有效四元数角速度

问题描述

我有一个用四元数表示的方向和一个用四元数或数字（围绕原始方向的弧度/秒）表示的角速度。我知道如何使用转换为轴角的方法来做到这一点，但是该方法在计算上相当昂贵，而且不是现实的选择。给定时间间隔（以秒为单位），我将如何修改方向四元数？我需要两种情况（四元数和数字）的解决方案。但是，将一种情况转换为另一种情况是可以接受的，并且可能更可取，这取决于转换所需的各种算法/公式的计算复杂性。

解决方案

要更新方向，您需要将当前方向乘以增量旋转。与轴角度转换相比，这是可比的昂贵操作。

表示角速度的常用方法是与旋转轴和旋转速度（弧度/秒）平行的3d向量中的指数图。转换为增量旋转四元数看起来像是

 四元数deltaRotation（const Vector3& em，double deltaTime）
 {
四元数q; 
 Vector3 ha = em * deltaTime * 0.5; //半角矢量
 double l = ha.norm（）; //幅度
 if（l> 0）
 {
 double ss = sin（l）/ l; 
 q =四元数（cos（l），ha.x（）* ss，ha.y（）* ss，ha.z（）* ss）; 
} else {
 q =四元数（1.0，ha.x（），ha.y（），ha.z（））; 
} 
 
返回q； 
} 
  

如果您的deltaTime小而旋转速度小，则可以使用第一泰勒级数乘数。但是您应该对结果四元数进行规范化处理，以避免更多的数值不稳定。 $ b {
Vector3 ha = em * deltaTime * 0.5; //半角矢量
return Quaternion（1.0，ha.x（），ha.y（），ha.z（））;
}

I have an orientation expressed with a quaternion and an angular velocity expressed as either a quaternion or a number (radians per second around the original orientation). I understand how to do this using conversion to axis-angle but that method is rather computationally expensive and is not a realistic option. How would I go about modifying the orientation quaternion given a time interval (in seconds)? I need a solution for both cases (the quaternion and the number). However, converting one case into the other is acceptable and may be preferable depending on the computational complexity of the various algorithms/formulae required for conversions.

解决方案

For update of orientation , you require to multiply current orientation by delta rotation. This is comparable expensive operation with axis angle conversion.

Common way to represent angular velocity is "exponential map"m the 3d vector parallel with rotation axis and magnitude of rotation velocity (radians per second). The conversion to delta rotation quaternion looks like

Quaternion deltaRotation(const Vector3& em, double deltaTime)
{
   Quaternion q;
   Vector3 ha = em * deltaTime *0.5; // vector of half angle
   double l = ha.norm(); // magnitude
   if( l > 0 )
   {
      double ss = sin(l)/l;
      q = Quaternion(cos(l), ha.x()*ss, ha.y()*ss, ha.z()*ss);
   }else{
      q = Quaternion(1.0, ha.x(), ha.y(), ha.z());
   }

   return q;
}

If your deltaTime is small and rotation speed is small , you can use approximation by 1st Taylor series multiplier. But you should normalize result quaternion to avoid numerical instability more often.

Quaternion deltaRotationAppx1(const Vector3& em, double deltaTime)
    {
       Vector3 ha = em * deltaTime *0.5; // vector of half angle
       return Quaternion(1.0, ha.x(), ha.y(), ha.z());
    }

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