Symbolic Computations and Dynamic Visualisation in Robotics Research and Education Title

Sandipan Bandyopadhyay,
 Indian Institute of Technology Madras, India

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

ABSTRACT

Symbolic computations and dynamic visualisations play important roles in doing research as well as teaching in the domain of robotics . In this talk, we shall see a few examples of from the domain of robotic manipulators.

A brief history of me and Mathematica!

1998: Started off as a “bypass” for doing matrix multiplications by hand!

2001: A major tool for research

2004: Developed own library of modules, got completely into it

2007 -  present:  An indispensable partner in teaching and research

Samples from teaching:  kinematics of parallel robots

Geometric formulation of forward kinematics: planar 3-RRR manipulator

Concurrence of three circles as well as three lines

Visualisations in space: spatial parallel manipulators

Geometric formulation of forward kinematics: spatial 3-RPS manipulator

Concurrence of 3 circles in space!

Applications in design: DIY tools for manipulator design!

A multi-objective design tool for a spatial parallel manipulator

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Consider a very simple expression involving a single trigonometric variable.

f = a Cos[θ]+ b Sin[θ];
g = a Sin[θ]- b Cos[θ];

h = f^2 + g^2
hSimplified = FullSimplify[h]

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Now, add a small “disturbance”!

f1 = a Cos[θ]+ b Sin[θ] + c;
g1 = a Sin[θ]- b Cos[θ];

h1 = f1^2 + g1^2
h1Simplified = FullSimplify[h1]

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Add just one more term!

f2 = a Cos[θ]+ b Sin[θ] + c;
g2 = a Sin[θ]- b Cos[θ] -c;

h2 = f2^2 + g2^2
h2Simplified = FullSimplify[h2]

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

A possible remedy: expand first

f2 = a Cos[θ]+ b Sin[θ] + c;
g2 = a Sin[θ]- b Cos[θ] - c;

h2 = f2^2 + g2^2
h2ExpandedSimplified = FullSimplify[Expand[h2]]

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Why did FullSimplify match the human notion of simplification for h1?

TreeForm /@{h1, h1Simplified}

LeafCount[TreeForm[#]]& /@{h1, h1Simplified}

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Why did it not expand h2 internally before simplification?

TreeForm /@{h2, h2ExpandedSimplified}

LeafCount[TreeForm[#]]& /@{h2, h2ExpandedSimplified}

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

There is a problem with “Expand”-ing first!

TreeForm[Expand[h2]]

LeafCount[Expand[h2]]

Tools for symbolic computations

Simplification of algebraic-trigonometric expressions

Proposed solution: divide and conquer!

Applications in research

Singularity of Stewart platform manipulators

Stewart platform manipulator

Applications in research

Derivation of the singularity surface in the closed form

A reduction the final expression size from 2.87 MB to about 45 KB
 (Badyopadhyay and Ghosal, 2006;
 https://doi.org/10.1016/j.mechmachtheory.2005.12.006)

Singularity of Stewart platform manipulators

Singularity-free sphere

(Nag and Bandyopadhyay, 2021; https://doi.org/10.1016/j.mechmachtheory.2020.104041)

Singularity of Stewart platform manipulators

Singularity-free path-planning: the ultimate goal (WIP)!

Creating a safe “corridor” by stringing together a family of singularity-free spheres

Conclusions

Robotics needs a lot of analytical computations but is fundamentally dependent on geometry

Visualisation aids understanding of complex motions and geometric relations; inspires new solution strategies

A robust and capable computation and visualisation platform like Mathematica makes lives easier, for students, teacher as well as researchers!

Thank you for your attention!

Questions? 
Comments?