In previous lesson, we have seen how we can solve the issue of requiring too much velocity and acceleration
from actuators that can actually saturate them.
We have solved it by using kinematic scaling, but during trajectory planning we have to consider also
the torque limits and not to saturate actuators by exceeding torque limits.
If we exceeded, we can then use dynamic scaling to solve the issue.
While planning trajectory, you have to consider a nonlinear TS and couplings in robot dynamics.
If your trajectory abides by the torque limits of each individual actuator separately, it doesn't mean
that it will do the same when they are considered together because of non minorities and couplings.
OK, let's start with noting down the robot dynamic equation.
We can separate it for each individual joint as this way.
Now, let's assume that we are given trajectory cue to be executed in time interval of zero two T.
Then we can rewrite dynamic equation Abbu as this way by plugging in the trajectory.
Let's call this part of the equation as tall as I t, because this part depends on not only by configuration,
but also velocity and acceleration.
However, granted, the term depends on the the configuration.
This is the new trajectory v where it is parameterized by function of Sigma T Dash.
As you can see, Sigma Zero, Sigma zero zero and Sigma T Dash is T limbless sigma is just the scaling
function of time.
We will see what's useful for us if we plug this trajectory into the dynamic model of the robot manipulator.
We will get this equation as Q Tilde T Dash is nothing but Q Sigma T Dash.
We can find its derivatives very easily by using chain formula we have seen before.
Here is how we find velocity name live first.
The first derivative of QTL, not T Dash as Sigma T Dash is nothing but t.
We can write the derivative of Q with respect to sigma as Q T.
Now let's find the second derivative of the QTL that he dash.
It is easy to find that you have to just take another derivative of Q Tilde, the dhoti dash.
As you can see, we have utilized the chain rule again.
I specified corresponding terms with colors so you can easily understand what's happening.
No, the equation can be rewritten in this way.
As we have formed velocity and acceleration, we can plug them into the equation 1.0 and obtain this
new equation.
Don't let it scare you.
It's very easy that you can stop video and try to follow the each step.
You can easily notice, though, as I term here again.
This will help us to ride the equation in a much simpler way.
And not this.
Also that gravity will not be affected by time scaling as it depends on the configuration.
Indeed, in previous slide, you have seen that Sigma only appears in and after the first derivatives
of the trajectory, so we can write equation in a much simpler way, as in this equation 1.1.
As you can see, we did not consider gravity torque because it is not affected by the scaling eval appended
at the end.
Now it's time to use scaling function of sigma.
Let's determine it as a linear function.
Keep this in mind you don't have to choose this function as this only you can choose it as your desire.
OK, now let's fund derivatives of the sigma so we can plug them in Eq. 1.1 and obtain this very simple
equation.
Now you can easily see how we scale the trajectory.
Now you can also include gravity terms.
Also, we can choose scaling coefficients of lambda in two ways either it will be less than one.
In this case, timespan of the trajectory will increase because T equals the number T dash and required
talks to execute the trajectory will decrease.
Otherwise opposite is happening.
Time duration increases and we have to apply more talks to find the trajectory at the right time to
excuse me.
We have to apply more torque in order to execute the trajectory through time.
OK, OK, I know you want to see it in an example, so let's see an example.
So assume that we.
Are given the trajectory to be executed by third degree of freedom, robot arm in two time units with
torque limits of tall, one bar and tall to bar for each one.
However, after we have calculated the trajectory, we see that the maximum torque while is required
to execute the trajectory from the joint actuators are higher than the thresholds.
What we have to do?
Surely we have to dynamically scale trajectory.
This is how we do it.
First, let me not bomb the equation here again.
From this equation, it's obvious that we can find lambda by just dividing torque limits with maximum
torque values because we want to decrease talks to the threshold values.
And surely we have to find the minimum, not maximum, because if we find the maximum, then one of
the joints will saturate because of high torque value.
After we have formed the lambda, we can change time to T Dash by scaling the original time span.
You can also notice that we didn't have to plan the trajectory again.
We just scaled a time span, and that's it.
We are in the limits of the torch by dynamic scaling.
You can also calculate optimal trajectory in terms of execution time, namely, if required, talks
are lower than the thresholds.
This means that you don't use full power of your actuators so you can scale down the time span and get
faster trajectory execution without surpassing the limits.
You can investigate this problem and more on trajectory planning from this book that I have used for
this course.
It explains everything in a simple manner.
So for more examples, you can refer this book for the problem.
I have set a minute before I go to the page 240.
An example 5.6.