![]() ![]() A common convention for an aircraft is to define the yaw or heading angle from north New Spherical(vSphr.Clock + Math.PI / 6.0, // The angle rotated by the "B" rotationĪnother common use of rotational coordinates is to express the attitude of a vehicle, with respect toĪ reference set of axes. Spherical vSphr = new Spherical(vAexpected) This is because // here we are doing an 'alibi' rotation in spherical coordinates, // as opposed to the 'alias' rotation of the unit quaternion. Notice how the angle we're adding here is the reverse of // the angle we specified in the rotation. Now compare against the expected value of the final vector. The new // Cartesian is then expressed in the basis resulting from rotating // the basis of "v" by "qA" and then "qB".Ĭartesian vqAB = v.Rotate(qB.Multiply(qA)) The following statements are equivalent and are intended // to follow the rotation matrix order of operations. The following statements are equivalent The following two statements are equivalent to "vA" The new Cartesian will be the equivalent of "v" expressed in // the coordinate basis represented by the quaternion rotation // with respect to the coordinate basis of "v"Ĭartesian vAexpected = new Cartesian( 1, Math.Sqrt( 2), 0) UnitQuaternion qB = new UnitQuaternion( new AngleAxisRotation(-Math.PI / 6.0, UnitCartesian.UnitZ)) UnitQuaternion qA = new UnitQuaternion( new AngleAxisRotation(Math.PI / 4.0, UnitCartesian.UnitX)) Two simple unit quaternion rotations about different axes. The relationship between common rotation matrices and unit quaternions, the following example shows how quaternion operations can beįormed to be equivalent to matrix rotations. DME Component Libraries generally does not use 'alibi' rotations. Is represented in the same basis but reoriented by the given amount. This is as opposed to an 'alibi' rotation which modifies the coordinates of the vector so that it (as in DME Component Libraries) the coordinate vector is modified so that the new coordinates represent the coordinates of the same vector as Unfortunately, 'rotate' can sometimes be ambiguous since there are two senses of a rotation. The quaternions use 'alias' rotations which transform (or rotate) a coordinate vector from DME Component Libraries uses normalized (unit) quaternionsĪs a standard for specifying orientation. ![]() Often, quaternions are used to represent rotations in a compact and efficient way. Perform the rotational transformation on a vector to produce its new representation.Ĭartesian vector = new Cartesian( 2.0, 1.0, -3.0) Ĭartesian rotatedFromMatrix = vector.Rotate(rotation) ![]() Matrix3B圓 rotation = rotationTwo.Multiply(rotationOne) ![]() double angleOne = Math.PI / 2.0 ĮlementaryRotation rotationOne = new ElementaryRotation(AxisIndicator.First, angleOne) ĮlementaryRotation rotationTwo = new ElementaryRotation(AxisIndicator.Second, angleTwo) Using the relation, $$ \hat e_\rho = \frac$ standard for cylindrical coordinates.// Create a rotational transformation from two elementary rotations. Let $ (\hat i, \hat j, \hat k) $ be unit vectors in Cartesian coordinate and $ (\hat e_\rho, \hat e_\theta, \hat e_z)$ be on spherical coordinate. ![]()
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