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Levelsets in Workspace Analysis

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F(X,Y,Z) = S1(Z). S2(X,Y,Z) = 0 S2: algebraic surface of degree 12

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Posture change without passing through a singularity

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Singularities of parallel manipulators

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rational parametrization M. Noether

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Architecture Singularity Selfmotion 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

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Architecture Singularity Selfmotion 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

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Architecture Singularity Selfmotion 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“

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Architecture Singularity Selfmotion 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

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Architecture Singularity Selfmotion 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.

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GD-platforms

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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 ?????

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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,…..

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Discussion of the Inverse Kinematics of General 6R-Manipulators

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Constraint manifolds of 3R-chains

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H. et. al. (2007), Pfurner (2006)

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Example Puma

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Exceptional 6R-chains (overconstrained chains)

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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 case Tsai and Roth (1973) cubic polynomial McCarthy and Perez (2000) finite displacement screws

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Synthesis of Bennett mechanims intersection of two three-spaces L 1 3, L 2 3 in a seven dimensional space P 7 can be: dim(L 1 3 Å L 2 3 )= -1, ) intersection is empty, dim(L 1 3 Å L 2 3 )= 0, ) intersection is one point, dim(L 1 3 Å L 2 3 )= 1, ) intersection is a line, dim(L 1 3 Å L 2 3 )= 2, ) intersection is a two-plane dim(L 1 3 Å L 2 3 )= 3 ) L 1 3 and L 2 3 coincide. Linear 3-space

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Synthesis of Bennett mechanims 3 Conclusions: The kinematic image of the Bennett motion is the intersection of a two-plane with the Study-quadric S 6 2. Bennett motions are represented by planar sections of the Study-quadric and vice versa. Bennett linkages are the only movable 4R-chains.

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Synthesis Algorithm 1.Given three poses of a frame 1, 2, 3 compute the Study parameters ! A,B,C (three points on S 6 2 ) 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

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Synthesis Algorithm (example) Computing the torsionfree paths to obtain the axes leads to four cubic surfaces having six lines in common.

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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.

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Section 5.5 The Real Zeros of a Polynomial Function.

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