Before diving into the details of the forward and inverse kinematics equation of a robot manipulator.
And before studying the motion of the robot in space based on the position and angle of its motors,
let's clarify what does it mean, the term pose of a robot manipulator and how to mathematically describe
it?
We can represent the pose of our manipulator robot by focusing uniquely on the position of the center
of the robot's gripper.
This point of the robot can freely move in the three dimensional space as it joint rotate.
We define a fixed reference frame, which we will call W that stands for word.
And therefore the problem of defining the robot's pose translate into the problem of defining the position
and orientation of the robot in the reference frame w of the word.
To do this, we can introduce also a new reference frame, this time a mobile reference frame that is
attached to the robot.
And we are going to call this one R.
This reference frame is mobile because it moves with the robot and so therefore is called inertial reference
frame.
As we can see, while the word reference frame remains fixed.
And so it is called Non-inertial reference frame.
The R reference frame instead moves along with the robot and is therefore called an inertial reference
frame.
Thus, the problem of determining the robot's pose can be reformulated as finding the position of the
reference frame R, which is mobile relative to the reference frame W, which is fixed by significantly
simplifying the way we express our problem.
We can now intuitively say that three variables are enough to define the pose of the reference frame
r relative to the reference frame w.
These are the X, y and z coordinates that express the position of the center of the reference frame
R with respect to the center of the reference frame w.
Together, these three variables express the pose of one reference frame relative to another.
Therefore, since the reference frame R is attached to the robot and so it moves with it.
These three variables also represents the pose of the robot as it moves in the three dimensional space.
However, with this we are intentionally omitting something.
With the variables X, Y, and Z.
We are only expressing the position of the reference frame R with respect to the reference frame W of
the world and we are not expressing its orientation.
In fact, in addition of being translated by a certain amount X, Y, and z, the two reference frames
so R and W are also rotated by a certain angle around each axis.
Therefore, for completeness, the pose of a manipulator robot is defined by six variables, three of
which refer to its position and the other three that refers to its orientation.
Pay attention to the number of variables that make up the robot's pose and to the number of motors and
so to the number of joints that our robot has.
Our robot has only three motors which can activate three joints.
Meaning that our robot only has three degrees of freedom to completely define the robot's pose.
As we can see.
And so including its position and orientation, we need six variables.
And so therefore we need six degrees of freedom.
This means that with our robotic arm, we cannot reach any position and orientation of the gripper within
the robot's workspace.
Specifically, if we fix a position for the robot, for example, if we want the robot to reach a specific
position in space, let's say a certain position, X, Y, Z, we cannot also determine the orientation
with which the robot will reach that position.
The orientation will be automatically determined by its position.
In other words, our robot has no control over the orientation and we cannot change it.
We don't have enough number of degrees of freedom, and so we don't have enough motors to define also
the orientation.
For this reason, especially for industrial applications, manipulator robots usually have six or even
seven degrees of freedom.
This allows reaching any position within the robot workspace also with any orientation.
Moreover, manipulator robots that have more than six degrees of freedom as they have more degrees of
freedom than are actually needed to define the robot's position and orientation can reach the same position
with the same orientation, with different configuration of the arms and with different trajectories.
And for this reason they are more flexible.