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Levelsets in Workspace Analysis

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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

M. Noether

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

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

Gough Platform''

Blaschke‘s movable octahedron

Topic is closely related to an old theorem of Cauchy (1813):

„Every convex polyhedron is rigid“

- 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

- 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.

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)

Wampler ,………….

Veldkamp, Roth, Tsai,

McCarthy, Perez,…..

Discussion of the Inverse Kinematics of General 6R-Manipulators

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

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

Linear 3-space

- intersection of two three-spaces L13, L23 in a seven dimensional space P7 can be:
- dim(L13Å L23)= -1, ) intersection is empty,
- dim(L13Å L23)= 0, ) intersection is one point,
- dim(L13Å L23)= 1, ) intersection is a line,
- dim(L13Å L23)= 2, ) intersection is a two-plane
- dim(L13Å L23)= 3 ) L13 and L23 coincide.

3 Conclusions:

- The kinematic image of the Bennett motion is the intersection of a two-plane with the Study-quadric S62.
- Bennett motions are represented by planar sections of the Study-quadric and vice versa.
- Bennett linkages are the only movable 4R-chains.

- Given three poses of a frame 1, 2, 3 compute the Study parameters ! A,B,C (three points on S62)
- Compute the conic f(s) passing through A,B,C
- Apply inverse kinematic mapping (-1): Substitute the parametric representation of f into transformation matrix ! parametric representation M(s) of Bennett motion.
- 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

Computing the torsionfree paths to obtain the

axes leads to four cubic surfaces having six

lines in common.

- 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.