levelsets in workspace analysis. f(x,y,z) = s1(z). s2(x,y,z) = 0 s2: algebraic surface of degree 12
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Levelsets in Workspace Analysis
Levelsets in Workspace Analysis
22 rX zY
F(X,Y,Z) = S1(Z) . S2(X,Y,Z) = 0
S2: algebraic surface of degree 12
Posture change without passing through a singularity
Singularities of parallel manipulators
rational parametrization
M. Noether
Architecture SingularitySelfmotion
Definition: a parallel manipulator is called architecture singular if it is singular in every position and orientation.
Det(J) ≡ 0
H. and Karger, A., 2001
Karger, A., 1998
Architecture SingularitySelfmotion
System of equations describing the kinematical constraints has to be
redundant affine variety is no longer zero dimensional
Planar case: only one possibility parallel bar mechanism
Parallel Manipulator performs a self motion when it moves with locked actuators
Architecture SingularitySelfmotion
H. , Zsombor-Murray, P., 1994, ''A Special Type of Singular Stewart Gough Platform''
Blaschke‘s movable octahedron
Topic is closely related to an old theorem of Cauchy (1813):„Every convex polyhedron is rigid“
Architecture SingularitySelfmotion
• Self motions of 3-3-platforms are well known since more than 100 years (Bricard 1897)
• Self motions of platform mani-pulators are closely related to motions with spherical paths
• Motions with spherical paths were the topic of the 1904 Prix Vaillant of the French Academy of Science
• E. Borel and R. Bricard won the prestigeous competition and gave many examples
• All architecture singular manipulators have self motion
• Still many open questions
Architecture SingularitySelfmotion
• Bricard‘s octahedron• Griffis-Duffy platform
Griffis, M., Duffy, J., 1993, "Method and Apparatus for Controlling Geometrically Simple Parallel Mechanisms with Distinctive Connections", US Patent # 5,179, 525, 1993.
GD-platforms
6-AS-mechanisms
Inverse kinematics of concatenations of AS-joints
4 AS-mechanism: dimension of the problem 48
6 AS-mechanism: dimension of the problem ?????
Analysis and Synthesis of Serial Chains
Lee and Liang (1988)
Raghavan and Roth(1990)
Wampler ,………….
Inverse Kinematics of general 6R-chains
Synthesis of Bennett mechanism
Veldkamp, Roth, Tsai,
McCarthy, Perez,…..
Discussion of the Inverse Kinematics of General 6R-Manipulators
Constraint manifolds of 3R-chains
Constraint manifolds of 3R-chains
Constraint manifolds of 3R-chains
Constraint manifolds of 3R-chains
Constraint manifolds of 3R-chains
H. et. al. (2007), Pfurner (2006)
Example Puma
Example Puma
Example Puma
Exceptional 6R-chains (overconstrained chains)
Exceptional 6R-chains (overconstrained chains)
Exceptional 6R-chains (overconstrained chains)
Synthesis of Bennett mechanims
Previous work:
Veldkamp (1967) instantaneous case: 10 quadratic equations elimination yields univariate cubic polynomial with one real solution
Suh and Radcliffe (1978) same result for finite caseTsai and Roth (1973) cubic polynomialMcCarthy and Perez (2000) finite displacement screws
Synthesis of Bennett mechanims
intersection of two three-spaces L13, L2
3 in a seven dimensional space P7 can be:
• dim(L13Å L2
3)= -1, ) intersection is empty,• dim(L1
3Å L23)= 0, ) intersection is one point,
• dim(L13Å L2
3)= 1, ) intersection is a line,• dim(L1
3Å L23)= 2, ) intersection is a two-plane
• dim(L13Å L2
3)= 3 ) L13 and L2
3 coincide.
Linear 3-space
Synthesis of Bennett mechanims
3 Conclusions:
• The kinematic image of the Bennett motion is the intersection of a two-plane with the Study-quadric S6
2.
• Bennett motions are represented by planar sections of the Study-quadric and vice versa.
• Bennett linkages are the only movable 4R-chains.
Synthesis Algorithm
1. Given three poses of a frame 1, 2, 3 compute the Study
parameters ! A,B,C (three points on S62)
2. Compute the conic f(s) passing through A,B,C
3. Apply inverse kinematic mapping (-1): Substitute the parametric representation of f into transformation matrix ! parametric representation M(s) of Bennett motion.
4. Compute the axes of the motion
– Following Bottema-Roth (1990)
– Computing planar paths
– Using the fact that the points of four planes have rational represenations with elevated degree
Synthesis Algorithm (example)
Computing the torsionfree paths to obtain the axes leads to four cubic surfaces having six lines in common.
Conclusion
• Mechanisms can be represented with sets of algebraic equations.
• Constraints map to algebraic varieties in the image space.
• Geometric preprocessing and symbolic computation allow to solve kinematic problems
• The algebraic constraint equations are highly sparse.