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Mathematical model of
fencer’s hand
Dainis Dzenushko
Student, SPb State Polytechnical University, Faculty for
Physics and Mechanics, Department for Theoretical
Mechanics
Task
Create mathematical model, that finds the position of a hand and
comes to this position from any previous position with
visualization on C# language, using given parameters (changes of
angles in joints and coordinates of a shoulder)
Degrees of freedom
This model has 9 degrees of freedom:
2 — translation movement of a shoulder (front/back; up/down)
3 — shoulder joint as a ball joint
1 — elbow joint
1 — rotation of wrist round it’s axis
2 — wrist (2 perpendicular cylindrical joints)
Hand rotation
Rotation of hand parts is realized by the rotation of corresponding vector round
the corresponding axis. This is realized by scalar multiplication of rotation tensor
and this vector. The rotation tensor which is calculated using following formula:
P (пЃЄ , e ) пЂЅ e e пЂ« ( E пЂ­ e e ) cos( пЃЄ ) пЂ« e п‚ґ E sin( пЃЄ )
~
~
~
~
a пЂЅ P (пЃё , k ) пѓ— P (пЃ№ , j ) пѓ— P (пЃЄ , i ) пѓ— a
After that it is needed to rotate other vectors and corresponding basis’ that are
farther from shoulder than considered one. It is also done by scalar multiplication
with the same rotation tensor. By the same way the rotation of all 3 vectors on
needed angles is done. Then we calculate new coordinates of joints.
Realization of rotation
For visualization of rotation we have to know the positions of hand between
initial and next positions. So the problem is that the rotation tensors are not
additive. Then we don’t rotate hand on additional angles from each new
position, instead of this we rotate hand on bigger angles on every step from
initial position.
Realization in MATLAB
Using the MATLAB this algorithm of rotation was realized in order to
calculate path of hand movement and check this algorithm. The
examples of such paths are shown below.
Structure of program interface in C#
The program interface is based on tabs (TabControl). And consists of two layers:
outer (Graph, Tools, Settings, About) and sub-layer for Settings tab:
(Color Settings, Graph Settings)
Then we will consider on all this tabs.
Graph tab
This tab contains visualization of ongoing position of a hand and has
functional: camera rotation and Zoom.
In this tab works following keys:
Up/Down – rotation round the horizontal axis
Left/Right – rotation round the vertical axis
Ctrl + Up/Down/Left/Right – faster rotations
+/- (OemPlus/OemMinus) – is Zoom
Settings tab
This tab is responsible for setting displaying parameters of hand model. It
contains two sub-tabs.
The first one (Color Settings) allows to set color and transparency of model.
Second one (Graph Settings) allows to set quality of displayed model. Also it
allows to set coefficients of camera rotation and camera zoom.
Program realization of 3D-graphics
Program interface is created by using special library WPF (Windows
Presentation Foundation) and built-in XAML redactor in Microsoft Visual
Studio.
What is WPF:
“The Windows Presentation Foundation is Microsoft’s next generation
UI framework to create applications with a rich user experience. It is part
of the .NET framework 3.0 and higher.
WPF combines application UIs, 2D graphics, 3D graphics, documents and
multimedia into one single framework. Its vector based rendering engine
uses hardware acceleration of modern graphic cards. This makes the UI
faster, scalable and resolution independent.”
Viewport3D element
Graphics contents in 3-D WPF application is encapsulated in Viewport3D
element, which can be placed on two-dimensional element such as program
window. Graphics system identifies Viewport3D as two-dimensional visual
element such as many in WPF. Viewport3D function as preview window of
three-dimensional scene. More precisely it is projection of 3-D scene on plane
(screen).
Drawing 3-D objects
Drawing of objects is realized by triangulation, that means that
we need to build it of triangles.
We will notice that back side of triangles is transparent as default.
Using high-speed camera
In order to capture the way fencer fights was used high-speed camera. It
was used because using only human eyes it is not able to see in details
what is happening, what fencer does. By the way mainly fencer have to
implement movements on high speed and he can not do it slower.
Results
The ongoing results of this project:
1) The simple model that moves hand from one point to another is
implemented in MATLAB.
2) MATLAB model was optimized selecting the closest way to new
position
3) The program interface containing main functionality is written in C#
4) Videos of fencer’s movements prepared using high-speed camera
What is going to be done:
1) Rewriting operating part from MATLAB to C# program
2) Usage of limitations in joints
3) Creating of database of kinds of movement using videos from highspeed camera
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