Rational Trigonometry Applied to Robotics

1
Rational Trigonometry Applied to Robotics
Content
Final Year Project Rational Trigonometry
Applied to Robotics João Pequito Almeida
2
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
3
Rational Trigonometry Applied to Robotics
1.1. Definition of Rational Trigonometry
What is it?
Rational Trigonometry is the study of triangles
using quadrances and spreads.
Quadrance Distance²
Spread Sin²(separation angle)
Or
4
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).

5
Rational Trigonometry Applied to Robotics
1.3. Application examples
A simple triangle
6
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.

7
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?
8
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)
Line choice matrix
Base value matrix
Reference frame matrix
Or, more generically We can also obtain R
matrices for other representations like the Euler
model for rotation (ideal for attitude
representation).
A new coordinate system
9
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?
10
Rational Trigonometry Applied to Robotics
2.2. Approach using Rational Trigonometry
If we consider each solution Si as one unique
tuple, then the set of all solutions can be
obtained by their Cartesian Product of solutions
11
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
if
otherwise
Solutions of previous (I) and next joints (O)
Counts total of combinations of coordinates from
a to b
The final swapped point
12
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
Aa2,Bb2
Where a and b are distances
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.
13
Rational Trigonometry Applied to Robotics
2.2. Approach using Rational Trigonometry
If each Pi 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 robots structure. This exposes
clearly the structure as a matrix revealing the
nature of both the direct and the inverse
kinematics problem
14
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
15
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)
16
Rational Trigonometry Applied to Robotics
2.2. Approach using Rational Trigonometry
17
Rational Trigonometry Applied to Robotics
2.2. Approach using Rational Trigonometry
Parallel manipulator example (P3 is the midpoint)
18
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
Again, as known functions (2D, r have the
appropriate signs)
Combining successive series and parallel
manipulators this way is, as you can see, quite
simple
19
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
• Its 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/

20
Rational Trigonometry Applied to Robotics
3. Questions, demonstrations
?
?