Lecture «Robot Dynamics»: Dynamics 1

Transcription

1 Leture «Robot Dynamis»: Dynamis V leture: CAB G11 uesday 10:15 12:00, every week exerise: HG E1.2 Wednesday 8:15 10:00, aording to shedule (about every 2nd week) offie hour: LEE H303 Friday Maro Hutter, Roland Siegwart, and homas Stastny Robot Dynamis - Dynamis

2 oi itle Intro and Outline L1 Course Introdution; Reaitulation Position, Linear Veloity, ransformation Kinematis 1 L2 Rotation Reresentation; Introdution to Multi-body Kinematis Exerise 1a E1a Kinematis Modeling the ABB arm Kinematis 2 L3 Kinematis of Systems of Bodies; Jaobians Exerise 1b L3 Differential Kinematis and Jaobians of the ABB Arm Kinematis 3 L4 Kinemati Control Methods: Inverse Differential Kinematis, Inverse Kinematis; Rotation Error; Multi-task Control Exerise 1 E1b Kinemati Control of the ABB Arm Dynamis L1 L5 Multi-body Dynamis Exerise 2a E2a Dynami Modeling of the ABB Arm Dynamis L2 L6 Dynami Model Based Control Methods Exerise 2b E2b Dynami Control Methods Alied to the ABB arm Legged Robots L7 Case Study and Aliation of Control Methods Rotorraft 1 L8 Dynami Modeling of Rotorraft I Rotorraft 2 L9 Dynami Modeling of Rotorraft II & Control Exerise 3 E3 Modeling and Control of Multioter Case Studies 2 L10 Rotor Craft Case Study Fixed-wing 1 L11 Flight Dynamis; Basis of Aerodynamis; Modeling of Fixed-wing Airraft Exerise 4 E4 Airraft Aerodynamis / Flight erformane / Model derivation Fixed-wing 2 L12 Stability, Control and Derivation of a Dynami Model Exerise 5 E5 Fixed-wing Control and Simulation Case Studies 3 L13 Fixed-wing Case Study Summery and Outlook L14 Summery; Wra-u; Exam Robot Dynamis - Dynamis 1 2

3 Reaitulation of Kinematis Kinematis = desrition of motions ranslations and rotations Various reresentations (Euler, quaternions, et.) Instantaneous/Differential kinematis Jaobians and geometri Jaobians Inverse kinematis and ontrol Floating base systems (unatuated base and ontats) Robot Dynamis - Dynamis

4 Dynamis in Robotis Robot Dynamis - Dynamis

5 Dynamis in Robotis Robot Dynamis - Dynamis

6 Dynamis Outline Desrition of ause of motion Inut τ Fore/orque ating on system Outut q Motion of the system Prinile of virtual work Newton s law for artiles Conservation of imulse and angular momentum 3 methods to get the EoM Newton-Euler: Free ut and onservation of imulse & angular momentum for eah body Projeted Newton-Euler (generalized oordinates) Lagrange II (energy) Introdution to dynamis of floating base systems External fores, M q qb q q g q τ JF q Generalized oordinates Mq Mass matrix bq, q Centrifugal and Coriolis fores gq Gravity fores τ Generalized fores F External fores J Contat Jaobian Robot Dynamis - Dynamis

7 Prinile of Virtual Work Prinile of virtual work (D Alembert s Prinile) Dynami equilibrium imoses zero virtual work variational arameter Newton s law for every artile in diretion it an move d F external fores ating on element i r ext aeleration of element i dm mass of element i r virtual dislaement of element i F ma r Frma fore d F m v dt Imulse or linear momentum dm r df r S moment d Γ rm dt v N N angular momentum Robot Dynamis - Dynamis

8 Virtual Dislaements of Single Rigid Bodies Rigid body Kinematis Alied to rinile of virtual work Robot Dynamis - Dynamis

9 Imulse and angular momentum Use the following definitions Conservation of imulse and angular momentum Newton Euler A free body an move In all diretions External fores and moments Change in imulse and angular momentum Robot Dynamis - Dynamis

10 1 st Method for EoM Newton-Euler for single bodies Cut all bodies free Introdution of onstraining fore Aly onservation and to individual bodies Ψ Ω F i2 m i System of equations 6n equation Eliminate all onstrained fores (5n) F i1 a i v i F g r OSi Pros and Cons + Intuitively lear + Diret aess to onstraining fores Beomes a huge ombinatorial roblem for large MBS {I} Robot Dynamis - Dynamis

11 Free Cut Cart endulum examle Find the equation of motion {I} m, l g m, Robot Dynamis - Dynamis

12 Free Cut Cart endulum Imulse / angular momentum art Imulse / angular momentum endulum Kinematis mx Fx my Fy Fl Fr mg FbFb r l mx Fx my F mg y Fl (1) (2) (3) (4) (5) Flos sin (6) x y x x (7) y 0 (onstraint) (8) 0 (onstraint) (9) 2 x xlsin (10 a) x x los lsin (10) 2 y los (11 a) y lsin los (11) (12 a) (12) 6 equations, 6 unknowns res. 12 equations, 12 unknowns How many dimensions does the EoM have? {I} y x Fl x m, F y mg b F y mg l F x Fr F x m, g Robot Dynamis - Dynamis

13 Free Cut Cart endulum (7),(10-12) in (1) and (4-6) x m, F y F x mx Fx (13) m x los lsin F (14) 2 2 Fl m lsin los F m g (15) y Flos sin (16) x y x Fl mg Fr From (13) and (14) remove F x 2 m m x ml lm os sin 0 {I} F y l F x g Insert (13) and (15) in (16) to remove F x and F y 2 ml ml x glm os sin 0 mg m, Robot Dynamis - Dynamis

14 Newton-Euler in Generalized Motion Diretions For multi-body systems Exress the imulse/angular momentum in generalized oordinates Virtual dislaement in generalized oordinates With this, the rinile of virtual work transforms to 0 W= q q Mq bq, q g q

15 Projeted Newton-Euler Equation of motion, M q q b q q g q 0 Diretly get the dynami roerties of a multi-body system with n bodies For atuated systems, inlude atuation fore as external fore for eah body If atuators at in the diretion of generalized oordinates, orresonds to staked atuator ommands Robot Dynamis - Dynamis

16 Projeted Newton-Euler Cart endulum examle Find the equation of motion {I} m, l g m, Robot Dynamis - Dynamis

17 Projeted Newton-Euler Cart endulum examle Kinematis art and endulum x q r OS J J x 0 dros 1 0 dq 0 0 dj 0 0 P dt 0 0 Equation of motion P P r OS J J P P x lsin l os dros 1 l os d 0 l sin q dj P 0 lsin dt 0 l os J R q m m lmos M JP m i ijp J i R i ijr J i PmJP JPmJP JR JR 2 lmos ml b JPm i ij Pq J i RΘ i ij Rq J i Rq Θ i ijrq 0 (lanar system) 2 lm sin i Pm J J Pq n g g JsF i i JP P i1 mg J mg mgl sin {I} x m, l m, Robot Dynamis - Dynamis g

18 3 rd Method for EoM Lagrange II Lagrangian Lagrangian equation kineti energy otential energy Sine U U q, qq inertial fores gravity vetor d dt U τ q q q 1 2 qmq with q Mq M q q q 1 1 Mq M q g Mqq b q, q gq τ 2 M q q q n Robot Dynamis - Dynamis

19 Lagrange II Kineti energy Kineti energy in joint sae Kineti energy for all bodies 1 2 qmq From kinematis we know that Hene we get Robot Dynamis - Dynamis

20 Lagrange II Potential energy wo soures for otential fores Gravitational fores Sring fores E j 0 0 d0 F k rr r r r r 0 Robot Dynamis - Dynamis

21 Lagrange II Cart endulum examle Find the equation of motion {I} m, l g m, Robot Dynamis - Dynamis

22 Lagrange II Cart endulum examle Kinematis art and endulum r r OS S x 0 x 0 OS Kineti and otential energy Equation of motion r r S x lsin l os x los l sin r os 2 S m i irs ω i i iωi mx mx ml mxl U mgl os 0-level an be hosen d 0 dt q q U q mx mx ml os 2 q ml mxl os 2 d mx mx ml os ml sin 2 dt q ml mxl osmxl sin 0 mxl sin q {I} 2 sin sin mx mx ml os ml 0 2 ml mxl os mgl x m, l m, g U 0 mgl sin q Robot Dynamis - Dynamis

23 External Fores Given: Generalized fores are alulated as: Given: Generalized fores are alulated For atuator torques: Robot Dynamis - Dynamis

24 External Fores Cart endulum examle Equation of motion without atuation Add atuator for the endulum Ation on endulum a 0 1 Reation on art a R 0 0 Add sring to the endulum 2 m 0 m lmos lm sin 2 lm os 0 mgl sin ml q 0 M b a J R (world attahment oint P, zero length 0, stiffness k) Ation on endulum x 2sin l r Fx 2os l Fx Fs F τ JFs y 1 2l os r 2lFxos Fysin s J s 0 2l sin q g 0 τ J J J a J Ri, i R R {I} x m, a l F k x2sin l x x l P x, y m, Robot Dynamis - Dynamis s s s g F k 2os l y y s

25 External Fores Cart endulum examle What is the external fore oming from the motor {I} x m, g M l m, l Robot Dynamis - Dynamis

26 External Fores Cart endulum examle What is the external fore oming from the motor Ation on art F at F F P 1 0 at τ F at J F J 0 {I} F at x m, l g m, l Robot Dynamis - Dynamis

27 Dynamis of Floating Base Systems Robot Dynamis - Dynamis

28 Dynamis of Floating Base Systems Quaternions Euler angles r EoM from last time Mq b g τ j b F s Not all joint are atuated Mq b g S τ Seletion matrix of atuated joints S 0 I q j Sq n6 nn Contat fore ating on system Mqbg S τ JF s s, ating on system M b F q gj s s, exerted by robot S τ q q q b j Un-atuated base Atuated joints Maniulator: Legged robot: UAV: interation fores at end-effetor ground ontat fores lift fore Note: for simliity we don t use here u but only time derivatives of q Robot Dynamis - Dynamis

29 External Fores Some notes External fores from fore elements or atuator E.g. soft ontat Aerodynamis k F k r r r F s des d 1 A 2 s v L External fores from onstraints Mq b g J F S τ Equation of motion (1) r J q 0 Contat onstraint (2) s 1 Substitute in (2) from (1) r J M S τ b g JF Jq 0 (3) s s s r J q J q 0 s s s s s s s s Solve (3) for ontat fore F J M J J M S τ b g Jq s s s s s Robot Dynamis - Dynamis

30 Suort Consistent Dynamis Equation of motion Mq b g JsFs S τ (1) Cannot diretly be used for ontrol due to the ourrene of ontat fores Contat onstraint r J q J q 0 s s s Contat fore Bak-substitute in (1), relae Jq s Jq s and use suort null-sae rojetion Suort onsistent dynamis F J M J J M S τ bg J q s s s s s 1 N IM J J M J J 1 1 s s s s s N Mq N b g N S τ S S S JN 0 s s Robot Dynamis - Dynamis