OK.
In previous video, we have gained some insight about manipulator dynamic modelling before jumping directly
to the concept.
I want to give you some insight about the method and the related terminology so you can easily digest
next lessons.
Warning I will not teach everybody the scenes, which is not possible.
Please, if you have any lack of basic terminologies here, make sure you have served and learned them.
So let's get started before anything else.
I would like to start with the notion of generalized coldness, which will be very useful for us.
During dynamic modelling.
A set of position and orientation variables that describe the configuration of a system is defined as
a set of generalized coordinates.
Generalized coordinates that we choose have to be able to describe the system completely.
The set of genera last quarters is not unique.
You can sometimes use rectangular coordinates x y to describe the system or cylindrical coordinates,
such as rotation angle of Titan.
It doesn't matter.
The key thing is that you have to choose such coordinates that makes your equation of motions simpler.
Additionally, the number of generalized coordinates has to be larger than the number of degree of freedom
of the system, because, as I have said before, systems should be complete described by generalized
coordinates.
However, in general, the number of generalized coordinates is the same with the degree of freedom
of the system.
Let's take this situation as an example, which is nothing but a simple one degree of freedom pendulum.
The system has three dimensions as you can see motion along x axis, motion along y axis and rotation
about that access.
However, we have two constraints that are indicated in equation, and one point mass can only move
in circular fashion, which is indicated to be the first equation.
And there is no motion along zip axes.
So three dimensions for which two dimensions are constrained.
We have one degree of freedom, so the number of generalized coordinates has to be either one or greater
than one.
In this case, we have two sets of generalized coordinates.
The first is X and Y, but these are not independent corners because X and Y is related by Eq. one.
So it is enough to know one of them and the other will be known automatically.
The second set is rotation angle of TITA.
You can see what about X anteater or Y anteater?
But if you think carefully, you will see that X, Y and Z are related with each other, especially
if we know Titov.
We can know both X and Y because they are related as given in these equations.
So one coordinate is enough to complete the described the position of rigid body, and this can be chosen
as to which makes equations of motion easier.
Let's not talk about the kinetic energy of a system which is given Equation 1.1.
As you can see, it consists of two parts, namely kinetic energy due to translational motion and the
one due to the rotational motion.
Here is the interesting thing kinetic energy due to the translational motion, namely, the first part
of the equation doesn't change with respect the reference frame.
Let's analyze it here.
The velocity is given with respect to the Baz's frame.
Let's say we want to convert it to another reference frame, so apply a rotation matrix of R from the
transportation transposition rule.
Excuse me, we can write equation in this way.
This equation is valid for all rotation mattresses of R as the transpose of rotation matrix is equal
to its inverse.
So the given product of rotation mattresses will be identity and we will get the same equation as before.
Interestingly, this property is also valid for the second term.
Let's analyze the second part of the equation.
Also, let's say in the equation, both inertia matrix both ie and angular velocity of omega are given
with respect to the base frame.
We say that it is not important with respect to which reference frame we write this equation.
It will not change.
So then we can write this here.
Angular velocity and inertia matrix are written with respect to F Dash reference frame.
We know that the relation between the old inertia matrix and new inertia matrix is given by the rotation
mattresses between the two friends.
If you cannot remember these, please revise energy matrix concept so we can avoid the equation in this
way.
Here are is the rotation matrix between base frame and F Typekit reference frame.
If we manipulate the equation by using the transposition rule, we will get.
This new equation where our transports are is identity, and we can exactly we we get the exactly the
same equation at the beginning.
So we proved that the second part also is invariant with respect to the reference frame.
So kinetic energy is invariant with respect to the reference frame we choose.
So what is the benefit of this?
This means that we can choose such a reference frame that makes our measurements and calculations easier.
For example, we chose by the frame, which is placed in the center of mass of the region, the body
and moves with it for inertia matrix right by the frame, but not based frame.
Because if we adjust based frame as a reference frame for the inertia matrix, it will change as the
body moves.
And we don't want that because it will just make calculations complicated at every instant.
We have to calculate new inertia matrix.
However, if we chose by the frame, which moves with body as a reference frame, then inertia matrix
will not change and calculations will be simpler.
But be careful here if you change your reference frame for inertia matrix, we have to do the same for
angular velocity as angular velocity is measured with respect to the base frame based reference frame.
We have to multiply it with the rotation matrix of R to convert it to the body reference frame.
So all in all, the kinetic energy form look becomes like that.
Here I indicates rigid body I, in our case of manipulators, link I let's not these down.
Also to clarify everything linear velocities and angular velocities are measured with respect to the
base frame inertia matrix is defined with respect to the body reference frame R is the rotation matrix
between base frame and body frame.
Total kinetic energy is given by the sum of kinetic energies of each ring.
OK, now let's see potential energy before we have said that before we have said that we will accept
leaks as very rigid bodies, so no elasticity.
Due to that assumption, the potential energy will be only due to the gravitational field.
Here's the scenario We are rigid.
The body is under the ground force of gravity and this is the potential energy of lint I am.
I is nothing but the mass of Link I PC.
I is position vector of the center of mass of Brigida by the eye given by coordinates of X Y and Z G
is the gravity vector.
Be careful it is vector, so only z component of it as value other than zero, because in X and Y directions
there is no gravity force.
Total potential energy is given by the sum of potential energies of each of the links.
After we get the formulas for kinetic and potential energy are ready for getting dynamic model of the
robot manipulator using the Lagrange formula, the Lagrange formula is given by this equation.
As you can see, it contains partial derivatives and also derivatives with respect to time here.
In the case, the number of rigid bodies or links in this case, you can ask What is it?
Here it is the popular Lagrange variable.
Nothing but the difference between kinetic and potential energies.
If L is written in this way, then we can write electrons for us in Eq..
One point five.
Be careful In the first term, partial derivative of potential energy with respect to joint velocity
is zero because potential energy does not depend on joint velocities, but only joint positions.
C k in this formula is nothing but that it calls conservative forces.
These forces include friction forces, damping forces, contact forces between the robot gripper and
the environment, and joint actuator talks.
Let's take this example in order to clarify the notion of dynamic modeling where it will grant formula.
As you can see, a rigid body of mass and moves along z axis under the gravity force and pulled by external
force of F friction coefficient is given by D in order to model this dynamically.
We need first to calculate kinetic energy, which is given by this formula.
This is very simple and comes from basic physics.
Potential energy is also very simple, which is due to only the gravity.
After we calculate the potential energy and kinetic energy, we can write Lagrange variable.
Let's remember the formula again.
If we calculate the first part we will get, this is it will not have any contribution because it doesn't
contain velocity term for the second part.
We will get this.
In this case, kinetic energy will not have any contribution because.
It is on the velocity dependent.
If we consider non conservative forces of ebb and friction force, which is given that thought, we
can obtain dynamic model of the given system.
As this, as you can see, this is the correct model and you can verify that by use just using Newton's
formulation.
So you can see that by the help of a little Legrand's formulation, we can get dynamic model of the
system from potential and kinetic energies.