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In previous lesson, we have seen how we can solve the issue of requiring too much velocity and acceleration

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from actuators that can actually saturate them.

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We have solved it by using kinematic scaling, but during trajectory planning we have to consider also

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the torque limits and not to saturate actuators by exceeding torque limits.

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If we exceeded, we can then use dynamic scaling to solve the issue.

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While planning trajectory, you have to consider a nonlinear TS and couplings in robot dynamics.

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If your trajectory abides by the torque limits of each individual actuator separately, it doesn't mean

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that it will do the same when they are considered together because of non minorities and couplings.

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OK, let's start with noting down the robot dynamic equation.

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We can separate it for each individual joint as this way.

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Now, let's assume that we are given trajectory cue to be executed in time interval of zero two T.

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Then we can rewrite dynamic equation Abbu as this way by plugging in the trajectory.

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Let's call this part of the equation as tall as I t, because this part depends on not only by configuration,

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but also velocity and acceleration.

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However, granted, the term depends on the the configuration.

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This is the new trajectory v where it is parameterized by function of Sigma T Dash.

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As you can see, Sigma Zero, Sigma zero zero and Sigma T Dash is T limbless sigma is just the scaling

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function of time.

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We will see what's useful for us if we plug this trajectory into the dynamic model of the robot manipulator.

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We will get this equation as Q Tilde T Dash is nothing but Q Sigma T Dash.

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We can find its derivatives very easily by using chain formula we have seen before.

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Here is how we find velocity name live first.

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The first derivative of QTL, not T Dash as Sigma T Dash is nothing but t.

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We can write the derivative of Q with respect to sigma as Q T.

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Now let's find the second derivative of the QTL that he dash.

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It is easy to find that you have to just take another derivative of Q Tilde, the dhoti dash.

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As you can see, we have utilized the chain rule again.

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I specified corresponding terms with colors so you can easily understand what's happening.

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No, the equation can be rewritten in this way.

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As we have formed velocity and acceleration, we can plug them into the equation 1.0 and obtain this

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new equation.

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Don't let it scare you.

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It's very easy that you can stop video and try to follow the each step.

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You can easily notice, though, as I term here again.

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This will help us to ride the equation in a much simpler way.

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And not this.

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Also that gravity will not be affected by time scaling as it depends on the configuration.

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Indeed, in previous slide, you have seen that Sigma only appears in and after the first derivatives

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of the trajectory, so we can write equation in a much simpler way, as in this equation 1.1.

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As you can see, we did not consider gravity torque because it is not affected by the scaling eval appended

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at the end.

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Now it's time to use scaling function of sigma.

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Let's determine it as a linear function.

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Keep this in mind you don't have to choose this function as this only you can choose it as your desire.

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OK, now let's fund derivatives of the sigma so we can plug them in Eq. 1.1 and obtain this very simple

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equation.

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Now you can easily see how we scale the trajectory.

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Now you can also include gravity terms.

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Also, we can choose scaling coefficients of lambda in two ways either it will be less than one.

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In this case, timespan of the trajectory will increase because T equals the number T dash and required

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talks to execute the trajectory will decrease.

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Otherwise opposite is happening.

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Time duration increases and we have to apply more talks to find the trajectory at the right time to

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excuse me.

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We have to apply more torque in order to execute the trajectory through time.

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OK, OK, I know you want to see it in an example, so let's see an example.

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So assume that we.

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Are given the trajectory to be executed by third degree of freedom, robot arm in two time units with

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torque limits of tall, one bar and tall to bar for each one.

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However, after we have calculated the trajectory, we see that the maximum torque while is required

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to execute the trajectory from the joint actuators are higher than the thresholds.

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What we have to do?

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Surely we have to dynamically scale trajectory.

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This is how we do it.

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First, let me not bomb the equation here again.

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From this equation, it's obvious that we can find lambda by just dividing torque limits with maximum

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torque values because we want to decrease talks to the threshold values.

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And surely we have to find the minimum, not maximum, because if we find the maximum, then one of

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the joints will saturate because of high torque value.

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After we have formed the lambda, we can change time to T Dash by scaling the original time span.

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You can also notice that we didn't have to plan the trajectory again.

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We just scaled a time span, and that's it.

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We are in the limits of the torch by dynamic scaling.

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You can also calculate optimal trajectory in terms of execution time, namely, if required, talks

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are lower than the thresholds.

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This means that you don't use full power of your actuators so you can scale down the time span and get

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faster trajectory execution without surpassing the limits.

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You can investigate this problem and more on trajectory planning from this book that I have used for

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this course.

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It explains everything in a simple manner.

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So for more examples, you can refer this book for the problem.

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I have set a minute before I go to the page 240.

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An example 5.6.