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After we have seen stiffness control, now it's time to switch to the impedance control, which is very

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widely known and use force control technique.

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Indeed, by using impedance control and its dual admittance control, you can achieve pretty interesting

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results.

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So without further ado, let's jump into the concept based on our previous knowledge.

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We somehow understand that modeling the environment is very important for force control techniques because

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based on the characteristic of the environment, we adjust the properties of the robot manipulator in

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order to achieve desired results or await dangerous situations by stiffness control.

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We accept the robot and the environment as just springs, which was not totally correct because environment

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doesn't only have stiffness properties, but also inertial and resistive properties, so environment

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can be in different categories as well.

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Let's first see this equation, which gives us the relation between the interaction forces and the motion

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of the robot manipulator.

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This relation is provided by the mechanical impedance, which is designated by Zip e in this equation.

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Let's see why this is called mechanical impedance.

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While modeling the systems, we separate two kinds of variables, namely effort and flows, which help

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us represent instantaneous flow of energy or power.

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So while the system can be different in nature like either mechanical or electrical or hydraulic, there

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are always the power variables, namely effort and flow in electrical domain.

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Effort is force, which corresponds to the four, which corresponds to the force, excuse me, in electrical

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domain.

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If it is voltage, which corresponds to the force in mechanical domain and the current is flow variable

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in electrical domain, which corresponds to the velocity in mechanical domain.

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So by the same way, we can write electrical impedance, which is the ratio between voltage and the

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current, which corresponds to mechanical impedance, namely the ratio between force and velocity.

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Based on these mechanical impedance, we can categorize or model the environment more precisely.

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Namely, the environment is inertial if and only if the impedance has value of zero when, as is zero.

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For example, this is an inertial environment in which force of F is given to the object.

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This is the mass H which cause velocity of export at the output.

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We can write the equation of the system and get the mechanical impedance.

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As you can see, if you set s to zero, then mechanical impedance will be zero in one.

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When these resistive if and only if z zero is constant, which is between zero and infinity.

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This is the example case where the robot is moving or the surface.

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The less friction system equations can be written this way, and we can get mechanical impedance easily

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from that system equation.

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As you can see, even we said s to zero.

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We get constant volley of D and environment can be capacitive if unduly if z zero is infinite.

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Below is the sample case where environment has mass of h, dumping of B and stiffness of key system.

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Equation can be rewritten in this way and we can get impertinence from this equation easily.

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As you can see, if you set s to zero, you will get infinite impedance from above examples.

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It's clear that how effectively an environment can be modelled by using mechanical impedance and how

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mechanical impedance determines the behavior of the system in this case environment.

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No, let's start to impedance control now.

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It's important for us to understand what we want to achieve by impedance control.

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The goal in utilizing impedance control is not to get exact position or velocity control or not to achieve

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desired force, namely pure force control.

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The goal is to control that dynamic behavior of the robot manipulator, namely the relationship between

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the motion of the robot and the interaction forces.

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Why do we want to achieve that?

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Because, as you know, from stiffness control periods, we need to have sufficiently good model of

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the environment to achieve the desired control on interaction forces.

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However, as we have said previously, it's very difficult to model the environment.

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But if we can find the relation between the motion of the robot and the interaction forces, then we

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will not need to model the environment because it will be sufficient for us to adjust just to measure

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interaction forces at.

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And the factor.

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And based on that, adjust the behavior of the robot, we humans do also the same when there is no interaction

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force at our hands, namely we move our hands in free space.

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We do that in the fast and rigid way.

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However, if we are in contact with some object, then we are slow and careful near make compliant the

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object.

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Do you want to achieve that result in robot manipulator or.

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Let's investigate how they can do that.

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Let's first see why it is called impedance control.

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As you know, generally when the robot interacts with some environment, it gives acceleration velocity

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to the environment by applying force into it.

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So environment has admittance while the robot has impedance.

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And we want to control that impedance.

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Let's ride the formula of mechanical impedance again.

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Based on Le Plus formulation, we can write equation in this form also.

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As you can see, this is nothing but a transfer function.

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If you remember in stiffness control, stiffness was constant, while in this case, impaired is a transfer

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function.

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Now let's define transfer function in this way, as you can see, impedance control is more general

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than the stiffness control because you don't only control the stiffness, but also the effective inertia

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and damping.

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Also, it will be more clear to you further in the presentation.

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So if we plug the definition of impairments in the formula, we will get this equation, which is desired

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impedance or desired dynamical behavior more correctly.

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We want to impose to our robot.

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This is done by fixing the values of inertia, stiffness and damping mattresses.

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If the right to transfer function in this way, we can see that by fixing the values of damping, stiffness

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and inertia mattresses, we can define the values of natural frequency and damping ratio, which characterizes

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our system response, which is second order system response here.

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Let's ride the dissolved dynamic behavior if you want to impose to our robot.

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However, there is a problem here.

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There is a huge difference between the original dynamics of our robot arm and desired dynamics we want

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to impose.

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Additionally, original dynamics of our robot is coupled, and this also creates a problem because we

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want to impose different dynamics based on the situation in different directions.

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So in the first step, we have to cancel the original dynamics of our robot in order to get leaner and

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the coupled dynamics.

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We will do that using inverse dynamics control.

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Then we impose the desired dynamics to our robotic manipulator.

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Let's first try the dynamics equation of our robot manipulator in Workspace Y workspace because the

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desired dynamics we want to impose is defined in workspace.

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Also, the interactions in workspace also the interactions happen in the workspace.

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Be careful in these formulations.

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We use analytical Jacobean because the rotation angles are defined either as earlier or are p y.

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It's not defined the input force of F as this, which will help us to cancel the original dynamics of

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our robotic manipulator.

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Don't forget, we have to consider interaction forces of f a.

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In order to cancel them.

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Also, these interaction forces in Eq. one point two will be obtained by force torque sensors.

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So we replace Eq. 1.0 in the equation 1.1 and get linear and decoupled dynamics as below.

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As we have said before, it's important for us to decouple the equation in order to impose desired behavior

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in each direction freely.

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Here you can see that, but we have an obvious and common problem.

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Here you can say that we have an obvious and common problem with inverse dynamics, which is not being

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able to model the dynamics.

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Exactly, exactly.

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Additionally, force torque sensors cannot exact the council interaction forces.

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However, robustness is not a big issue here because we just want to impose behavior to our robot arm.

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We just wanted these impose desired behavior to our robot arm.

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We don't want exact position or velocity tracking or achieving exact design forces.

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Now we have decoupled and linear ized our system dynamics.

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We need to switch to the second step, namely imposing desired dynamics that our robot manipulator.

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So we define our Y as this way in order to achieve desired dynamic behavior.

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When we replace Eq. 1.4 in 1.3 here of here, I want to note something important.

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As you have seen moment before, we have canceled interaction forces.

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We have done that in order to decouple our original dynamics equation.

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Now we are adding it into y again y.

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Let's see what would happen if we didn't add it.

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If we didn't do that, then the imposed by that mix equation would be in this way.

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What does this dynamic equation say to us?

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Firstly, we know that our system is asymptotically stable because of non-zero damping term.

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Secondly, from this equation?

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You see, that robot always tries to achieve exact the desired position without considering environment

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or without either an introduction forces.

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You can imagine that this can cause very serious issues, both for the robot manipulator and the environment.

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We need to adjust dynamic behavior of our robot based on interaction forces, and we can achieve that

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by adding interaction forces in Equation 1.4.

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So after defining Variable Y, let's see the full formula for control input F, and this is given as

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below.

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Here, I want to note something if you analyze equation one point six carefully, you will see that

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if we define our desired inertia matrix as the original robot inertia matrix, then we can avoid utilizing

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force.

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Torque sensors and control equation becomes simpler, but we don't want to do that.

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Why?

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Because original inertia matrix is coupled, but we want to decouple the equations, so we prefer positive,

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definite and decoupled inertia matrix.

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Now, after all, how do we how do we have to choose the desired impedance in order to get desired dynamics

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behavior?

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We want to avoid high impacts due to the uncertainties in environment, position and shape.

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And also, we want to adapt the stiffness properties of the stiffness properties of the environment.

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We can achieve that by mimicking human behavior, namely fast and rigid behavior in free space motions,

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which corresponds to the low values of four inertia and high values for stiffness via slow and compliant

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behavior during approaching the environment or during interaction with the environment.

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This corresponds to higher values for inertia matrix, while low values for stiffness matrix.

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We want to assign small values for stiffness, matrix during interaction with stiff environment, and

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in order to get desired trends in behavior such as control and oscillations, we have to adjust damping

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matrix.