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Until now, you should have a solid knowledge about robust control, especially variable tribal structure

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controller and chattering them after we have learned all of these concepts.

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It's time to apply them on industrial robot manipulators before continuing.

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Let's remind us some concepts that we have seen during introduction to robust control.

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Firstly, we have said that we couldn't achieve exact compensation in inverse dynamics as we have uncertainties

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in the parameters of the dynamic model of robot arm.

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So we couldn't achieve exactly an organization and decoupling which have seen by this formula, which

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we have seen by this form.

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In this formula, as you can see, which represents uncertainties in the dynamic model doesn't let exactly

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unionisation and decoupling because data itself is non-linear time learning and coupled.

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We have also say that as linear and decoupling aren't exact, it's not enough to apply this simple linear

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input control to obtain asymptotically stable aerodynamics, namely by choosing this as input control.

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We will get this aerodynamics, which is stable but not converges to zero, which we wanted, which

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we wanted to convert to zero.

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But it converges to ETA.

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This means that we will not have zero error in trajectory tracking at the end.

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If we remember that due to the double dot is nothing but the difference between desire desired double

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and triple double talk, then we can manipulate the equation 1.0 and obtain this equation, which will

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help us to determine our dynamic states.

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So we can write this equation in matrix form like this, and from here, you can see that the state

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vector of aerodynamics is seen.

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If we called these two mattresses as H.A., we can write equations simpler.

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As one point two, as we have said, C is nothing but state vector of aerodynamics.

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So Equation 1.3 is 1.2 is the error of dynamics like 1.1 but written in matrix form.

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Our aim is to design the control y to make aerodynamics in Equation Point Asymptotically stable, namely

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see if it'll go to zero.

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Be careful aerodynamics is nonlinear and time varying because of ETA.

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So we cannot apply simple linear control.

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We will instead develop the robust controller to make the inner dynamics asymptotically stable.

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So our purpose is to suppress ETA uncertainty term to make the dynamics asymptotically stable by using

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high energy input control, as we have seen before.

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In order to be able to do that, we have said that we have to obey some conditions like condition A.

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Where we have to note that ETA is bounded because we have said before that robust control can be applied

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successfully.

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If we know the bond on disturbances, know the equation is that and now the equation, the question

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is that is eta bonded.

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And the answer is fortunately yes.

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ETA uncertain to term is bound because it includes dynamic terms like inertia, Coriolis gravity friction

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terms, and we know that they are bonded.

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We have seen this while discussing robot manipulator dynamics and also acceleration velocity and generalized

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coordinates like rotation angles.

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Interval of chance and length in prismatic joints are all bonded because they cannot be in finite.

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So, all in all, uncertain the term ETA is bounded and this means that we can apply a robust controller

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to solve the problem.

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Here are bonds on the terms that constitute uncertain the term ETA Bond film.

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While these equations seems intimidating, they just mean and that desired acceleration is bounded by

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cu m until the term is bounded by.

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And as it contains dynamic model terms and Eq. one point four is bounded as it includes inertia matrix,

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which is not bonded.

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Now we have design, control action y to be able to get asymptotically stable aerodynamics.

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Let's choose it like that.

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As you can see, it consists of previous y with the new term W.

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This W term is added to compensate for ETA uncertainties.

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You will see that we will design W such that to get asymptotically stable system and as we have seen

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before, q the table is just acceleration feedforward term to the system.

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Them PD control part is to stabilize aerodynamics as Cady and keep mattresses are positive, definite

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and can be designed based on dynamic system requirements, namely initial frequency and damping coefficient.

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We will do some manipulations or aerodynamics we have seen before, so let's plug Eq. 1.6 into 1.2 to

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get this equation.

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We can manipulate this and obtain this equation where KP and Cady are combined into one game matrix.

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If we can gain Matrix as a tilde, we can write equation in simple form as in one point seven.

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As you can see, H Tilde is negative definite matrix with negative eigenvalues because it has triangular

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structure, and so CP and key determines define its eigenvalues as h tilde has negative eigenvalues.

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This means that if atom would be zero, then we would get asymptotically stable aerodynamics by setting

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a term to zero.

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But we have uncertainties in the parameters, so not p zero.

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So we have to choose W such that it will take into account also atatu and will make dynamic system asymptotically

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stable.

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We can determine such rule for W by the optimal analysis method.

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Let's choose of function like this.

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As you can see, it is positive definite except origin where C is zero, and this means that the optimal

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function we have chosen is valid.

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We also included Term C because we won't when we thought becomes zero.

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C also becomes zero.

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And surely, as always, Q metrics will be chosen, can be chosen by us and it is positive, definite

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and symmetric matrix.

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Now let's analyze the derivative of the optimal function in order system to be asymptotically stable,

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it has to be negative, definite, as you know.

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Let's plug Eq. 1.7 into this equation and manipulate it and write in this way and know if the group

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returns together and green terms together by taking into account the transpose of Q is equal to itself.

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Equation can be written like that.

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As we have noted before, each tilde is negative definite matrix and Q is positive definite.

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So we can write this equation for positive definite p matrix.

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This is valid because h tilde is negative definite and the Q is positive definite matrix.

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The sum will always be negative definite.

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So for a positive definite matrix p, we will have a unique solution of Q so we can plug P into the

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equation and write it in this almost final form.

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We will adjust to a simplification by calling D Transpose C term, uh, as it perfect.

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No, analyze the terms that constitutes.

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We don't.

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We know that first term is negative definite because it consists of positive, definite matrix of p

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and quadratic relation of C with minus in-front.

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But what about the second term?

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We don't have any idea about it signed.

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It can be zero.

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It can be positive or negative as it is unknown and changing.

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But we want to make the negative definite.

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We have several options for that.

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Let's analyze them.

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The first is that C will be in the null space of D Transpose Q, which will make z zero.

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So second term, the total will be zero.

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But is that realizable?

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No, because C is changing, while d transpose Q is a constant matrix and we cannot always ensure that

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C is in the space of D transpose Q matrix.

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The second option is making W equals to ETA, which we'll make again the second term of the total zero.

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But this is not possible also because in order to do that, we have to move ETA on certain to term.

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Otherwise, how we can say it w equals the ETA.

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However, we don't know ETA term, so this option is also unrealistic.

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And the last and valid option is that to make the second term of it not negative, definite also, then

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we will become some of two negative definite terms and itself will become negative definite.

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We can make sure that the second term is negative definite by choosing W in this way.

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This is indeed very similar to the equation we have seen before in the huddle structure control, namely

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k times sine of ethics, which was sliding surface.

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Here, sliding surface is Z and the bi driving it by right.

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It excuse me, by dividing it with its more the obtained unit are rich in Vector, which is like signs,

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signs seeking them function OK, not Red Cross and 3km function, but signal function because seeing

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them function was needed for us to get the knowledge of in which the action, we have to apply input

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control action.

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And this this is provided by zero divided by law of Z because it gives us unit direction.

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Vector and the row is the same as gain of K, which has to be higher than zero.

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What I want to say is that this equation shouldn't be strange to you because we have analyzed it deeply.

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We have said before that in order to achieve asymptotically convergence with this, input control key

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has to be higher than the magnitude of the disturbance.

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Remember condition eight?

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So in this case, we can apply.

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The same Sor rule has to be higher than the upper bound of ETA uncertain to term.

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Let's see that here we have written previous equation with the new definition of W by using Triangle

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Inequality Row, we can write equation like this.

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We can take, normalize it as common, multiply all side and write equation in this way.

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From here, you can immediately see that if we choose the wrong greater than the normal ETA Z transpose

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atom minus W will be always negative definite.

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So it's not.

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So we thought now let's do final manipulation and describe a row with the bounce we have seen earlier

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because we can determine these bounds from which we can determine bound for ETA.

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And if we choose to greater than this bound, we will ensure that we don't is negative definite.

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So if we apply Eq. one point six in the equation of ETA, we have seen before.

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And by using triangle inequality rule, we can write this as you can see clearly v all of these terms

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are bounded so we can remember the bounds we have defined in Eq. one, one point 1.4 and one point five

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and plug them here and obtain this equation.

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From that equation, we can easily obtain minimum value for rule.

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So if beeches row in this way, we ensure that we don't, we'll be negative definite.

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So aerodynamics will be asymptotically stable, even with uncertainties in the parameters or ETA.

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So all of the control input to the manipulator consists of inverse dynamics control, which consists

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of uncertain dynamic parameters or provide partial compensations or partial generalization and coupling

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the coupling.

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Excuse me, feedforward and PD control together to stabilize aerodynamics.

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Be careful not to make aerodynamics asymptotically stable, but simply stable.

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And finally, robustness term, which ensures asymptotically the asymptotic stability.

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Finally, we can apply boundary layers method simply to avoid or more precisely reduce the effect of

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chattering.

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As you know here, Epsilon is nothing but bits of boundary layers.

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And here is the overall picture of the controller, which will make it easier for you to understand

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the whole control.

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You can see the robustness time is added to the overall controller.

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Before closing, let me not something, as we have said, is it is our sliding surface which contains

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cue tricks that can be chosen by us.

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So a sliding surface dynamics and the aerodynamics can be adjusted by us as we have seen previous lessons

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by adjusting Q metrics.

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Additionally, aerodynamics also will be affected by Cheryl CP and CD terms, which are for stabilizing

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aerodynamics.