Differential Motion  n n Forward Kinematics Instantaneous Kinematics TG Relationship GT Linear Velocity Angular Velocity iiiii qd TH Joint Coordinates revolute d prismatic coordinate i Joint coordina
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Differential Motion n n Forward Kinematics Instantaneous Kinematics TG Relationship GT Linear Velocity Angular Velocity iiiii qd TH Joint Coordinates revolute d prismatic coordinate i Joint coordina

qqqq brPage 2br Jacobians Direct Differentiation fq fq fq mm mm mm 1 1 mmn n Jqq uuu 1 1 mmnn Jqq uu ij i Jq 1 1 mmn n Jqq uu xJ TGT Example xy lc lc yls ls 11 212 11 212 GT GT GGTGT ylclc lc 11 212 1 212 2 11 212 1 212 2 GT GT

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Differential Motion n n Forward Kinematics Instantaneous Kinematics TG Relationship GT Linear Velocity Angular Velocity iiiii qd TH Joint Coordinates revolute d prismatic coordinate i Joint coordina




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Presentation on theme: "Differential Motion n n Forward Kinematics Instantaneous Kinematics TG Relationship GT Linear Velocity Angular Velocity iiiii qd TH Joint Coordinates revolute d prismatic coordinate i Joint coordina"— Presentation transcript:


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Differential Motion {0} {n} {n} Forward Kinematics Instantaneous Kinematics TG Relationship: GT Linear Velocity Angular Velocity iiiii qd TH Joint Coordinates revolute d prismatic coordinate i Joint coordinate-i: 0 revolute 1 prismatic with ii and Joint Coordinate Vector: 12 ( .... ) qqqq
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Jacobians: Direct Differentiation (); fq fq fq mm ## () () () mm mm ### (1) ( ) (1) () mmn n Jqq uuu (1) ( ) (1) () mmnn Jqq uu () () ij i Jq (1) ( ) (1) () mmn n Jqq uu () xJ TGT Example (x,y) lc lc yls ls 11 212 11 212 GT GT GGTGT ylclc lc () () 11 212 1 212 2 11 212 1 212

2 GT GT yls xlc 212 212 xx yy wT wT wT wT 12 12 () xJ 212 212 ls lc {0} i-1 i-1 000 -90 0 d 90 0 d 00 0 -90 0 0 90 0 0 {0} C C CCS SC SSS S SCS CC SCCCS SC SSS C 1 2 4 56 4 6 256 1 4 56 4 6 124 646 2 61 [( ) ]( ) [( ) ] () () SCS CC SCCS SC CSS 646 2456 46 256 CC CCC SS SSC SSCC CS SC CCC SS SSC CSCC CS SCCC SS CSC 1 2 456 46 256 1 456 46 12 4 646 2 614 646 2456 46 256 [( ) ] ( ) [( ) ] ( ) () () () CCCS SC SSS SCCS SC CSS SCS CC 1245 25 145 1245 25 145 245 25 CSd Sd SSd Cd Cd 12 3 12 12 3 12 23 xr
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Stanford Scheinman Arm csd sd ssd cd cd 12 3 1 2 12 3 1 2 23 (3 1) (3 6) (6 1)

() xx Jqq uu yccd cs xscdss sd c 12 3 12 12 3 12 23 2 000 000 0000 Linear Velocity Linear Velocity Position rq rq rq () () () () 91) 96 61) () qq RX Orientation: Direction Cosines C C CCS SC SSS S SCS CC SCCCS SC SSS C 1 2 4 56 4 6 256 1 4 56 4 6 124 646 2 61 [( ) ]( ) [( ) ] () () SCS CC SCCS SC CSS 646 2456 46 256 CC CCC SS SSC SSCC CS SC CCC SS SSC CSCC CS SCCC SS CSC 1 2 456 46 256 1 456 46 12 4 646 2 614 646 2 4 5 6 46 25 6 [( ) ] ( ) [( ) ] ( ) () () () CCCS SC SSS SCCS SC CSS SCS CC 1245 25 145 1245 25 145 245 25 () 91) 96 61) Representations Cartesian Spherical Cylindrical .

Euler Angles Direction Cosines Euler Parameters Jacobian for X () () qq xJqq PX RX () () Jq () (12 (12 ) ( XJqq xXxx 1) 6 6 1) The Jacobian is dependent on the representation Cartesian & Direction Cosines xExv xEx PPP RRR () () (6 1) Jq q xn nx ()( () 06 1)
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xEx sc cc cs Ex RRR PPP DD ;() .. ;() 100 010 001 Jacobian for X () Jqq () () () Jq ExJq Given a representation () Jqq Basic Jacobian Jqq vJq Jq (). (.) (.) Ev xE x EJq PP P Pv RR R R JEJ XPv XR XP XR w JJ () ( ) () Jq EX J q () Jqq ;; PXPv IJ J () XI (, ,) yz cos sin 0 () sin cos 0 001 EX TT TT UU (,,) ()(cossin) TT yz

z UTUT cos sin sin sin cos () sin cos 0 sin sin cos cos sin cos sin EX ITII TT UIUI TITII UU (,,) TI ( ) ( cos sin sin sin cos ) TT xyz TIUTIUT
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.. ;() 0 RRR sc cc ss xExcs sc ss DEDE EE EDD JDD EE Jacobian for X () Jqq () () () Jq ExJq Given a representation () Jqq Basic Jacobian qq xn nx ()( () 61) 61) Linear & Angular Velocities Linear Velocity PA {A} {B} {C} PA PA PA Pure Translation {B} PA {A} AB
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PB {B} PA {A} AB vvv PB AB PA /// Pure Translation Rotational Motion rigid body fixed points on the rigid body Axis of rotation Angular Velocity Rotational Motion

Angular Velocity Rotational Motion Angular Velocity Rotational Motion Angular Velocity fixed point Rotational Motion
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Angular Velocity fixed point Rotational Motion Angular Velocity Psin fixed point is proportional to: || || ||Psin || and v _| : v _| vP Rotational Motion cab u Cross Product Operator zyx zxy xz aab cab a a b aa b cab : a skew-symmetric matrix vectors matrices zz ab aabb ab cab vPvP PP u :: Cross Product Operator vP zy zx yx :: :: :: vP : a skew-symmetric matrix Simultaneous linear and angular motion {A} {B} BA PB PA BA B AB PB BB AB vvRv RP // / ..

u vvv P PA BA PB B ///
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Spatial Mechanisms {2} {0} {1} {n} v (). xJ : linear velocity Z angular velocity Propagation of velocities {i} i+1 i+1 L {i+1} i+1 i+1 iii iii 11 111 vv PdZ iiiiii u 1111 Velocity propagation Linear Angular ii ii ii vRv PdZ u 11 .( ) Velocity propagation Joint 1 and in frame {1} Joint i+1 ii ii ii RZ ZZT and vR ZZ {n-1} {n} Example {0} vv P iiii u 11 vv P vv P PP PP 21 32 12 23 u u 11 11 11 11 11 1 00 00 0000 0 lc ls ls lc .. TT 000 32 vv P PP 11 11 1 1 2 010 100 000 ls lc P .. .( ). TTT ls lc ls lc 11 11 1

212 212 1 2 00 .. ..( TTT 3123 ). lc ls 212 212
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11 212 212 11 212 212 000 ls ls ls lc lc lc () 000 000 111 iii : iii VZq iin in iin :u [()] iiiiin vV P :u ii iii VZq iii : iin in iin :u [()] ii i i in i vZZPq u () ii i iii VZq iii 11 1 1 1 1 11 1 1(1) 1 [()] [()] nn n n nnn nnn vZ ZPq ZP q Zq u u >@ 11 1 1 1 2 2 2 2 2 () ( ) nn vZ ZP Z ZP uu vJq 111 2 22 nnn qZq Zq >@ 11 2 2 nn ZZ Z PP P 12 PP P xx qq q www www
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10 ZZ Z PP P nn 12 11 2 2 .. . 12 ZZ Z PP P nn .. . 00 11 22 00 RZ RZ RZ PP nn () () () .( . ) .( . ) .( . ) ii ZZ 113 223 000 i-1 i-1 000 -90 0 d 90 0 d 00 0 -90 0 0 90 0 0 {0} i-1 i-1 000 -90 0 d 90 0 d 00 0 -90 0 0 90 0 0 T = i - 1 -s 0a i-1 i-1 i-1 -s i-1 -s i-1 i-1 i-1 i-1 i-1 0 0 0 1 T = T T ... T N-1 Forward Kinematics:
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11 11 11 00 00 0010 0001 cs sc 22 22 00 001 00 0001 cs sc 10 0 0 00 1 01 0 0 00 0 1 44 44 00 00 0010 0001 cs sc 55 55 00 0010 00 0001 cs sc 66 66 00 00 10 00 0001 cs sc 12 12 1 1 2 12 12 1 1 2 22 00 0001 cc c sc ss c cd sc 12 1 12 1 32 1 2 12 1

12 1 32 1 2 2232 000 1 cc s cs cds sd sc c ss sds cd scdc 11 11 00 00 0010 0001 cs sc 124 14 124 14 12 1 32 1 2 124 14 124 14 12 1 32 1 2 24 24 2 32 0001 ccc ss ccs sc cs cds sd cc cs scs cc ss sds cd sc ss c dc 124 14 1 32 1 2 124 14 1 32 1 2 24 32 00 0 1 XX ccs sc cds sd XX scs cc sds cd XX ss dc 1245 145 125 1 32 1 2 1245 145 125 1 32 1 2 24 55 232 00 0 1 XXcccs sss css cds sd XXsccscsssscsdscd XX scs cc dc C C CCS SC SSS S SCS CC SCCCS SC SSS C 1 2 4 56 4 6 256 1 4 56 4 6 124 646 2 61 [( ) ]( ) [( ) ] () () SCS CC SCCS SC CSS 646 2456 46 256 CC CCC SS SSC SSCC CS SC CCC SS SSC CSCC CS SCCC

SS CSC 1 2 456 46 256 1 456 46 12 4 646 2 614 646 2 4 5 6 46 25 6 [( ) ] ( ) [( ) ] ( ) () () () CCCS SC SSS SCCS SC CSS SCS CC 1245 25 145 124 14 245 25 CSd Sd SSd Cd Cd 12 3 12 12 3 12 23 xr 1245 145 125 132 12 124 14 12 132 12 24 55 232 00 0 1 X X cccs sss css cds sd XXsccs css ssc sds cd XX scs cc dc 000 123 00 000 12 456 000 PPP xxx qqq ZZZZ www www cd dccdc sd csd scd ss sd c scsccssccccsssscsc cssscsccsccscssssc css scscc 1 2 12 3 12 3 12 1 2 12 3 12 3 12 23 2 1 12 124 14 1245 145 125 11212414124 14 12 224 24 55 00 0 00 0 0000 00 00 100
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12 12 JJ J det 0 det det ij JJ

det[ ] det[ ] BA det det ij JJ Singular Configurations det[ ( )] 0 Jq Singular Configurations 12 det[()] () ()... () 0 Jq S qS q S q () 0 () 0 () 0 Sq Sq Sq 12 2 12 2 112 12 112 12 lS lS lS lC lC lC 12 det 2 JllS 12 112 lC lC 12 112 ylS lS qk {0} {1} (x,y) 22 12 2 22 11 22 11 lS lS CS llC lC SC 110 RJ 12 2 00 ll l {0} {1} (x,y) 12 122 GGTGT yll l () (x,y) 12 1 (1) 12 12 2 1 11 ll ll ll l ' ' qJ X qX '' ' {1}
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13 (1) (1) 12 1 '' ' ll 12 (1) (1) 12 2 1 ' ' ll x ll l qX '' ' {1} (1) (1) (1) (1) 12 (1) 2(1) () ll ly {1} 12 2 12 2 112 12 112 12 lS lS lS lC lC lC 12 det 2

JllS qk {0} {1} (x,y) 12 2 12 2 112 12 112 12 00 00 11 lS lS lS lC lC lC 1 1 2 12 3 123 2 12 3 123 3 123 1 1 2 12 3 123 2 12 3 123 3 123 000 000 000 111 lc lc lc lc lc lc 1 1 2 12 3 123 2 12 3 123 3 123 1 1 2 12 3 123 2 12 3 123 3 123 111 ls ls ls ls ls ls J lclclc lclc lc 1 1 2 12 3 123 2 12 3 123 3 123 11 212 3123 212 3123 3123 000 000 000 111 lc lc lc lc lc lc
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14 {0} i-1 i-1 000 -90 0 d 90 0 d 00 0 -90 0 0 90 0 0 T = i - 1 -s 0a i-1 i-1 i-1 -s i-1 -s i-1 i-1 i-1 i-1 i-1 0 0 0 1 T = T T ... T N-1 Forward Kinematics: 000 123 00 000 12 456 000 PPP xxx qqq ZZZZ www www cd

dccdc sd csd scd ss sd c scsccssccccsssscsc cssscsccsccscssssc css scscc 1 2 12 3 12 3 12 1 2 12 3 12 3 12 23 2 1 12 124 14 1245 145 125 11212414124 14 12 224 24 55 00 0 00 0 0000 00 00 100 12 123 123 12 12 123 123 12 23 2 1121241412 1121241412 2242 000 000 0 000 00 00 100 cd ssd ccd cs sd csd scd ss sd c cs ccs sc cs cssscsccss cssc Jacobian at the End-Effector ne {n} {e} vv en nne u vv enne n en u ZZ
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15 vv ZZ ne JJ en ne vv enne n en u ZZ Jq Jq en ne IP ne Cross Product Operator (in diff. frames) 00 0 z (.) . 00 0 PRP nn u u ZZ .( ) 00 0 0 0 0 ..( .)

.( ..) PRP RPR nn ZZZ 00 0 nT nn PRPR 00 PRP ne {n} {e} 00 0 nT nnnen en RRPR JJ Wrist Point lc lc yls ls 11 212 11 212 End-Effector Point yls ls ls 11 212 3123 1 1 2 12 3 123 Wrist Point lc lc yls ls 11 212 11 212 End-Effector Point yls ls ls 11 212 3123 1 1 2 12 3 123 Jacobian (W) ls ls ls lc lc lc 11 212 212 11 212 212 000 000 000 111 IP WE Wrist Point lc lc yls ls 11 212 11 212 End-Effector Point yls ls ls 11 212 3123 1 1 2 12 3 123 ls ls ls lc lc lc 11 212 212 11 212 212 000 000 000 111 1 1 2 12 3 123 2 12 3 123 3 123 1 1 2 12 3 123 2 12 3 123 3 123 000 000 000 111 lc lc lc lc lc lc


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16 Wrist Point lc lc yls ls 11 212 11 212 End-Effector Point yls ls ls 11 212 3123 1 1 2 12 3 123 ls ls ls lc lc lc 11 212 212 11 212 212 000 000 000 111 IP WE 3123 3 123 3 123 3123 3 123 3 123 00 00 lc ls P ls lc ls lc WE WE Resolved Motion Rate Control (Whitney 72) GT () Outside singularities GT Jx () Arm at Configuration () GT GT Resolved Motion Rate Control J -1 Control Control Control Joint n Joint 2 Joint 1 Forward Kinematics Angular/Linear Velocities/Forces u vp vp Angular/Linear Velocities/Forces yx pp xy vp ()
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17 Velocity/Force Duality

Propagation Elimination of Internal forces Energy Analysis Virtual Work Static Equilibrium -f -n link 3 -n -f -f -n link 1 -n link 2 -f Link i i+1 -f i+1 -n i+1 forces = 0 moments / a point = 0 About origin {i} nnP f ii iii i u () () () 11 1 nn P f ii ii i i u 111 Static Equilibrium Link i i+1 -f i+1 -n i+1
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18 Prismatic Joint Revolute Joint ii fZ ii nZ Algorithm nn ii ii ii ii ff nnP f fRf nRnPf u u 11 Virtual Work Principal TT qFx GG Jq TT FJ Static Equilibrium: If the virtual work done by applied forces is zero in

displacements consistent with constraints Internal forces are workless ii wfx applied forces virtual displacements () 0 Fx using WG Velocity/Force Duality 12 2 12 2 112 12 112 12 lS lS lS lC lC lC {0} {1} (x,y) 12 1 2 22 112 112 12 12 lS lS lC lC lS lC 12 12 1 2 22 112 112 112 112 12 12 lC lC lS lS lC lC lC lS lC 12 1 2 1; 0; 60 ll TT 3/2 1/2 {0} {1} (x,y) 12 12 1 2 22 112 112 112 (1 ) 112 12 12 lC lC lS lS lC lC KlC lS lC 12 1 2 1; 90; 0 ll TT {1} (x,y)