NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
4. Moton Knematcs 4.2 Angular Velocty Knematcs Summary From the last lecture we concluded that: If the jonts are all rotary then: ω () t zˆ zˆ zˆ J n 2 n 2 n However, f Jont "" s Prsmatc then the correspondng column n the angular velocty Jacoban (.e. J ) wll be ZERO!! For example f jont #2 s prsmatc we get: ω ( ) ˆ ˆ d d t z z J n n 2 2 n n Asde: What s the dfference between Tp B B A and Rp B B A? Pure translaton wll NOT gve rse to a change n orentaton 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Snce the jont varables could be θ (rotary jont) or d (prsmatc jont), let's represent the jont varable by the symbol q, whch could be ether rotary or prsmatc. If Jont "" s Prsmatc: (Assume that only one jont s movng) q, j j d n d d () t o Remember, only jont "" s movng, the others are not movng (q j =, j ). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 3 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs From the dagram, dn t d d q t dn ( ) ( ( )) ( ) d R d ( q ( t)) R d n (4.) CONSTANT Tme Varyng CONSTANT d n d n d d () t 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 4 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs But, (neglectng the sub-sub-scrpt) d ( q ( t)) q ( t) z d Note that R d d as d T d = R refers to the orgn of {n} as seen from {}. d + d Substtutng equaton (4.2) nto (4.) and dfferentatng gves, q (4.2) d dt d dn t R d q t ( ) ( ( )) dt v ( t) R z q ( t) z q ( t) n (4.3) Note that R R v () t z = z as z means z, the z-axs of coordnate frame {} descrbed n terms of {}. The vector z represents the mappng of a "free vector". 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 5 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs If Jont "" s Rotary: (Assume that only one jont s movng) d n d Agan only jont s movng, the others are not movng. 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 6 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs From the dagram, d ( t) d ( d ) ( d ) n n d ( ( )) R d R t d n (4.4) Dfferentatng (4.4) gves, d n d dt RR d ( t) R R( ( t)) d n n R ( t) R( t) d n (4.5) a3 a 2 b a2b3 a3b2 R ( t) ( R( t) d ) a b n a3 a b 2 a3b ab 3 a b ( ar 2 a ( t)) ( R ( t b ) d n 3 a ) b2 a2b 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 7 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Now, recallng that ( t) ˆ k ( t) z ( t) allows us to smplfy the expresson R ( t) R z ( t) and z () t d n d n R( t) d d ( t) d n n d 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 8 / 27
4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Substtutng back nto (4.5) yelds, (4.6) Note that R z = z as z means z, the z-axs of coordnate frame {} descrbed n terms of {}. The vector z represents the mappng of a "free vector". Equaton (4.6) s effectvely where and vn z t dn t d ( ) ( ( ) ) v r z r d d ( n ) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 9 / 27
4. Moton Knematcs 4.2. Compoundng of Translatonal Veloctes (all jonts movng) As n the prevous case wth angular veloctes, f all the jonts are movng (prsmatc or rotary) the translatonal velocty at the tp becomes: v ( t) v ( t) v ( t) v ( t) n 2 n where agan the v 's are the ndvdual contrbutons compounded by equaton (4.3) as or v ( t) z q( t) f the th jont s prsmatc ( ) ( n ( ) ) ( ) v t z d t d q t f the th jont s rotary where the jont varable q (t) could be an angle (θ (t)) or a dsplacement (d (t)). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde / 27
4. Moton Knematcs 4.3. The Drect Method To account for the fact that a jont varable could be θ(t) (rotary jont) or d(t) (prsmatc jont) we can use q(t) as a "generc" jont varable to represent both possbltes. In lght of our orgnal ntent, whch was to descrbe how the rate of change of jont varables effects the rate of change of tool pose (.e., n th coordnate frame moton), we can now wrte: vn () t n() t J q( t) q( t) J J v q() t q() t qt () J J2 J n q() t (4.7) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde / 27
4. Moton Knematcs 4.3. The Drect Method If the th jont s rotary, then: If the th jont s prsmatc, then: ˆ z dn () t d org org J,,, n zˆ (4.8) J,,, n z ˆ (4.9) Thus, usng equatons (4.8) and (4.9) we can buld J. Ths s the Drect Method of Constructng the Jacoban 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
4. Moton Knematcs 4.3.2 An Alternatve Method As noted earler the tool poston s d n () t and therefore the rate of change of tool poston s d dn () v ( t) dn ( q( t)) q( t) dt q n 2 Therefore, the translatonal part of the Jacoban can be obtaned as Jv q( t) q( t) Jv Jv Jv n q( t) J v dn () q (4.2) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 3 / 27
4. Moton Knematcs 4.3.2 An Alternatve Method For the orentatonal part of the Jacoban we could obtan the angular velocty by recallng equaton (4.4), whch allows us to wrte.e., ( t) R( t) R T ( t) n n n (4.2) z y ωn z x n n T ( t) R( t) R ( t) y x (4.22) Now equatng the correspondng terms gves R( t) R ( t) T n n 3,2 x T y ωn ( t) n R( t) n R ( t),3 z T nr( t) nr ( t) 2, 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 4 / 27
4. Moton Knematcs 4.3.2 An Alternatve Method Then factorng out terms n q,..., q n from the rhs of the above gves rse to ( t ωn ) q ( t ) o Choosng between the approaches descrbed n Sectons 4.3. and 4.3.2 s somewhat dependent on the problem at hand! 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 5 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Example: The ADEPT robot s a so called SCARA (Selectvely Complant Assembly Robot Arm) type robot. VRML Smulaton 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 6 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example The Denavt-Hartenberg table: D-H params. - a - d d a 2 a 2 3 8 d t 4 3 () d 4 () t () t 2 () t 4 Usng MATHEMATICA to generate the T-matrces. 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 7 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example C S S C T(,,, ( t)) C2 S2 a S2 C2 2T(, a,, 2( t)) a2 T(8, a, d ( t),) -d3( t) C4 S4 3 S4 C4 4T(,, d4, 4( t)) d4 2 3 2 3 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 8 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Note that: q( t) ( t), ( t), d ( t), ( t) T 2 3 4 (4.23) C2 S2 Ca S C S a R d xˆ yˆ zˆ d 2 2 2 2 2 2 2 2 2T T 2T C2 S2 C2a2 Ca S C S a S a R d 2 2 2 2 2 3 3 3T 2T 3T d3( t) C2C4 S2S4 S2C4 C2S4 C2a2 Ca S C C S C S S S a S a R d 3 2 4 2 4 2 2 4 2 2 4 4 4T 3T 4T d4 d3( t) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 9 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Usng the Drect Method of Secton 4.3. Snce jont s rotary then: J C2a2 Ca S2a2 Sa S S a a C a Ca zˆ 2 2 2 2 ˆ ( ) z d4 d d4 d3 t Snce jont 2 s rotary then: J 2 C2a2 S2a2 S C a a zˆ 2 2 2 2 2 ˆ ( ) z2 d4 d2 d4 d3 t 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Snce jont 3 s prsmatc then: J 3 zˆ 3 Snce jont 4 s rotary then: J 4 ˆ z4 d4 d4 zˆ 4 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Puttng ths all together yelds the Jacoban as Hence, 2 2 2 2 J v 2 3 4 J J q() t J J J J () t v4() t 2() t J q( t) J q( t) q( t) 4() t d3() t 4() t S2a2 Sa S2a2 C a C a C a (4.24) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 22 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Asde: Recall that ω4(t) = J ω q and v4(t) = J v q, hence 4( t) ( t) 2( t) 4( t) ( t) 2( t) 4( t) Usng the Alternatve Method of Secton 4.3.2 Let us frst look at the translatonal part of the Jacoban. Now, d v ( t) d ( q( t)) dt 4 4 d 4 ( ) d4 ( ) d4 ( ) d4 ( ) q ( t) q ( t) q ( t) q ( t) q q q q 2 3 4 2 3 4 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 23 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example d Evaluatng the ndvdual partals gves: 4 2 2 C2a2 Ca S S a a d4 d3() t S2a2 Sa d4 () C2 2 C a a d 2 2 4 d d 4 4 2 d 3 4 () () () S a C 2a2 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 24 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Therefore, J v S2a2 Sa S2a2 C2a2 Ca C2a2 (4.25) Let us next look at the rotatonal part of the Jacoban. Now, T ( t) ( t) R( t) R ( t) 4 4 4 4 (4.26) C C S S S C C S S C C S C C S S 2 4 2 4 2 4 2 4 4 R 2 4 2 4 2 4 2 4 C24 S24 S24 C24 where C 2 4 Cos(θ + θ 2 θ 4 ) and S 2 4 Sn(θ + θ 2 θ 4 ). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 25 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example Usng the substtuton θ 2 4 = (θ + θ 2 θ 4 ) for the sake of clarty: S24 C24 d 4 R C2 4 S2 4 24 dt S24 C24 C24 S24 T ωn ( t) n R( t) n R ( t) C S S C 2 4 24 24 24 24 24 3 2 3 2 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 26 / 27
4. Moton Knematcs 4.3.3 A Jacoban Example ( ) ( ) 2 Jq( t) d 3 4 ω t n 24 2 4 (4.27) Puttng together the results of equaton (4.25) for J v and equaton (4.27) for J ω we get, J q() t S2a2 Sa S2a2 C2a2 Ca C2a2 Jv q() t J q() t Both methods agree!!! 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 27 / 27