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We have seen some trajectory planning techniques.

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However, during the planning of these trajectories, we did not consider one thing which is really

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important during trajectory planning.

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We have to ensure that activity velocity and acceleration have to be in the limits of the maximal velocity

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and acceleration that actuator can output.

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Otherwise, actuators will saturate and throughput will not be able to execute the trajectory correctly.

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Let's assume that we have planned the trajectory and we noticed that during execution of the trajectory

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on realisable velocity and acceleration are required.

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What we have to do, we will use kinematics scaling in this case by kinematics scaling.

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We will obtain feasible trajectory that can be actually executed by the robot actuators in order to

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do kinematic scaling on the trajectory.

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It is better to write it in normalized form.

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Let's assume that this is the given trajectory we have to execute with length of age and time lengths

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of T. Let's ride normalized trajectory now, which is denoted by and uh, in the index.

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It changes between zero and one and time lengths is also one.

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It's very easy to obtain the trajectory from normalized trajectory.

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Indeed, we can ride the given trajectory in this way with respect to the normalized trajectory.

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Let's see the logic behind this structure.

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This is normalized trajectory.

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As you can see, it starts from zero and goes the configuration of one, while the time changes from

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zero to one.

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However, we know that the lengths of the given trajectory is not always one, but rather h.

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So we multiply and normalized trajectory with age and get the desired length of the trajectory.

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As you can see, while we get the desired trajectory, length time span stays the same.

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Now we have to consider the initial condition of Q zero because trajectory not always starts from zero,

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but from Q zero.

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So we add Q zero also, OK, we are close to regaining our trajectory.

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We have to consider just time span and initial time of two zero.

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And these features are encoded inside Q and tilde.

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Here is the Q and till the T, as you can see, it is equal to Q and toll where tall is given as below.

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As you can see, by subtracting T zero from T v, consider initial condition, namely T zero mem is

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slow the trajectory t zero ahead, then by dividing T minus T zero by T T v complete the obtained original

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trajectory.

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Here is the way we have gone so far.

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We started from normalized trajectory and obtained the given trajectory.

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OK, this is perfect because this will be very helpful for us.

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Now let's try to find the derivative of the trajectory.

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As you can see, the derivative of initial condition becomes zero as it is constant and we have to find

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derivative of Q.

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And till the T, I know that the formulation of Q and tilde t here again, we will use the chain rule

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of derivative to find this differentiation by considering this rule.

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We cannot differentiate in in this way.

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Derivative of tar with respect to T is nothing but one over T.

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You can see it easily from the formula of tongue.

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In the same way you can find a second derivative of the Q, so we have velocity and acceleration in

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our hands.

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Let's call this as equation one from equation 1.0.

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It's obvious that both velocity and acceleration will get their maximal values when Q and one tau four

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and two and two to reach their maximum.

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We can also determine at which time instant of toll we will get maximum velocity and acceleration from

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equation one point zero.

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You can easily observe that we can change maximal velocity and acceleration.

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Output during execution of the trajectory can be adjusted by change of T.

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Additionally, optimal trajectory in terms of time, namely the trajectory that's executed in minimum

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time, can be determined by setting Q1 and Q2 or T to maximum permissible velocity and acceleration

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limits and determined.

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10.

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OK.

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OK.

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Let's see it in the example.

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Let's assume that we are given this quick trajectory in normalized form.

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So it's zero in time, zero one in time one and velocity of the trajectory is zero at the beginning

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and the end.

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Is on these boundary conditions, we can ride the system of equations and determine coefficients for

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the normalized trajectory after a more efficient have been determined, coefficients have been determined,

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normalized trajectory can be constructed and velocity acceleration and other higher order derivatives

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can be found if required.

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Now let's determine the maximal values of velocity and acceleration for the maximum value of the velocity,

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we have to determine the instant total we are.

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QM one will be maximum.

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We can find it by making its derivative equal to zero, which gives value of zero point five for thought.

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If you plug this a volley in velocity formula, we will get this for the maximum value of the normalized

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trajectory velocity.

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From this, we can easily find the maximum velocity of the original trajectory.

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This can be determined from Equation 1.0 zero.

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We can do the same for acceleration also.

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And here is the value of it.

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Now, let's assume that the initial point of the trajectory is 10 and final point is 50, which gives

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lengths of h or 40.

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Maximum permissible velocity and acceleration values are given a certain age, respectively.

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By plugging them into the above formulas we have obtained, we can determine timespan of D that will

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help us to execute trajectory while abiding by limits.

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As you can see, we have formed two different values for T.

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We will surely find their maximum, namely two, because if you find their minimum, then while acceleration

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will be in the limits velocity, what sets read which velocity would cause the mothers to saturate actuators

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to saturate, which we don't want.

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Indeed, you can do this exact calculation for even higher degrees of polynomials, and you will notice

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that while the degree of polynomial increases name, smoothness of the trajectory increases.

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This time Ti also increases.

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So if you want to achieve smoother trajectory, you will have to execute it in larger time span.

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Why?

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Because your velocity acceleration becomes smoother.

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So a robot cannot accelerate instantly, so or decelerate instantly.

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So it takes more time to execute smoother trajectory.