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In the last lesson, we have seen how we can calculate trapezoidal trajectory over during calculation

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of the trajectory, we have imposed constant velocity, which was given by Cuvee Dot and obtained constant

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acceleration based on the given acceleration time.

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Now we can impose inability.

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No.

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Can we impose any velocity and acceleration time we want?

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Indeed, we have some constraints to consider, such as the actuator velocity and acceleration limits.

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Namely, if the impulse and the velocity and acceleration time we want, we can get velocity and acceleration

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that are higher than the capabilities of the actuator.

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And this will saturate the actuators and the trajectory will not be executed properly.

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So I move variables, namely constant velocity and acceleration and acceleration time are related with

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each other, and we have to obey some constraints.

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Let's determine these constraints.

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The first one is the first constraint is on acceleration time to.

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It has to be less than or equal to the half of the total time span.

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Venti is exactly T overture, then the linear segment will not appear.

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Then we have to consider the relation between constant acceleration and acceleration time to in order

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to get the relation between them, we have to analyze this graph.

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This is the usual orientation graph of the trapezoidal trajectory.

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Q m is nothing but the position point for the middle of the whole trajectory.

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We can find a easily, as we have seen before.

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Q A Excuse me, just take whatever formula for M first phase and plug to instead of T, then we can

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ride constant velocity equation in this way.

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Constant velocity is obtained at the end of the first phase, namely at time to by using Eq. from simple

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physics.

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You can calculate it easily.

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Of course, either that initial velocity is zero.

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And then if we denote the length of the trajectory as L and time span as T, we can write equations

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for CU m, a. M. very easily.

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Now we will take Eq. 1.0 and plug it into one point one.

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With some algebraic manipulations, we can get Equation one point two, which denotes the constraint

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on the choice of constant acceleration and acceleration time.

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So if you impose some velocity and acceleration time and obtained acceleration, you will have to check

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whether THC and acceleration fulfills this constraint.

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The third one is the constant velocity equation.

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This is not exactly a constraint, but formulation for constant velocity.

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We will find it by using Velocity Graph.

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As you know, the length of the trajectory is can because you can be fined by.

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It can be calculated by finding the area of the trapezoid.

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We can find the area of the trapezoid by summing individual areas for each phase.

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And this is the result.

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It's very easy to calculate because two shapes are right triangle, while the middle shape is just a

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rectangle.

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By manipulating this equation, we can get equation for velocity.

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After we have seen those three constraints, we can do some useful things with them during trajectory

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planning.

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We can assign desired acceleration time of T a and obtain constant acceleration from Eq..

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One point two As stay is specified, we can obtain constant velocity and check whether a constant velocity

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and acceleration values are compatible with actuator limits.

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Or we can assign constant acceleration value and obtain acceleration time to again from Eq. one point

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two then you can calculate constant velocity also.

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This case can be useful when you know maximum acceleration value for your actuator, so you can impose

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it and find acceleration time for the trajectory.

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Well, you can impose maximum velocity and acceleration.

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That actuator can output and then calculate T and acceleration time.

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This is very useful to obtain optimal trajectory in terms of minimizing time span of T, because as

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you impose maximum velocity and acceleration, you will get minimal execution time.

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Let's not analyze each case.

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In case one, acceleration time is specified from below.

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That time span has to be at least two times bigger than acceleration time.

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Then you can easily calculate velocity and acceleration values.

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Surely, you could use Equation one point two to find a compatible acceleration.

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Will get to verify that the velocity and acceleration values are compatible with actuator limits for

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keys to constant an acceleration imposed by us.

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We can impose it as maximum acceleration that actuator can output, then we can obtain acceleration

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time from Eq. one point to easily in case three constant velocity and acceleration are imposed.

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Indeed, by imposing the maximum permissible velocity and acceleration, we can obtain optimal trajectory

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in terms of execution time.

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We will get acceleration time by dividing velocity and acceleration because a time table or city will

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be the constant velocity we imposed and acceleration will be the acceleration we imposed.

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In order to obtain execution time of T, we can ride the form a low velocity and extract T from it by

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algebraic manipulation.

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This is the final expression for the T after we have plugged into the formula for Tay in the third and

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last part of the trapezoidal trajectory.

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We will talk about how to implement this trajectory method for multiple joints.