Assignment-2

# Assignment-2 - 2 Use sinusoidal inputs to drive the...

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MECH646 Wheeled Mobile Robots Danielle Sami Nasrallah Department of Mechanical Engineering Faculty of Engineering and Architecture American University of Beirut 2010-10-28

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ASSIGNMENT 2 For questions 1 and 2 we consider the unicycle shown below: Figure 1: Unicycle whose two inputs are the steering rate u 1 = ˙ φ and the heading velocity u 2 = r w ˙ θ . The kinematic model is given by ˙ q = ˙ x ˙ y ˙ φ = 0 0 1 u 1 + cos φ sin φ 0 u 2 Question 1: Chained form The change of coordinates is given by: x 1 = - φ x 2 = x cos φ + y sin φ x 3 = - x sin φ + y cos φ while the input transformation is, in turn, v 1 = - u 1 v 2 = ( - x sin φ + y cos φ ) u 1 + u 2 1. Compute ˙ x 1 , ˙ x 2 and ˙ x 3 2. Show that the new state vector combined to the new inputs corresponds to one-chain system
Question 2: Sinusoidal Inputs Given the initial conﬁguration x 0 y 0 φ 0 = 3 2 π/ 4 and the ﬁnal conﬁguration x f y f φ f = 0 0 0 1. Apply necessary transformations to change initial and ﬁnal conﬁgurations to their corresponding one in chained form
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Unformatted text preview: 2. Use sinusoidal inputs to drive the unicycle to the ﬁnal goal in 10 seconds. 3. Retrieve x and y coordinates and plot the x-y graph showing the steering of the unicycle Question 3: Piecewise Constant Inputs We consider the following chained-form system: ˙ q 1 = u 1 ˙ q 2 = u 2 ˙ q 3 = q 2 u 1 ˙ q 4 = q 3 u 1 (1) Write an algorithm that computes the piecewise constant inputs needed to drive the system from any initial conﬁguration:     q 1 q 2 q 3 q 4     to the ﬁnal conﬁguration:     q 1 f q 2 f q 3 f q 4 f     in T seconds. For veriﬁcation purposes, to steer the system from [ 4 0 1 1 ] T to [ 0 0 0 0 ] T in 12 seconds the inputs must be as follows: v 1 =-1 3 , v 21 = 15 64 , v 22 =-9 32 , v 23 = 3 64...
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Assignment-2 - 2 Use sinusoidal inputs to drive the...

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