I hope that you have enough information about a positive trajectory until now, we analyze in detail
how we can calculate the trajectory for single joint.
But we know that robot arms don't consist of one joint, but instead of instead more than one joint.
So what will happen in this case?
Do we have to find new equations?
Surely not.
What we want is only what we want.
Only is all the joints to start at the same time and finish at the same time because otherwise it would
be very odd.
Just think it like that you want to take a glass of water in front of you.
So you first move your shoulder after you have finished, you move your elbow.
After you have finished, you move your wrist, then your fingers and so on.
You don't do it like that in real life.
Instead, you move all of them together in a synchronized way.
Yeah.
This is called synchronization.
So we want synchronization between joints.
How will we achieve that?
It's very easy and logical indeed.
Namely, we have to determine first the joint that will move the mosque.
So has the largest displacement.
Let's say this is the kids joint.
Then we will co-create acceleration, time and execution time for that joint base on the given acceleration
and velocity values.
Then we will use these time parameters for the whole trajectory and calculate corresponding velocity
and acceleration values for the other joints, except Keith one.
In this way, all the trajectories will start and end at the same time, so they will be synchronized
after we have obtained velocity, acceleration values and also acceleration time and execution time.
We can calculate the trajectory very easily for each joint.
Let's jump to the math club and see how it will be done.
OK.
Here is how we will apply trapezoidal trajectory in six degrees of freedom robotic manipulator.
Um, OK.
We will just think that it's six degrees of freedom, Robert.
Apparently there.
Indeed, I have taken the example as you are, as you can see, you are five robotic manipulator.
This is from Peter Cook his library.
OK.
So we just import this model.
Then we are sitting for maximal velocity and maximal acceleration for each of the giants time, then
maximal velocity.
I found it from Datasheet Mart.
I myself assigned for maximal acceleration.
Anyway, it's just an example.
Let's assume that we are given with maximum velocity and maximum acceleration that our robot's actuators
can output.
This is the initial OK.
We have made it equal to the Q.
Is it because this cuz it comes from this, then we import the model.
You are five.
You will see.
Yes, this is cuz it is for each of the joints and this is zero.
So every joint starts from zero orientation.
OK, get your zero angle and the final is q r.
We want to reach this final position or orientation.
OK.
What we have see there in the lesson, we have said that as we have more than one joint, we have to
do a synchronization.
What was the first step for synchronization?
We have to find the maximal displacement, OK and the join with the maximal displacement, how we will
do that.
We first find the difference between the final position, find the orientation and the initial orientation,
and we think it's absolutely because displacement is needed for us.
OK.
And then what we are doing, we find it's the maximum displacement in that displacement vector.
OK, and tackles its index.
Then what we are doing, we are doing synchronization v calculating acceleration time and time duration
to execute the trajectory.
We are calculating it for their maximal displacement.
OK, as you can see, L Max, we have seen it in the presentation then after we have obtained T A. Acceleration
time and execution time, we can apply these two for the whole trajectory sort.
So for the whole other joints.
OK.
So what we are doing, we are first.
So this is for sampling.
No, because as you know, our trajectory trapezoidal trajectory consists of what?
Three phases?
Yes.
Acceleration.
Constant velocity deceleration.
OK.
And for each.
For each phase, we take the time, for example, this is the time from zero to a for the first phase,
from T to T minus to for the second phase from T minus T a two T for the third phase.
OK.
And then we are creating by six by.
This is the sampling number of matrix.
This is three dimensional Matrix Y because in each phase, OK, we will talk with four different people,
use different formulation to calculate the Q and the same for velocity and the same for acceleration.
This six is because we have six degrees of freedom, a robot manipulator.
I mean, six joint.
OK, so this matrix will have all these mattresses are three dimensional, so they have six rows and
some called arms.
But three of these, OK.
I mean the dimension or does it axis also?
OK?
Three of these three or six by s and two Matrix City of six by S and the Q Matrix and three of six by
S and D D Q Matrics acceleration matrix.
Then we are doing for each phase.
OK.
What we are doing for each phase we are calculate and for each this is for each joint.
OK.
We are looping in each phase and in each phase.
We calculate for all the joints what except the joint keys joint.
Do you remember the one with the maximum maximum
displacement?
OK, except this what we are doing.
We are calculating for the others the velocity and acceleration from their calculated T.
A. A. From here.
OK.
And if it is phase one, we apply the formulation for phase one velocity and acceleration.
We have seen this from the presentation.
You can revise it.
You can check it.
Or for the second phase, if you remember, this was the, um uh, for the cure and for this velocity
and for this acceleration.
As you can see, the velocity is constant and acceleration is zero in the first phase.
This is the acceleration, OK, because it's positive in this case, as you can see this variation because
it's negative or constant deceleration.
OK.
And uh, for the doing, this is the joint.
This is the case joint key for case joint.
We assign in the velocity and, um, acceleration as maximum as maximum.
And then in the first phase, we are doing calculation.
OK.
In the second phase, we are doing calculation and for the third phase.
OK, I think this is easy.
The important thing is how we have done the synchronization.
As you can see, if you want a maximal displacement joint, if you calculated for it, from it to A..
And we apply this to A. For all their joints except case joint OK, and find velocity and acceleration
for them.
And then what we have done, we calculated for maximal maximum acceleration for them and we have calculated
the trajectory for them.
OK.
This is for concurrent creating the time vector.
OK, let's do it in this way.
So you can see first.
Let's do that part.
Uh, run on advanced.
OK, let's if you run this again, if you run this, you will see.
Indeed, I can do that.
Let me just to evaluate in this way, in order to show you evaluate this section, this whole could,
as you can see, what is the dimension of T?
Let's let's see size of T size of T size of T is three by one end.
Yeah.
Well, what I want to do is I want to make it.
I want to control the get all the time slices.
OK, so our time thesis into one vector so we can plot all of them in one plot.
So I want to make it, what, three thousand by one or one by three thousand?
OK.
And I'm doing it in this way first and transposing it.
Then I'm doing to control communication.
OK, let's run it again around the section, around this section also.
OK.
As you can see now, we can check the size of T.
OK, as you can see one by three thousand.
Yeah, then what we are doing this will make for us if this will make for us easier, uh, for plotting.
OK, but then you are plotting the position velocity and.
Acceleration for each axis, OK, for each of you will draw for old accesses, OK?
Let's do this plotting around and advance.
Okay.
I think we have to see the plots, OK?
As you can see, this is position plot.
Yes.
You can see that.
First of all, let me show you that as you can see from Let Me Just We started from Q.
Is it OK you initial was this one from zero zero zero zero?
What the hell is that?
Why does I dad queue initial?
Is he OK?
Excuse me, it's cute.
I yeah.
Four Q Initial what we have done.
Q Zero.
Oh yeah, it's Q zero.
We will start from this configuration.
And Q R, as you can see on the first and fifth joint, are moving.
So that's why only we can see a position in first and fifth.
Um, excuse me, first and fifth.
Yeah.
Which one was three point?
OK, then this is the first one.
This is the fifth one fifth.
Stewart and all others are zero.
As expected, OK?
You can see the trapezoidal trajectory.
You can see blends, OK?
And you can see a linear segment and you can see deceleration.
That's perfect.
Then let's see it.
As you can see, this is totally the trapezoidal velocity profile.
You can see a linear segment.
Excuse me, you can see, yeah, linear function.
This is constant function and this is the deceleration phase.
OK.
And you can see the beautiful acceleration.
Yeah, this is just because as as you can see, as you can see here, the synchronization, all of the
acceleration time is the same for both of the joints.
So for the first one and the fifth one, the same is deceleration phase as you can see as T A and T
I mean s acceleration time and deceleration time are the same.
The figures are symmetric.
OK, and as you can see, the acceleration profile is discontinuous, which is not desired too much.
But for that simple trajectory, we can accept it.