Download The Simplest Truly Spatial Motion

The Simplest Truly Spatial Motion
Hans-Peter Schröcker1
Bert Jüttler2
1 Unit
Geometry and CAD
Universität Innsbruck
2 Institute of Applied Geometry
Johannes Kepler University Linz
Conference on Geometry: Theory and Applications
Pilsen, June 29 2009
Overview
Dual quaternion representation
Fundamental properties
Biarc motion interpolation
Dual quaternion cross-ratios
Simplest coordinatization of the motion group
Requirements
• bilinear composition law
• minimal number of parameters
Simplest coordinatization of the motion group
Requirements
• bilinear composition law
• minimal number of parameters
Solution (Study, ≈ 1890)
• dual quaternions X + εY with X, Y ∈ H and ε2 = 0
• composition law ≈ dual quaternion multiplication
• essentially unique
(a0 , a) 7→ (εX) ? (a0 , a) ? (X + εY)−1
with kXk2 = 1,
XT Y = 0
Simplest coordinatization of the motion group
Requirements
• bilinear composition law
• minimal number of parameters
Solution (Study, ≈ 1890)
• dual quaternions X + εY with X, Y ∈ H and ε2 = 0
• composition law ≈ dual quaternion multiplication
• essentially unique
(a0 , a) 7→ (εX) ? (a0 , a) ? (X + εY)−1
with kXk2 = 1,
XT Y = 0
Euclidean displacements ⇐⇒
points (X, Y) on Study quadric S ⊂ P7 : XT Y = 0 (and X 6= 0)
The simplest motions
straight lines ⊂ S ⇐⇒
• rotations about fixed axis or
• translations in fixed direction
The simplest motions
straight lines ⊂ S ⇐⇒
conics ⊂ S
• rotations about fixed axis or
• translations in fixed direction
⇐⇒ coupler motions of Bennett linkages
Fundamental properties
Line-symmetric motion
• The Bennett motion is line-symmetric with respect to one
family of rulings on a hyperboloid (Krames 1937).
Inverse and dual motion
• The inverse motion (fixed and moving frame
interchanged) and the dual motion (transformation of
planes) are Bennett motions.
Instantaneous screws
• The instantaneous screws of a Bennett motion are always
of the same chirality (Baker, 1998).
• The tangents of a planar section of a quadric Q always lie
on the same side of Q.
Trajectories
Trajectories of points
• (a0 , a) 7→ (εX) ? (a0 , a) ? (X + εY)−1
• rational curves of degree four
Trajectories of planes
• (u0 , u) 7→ (εX) ? (u0 , u) ? (X + εY)−1
• rational torses of class four
Trajectories of lines
• (X − εY) ? (p + εq) ? ((x0 , −x) − ε(y0 , −y))
• rational ruled surfaces of degree four
Biarcs on quadrics
Biarcs on quadrics (proof)
J
• unique rotation such that
(A, a) 7→ (B, b)
B
b
a
• equal blue angles (red angles)
(for reasons of symmetry)
A
• equal angles at A and B
(by construction)
• proof remains valid in the elliptic plane
(Fuhs, Stachel 1988) =⇒ biarcs on oval quadrics
• proof remains valid (with restrictions) in the hyperbolic
plane =⇒ biarcs on ring quadrics.
• computational proofs by Wang and Barry (1997) or
Pottman and Wallner (2001)
Bennett biarc motions
Properties of Bennett biarc motions
• constant chirality of instantaneous screw
• simple collision tests (equations of degree four):
I moving point vs. fixed plane
I moving plane vs. fixed point
I moving line vs. fixed line
• possible joint poses form a Bennett motion
• simple bi-invariant choice of joint pose is possible
H.-P. Schröcker, B. Jüttler
Motion interpolation with Bennett biarcs
Proceedings of Computational Kinematics, Springer 2009
Dual quaternion cross-ratios (preliminary results)
Cross-ratio for dual quaternions
CR(A, B, C, D) = (A − B) ? (B − C)−1 ? (C − D) ? (D − A)−1
Three poses A, B, C on a Bennett motion B
• In general there exists exactly one D ∈ B such that the
cross-ratio is scalar: CR(A, B, C, D) ∈ R + εR.
• The cross-ratio is scalar for all poses D ∈ B if B is
line-symmetric with respect to an equilateral hyperboloid:
1
1
1
+ 2 − 2 = 0.
2
a
b
c
Conclusion
The simplest truly spatial motion lends itself to
• biarc motion interpolation and
• basic collision tests.
Particular features are
• constant chirality during the motion and
• low-degree trajectories of points, lines and planes.
Future research
Interpretation of dual quaternion cross-ratio:
• scalar cross-ratio
• cross-ratio with real norm