在c ++中，如何根据最新一个的四元数旋转在三维空间中附着到另一个点 [英] In c++, how to traslate a point attached to another in 3D space according to the latest one's quaternion rotations

问题描述

p1和p2，p2连接到p1，不仅连接到p1的位置，而且连接到它的旋转，因此q1是代表p1旋转的四元数。
如果q1旋转，则p1的位置也必须围绕p1旋转。
我只需要计算p2的位置，而不是旋转，我已经完成了旋转。
所以基本上是一个飞船停靠到一个车站，我需要移动和旋转站与周围的船停靠到它。
我如何做？

我写的代码工作，只要车站在对接期间不旋转​​：

  bool docked [100]; 
 Quaternion quatTarget [100]; 
 double distance_dock [100]; 
 
 vector3 docking_position（int ship，int station）
 {
 if（！docked [ship]）
 {
 docked [ship] = true; 
 distance_dock [ship] = distance（position [ship]，position [station]）; 
 vector3 direcc = normalized（position [station]  -  position [ship]）; 
 quatTarget [ship] = vecToVecRotation（direcc，{0，0，1}）; 
 QuaternionNormalize（& quatTarget [ship]，& quatTarget [ship]）; 
} 
 
四元方位= total_rotation [station] * quatTarget [ship]; 
 
 Matrix docking_place; 
 
 MatrixRotationQuaternion（& docking_place，& orientation）; 
 
 vector3 axis_z = {docking_place（0，2），docking_place（1,2），docking_place（2,2）}; 
 
 return position [station] + -axis_z * distance_dock [ship]; 
} 
  

我在这里做的是从船到车站采取四元数停靠时间，然后沿着定向的负z轴翻转船distance_dock单位，因此船舶将总是相应地移动，但是如果我在船只已经旋转的情况下停泊船舶，则我得到初始停靠位置

解决方案

如果我正确理解你，你有两个对象，有一个它们之间的刚性变换。问题是，你想计算一个姿势（位置+方向），给定另一个的姿势。

假设您有三个框架;车架S，车架V和全局框架G（我假设您的图形环境具有全局3D笛卡尔坐标系）。
帧S和V之间的变换是完全已知的（平移和定向）和常数，并且表示为S_p_SV（车站位置，以Station帧表示）和SV_q（四元数方向车辆，在车站框架中表示）。

如果你没有刚体力学的经验，这将是令人困惑的，在这种情况下，你应该阅读一些介绍性的笔记/

我已在LATEX中编写了表达式，但不幸的是StackOverflow不支持它，所以我附上它作为一个图像。可以在此处找到原始的LATEX。
在下面的符号中，例如 Sp_SV 的第一行是 V 车辆w.r.t.的 p 位置。 S （旋转）框架中表示的 S 。
前缀上标表示旋转框架。例如，对于四元数 G_Sq ，这表示 S 圆形框架中 S 框架的方向。

在C ++中实现这一点，我不确定你使用的是什么库，但是你需要以下函数：

• 将欧拉转换为四元数 - 如果您要手动指定旋转SV q （Vehicle wrt Station的旋转）


• 将四元数转换为DCM - 适用于LATEX中的第一种方法


• 四元数乘法 - 适用于LATEX中的第二种方法

< Conjugate - 对于LATEX

中的第二种方法

I have to possitions, p1 and p2, p2 is attached to p1, not only to p1's position but also to it's rotation, so q1 is a quaternion which represents p1's rotation. If q1 rotates, then p1's position must also rotate around p1 accordingly. I only need to calculate p2's position, not it's rotation, I worked the rotation out already. So basically is a spaceship docked to a station, I need to move and rotate the station around with the ship docked to it. How do I do it?

the code i wrote for it works as long as the station is not rotated during the time of docking:

bool docked[100];
Quaternion quatTarget[100];
double distance_dock[100];

vector3 docking_position(int ship, int station)
{
    if (!docked[ship])
    {
        docked[ship] = true;
        distance_dock[ship] = distances(position[ship], position[station]);
        vector3 direcc = normalized(position[station] - position[ship]);
        quatTarget[ship] = vecToVecRotation(direcc, { 0, 0, 1 });
        QuaternionNormalize(&quatTarget[ship], &quatTarget[ship]);
    }

    Quaternion orientation = total_rotation[station] * quatTarget[ship];

    Matrix docking_place;

    MatrixRotationQuaternion(&docking_place, &orientation);

    vector3 axis_z = { docking_place(0, 2), docking_place(1, 2), docking_place(2, 2) };

    return position[station] + -axis_z * distance_dock[ship];
}

What I do here is take an orientation quaternion from the ship to the station at the time of docking and then traslate the ship "distance_dock" units along the negative z axis of the orientation, so the ship will always move accordingly, but somehow if I dock the ship when the station is already rotated then I get the initial docking position wrong, though it still rotates perfectly along with the station.

解决方案

If I understand you correctly, you have two objects that have a rigid transformation between them. The problem is that you want to calculate the pose (position + orientation) of one, given the pose of the other.

Let's say you have three frames; the Station frame "S", the Vehicle frame "V" and the Global frame "G" (I assume your graphics environment has a global 3D Cartesian frame). The transformation between frames S and V is fully known (translation and orientation) and constant, and is denoted S_p_SV (the position of the Vehicle w.r.t the Station, expressed in the Station frame) and SV_q (the quaternion orientation of the Vehicle, expressed in the Station frame).

This will be confusing if you have not had experience in rigid-body mechanics, in which case you should read some introductory notes/slideshows on "Rigid-Body Mechanics" which are plentiful on Google results.

I have written the expression in LATEX but unfortunately StackOverflow does not support it, so I have attached it as an image. The original LATEX can be found here. In my notation below, for example on the first line Sp_SV , is the position of the Vehicle w.r.t. the Station, expressed in the Station frame (of rotation). The prefixed superscript indicates the rotation frame. For the quaternion G_Sq for example, this represents the orientation of the Station frame from the Ground Frame.

In terms of implementing this in C++, I am unsure of what library you are using for Quaternions, but you will need the following functions:

• Convert Euler to Quaternions - If you are going to manually specify the rotation SVq (rotation of Vehicle w.r.t Station)
• Convert Quaternion to DCM - For the first method in the LATEX
• Quaternion Multiply - For the second method in the LATEX
• Quaternion Conjugate - For the second method in the LATEX

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