8/12/2019 Feedback Control Project

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EE 141 ProjectWinter 2014

Due on March 14 by 5pm

Consider the segway type robot in the gure above. Its dynamics is given by the followingnonlinear differential equations:

(I c + R (m c + m s )) m s dR cos2 x = R F + m s sin (d 2 g cos )

d I c + R (m c + m s ) + m s d2 R cos2 = g sin I c + R (m c + m s )) R cos (F + m s d 2 sin

where we assume that both wheels move with the same velocity so that the robots trans-lational motion is restricted to a line. We denote the position of the robot on this line byx . The angle between the robots body and its natural upright position is . The followingparameters are used in the equations:

Symbol Value Descriptiong 9.81 ms 2 gravitational accelerationR 0.062 m wheel radius

d 0.035 m distance from center of wheel to center of massm c 2 0.14 kg wheels mass (2 wheels)m s 2.6 m c kg bodys mass (2.6 kg is the total robot mass)I c 12 m c R

2 kg m 2 wheel inertia

and F is the force applied on the robot through its wheels. Since we can control this force byregulating the current sent to the motors attached to the wheels, we consider F as the input.

We seek to design a controller that balances the robot in its upright position.

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Problem I: Choose an equilibrium point compatible with = 0 and x = 0. Linearize thedifferential equations around that equilibrium.

Problem II: Consider the linearized equations governing the angle . Through simulation,determine how much the initial conditions can deviate from the equilibrium while keeping theerror between the trajectories of the linearized and the nonlinear model below 10%.

Problem III: Do the linearized equations governing the angle dene a stable system?

Problem IV: The region of attraction of an equilibrium is the largest set of initial conditionsS satisfying the following property: any trajectory starting from S asymptotically converges

to the equilibrium. Design a controller that stabilizes the linearized model and that rendersthe region of attraction of the equilibrium for the nonlinear model as large as possible.

Problem V: Solve again Problem II but using the controller you found in Problem IV.Comment on what you observe.

Problem VI: Simulate the evolution of x using the nonlinear equations and the controlleryou found on Problem IV. Does x converge to a specic location? Comment on what youobserve.

Problem VII: Test how robust your controller is by simulating the nonlinear model start-ing from the angle = 0 and angular velocity = . How large can you make whileguaranteeing that the robot returns to its upright position?

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