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hw2 - ECE 594D Robot Locomotion Homework 2 Winter 2010(due...

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ECE 594D Robot Locomotion Winter 2010 Homework 2 (due 1/27) 2.1 Rimless Wheel (RW) return map. In this problem, you are asked to annonate Figure 1, in which the solid (blue) lines show the return map relating the post-collision angular velocity after a given step, Ω + n , to the velocity after the next step, Ω + n +1 . (Note that Coleman02 [1] plots return maps on “ Z ”; Figure 1 maps Ω itself.) Hint: Grab a ruler! Also, download the “RW cheatsheet” from the homework website. (Assume a unit length leg ( L = 1 [m]), point mass ( J = 0 ), and 8 spokes ( 2 α = 2 π/ 8 ).) -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 n + (rad/s) n+1 + (rad/s) Figure 1: Rimless Wheel return map. (Last revised January 23, 2010) 1 Homework 2
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ECE 594D Robot Locomotion Winter 2010 2.1 Rimless Wheel (RW) return map. (continued...) a) From Figure 1, what (approximately) is the value of the rolling fixed point, + * ? b) Given your estimate of + * (from part a), estimate the angle of the ground, γ . c) What is the slope of the return map near + * ? d) Given: “ + n = Ω + * + . 01 ”, estimate + n +1 . Hint: Use part c. e) Graphically find the basins of attraction for the standing and rolling fixed points, using a “stair step” approach. Shade in the portions corresponding to the “rolling” fixed point. Hint: down- load the m-files called “RW_return.m” and “RW_stairstep.m” from the homework webpage. Run RW_return to create a return map. Then run RW_stairstep to graphically click on the fig- ure to create stair steps to the appropriate fixed point for this initial condition. You can continue to run RW_stairstep again and again, to overlay many stair step traces. Note that the return map shown will (intentionally) be for a different value of ground slope, γ .
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