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Before we have seen Kubik and fifth order polynomial trajectories based on the calculations of these

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two trajectories, we can get higher order polynomial trajectories very easily.

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Additionally, we have noted that other degree of polynomial increases smoothness of the trajectory

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also increases, but the number of coefficients to be calculated also increases.

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So in this lesson, we will try to get benefits of both of these techniques.

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How we will mix different degree of polynomial trajectories.

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The simplest and popular one is trapezoidal trajectory or with other name linear segment with parabolic

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blends.

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You will see the reason of this naming as we go further.

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Here is the position, velocity and acceleration profiles for this trajectory.

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As you can see, it consists of three phases.

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The first and third phases are second order polynomials, namely parabola, while the second phase is

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just the line.

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That's why it is called linear segment with parabolic blends.

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If you look to the velocity profile, it is in the shape of trapezoid.

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So another name for this trajectory is trapezoidal trajectory.

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Another thing you can immediately notice is that while this trajectory provides smoothness in velocity,

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the acceleration profile is not continuous.

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OK, now let's analyze each segment one by one.

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The first phase is second order polynomial function.

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It represents constant acceleration.

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The time required to finish the first phase is named as acceleration time and denoted by 2A.

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You can notice it from acceleration profile, while the second phase is a deceleration phase.

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Excuse me, while the second phase is a linear function.

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So zero acceleration, but constant velocity.

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The third phase is caused them deceleration phase.

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It's also a second order polynomial function, and the time required to finish this phase is called

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deceleration time, which is generally chosen the same as acceleration time.

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So trajectory becomes symmetric.

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OK.

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As we have see, the 8A is nothing but acceleration time and chosen the same as this celebration time

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and T is the overall time required for tracking or executing the trajectory.

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T is equal to the difference between final time and the initial time.

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We will assume initial time as zero.

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Otherwise, you can include them very easily.

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Let's deep dive into each phase and analyze equations.

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Let's start with acceleration phase.

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Namely, time range between zero and T, as this is second order polynomial, orientation, velocity

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and acceleration.

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We'll have this structure.

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The boundary conditions are given as initial position of Q and initial velocity of zero.

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But these two conditions are not enough for us.

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We need stored condition to be able to solve for a long time coefficients.

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Let's analyze this velocity graph as you can see a final velocity for the first phase, namely velocity.

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That time T A is equal to the velocity of constant velocity phase or second phase.

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So we can write this third and final boundary condition.

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Now, as we have enough boundary conditions to solve for unknown coefficients, we can write below equations

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in matrix form.

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And so for Vector A..

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Let's continue with constant velocity phase in constant velocity, phase orientation, velocity and

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acceleration.

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We'll have this form.

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As you can see, acceleration is zero in this phase.

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Now let's determine boundary conditions.

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We can determine the initial position from Equation 1.0.

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This formula is from basic physics.

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Well, you can get this equation even from the first phase orientation equation.

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Just plug it instead of tea, because the initial time for a second phase is to say no.

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Again, analyze this graph.

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We need a second boundary condition also to solve for unknown coefficients.

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And this is the initial velocity for this phase, namely Cuvee Dot, which is imposed by us.

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OK, after we have obtained the boundary conditions, let's go for our long coefficients again.

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Not the last phase is deceleration phase, which is similar to the acceleration phase.

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Again, let's determine the boundary conditions.

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The first one is final position or orientation.

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The second one is final velocity, which is zero.

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And the third one is initial velocity, which is nothing but the velocity at time.

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T a t minus T.

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Excuse me.

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This velocity is the same as the velocity at the end of the second phase, and this is constant the

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velocity that we have imposed.

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We cannot determine the unknown coefficients easily after we have analyzed each phase individually.

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In the second part, we will take into account some other issues also.