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CHAPTER 12 CALIBRATION
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Axis of Rotation
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Rotation can also be specified as a counterclockwise (right-handed) rotation about the axis specified by the unit vector (w x , wy , wz ) This is a very intuitive way of viewing rotation, but it has the same problems as Euler angles in numerical algorithms The angle and axis representation can be converted into a rotation matrix for use in the formula for rigid body transformation (Equation 129), but it would be nice to have a scheme for working directly with the angle and axis representation that produced good numerical algorithms This is part of the motivation for the quaternion representation for rotation, discussed in the next section
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Unit Quaternions
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The quaternion is a representation for rotation that has been shown through experience to yield well-conditioned numerical solutions to orientation problems A quaternion is a four-element vector, (1217) To understand how quaternions encode rotation, consider the unit circle in the x-y plane with the implicit equation (1218) Positions on the unit circle correspond to rotation angles In three dimensions, the unit sphere is defined by the equation (1219) Positions on the unit sphere in three dimensions encode the rotation angles of wand <p about the x and y axes but cannot represent the twist K, about the z axis One more degree of freedom is required to represent all three rotation angles The unit sphere in four dimensions is defined by the implicit equation (1220)
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points on the unit sphere in four dimensions
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122 RIGID BODY TRANSFORMATIONS
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Rotation is represented by unit quaternions with
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(1221) Each unit quaternion and its antipole -q = (-qo, -ql, -q2, -q3) represent a rotation in three dimensions The rotation matrix for rigid body transformation can be obtained from the elements of the unit quaternion:
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2(qlq3 + qOq2) 2(q2q3 - qOql) q5 + q~ q~
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qr -
(1222) After the unit quaternion is computed, Equation 1222 can be used to compute the rotation matrix so that the rotation can be applied to each point using matrix multiplication The unit quaternion is closely related to the angle and axis representation for rotation, described in Section 1222 A rotation can be represented as a scalar (j for the amount of rotation and a vector (w x , wy , wz ) for the axis of rotation A quaternion has a scalar part, which is related to the amount of rotation, and a vector part, which is the axis of rotation Let the axis of rotation be represented by the unit vector (w x , W y , wz ) and use i, j, and k to represent the coordinate axes so that the unit vector for the rotation axis can be represented as (1223) The unit quaternion for a counterclockwise rotation by
about this axis is (1224) (1225)
cos 2"
(j + sm 2" ( + wyJ + W z k) Wxl qo + qx i + qyj + qzk (j
The first term is called the scalar (real) part ofthe quaternion, and the other terms are called the vector (imaginary) part A point p = (x, y, z) in space has a quaternion representation r which is the purely imaginary quaternion with vector part equal to p, r = xi + yj
+ zk
(1226)
CHAPTER 12 CALIBRATION
Let p' be point p rotated by matrix R( q),
p' = R(q)p
(1227)
If r is the quaternion representation for point p, then the quaternion representation r' for the rotated point can be computed directly from the elements of quaternion q, r' = qrq*, (1228)
where q* = (qo, -qx, -qy, -qz) is the conjugate of quaternion q and quaternion multiplication is defined as
rq =
(roqo - rxqx - ryqy - rzqz,
(1229)
+ rxqo + ryqz - rzqy, roqy - rxqz + ryqo + rzqx, roqz + rxqy - ryqx + rzqo)
roqx
Rigid body transformations can be conveniently represented using a sevenelement vector, (qO, ql, q2, q3, q4, q5, q6), in which the first four elements are a unit quaternion and the last three elements are the translation If we let R( q) denote the rotation matrix corresponding to the unit quaternion in this representation, then the rigid body transformation is
(1230)
We will use quaternions in the next section to present an algorithm for solving the absolute orientation problem
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