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So now let's have a look at the pendulum, the system, a simple pendulum system has a mass that suspended

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from a pivot point that allows it to freely swing back and forth.

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Breaking the system down, that's a shame that we have a weight of mass m at the end of this weightless

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arm of length L from a pivot point, the angle that the arm is can be described by the angle theta and

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its velocity is described by the angular right theta dot.

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The forces tau can be applied to the system as the input or just left as zero for the free response

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or unforced response of the system.

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The equation of motion, of the angle of motion can be derived from the forces and inertia which results

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in the nonlinear differential equations that describe the motion of the dynamic system as shown here.

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You can see from the equation that is a non-linear due to this time here, ordinary differential equation.

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So the first step in modeling this system is to break it down into the state's best representation of

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two first order systems.

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So the states X1 and x2 of the system become theta and theta dot with the state right dynamics for X1

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and x2 described by using these equations here.

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Now, if we assume that our state X1 or Theta is always going to be a small angle, so say within 15

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degrees, then we can approximate this sine Theta X here using this small angle approximation and we

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just can approximate it to be theta or X1.

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So the system expressed in the linear continuous space form is approximately given by this linear states

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based system here.

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So we can say we replace the sign X with just X or Theta and we've ended up with this date space from

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here.

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Now we can turn this continuous linear system into a discrete form using the matrix exponential relationship

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for filtering an estimation as we've done in the past.

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So let's build a path, a model for the non-linear and linear approximation of the system and simulate

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the dynamics of both systems and see the responses.

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So here we have the Python script called Assignment to underscore Sim's Peiyi.

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Now this simulation script simulates a free or unforced response of the linear and nonlinear versions

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of the pendulum dynamic system.

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So we have a look at the code here, we can see.

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We first set up the simulation properly, so we set up a time set up date of our point, zero one seconds,

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a simulation time of 10 seconds.

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So we're going to simulate the dynamics for over ten seconds.

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We set up our system properties for the pendulum.

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So we define our gravitational constant being a nine point eight one, we have the length of the pendulum

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pain point five meters.

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And then we also set up the initial theta for the initial condition of the pendulum.

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And we've said it to be ten degrees.

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This part of the curve here, this is where we define the linear system.

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So this is the linear approximation and on the system and we use the continuous form to work out the

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straight matrix using our matrix exponential.

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And then lower in the code, we can actually see where we do the integration.

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So this is a simulation code that steps through time.

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We work at the linear system dynamics by basically propagating our X linear state, using our F matrix

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multiplied by our current X matrix.

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Our nonlinear equations are calculated here.

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So we work out our theta acceleration and we integrate that to get our theta dot.

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And then we do the same thing for Theta and then we have some code down here that just does animation.

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So let's run this simulation as is.

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So we're starting off our simulation with the pendulum being at 10 degrees.

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So let's run this.

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So you can see we have an animation pop up and there's actually a red line and a blue line in this animation,

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the red line is the linnaeus' system, the blue line is the nonlinear system.

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So we can see when we started this simulation off at 10 degrees, both systems are fairly well aligned.

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They both agree there's not too much variation between the two different systems.

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So we'll run us again.

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So you can see that both of the non-linear and linear responses pretty much track each other.

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So there's little error between the two.

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Now, if we were to run this with a different initial angle, so let's say instead of 10 degrees, we

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set this to be 45 degrees.

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Now I angle of 45 degrees is going to be definitely outside the linear zone of the sine function.

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So it will be a bad approximation to use a small angle approximations on angles such as 45 degrees.

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So now let's run the simulation.

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OK, now we're getting a very big difference between a linear system and nonlinear system.

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So, again, the red line here is the linear system.

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The blue line here is the nominee system.

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So you can see quite a dramatic difference between the two.

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And the longer you run, this is just going to continuously be out of phase and offset and drift.

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So we have to be careful when we're using a litany, linear approximation of a nominee system.

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We have to make sure that we are operating inside the linear range for the linear approximation to be

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valid.

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So when we use a common Thawra to estimate our angle theta, we just have to be wary of these limitations.