Basic Dynamic Model. Basic Dynamic Model. Unified Motion/Force Control. Unified Motion/Force Control ( ) Operational Space Dynamics.

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1 Moble Robot Manpulaton Oussaa Khatb Robotcs Laboratory Departent of Coputer Scence Stanford Unversty Wth the Stanford-Schenan Ar Wth the PUMA 6 Stanford Free-Flyng Robot Stanford Robotc Platfors (993) ask-orented Control Γ ask-orented Dynacs V ( xgoal ) F x F V ( xgoal ) Γ F F x + µ + p F F F (dynacs)

2 Operatonal Space Dynacs Unfed Moton/Force Control ( x) x + µ ( x, x ) + p( x) F x : End-Effector Poston and Orentaton ( x): End-Effector Knetc Energy Matrx µ ( x, x ): End-Effector Centrfugal and Corols forces p( x): End-Effector Gravty forces F : End-Effector Generalzed forces F contact F oton F F + F oton contact Unfed Moton/Force Control Basc Dynac Model Operatonal force? Forces & Moents Generalzed Selecton Matrx X qq ( ) Dynac Model (Hoogenety) ( ) ( x) ϑ + µ x, ϑ + p ( x) + Fcontact F o Γ ( q) F Forces & Moents Basc Dynac Model v ( ) q q F f ; q E( X) q v X E F E F f E F E E x + µ ( xx, ) + px ( ) F E ϑ + µ ( x, ϑ) + p ( x) F wth v ϑ ˆ

3 End-Effector/Sensor Syste ϑ + µ ( x, ϑ) + p ( x) + Fcontact F Unfed Control F Foton + Fforce F ˆ ˆ ˆ oton Ω F + µ + P * oton F ˆ Ω + * force Fforce Fsensor End-Effector/Sensor Syste ϑ + µ ( x, ϑ) + p ( x) + Fcontact F Unfed Control F Foton + Fforce F ˆ ˆ ˆ oton Ω F + µ + P * oton F ˆ Ω F +ΩF * force force desred Unfed Moton & Force Control wo decoupled Subsystes Ω ϑ Ω Ω ϑ Ω * F oton * F force Redundancy Equatons of Moton ont Space Aqq ( ) + bqq (, ) + gq ( ) Γ Operatonal Space ( x) x+ µ ( x, x ) + p( x) F Relatonshps Γ F ( Aq + b+ g) ( x + µ + p) ( Aq + b) ( x + µ ) Inertal forces Non Redundancy Aq + b+ g Γ x + µ + p F (jont dynacs) (ask dynacs) 3

4 projecton Aq + b+ g Γ Redundancy x + µ + p F (jont dynacs) (ask dynacs) where A and A : dynacally consstent generalzed nverse Dynac Consstency Γ A jont torques task acceleraton Γ Relatonshp # Γ F + I Γ Dynac Constrant ( ) A I ( ) # Γ ( A A ) # A A # ( ) Dynac Consstency ( q) s the Dynacally Consstent Generalzed Inverse heore (Consstency) s unque and Non-redundant A 4

5 ask dynacs ( qx ) + µ ( qq, ) + pq ( ) F ( A ) µ ( q, q ) b( q, q ) ( q) ( q) q Inertal Propertes p( q) g( q) Effectve Mass/Inerta Redundancy Redundancy of a Manpulator ask Redundancy n > Redundancy wth respect to a ask n> ask () ask Redundancy y y y q q qn y q y v y v A y vy I z A z z vy z y z v v x x q qn y y q qn z z q q n vy vy y y ( ) v q qn y v 5

6 Effectve Mass/Inerta y y y v I z z z Effectve Mass/Inerta vu u u v u Effectve Mass u( v) u v u Effectve Inerta I u ( ) u u u Effectve Mass/Inerta Effectve Mass/Inerta v A ( v ) A v v A v v A A v A v Effectve Inertal Property σ ( ) w w w : A unt vector n the -densonal space w Exaple y A q x () S S + C C + (+ C) (+ C) (+ C) 6

7 Redundancy y q x wth respect to otons n the drecton y y ( ) + C C A y y y ( + C) + C + y + S (+ C) + ( + C) C ( ) [ C C ] C + S S * Y ( + ) * y ( + ) S C + S Effectve Mass & Inerta v y x * x ( + ) S S ( + ) Belted Ellpsod x y z a b c + + x y z + + a x + y + z b x + y + z c x + y + z 7

8 Macro/Mn Structures Moble Manpulators Inertal Propertes Inertal Propertes Reduced Effectve Inerta Effectve Inerta: Equalty 8

9 Reduced Effectve Inerta Inerta Property Effectve Inerta/ass perceved n a drecton w σ w( ) w w heore: Reduced Effectve Inerta he nertal propertes of a acro/n robot are bounded above by the propertes of ts n structure σ ( ) σ ( ) w w n n any drecton w n Macro/Mn Structure v v + v + p M M + M [ ] V M V I pˆ 3 I 3 M v M v Macro/Mn Structure M v M v A and A M v M v +c α σ ( ) σ ( ) + w w A A A A A α w c w A A 9

10 and + c c ( VM A A) ( A AA A ) ( V M A A) M v M v σ and σ ( ) w ( ) σ ( ) w w w heore σw( ) σw( ) for any w n Corollary I u u ( v) u( ( v) ) ( w) u( ( w) ) v [ ] [ ] ( I ) I I I cv ( ) c n + η 3 {} 3 o η ; I + q η I + q + q ( ) 3 Control Structure Γ F + where and N N I posture Γ Γ V posture posture Robot Cooperaton

11 Cooperatve Manpulaton Dynacs: Augented Object Mult-robot Control Leader/Follower δ x δ x δ x Internal Forces: Vrtual Lnkage Centralzed Control: Fxed-Base Manpulaton Decentralzed Control: Moble-Base Manpulaton δ q δ q Augented Object Model Effect of a Load v F x effector x effector x load v l v+ Il ( ) I l 3 load I l Effect of a Load v F x Mult-Ar Control F x Knetc Energy ( xx, ) effector + load Lea + effector load Lea + load

12 heore (Augented Object) Augented Object Model µ µ µ p p p F F x+ µ ( x, x ) + p ( x) F load load load x+ µ ( x, x ) + p ( x) F Allocaton of Forces τ requred τ ax F F α α F Measure of actuator effort τ requred ax( ) r τ ax α r α r τ requred ax( ) r τ ax τ requred τ ax Allocaton of Forces For N robots α r α r α r α : N N Mnzed overall effort β rr r α ; where β β + β + + β r N N Allocaton of Forces Γ F Γ F

13 Internal Forces f Vrtual Lnkage fr W f # f W f r f 3 # ( I WW) f + f I W W f # nternal ( ) f Actuator DOF: 8 Resultant Force: 6 Actuator Redundancy: Internal Moents (3N): 9 Internal Forces (3N-6): 3 Vrtual Lnkage A Four-Grasp Vrtual Lnkage For an N-grasp anpulaton task, the vrtual lnkage s a 6(N-) DOF echans, whose actuated jonts characterze the nternal forces and oents. Actuaton: 4DOF Internal: 8DOF Vrtual Lnkage fr f W(6 8) r W(6 8) Wf (6 9) W (6 9) W f W Resultant Forces I3 I3 I3 rˆ ˆ ˆ r r 3 I3 I3 I 3 Vrtual Lnkage Internal Forces f E t e E e e e t E f (3 9) (9 3) 3 3 e e 3 3 3

14 Vrtual Lnkage Internal Moents τ 3 3 Grasp Descrpton Matrx fr r f G(8 8) t τ Wf W G E 39 9 I 9 full rank G W f E WfW I9 f 3 f 3 f Exaple Augented Object x+ µ ( x, x ) + p ( x) F Vrtual Lnkage: Grasp Matrx fres G (6N 6 N) f nt f N f res G( 6N 6N) f nt f N f f Centralzed Control Structure Exaple 4

15 Decentralzed Control Structure Dancng wth ulet Autonoous asks 5

16 rackng Vsual Servong Depth 6