Levelsets in Workspace Analysis. Levelsets in Workspace Analysis. 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.
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Levelsets in Workspace Analysis
F(X,Y,Z) = S1(Z) . S2(X,Y,Z) = 0
S2: algebraic surface of degree 12
Definition: a parallel manipulator is called architecture singular if it is singular in every position and orientation.
H. and Karger, A., 2001
Karger, A., 1998
Parallel Manipulator performs a self motion when it moves with locked actuators
System of equations describing the kinematical constraints has to be redundantaffine variety is no longer zero dimensional
Planar case: only one possibility parallel bar mechanism
H. , Zsombor-Murray, P., 1994, ''A Special Type of Singular Stewart
Blaschke‘s movable octahedron
Topic is closely related to an old theorem of Cauchy (1813):
„Every convex polyhedron is rigid“
Griffis, M., Duffy, J., 1993, "Method and Apparatus for Controlling Geometrically Simple Parallel Mechanisms with Distinctive Connections", US Patent # 5,179, 525, 1993.
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
Inverse Kinematics of general 6R-chains
Synthesis of Bennett mechanism
Lee and Liang (1988)
Raghavan and Roth(1990)
Veldkamp, Roth, Tsai,
Discussion of the Inverse Kinematics of General 6R-Manipulators
H. et. al. (2007), Pfurner (2006)
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
Computing the torsionfree paths to obtain the
axes leads to four cubic surfaces having six
lines in common.