Rational Trigonometry Applied to Robotics

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Rational Trigonometry Applied to Robotics

Content


Final Year Project




Rational Trigonometry Applied to Robotics





João Pequito Almeida

Rational Trigonometry Applied to Robotics

Content

Content


1. Introduction

1.1. Definition of Rational Trigonometry

1.2. Main advantages

1.3. Application examples


2. Application to Robotics

2.1. Problems using standard approaches to kinematics modeling

2.2. Approach using Rational Trigonometry

2.2.1. Fundamental principle

2.2.2. Equations

2.2.3. Application examples

2.3. Conclusions, perspectives and future work


3. Questions, demonstrations


Rational Trigonometry Applied to Robotics

1.1. Definition of Rational Trigonometry

Rational Trigonometry
is the study of triangles using
quadrances
and spreads.

Quadrance = Distance²

Spread = Sin²(separation angle)

What is it?

Or

Rational Trigonometry Applied to Robotics

1.2. Main advantages

1.

Defines separation of lines in an unambiguous way;

2.

Avoids the use of circular functions for the study of triangles;

3.

Privileges an algebraic approach instead of a functional approach;

4.

Allows the solving of complex geometrical problems using rational expressions;

5.

Simplifies the computational implementation of geometrical problems (namely avoids
error propagation in series expansions).

Rational Trigonometry Applied to Robotics

1.3. Application examples

A simple triangle

Rational Trigonometry Applied to Robotics

2.1. Problems using other models

1.

The simplicity in obtaining the solutions depends on the choice of reference frames
(i.e. some choices make calculations easier than others);

2.

Difficulty in applying solutions obtained for one robot to another with a different
kinematics structure, i.e., the structure of a solution is in general not kept for a different
problem

3.

Difficulty in solving inverse problems


RT will be shown to provide a generic solution
to the inverse problem as a projection map between the
joint space

and task space

4.
Forward kinematics for parallel manipulators are in general complex and usually
obtained through numeric methods.

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

By using Rational Trigonometry, a difficulty immediately comes up for rotation, better
illustrated in the figure. In which of these situations is the point
P
defined only by its
quadrance
Q
and spread s?

Why not use this, then, as an advantage?

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

By separating the
sign

and the
value

at stake we can have a clearer description of the
point location and take advantage of the natural redundancy of an axis system with
reflections (example [+
x

-
y

+
z
])
:

Or, more generically


We can also obtain R matrices for other representations like the Euler model for rotation
(ideal for attitude representation).

Line choice matrix

Reference frame matrix

Base value matrix

A new coordinate system

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Then, using these new coordinate system (L,R,B) allows for a description of the point as
a set of values:


1. A line choice matrix;



2. Reference frame matrix;



3. Base value matrix.



This notation also allows the calculation of all reflected points using just one value
calculation.



Now, how to combine them to obtain a description of the motion of a set of points?

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

If we consider each solution
S
i

as one unique tuple, then
the set of all solutions can
be obtained by their
Cartesian Product of
solutions

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Hypothesis: generate matrices than can be combined uniquely, similarly to the Cartesian
product of sets where each Tuple is one of these coordinate points (a row of the P
matrix), i.e. if we take the same row of each matrix it will be a unique combination of
coordinates. For that we need to repeat and swap rows, a possible solution (using some
algebra “tricks”) is the following set of equations:

Coordinate selection matrix

Solutions of previous (
I
) and next joints (
O
)

The final swapped point

Counts total of combinations of coordinates from
a

to
b

if

otherwise

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Now, how to relate these swapped points to one another? When we add Quadrances
using standard addition, they give the Quadrance of the diagonal, not the Quadrance of
the added distance. In order to get the actual final Quadrance we need to use a special
sum operator that turns quadrances into distances and back into quadrances:

Note that these are just Quadrances converted to distances, added (thanks to the
linearity of distance) and converted back to Quadrances. This allows the common sum
operator to be used to combine solutions, as long as the square root is used. Note that
the sign of the square root is ignored, since in this framework we use extra matrices for
sign representation.

A=a
2
,B=b
2

Where a and b are distances

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

If each
P
i

has all points of joint
i
, P* (the swapped solutions) can be added (provided that
the previous rules are respected). By using swapped coordinates we can simply add
them according to the robot’s structure. This exposes clearly the structure as a matrix
revealing the nature of both the direct and the inverse kinematics problem:

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Series manipulator example

As you could see in the previous equation, this model generalizes very well for any number
of joints, so, as an example, the 2D hiper
-
redundant snake:

Reference Frame Matrices

Base Matrices

Combined Reference Frame

Combined Base Matrix

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Series manipulator example

Solution with known
L
, (this will be the case for the most common uses as
L
matrices can be
obtained directly from the data)

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

Parallel manipulator example (P3 is the midpoint)

Rational Trigonometry Applied to Robotics

2.2. Approach using Rational Trigonometry

The solution is similar to the series example but P3 can be obtained from P1 and P2, so,
assuming P1 and P2 are determined:

Combining successive series and parallel manipulators this way is, as you can see,
quite simple

Again, as known functions (2D,
r

have the appropriate signs):

Rational Trigonometry Applied to Robotics

2.3. Conclusions, Perspectives and Future Work

This model provides


A straightforward way to approach any robotic structures in a generic way;


An algebraic representation that allows a global perspective of the problems at hand
(e.g. the existence of redundant solutions).


Perspectives


It’s clear that using this framework the inversion of cinematic models might be
simplified for many cases and current work involves optimizing this inversion in a
global way;


Computational versions are easy to obtain and implement.


Future Work


Current plans include solving the inverse model in an optimal way (hopefully globally
optimal)


Implement a working toolbox for MatLab/Octave and/or a library for C/C++/Java


Developments available online at
http://web.ist.utl.pt/ist152027/content/tfc/



Rational Trigonometry Applied to Robotics

3. Questions, demonstrations