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Assignment-4 - I =   I a I e symm I b-I e I a I e I...

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MECH646 Wheeled Mobile Robots Danielle Sami Nasrallah Department of Mechanical Engineering Faculty of Engineering and Architecture American University of Beirut 2011-01-14
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ASSIGNMENT 4 Question 1: Inertia matrix Given the twist shaping matrix T 1 and the 6 × 6 mass dyad M 1 as follows: T 1 = [ r w b k + l - r w b k l r w h 0 3 r w h ] and M 1 = [ I w O 3 O 3 m w 1 3 ] show that: T T 1 M 1 T 1 = ( r w /b ) 2 I wu + I wl + m w r 2 w symm - ( r w /b ) 2 I wu ( r w /b ) 2 I wu I wl + m w r 2 w 0 I wl + m w r 2 w Question 2: Coriolis and centrifugal terms Given the 6 × 6 angular-velocity dyad W 1 as follows: W 1 = [ 1 O 3 O 3 O 3 ] Show that the contribution of wheel 1 to the Coriolis and centrifugal terms is zero, i.e., 1. T T 1 M 1 ˙ T 1 = O 3 2. T T 1 W 1 M 1 ˙ T 1 = O 3 Question 3: Active and gravity forces Knowing that the active and gravity wrenches for wheel 1 are given by: w A 1 = [ τ 1 l 0 3 ] and w G 1 = [ 0 3 - m 1 g k ] compute 1. T T 1 w A 1 2. T T 1 w G 1 Question 4: Inversion of the inertia matrix The generalized inertia matrix for a MWP moving on a horizontal plane is given by:
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Unformatted text preview: I =   I a + I e symm I b-I e I a + I e I c-I f I c-I f 2 I c + I d-4 I f   The goal of this exercise is the computation of I-1 . For that we start by introducing matrix A and vectors b and v as follows: A =   1-1 0 1 1 1   , b =   b 1 b 2 b 3   and v =   v 1 v 2 v 3   Knowing that Iv = b 1. Multiply the equation above by matrix A on both sides 2. Using the fist row compute ( v 1-v 2 ) 3. Using the second and third row compute ( v 1 + v 2 ) as well as v 3 4. Deduce then v 1 and v 2 separately...
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