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The Robust Design of RoboticMechanical Systems

Jorge Angeles, ing., Ph.D., [email protected]

Department of Mechanical Engineering &Centre for Intelligent Machines

McGill University

Jorge Angeles – p.1

Overview• Introduction: main issues & proposed solutions• Design Paradigms

– Robust– Axiomatic– The German School

• A Model-Based Theoretical Framework forRobust Design

• The Design of Robotic Mechanical Systems– Kinetostatic– Elastostatic– Elastodynamic

• ConclusionsJorge Angeles – p.2

Introduction• Broadly applicable design principles and design

methodologies are inexistent in connection withrobotic mechanical systems (RMS)

• The common feature of the tasks to be performedby RMS is uncertainty: A multi-dof machine isdesigned for a class of tasks, rather than for aspecific task

• Other sources of uncertainty: manufacturing;assembly; measurement; and computation errors

• Proposed solutions (mainly three schools):- Taguchi’s robust engineering- Suh’s axiomatic design- The German School


Introduction- Rodenacker (1976): Systematic Design

(Methodisches Konstruieren)- Taguchi’s work (1987): The philosophy of robust

design- Pahl and Beitz (1991): Systematik- Gadallah and ElMaraghy (1999): Robust

mechanical design of a clutch- Zhu and Ting (2000): Robust design of linkages- Portman, Sander, and Zahavi (2000): Kinetostatic

and elastodynamic design of a six-dof parallelmicro-manipulator

- Suh (2001): Axiomatic designJorge Angeles – p.4

Robust Design• A design is called robust when its performance is

least sensitive to variations in environmentconditions

• Since the environment cannot be controlled, theproduct should be designed to be least sensitive toenvironmental changes

• Products robustly designed can perform well overa broad range of variations of operation andenvironment conditions, i.e., under extremeconditions


Robust DesignTaguchi’s philosophy relies on two key concepts:

• The Signal-to-Soise (S/N) ratio, that measures thesensitivity of a design to changes in environmentconditions:the larger the S/N ratio, the more robust thedesign

• The Loss Function, that measures the loss ofsociety by virtue of a flawed design


Robust Design Examples

The catastrophe of the Concorde on July 25th, 2000


Robust Design ExamplesThe International Communications Alphabet (ICA)

A alpha J Juliette S SierraB Bravo K Kilo T TangoC Charlie L Lima U UniformD Delta M Mike V VictorE Echo N November W WhiskeyF Foxtrot O Oscar X XrayG Golf P Papa Y YankeeH Hotel R Romeo Z ZuluI India Q Quebec

The ICA was designed to reduce the potential of mis-understanding spoken letters in aviation and militaryoral communication Jorge Angeles – p.8

Axiomatic DesignSuh’s axiomatic design methodology relies on twoaxioms:

• The Independence Axiom:The best design is one in which all designfunctional requirements are independentlysatisfied

• The Minimum-Information Axiom:The best design is the one containing theminimum amount of information


A Framework for Robust DesignClassification of Design Quantities

• Design Variables (DV): to be decided on by the designerwith the purpose of meeting performance specifications

x ≡ [ x1 x2 · · · xn ]T

• Design Environment Parameters (DEP): randomparameters over which the designer has no control

p ≡ [ p1 p2 · · · pν ]T

• Performance Functions (PF): functions that represent theperformance of the design in terms of DV and DEP

f ≡ [ f1 f2 · · · fm ]T , f = f(x;p)

Main assumption: All variables, x, p and f are normalized bydividing them by nonzero reference values

⇒ F is dimensionless


A Framework for Robust DesignPerformance design matrix: a m × ν matrix, denoted by F,mapping the space of relative variations of p into that of relativevariations of f : f = f(x;p)

=⇒ f(x;p0 + ∆p) = f(x;p0) +

(∂f

∂p

)∣∣∣∣p=p0

∆p + · · · + HOT

1st-order approximation: ∆f = F∆p, F(x;p0) ≡(

∂f

∂p

)∣∣∣∣p=p0

p0 : Nominal design-environment parameter vector

∆p ≡ p − p0

∆f ≡ f(x; p0 + ∆p) − f(x;po)

F : Jacobian matrix of f with respect to p (dimensionless)Jorge Angeles – p.11

A Framework for Robust Design• Main assumption:

All quantities, x, p and f are normalized by dividing themby nonzero nominal values⇒ F is dimensionless


A Framework for Robust DesignLet µp and P be the expected and the covariancematrix of ∆p, respectively. The expected value of ∆fis

µf ≡ E[∆f ] = Fµp

The corresponding covariance matrix Φ of ∆f is

Φ ≡ V [∆f ] = E[(∆f − µf)(∆f − µf)T ] = FPFT

Robust design is achieved by minimizing any norm‖Φ‖ of Φ


A Framework for Robust Designz(x) ≡ ‖FPFT‖ −→ min

x

subject tof(x;p0) = f0

Hence, the general robust design problem can beformulated as one of the minimization of a norm ofthe covariance matrix of the variations in the PF uponvariations in the DEP.


A Framework for Robust Design• If 2-norm is used, then ‖Φ‖ = maxφim

1; φi: ith

eigenvalue of Φ

• Numerical difficulties with 2-norm: Non-analyticfunction of Φ

• Alternatives: Frobenius norm; isotropic design


A Methodology for Robust DesignPossible methodology to obtain a robust design:(1) render the performance matrix isotropic—all its

singular values identical—and then(2) minimize its resulting common singular value.Advantages:

• the performance of an isotropic design ishomogeneous over the DEP space, i.e., any DEPdisturbance of a given norm will have identicalimpact on the PF regardless of the direction of theDEP disturbance.

• the design performance variation will be both aminimum and independent of a particular ∆p atthe same time


A Methodology for Robust Design• An isotropic matrix has identical nonzero

singular values.• If σi = λ, for i = 1, . . . , r,

r = rank(F) = minm, ν, and λ > 0, then

S ≡ FTF = λ21

where λ is the multiple singular value of F, ofalgebraic multiplicity r.

• Matrix S is termed here the sensitivity matrix,which is apparently square, symmetric, andpositive-definite.


A Methodology for Robust DesignThe Design Sensitivity EllipsoidLet us assume that FTF is in diagonal form:

FTF = diag(λ2

1, · · · , λ2

ν), ∆p = [y1 · · · yν]T

The eigenvalues of S: λ2

1≤ λ2

2≤ · · · ≤ λ2

ν

S2 =λ2

1y2

1+ · · · + λ2

νy2

ν

y2

1+ · · · + y2

ν

Let ‖y‖ = 1 and S = So:y2

1

S2o/λ

2

1

+y2

2

S2o/λ

2

2

+ · · · + y2

ν

S2o/λ

2ν

= 1

⇒ ν-dimensional ellipsoid centered at the origin ofthe ∆p-space Jorge Angeles – p.18

Geometric Interpretation

A three-dimensional sensitivity ellipsoid:λ1 < λ2 < λ3


The Design Sensitivity Ellipsoid• The largest and the smallest values of ‖∆p‖ occur in the

principal directions of S, for a given sensitivity So.

• For a given sensitivity So, a small variation in the third DEP(y3) exerts the same influence in the performance, ∆f , as alarge variation in the first DEP (y1)

the largest eigenvalue of S is a measure of the designsensitivity to changes in the DEP

• The least sensitive design is, then, one with an ellipsoidwhose largest semiaxis is a minimum⇒ Robust design is a minimax problem


The Isotropic Design• A design is isotropic if its design performance

matrix is isotropic, i.e., if

λ = λ1 = λ2 = · · · = λr

• Minimax problem reduces to minimizingcommon eigenvalue λ of sensitivity matrix

• Eigenvalues λir1

of S are the squares of thesingular values σir

1of F

κ2(F) =σmax

σmin

=

√

λmax

λmin

where σmax and σmin are the maximum and theminimum singular values of the given matrix


Bandwidth of the Sensitivity MatrixThe bandwidth of a matrix A, measured in decades,

b = log10

(σmax

σmin

)

=1

2log10

(λmax

λmin

)

The bandwidth of a matrix A, measured in decades,

b = log10

(σmax

σmin

)

Since the eigenvalues of S are the squares of thesingular values of F,

b =1

2log10

(λmax

λmin

)


Geometric Interpretation ofthe Condition Number

• Polar decomposition of any square matrix A:

A = RU = VR

R orthogonal; U & V symmetricpositive-(semi)definite

• κ(A) indicates the distortion occurring to a unitsphere under the mapping A

• The larger the distortion, the larger the conditionnumber


Mechanical Interpretation of the PDTThe Polar-Decomposition Theorem (PDT) incontinuum mechanics:A small ball of a deformable material distorts into arotated small ellipsoid


Visualization of the PDT


Kinetostatic Design• Kinetostatics: Force-and-motion relations among the rigid

bodies of a mechanical system under static, conservativeconditions

• Kinetostatic relations for serial manipulators:t = Jq τ = JTw

• Jacobian maps joint rates into end-effector (EE) twist

• Transpose Jacobian maps wrench acting on EE into jointtorques

• Kinetostatic relations for serial RMSs:q = J−1t w = J−T τ

• Parallel RMSs entail two Jacobians


Kinetostatic Model (Cont’d)• The design-variable vector x corresponds to the

manipulator architecture, i.e., to the constantgeometric parameters of the manipulator,appearing in matrix J.

• The DEP vector apparently corresponds to thewrench w,

• the performance vector corresponds to the vectorof actuator forces

⇒ ∂τ

∂w= J−T ≡ F


A Human Arm Postured Isotropically


Elastostatic DesignFor the elastostatic model we base our analysis on theknown relation

Ku = φ

in which u and φ are the vectors of generalizeddeflection and generalized force, while K is thestiffness matrix.


Elastostatic DesignIn designing manipulators for high precision, we aimat a structure with minimum elastic deformation,which indicates that u is the performance vector x,while φ is the vector p of DEP. We can thus express

u = Cφ

C being termed the compliance matrix. Whence it isapparent that C plays the role of the performancematrix F.

C =∂u

∂φ

Elastostatic performance matrix: C


Elastostatic Model (Cont’d)• The potential energy V of RMSs is given by

V = 1

2qTKqq

Kq: the stiffness matrix in joint-coordinate space

• If all legs have identical stiffness k, then Kq is isotropic:Kq = k1 ⇒ V = 1

2k‖q‖2

• Hence, RMSs can be readily designed with an isotropicstiffness matrix k1 that is proportional to the ID matrix


Elastodynamic ModelThe linearized elastodynamic model of a robot at agiven posture:

Mqq + Kq q︸︷︷︸√M−1

q y

= 0

⇒ y + Ω2y = 0

Ω2 =√

M−1q Kq

√

M−1q

Mq: mass matrix in joint-coordinate spaceKq: stiffness matrix in joint-coordinate spaceq: vector of elastic joint displacementsΩ: frequency matrix


Elastodynamic Model (Cont’d)The total energy E of the system:

E =1

2qTMqq +

1

2qTKqq

E = 1

2zTWz: quadratic form in state-variables z

W =

[Ω2 O

O Ω2

]

z =

[y

Ω−1y

]

Elastic energy is bounded as:

1

2ω2

min‖z‖2 ≤ 1

2zTWz ≤ 1

2ω2

max‖z‖2


Elastodynamic Criterionz represents the deviation of the state of a manipulatordue to variations in the DEP p

Emax → min

p

Emax = Emin

ωmax = ωmin

⇒ κ(Ω) = 1

Elastodynamic performance matrix: Ω


Kinetostatic → Elastodynamic

J =

(e1 × r1)T eT

1

... ...(en × rn)

T eTn

Theorem: A kinetostatically isotropic parallelmanipulator, with identical legs, is elastodynamicallyisotropic if and only if the inertia tensor of its movingplatform is isotropic.


Three-DOF Planar Manipulator• one rotation + two translations

J =

−eT1Er1) eT

1

−eT2Er2) eT

2

−eT3Er3) eT

3


3RPR Manipulator-ElastodynamicLet ρ be the radius of gyration of the platform. Forelastodymanic isotropy, ρ =

√2r

Elastodynamic modelJorge Angeles – p.37

The Multi-Modular Manipulator (M3)• 11- dof manipulator for long-reach tasks, e.g., for

aircraft, repair, maintenance and servicing• Tasks: cleaning, brushing, stripping, and painting


Long-Reach Manipulation Tasks

Skywash, a robot for aircraft-cleaning

A Russian Antonov cargo-jet under de-icingat Dorval Airport Jorge Angeles – p.39

The Three Modules of M3


Gross Manipulator + Cuatro Arm

A Hybrid (Serial-Parallel) RMS


Kinematic Chain ofGross Manipulator


Jacobian of Gross Manipulator

J =

[1 0 1 0

e1 × p1 e2 × p2 e3 × p3 e4 × p4

]

The twist of the EE of the gross manipulator

t =

[φ

p

]


Isotropy Locus


Isotropy Surface


The McGill UniversitySchonflies-Motion Generator


Conclusions• Task-uncertainty poses a major challenge to the

designer of modern products, notably multi-dofRMSs

• As a means to cope with uncertainty, three maindesigns schools have been scrutinized: robustengineering; axiomatic design; and the GermanSchool

• All schools offer valuable elements to cope withmodern design demands, but robust design hasbeen adopted here

• Within robust design, a means to cope withuncertainty has been found to be isotropy

• An isotropic design exhibits equal sensitivities inall directions of the environment space

• Isotropic design has been implemented in variousprototypes, of which a sample was outlined


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