude direction;
为受力点绕纬度方向转动的角度； force point The angle of rotation around the latitude direction;
和分别为车辆质心所在的笛卡尔坐标系中，受力点在x轴、y轴和z轴上的坐标； and Respectively, in the Cartesian coordinate system where the center of mass of the vehicle is located, the force point coordinates on the x-axis, y-axis and z-axis;
所述车辆质心极坐标系下受力点的旋转矩阵为：The point of force under the polar coordinate system of the center of mass of the vehicle The rotation matrix of for:
本发明的有益效果为：The beneficial effects of the present invention are:
本发明提供的极坐标系下的车辆动力学仿真方法，针对车辆动力学系统中车辆旋转运动存在奇异点的问题，采用空间极坐标系结合牛顿定律、离心力和科氏力计算方法，给出了车辆在极坐标系下的六自由度动力学仿真方法，用于描述车辆在空间中的运动位置和姿态。利用车辆在转动过程中各瞬时转轴会产生变化、空间旋转具有继承性的原理，将传统动力学模型中的三个平动和三个转动自由度转换为一个平动五个转动自由度，能够很好地描述车辆在空间中的运动，同时能够有效解决车辆动力学系统中车体等部件空间旋转奇异点的问题。The vehicle dynamics simulation method under the polar coordinate system provided by the present invention aims at the problem that there are singular points in the vehicle rotational motion in the vehicle dynamic system, and adopts the space polar coordinate system combined with Newton's law, centrifugal force and Coriolis force calculation method, and provides The six-degree-of-freedom dynamics simulation method of the vehicle in the polar coordinate system is used to describe the moving position and attitude of the vehicle in space. Utilizing the principle that each instantaneous rotation axis will change during the rotation of the vehicle and the spatial rotation is inherited, the three translation and three rotation degrees of freedom in the traditional dynamic model are converted into one translation and five rotation degrees of freedom, which can It can well describe the movement of the vehicle in space, and at the same time, it can effectively solve the problem of the singular point of the space rotation of the vehicle body and other components in the vehicle dynamics system.
附图说明Description of drawings
图1为本发明中极坐标坐标系下的车辆动力学仿真方法流程图。Fig. 1 is a flow chart of the vehicle dynamics simulation method under the polar coordinate system in the present invention.
图2为本发明中构建车辆动力学方程方法流程图。Fig. 2 is a flow chart of the method for constructing vehicle dynamics equations in the present invention.
图3为本发明提供的实施例中笛卡尔坐标系下受力点的位移示意图。Fig. 3 is the stressed point under the Cartesian coordinate system in the embodiment provided by the present invention The displacement diagram.
图4为本发明提供的实施例中笛卡尔坐标系中旋转角度示意图。Fig. 4 is a schematic diagram of the rotation angle in the Cartesian coordinate system in the embodiment provided by the present invention.
图5为本发明提供的实施例中极坐标系下受力点的位移示意图。Fig. 5 is the stressed point under the polar coordinate system in the embodiment provided by the present invention The displacement diagram.
图6为本发明提供的实施例中极坐标系下旋转角度示意图。Fig. 6 is a schematic diagram of the rotation angle in the polar coordinate system in the embodiment provided by the present invention.
图7为本发明提供的实施例中笛卡尔坐标系下受力点的轨迹示意图。Fig. 7 is the stressed point under the Cartesian coordinate system in the embodiment provided by the present invention The schematic diagram of the trajectory.
图8为本发明提供的实施例中极坐标系下受力点的轨迹示意图。Fig. 8 is the stress point under the polar coordinate system in the embodiment provided by the present invention The schematic diagram of the trajectory.
具体实施方式Detailed ways
下面对本发明的具体实施方式进行描述，以便于本技术领域的技术人员理解本发明，但应该清楚，本发明不限于具体实施方式的范围，对本技术领域的普通技术人员来讲，只要各种变化在所附的权利要求限定和确定的本发明的精神和范围内，这些变化是显而易见的，一切利用本发明构思的发明创造均在保护之列。The specific embodiments of the present invention are described below so that those skilled in the art can understand the present invention, but it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, as long as various changes Within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are included in the protection list.
如图1所示，一种极坐标系下的车辆动力学仿真方法，包括以下步骤：As shown in Figure 1, a vehicle dynamics simulation method in a polar coordinate system includes the following steps:
S1、设置车辆动力学的仿真总时长T和仿真步长dt；S1. Set the total simulation time T and simulation step dt of vehicle dynamics;
S2、确定待仿真车辆的参数，并根据其搭建极坐标系下的车辆动力学方程；S2. Determine the parameters of the vehicle to be simulated, and build the vehicle dynamics equation in the polar coordinate system according to them;
S3、设置初始仿真时间t0为0，对应的初始积分步数为1；S3. Set the initial simulation time t0 to be 0, and the corresponding initial integration step number to be 1;
S4、根据当前仿真时间，求解搭建的车辆动力学方程，获得对应的极坐标系下的车辆自由度数据；S4. According to the current simulation time, solve the built vehicle dynamics equation, and obtain the vehicle degree of freedom data in the corresponding polar coordinate system;
S5、使仿真时间增加仿真步长dt，对应的积分步数增加1，并判断增加仿真步长dt后的当前仿真时间t是否大于仿真总时长T；S5. Increase the simulation time by the simulation step size dt, increase the corresponding integration step by 1, and judge whether the current simulation time t after increasing the simulation step size dt is greater than the total simulation time T;
若是，则进入步骤S6；If so, proceed to step S6;
若否，则返回步骤S4；If not, return to step S4;
S6、输出极坐标系下车辆动力学方程当前对应的车辆自由度数据，作为车辆动力学仿真结果。S6. Outputting the current vehicle degree of freedom data corresponding to the vehicle dynamics equation in the polar coordinate system as a vehicle dynamics simulation result.
上述步骤S2中，待仿真车辆的参数包括车辆质量m、笛卡尔坐标系下的转动惯量J＝[Jx,Jy,Jz]T、车辆在笛卡尔坐标系下受力点所受的力力矩车辆的空间初始位置和车辆的初始姿态R0＝[α0,β0,γ0]T；In the above step S2, the parameters of the vehicle to be simulated include vehicle mass m, moment of inertia J=[J x , J y , J z ] T in the Cartesian coordinate system, force point of the vehicle in the Cartesian coordinate system Force torque The initial position of the vehicle in space and the initial pose of the vehicle R 0 =[α 0 ,β 0 ,γ 0 ] T ;
其中，i为各受力点的编号，且i＝1,2,3,...M，M为受力点的总数；Among them, i is the serial number of each stress point, and i=1,2,3,...M, M is the total number of stress points;
α0,β0,γ0分别笛卡尔坐标系下绕x,y,z轴旋转时的初始角度，且旋转顺序为先绕z轴旋转γ0，再绕y轴旋转β0，最后绕x轴旋转α0；α 0 , β 0 , and γ 0 are the initial angles when rotating around the x, y, and z axes in the Cartesian coordinate system, and the rotation sequence is to first rotate γ 0 around the z axis, then rotate β 0 around the y axis, and finally rotate around x axis rotation α 0 ;
步骤S2中的极坐标系包括三维空间极坐标系和车辆质心极坐标系；三维空间极坐标系用于描述车辆质心在三维空间中的位置，车辆质心极坐标系用于描述车辆在三维空间中的运动姿态。以地球为例，单个质点在空间极坐标系中的运动包含沿地球半径方向的平动、沿经度和纬度方向的转动。The polar coordinate system in step S2 includes a three-dimensional space polar coordinate system and a vehicle center-of-mass polar coordinate system; the three-dimensional space polar coordinate system is used to describe the position of the vehicle center of mass in three-dimensional space, and the vehicle center-of-mass polar coordinate system is used to describe the position of the vehicle in three-dimensional space movement posture. Taking the earth as an example, the movement of a single particle in the space polar coordinate system includes translation along the radius of the earth and rotation along the longitude and latitude.
如图2所示，步骤S2具体为：As shown in Figure 2, step S2 is specifically:
S21、将车辆在笛卡尔坐标系下的初始位置转换至三维空间极坐标系下，并根据其计算三维空间极坐标系下车辆位置旋转矩阵 S21. Convert the initial position of the vehicle in the Cartesian coordinate system to the three-dimensional space polar coordinate system, and calculate the vehicle position rotation matrix in the three-dimensional space polar coordinate system according to it
其中，将车辆在三维空间坐标系下的初始位置转换至三维空间极坐标系下的转换公式为：Among them, 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:
式中，在三维空间极坐标系下，L0为车辆初始位置P0沿地球半径方向的平动距离；In the formula, in the three-dimensional space polar coordinate system, L 0 is the translational distance of the initial position P 0 of the vehicle along the radius of the earth;
θ0为车辆初始位置P0绕经度方向的转动角度；θ 0 is the rotation angle of the initial position P 0 of the vehicle around the longitude direction;
为车辆初始位置P0绕纬度方向的转动角度； is the rotation angle of the initial position P0 of the vehicle around the latitude direction ;
和分别为笛卡尔坐标系中车辆初始位置的x轴、y轴和z轴坐标； and are the x-axis, y-axis and z-axis coordinates of the initial position of the vehicle in the Cartesian coordinate system;
三维空间极坐标系下车辆位置旋转矩阵为：Vehicle position rotation matrix in three-dimensional space polar coordinate system for:
S22、将车辆的初始姿态转换至车辆质心极坐标系中，并根据其计算车辆质心极坐标系下车辆姿态旋转矩阵 S22. Transform the initial attitude of the vehicle into the vehicle center-of-mass polar coordinate system, and calculate the vehicle attitude rotation matrix in the vehicle center-of-mass polar coordinate system based on it
在笛卡尔坐标系下定义的绕x,y,z轴的旋转的初始角度与极坐标系下定义的初始角度一致；因此，将车辆的初始姿态转换至车辆质心极坐标系中的转换公式为：The initial angle of rotation around the x, y, and z axes defined in the Cartesian coordinate system is consistent with the initial angle defined in the polar coordinate system; therefore, the conversion formula for converting the initial attitude of the vehicle to the vehicle center-of-mass polar coordinate system is :
式中，在三维空间及坐标系中，为车辆绕经度方向的转动角度；In the formula, in the three-dimensional space and the coordinate system, is the rotation angle of the vehicle around the longitude direction;
ψ0为车辆绕纬度方向的转动角度；ψ 0 is the rotation angle of the vehicle around the latitude direction;
φ0为车辆绕地球半径方向的转动角度；φ 0 is the rotation angle of the vehicle around the radius of the earth;
车辆质心极坐标系下车辆姿态旋转矩阵 Vehicle attitude rotation matrix in the vehicle center of mass polar coordinate system
此时，在车辆质心极坐标系下车辆的转动惯量其计算公式为：At this time, the moment of inertia of the vehicle in the polar coordinate system of the vehicle center of mass Its calculation formula is:
式中，J车辆在笛卡尔坐标系下的转动惯量，且J＝[Jx,Jy,Jz]T；In the formula, J is the moment of inertia of the vehicle in the Cartesian coordinate system, and J=[J x ,J y ,J z ] T ;
S23、根据车辆位置旋转矩阵计算三维空间极坐标系下车辆所受合力F0；S23. Rotate the matrix according to the vehicle position Calculate the resultant force F 0 on the vehicle in the three-dimensional space polar coordinate system;
其中，三维空间极坐标系下车辆所受合力F0为：Among them, the resultant force F 0 on the vehicle in the three-dimensional space polar coordinate system is:
式中，为三维空间极坐标系下车辆位置旋转矩阵；In the formula, is the vehicle position rotation matrix in the three-dimensional space polar coordinate system;
为车辆在笛卡尔坐标系下受力点所受的力； is the force point of the vehicle in the Cartesian coordinate system the force received;
S24、根据车辆姿态旋转矩阵和车辆所受合力F0，计算车辆质心极坐标系下车辆所受合力矩M0；S24. Rotate the matrix according to the vehicle attitude and the resultant force F 0 on the vehicle, calculate the resultant moment M 0 on the vehicle in the polar coordinate system of the center of mass of the vehicle;
其中，车辆质心极坐标系下车辆所受合力矩M0：Among them, the resultant moment M 0 of the vehicle in the polar coordinate system of the vehicle center of mass:
式中，为车辆在笛卡尔坐标系下受力点的力矩。In the formula, is the force point of the vehicle in the Cartesian coordinate system moment.
S25、根据合力F0和合力矩M0，搭建车辆动力学方程。S25. Establish a vehicle dynamics equation according to the resultant force F 0 and the resultant moment M 0 .
根据牛顿运动定律，结合合力F0和合力矩M0得到搭建的车辆动力学方程为：According to Newton's law of motion, the vehicle dynamics equation obtained by combining the resultant force F 0 and the resultant moment M 0 is:
式中，m为车辆质量；In the formula, m is the mass of the vehicle;
为三维空间极坐标系中移动自由度L的二阶导数； is the second derivative of the degree of freedom L of movement in the three-dimensional space polar coordinate system;
为三维空间极坐标系中车辆所受合力F0沿移动自由度L方向的分量； is the component of the resultant force F 0 on the vehicle along the direction of the moving degree of freedom L in the polar coordinate system of the three-dimensional space;
FQ为三维空间极坐标系中车辆绕经度方向的转动θ和绕纬度方向的转动产生的离心力的合力；F Q is the rotation θ of the vehicle around the longitude direction and the rotation around the latitude direction in the three-dimensional space polar coordinate system the resulting centrifugal force;
为三维空间极坐标系中第一旋转自由度θ的二阶导数； is the second derivative of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中车辆所受合力F0沿第一旋转自由度θ方向的分量； is the component of the resultant force F 0 on the vehicle along the direction of the first rotational degree of freedom θ in the polar coordinate system of the three-dimensional space;
FCθ为三维空间极坐标系中车辆绕经度方向的转动θ产生的科氏力；F Cθ is the Coriolis force generated by the rotation θ of the vehicle around the longitude direction in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度的二阶导数； is the second rotational degree of freedom in the three-dimensional space polar coordinate system The second derivative of ;
为三维空间极坐标系中车辆所受合力F0沿第二旋转自由度方向的分量； is the resultant force F 0 on the vehicle in the three-dimensional space polar coordinate system along the second rotational degree of freedom component of direction;
为三维空间极坐标系中车辆绕纬度方向的转动产生的科氏力； is the rotation of the vehicle around the latitude direction in the three-dimensional space polar coordinate system The resulting Coriolis force;
为车辆质心极坐标系中车辆绕第三旋转自由度的转动惯量； is the third rotational degree of freedom of the vehicle in the polar coordinate system of the vehicle center of mass moment of inertia;
为车辆质心极坐标系中第三旋转自由度的二阶导数； is the third rotational degree of freedom in the polar coordinate system of the vehicle center of mass The second derivative of ;
为车辆质心极坐标系中车辆所受合力矩M0沿第三旋转自由度方向的分量； is the resultant moment M 0 of the vehicle in the polar coordinate system of the vehicle center of mass along the third rotational degree of freedom component of direction;
Jψ为车辆质心极坐标系中绕第四旋转自由度ψ的转动惯量；J ψ is the moment of inertia around the fourth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中第四旋转自由度ψ的二阶导数； is the second derivative of the fourth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
车辆质心极坐标系车辆所受合力矩M0沿第四旋转自由度ψ方向的分量； The component of the resultant moment M0 of the vehicle in the polar coordinate system of the vehicle's center of mass along the direction of the fourth rotational degree of freedom ψ;
Jφ为车辆质心极坐标系中绕第五旋转自由度φ的转动惯量；J φ is the moment of inertia around the fifth rotational degree of freedom φ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中第五旋转自由度φ的二阶导数； is the second derivative of the fifth rotational degree of freedom φ in the polar coordinate system of the vehicle center of mass;
为车辆质心极坐标系中车辆所受合力矩M0沿第五旋转自由度ψ方向的分量； is the component of the resultant moment M 0 on the vehicle along the direction of the fifth rotational degree of freedom ψ in the polar coordinate system of the vehicle center of mass;
为三维空间极坐标系中第一旋转自由度θ的一阶导数； is the first derivative of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度的一阶导数； is the second rotational degree of freedom in the three-dimensional space polar coordinate system The first derivative of ;
为三维空间极坐标系中移动自由度L的一阶导数； is the first derivative of the degree of freedom L of movement in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第一旋转自由度θ上一积分步数时的值； is the value of one integration step on the first rotation degree of freedom θ in the three-dimensional space polar coordinate system;
为三维空间极坐标系中第二旋转自由度上一积分步数时的值。 is the second rotational degree of freedom in the three-dimensional space polar coordinate system The value at the last number of integration steps.
上述步骤S4具体为：根据当前仿真时间，通过四阶龙格库塔法对车辆动力学方程进行求解；The above step S4 is specifically: according to the current simulation time, solve the vehicle dynamics equation by the fourth-order Runge-Kutta method;
车辆自由度数据包括当前仿真时间t对应的积分步数下，三维空间极坐标系下车辆移动及转动自由度数据及车辆质心极坐标系下车辆转动自由度数据；The vehicle degree of freedom data includes the vehicle movement and rotation degree of freedom data in the three-dimensional space polar coordinate system and the vehicle rotation degree of freedom data in the vehicle centroid polar coordinate system under the integration steps corresponding to the current simulation time t;
其中，三维空间极坐标系下车辆移动及转动自由度数据包括Lj,θj, 由四阶龙格库塔法所建立的车辆动力学方程进行迭代求解得到；Among them, the vehicle movement and rotation degrees of freedom data in the three-dimensional space polar coordinate system include L j , θ j , It is obtained by iteratively solving the vehicle dynamics equation established by the fourth-order Runge-Kutta method;
其中，j为车辆动力学仿真过程中，积分步数的编号，且j＝1,2,3,...J，J为积分步数总数；Wherein, j is the number of integration steps in the process of vehicle dynamics simulation, and j=1,2,3,...J, J is the total number of integration steps;
Lj为第j积分步数下，三维空间极坐标系下移动自由度L的值；L j is the value of the degree of freedom L of movement in the three-dimensional space polar coordinate system under the jth integration step;
θj为第j积分步数下，三维空间极坐标系下第一旋转自由度θ的值；θ j is the value of the first rotational degree of freedom θ in the three-dimensional space polar coordinate system under the jth integration step;
为第j积分步数下，三维空间极坐标系下第二旋转自由度的值； is the second rotational degree of freedom in the polar coordinate system of the three-dimensional space under the jth integration step value;
为第j积分步数下，移动自由度L的一阶导数值； is the first derivative value of the moving degree of freedom L under the jth integration step;
为第j积分步数下，第一旋转自由度θ的值； is the value of the first rotational degree of freedom θ under the jth integration step;
为第j积分步数下，第二旋转自由度的一阶导数值； is the second rotational degree of freedom under the jth integration step The value of the first order derivative;
为第j积分步数下，移动自由度L的二阶导数值； is the second-order derivative value of the moving degree of freedom L under the jth integration step;
为第j积分步数下，第一旋转自由度θ的二阶导数值； is the second derivative value of the first rotational degree of freedom θ at the jth integration step;
为第j积分步数下，第二旋转自由度的二阶导数值； is the second rotational degree of freedom under the jth integration step The second derivative value of ;
车辆质心极坐标系下车辆转动自由度数据包括ψj,φj、由四阶龙格库塔法对所建立的车辆动力学方程进行迭代求解得到；The vehicle rotation degree of freedom data in the polar coordinate system of the vehicle center of mass includes ψ j ,φ j , It is obtained by iteratively solving the established vehicle dynamics equations by the fourth-order Runge-Kutta method;
其中，为第j积分步数下，车辆质心极坐标系下第三旋转自由度的值；in, is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step value;
ψj为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的值；ψ j is the value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
φj为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的值；φ j is the value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第三旋转自由度的一阶导数值； is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step The value of the first order derivative;
为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的一阶导数值； is the first derivative value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的一阶导数值； is the first derivative value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第三旋转自由度的二阶导数值； is the third rotational degree of freedom in the polar coordinate system of the center of mass of the vehicle under the jth integration step The second derivative value of ;
为第j积分步数下，车辆质心极坐标系下第四旋转自由度ψ的二阶导数值； is the second derivative value of the fourth rotational degree of freedom ψ in the polar coordinate system of the center of mass of the vehicle at the jth integration step;
为第j积分步数下，车辆质心极坐标系下第五旋转自由度φ的二阶导数值。 is the second derivative value of the fifth rotational degree of freedom φ in the polar coordinate system of the center of mass of the vehicle at the jth integration step.
具体地，获得车辆在三维空间极坐标系下，车辆移动及转动自由度数据的方法具体为：Specifically, the method of obtaining vehicle movement and rotation degree of freedom data in a three-dimensional space polar coordinate system is as follows:
A1、根据当前仿真时间，确定当前积分步数j下的车辆位置旋转矩阵为；A1. According to the current simulation time, determine the vehicle position rotation matrix under the current integration step j for;
式中，为第j积分步数时，车辆的第一旋转自由度θj和第二旋转自由度变化产生的旋转矩阵变化量；In the formula, When is the jth integration step, the first rotational degree of freedom θ j and the second rotational degree of freedom of the vehicle The amount of change in the rotation matrix generated by the change;
为第j-1积分步数时在三维空间极坐标系下车辆位置的旋转矩阵； is the rotation matrix of the vehicle position in the three-dimensional space polar coordinate system at the j-1th integration step;
上述公式体现了车辆自由度数据求解过程中旋转矩阵的继承性；The above formula reflects the inheritance of the rotation matrix in the process of solving the vehicle degree of freedom data;
表示为： Expressed as:
式中， In the formula,
A2、根据三维空间极坐标系中车辆位置旋转矩阵确定当前积分步数下车辆所受合力Fj为：A2. According to the vehicle position rotation matrix in the three-dimensional space polar coordinate system Determine the resultant force F j on the vehicle under the current integration steps as:
式中，当j＝1时， In the formula, when j=1,
A3、将每一积分步数下的合力Fj代入车辆动力学方程，并通过四阶库塔求解器对车辆动力学方程进行求解，输出三维空间极坐标系下车辆移动及转动自由度数据。A3. Substitute the resultant force F j under each integration step into the vehicle dynamics equation, and solve the vehicle dynamics equation through the fourth-order Kutta solver, and output the vehicle movement and rotation degree of freedom data in the three-dimensional space polar coordinate system.
具体地，获得车辆在车辆质心极坐标系下的车辆转动自由度数据的方法具体为：Specifically, the method of obtaining the vehicle rotation degree of freedom data of the vehicle in the polar coordinate system of the vehicle center of mass is as follows:
B1、根据当前仿真时间，确定当前步长下的车辆姿态旋转矩阵 B1. According to the current simulation time, determine the vehicle attitude rotation matrix under the current step size
式中，为第j积分步数时，车辆转动产生的角度自由度变化产生的旋转矩阵变换量；In the formula, When is the j-th integration step, the transformation amount of the rotation matrix generated by the change of the angular degree of freedom caused by the vehicle rotation;
为第j-1积分步数时的车辆姿态旋转矩阵； is the vehicle attitude rotation matrix at the j-1th integration step;
上述计算公式体现了旋转矩阵的继承性；The above calculation formula reflects the inheritance of the rotation matrix;
表示为： Expressed as:
式中， In the formula,
B2、根据车辆位置姿态矩阵确定当前积分步数下车辆质心极坐标系下车辆所受合力矩Mj为；B2. According to the vehicle position attitude matrix Determine the resultant moment M j of the vehicle under the polar coordinate system of the center of mass of the vehicle under the current integration steps;
式中，为车辆质心极坐标系下车辆姿态旋转矩阵；In the formula, is the vehicle attitude rotation matrix in the polar coordinate system of the vehicle center of mass;
为车辆在笛卡尔坐标系下的受力点； is the stress point of the vehicle in the Cartesian coordinate system;
Fj为第j积分步数时车辆所受合力；F j is the resultant force on the vehicle at the jth integration step;
B3、将每一积分步数下的合力矩Mj代入车辆动力学方程，并通过四阶龙格库塔求解器求解车辆动力学方程，输出车辆质心极坐标系下的车辆转动自由度数据。B3. Substitute the resultant moment M j under each integration step into the vehicle dynamics equation, and solve the vehicle dynamics equation through the fourth-order Runge-Kutta solver, and output the vehicle rotation degree of freedom data in the polar coordinate system of the vehicle center of mass.
在上述计算三维空间极坐标系中的车辆转动及移动自由度数据和计算车辆质心极坐标系中的车辆转动自由度数据时，将第j积分步数得到的车辆所受合力Fj和受合力矩Mj，代入车辆动力学方程中，通过四阶龙格库塔求解器积分得到下一积分步数j+1下的车辆各移动和转动自由度及其一阶、二阶导数值。又重新更新j+1积分步数得到的车辆所受合力Fj+1和受合力矩Mj+1，重新进行迭代运算。最终达到第J积分步数，则停止迭代运算，输出最终的自由度数据，作为车辆动力学的仿真结果。When calculating the vehicle rotation and movement degree of freedom data in the polar coordinate system of the three-dimensional space and calculating the vehicle rotation degree of freedom data in the vehicle center-of-mass polar coordinate system, the resultant force F j and the resultant force The moment M j is substituted into the vehicle dynamics equation, and the four-order Runge-Kutta solver is integrated to obtain the vehicle's movement and rotation degrees of freedom and their first-order and second-order derivative values at the next integration step j+1. The resultant force F j+1 and resultant moment M j+1 received by the vehicle obtained by j+1 integration steps are updated again, and the iterative calculation is performed again. When the Jth integration step is finally reached, the iterative operation is stopped, and the final degree of freedom data is output as the simulation result of vehicle dynamics.
在上述计算过程中，需要说明的是，与其他任何方法不同的是，本发明中当前积分步数的旋转矩阵，不仅与当前步长下各旋转自由度变化量有关，而且与上一步旋转矩阵有关。即在上一步旋转矩阵的基础上，叠加当前各旋转自由度变化量的旋转矩阵，只要通过控制积分步数，使当前各旋转自由度变化量小于90度，即可避免旋转奇异点的问题。In the above calculation process, it should be noted that, unlike any other method, the rotation matrix of the current integration step in the present invention is not only related to the variation of each rotation degree of freedom under the current step size, but also related to the rotation matrix of the previous step related. That is, on the basis of the rotation matrix in the previous step, superimpose the rotation matrix of the current variation of each rotation degree of freedom. As long as the number of integration steps is controlled so that the current variation of each rotation degree of freedom is less than 90 degrees, the problem of rotation singularity can be avoided.
在本发明的一个实施例中，当需要输出当前积分步数j下，车辆各受力点在笛卡尔坐标系下的绝对坐标位置时，通过以下公式计算：In one embodiment of the present invention, when it is necessary to output the current integration step number j, each force point of the vehicle In the absolute coordinate position under the Cartesian coordinate system, it is calculated by the following formula:
式中，为在笛卡尔坐标系中，受力点的位置矩阵；In the formula, In the Cartesian coordinate system, the force point The position matrix;
为车辆在笛卡尔坐标系下的受力点转换到车辆质心极坐标系中时，对应的旋转矩阵； is the force point of the vehicle in the Cartesian coordinate system The corresponding rotation matrix when converting to the polar coordinate system of the vehicle center of mass;
为第j积分步下，车辆质心极坐标系中车辆姿态转置矩阵； is the vehicle attitude transposition matrix in the polar coordinate system of the vehicle center of mass at the jth integration step;
为第i个受力点的初始方向矢量，其中，为第i个受力点在三维空间极坐标系下的移动自由度值，上标T为转置运算符； is the i-th stress point The initial direction vector of , in, is the i-th stress point The value of the degree of freedom of movement in the three-dimensional space polar coordinate system, the superscript T is the transpose operator;
为第j积分步下，三维空间极坐标系中车辆位置旋转矩阵； is the vehicle position rotation matrix in the three-dimensional space polar coordinate system under the jth integration step;
Oj为第j积分步数下，车辆中心点的矢量方向矩阵，且Oj＝[Oj,0,0]，Lj为第j积分步数下，三维空间极坐标系下的移动自由度L的值；O j is the vector direction matrix of the center point of the vehicle at the jth integration step, and O j = [O j ,0,0], L j is the freedom of movement in the three-dimensional space polar coordinate system at the jth integration step the value of degree L;
其中，车辆在笛卡尔坐标系下的受力点转换到车辆质心极坐标系中时的转换公式为：Among them, the force point of the vehicle in the Cartesian coordinate system The conversion formula when converting to the polar coordinate system of the vehicle center of mass is:
式中，在车辆质心极坐标系中，为受力点沿地球半径方向的平动距离；In the formula, in the polar coordinate system of the vehicle center of mass, force point Translational distance along the radius of the earth;
为受力点绕经度方向转动的角度； force point The angle of rotation around the longitude direction;
为受力点绕纬度方向转动的角度； force point The angle of rotation around the latitude direction;
和分别为车辆质心所在的笛卡尔坐标系中，受力点在x轴、y轴和z轴上的坐标； and Respectively, in the Cartesian coordinate system where the center of mass of the vehicle is located, the force point coordinates on the x-axis, y-axis and z-axis;
车辆质心极坐标系下受力点的旋转矩阵为：The point of force under the polar coordinate system of the center of mass of the vehicle The rotation matrix of for:
在本发明的一个实施例中，提供了验证本发明中极坐标系无奇异点六自由度车辆动力学仿真方法正确性的实例：In one embodiment of the present invention, an example is provided to verify the correctness of the six-degree-of-freedom vehicle dynamics simulation method in the polar coordinate system in the present invention:
令在笛卡尔坐标系下，车辆的质量为车辆的质量m＝100kg，转动惯量Jx＝Jy＝Jz＝120kg·m2，受到两个方向的力，受力点坐标为(3m，4m，3m)，力 坐标为(0m，1m，1m)，力车辆初始位置和姿态均为0，同时约束车辆的移动自由度仿真时间为40s，采用龙格库塔积分算法分别计算车辆的运动，并与SIMPACK软件计算结果进行对比，结果如图3-图8所示。Let in the Cartesian coordinate system, the mass of the vehicle is the mass of the vehicle m = 100kg, the moment of inertia J x = J y = J z = 120kg·m 2 , the force in two directions, the force point Coordinates are (3m, 4m, 3m), force Coordinates are (0m, 1m, 1m), force The initial position and attitude of the vehicle are both 0, and the freedom of movement of the vehicle is constrained The simulation time is 40s, and the Runge-Kutta integral algorithm is used to calculate the motion of the vehicle respectively, and compared with the calculation results of SIMPACK software, the results are shown in Figure 3-Figure 8.
从图3-图8中可以看出，本发明中的无奇异点六自由度车辆动力学仿真方法能够很好地描述车辆在空间中的运动，同时能够准确地求解车辆在空间中的运动轨迹。求解的结果与在传统笛卡尔坐标系建立的车辆动力学模型完全一致。It can be seen from Fig. 3-Fig. 8 that the six-degree-of-freedom vehicle dynamics simulation method without singularity in the present invention can well describe the motion of the vehicle in space, and at the same time can accurately solve the motion trajectory of the vehicle in space . The solution results are completely consistent with the vehicle dynamics model established in the traditional Cartesian coordinate system.
但极坐标系下三个旋转自由度与笛卡尔坐标系的三个旋转自由度结果完全不同。笛卡尔坐标系下的旋转自由度(α,β,γ)的角度值均较大，存在多个自由度的旋转轴同时旋转90°的风险，引起旋转奇异问题。而在极坐标系下，由于当前积分步的旋转矩阵只与当前自由度的旋转角度变化量和上一步的旋转矩阵有关。只要满足当前自由度的旋转角度变化量小于90°，就不会存在旋转奇异问题。But the three rotational degrees of freedom in the polar coordinate system The three rotational DOF results are completely different from the Cartesian coordinate system. The rotation degrees of freedom (α, β, γ) in the Cartesian coordinate system have large angle values, and there is a risk that the rotation axes of multiple degrees of freedom rotate 90° at the same time, causing the rotation singularity problem. In the polar coordinate system, the rotation matrix of the current integration 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 variation of the rotation angle satisfying the current degree of freedom is less than 90°, there will be no rotation singularity problem.
本发明的有益效果为：The beneficial effects of the present invention are:
本发明提供的极坐标系下的车辆动力学仿真方法，针对车辆动力学系统中车辆旋转运动存在奇异点的问题，采用空间极坐标系结合牛顿定律、离心力和科氏力计算方法，给出了车辆在极坐标系下的六自由度动力学仿真方法，用于描述车辆在空间中的运动位置和姿态。利用车辆在转动过程中各瞬时转轴会产生变化、空间旋转具有继承性的原理，将传统动力学模型中的三个平动和三个转动自由度转换为一个平动五个转动自由度，能够很好地描述车辆在空间中的运动，同时能够有效解决车辆动力学系统中车体等部件空间旋转奇异点的问题。The vehicle dynamics simulation method under the polar coordinate system provided by the present invention aims at the problem that there are singular points in the vehicle rotational motion in the vehicle dynamic system, and adopts the space polar coordinate system combined with Newton's law, centrifugal force and Coriolis force calculation method, and provides The six-degree-of-freedom dynamics simulation method of the vehicle in the polar coordinate system is used to describe the moving position and attitude of the vehicle in space. Utilizing the principle that each instantaneous rotation axis will change during the rotation of the vehicle and the spatial rotation is inherited, the three translation and three rotation degrees of freedom in the traditional dynamic model are converted into one translation and five rotation degrees of freedom, which can It can well describe the movement of the vehicle in space, and at the same time, it can effectively solve the problem of the singular point of the space rotation of the vehicle body and other components in the vehicle dynamics system.
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