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Cds202-hw7_wi09 - and φ(a Let Q = SE(2 × S 1 represent...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #7 Issued: 19 Feb 09 Due: 26 Feb 09 Reading: Abraham, Marsden, and Ratiu (MTA), Section 5.3; and Kelly and Murray (1994) [from web site] Problems: 1. [Boothby, page 151, #4] Find the one-parameter subgroups of GL (2 , R ) generated by A = b 0 1 - 1 0 B B = b 0 1 0 0 B . Find the corresponding actions on R 2 and their in±nitesimal generators, starting from the natural action of GL (2 , R ) on R 2 . 2. MTA 5.3-1: semidirect product groups 3. MTA 5.3-2: SE(3) 4. MTA 5.3-4: tangent bundle of a Lie group 5. Consider the locomotion system given by a disk rolling on the plane, θ φ ( x,y ) where we assume we can control the angles θ
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Unformatted text preview: and φ . (a) Let Q = SE (2) × S 1 represent the con±guration space for the system. Compute the Lagrangian for the sytem and show that is invariant under the action of SE (2) given by translation and rotation as well as the subgroup of actions given just by translation. (b) Compute the kinematic connection for the system A : TQ → g corresponding to the system rolling without slipping. (c) Determine if the system is totally controllable and/or ±ber controllable. (d) (Optional) Construct an explicit trajectory that moves the system from an arbitrary initial con±guration q = ( x ,y ,θ ,φ ) to the origin....
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