lec15 - Outline Motivation Differential Geometry Tools...

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1 Outline ± Motivation ± Differential Geometry Tools stems ± Vector field ± Lie Derivative ± Lie-Bracket ± Integrability ± Controllability of a class of Nonlinear ME6402, Nonlinear Control Sys Systems ± Illustrative Examples Motivation ± Feedback Linearization has received increasing attention in recent years ± It can be used to transform a nonlinear system to one that can be exactly linearized. ± It has been successfully applied to many nonlinear systems like helicopters, aircrafts, and robots. ± There are still several shortcomings and limitations that needs to be further investigated
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2 A Motivating Example: A Revolving Pendulum τ τ = θ + θ b a sin & & Eq. Of Motion: stems θ ME6402, Nonlinear Control Sys What about more general systems? Example of an underactuated system: Acrobot. q 1 q 2 u ( ) = + β β + γ β + γ β + γ β + α u q l m q l +m q l m q q q q q q q q q q q c c 0 sin sin sin sin sin 2 cos cos cos 2 2 2 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 2 2 2 & & & & & & & & Eq. Of Motion : Input u cannot directly cancel nonlinearities!
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3 Mathematical Tools ± Vector field Flow stems ± ± Lie Derivative ± Lie-Bracket ± Integrability of vector fields ME6402, Nonlinear Control Sys Vector Field A vector field is vector valued function f(x) that assigns a vector to each point x. Example: Revolving Pendulum f(x)=[x sin(x) T Example: Revolving Pendulum 2 –sin(x 1 ) ] 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 x1 x2
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4 Flow of a Vector Field The flow of a vector field f at time t, denoted by Φ f t , is a function whose value x(t)= Φ f t (x 0 ) is the stems solution of dx/dt=f(x) at time t starting with initial condition x(0)=x 0 . • The flow starting at x 0 is simply the curve tangent to the vector field f at every point. • The flow always exists (at least locally) for a smooth vector field ME6402, Nonlinear Control Sys • The flow, in most cases, is determined numerically. • The subscript ‘0’ is often dropped from x 0 in Φ f t (x 0 ) to express flow as a function of x: Φ f t (x). Graphical Illustration of Flow 3 4 -1 0 1 2 x Φ f t (x) -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2
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5 Example: Differential Wheeled robots ⎡θ = L R R R R v & & & 2 2 0 Forward Velocity Kinematics: stems θ θ R L L 2 2 Coordinate transformation : θ = θ = sin , cos o o v y v x & & State eq ME6402, Nonlinear Control Sys x y : () θ θ θ θ + θ θ θ + θ = θ L R R L R L L R R R y x & & & & & & & & & 2 sin 2 cos 2 Vector Field/Flow Example Determine the vector field and flow corresponding to ω L =1 and ω R =3 rad/sec. Let R=L=0.5 m.
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This note was uploaded on 09/22/2011 for the course ME 6402 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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lec15 - Outline Motivation Differential Geometry Tools...

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