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OK, after we have seen one of the simpler, centralized control techniques, namely PD plus gravity

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compensation control, let's not not learn inverse dynamics control.

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This is very important control technique indeed, which constitutes just of many other control algorithms.

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So let's try to understand this as much as possible.

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If everything OK, let's continue.

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Let's ponder about why the control of the robot manipulator is complicated.

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What makes what makes it difficult?

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Indeed, we have seen robot manipulator dynamic model before.

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We have seen that it is coupled and nonlinear.

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Coupled means when we try to control one joint, it affects others also.

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So this makes control complex because while we are trying to control one joint, you have to take into

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account the other, the joints also.

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Additionally, the nonlinear the robot dynamic model requires nonlinear control algorithm to be implemented,

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which is surely harder than the simple linear control algorithm.

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So it would be better and easier for us to get rid of the nonlinearity in the robot dynamics, decouple

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it and apply linear control or separately for each joint.

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The big question is that can we do that?

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Ideally, yes, and we will see how we will accomplish this mission.

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In this lesson, we will see at the end of the lesson why I have said ideally, OK, here is a dynamic

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model of the robot manipulator that we have seen before.

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We can write it in more compact form in this way, as we have seen it before.

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Robot dynamics is non-linear in terms of joint positions velocities.

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However, if you look at the equation 1.0, you will see that the dynamics equation indeed linear in

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terms of input control.

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Paul, just accept M and M as some variables and you will see that you are getting linear equality.

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So what does this mean?

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This means that we can indeed linear ize our dynamic model.

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By specifying input control of toll after linear transition and decoupling, we can simply apply linear

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control.

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Let's see how we will do that.

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Let's choose Tal like that where y is new control input to the nearest and decoupled system?

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If we plug Eq. 1.1 into 1.0, we will get this equation, which represents new system dynamics.

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This new dynamics has been obtained by exact compensation because we have assumed that we know the values

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of inertia, matrix M and other dynamic terms.

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And exactly so we construct the tunnel as in Equation 1.1.

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Keep this in mind.

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This is key concept, and the new system dynamics is decoupled and linear, which is what we wanted.

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As you can see, each y i component influences with a double integral relationship only to the Q I joined

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variable and there is no end dependency, so we have accomplished first task.

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Now we have to design new control input y as linear controller.

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Let's do that.

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Here is the definition of Y.

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As you can see, it's simple p d linear control with reference input of R, we will see that what is

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R exactly?

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Now, if you plug Eq. one point three back to the Equation 1.2, you will obtain this second order dynamic

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system, which is asymptotically stable because CP and K the mattresses are positive.

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Definite you can design, copy and KDE Matters is indeed based on the specification of your required

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system parameters like natural frequency and damping ratio.

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Now we can obtain error dynamics and analyze it.

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In order to do that, let's assume that we are on a given trajectory with desired position of cure the

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velocity of the Q.

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But the acceleration of double boat.

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The then you can write a reference are like that.

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And if you plug our back into the equation 1.4, you will obtain aerodynamics.

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As you can see, it is asymptotically stable because of positive, definite Cadee and KP mattresses,

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which is what we wanted.

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This means that QTL and Cure Tilt the board will convert to zero irrelevant of the initial project through

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error, so we will get the desired orientation and, excuse me, desired configuration and velocity

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tracking.

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Desired velocity does not have to be constant as in the case of PD and gravity compensation.

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Additionally, I want to note that aerodynamics convert.

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It is complete.

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Convergence is controlled by CP and Keady mattresses, as you can see.

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So by tuning them, you can get different system response in terms of aerodynamics.

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Perfect.

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We have obtained what we wanted.

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Here's a complete picture of the control algorithm, namely in inner loop.

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We are linear rising a decoupling system, dynamics nonlinear and coupled system dynamics by nonlinear

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state feedback as we are using inverse dynamics during comp. This control algorithm is called inverse

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dynamics control after system dynamics as being the nearest and decoupled we can apply all through look

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are to loop linear control, which is a combination of PD control, action and desired acceleration.

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Fit for the purpose of ultra loop is to stabilize the aerodynamics.

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OK, indeed, we get very good result.

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Ideally, we have some problems with inverse dynamics in practice.

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My friends, first of all, we have to compute inverse dynamics in real time, which requires many computing

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resources.

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Secondly, as I have said before, we have assumed exactly innovation in inverse dynamics control.

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Indeed, it was the fundamental concept of inverse dynamics in which the whole control method is based

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on.

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However, in reality, we cannot get exactly near position and decoupling because of uncertainties in

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the dynamic parameters of robot dynamics on land and the fact that a payload that can change dynamic

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parameters and the existence of a model dynamics as we can not model every dynamics of robot manipulator

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exactly like fiction, we cannot do exact compensation, which caused non-linear rotors and couplings

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in the final dynamics, namely that this control method is not robust against disturbances and uncertainties.

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But don't worry, in the future lessons, we will see how we will make this control method robust against

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disturbances and uncertainties by using robust control techniques.