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Therefore the original rigid motion f satisfies rot('''v''') = f('''v''') - '''v'''<sub>0</sub> for all '''v''' in '''R'''<sup>3</sup>, and hence |
Therefore the original rigid motion f satisfies rot('''v''') = f('''v''') - '''v'''<sub>0</sub> for all '''v''' in '''R'''<sup>3</sup>, and hence |
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:f('''v''') = rot('''v''') + '''v''' |
:f('''v''') = rot('''v''') + '''v'''<sub>0</sub> |
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for all '''v''' in '''R'''<sup>3</sup>. |
for all '''v''' in '''R'''<sup>3</sup>. |
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We may now add the translation vector '''v'''<sub>0</sub> by first adding '''v'''<sub>'''⊥'''</sub>, and next adding '''v'''<sub>'''L'''</sub>, as follows: |
We may now add the translation vector '''v'''<sub>0</sub> by first adding '''v'''<sub>'''⊥'''</sub>, and next adding '''v'''<sub>'''L'''</sub>, as follows: |
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:f('''v''') = rot('''v''') + '''v'''<sub>'''⊥'''</sub> + '''v'''<sub>'''L'''</sub>, |
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or |
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:f('''v''') = (rot('''v''') + '''v'''<sub>'''⊥'''</sub>) + '''v'''<sub>'''L'''</sub> |
:f('''v''') = (rot('''v''') + '''v'''<sub>'''⊥'''</sub>) + '''v'''<sub>'''L'''</sub> |
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Then, there is a unique point '''x''' in the plane '''P''' such that f<sub>1</sub>('''x''') = rot('''x''') + '''v'''<sub>'''⊥'''</sub> = x. Or in other words, a unique point '''x''' in '''P''' that is fixed by the mapping f<sub>1</sub>. |
Then, there is a unique point '''x''' in the plane '''P''' such that f<sub>1</sub>('''x''') = rot('''x''') + '''v'''<sub>'''⊥'''</sub> = x. Or in other words, a unique point '''x''' in '''P''' that is fixed by the mapping f<sub>1</sub>. |
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Since we want rot('''x''') + '''v'''<sub>'''⊥'''</sub> = '''x''', we may easily solve this uniquely for '''x''' by writing |
Since we want rot('''x''') + '''v'''<sub>'''⊥'''</sub> = '''x''', or in other words ('''I''' - rot)'''x''' = '''v'''<sub>'''⊥'''</sub> (where '''I''' is the identity on '''P'''), we may easily solve this uniquely for '''x''' by writing |
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:'''x''' = ('''I''' - rot)<sup>-1</sup>('''v'''<sub>'''⊥'''</sub>). |
:'''x''' = ('''I''' - rot)<sup>-1</sup>('''v'''<sub>'''⊥'''</sub>). |
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This makes sense as long as ('''I''' - rot) is invertible, or in other words as long as rot is not equal to the identity '''I''' |
This makes sense as long as ('''I''' - rot) is invertible, or in other words as long as rot is not equal to the identity '''I''', and we excluded that case above. |
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Since f<sub>1</sub>: '''P''' → '''P''' is a rigid motion of a plane with a fixed point ( f<sub>1</sub>('''x''') = '''x'''), it must be a rotation of '''P'''. |
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| ⚫ | |||
Hence the effect on '''R'''<sup>3</sup> of mapping any '''v''' to rot('''v''') + '''v'''<sub>'''⊥'''</sub> is just a rotation (by the same angle as rot) about the line '''L'''<sub>1</sub> that is parallel to '''L''', and passes through the point '''x'''. |
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Recall that he original rigid motion f on '''R'''<sup>3</sup> is given by f('''v''') = (rot('''v''') + '''v'''<sub>'''⊥'''</sub>. |
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| ⚫ | |||
Conclusion: every rigid motion of '''R'''<sup>3</sup> is the result of a rotation of '''R'''<sup>3</sup> followed by a translation along the axis of the rotation — which is the definition of a screw motion. |
Conclusion: every rigid motion of '''R'''<sup>3</sup> is the result of a rotation of '''R'''<sup>3</sup> followed by a translation along the axis of the rotation — which is the definition of a screw motion. |
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Revision as of 18:30, 17 September 2014
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis.[1][2]
Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position. The special case when the first vector is zero is interpreted as a pure translation in the direction of the second vector. A screw axis is associated with each pair of vectors in the algebra of screws, also known as screw theory.[3]
The spatial movement of a body can be represented by a continuous set of displacements. Because each of these displacements has a screw axis, the movement has an associated ruled surface known as a screw surface. This surface is not the same as the axode, which is traced by the instantaneous screw axes of the movement of a body. The instantaneous screw axis, or 'instantaneous helical axis' (IHA), is the axis of the helicoidal field generated by the velocities of every point in a moving body.
When a spatial displacement specializes to a planar displacement, the screw axis becomes the displacement pole, and the instantaneous screw axis becomes the velocity pole, or instantaneous center of rotation, also called an instant center. The term centro is also used for a velocity pole, and the locus of these points for a planar movement is called a centrode.[4]
History
The proof that a spatial displacement can be decomposed into a rotation and slide around and along a line in space is attributed to Michel Chasles in 1830.[5] Recently the work of Gulio Mozzi has been identified as presenting a similar result in 1763.[6][7]
Screw axis symmetry
A screw displacement (also screw operation) is the composition of a rotation by an angle φ about an axis (called the screw axis) with a translation by a distance d along this axis. A positive rotation direction usually means one that corresponds to the translation direction by the right-hand rule. Except for φ = 180°, we have to distinguish a screw displacement from its mirror image. Unlike for rotations, a righthand and lefthand screw operation generate different groups.
The combination of a rotation about an axis and a translation in a perpendicular direction is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation. a pure rotation, and the identity. Together these are all the direct isometries in 3D.
In crystallography, a screw axis symmetry is combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged. If φ = 360°/n for some positive integer n, then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw displacement.
Applicable for space groups is a rotation by 360°/n about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n. This multiple is indicated by a subscript. So, 63 is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.[8]
A non-discrete screw axis isometry group contains all combinations of a rotation about some axis and a proportional translation along the axis (in rifling, the constant of proportionality is called the twist rate); in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.
Screw axis of a spatial displacement
Geometric argument
Let f: R3 →R3 denote a rigid motion of R3, and let v denote an arbitrary vector of R3.
Now let v0 denote the vector f(0). We may now define the rigid motion rot: R3 →R3 via rot(v) := f(v) - v0 for all v in R3. This guarantees that rot(0) = 0.
Since rot(0) = 0 (and being a motion of R3, rot preserves orientation), rot must be a rotation.
If rot is equal to the identity I, then in this very simple case, the screw motion is merely translation by the original vector v0 and does not involve a rotation. From here on we assume rot is not the identity I.
We know that each rotation of R3 must have an axis (a bi-infinite straight line) that is pointwise fixed by the rotation. Call that line L.
Therefore the original rigid motion f satisfies rot(v) = f(v) - v0 for all v in R3, and hence
- f(v) = rot(v) + v0
for all v in R3.
This means that f is the result of applying a rotation rot, followed by a translation by v0.
Finally, we may uniquely resolve the vector v0 into a sum of two vectors, one parallel to the line L, and the other perpendicular to L, as follows:
- v0 = vL + v⊥.
We may now add the translation vector v0 by first adding v⊥, and next adding vL, as follows:
- f(v) = (rot(v) + v⊥) + vL
Now, the first portion (rot(v) + v⊥) of f(v) may be thought of a taking place purely within any given plane P perpendicular to L, where v⊥ is a vector in the plane P. In fact, define
- f1(v) := rot(v) + v⊥, where both the rotation rot and the vector v⊥ are thought of as lying in the plane P.
Then, there is a unique point x in the plane P such that f1(x) = rot(x) + v⊥ = x. Or in other words, a unique point x in P that is fixed by the mapping f1.
Since we want rot(x) + v⊥ = x, or in other words (I - rot)x = v⊥ (where I is the identity on P), we may easily solve this uniquely for x by writing
- x = (I - rot)-1(v⊥).
This makes sense as long as (I - rot) is invertible, or in other words as long as rot is not equal to the identity I, and we excluded that case above.
Since f1: P → P is a rigid motion of a plane with a fixed point ( f1(x) = x), it must be a rotation of P.
Hence the effect on R3 of mapping any v to rot(v) + v⊥ is just a rotation (by the same angle as rot) about the line L1 that is parallel to L, and passes through the point x.
Recall that he original rigid motion f on R3 is given by f(v) = (rot(v) + v⊥.
This implies that the original rigid motion f is the same as first rotating R3 about the line L1, and following this by translation by the vector vL (which is in the same direction as L1).
Conclusion: every rigid motion of R3 is the result of a rotation of R3 followed by a translation along the axis of the rotation — which is the definition of a screw motion.
Matrix argument
A spatial displacement, T=([A],d), is the Euclidean transformation consisting of the spatial rotation defined by the matrix [A] and spatial translation by the vector d. Coordinates x in a moving frame have coordinates X in the fixed frame given by:
![{\displaystyle \mathbf {X} =[A]\mathbf {x} +\mathbf {d} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbfd05e68a1d9a45275ed3876c63a17e24eb374c)
The screw axis of this displacement is the line through a point C in the direction S,
that has the property that it is remains in place before and after the spatial displacement. The vector S is the unit vector in the direction of the rotation axis of [A]. Points on this line translate along the line in the direction S by the amount k=(d·S).
A point on the screw axis satisfies the equation:
![{\displaystyle \mathbf {C} =[A]\mathbf {C} +\mathbf {d} -(\mathbf {d} \cdot \mathbf {S} )\mathbf {S} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca3a6df28826548d205bd518c3b8a7708b1b289)
The term (d·S)S subtracts the slide along the screw axis in order to ensure that C exists.
Solve this equation for C using Cayley's formula for a rotation matrix
![{\displaystyle [A]=[I-B]^{-1}[I+B],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1fc95267a00ede6614b99b00fc650a081ee604)
where [B] is the skew-symmetric matrix constructed from Rodrigues' vector
such that
![{\displaystyle [B]\mathbf {y} =\mathbf {b} \times \mathbf {y} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c59017f8f8d1d59331b37d816b249816d8b115ab)
The result is that the point C on the screw axis P(t) is given by,
The screw axis P(t)=C+tS of this spatial displacement has the Plücker coordinates S=(S, C×S).[9]
Dual quaternion
The screw axis appears in the dual quaternion formulation of a spatial displacement D = ([A],d). The dual quaternion is constructed from the dual vector S=(S, V) defining the screw axis and the dual angle (φ, d) where φ is the rotation about and d the slide along this axis, which defines the displacement D to obtain,
A spatial displacement of points q represented as a vector quaternion can be defined using quaternions as the mapping
where d is translation vector quaternion and S is a unit quaternion, also called a versor, given by,
that defines a rotation by 2θ around an axis s.
In the proper Euclidean group E+(3) a rotation may be conjugated with a translation to move it to a parallel rotation axis. Such a conjugation, using quaternion homographies, produces the appropriate screw axis to express the given spatial displacement as a screw displacement, in accord with Chasles’ theorem.
Mechanics
The motion of a rigid body may be the combination of rotation about an axis (the screw axis) and a translation along that axis. This screw move is characterized by the velocity vector for the translation and the angular velocity vector in the same or opposite direction. If these two vectors are constant and along one of the principal axes of the body, no external forces are needed for this motion (moving and spinning). As an example, if gravity and drag are ignored, this is the motion of a bullet fired from a rifled gun.
Biomechanics
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.
See also
- Helical symmetry
- Euler's rotation theorem – rotations without translation
- Screw theory
- Line group
- Space group
- Corkscrew (roller coaster element)
- Glide reflection
References
- ^ Bottema, O, and B. Roth, Theoretical Kinematics, Dover Publications (September 1990), link to Google books
- ^ Hunt, K. H., Kinematic Geometry of Mechanism, Oxford University Press, 1990
- ^ R.S. Ball, A Treatise on the Theory of Screws, Hodges, Dublin, 1876, Appendix 1, University Press, Cambridge, 1900, p. 510
- ^ Homer D. Eckhardt, Kinematic Design of Machines and Mechanisms, McGraw-Hill (1998) p. 63 ISBN 0-07-018953-6 on-line at Google books
- ^ M. Chasles, Note sur les Proprietes Generales du Systeme de Deux Corps Semblables entr'eux, Bullettin de Sciences Mathematiques, Astronomiques Physiques et Chimiques, Baron de Ferussac, Paris, 1830, pp. 321±326
- ^ G. Mozzi, Discorso matematico sopra il rotamento momentaneo dei corpi, Stamperia di Donato Campo, Napoli, 1763
- ^ M. Ceccarelli, Screw axis defined by Giulio Mozzi in 1763 and early studies on helicoidal motion, Mechanism and Machine Theory 35 (2000) 761-770
- ^ Walter Borchardt-Ott (1995). Crystallography. Springer-Verlag. ISBN 3-540-59478-7.
- ^ J. M. McCarthy and G. S. Soh, Geometric Design of Linkages, 2nd Edition, Springer 2010
- ^ Woltring HJ, de Lange A, Kauer JMG, Huiskes R. 1987 Instantaneous helical axes estimation via natural, cross-validated splines. In: Bergmann G, Kölbel R, Rohlmann A (Editors). Biomechanics: Basic and Applied Research. Springer, pp 121-128. full text