Screw axis

Let D : R³ → R³ be an orientation-preserving rigid motion of R³. The set of these transformations is a subgroup of Euclidean motions known as the special Euclidean group SE(3). These rigid motions are defined by transformations of x in R³ given by

${\displaystyle D(\mathbf {x} )=A(\mathbf {x} )+\mathbf {d} }$

consisting of a three-dimensional rotation A followed by a translation by the vector d.

A three-dimensional rotation A has a unique axis that defines a line L. Let the unit vector along this line be S so that the translation vector d can be resolved into a sum of two vectors, one parallel and one perpendicular to the axis L, that is,

${\displaystyle \mathbf {d} =\mathbf {d} _{L}+\mathbf {d} _{\perp },\quad \mathbf {d} _{L}=(\mathbf {d} \cdot \mathbf {S} )\mathbf {S} ,\quad \mathbf {d} _{\perp }=\mathbf {d} -\mathbf {d} _{L}.}$

In this case, the rigid motion takes the form

${\displaystyle D(\mathbf {x} )=(A(\mathbf {x} )+\mathbf {d} _{\perp })+\mathbf {d} _{L}.}$

Now, the orientation preserving rigid motion D* = A(x) + d_⊥ transforms all the points of R³ so that they remain in planes perpendicular to L. For a rigid motion of this type there is a unique point c in the plane P perpendicular to L through 0, such that

${\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}$

The point C can be calculated as

${\displaystyle \mathbf {C} =[I-A]^{-1}\mathbf {d} _{\perp },}$

because d_⊥ does not have a component in the direction of the axis of A.

A rigid motion D* with a fixed point must be a rotation of around the axis L_c through the point c. Therefore, the rigid motion

${\displaystyle D(\mathbf {x} )=D^{*}(\mathbf {x} )+\mathbf {d} _{L},}$

consists of a rotation about the line L_c followed by a translation by the vector d_L in the direction of the line L_c.

Conclusion: every rigid motion of R³ is the result of a rotation of R³ about a line L_c followed by a translation in the direction of the line. The combination of a rotation about a line and translation along the line is called a screw motion.

A point C on the screw axis satisfies the equation:[9]

${\displaystyle D^{*}(\mathbf {C} )=A(\mathbf {C} )+\mathbf {d} _{\perp }=\mathbf {C} .}$

Solve this equation for C using Cayley's formula for a rotation matrix

${\displaystyle [A]=[I-B]^{-1}[I+B],}$

where [B] is the skew-symmetric matrix constructed from Rodrigues' vector

${\displaystyle \mathbf {b} =\tan {\frac {\phi }{2}}\mathbf {S} ,}$

such that

${\displaystyle [B]\mathbf {y} =\mathbf {b} \times \mathbf {y} .}$

Use this form of the rotation A to obtain

${\displaystyle \mathbf {C} =[I-B]^{-1}[I+B]\mathbf {C} +\mathbf {d} _{\perp },\quad [I-B]\mathbf {C} =[I+B]\mathbf {C} +[I-B]\mathbf {d} _{\perp },}$

which becomes

${\displaystyle -2[B]\mathbf {C} =[I-B]\mathbf {d} _{\perp }.}$

This equation can be solved for C on the screw axis P(t) to obtain,

${\displaystyle \mathbf {C} ={\frac {\mathbf {b} \times \mathbf {d} -\mathbf {b} \times (\mathbf {b} \times \mathbf {d} )}{2\mathbf {b} \cdot \mathbf {b} }}.}$

The screw axis P(t) = C + tS of this spatial displacement has the Plücker coordinates S = (S, C × S).[9]

This parameter is often used in biomechanics, when describing the motion of joints of the body. For any period of time, joint motion can be seen as the movement of a single point on one articulating surface with respect to the adjacent surface (usually distal with respect to proximal). The total translation and rotations along the path of motion can be defined as the time integrals of the instantaneous translation and rotation velocities at the IHA for a given reference time.[10]

In any single plane, the path formed by the locations of the moving instantaneous axis of rotation (IAR) is known as the 'centroid', and is used in the description of joint motion.