ntroducton To obotcs Kneatcs, Dynacs, and Desgn SESSON # 6: l Meghdar, Professor School of Mechancal Engneerng Sharf Unersty of Technology Tehran, N 365-9567 Hoepage: http://eghdar.sharf.edu
So far we hae only studed oton of anpulators wthout regard to forces causng the oton. Let us now dere the equatons of oton for anpulator ars. n dynacs, we generally consder the followng ssues: Forward Dynacs: oputng the resultng oton of the anpulator ar,, under the applcaton of a set jont torques. Ths s useful for sulaton of the ar. nerse Dynacs: oputng the ector of jont torques for the gen jont oton trajectory,,. Ths s useful for controllng of the ar.
obotc rs Dynac Forulaton Hstory Lagrangan Dynacs Newton-Euler Dynacs Kane Dynacs Ucker/Kahn Waters Hollerbach Kane/Lenson 4x4 Matrces ackward ecurson 4x4 Matrces Forward eurson Euler's Paraeters and elate oordnates 4x4 Matrces 3x3 Matrces
uthor Method Multplcatons ddtons Ucker/Kahn Lagrangan Dyn. 4 4 Matrces 66,7 5,548 Waters Lagrangan Dyn. Hollerbach Lagrangan Dyn. Hollerbach Lagrangan Dyn. 4 4 ackward ecurson 4 4 Forward ecurson 3 3 Forward ecurson 7,5 5,65 4,388 3,586,95,79 Newton-Euler ecurse 85 738 Kane/Lenson Kane Dynacs 646 394 abert/horn Yang/Tzeng onfguraton Space Method SM Dyn. Splfcaton by Desgn 468 64 7 34 + 4 Trg. Functons.
Lnear cceleratons of gd odes: onsder a pont n space, and descrbe ts kneatcs n two fraes {} and {}. {}={fxed} OG Fro hapter-5 we hae: OG OG {}={gd ody Frae} Dfferentatng the elocty equaton wth respect to the te we hae:
Lnear cceleratons of gd odes: Notng that: f s constant on the.., then: OG OG
ngular cceleraton of gd odes: onsder: - Frae {} rotatng relate to {} wth: - Fraes {} rotatng relate to {} wth: Then: Su the ectors n frae {} {}={fxed} OG {} {}={gd ody Frae}
Newtonan Mechancs: For a gd ody whose center of ass s acceleratng wth a, the Force F actng at the ass center s gen by: The Newton s Law of Moton: F f P a = Te rate of change of oentu a F
Newtonan Mechancs: For a gd ody rotatng wth an angular elocty, and an angular acceleratng, the Moent N whch ust be actng on the body to cause ths oton, s gen by: The Euler s Equaton: N where: = nerta Tensor of the.. wrtten n frae {} The rotatonal analogy of the Newton s nd law coes fro the Prncple of Moent of Moentu N
Mass Dstrbuton: The nerta Tensor of an object descrbes the object s ass dstrbuton a generalzaton of the scalar oent of nerta. elate to a frae {} s expressed as: where: xx yy zz xx xy xz yy y x x xy yz z z y zz xz yz d; d; d; xy X xz yz {} Z P xyd xzd yzd d Y P=[x y z] T
terate Newton-Euler Dynac Forulaton: Let us now study the proble of coputng the ector of jont torques for the gen jont oton trajectory,,. The nerse Dynacs proble useful for controllng of the ar. Outward teratons to opute eloctes and cceleratons: To study dynacs fro Newton & Euler equatons, t s obous that we need propagaton equatons for &. Fro hapter-5, the angular elocty equaton for eery nstant s: ˆ Dfferentatng wth respect to te we hae: Z
Where: P + + + + + ˆ ˆ Z Z ˆ ˆ Z Z
lso fro hapter-5, the lnear elocty equaton for eery nstant s: Dfferentatng wth respect to te we hae: Snce at eery nstant: P tan t cons P P P P
To fnd the lnear acceleraton of the center of ass, we hae: P Dfferentatng wth respect to te we hae: P P xs- Z Y Lnk- {} X P
Hang coputed all acceleraton equatons, we shall now apply the Newton-Euler Equatons as follows: Frst copute the nertal Force and Torque actng at the ass center of each lnk; F N a = nerta Tensor of the lnk- wrtten n frae { } wth t s orgn at the ass center, and hang the sae orentaton as frae {}. Then, perfor nward teratons to copute forces and torques;
nward teratons to opute Forces and Torques: Wrte the force balance on lnk-: f + f f F F f P + n + n Wrte the oent balance about the orgn of lnk frae-: N n N n P F P f Note: The requred jont torques are found by takng the Z-coponent of the torque appled by one lnk on t s neghbor.
nward teratons to opute Forces and Torques: Therefore, for eolute Jonts we hae: T n Z ˆ n F f P + f + n + Therefore, for Prsatc Jonts we hae: T f Zˆ N Note: For a robot ong n free space, we ay hae: N f N N n N where as for a robot beng n contact wth the enronent, we ay hae: N N N f n N
terate Newton-Euler Dynac lgorth: Frst: opute lnk eloctes and acceleratons terately fro lnk- to lnk-n, and apply the Newton-Euler equatons to each lnk. Second: opute the forces and torques of nteracton recursely fro lnk-n back to lnk-.
losed-for Sybolc For Dynac Equatons: Exaple: The -DOF Manpulator r. ssuptons: Pont asses at the dstal end of each lnk, X Y L L nt, ˆ ass po ter graty g gy g S g g S S ctuator torques as a functon of jonts poston, elocty, and acceleraton.