Mechanical Systems: Stabilization, Trajectory Tracking, and Path Tracking - Prof. Harry G., Study notes of Mechanical Engineering

Download Mechanical Systems: Stabilization, Trajectory Tracking, and Path Tracking - Prof. Harry G. and more Study notes Mechanical Engineering in PDF only on Docsity! Variable Structure Control ~ Mechanical Systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Mechanical Systems Stabilization Trajectory Tracking Path Tracking Example: wheelset Example: Double Pendulum 2θ m1 m2 1 2 1θ ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) 2 2 1 1 2 2 2 1 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 1 1 2 1 2 2 1 2 1 2 2 1 2 2 , 2 cos cos cos sin sin , , sin V q I m m m m m M q m m g m m g m T F p q u g m T θ θ θ θ θ θ θ θ = ⎞⎛ + + + + = ⎟⎜⎜ ⎟+⎝ ⎠ ⎞⎛− + − + + = ⎟⎜⎜ ⎟− + +⎝ ⎠ Mechanical Systems ~ Path Tracking ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , ~ , ~ Given: a path : Problem: find such that n q V q p kinematics M q p F p q G p q u dynamics q R p R q R R u t q t q ε ε = = + ∈ ∈ → → Q P Example: Auto θ Vs = V β rF lF m J, [ ] ( ) ( )2 2 1 0 0 0 cos sin 0 sin cos 0 0 0 0 0 0 2( ) 2 2 0 0 0 2 0 2 0 0 4 2( ) x y y x x y x x x q X Y p v v V J M m m v a b a b v v a f F v F mv a b v v θ ω θ θ θ θ θ ωκ κ κ δ κωω κ κ ⎡ ⎤= = ⎣ ⎦ ⎡ ⎤ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎡ ⎤ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎡ ⎤ − − +⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥= + ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ⎣ ⎦ ⎣ ⎦⎢ ⎥− + − −⎢ ⎥⎣ ⎦ Obstacle Avoidance ( ) ( )2 Create an obstacle security circle centered at with radius : cos 1, ln ln , 2 2 cos D R r D RU r r R Dr D D ρ ϕϕ ρ ϕ ⎛ ⎞− ⎛ ⎞⎜ ⎟= − =⎜ ⎟⎜ ⎟ +− + ⎝ ⎠⎝ ⎠ -1 0 1 2 x -1 -0.5 0 0.5 1 y -2 -1 0 1 U -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 Wheelset ~ Setup ( ) ( ) ( ) ( ) min max min max max max cos sin , , , , x x x x x x x x v y v Mv N v F J K v T M M M J J J N v N K v K θ θ θ ω ω ω ω ω ω = = = + = + = < < < < < < xv Wheelset ~ Sliding Mode ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 2 For perfect tracking with , ln we need speed: ( , ) ,direction: arctan , Switching surfaces ,, arctan , arctan What is the mo x y x x y e e e x U x y x y v v x y U x y U x y s v v U x ys U x y xs y θ λθ θ θ θ ω λ θ = + = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ = − ⎛ ⎞= + = − ⎜ ⎟ ⎝ ⎠ ⎛ ⎞−⇒ = + −⎜ ⎟−⎝ ⎠ ( ) tion like in sliding? Solve kinematics along with , arctanx xv v yω λ θ ⎛ ⎞−= = − −⎜ ⎟−⎝ ⎠