In the last lesson, we have seen how we can calculate trapezoidal trajectory over during calculation
of the trajectory, we have imposed constant velocity, which was given by Cuvee Dot and obtained constant
acceleration based on the given acceleration time.
Now we can impose inability.
No.
Can we impose any velocity and acceleration time we want?
Indeed, we have some constraints to consider, such as the actuator velocity and acceleration limits.
Namely, if the impulse and the velocity and acceleration time we want, we can get velocity and acceleration
that are higher than the capabilities of the actuator.
And this will saturate the actuators and the trajectory will not be executed properly.
So I move variables, namely constant velocity and acceleration and acceleration time are related with
each other, and we have to obey some constraints.
Let's determine these constraints.
The first one is the first constraint is on acceleration time to.
It has to be less than or equal to the half of the total time span.
Venti is exactly T overture, then the linear segment will not appear.
Then we have to consider the relation between constant acceleration and acceleration time to in order
to get the relation between them, we have to analyze this graph.
This is the usual orientation graph of the trapezoidal trajectory.
Q m is nothing but the position point for the middle of the whole trajectory.
We can find a easily, as we have seen before.
Q A Excuse me, just take whatever formula for M first phase and plug to instead of T, then we can
ride constant velocity equation in this way.
Constant velocity is obtained at the end of the first phase, namely at time to by using Eq. from simple
physics.
You can calculate it easily.
Of course, either that initial velocity is zero.
And then if we denote the length of the trajectory as L and time span as T, we can write equations
for CU m, a. M. very easily.
Now we will take Eq. 1.0 and plug it into one point one.
With some algebraic manipulations, we can get Equation one point two, which denotes the constraint
on the choice of constant acceleration and acceleration time.
So if you impose some velocity and acceleration time and obtained acceleration, you will have to check
whether THC and acceleration fulfills this constraint.
The third one is the constant velocity equation.
This is not exactly a constraint, but formulation for constant velocity.
We will find it by using Velocity Graph.
As you know, the length of the trajectory is can because you can be fined by.
It can be calculated by finding the area of the trapezoid.
We can find the area of the trapezoid by summing individual areas for each phase.
And this is the result.
It's very easy to calculate because two shapes are right triangle, while the middle shape is just a
rectangle.
By manipulating this equation, we can get equation for velocity.
After we have seen those three constraints, we can do some useful things with them during trajectory
planning.
We can assign desired acceleration time of T a and obtain constant acceleration from Eq..
One point two As stay is specified, we can obtain constant velocity and check whether a constant velocity
and acceleration values are compatible with actuator limits.
Or we can assign constant acceleration value and obtain acceleration time to again from Eq. one point
two then you can calculate constant velocity also.
This case can be useful when you know maximum acceleration value for your actuator, so you can impose
it and find acceleration time for the trajectory.
Well, you can impose maximum velocity and acceleration.
That actuator can output and then calculate T and acceleration time.
This is very useful to obtain optimal trajectory in terms of minimizing time span of T, because as
you impose maximum velocity and acceleration, you will get minimal execution time.
Let's not analyze each case.
In case one, acceleration time is specified from below.
That time span has to be at least two times bigger than acceleration time.
Then you can easily calculate velocity and acceleration values.
Surely, you could use Equation one point two to find a compatible acceleration.
Will get to verify that the velocity and acceleration values are compatible with actuator limits for
keys to constant an acceleration imposed by us.
We can impose it as maximum acceleration that actuator can output, then we can obtain acceleration
time from Eq. one point to easily in case three constant velocity and acceleration are imposed.
Indeed, by imposing the maximum permissible velocity and acceleration, we can obtain optimal trajectory
in terms of execution time.
We will get acceleration time by dividing velocity and acceleration because a time table or city will
be the constant velocity we imposed and acceleration will be the acceleration we imposed.
In order to obtain execution time of T, we can ride the form a low velocity and extract T from it by
algebraic manipulation.
This is the final expression for the T after we have plugged into the formula for Tay in the third and
last part of the trapezoidal trajectory.
We will talk about how to implement this trajectory method for multiple joints.