Rigid Part Mating. Goals of this class

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Horatio Mitchell Hodges
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1 Rigid Part Mating Goals of this class understand the phases of a typical part mate determine the basic scaling laws understand basic physics of part mating for simple geometries relate forces and motions arising from geometric errors compare logic branching and direct error-feedback part mating strategies 1

2 Basic Bandwidth Issues and Time-Mass- Distance Scaling Laws Torque required to move a mass M at the end of an arm of length L in time T is proportional to M L 2 /T 2 This implies that really fast motions must be really small or use a small arm with small mass I estimated my hand s mass = 250g, effective length = 10cm my arm + hand s mass = 1700g, effective length = 35 cm ratio arm:hand of ML 2 = T 2 = 85 Don t forget: arm mass+payload mass=m 2

3 Main Phases of a Part Mating Event contact force Approach Chamfer One-point Two-point Line Crossing Cont act Cont act Cont act 3

4 Required Bandwidth for Chamfer Crossing lateral motion E 0.5E E/2 0.5E/V E/2V τ 10E/V 20E/V T time Fourier coefficient = 2 π T / (n2 π2 τ) si n (2 n π τ /T) T = 20 E / V; τ = E / 2 V; T / τ = 40 Period = 2π = ωt=ω20e/v ω = π V /10 E f = V / 20 E If V = 10 in/s and E = 0.05", f = 10 Hz If 5th harmonic must be adhered to, bandwidth needed = 50Hz 4

5 Trapezoidal Wave Harmonics Image removed for copyright reasons. Source: Figure 9-7 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN: rigid part mating 9/13/2004 5

6 Conclusions Gross motions can be (must be) done by large arms that necessarily will move slowly No robot arm with practical reach can make fine motion error removal adjustments at 50 Hz Fine motions can be fast if they are done by small arms, and must be fast to absorb typical errors at economical speeds Big tasks with big parts will take a long time compared to small tasks with small parts What we see: small parts cycle times are ~5s while big parts cycle times are ~ 60s. 6

7 Essentials of Part Mating Theory for Fine Motions Quasi-static assumption Geometry of pegs and holes Applied forces Normal reaction forces and friction reaction forces Entry geometry limits Wedging conditions Jamming conditions Alternate strategies for accomplishing fine motion 7

8 The Basic Idea In gross motions, it pays to pre-plan to prevent errors In fine motion, it does not pay to try to prevent errors So the principle is to anticipate errors and figure out how to make assembly happen anyway This requires us to understand three factors: Geometry Compliance Friction 8

9 Geometry of Peg-Hole Mates 10 TYPICAL MACHINED PARTS l /d = c /θ l PRECISION PARTS Image removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN: l /D l/d = c/θ C = C = C = θ C = 0.1 θ m FOR EACH C θ m = 2 C RADIANS θm = 2c rigid part mating 9/13/2004 9

10 Dimensioning Practice Image removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

11 Geometry Definitions θ o Insertion Direction r c = (D-d)/D d α ε o R W D 11

12 Insertion History 12

13 Insertion History 13

14 Insertion History 14

15 Insertion History 15

16 Insertion History 16

17 Insertion History 17

18 Insertion History 18

19 Insertion History 19

20 Life Cycle of a Part Mate Image removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

21 Model of a Compliant Support Images removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN: All support is assumed concentrated at one point and consists of one lateral stiffness and one angular stiffness 21

22 How Compliance Center Reacts to Force Images removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN: Force away from C. C. causes rotation and translation Force on C. C. causes only translation 22

23 Forces and Moments - Two Point Contact Case Images removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN: All applied and reaction forces are expressed in coordinates at peg s tip 23

24 Forces Applied During Two-point Contact K x K θ g L g When L >> 0 K x When L K g ~ 0 θ K x K θ Big Big K x K θ Small 24

25 Making L g Small is Good How to do it? Active Robot Force Feedback Costly Slow Some way that acts by itself It was invented almost 30 years ago Called Remote Center Compliance Reduces assembly force Avoids one of two main failure modes 25

26 INSERTION FORCE F, NEWTONS z CHAMFER CROSSING ONE-POINT CONTACT TWO-POINT CONTACT Insertion Force History l 1 l 2 l* INSERTION DEPTH l, mm 26

27 Assembly Failure Modes Both occur during two-point contact Wedging sets up compressive forces inside the parts Jamming results from incorrect insertion forces We can derive the requirements to avoid both of these failure modes 27

28 Wedging: Compressive Friction Forces Prevent Insertion Regardless of Insertion Force d d φ = tan -1 µ φ φ = tan -1 µ φ l θ f 1 D f 2 l - d θ l θ f 1 D f 2 l - d θ Wedging can happen if θ > c/µ when two-point contact occurs Wedging can be avoided if µ is small enough or if two-point contact occurs deep enough in the hole 28

29 What s a Friction Cone? Friction cone F θ = tan -1 µ F F N F F T µf N Sliding will occur if F T > µf N F T /F N = tan θ So, sliding will occur if tan θ > µ and F will lie on the boundary of the cone If F is inside the cone then sliding will not happen because F T < µf N and F can be any value 29

30 Conditions for Avoiding Wedging S = L g 2 L +K /K g θ x Images removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

31 Jamming: Insertion Force Directed the Wrong Way - Can t Overcome Friction COMPONENT NORMAL TO PEG AXIS COMPONENT PARALLEL TO PEG AXIS INSERTION FORCE FRICTION CORRESP. TO NORMAL COMPONENT COMPONENT PARALLEL TO PEG AXIS COMPONENT NORMAL TO PEG AXIS INSERTION FORCE REACTION TO NORMAL COMPONENT Component of Insertion force Along insertion direction Not big enough: Peg Is Jammed Component of Insertion force Along insertion direction Is big enough: Peg Goes In 31

32 Conditions for Avoiding Jamming Image removed for copyright reasons. Source: Figure in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

33 Jamming Examples COMPONENT NORMAL TO PEG AXIS COMPONENT PARALLEL TO PEG AXIS INSERTION FORCE FRICTION CORRESP. TO NORMAL COMPONENT COMPONENT PARALLEL TO PEG AXIS COMPONENT NORMAL TO PEG AXIS INSERTION FORCE M REACTION TO NORMAL COMPONENT M F x F z F x /F z is big. M/rF z is big. F z F x F x /F z is small. M/rF z is small. 33

34 Target Expands as Depth Increases M rf z This point moves As λ increases l λ = 2r µ 2λ + 1 λ 1 1/µ 1/µ -1 F x F z λ -( 2λ+ 1 ) This point moves As λ increases rigid part mating 9/13/

35 Experimental Data rigid part mating 9/13/

36 Experimental Data -2 36

37 Test Your Understanding Why does insertion force rise and then fall during two-point contact? 37

38 Experimental Data - 3 When L g = g 0 there there is isbarely any any insertion force. All All that s left left is is chamfer crossing force. 38

39 Test Your Understanding Again Why does the insertion force not rise after chamfer-crossing is finished? 39

40 Review of Force Feedback Strategy Create a coordinate frame at the working point of the part or tool Separate lateral and angular sensing and response motions in that frame Devise a response strategy The Remote Center Compliance is a purely passive implementation of one such strategy 40

41 Simplified Explanation of the Remote Center Compliance (RCC) Images removed for copyright reasons. Source: Figure 9-8 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

42 RCC Response to External Loads (d) RCC UNDER LATERAL LOAD (e) RCC UNDER ANGULAR LOAD (f) LINKAGE RCC UNDER LATERAL AND ANGULAR DEFORMATION 42

43 (d) RCC UNDER LATERAL LOAD 43

44 (d) RCC UNDER LATERAL LOAD 44

45 (e) RCC UNDER ANGULAR LOAD 45

46 (e) RCC UNDER ANGULAR LOAD 46

47 Angular Error r = 3 First RCC Experiment - 1 rigid part mating 9/13/2004 Daniel E Whitne y

48 First RCC Experiment - 2 rigid part mating 9/13/2004 Daniel E Whitne y

49 Commercial Remote Center Compliances Images removed for copyright reasons. Source: Figure 9-9 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture, and Role in Product Development. New York, NY: Oxford University Press, ISBN:

50 rigid part mating 9/13/

51 Assembly is a matter of geometry Geometry contains errors in parts in equipment Ignore errors (Matt Mason approach) Taxonomy of Fine Motions Accept errors Detect errors indirectly by sensing collisions Detect errors directly by sensing them geometrically Seek perfection Costs too much Collisions Collisions But they are small are bad contain info and usually obscured Small stiffness small signals and oscillations and position uncertainty Collisions + But: friction limits stiffness matrix the available information = force signals in force data Large stiffness large signals but possible instability Active force feedback Passive accommodation (incl small stiffness) Can be smart Will be stable can be unstable can be "strange" Engineered Accidental, can be slow contextual Will avoid jamming May not avoid jamming taxonomy of fine motions With sensors IRCC Without sensors RCC basically Lyapunov stable 51

Mobile Manipulation: Force Control

8803 - Mobile Manipulation: Force Control Mike Stilman Robotics & Intelligent Machines @ GT Georgia Institute of Technology Atlanta, GA 30332-0760 February 19, 2008 1 Force Control Strategies Logic Branching