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In the previous lesson, we have seen Quebec polynomial trajectory, namely trajectory that's parameterized

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by polynomial with degrees three.

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We have also noted that while this trajectory provides smoothness in velocity, it doesn't provide smoothness

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in acceleration, and this sudden changes in acceleration can cause stress on the body of the robot

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manipulator.

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So we need trajectory with smooth acceleration profile.

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We are given these boundary conditions, including acceleration.

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As you can see, we have six boundary conditions in total, which makes us in total, which makes us

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said the degree of polynomial at least five.

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Surely we can choose higher for smoother trajectory, but let's choose the minimum possible, namely

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five.

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So we have fifth for the trajectory.

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Let's plug boundary conditions and get this system of equations and write them in matrix form to determine

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the unknown coefficient vector a bi inversion of Matrix M.

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Here are the coefficients that have been calculated.

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Notice that the number of coefficients to determine increased as the degree of polynomial increase.

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Let's jump into the MATLAB and see the code.

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OK, now let's try to find.

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Let's try to plot the order trajectory.

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As you can see, if you have specified the boundary conditions again and now calculating coefficients

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for the polynomial.

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This is good for calculating coefficients.

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As you can see, we first extracted the boundary conditions.

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OK, then the rate and the P vectors OK, which is binary conditions vector in matrics.

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We know that from the slides and a non-conscious are coefficients are calculated by inverting and matrix

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two and multiplying it with P vector.

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OK, and then we are just, uh, creating our time and time span.

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OK.

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And this is the position.

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The velocity, an acceleration points.

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OK.

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At each time sample, then we are just again Viacom concreting for the given range, the trajectory

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and before the initial time position will be, it will be equal to initial position.

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Our velocity and next version will be zero.

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The same is for the after the final time.

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After the final time position will be equal to Q final q final thought velocity and acceleration will

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be zero.

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OK.

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Then we are just plotting OK.

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This is very similar to the, uh, cubic polynomials called.

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So let's just arrange it and see what's happening.

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OK?

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As you can see now, we have a much more smoother trajectory.

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You can even see the smoothness here.

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If you remember at cubic polynomial, the velocity would instantly change and come like that.

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So these were like a 90 degrees angle.

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But in this case, as you can see, these are much more smoother because in fifth degree, polynomial

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velocity and acceleration is also smoother.

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As you can see, they they didn't change it in an instant way.

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OK.

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However, for the acceleration, you can see that as you can see at this point, acceleration changes

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instantly.

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Like it, like instantly, OK?

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Um, however, if you have a continuous acceleration, continuous velocity.

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OK.

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And if you compare also, you will see that the velocity limit is higher for the velocity is higher

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than the cubic polynomial because for the given time range, because in cubic, probably it could be

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a trajectory.

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And and fine, excuse me if you or the trajectory is executed for one second.

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OK.

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So in order to finish in one second, but which would be the smoother way.

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So there's a discontinuous discontinuities.

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So velocity cannot change in an instant way.

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So it should be faster than the cubic trajectory.

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OK, so that's why the velocity is high and it's the general rule.

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If you increase to the same degree polynomial, then the velocity will be higher than a velocity of

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fifths or cubic pulling.

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It could be trajectory here.

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I just want to compare the results of cubic polynomial and fifth order polynomial.

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You can clearly see that in cubic polynomial, maximum velocity is thirty, while in fifth order polynomial,

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it's 37.

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What does this mean?

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This means that as the degree of polynomial increases name, the trajectory becomes smoother maximal

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velocity.

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So variation increases for a given time, because in cubic trajectory, velocity and acceleration can

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change faster compared to fifths or the trajectory, because it's or the trajectory smoother.

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So it needs higher velocity and acceleration to finish the trajectory in a given time.

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Let's summarize some important models, as we have said before, as the degree of polynomial increases,

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smoothness of the trajectory increases, while increasing degree of polynomial is desired because it

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generates smoother trajectory.

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Don't forget that the number of unknown convictions to be calculated also increases additional and other

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smoothness increases.

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We have seen that the required velocity and acceleration to execute the trajectory also increases,

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which requires more capable actuator or driver that can output required velocities and accelerations.