The pushover manipulator robot, or rather the position and the orientation of the gripper of a manipulator
robot is defined by six variables, three that define its position and three that define its orientation.
This is a very complex problem to solve, as is the mathematics and the equations that express the pose
of a robot.
And therefore the relationship between the reference frame are attached to the robot's gripper and the
fixed reference frame of the word that we called W.
When engineers and mathematicians are faced with a complex problem to solve, the approach they often
use is to divide it into smaller, more easily solvable sub problems.
That's what we are doing in this case as well.
The problem of determining the pose of a robot in space can be decomposed into two sub problems.
The first one consists of determining only its position and therefore the translation between the reference
frame attached to the robot and the fixed reference frame attached to the award.
The second sub problem consists of determining only the orientation that is the rotation between the
robot's reference frames and the fixed world reference frame.
Once these two simple sub problems are solved, we can combine them and solve the general case of determining
the pose of a robot, which includes the translation and rotation of the reference frame attached to
the robot with the respect to the world reference frame.
Let's start with the study of the translation and see how to mathematically express the X, Y, and
Z coordinates of a robot pose.
The simplified problem we want to solve in this case is to determine only the position of the reference
frame R, which is inertial and is attached to the robot relative to the fixed frame of the word W.
So we are assuming that the irrotational component is zero and thus the reference frame R is always
oriented the same way as the reference frame w.
In fact, we can see that the x, y and z axis of the two reference frame are oriented in the same way
without any rotation among them.
In other words, the problem we want to solve consists of calculating the coordinates of the center
of the reference frame are relative to the world reference frame w.
To describe the three components of the translation.
So X, y, and z, we can draw a vector t that connects the center of the reference frame w with the
center of the reference frame r.
The projection of this vector along the X, y, and z axis of the reference frame W are precisely the
X, y, and z.
Coordinates of the center of the reference frame are relative to the reference frame w.
Mathematically, we can say that the projection of a vector T along the X, Y, and Z axis of the reference
frame w are equal to the coordinates of the center of the reference frame R relative to the world reference
frame.
These three equations can also be rewritten in a matrix form while maintaining the same logical meaning.
Now, if we consider a real case with real values and assign a numerical value to the X, Y, and Z
coordinates, we can easily say that the center of the reference frame are as coordinates three, six
and five within the reference frame.
W.
Let's remember this result and set it aside for a moment.
We will soon see why this vector is useful.
Since the arm of our robot is a certain length, we might also want to know the position in the world
of any point on the robot's arm.
In order, for example, to avoid collision with obstacles, as we know the dimensions of the robot's
arm so we know its length with a bit of trigonometry, we can easily calculate the position of the point
p relative to the reference frame R and therefore we can know its coordinates x, p, y, P and z p.
This information.
So these coordinates are relative coordinates.
They tell us where the point P is located on the robot's arm.
So relative to the our reference frame, which is always attached to the robot's arm and so moves along
with it.
However, we might be interested in another piece of information.
Still, in the case of developing an algorithm that prevents parts of the robot from colliding with
other objects in the world.
We might be interested in knowing the position of the point P and thus the end of the robot's arm relative
to the world.
That is, we might be wondering which are the coordinates of the point P relative to the world reference
frame.
This problem is quite simple and also is intuitive to solve, and it simply involves adding coordinates.
In fact, the position of the point p in the reference frame w is given by the sum of the position of
the point P in the robot reference frame R plus the position of the reference frame R relative to the
reference frame w, which is defined in the vector t.
We can also express this equation in a matrix form by decomposing each vector into its x, y, and z
components.
With this equation we can solve the problem and so determine the position of the point p in the reference
frame W by knowing the position of the point P in the reference frame r.
Once again, if we assign some real values and assume that we know the coordinates of the point P relative
to the reference frame R which are equal to two, four and four, we just need to add these coordinates
to those of the vector t which we calculated previously and which is three, six and five, and we will
have the position of the point P in the reference frame W.
This vector is called translation vector and indicates the distance between the reference frame R and
the reference frame W and thus three components which are its projections along the X, y, and Z axis.