Mechanism and Robot Kinematics, Part I:

Algebraic Foundations

Charles WamplerGeneral Motors R&D Center

In collaboration with

Andrew SommeseUniversity of Notre Dame

2

Overview

n Why kinematics is (mostly) algebraic geometryn Rigid bodies with algebraic surfaces in contact

n A good approximation to the most common devices

n Mechanism & robot typesn Families and sub-familiesn Classification via type graphs

n Types of kinematic problemsn Analysis: how does this mechanism move?

n Motion paths, workspace limits, singularities

n Synthesis: what mechanism will move this way?n Exceptional (“overconstrained”) mechanisms

3

Kinematics: Then & Now

Model of Watt Engine, 1784(http://kmoddl.library.cornell.edu/tutorials/05/)

Stewart-Gough robot(Fanuc F200)

4

Griffis-Duffy Platform

This is an algebraic curve of

degree 40

5

Definition of a Mechanismn A collection of links connected by joints.

n Links are rigid bodiesn In reality, not quite, but a good approximationn “Compliant mechanisms” are another story

n Joints are mechanical contacts between surfaces of two linksn Pin joint (hinge), Ball-and-socket, etc.

6

Rigid-Body Motion

n A rigid body has two defining properties:n Preservation of distancen Preservation of chirality (handedness)

7

Rigid-Body Motion (cont.)

n Suppose a,b,c,d ∈ R3 are given, non-coplanarn What is the set a’,b’,c’,d’ ∈ R3 such that

homologous distances are preserved?

a

c

d

b

(b’-a’)T(b’-a’) = (b-a)T(b-a)

(c’-b’)T(c’-b’) = (c-b)T(c-b)

Etc.

à6 polynomials in 12 variables

àHas 2 components, each 6 dimensional.àOne component contains (a,b,c,d)

àOther component: mirror image

8

Rigid Body Motion (Spatial)

n SE(3) = R3×SO(3) (Lie Group R3×SO(3))n Preserves distance & chiralityn Let T∈SE(3) be given by (p,Q)

n p∈R3, Q∈SO(3)⊂ R3×3

n QTQ=I, det Q = 1.

n T: R3 → R3

n operates on point x∈R3 as T(x) = p + Q⋅x

Picks out the component that preserves chirality

n Factsn SE(3) is a 6-dimensional, algebraic subset of R3×4

n 3 translations, 3 rotations

n T: R3 → R3 is an algebraic map

O(3) preserves distance

9

Isomorphisms of SE(3)

n 4x4 Homogeneous matrix groupn Products and inverses are in the

same formn Transforms may be written as

n SO(3) isomorphic to P3

n Quaternionsn Unit quaternions double cover SO(3)

n Study’s soma (Study coordinates)n Study quadric in P7

10pQ

=

110)(

xpQxT

[ ]0

7

=+++∈

dtcsbraqtsrqdcba P

10

Subgroups of SE(3)

n Planar rigid-body motionn SE(2) = R2×SO(2) ⊂ SE(3)n dim SE(2) = 3

n Spherical rigid-body motionn SO(3) = 0×SO(3) ⊂ SE(3)n dim SO(3) = 3

n These are both of interest in kinematics too.

11

Definition: Link and Framed Link

n Let a link be a collection of featuresn Assume features are algebraic sets in R3.

n Points, curves, surfaces

n Call an element of SE(3) a frame.n Let framed link = link plus a frame.

link

framedlink

12

Transformed features

n Let A be a linkn Let A ={T,A}, T∈SE(3), a framed linkn Feature point x∈A transforms to x’=T(x)n Feature given by f(x)=0 transforms to

f(T-1(x’))=0

Link A

Framed link A

x

x

x’

13

Joints

n Joint = tangential contact between transformed features of two framed links.n Tangential contact =

n Share at least one point, called “contact” point(s)n Surface tangent plane contains the tangent space

of the opposing feature at contact point

14

Contact types

equalitycurve contactpoint contact

surface

inclusionpoint contact

equalitymeetcurve

inclusioninclusionequalitypoint

surfacecurvepointA ? B

BA

15

Linear contact types

equalityplane

inclusionequalitymeetline

inclusioninclusionequalitypoint

planelinepointA ? B

BA

16

Example: Planar 7-bar Structure

Problem:

Assemble these 7 pieces, as labeled.

17

Example 2 Preparation

n Ball bar çè Spherical surface

A

C

B B

A

18

Example: Stewart-Gough platforms

n Top link, Bn 6 feature points

n b1, b2, b3, b4, b5, b6

n Base link, An 6 feature spheres

n Centers a1, a2, a3, a4, a5, a6

n Radii r1, r2, r3, r4, r5, r6

n Joints: point in surfacen Point i of top link lies on

sphere i of base link, i=1,…,6

General S-G platform

Each leg is a spherical constraint

A

B

19

Example (cont): Forming equations

n For i=1,…,6n Sphere equation:

fi(x)=(x-ai)T(x-ai)-ri2=0

n Point inclusions:fi(T(bi))=0 (*)

Where T is the transform for framed link B

n For given links, the T that satisfy (*) are the possible locations of B w.r.t. A

20

Joints: Lower-order pairs

n What surface=surface joints allow relative motion?n Answer given by Reuleaux, 1875n Rigorous proof by Selig, 1989, IJRR

n Lie groups

n Why engineers care:n Distributed stress = less wear

n And the answer is…

21

Joints: Lower-order pairs

f=1, c=5

P R H

Prismatic Rotational Helical (Screw)

f=2, c=4

C

Cylindrical

f=3, c=3

E S

Plane Sphere

f = freedom

c = constraint in SE(3)

Not Algebraic

22

5 of 6 lower-order pairs are linear

P

Prismatic

Line-in-plane

R

Rotational

Point-in-line

C

Cylindrical

Line

E

Plane

S

Sphere

Point

Equality between

flags

23

Linear link types

n Let link type index = integer triplet(# feature points, # feature lines, # f. planes)

n For each link type index, (a,b,c), there is an associated link type universal spaceLabc = Aff(0,3)a × Aff(1,3)b × Aff(2,3)c

n A link type is an algebraic subset of a link type universal space, say

Y ⊂ Labcn Inclusions, parallelism, perpendicularity, etc.

24

Mechanism Definition Revisited

n Limit our attention to linear featuresn Mechanism family is given by

n List of N framed linksn Each link i has a type Yi ⊂ L(abc)i

n (a,b,c)i is link type indexn Each linear element is a feature

n Framed link i has a transform Ti∈SE(3), n List of joints, each consisting of

n A pair of features of two distinct linksn A contact operation (equality, inclusion, meet)

n Family is M ⊂ SE(3)N × Y , n where Y is a given algebraic set Y ⊂Y1 × … × YNn Equations for M derive from the contact operations

applied to transformed feature pairs.

25

Some Mechanism Families

n Native linear familiesn Those for which Y= L(abc)1

× … × L(abc)Nn Lower-order pair families

n All joints are one of P,R,C,E,Sn Planar mechanisms

n Frames are in SE(2)n Joints are either:

n P with lines parallel to reference planen R with axes orthogonal to reference plane

n Spherical mechanismsn Frames are in SO(3)n Joints are all R with axes through the origin

26

Spherical four-bar

Courtesy of J. M. McCarthy, UC IrvineGraphic by Hai-Jun Su

27

Mechanism Type Graphs

n For lower-order pair mechanismsn Joint type (P,R,C,E,S) implies feature type

n If there are no other conditions imposed(parallelism, orthogonality, etc.)

then a colored graph defines the familyn Links are nodesn Joints are colored (labeled) edgesn Simple example: house door

n Extra conditions can beannotated separately

Rhinge

door

house

28

Examples

connecting rod

engine block

crankshaft

rod1 rod2

piston1 piston2

C CR

R

R

R

R

base plate

top plate

P

U

S

lower leg

upper leg

29

Type Enumeration

n For planar & spherical generically 1 DOF lower-pair mechanisms, we can enumerate all possible mechanism types up to N=12n Graph isomorphism is at issue

n For spatial, there are too many to bother

Two-bar Four-bar Watt six-bar

Stephenson six-bar

16 distinct eight-bars

30

Grounded Mechanism

n A mechanism, as defined above, floats freely in 3-space

n A grounded mechanism is a mechanism with one link held stationary.n Say, TN=In Configuration space becomes SE(3)N-1

n Different grounded mechanisms derived from the same mechanism are called inversions of the mechanism.

Watt I Watt II

31

Input Jointsn Typically, some joints are actuated

n These are the input jointsn We may directly command the input angle (R joint) or input

translation (P joint)n The relative motion is parameterizable

Bcssc

ATT ij

−

=

100001000000

θθ

θθ

BATT ij

=

1000100

00100001

δconstant ),3(SE, ∈BA

R joint P joint

n This defines a map JRR =×××××→ 1111NSE(3) : LL SSJ

R joints P joints

32

Output Link(s) or Output Joint(s)

n There is usually an output link or output joint(s) that directly interacts with the environment to achieve the purpose of the mechanismn Examples: robot hand, automobile wheeln This gives an output function:

K: SE(3)N à Kn Output link, K: SE(3)N à SE(3)n Rotational output joint, K: SE(3)N àS1

33

Big Picture

1π 2π

SE(3)N Y

natural projections

M ⊂ SE(3)N × Y

J

J

K

Kinput output

mechanism typeconfiguration space

mechanism

34

Overview (revisited)

n Why kinematics is (mostly) algebraic geometryn Rigid bodies with algebraic surfaces in contact

n A good approximation to the most common devices

n Mechanism & robot typesn Families and sub-familiesn Classification via type graphs

n Types of kinematic problemsn Analysis: how does this mechanism move?

n Motion paths, workspace limits, singularities

n Synthesis: what mechanism will move this way?n Exceptional (“overconstrained”) mechanisms

35

Analysis

n How does a given mechanism move?n Family M ⊂ SE(3)N × Y

1π 2π

SE(3)N Y

n is a mechanism in the familyn is its motionn is called its “degrees of freedom”

YMy ⊂∈ )(20 π))(( 0

121 y−ππ

))(( dim 01

21 y−ππ

36

Analysis

n The motion may have several components.n Find the DOF of each componentn Real dimension = complex dimension

n The DOF’s may be different for different pointsn For each irred. component of find the generic DOF’s

))(( 01

21 y−ππ

)(20 My π∈YM ⊂)(2π

37

Example: 7-bar Structure

Problem:

Assemble these 7 pieces, as labeled.

38

Result for Generic Links

18 rigid structures

• 8 real, 10 complex for this set of links.

•All isolated – can be found with traditional homotopy

39

Special Links (Roberts Cognates)

Dimension 1:

6th degree four-bar motion

Dimension 0:

1 of 6 isolated (rigid) assemblies

40

Robot Workspace Analysis

n Find the workspace boundary (in the reals)n Real points of the singular set

of a complex component of

n The set being analyzed here is the position of the handn i.e., the image of a projection

map from SE(3)4 to R3

A. Malek, U. Iowa

ground up’r_arm lwr_arm hand

R R R

)))((( 01

21 yK −ππ

41

Forward and inverse kinematics

n Suppose

n Forward kinematicsn Find

n Inverse kinematicsn Find

n Find singularities of these maps

1π 2π

SE(3)N Y

J

J

K

Kinput output

M

)( )),((

,dim)(dim)(dim

2001

21 MyyC

CCKCJ

πππ ∈=

==−

))(( 1 xJK −

))(( 1 xKJ −

42

Synthesis

n Find a mechanism to approximate a given motionn Finite position synthesis = interpolate a

finite set of given outputs

n Input-output synthesis

{ }{ }given are ,, where

,,)))((( )(Z

1

11

212

K∈

⊃∈= −

m

m

oo

ooyKMy

K

Kπππ

[ ] [ ] [ ]{ }{ }[ ] [ ] given are ,,,, where

,,,,)))((())),((( )(Z

11

111

211

212

KJ ×∈

⊃∈= −−

mm

mm

onon

ononyKyJMy

K

Kπππππ

43

Another view: Fiber Product

n Let be the parameter spacen The fiber product

is the space of multiple instances of the same mechanism in j different configurations

n Let π1i be the projection onto the ithconfiguration

n Output synthesis sets

)(2 MP π=

44 344 21 L timesj

PPjP MMM ××=Π

mioMK ijPi ,...,1 ,))(( 1 ==Ππ

44

Synthesis Example

n 9-Point Path Generation for Four-barsn Problem statement

n Alt, 1923

n Bootstrap partial solutionn Roth, 1962

n Complete solutionn Wampler, Morgan & Sommese,

1992n m-hom. Continuation

n 143,360 paths (2-way symmetry)n 4326 finite, isolated solutions

n 1442 Robert cognate triples

45

Spherical four-bar

Courtesy of J. M. McCarthy, UC IrvineGraphic by Hai-Jun Su

46

Exceptional Mechanisms

n A.K.A. “overconstrained mechanisms”n For each component of M, there is a

generic fiber dimension, M’s DOF

n M may have algebraic subsets where the fiber dimension is larger than d*n These are exceptional mechanismsn Finding such mechanisms is challenging

Mxxd x ∈= − genericfor )))((dim 21

2* ππ

47

Exceptional 7-bar (Roberts Cognates)

Dimension 1:

6th degree four-bar motion

48

Bennett four-bar

Courtesy of J. Michael McCarthy, UC Irvine

49

Exceptional Stewart-Gough Platform

50

Summary

n Rigid-body mechanisms with algebraic joint features are algebraicn Linear features include all lower-order pairs

n P,R,C,E,S

n Linear mechanism spaces are a rich source of interesting algebraic sets

n Kinematic problems fall into three main classesn Analysisn Synthesisn Discovery of Exceptional Mechanisms

n Many challenges await!