Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion KF: from join space variables o ask space pose Inverse Posiion KF: from ask space pose o join space variables Direc Velociy KF: from join space velociies o ask space velociies Inverse Velociy KF: from ask space velociies o join space velociies
Kinemaic funcions Posiions and velociies of wha? I can be he posiion or velociy of any poin of he robo, bu, usually, is he las RF posiion and velociy LAST RF R P R BASE 0? Wha is he relaion beween hese wo RF?
Kinemaic funcions The firs sep for defining KFs is o fix a reference frame on each body In general, o move from a RF o he following one, i is necessary o define 6 parameers (hree ranslaions of he RF origin + hree angles of he RF roaion) A number of convenions were inroduced o reduce henumber of parameers and o find a common way o describe he relaive posiion of reference frames Denavi Harenberg convenions (1955) were he firs o be inroduced and are widely used in indusry (wih some minor modificaions)
A procedure o compue he KFs 1 1. Selec and idenify bodies and he relaive consrains 2. Se he RF on each body 3. Compue he homogeneous ransformaion beween he base RF and he final RF 4. Exrac he direc posiion KF from he homogeneous ransformaion 5. Compue he inverse posiion KF 6. Inverse velociy KF: analyical or geomerical approach
A procedure o compue he KFs 2 1. Selec and idenify links and joins 2. Se all RFs using DH convenions 3. Define consan geomerical parameers q () 4 q () 5 q () 2 q () 6 q () 1 R P FINAL R 0 BASE 0 T P 0 0 0 () () q () q T q = R P P P T 0 1
Join and caresian variables Join variables q1() q2() q 3() q() = q4() q 5() q6() Direc KF Task/caresian variables/pose x 1() x2() x 3() p() = α1() α2() α3() Inverse KF posiion orienaion 1 p() = f ( q () ) q() = f ( p() )
Direc posiion KF 1 q1() p1() q 2() p2() q ( ()) 3() p3() x q q() = ( ()) = = p q q 4() p4() q ( ()) 5() p5() α q q6() p6() posiion orienaion 0 0 0 0 1 2 3 4 5 6 P P P 1 ( q1) 2 ( q2) 3 ( q3) 4 ( q4) 5 ( q5) 6 ( q6) R T = T T T T T T T P = 1 0 T orienaion posiion 0 0 0 0 R = R ( q () ) = ( q () ) P P P P
Direc posiion KF 2 T 0 P 0 0 RP P = 0 T 1 Direc caresian posiion KF: easy x x 1 1 x 0 2 P 2 x 3 3 Direc caresian orienaion KF: no so easy o compue, bu no difficul We will solve he problem in he following slides
Direc posiion KF 3 ( () ) α( q() ) Rq We wan o compue angles from he roaion marix. Bu i is imporan o decide which represenaion o use α ( q() )? Euler angles RPY angles Quaernions Axis angle represenaion
q () ( ) 2 () x q q () 3 pq ( ( )) = = q( ) q () α ( ) 4 () q q () 5 q () 6 Inverse posiion KF 1 q () 1 This KF is imporan, since conrol acion are applied o he join moors, while he user wans o work wih caresian posiions and orienaions q () 3 q () 1 q () 2 ( () ) x q α( q() )
Inverse posiion KF 2 1. The problem is complex and here is no clear recipe o solve i 2. If a spherical wris is presen, hen a soluion is guaraneed, bu we mus find i... 3. There are several possibiliies Use brue force or previous soluions found for similar chains Use inverse velociy KF Usesymbolicmanipulaion symbolic manipulaion programs (compuer algebra sysems as Mahemaica, Maple, Maxima,, Lisp) Ieraively compue an approximaed numerical expression for he nonlinear equaion (Newon mehod or ohers) ( ) 1 ( ) { 1 q() f ( p() ) } q f p 1 () = () q() f p() = 0 min () ()
Direc velociy KF 1 Linear and angular direc velociy KF q () p () 1 1 q () p () 2 2 (), () x q q q () p () 3 3 q () = () q () p = p () = 4 4 (), () q q q () p () 5 5 q () p () 6 6 ( ) Linear velociy ( (), () ) vq q = (), () ω q q α ( ) ( ) Angular velociy
Direc velociy KF 2 A bi brief review of mahemaical i noaions i General rule d pq (()) d d ( () ) x q d = d ( () ) α q d d d f ( q () ) = f ( q (),, q (),, q () ) i i 1 j n d d f i f i f i = q () + + q () + + q () 1 j n q q q 1 j n
Direc velociy KF 3 q () 1 d f f f ( ()) i i i f q () T q = ( () ) () i = j fi J q q d q q q 1 j n q () n f f f 1 1 1 JACOBIAN q q q 1 j n q () 1 d ( ) f f f i i i f q() = q () ( () ) () j d = q q q J q q 1 j n q () n f f f m m m q q q 1 j n
Furher noes on he Jacobians Velociykinemaics is characerizedby Jacobians There are wo ypes of Jacobians: Geomerical Jacobian Analyical ljacobian J g J a ( ) p () = J q() q () also called Task Jacobian The firs one is relaed o Geomerical Velociies v p x = = ω J q g The second one is relaed o Analyical Velociies p x = = α α J q a
Geomerical and Analyical velociies Wha is he difference beween hese wo angular velociies? On he conrary, linear velociies do no have his problem: analyical and geomerical velociies are he same vecor, ha can be inegraed o give he caresian posiion
Derivaive of a roaion marix
Some useful formulas