# hw2 - ECE194D HW2:DueThursday,April28,2011at5pm(indropbox)...

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ECE 194D HW 2: Due Thursday, April 28, 2011 at 5pm (in dropbox) Spring 2011 Page 1 of 3 Homework 2 This assignment focuses on the Jacobian matrix and on motion planning for nonholonomic wheeled systems. Note that although the omnibot discussed throughout has 3 independent degrees of freedom (locally), we actually require 5 total generalized coordinates (globally) to describe the position and orientation of the robot, ( x b , y b , and ߶ ), and the rotations of the wheels, ( q 1 , q 2 , and q 3 ). (Because ߶ can be written directly as a function of q 1 , q 2 , and q 3 , there are only 5 independent generalized coordinates required to describe the system completely, rather than 6.) Problem 1) Derive the Jacobian matrix that relates joint velocities, ݍሶ ൌ ሾݍሶ ݍሶ ݍሶ (i.e., wheel rotations,) to velocities of the center of body of the vehicle chassis, ߦ ൌ ሾ ݔሶ ݕሶ ߶ , where the subscript I indicates that coordinates are with respect to the global (aka “inertial”) reference frame. (Recall the Jacobian is defined by: ߦ ൌܬݍሶ .) For full credit, your derivation should include steps such as 1A through 1C below; parts 1D and 1E are required MATLAB. Pages 47-60 in the Siegwart handout (from Lecture 7) may be helpful; equation (3.19) has the essential relationship you are trying to derive! Figure 1. Left: Omnibot, with global (aka Inertial) and local (aka Relative) coordinate frames aligned. Right: Relative frame is now offset and rotated, wrt inertial frame.

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hw2 - ECE194D HW2:DueThursday,April28,2011at5pm(indropbox)...

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