Ch. 6 Single Variable Control ES159/259

Transcription

1 Ch. 6 Single Variable Control

2 Single variable control How o we eterine the otor/actuator inut o a to coan the en effector in a eire otion? In general, the inut voltage/current oe not create intantaneou otion to a eire configuration Due to ynaic (inertia, etc) Nonlinear effect Backlah Friction Linear effect Coliance Thu, we nee three baic iece of inforation: 1. Deire joint trajectory. Decrition of the yte (ODE Orinary Differential Equation) 3. Meaureent of actual trajectory

3 SISO overview Tyical ingle inut, ingle outut (SISO) yte: We want the robot track the eire trajectory an reject external iturbance We alreay have the eire trajectory, an we aue that we can eaure the actual trajectorie Thu the firt thing we nee i a yte ecrition

4 SISO overview Nee a convenient inut-outut ecrition of a SISO yte Two tyical rereentation for the lant: Tranfer function State-ace Tranfer function rereent the yte ynaic in ter of the Lalace tranfor of the ODE that rereent the yte ynaic For exale, if we have a 1DOF yte ecribe by: τ ( t ) J θ ( t ) B θ ( t ) We want the rereentation in the Lalace oain: τ ( ) Jθ ( ) Bθ ( ) ( J B) θ ( ) Therefore, we give the tranfer function a: θ ( ) ( ) 1 1/ J P τ J B B / J ( ) ( ) ( )

5 Review of the Lalace tranfor Lalace tranfor create algebraic equation fro ifferential equation The Lalace tranfor i efine a follow: x t ( ) e x( t ) For exale, Lalace tranfor of a erivative: { x( t )} L x L t 0 t ( t ) x( t ) 0 e t t t Integrating by art: x L t ( t ) e x t x t ( t ) e x( t ) 0 ( ) x( 0) 0 t

6 Review of the Lalace tranfor Siilarly, Lalace tranfor of a econ erivative: L { x ( t )} x L t 0 ( t ) x( t ) e t ( ) x( 0) x ( 0) Thu, if we have a generic n orer yte ecribe by the following ODE: x t bx t kx t F t An we want to get a tranfer function rereentation of the yte, take the Lalace tranfor of both ie: L t t ( ) ( ) ( ) ( ) { x ( t )} bl{ x ( t )} kl{ x( t )} L{ F( t )} x( ) x( 0) x ( 0) b( x x 0 ) kx( ) F( ) ( ) ( ) ( ) x

7 Review of the Lalace tranfor Continuing: ( b k ) x( ) F( ) x ( 0) ( c) x( 0) The tranient reone i the olution of the above ODE if the forcing function F(t) 0 Ignoring the tranient reone, we can rearrange: ( ) x 1 F( ) b k Thi i the inut-outut tranfer function an the enoinator i calle the characteritic equation

8 Review of the Lalace tranfor Proertie of the Lalace tranfor Take an ODE to a algebraic equation Differentiation in the tie oain i ultilication by in the Lalace oain Integration in the tie oain i ultilication by 1/ in the Lalace oain Conier initial conition i.e. tranient an teay-tate reone The Lalace tranfor i a linear oerator

9 Review of the Lalace tranfor for thi cla, we will rely on a table of Lalace tranfor air for convenience Tie oain Lalace oain x ( t ) x( ) L x( t ) 0 t { } e x( t ) x ( t ) x( ) x( 0) x ( t ) x( ) x( 0) x ( 0) Ct C te 1 co( ωt ) in( ωt ) ω ω ω t

10 Review of the Lalace tranfor Tie oain Lalace oain x ( t α ) H( t α ) e α x( ) ( t ) x ( a ) e at x x( at ) Cδ ( t ) 1 x a a C

11 SISO overview Tyical ingle inut, ingle outut (SISO) yte: We want the robot track the eire trajectory an reject external iturbance We alreay have the eire trajectory, an we aue that we can eaure the actual trajectorie Thu the firt thing we nee i a yte ecrition

12 SISO overview Nee a convenient inut-outut ecrition of a SISO yte Two tyical rereentation for the lant: Tranfer function State-ace Tranfer function rereent the yte ynaic in ter of the Lalace tranfor of the ODE that rereent the yte ynaic For exale, if we have a 1DOF yte ecribe by: τ ( t ) J θ ( t ) B θ ( t ) We want the rereentation in the Lalace oain: τ ( ) Jθ ( ) Bθ ( ) ( J B) θ ( ) Therefore, we give the tranfer function a: θ ( ) ( ) 1 1/ J P τ J B B / J ( ) ( ) ( )

13 Syte ecrition A generic n orer yte can be ecribe by the following ODE: x ( t ) bx ( t ) kx( t ) F( t ) An we want to get a tranfer function rereentation of the yte, take the Lalace tranfor of both ie: L x t bl x t kl x t L F t { ( )} { ( )} { ( )} { ( )} ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x x 0 x 0 b x x 0 kx F ( b k ) x( ) F( ) x ( 0) ( c) x( 0) Ignoring the tranient reone, we can rearrange: ( ) x 1 F( ) b k Thi i the inut-outut tranfer function an the enoinator i calle the characteritic equation

14 Exale: otor ynaic DC otor are ubiquitou in robotic alication Here, we evelo a tranfer function that ecribe the relationhi between the inut voltage an the outut angular ilaceent Firt, a hyical ecrition of the ot coon otor: eranent agnet torque on the rotor: φ τ 1 i a

15 tator Phyical intantiation coutator rotor (arature)

16 Motor ynaic When a conuctor ove in a agnetic fiel, a voltage i generate Calle back EMF: V b φω Where ω i the rotor angular velocity arature inuctance arature reitance i L t a Ri a V V b

17 Motor ynaic Since thi i a eranent agnet otor, the agnetic flux i contant, we can write: τ 1φi a ia Siilarly: V b φω θ t an b are nuerically equivalent, thu there i one contant neee to characterize a otor b torque contant back EMF contant

18 Motor ynaic Thi contant i eterine fro torque-ee curve Reeber, torque an ilaceent are work conjugate τ 0 i the blocke torque

19 Single link/joint ynaic Now, let take our otor an connect it to a link Between the otor an link there i a gear uch that: θ rθ L Lu the actuator an gear inertia: J J J Now we can write the ynaic of thi echanical yte: a g J θ t B θ t τ τ L r i a τ L r

20 Motor ynaic Now we have the ODE ecribing thi yte in both the electrical an echanical oain: In the Lalace oain: r i t B t J L a τ θ θ t V Ri t i L b a a θ ( ) ( ) ( ) ( ) r I B J L a τ Θ ( ) ( ) ( ) ( ) V R I L b a Θ

21 Motor ynaic Thee two can be cobine to efine, for exale, the inut-outut relationhi for the inut voltage, loa torque, an outut ilaceent:

22 Motor ynaic Reeber, we want to exre the yte a a tranfer function fro the inut to the outut angular ilaceent But we have two otential inut: the loa torque an the arature voltage Firt, aue τ L 0 an olve for Θ (): ( J B ) Θ ( ) Θ V I a ( ) ( L R)( J B) Θ ( ) V ( ) Θ ( ) ( ) ( ) [ ( L R)( J B ) ] b b

23 Motor ynaic Now conier that V() 0 an olve for Θ (): I a ( ) b L R Θ τ L ( ) Θ ( J B ) Θ ( ) ( ) ( ) ( L R) / r [( L R)( J B ) ] b bθ L R ( ) τ ( ) L r Note that thi i a function of the gear ratio The larger the gear ratio, the le effect external torque have on the angular ilaceent

24 Motor ynaic In thi yte there are two tie contant Electrical: L/R Mechanical: J /B For intuitively obviou reaon, the electrical tie contant i aue to be all coare to the echanical tie contant Thu, ignoring electrical tie contant will lea to a iler verion of the reviou equation: Θ V Θ τ L ( ) / R ( ) [ J B / R] ( ) 1/ r ( ) [ J B / R] b b

25 Motor ynaic Rewriting thee in the tie oain give: By ueroition of the olution of thee two linear n orer ODE: ( ) ( ) [ ] R B J r b L / 1/ τ Θ ( ) ( ) [ ] R B J R V b / / Θ ( ) ( ) ( ) ( ) ( ) t R V t R B t J b / / θ θ ( ) ( ) ( ) ( ) ( ) t R t R B t J L b τ θ θ / / 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) t R t R V t R B t J L b τ θ θ / 1 / / J B ( ) t u ( ) t

26 Motor ynaic Therefore, we can write the ynaic of a DC otor attache to a loa a: J θ t B θ t u t t ( ) ( ) ( ) ( ) Note that u(t) i the inut an (t) i the iturbance (e.g. the ynaic couling fro otion of other link) To rereent thi a a tranfer function, take the Lalace tranfor: ( J B) Θ ( ) U( ) D( )

27 Setoint controller We will firt icu three initial controller: P, PD an PID Both attet to rive the error (between a eire trajectory an the actual trajectory) to zero The yte can have any ynaic, but we will concentrate on the reviouly erive yte

28 Proortional Controller Control law: u Where e(t) θ (t) - θ(t) in the Lalace oain: U ( t) e( t) ( ) E( t) Thi give the following cloe-loo yte:

29 PD controller Control law: Where e(t) θ (t) - θ(t) in the Lalace oain: U u ( t ) e( t ) e ( t ) Thi give the following cloe-loo yte: ( ) ( ) E( t )

30 PD controller Thi yte can be ecribe by: ( ) ( ) U D Θ ( ) J B Where, again, U() i: ( ) ( )( ( ) Θ ( ) ) U Θ Cobining thee give u: Θ Θ ( ) J B Solving for Θ give: ( )( ( ) Θ ( ) ) D( ) ( J B) Θ ( ) ( ) Θ ( ) ( ) Θ ( ) D( ) ( J ( B ) ) Θ ( ) ( ) Θ ( ) D( ) ( ) Θ ( ) D( ) Θ ( ) J ( B )

31 PD controller The enoinator i the characteritic olynoial The root of the characteritic olynoial eterine the erforance of the yte ( B ) 0 J J If we think of the cloe-loo yte a a ae econ orer yte, thi allow u to chooe value of an ζω ω 0 Thu an are: ω J ςωj B A natural choice i ζ 1 (critically ae)

32 PD controller Liitation of the PD controller: for illutration, let our eire trajectory be a te inut an our iturbance be a contant a well: C D Θ ( ), D( ) Plugging thi into our yte ecrition give: ( ) C D Θ ( ) ( J ( B ) ) For thee conition, what i the teay-tate value of the ilaceent? ( ) C D ( ) C D C D D θ li li C 0 ( J ( B ) ) 0 J ( B ) Thu the teay tate error i D/ Therefore to rive the error to zero in the reence of large iturbance, we nee large gain o we turn to another controller

33 PID controller Control law: In the Lalace oain: ( ) ( ) ( ) ( ) t t e t e t e t u i ( ) ( ) E U i ( ) ( ) ( ) ( ) ( ) i i B J D 3 Θ Θ

34 PID controller The integral ter eliinate the teay tate error that can arie fro a large iturbance How to eterine PID gain 1. Set i 0 an olve for an. Deterine i to eliinate teay tate error However, we nee to be careful of the tability conition i < ( B ) J

35 PID controller Stability The cloe-loo tability of thee yte i eterine by the root of the characteritic olynoial If all root (otentially colex) are in the left-half lane, our yte i table for any boune inut an iturbance A ecrition of how the root of the characteritic equation change (a a function of controller gain) i very valuable Calle the root locu

36 Proortional Suary A ure roortional controller will have a teay-tate error Aing a integration ter will reove the bia High gain () will rouce a fat yte High gain ay caue ocillation an ay ake the yte untable High gain reuce the teay-tate error Integral Reove teay-tate error Increaing i accelerate the controller High i ay give ocillation Increaing i will increae the ettling tie Derivative Larger ecreae ocillation Irove tability for low value of May be highly enitive to noie if one take the erivative of a noiy error High noie lea to intability