Robot Dyn Heheng Wng Dept. of Autoton Shngh Jo Tong Unverty
Wht Robot Dyn? Robot dyn tude the reton between robot oton nd fore nd oent tng on the robot. x
Rotton bout Fxed Ax The veoty v n be deterned fro the ro produt of nd r p. Here r p vetor fro ny pont on the x of rotton to P. v = x r p = x r The dreton of v deterned by the rght-hnd rue.
Rotton bout Fxed Ax ontnued The eerton of P n o be defned by dfferenttng the veoty. = dv/dt = d/dt x r P + x dr P /dt = x r P + x x r P Tngent e Nor/entrpet eerton t n be hown tht th equton redue to = x r w r = t + n The gntude of the eerton vetor = t + n
Rotton of Vetor Conder rotton of vetor bout x. Pont P rottng bout x u. r the poton vetor of pont P. : the peed of rotton The ngur veoty : ω u r u P The rte of hnge of poton vetor d r ωr dt r :
B Rotton of Fre Conder fre B rottng bout n unt vetor u., jb, k B d dt B y B : unt dreton vetor of the xe of fre B w.r.t. the referene fre A ω B The dervtve of R d B B dt R, d dt j B ω j B, d dt the rotton trx of k B z B Fre A ωk j k ω ω j ωk B ω R B B fre B : B x B u B
Gener Moton A gener oton n be ondered obnton of trnton wth pont nd oton bout the pont. B A O xo yozo : A fxed referene fre z y A x y : A fre tht trnte wth A z The veoty reton: V B V A ωr The eerton reton: x O z x y B A α r ω ωr
ntroduton to Dyn
Newton Lw of Moton Frt Lw: A prte orgny t ret, or ovng n trght ne t ontnt veoty, w ren n th tte f the reutnt fore tng on the prte zero. Seond Lw: f the reutnt fore on the prte not zero, the prte experene n eerton n the e dreton the reutnt fore. Th eerton h gntude proporton to the reutnt fore. F F : the : the net fore F : the eerton Thrd Lw: Mutu fore of ton nd reton between two prte re equ, oppote, nd oner. F F
Expe Fnd the eerton of the b nd the wedge. Souton: Drw the free-body dgr of the prte Appy Newton nd Lw For the b x b y b N n N o g R Frtone urfe N For the wedge x w N n 3 N g y x g
Expe ontnued Conder eerton retonhp between the b nd the wedge b b / w w b / w : retve veoty of w.r.t. the wedge the b Note tht the retve veoty ong the urfe of the wedge Fro the dgr, we hve x b y b x w b / w b / W n o 4 5 b w b/w Fro eq. -5, we n ove the fve unknown,.e. the eerton
Lner Moentu Lner oentu: produt of nd veoty: L V t vetor, n the e dreton veoty - S Unt : Kg / V Fro Newton' nd Lw : d F V L dt Prnpe of Lner Moentu: The rte of hnge of the ner oentu of prte equ to the reut fore tng on the prte V
Angur Moentu r: the poton vetor of prte w.r.t. referene pont O. V: the veoty of the prte : the of the prte The ngur oentu of the prte bout referene O: O r V H o r V H o vetor perpendur toboth V ndr. t dreton deternedby therght- hnd rue t unt Kg /
Prnpe of Angur Moentu Conder fore F tng on the prte V r O F o o o M F r H F r V r V r H Prnpe of ngur oentu: The rte of hnge of the ngur oentu of prte bout fxed pont O equ to the reutnt oent of fore tng on the prte bout pont O Dfferenttng the ngur oentu
Dyn of Syte of Prte Conder yte of n prte. f : the extern fore exerted on prte e j : the ntern fore exerted on prte by prte j : of prte. r : poton vetor of prte d Lner oentu : L V r dt r The enter of :r L The ner oentu of yte of prte equ to the produt of the tot nd the veoty of the enter of f f Tot r MV C e j f e j f j j Veoty of enter of
Dyn of Syte of Prte Dfferenttng the ner oentu d L r f ej dt e j L j The rte of hnge the ner oentu of yte of prte equ to the reutnt of EXTERNAL fore tng on the prte L MV C j M f The enter of of the yte ove f the fore nd e re onentrted t the enter of f f f e j e f f j : eerton of enter of j Equton of oton of Center of
Angur Moentu of Syte of Prte Sry, by dfferenttng the ngur oentu H o d dt r V M O The reutnt oent of EXTERNAL fore tng on the yte bout O f f O f e e j f j j The referene pont ut be fxed pont. However, the enter of of the yte n be the referene pont even when t ovng
Expe: Expe: Cute the ngur eerton of the e nk Souton Conder the two prte nd the nk yte Anyze the extern fore Conder the ngur oentu bout O g H o M o d dt o g g o R y O o R x g
Lner nd Angur Moentu for Rgd Body Sne rgd body n be ondered yte wth nfnte nuber of prte, the ner oentu V C L V C The ngur oentu bout the enter of H nert tenor trx bout C. ω : C ω :ngur veoty of the body Center of ω
Newton Equton nd Euer Equton A gener oton n be ondered obnton of trnton wth the enter of nd oton bout the enter of O C x x y z o o o yz : : A fxed referene fre The equton of trnton: d dt A A fre tht trnte wth C F f The equton of oton bout C: A A H M ω ω ω C C z x n O z C x y r Newton equton Euer equton r f n f f y
Newton Equton nd Euer Equton Cont The ngur veoty wth repet to the trntng fre. The nert tenor trx wth repet to the trntng fre, o t w hnge t vue wth rotton of the body. The fore the reutnt of the EXTERNAL fore The oent the reutnt oent of the EXTERNAL fore nd oent.
Expe Derve the dyn equton of the DOF nputor. Here, the e of nk nd re nd repetvey. Aue tht the unfory dtrbuted over the nk. Souton: Anyzng fore tng on the nk R y N x R x O x R x R y g N y g
Expe ontnued Dyn of nk. A nk rottng bout O, N x R x R y O M o N y g Moent of nert bout O Reutnt oent bout O O g o Ry o Rx n
Expe 3 Dyn of nk. A nk n gener pne oton C C C C M ω ω ω F g R x R y C B Aeerton of C: Bn C Bt C B C / /... Bt Bn B Bn Bt Bn C / Bt C / / Bn C / Bt C C Newton equton: 3 g R R y y x x
Expe ontnued Conder Euer equton. A the n unfory dtrbuted nd the nk yetr, the nert tenor trx dgon. R R y x zz yy xx zz yy xx 4 R R y x zz Fro nd 3, we n ove R x nd R y. Subttutng R x nd R y n eq. nd 4 ed to the dyn equton of the robot r.
Foruton of Robot Dyn
Reurve Newton-Euer Foruton We onder nputor wth revoute jont ony. Lnk - z - y z Jont Jont - y - - O r - x - O - x C : enter of Angur veoty reton between nk - nd : Dfferenttng ω ω ω ω ω ω ω / z z ω z Retve ngur veoty f nk to nk -
Reurve Newton-Euer Equton Cont Conder veoty nd eerton of O. - / - to O O retve veoty of : O veoty of : O veoty of : V V V / V V V A the retve oton of O w.r.t. O - oton bout O -, / ω V 3 ω V V Aeerton t O : 4 Rn Rt ω ω ω Aeerton t the enter of : 5 / O C C r ω ω r ω
Forwrd Equton ω ω C ω ω z z ω z ω ω ω ω r ω ω r 4 3 = Fro,, 4 nd 5, we n reurvey ute the ngur veoty nd eerton of the nk, nd the eerton t the enter of. The nt ondton: When, ω, V, ω =+ Cute + fro. Cute + fro Cute + fro 3 Cute + fro 4 No =n- YES END
Dyn equton of robot nk Derve the dyn by ppyng Newton-Euer equton to nk. -n + Drw the free-body dgr of nk O. Aue tht the nk re rgdy + onneted. Cut nk fro the r: f : the fore tng on nk by nk - C n n : the oent pped on nk by nk - O r -f + Appyng Newton Lw f f f g f g f 7 g
Dyn of robot nk Cont Appyng the Euer equton ω n ω ω ω ω ω n n n r r f r f f Eq. 7 nd 8 gve the reurve bkwrd equton for utng the nterton fore nd oent Reton between n nd jont torque z T n r f tutor produe torque bout the jont x ony -n + n O r -f + f C g 8 O +
Bkwrd Cuton nt ondton: When k for the t nk : fk, nk k : the nuber of nk =K Cute f fro 7 Cute n fro 8 =- NO YES = END
Lgrnge Foruton of Robot Dyn Lgrnge foruton n nyt ethod for dervng the robot dyn. t bed on the energy nd work prnpe Energy of Mehn Syte Knet energy: energy due to oton of prte or body Potent energy: due to grvtton fore, deforton of ehn yte, et.
Knet Energy A prte body h knet energy when t ove. Knet energy wy greter thn zero For For For prte : yte of rgd body : K prte : K K r V V V T r d ω T V ω C ω V : nert tenor V
Potent Energy We here onder the grvtton potent energy ony. For prte : U gz z: the heght of the prte w.r.t. referene eve For yte of prte : U gz z For rgd body : U gz The heght of the enter of Potent energy vue retve to the referene. t oud be potve, zero nd negtve
Work When prte underwent dpeent r under ontnt fore f, the work done by the fore on the prte W f T r Work r t oud be potve, zero nd negtve S unt: N r f
Work done by te-vryng fore Conder the work done by te-vryng fore on prte tht oved fro one poton to nother. T V V Poton W f t dr ft dv dr VdV dt Poton V V Work-energy prnpe: The work done K by fore tng on prte equ to the hnge of t knet energy
Conervtve fore nd Non- Conervtve fore The fore oted to the potent energy ed onervtve fore. A fore tht not oted wth the potent energy ed non-onervtve fore. The work done by onervtve fore grvty fore W g g z z U z The work done by non-onervtve fore Wn Wg K W K U K U n g z The work done by non-onervtve fore equ to the hnge of tot energy
Conervton of Energy f no non-onervtve fore tng on yte prte, or yte of prte, or rgd body, doe not do ny work, the tot energy of the yte onerved. K U ontnt
Lgrnge Equton Generzed oordnte q: A et of preter for repreentng the onfgurton poton & orentton of yte. q ut pefy the onfgurton unquey One the vue of q re fxed, the yte nnot ove. The hoe of q not unque. Degree of freedo DOF: The denon of the generzed oordnte vetor q ed degree of freedo of the yte
Expe q DOF = x x,y q x y DOF=4
Generzed Fore Conder the work done by non-onervtve fore under dfferent dpeent of the yte r : dfferent dpeent t the ton pont of f due to dfferent hngeq of the generzed oordnte q The dfferent work W : W f r T The generzed fore F of the yte gven by F W q : generzed oordnte vetor q f
当前无法显示此图像 Expe Cute the generzed fore of the DOF r Generze d Coordnte : q For dfferent oton q The dfferent work W f x x : the dfferent oton of x J q q Generzed Fore: T W, the end pont T f T q J q f q T f J q q x
Lgrnge Equton q: the generzed oordnte of yte K: Knet energy of the yte U: the potent energy of the yte F: the generzed fore of the yte Defne L=K-U: Ced Lgrngn The dyn of the yte gven by d dt L q L q F
Dervton of Robot Dyn ung Lgrnge Equton Chooe the generzed oordnte q uuy ue the knet preter defned by the D-H ethod dentfy the non-onervtve fore tht re exerted t the yte nd do work 3 Cute the knet energy K nd the potent energy U, nd then L=K-U 4 Cute the prt dervtve 5 Cute the generzed fore F. 6 Appy the Lgrnge equton.
Expe Expe: Denote the of nk by. The unfory dtrbuted over the nk. Derve the dyn of the DOF r. Souton Generzed oordnte : q Non - onervtve fore tht do work :, 3 The knet energy: nk rotte bout O, K nk n gener pne oton, K Poton of the Center Of x y V o V x y 4 y Center of x
Expe ontnued The tot knet energy 8 o K K K The potent energy ue y= the referene g g gy gy U 4 Cute the prt dervtve 4 L 4 L g g L g L
Expe ontnued 5 The generzed fore: W F 6 Appy the Lgrnge equton: F q q L L dt d.5.5.5.5.5.5.5.5.5.5.5 g g g o
Expe Expe : The oent of nert of the frt nk. The of nk. The onentrted t the endpont. An extern fore f t t the endpont. Derve the dyn of the r. Souton: 3 9 Generzed oordnte Ue the D-H ethod to gn fre nd eet the jont nge the generzed oordnte d Generzed oordnte: q z z z y 3 y x x y z 3 y x x 3 f
Expe ontnued No-onervtve fore tht do work: 3 Knet energy, potent energy nd Lgrngn : Lnk K : Lnk V K To fnd the veoty V of the endpont, we need to ove the forwrd knet T T 3 T fore f nd the extern,, nput : jont Knet energy:
Expe ontnued Forwrd knet: T T T 3 3 T T T V dt d V The potent energy Aung tht U= when z =. g gh U g gh U K L Lgrngn:
Expe ontnued 4 Cute the prt dervtve L L L n g L
Expe ontnued 5 Generzed fore: f q J F x f T T W x: the poton of the endpont x q q x q J 6 Appyng the Lgrnge equton ed to the dyn: f q J n.5 n.5 T g
Struture of Robot Dyn The dyn of the DOF nputor: H q : C q, q.5.5.5 o.5.5.5.5.5 g.5g.5.5g C q, q H q ntert trx of the nputor, dependng on the jont poton : Centrgufore Coro Gq : theentrfug fore G q H T nd Corofore :Tertht dependon thequreof :Ter of the tht depend jont veote The grvty fore on the of produt two q H q yetr trx the jont veoty of jont. jont.
Struture of Robot Dyn The entrfug nd Coro ter n be re-wrtten,.5.5.5.5.5.5.5.5, q q S q H q C q,, For ny,,.e. yetr trx, kew -, T x q q S x x q q S q q S q q S T
Struture of Robot Dyn n gener, the dyn of robot nputor h the foowng for: H q q H q S q, q q G q τ nert fore Hq : S q, q : Syetr nd potve - defnte nert q T H q q the knet energy kew - yetr trx x T Centrfug nd Coro fore S q, q x, x R n Grvty trx. Jont nput
Lner Preterzton of Robot Dyn The dyn of the DOF r:.5.5.5.5.5.5.5.5.5.5.5 g g g o 5 4 3.5.5.5.5.5 o Defne The preter depend on, ength, oent of nert. They re the phy preter 5 4 3,,,.5 β q q q q Y g θ θ θ θ g g θ θ θ θ θ θ θ Preter vetor The reut n be generzed to n DOF robot nputor