Ch. : Invee Kinemati Ch. : Velity Kinemati The Inteventinal Cente
eap: kinemati eupling Apppiate f ytem that have an am a wit Suh that the wit jint ae ae aligne at a pint F uh ytem, we an plit the invee kinemati pblem int tw pat:. Invee pitin kinemati: pitin f the wit ente. Invee ientatin kinemati: ientatin f the wit Fit, aume DOF, the lat thee inteeting at ( q,..., q ) ( q,..., q ) Ue the pitin f the wit ente t etemine the fit thee jint angle The Inteventinal Cente
The Inteventinal Cente eap: kinemati eupling Nw, igin f tl fame,, i a itane tanlate alng z (ine z an z ae llinea) Thu, the thi lumn f i the ietin f z (w/ epet t the bae fame) an we an wite: eaanging: Calling [ y z ] T, [ y z ] T z y z y
eap: kinemati eupling Sine [ y z ] T ae etemine fm the fit thee jint angle, u fwa kinemati epein nw allw u t lve f the fit thee jint angle euple fm the final thee. Thu we nw have Nte that: T lve f the final thee jint angle: T ( ) ( ) Sine the lat thee jint f a pheial wit, we an ue a et f Eule angle t lve f them The Inteventinal Cente
eap: Invee pitin kinemati Nw that we have [ y z ] T we nee t fin q, q, q Slve f q i by pjeting nt the i-, y i- plane, lve tig pblem Tw eample elbw () manipulat: lutin (left-am elbw-up, left-am elbw-wn, ight-am elbw-up, ight-am elbw-wn) pheial (P) manipulat: lutin (left-am, ight-am) The Inteventinal Cente
Invee ientatin kinemati Nw that we an lve f the pitin f the wit ente (given kinemati eupling), we an ue the eie ientatin f the en effet t lve f the lat thee jint angle Fining a et f Eule angle epning t a eie tatin mati We want the final thee jint angle that give the ientatin f the tl fame with epet t (i.e. ) The Inteventinal Cente
The Inteventinal Cente Invee ientatin: pheial wit Peviuly, we ai that the fwa kinemati f the pheial wit wee iential t a ZYZ Eule angle tanfmatin: A A A T
The Inteventinal Cente The invee ientatin pblem eue t fining a et f Eule angle (θ, θ, θ ) that atify: t lve thi, take tw ae:. Bth an ae nt ze (i.e. θ ) nningula. θ, thu ingula Nningula ae If θ, then ± an: ± ±, atan, θ Invee ientatin: pheial wit
Invee ientatin: pheial wit Thu thee ae tw value f θ. Uing the fit ( > ): θ atan θ atan (, ) (, ) Uing the en value f θ ( < ): θ atan θ atan Thu f the nningula ae, thee ae tw lutin f the invee ientatin kinemati (, ) (, ) The Inteventinal Cente
The Inteventinal Cente In the ingula ae, θ thu an Theefe, ha the fm: S we an fin the um θ θ a fllw: Sine we an nly fin the um, thee i an infinite numbe f lutin (ingula nfiguatin) Invee ientatin: pheial wit ( ) ( ),, atan atan θ θ
Invee Kinemati: geneal peue. Fin q, q, q uh that the pitin f the wit ente i:. Uing q, q, q, etemine. Fin Eule angle epning t the tatin mati: T ( ) ( ) invee pitin kinemati invee ientatin kinemati The Inteventinal Cente
The Inteventinal Cente Eample: am with pheial wit F the DH paamete belw, we an eive fm the fwa kinemati: We knw that i given a fllw: T lve the invee ientatin kinemati: F a given eie link a i α i i θ i 9 θ a θ a θ ( ) T
Eample: am with pheial wit Eule angle lutin an be applie. Taking the thi lumn f ( ) T Again, if θ, we an lve f θ : Finally, we an lve f the tw emaining angle a fllw: F the ingula nfiguatin (θ ), we an nly fin θ θ thu it i mmn t abitaily et θ an lve f θ ( ) θ atan, ± θ atan θ atan (, ) (, ) The Inteventinal Cente
Eample: elbw manipulat with pheial wit Deive mplete invee kinemati lutin lin k a i α i i θ i 9 θ a θ a θ -9 θ θ we ae given H T uh that: y z, θ The Inteventinal Cente
The Inteventinal Cente Eample: elbw manipulat with pheial wit Fit, we fin the wit ente: Invee pitin kinemati: Whee i the hule ffet (if any) an D i given by: z y z y ( ) ( ) ( ),,,, D D a a a z y y ± atan atan atan atan θ θ θ ( ) a a a a z y D
Eample: elbw manipulat with Invee ientatin kinemati: pheial wit Nw that we knw θ, θ, θ, we knw. nee t fin : T ( ) Slve f θ, θ, θ, Eule angle: θ atan(, ) θ atan θ atan ( ) (, ), ± The Inteventinal Cente
Eample: invee kinemati f We ae given T : SCAA manipulat T a a a a lin k a i α i i θ i a θ a 8 θ θ The Inteventinal Cente
Eample: invee kinemati f The Inteventinal Cente SCAA manipulat Thu, given the fm f T, mut have the fllwing fm: Whee α i efine a: T lve f θ an θ we pjet the manipulat nt the -y plane: Thi give tw lutin f θ : One θ i knwn, we an lve f θ : θ i nw give a: α Finally, it i tivial t ee that z θ θ α α α α θ θ θ ( ) atan, y a a a a θ atan, ± (, )- ( a a a ) atan y atan, ( ) θ θ atan,
Eample: numbe f lutin Hw many lutin t the invee pitin kinemati f a plana -link am? given a eie [ y ] T, the fwa kinemati an be witten a: a a a y a a Theefe the invee kinemati pblem i une-ntaine (tw equatin an thee unknwn) a lutin i i inie the wkpae lutin if i n the wkpae bunay lutin ele The Inteventinal Cente
Eample: numbe f lutin What if nw we eibe the eie pitin an ientatin f the en effet? given a eie [ y ] T, we an nw all the pitin f the wit ente. Thi pitin i given a: w a ( θ ) w y y a in( θ ) Nw we have eue the pblem t fining the jint angle that will give the eie pitin f the wit ente (we have ne thi f a D plana manipulat). Finally, θ i given a: θ θ ( θ ) θ lutin if the wit ente i n the igin lutin if wit ente i inie the -link wkpae lutin if wit ente i n the -link wkpae bunay lutin ele The Inteventinal Cente
Velity Kinemati Nw we knw hw t elate the en-effet pitin an ientatin t the jint vaiable Nw we want t elate en-effet linea an angula velitie with the jint velitie Fit we will iu angula velitie abut a fie ai Sen we iu angula velitie abut abitay (mving) ae We will then intue the Jabian Intantaneu tanfmatin between a vet in n epeenting jint velitie t a vet in epeenting the linea an angula velitie f the en-effet Finally, we ue the Jabian t iu numeu apet f manipulat: Singula nfiguatin Dynami Jint/en-effet fe an tque The Inteventinal Cente
Angula velity: fie ai When a igi by tate abut a fie ai, evey pint mve in a ile Let k epeent the fie ai f tatin, then the angula velity i: ω θkˆ The velity f any pint n a igi by ue t thi angula velity i: v ω Whee i the vet fm the ai f tatin t the pint When a igi by tanlate, all pint attahe t the by have the ame velity The Inteventinal Cente
Net la Deivatin f the Jabian The Inteventinal Cente