Before we have seen Kubik and fifth order polynomial trajectories based on the calculations of these
two trajectories, we can get higher order polynomial trajectories very easily.
Additionally, we have noted that other degree of polynomial increases smoothness of the trajectory
also increases, but the number of coefficients to be calculated also increases.
So in this lesson, we will try to get benefits of both of these techniques.
How we will mix different degree of polynomial trajectories.
The simplest and popular one is trapezoidal trajectory or with other name linear segment with parabolic
blends.
You will see the reason of this naming as we go further.
Here is the position, velocity and acceleration profiles for this trajectory.
As you can see, it consists of three phases.
The first and third phases are second order polynomials, namely parabola, while the second phase is
just the line.
That's why it is called linear segment with parabolic blends.
If you look to the velocity profile, it is in the shape of trapezoid.
So another name for this trajectory is trapezoidal trajectory.
Another thing you can immediately notice is that while this trajectory provides smoothness in velocity,
the acceleration profile is not continuous.
OK, now let's analyze each segment one by one.
The first phase is second order polynomial function.
It represents constant acceleration.
The time required to finish the first phase is named as acceleration time and denoted by 2A.
You can notice it from acceleration profile, while the second phase is a deceleration phase.
Excuse me, while the second phase is a linear function.
So zero acceleration, but constant velocity.
The third phase is caused them deceleration phase.
It's also a second order polynomial function, and the time required to finish this phase is called
deceleration time, which is generally chosen the same as acceleration time.
So trajectory becomes symmetric.
OK.
As we have see, the 8A is nothing but acceleration time and chosen the same as this celebration time
and T is the overall time required for tracking or executing the trajectory.
T is equal to the difference between final time and the initial time.
We will assume initial time as zero.
Otherwise, you can include them very easily.
Let's deep dive into each phase and analyze equations.
Let's start with acceleration phase.
Namely, time range between zero and T, as this is second order polynomial, orientation, velocity
and acceleration.
We'll have this structure.
The boundary conditions are given as initial position of Q and initial velocity of zero.
But these two conditions are not enough for us.
We need stored condition to be able to solve for a long time coefficients.
Let's analyze this velocity graph as you can see a final velocity for the first phase, namely velocity.
That time T A is equal to the velocity of constant velocity phase or second phase.
So we can write this third and final boundary condition.
Now, as we have enough boundary conditions to solve for unknown coefficients, we can write below equations
in matrix form.
And so for Vector A..
Let's continue with constant velocity phase in constant velocity, phase orientation, velocity and
acceleration.
We'll have this form.
As you can see, acceleration is zero in this phase.
Now let's determine boundary conditions.
We can determine the initial position from Equation 1.0.
This formula is from basic physics.
Well, you can get this equation even from the first phase orientation equation.
Just plug it instead of tea, because the initial time for a second phase is to say no.
Again, analyze this graph.
We need a second boundary condition also to solve for unknown coefficients.
And this is the initial velocity for this phase, namely Cuvee Dot, which is imposed by us.
OK, after we have obtained the boundary conditions, let's go for our long coefficients again.
Not the last phase is deceleration phase, which is similar to the acceleration phase.
Again, let's determine the boundary conditions.
The first one is final position or orientation.
The second one is final velocity, which is zero.
And the third one is initial velocity, which is nothing but the velocity at time.
T a t minus T.
Excuse me.
This velocity is the same as the velocity at the end of the second phase, and this is constant the
velocity that we have imposed.
We cannot determine the unknown coefficients easily after we have analyzed each phase individually.
In the second part, we will take into account some other issues also.