ME135 ADANCED OBOICS Independent oint Contro ee-hwan yu Schoo of Mechanica Engineering Introduction to Contro Contro: deterining the tie hitory of joint input to do a coanded otion Contro ethod are depend on hardware and appication Carteian v. Eow Motor with gear reduction v. High torque otor without gear Continuou path v. P-to-P Contro ethod have een advanced with the deveopent of copicated hardware he ore copicated hardware, the ore advanced contro ethod
Independent oint Contro Each Ai -> SISO Couping effect -> diturance Ojective: tracking and diturance rejection Actuator Dynaic DC Motor: Sipe and eay to ue F i φ τ 1φi a φω
Actuator Dynaic dt d i i i dt di L a a a a θ ω φω φ τ 1 orque Speed eation
orque Contant When otor i taed r a r i τ τ Independent oint Mode r i r dt d B dt d a / / τ τ τ θ θ
Independent oint Mode ( ) ( ) ( ) ( ) ( ) r I B I L a a / τ Independent oint Motion ( ) ( )( ) [ ] ( ) ( )( ) [ ] with, / with, B L r L B L τ τ [ ] [ ] B r B / 1/ / / τ Effect of the oad torque (diturance) i reduced y the gear reduction B L << Eectrica tie contant << Mechanica tie contant
Independent oint Motion & ( t) ( B / ) & θ( t) ( / ) ( t) τ ( t) t B & θ t u t d t & θ / r && θ B: effective daping PD Copenator for Set Point racking Set point tracking: tracking a contant or tep reference Ω 1 P D d Ω Ω ( B D ) P D
PD Copenator E d B Ω d Ω d d, D 1 Ω D D For a tep reference input and a contant diturance e i E D P Larger gear reduction and arge P-gain can reduce the teady-tate error PD Copenator Coed-oop characteritic poynoia ( B ) ω, P D P ζω ω ζω B D For rootic appication, criticay daped, fatet nonociatory repone ω ζ 1 deterined the peed of repone
Eape 6.1 Eape 6.1
Eape 6. PID Copenator Ω ( D P I ) 3 d Ω ( B D ) P I Ω D outh criteria I < ( B ) D P
PID Copenator Deign rue-of-thu for PID Copenator Firt, et _I ; Deing PD gain to achieve the deired tranient ehavior ie tie, etting tie, etc) Deign _I within the iit
he Effect of Saturation and Feiiity In theory, aritrary fat repone and aritrary a teady tate error to a contant diturance can e achieved y ipy increaing the gain In practice, however, there i a aiu peed of repone achievae fro the yte wo ajor factor Saturation oint feiiity Saturation
Feiiity Shoud avoid reonant frequency Can not increae w aritrary Feedforward Contro o track tie-varying trajectorie c a, H, F d q G p Y Feedforward yte and coed-oop yte houd e tae q ( ) ( c( ) ( ) a( ) d( ) ) ( p d q c )
Feedforward Contro 1 G F / ( ) Y ( ) Forward pant i tae -> yte i iniu phae Feedforward Contro E p q( ) d( ) d q c D
Feedforward Contro Differentiation of a actua igna i not required Independent of the reference trajectory With PID, teady-tate error to a tep diturance i zero d d d d t & θ B & θ ( θ θ ) ( θ θ ) f t e& t e t D D P P Drive rain Dynaic Popuar for ue in root due to ow ackah, high torque traniion, copact oint feiiity i ignificant
Haronic Drive he Fepine ha two e teeth than the Circuar Spine he gear ratio i cacuated y {#Fepine eeth} / {#Fepine eeth - #Circuar Spine eeth}. Drive rain Dynaic Feiiity i the iiting factor to the achievae perforance in any cae ( θ θ ) k( θ θ ) u & θ B & θ k && θ B & θ
Drive rain Dynaic p p k p B k k U p B k U ( ) k p p k Drive rain Dynaic In practice, the tiffne of haronic drive i arge and the daping i a Negect daping 4 k( ) Frequency of iaginary poe increae with increaing joint tiffne Difficut to Contro
Drive rain Dynaic Stae, ut undeirae ociation Drive rain Dynaic
State Space Deign θ θ θ θ & & 4 3 1 B k k k B k A u A 1, 1 1 & [ ] 1 c c y ( ) A I c U G 1 Poe of the G are eigenvaue of the atri A State Feedack Contro r k r k t u i i i 4 1 Copare with previou PD/PID? ( ) r k A & More free paraeter
Controaiity Definition 6.1: A inear yte i aid to e copetey controae, or controae for hort, if for each initia tate (t_) and each fina tate (t_f) there i a contro input u(t) that tranfer the yte fro (t_) at tie t_ to (t_f) at tie t_f. Lea 6.1: A inear yte of the for (6.5) i controae if and ony if n 1 det[ A A L A ] n n1 heore 1: Let α α n L α α e an aritrary poynoia of degree n with rea coefficient. hen there eit a tate feedack contro aw of the for Eq. (6.55) uch that det 1 ( I A k ) α( ) if and ony if the yte (6.5) i controae. We ay achieve aritrary coed-oop poe uing tate feedack Poe Aignent How to chooe an appropriate et of coed-oop poe aed on the deired perforance, the iit on the avaiae torque, etc. Optia Contro { t Q t u t }dt A u k k 1 P 1 P PA P P Q
Oerver Contro aw ut e a function of a of the tate Oerver: dynaica yte (contructed in oftware), attept to etiate the fu tate uing the yte ode and output. ˆ & Aˆ u ( y c ˆ ) Auption: given the yte ode, don t know the initia condition t e ˆ e& ( A c )e ( ) Oervaiity: the eignevaue of A c can e aigned aritrary Oervaiity Definition 6. A inear yte i copetey oervae, or oervae for hort, if every initia tate (t_) can e eacty deterined fro eaureent of the output y(t) and the input u(t) in a finite tie interva. heore the pair (A,c) i oervae if and ony if det[ ] 1 c A c L A n c
Seperation Principe & u A u k ˆ & e& A k k A c e Aow u to eparate the deign of the tate feedack contro aw fro the deign of the tate etiator Pace the oerver poe to the eft of the poe of feedack contro aw Drawack Large oerver gain can apify the eaureent noie Large gain of tate feedack contro aw can reut in aturation of the input Uncertaintie in the yte paraeter noninearitie