levelsets in workspace analysis. f(x,y,z) = s1(z). s2(x,y,z) = 0 s2: algebraic surface of degree 12

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.