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Nine-bar structure

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Page 1: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Nine-bar structure

Page 2: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Nine-bar structure

The univariate polynomial of degree 16 of this system can be computed in general!

It can be used

1. to find the maximum number of real solutions

2. To find all exceptional mechanisms

D. Walter et.al. 2007

Page 3: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Dixon‘s mechanisms

Theorem: Dixon‘s mechanisms are the only movable 9-bar structures

Page 4: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Nine-bar structure (3-D version)

Page 5: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real
Page 6: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Methods to generate the polynomials

1. Parameter elimination (implicitation)

2. Geometric properties of the mechanism

Page 7: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Left and right translations on S62

Page 8: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Left and right translations on S62

Page 9: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Direct Kinematics of Stewart-Gough Platforms

Page 10: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Direct Kinematics of Stewart-Gough Platforms

Page 11: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Direct Kinematics of Stewart-Gough Platforms

Page 12: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Direct Kinematics of Stewart-Gough Platforms

Page 13: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Singularity

Page 14: Nine-bar structure. The univariate polynomial of degree 16 of this system can be computed in general! It can be used 1.to find the maximum number of real

Workspace and singularities of 3-R robots


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