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Equation Sheets for final

Equation Sheets for final - EQUATION SHEETS Transformation...

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EQUATION SHEETS Transformation to discrete systems: Forward difference T z s 1 = Backward difference Tz z s 1 = Bilinear transformation 1 1 2 + = z z T s Final Value () 1 1 lim 1 ( ) ss z y zY z ⎡⎤ =− ⎣⎦ ). 1 sin 1 (cos 2 2 1 2 ζω ± = = = ± T j T e e e z n n T j sT n n n Control Algorithm for Desired Input-output Relationship () () () 1 1( ) ()1 d d Yz DzGz Tz R zD z G Dz Gz T z == + = z Controller Design Based on Error-Input Relationship 1 1 1 1 n de n de n nd e e rD z G z m eG zT z + Command generation: i i i B A R = or i i i B A R * * * = ()()( ) ( ) [] i i i i i t r t r t r t r R ± ± 1 1 = and ( ) ( )( ) [ ] i i i i i t b t b t b t b B ± ± 1 1 = Position ( ) ( ) i tt = rA b

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Velocity () i tt =⋅ rA b ± ± Normalized Spline () () ( ) ( ) () () 1 11 1 1 ** * * 01 i ii i i i i i tt u u u dt du u u du du du −− =+ = < < =− = = b r A b rb b A ± ± *3 2 * 2 1 32 1 0 T i i T i i uu u u du ⎡⎤ = ⎣⎦ = b b Three splines with constant acceleration max max 21 2 max max 1 max 2 i V ttt A xx x t VV x t V
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Equation Sheets for final - EQUATION SHEETS Transformation...

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