四元数转欧拉角算法-如何转换为“Y = Up”并在左右手之间转换？

Question

四元数转欧拉角算法-如何转换为“Y = Up”并在左右手之间转换？

9

我有一个算法可以在四元数和欧拉角之间进行转换。

    public static Vector3 ToEulerAngles(this Quaternion q)
    {
        // Store the Euler angles in radians
        Vector3 pitchYawRoll = new Vector3();

        double sqw = q.W * q.W;
        double sqx = q.X * q.X;
        double sqy = q.Y * q.Y;
        double sqz = q.Z * q.Z;

        // If quaternion is normalised the unit is one, otherwise it is the correction factor
        double unit = sqx + sqy + sqz + sqw;
        double test = q.X * q.Y + q.Z * q.W;

        if (test > 0.4999f * unit)                              // 0.4999f OR 0.5f - EPSILON
        {
            // Singularity at north pole
            pitchYawRoll.Y = 2f * (float)Math.Atan2(q.X, q.W);  // Yaw
            pitchYawRoll.X = PI * 0.5f;                         // Pitch
            pitchYawRoll.Z = 0f;                                // Roll
            return pitchYawRoll;
        }
        else if (test < -0.4999f * unit)                        // -0.4999f OR -0.5f + EPSILON
        {
            // Singularity at south pole
            pitchYawRoll.Y = -2f * (float)Math.Atan2(q.X, q.W); // Yaw
            pitchYawRoll.X = -PI * 0.5f;                        // Pitch
            pitchYawRoll.Z = 0f;                                // Roll
            return pitchYawRoll;
        }
        else
        {
            pitchYawRoll.Y = (float)Math.Atan2(2f * q.Y * q.W - 2f * q.X * q.Z, sqx - sqy - sqz + sqw);       // Yaw
            pitchYawRoll.X = (float)Math.Asin(2f * test / unit);                                             // Pitch
            pitchYawRoll.Z = (float)Math.Atan2(2f * q.X * q.W - 2f * q.Y * q.Z, -sqx + sqy - sqz + sqw);      // Roll
        }

        return pitchYawRoll;
    }

这种方法仅适用于右手笛卡尔坐标系，其中Z轴指向上方。

如果我想让Y轴指向上方而不是Z轴，我需要做哪些更改？（交换X和Z是否可行？）

如何适应左手坐标系？

编辑：

public static Quaternion CreateFromYawPitchRoll(float yaw, float pitch, float roll)
{
float num = roll * 0.5f;
float num2 = (float)Math.Sin((double)num);
float num3 = (float)Math.Cos((double)num);
float num4 = pitch * 0.5f;
float num5 = (float)Math.Sin((double)num4);
float num6 = (float)Math.Cos((double)num4);
float num7 = yaw * 0.5f;
float num8 = (float)Math.Sin((double)num7);
float num9 = (float)Math.Cos((double)num7);
Quaternion result;
result.X = num9 * num5 * num3 + num8 * num6 * num2;
result.Y = num8 * num6 * num3 - num9 * num5 * num2;
result.Z = num9 * num6 * num2 - num8 * num5 * num3;
result.W = num9 * num6 * num3 + num8 * num5 * num2;
return result;
}

- user1423893

四元数和欧拉角与坐标系的对齐和方向无关。偏航、俯仰和翻滚分别定义了绕z、y和x轴的旋转。轴的方向如何定位并不重要。 - Nico Schertler

如果我将其与XNA的Quaternion.CreateFromYawPitchRoll结合使用，那么我将无法获得原始的四元数。http://msdn.microsoft.com/en-us/library/microsoft.xna.framework.quaternion.createfromyawpitchroll.aspx - user1423893

那么这些函数可能使用不同的偏航/俯仰/翻滚与轴之间的关联。交换欧拉角，使定义匹配。 - Nico Schertler

请您能否提供一个基于我所做的修改的示例，其中现在包括Quaternion.CreateFromYawPitchRoll？ - user1423893

1个回答

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可以查看英文原文，
原文链接

- Nico Schertler · Accepted Answer

这里是使用相同的yaw、pitch、roll定义的已更改方法：

public static Quaternion CreateFromYawPitchRoll(float yaw, float pitch, float roll)
{
    float rollOver2 = roll * 0.5f;
    float sinRollOver2 = (float)Math.Sin((double)rollOver2);
    float cosRollOver2 = (float)Math.Cos((double)rollOver2);
    float pitchOver2 = pitch * 0.5f;
    float sinPitchOver2 = (float)Math.Sin((double)pitchOver2);
    float cosPitchOver2 = (float)Math.Cos((double)pitchOver2);
    float yawOver2 = yaw * 0.5f;
    float sinYawOver2 = (float)Math.Sin((double)yawOver2);
    float cosYawOver2 = (float)Math.Cos((double)yawOver2);
    Quaternion result;
    result.X = cosYawOver2 * cosPitchOver2 * cosRollOver2 + sinYawOver2 * sinPitchOver2 * sinRollOver2;
    result.Y = cosYawOver2 * cosPitchOver2 * sinRollOver2 - sinYawOver2 * sinPitchOver2 * cosRollOver2;
    result.Z = cosYawOver2 * sinPitchOver2 * cosRollOver2 + sinYawOver2 * cosPitchOver2 * sinRollOver2;
    result.W = sinYawOver2 * cosPitchOver2 * cosRollOver2 - cosYawOver2 * sinPitchOver2 * sinRollOver2;
    return result;
}

对于 ToEulerAngles（省略奇点）：

pitchYawRoll.Y = (float)Math.Atan2(2f * q.X * q.W + 2f * q.Y * q.Z, 1 - 2f * (sqz  + sqw));     // Yaw 
pitchYawRoll.X = (float)Math.Asin(2f * ( q.X * q.Z - q.W * q.Y ) );                             // Pitch 
pitchYawRoll.Z = (float)Math.Atan2(2f * q.X * q.Y + 2f * q.Z * q.W, 1 - 2f * (sqy + sqz));      // Roll

我进行了以下测试：

var q = CreateFromYawPitchRoll(0.2f, 0.3f, 0.7f);
var e = ToEulerAngles(q);
var q2 = CreateFromYawPitchRoll(e.Y, e.X, e.Z);

得到以下结果：

e = (0.3, 0.2, 0.7) //pitch, yaw, roll
q2 = q

来源：维基百科