practice-midterm1

# practice-midterm1 - P RACTICE M ID - TERM E XAM L INEAR S...

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P RACTICE M ID - TERM E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all your answers. 1. Consider the one-link robot in the following figure: θ τ m x y where denotes the angle of the link with the horizontal, the torque applied at the base, ( x , y ) the position of the tip, the length of the link, I its moment of inertia, m the mass at the tip, g gravity’s acceleration, and b a friction coefficient. This system evolves according to the following equation: I ¨ = - b ˙ - gm cos + . (a) Compute the state-space model for the system when u = is regarded as the input and the vertical position of the tip y is regarded as the output. Please denote the state vector by z to avoid confusion with the horizontal position of the tip x . Hint: Do not forget the output equation! (b) Show that ( t ) = π / 2, ( t ) = 0, 2 t 0 is a solution to the system and compute its linearization around this solution. From your answer, can you predict if there will be problems when one wants to control the tip position

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## This note was uploaded on 12/29/2011 for the course ME 243a taught by Professor Abamieh during the Fall '09 term at UCSB.

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practice-midterm1 - P RACTICE M ID - TERM E XAM L INEAR S...

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