The Development of a Fast Pick-and-Place Robot with an Innovative

IntroductionThe C-Drive Mathematical Model

The C-Drive PrototypeThe PMC Mathematical Model

The PMC PrototypeConclusions

The Development of a Fast Pick-and-Place Robotwith an Innovative Cylindrical Drive

Jorge Angeles

Department of Mechanical Engineering and Centre for Intelligent Machines,McGill University, Montreal, Quebec, Canada

Jorge Angeles The Development of a Fast Pick-and-Place Robot 1




Outline

1 Introduction

2 The C-Drive Mathematical Model

3 The C-Drive Prototype

4 The PMC Mathematical Model

5 The PMC Prototype

6 Conclusions





Schonflies Motion GeneratorsConventional Parallel SMGConventional Hybrid Parallel-Serial SMGIsostatic Schonflies Motion GeneratorDesign Principles of the C-drive

Schonflies Motion Generators

Schonflies Motion (SM)

I Three translations and onerotation about one axis of fixeddirection.

I Industrial pick-and-place robotsare capable of SM.

I The speed record is five testcycles per second.

25 mm 300 mm

0 deg 90 deg 180 deg Adept Quattro 650H






Conventional Parallel SMGAdept Quattro 650H (Delta-Like Robot)

MP

I 4 limbsI rotation amplification required I too many extra joints to accommodate hyperstaticity






Conventional Hybrid Parallel-Serial SMGABB IRB 360 FlexPicker

I parallel array of 3 limbsI telescopic Cardan shaft in series with end-effector I too many extra joints to accommodate hyperstaticity






Isostatic Schonflies Motion GeneratorCRRHHRRC Parallel PMC

C

C

R

R

R

R

H

H

I 2 limbsI

::::::isostatic

I::no

::::::rotation

::::::::::amplification

::::::required

I proposed by C.C. Lee and P.C. Lee (2010)






Design Principles of the C-drivemotor L

Lscrew L

(left-hand)

u

O

collar

nut R

screw R(right-hand)

support unitcoupling

rotor

motor R

R

nut L

1. The number of joints must be as low as possible;I C joint can be synthesized by means of a serial array of two simple LKP

2. A R joint is preferred over a P joint, and a P joint over a H joint;

3. Use as many identical joints as possible, as complexity grows with theirdiversity






Design Principles of the C-drive (Contd)motor L

Lscrew L

(left-hand)

u

O

collar

nut R

screw R(right-hand)


rotor

motor R

R

nut L

4. Reduce the diversity of the actuators; and

5. For purposes of reducing and uniformly distributing the load, fix the motors tothe base





KinematicsDynamicsState-Space Representations

Kinematics - Modelmotor L

Lscrew L

(left-hand)

u

O

collar

nut R

screw R(right-hand)


rotor

motor R

R

nut L

I u, : the translational and angular displacements of the C-drive collarL, R: the angular displacements of the motorspL, pR : the pitches of the left-hand screw and its right-hand counterpart

pL2

(L) = u,pR2

(R) = u (1)

I pL 6= pR in agreement with the condition from Lie groupsI we introduce symmetry as the sixth design principle

pL = p, pR =p (2)






Kinematics Jacobian

I Dimensionally homogeneousChange of variable v (p/2) the actuator Jacobian J is dimensionallyhomogeneous:

w = J (3)

w [

uv

], J p

4

[1 11 1

],

[LR

](4)

I Differential mechanismpure rotation: the actuators turn by the same amount in the same directionpure translation: the actuators turn by the same amount, but in opposite directions

I IsotropicJT J is proportional to the 22 identity matrix 1

JT J =p2

821 (5)

I Kinematic relation between rates

w = J (6)






Dynamics

I Mathematical model describing the dynamics of the C-drive

M +D + c = (7)I Generalized inertia matrix M (22):

M Mc +Mh (8)

Mc = JT IdriveJ, Idrive =[

m 00 (2/p)2Ic

], Mh = Ih1 (9)

Mc: generalized inertia matrix of the collarMh: generalized inertia matrix of the screws of the ballscrewsIc, Ih: collar and screws moment of inertia about C-drive axis






Dynamics: Friction

I Mathematical model describing the dynamics of the C-drive

M +D + c = (10)I Viscous and Coulomb friction forces:

D

+ 12 12 1

2 +

12

, c = [ sgn(L)+sgn(L R) sgn(R)+sgn(R L)]

(11)

, : viscous friction coefficients , : Coulomb friction coefficients






State-Space Representations

Standard Linear State-Space Form

x = Ax+Bu (12)

y = Cx (13)

with the output vector y defined as

y (14)

: vector of joint (motor) displacementsx: state vectoru: input vector, i.e., applied motor torques

In the following models, the state vectors are dimensionally homogeneous






State-Space Representations: Collar Coordinates as StatesI Second-order generalized-coordinate model:

M +D + c = (15)

I Substituting with its expression in terms of collar coordinates w:

MJ1w+DJ1w = (16)

I Upon premultiplication by Jacobian matrix J:

JMJ1 E

w+JDJ1 G

w = J (17a)

E

[22 0

0 21

], G

[2 0

0 1

], i = 1, 2 (17b)

I 2i and i are the eigenvalues of M and D, respectively






Dynamics in Monic FormI The dynamics is obtained in monic form, with Coulomb friction forces

neglected + = u (18)

via the substitutions

E1

G

E1,

Ew, u

E1

J (19)

where

[2 0

0 1

],

E1

J p4

[1/2 1/21/1 1/1

](20)

withi i/2i , i = 1, 2 (21)






State-Space Representations: 2D and 4Dx = Ax+Bu, y = Cx (22)

2D Representation

x (23a)

y (23b)

A, B 1 (23c)

C J1

E1

(23d)

4D Representation

x[T TT

]T(24a)

y (24b)

A

[O

O

], B

[O

1

](24c)

C[J1

E1

O]

(24d)

I 1 and O are the 22 identity and zero matrices, respectivelyI The 2D system is simpler; the response of the system is readily derivedI The 4D system has an observable state vector, as is measured





MechanismControl SystemControl ArchitecturePure TranslationPure RotationHelical Motion

Mechanismmotor L

motor R

screw Lcollar (left-hand)

screw R

support unit

(right-hand)






Control System






Control Architecture






Pure Translation

0 500 1000 1500 200050

0

50

100

150

200

Time (ms)

Pos

ition

(m

m)

Achieved uAchieved vDesired uDesired v

0 500 1000 1500 20000.1

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

Time (ms)

Err

or (

mm

)

Error in uError in v






Pure Rotation

0 500 1000 1500 200050

0

50

100

150

200

Time (ms)

Pos

ition

(m

m)


0 500 1000 1500 20000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

Time (ms)

Err

or (

mm

)







Helical Motion

0 500 1000 1500 2000100

50

0

50

100

150

200

250

Time (ms)

Pos

ition

(m

m)


0 500 1000 1500 20000.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Time (ms)

Err

or (

mm

)






PMC Kinematic ModelPMC Inverse Displacement AnalysisPMC Forward Displacement AnalysisPMC KinematicsSingularity Analysis

Kinematic Model of the PMC

O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

a

i

jk

2

v1

v2A2

B1

B2

P2

P1C

R

RH

HR

R

C

I The PepperMill-Carrier (PMC):CRRHHRRC parallel mechanism

I di and i (for i = 1, 2): the translational androtational displacement variables of the ithC joint

I The pose x of the PepperMill is given byc = [x, y, z]T , the position vector of point C,and angle :

x[

c

](25)







O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

a

i

jk

2

v1

v2A2

B1

B2

P2

P1C

R

RH

HR

R

C

I Points Oi: the intersection of the ith-driveaxis with the common normal to the two,for i = 1, 2

I The fixed origin O of the base frame is set atthe midpoint of segment O1O2

I i, j and k: unit vectors parallel to the x, yand z axes







O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

a

i

jk

2

v1

v2A2

B1

B2

P2

P1C

R

RH

HR

R

C

For i = 1, 2,

I i: the angle of rotation of the proximalpassive R joint

I ui and vi: unit vectors parallel to the axes ofthe proximal and the distal links

I ri and li: the lengths of the proximal and thedistal links

I The end of the distal link of the ith limb iscoupled with the passive R pair of thePepperMill at point Pi






PMC Inverse Displacement Analysis

O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

iu

iu ixcos= +

i

i

i iysini

x

iy

a

i

jk

2

v1

v2A2

B1

B2

P2

P1

I Displacement analysis performed vialoop-closure equations

I The translational displacements of eachC-drive are obtained directly from thehorizontal displacements of thePepperMill:

d1 = x, d2 = y (26)






PMC Inverse Displacement Analysis

O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

iu

iu ixcos= +

i

i

i iysini

x

iy

a

i

jk

2

v1

v2A2

B1

B2

P2

P1p1

p2

I The angular displacements i,i = 1, 2, of each C-drive arecomputed from

(qTi xi) cos i +(qTi yi)sini (27)

=qTi qi + r2i l2i

2ri

where

qi pidiai (28)






Forward Displacement AnalysisCRRHHRRC Parallel PMC

O xy

z

CO1

1

2

1

2

O2

A1r1

d2

b0

b0

d1

r2l2

l1u1

u2

a1

iu

iu ixcos= +

i

i

i iysini

x

iy

a

i

jk

2

v1

v2A2

B1

B2

P2

P1

I The horizontal displacements of thePepperMill are simply

x = d1, y = d2 (29)

I The vertical and angular displacementsof the PepperMill are computed from[

1 p11 p2

] [z

]= (30)[

r1 sin1 + l1 sin(1 +1)b0r2 cos2 + l2 cos(2 +2)+b0

]

I The leftmost matrix is invertible whenp1 6= p2






PMC Kinematics: Jacobian Analysis

I The PepperMill pose x and proximallink displacements are

x [x y z ]T (31)

[d1 d2 1 2]T (32)

I Vectors li are defined as shown in thefigure, for i = 1, 2






PMC Kinematics: Jacobian AnalysisI Kinematics relation

Ax = B (33)where Jacobian matrices A and B are

A =

1 0 0 00 1 0 00 vy1 v

z1 p1v

z1

vx2 0 vz2 p2v

z2

(34)

B =

1 0 0 00 1 0 00 0 vT1 l1 00 0 0 vT2 l2

(35)For i = 1, 2 and j = x, y, z, vji is thejth component of vi






The Singularity of the First Kind (inverse-kinematics singularity)

B =

1 0 0 00 1 0 00 0 vT1 l1 00 0 0 vT2 l2

I The singularity of the first kind occurs when the Jacobianmatrix B is singular, i.e., when

det(B) (vT1 l1)(vT2 l2) = 0 (36)

I Geometrically, this singularity occurs when v1 l1 or v2 l2,i.e., when the unit vector ui of the proximal link and the unitvector vi of the distal link are parallel in at least one limb






The Singularity of the Second Kind (direct-kinematics singularity)

ui

vk

i

vi =

vxivyivzi

I The singularity of the second kind occurs when the Jacobianmatrix A is singular:

det(A) = det

1 0 0 00 1 0 00 vy1 v

z1 p1v

z1

vx2 0 vz2 p2v

z2

= det([

1 00 1

])det([

vz1 p1vz1

vz2 p2vz2

])= vz1v

z2(p2p1) = 0 (37)

I The pitches of the two H joints of the PepperMill are distinct:

p1 6= p2 (38)

I Geometrically, these conditions occur when the unit vector ofthe distal link of at least one limb is normal to the z-axis






The Singularity of the Third Kind (complex-kinematics singularity)

I The singularity of the third kind occurs when the the twoJacobian matrices A and B become singular

det(A) = vz1vz2(p2p1) = 0

det(B) = (vT1 l1)(vT2 l2) = 0

I This singularity depends on the robot architecture. Given thesymmetries with which the PepperMill was designed, this robotadmits the singularity of the third kind, as illustrated in thefigure





Prototype: PepperMill MechanismPrototype (Contd)Magnetic GripperMechanical Interference Detection

PepperMill Motion






PMC Prototype






Magnetic Gripper Functionality

Wireless Magnetic Gripper Assembly

I 10W 24VDC electromagnet actuatedwirelessly

I Lithium-Ion battery recharged incircuit

I 1 mW Xbee radio communicates via2.4 GHz 802.15.4 protocol

I Digital I/O line passing betweencontrol module and embeddedmodule






Magnetic Gripper Features

Wireless Magnetic Gripper AssemblyI External 1/4 wave monopole antenna

I Uses gripper enclosure asground plane

I Robust 802.15.4 protocol supportspoint-to-point, point-to-multipointand peer-to-peer network topologies

I Scalable: multiple modulesmay be used in close proximity

I Can be configured to actuatedifferent types of gripper (e.g.clamp)






Mechanical Interference Detection

Figure: Single C-drive

I Robot dimensions chosen such that moving links cannot collideI Proximal linksthe C-drivescan collide with the base in four waysI C-drives present unusual challenges in detecting extreme positions






Mechanical Interference Detection Switches: Contactless

Translation Limit Switch

I Magnetic proximity switch detectsedge of C-drive cap

I Cap edge angled 30 to increasesensor sensitivity

Rotation Limit Switch

I Laser photo-electric switchI Laser beam interrupted by proximal

link when close to extreme position





ConclusionsAcknowledgements

Conclusions

I The development of an innovative two-DOF cylindrical drive wasdescribed

I The kinematics and dynamics of the C-drive were derived, as well asconvenient state-space representations thereof

I The C-drive mechanism and control system were described; test resultsyielded low error

I PMC prototype assembly shown, including safety featuresI The singularity analysis of the PMC was reportedI The embedded electromagnetic gripper was described





ConclusionsAcknowledgements

Ackowledgements

The work described here is the result of team work. Team members: DamienTrezieres, Masters Visitor from Lyon, France; Takashi Harada, Kinki U.,Japan; Thomas Friedlaender, M.Eng. student at McGill U.

The support from various sources is to be acknowledged: NSERC (CanadasNatural Sciences and Engineering Research Council), James McGillProfessorship to J. Angeles; Japans MEXT-supported Program for theStrategic Research Foundation to Private Universities; and THK Co., Ltd fortheir support.


IntroductionSchnflies Motion GeneratorsConventional Parallel SMGConventional Hybrid Parallel-Serial SMGIsostatic Schnflies Motion GeneratorDesign Principles of the C-drive

The C-Drive Mathematical ModelKinematicsDynamicsState-Space Representations

The C-Drive PrototypeMechanismControl SystemControl ArchitecturePure TranslationPure RotationHelical Motion

The PMC Mathematical ModelPMC Kinematic ModelPMC Inverse Displacement AnalysisPMC Forward Displacement AnalysisPMC KinematicsSingularity Analysis

The PMC PrototypePrototype: PepperMill MechanismPrototype (Cont'd)Magnetic GripperMechanical Interference Detection

ConclusionsConclusionsAcknowledgements

fd@Trans: mbtn@0: fd@Rot: mbtn@1: fd@Helix: mbtn@2: fd@pepper: mbtn@3:

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The Development of a Fast Pick-and-Place Robot with an Innovative