Ch. : Fowa an Invee Knemat Reap: The Denavt-Hatenbeg (DH) Conventon Repeentng eah nvual homogeneou tanfomaton a the pout of fou ba tanfomaton: pout of fou ba tanfomaton: x a x z z a a a Rot Tan Tan Rot a
Reap: the phyal ba fo DH paamete a : lnk length tane between the o an o (pojete along x ) : lnk twt angle between z an z (meaue aoun x ) : lnk offet tane between o an o (pojete along z ) : jont angle angle between x an x (meaue aoun z ) Geneal poeue fo etemnng fowa knemat. Label jont axe a z z n- (ax z jont ax fo jont ). Chooe bae fame: et o on z an hooe x an y ung ghthane onventon. Fo :n-. Plae o whee the nomal to z an z - nteet z. If z nteet z - put o at nteeton. If z an z - ae paallel plae o along z uh that. x the ommon nomal though o o nomal to the plane fome by z - an z f the two nteet. Detemne y ung ght-hane onventon. Plae the tool fame: et z n paallel to z n-. Fo :n fll n the table of DH paamete. Fom homogeneou tanfomaton mate 7. Ceate T n that gve the poton an oentaton of the en-effeto n the netal fame
Example : ylnal obot wth pheal wt DOF: nee to agn even oonate fame But we aleay th fo the pevou two example o we an fll n the But we aleay th fo the pevou two example o we an fll n the table of DH paamete: lnk a o o o ae all at the ame pont o Example : ylnal obot wth pheal wt Note that z (ax fo jont ) ollnea wth z (ax fo jont ) thu we an make the followng ombnaton: an make the followng ombnaton: z y x T T T z y x
Example : the Stanfo manpulato DOF: nee to agn even oonate fame: Chooe z ax (ax of otaton fo jont bae fame). Chooe z ax (ax of otaton fo jont bae fame). Chooe z -z axe (axe of otaton/tanlaton fo jont -). Chooe x axe. Chooe tool fame. Fll n table of DH paamete: lnk a Example : the Stanfo manpulato Now etemne the nvual homogeneou tanfomaton:
Example : the Stanfo manpulato Fnally ombne to gve the omplete epton of the fowa knemat: knemat: z y x T [ ] [ ] [ ] [ ] z y x Example : the SCR manpulato DOF: nee to agn fve oonate fame: Chooe z ax (ax of otaton fo jont bae fame). Chooe z ax (ax of otaton fo jont bae fame). Chooe z -z axe (axe of otaton/tanlaton fo jont -). Chooe x axe. Chooe tool fame. Fll n table of DH paamete: lnk a
Example : the SCR manpulato Now etemne the nvual homogeneou tanfomaton: a a a a a a a a a a T Fowa knemat of paallel manpulato Paallel manpulato: two o moe ee han onnet the enp effeto to the bae (loe-han) Gueble fomula (D): n j j L f n n #DOF #DOF fo jont numbe of lnk* *exlung goun numbe of jont #DOF fo jont
Fowa knemat of paallel manpulato Example (D): Invee Knemat Fn the value of jont paamete that wll put the tool fame at a ee poton an oentaton (wthn the wokpae) Gven H: R H o SE ( ) 7
8 Example: the Stanfo manpulato [ ] [ ] Fo a gven H: Fn :.7. H [ ] [ ] [ ] One oluton: π/ π/. π/ π/.7 Invee Knemat Fo the fowa knemat thee alway a unque oluton y q The nvee knemat may o may not have a oluton
Ovevew: knemat eouplng ppopate p fo ytem that have an am a wt Ovevew: knemat eouplng Now ogn of tool fame o a tane tanlate along z (ne g g ( z an z ae ollnea) 9
Invee poton Now that we have [x z T y ] we nee to fn q q q Bakgoun: two agument atan We ue atan( ) ntea of atan( ) to aount fo the full ange of angula oluton Calle fou-quaant atan ( y x) atan y < y π atan y x < x atan( y x) y atan y x x π y > x unefne y x
Example: RRR manpulato. To olve fo pojet the am onto the x y plane atan( x y ) Caveat: ngula onfguaton offet If x y unefne If thee an offet then we wll have two oluton fo : left am an ght am
Left am an ght am oluton Left am: Rght am: Left am an ght am oluton Theefoe thee ae n geneal two oluton fo fo :
Left am an ght am oluton The two oluton fo oepon to the elbow-own an elbow-up poton epetvely RRR: Fou total oluton In geneal thee wll be a maxmum of fou oluton to the nvee poton knemat of an elbow manpulato Ex: PUM
Spheal onfguaton Example: RRP manpulato Next la Complete the uon of nvee knemat p Invee oentaton Intouton to othe metho Intouton to veloty knemat an the Jaoban