Brief presentation of the thesis

Kinematostatic Modelling
of Compliant Parallel Mechanisms

Cyril Quennouelle

PhD Thesis - Mechanical Engineering

Wednesday, April 29th 2009

Outlines

1 Context and objectives of the PhD

2 Kinematostatic Modelling

3 Stiﬀness Matrix

4 Quasi-static Modelling

5 Conclusion

C.Quennouelle ( ) Kinematostatic Modelling 29/04/09 2 / 20

Context and objectives 1 - Parallel Mechanisms

Analysis and synthesis of robotic manipulator

How to keep on improving their performances ?
Mechanical clearance and wearing
Complexity and cost of assembly


Context and objectives 1 - Parallel Mechanisms

Analysis and synthesis of robotic manipulator

How to keep on improving their performances ?
Mechanical clearance and wearing
Complexity and cost of assembly
Use of compliant joints


Context and objectives 2 - Compliant Joints

Kinematics properties of compliant joints

Complex motion
More than one DOF
Kinematic / static coupling


Context and objectives 3 - Objectives of the PhD

Challenges of the modelling of CPMs

l l fy
m0 m0
γl l0 l1 fx
l2 b
(1 − γ)l l3
θp a θp

Determine additional mobilities
number of DOF of a joint
kinematically dependent
coordinates
Consider the effet of external loads
onto the mechanism configuration
static modelling
kinematic modelling
stiffness of the joints

Kinematostatic Modelling 1 - Deﬁnition of the challenge


M degrees of mobility
A(θ)
Pose of the end-eﬀector F DOF
x M≥F
A actuators
M≥A
How to determine the M generalized coordinates ψ ?


Kinematostatic Modelling 2 - Kinematic Modelling

Recalls

Kinematic modelling
• C dependent coordinates: λ = C(ψ)

Kab (ψ, λ)

Kac (ψ, λ)



Recalls

Kinematic modelling
• End-eﬀector’s pose: x = P(ψ)
x



Recalls

Kinematic modelling
x

Instaneous kinematic modelling
• λ’s variation: ˙ ˙
λ = Gψ
• End-eﬀector’s variation: ˙ ˙
x = Jψ



Recalls

Kinematic modelling
x

Instaneous kinematic modelling
• λ’s variation: ˙ ˙
λ = Gψ
• End-eﬀector’s variation: ˙ ˙
x = Jψ

The inverse models are not determined (M ≥ F)


Kinematostatic Modelling 3 - Static Modelling

Static Constraints

Minimization of the potential energy
• Elastic
ψ ψ
ξk = ξψ + ξλ = τ T dψ +
ψ τ T Gdψ
λ
ψ0 ψ λ0
• Associate with f
ψ
ξf = − f T Jdψ dξf = −f T Jdψ
ψ x0
• Gravitational
n ψ
ξw = − wiT Pi dψ with wi = mi g
i=1 ψ0



Static Constraints

• Elastic
ψ ψ
ψ τ T Gdψ
λ
ψ0 ψ λ0
ψ
ψ x0
• Gravitational
n ψ
i=1 ψ0

Static Equilibrium
n
dξ
= τ ψ + GT τ ψ − J T f − PT wi = 0M
i
dψ
i=1



Static Constraints

• Elastic
ψ ψ
ψ τ T Gdψ
λ
ψ0 ψ λ0
ψ
ψ x0
• Gravitational
n ψ
i=1 ψ0

Static Equilibrium
τψ − JT f = 0M


Kinematostatic Modelling 4 - Kinematostatic Modelling


Static Modelling
M independent equations: determination of the conﬁguration
dξ
= S(ψ) = 0M
dψ
Presence of variable external parameters (f et φ)
S(ψ, f, φ) = 0M ⇒ ψ = F(f, φ)


Model Constraints

K(λ, ψ) = 0C 
x = P (F(φ, f))
B(θ) = x
S(ψ, φ, f) = 0M

x = M(φ, f)


Stiffness Matrix 1 - Definition

Stiffness matrix

Definition
Hessian matrix of the potential energy
Jacobian matrix of the static equilibrium


Stiffness Matrix 1 - Definition

Stiffness matrix
n
S = τ ψ + GT τ ψ − J T f − PT wi = 0M
i
i=1

Definition
Hessian matrix of the potential energy
Jacobian matrix of the static equilibrium

Generalized stiffness matrix
KM = Kψ + KI + KE + KW

• Kψ : Stiffness of the generalized coordinates
• KI : Internal stiffness
• KE : Stiffness due to the external loads
• KW : Stiffness due to the weight


Stiﬀness Matrix 2 - Properties



Generalization of the
preexisting matrices KM = Kψ
if KE = KI = 0, ∃J−1
Symmetry KC = JT Kψ J (Salisbury, 1980 )
Stability





preexisting matrices KM = Kψ + KI
if KI = 0, ∃J−1
KC = JT KM J (Griﬃs and Duﬀy,1993 )
Symmetry
(Chen and Kao, 2000 )
Stability





preexisting matrices KM = Kψ + GT Kλ G
if KIG = KE = 0 (Zhang and Gosselin,2002 )
Symmetry
Stability





Generalization of the If coordinate basis
preexisting matrices (Schwarz’ Theorem:
Symmetry ∂2y ∂2y
∂u∂v = ∂v ∂u )
Stability





ρ
preexisting matrices φ

Symmetry
Stability depending
quot; #
1 0
KC = 2kρ − 2ρ0
on the semi-positivity 0


Quasi-static Modelling 1 - Definition of the QSM

Quasi-static Model

Differentiation of the kinematostatic model
∂M(φ, f) ˙ ∂M(φ, f) ˙
˙
xc = φ+ f
∂φ ∂f
Linear relationship between the variation of the external parameters and
that of the configuration of the mechanism

Simple formulation

˙ ˙ ˙
xc = JC φ + CC f


Quasi-static Modelling 1 - Deﬁnition of the QSM

Quasi-static Model

Cartesian Compliance Matrix
∂xc
CC = = JK−1 JT
M
∂f

Quasi-static Jacobian Matrix
∂xc
JC = = JK−1 Kφ = JT
M
∂φ


Quasi-static Modelling 2 - Transmission Matrix

Transmission Matrix

Deﬁnition
Relates the variation of the generalized coordinates with the variation of
the commanded value of the actuators
∂ψ
T= (dimension: M × A)
∂φ

Model of a Compliant Actuator: τ φj = kφj ψ j − φj

φj = ψ0j ∆ψj
∆ψj

φj = ψ0j ψj
ψj
Prismatic actuator Revolute actuator


Quasi-static Modelling 2 - Transmission Matrix

Transmission Matrix

Deﬁnition
Relates the variation of the generalized coordinates with the variation of
the commanded value of the actuators
∂ψ
T= (dimension: M × A)
∂φ

Model of a Compliant Actuator: τ φj = kφj ψ j − φj

φj = ψ0j ∆ψj
∂S ∂τ ψ ∂τ ψ ∂τ φ
= =
∂φ ∂φ ∂τ φ ∂φ

ψj = A (−Kφ ) = −Kφ
Prismatic actuator


Quasi-static Modelling 3 - Comments on the QSM

Comments on the Quasi-static Model

Transmission Matrix
    
 T = K−1
M
  Kφ 




Transmission Matrix
    
 T = K−1
M
  Kφ 

Equivalency Between Actuators and External Loads

xc = JK−1 AKφ φ + JT f = JCM τ e
˙ M
˙ ˙ ˙




Transmission Matrix
    
 T = K−1
M
  Kφ 

Equivalency Between Actuators and External Loads

xc = JK−1 AKφ φ + JT f = JCM τ e
˙ M
˙ ˙ ˙

Valid for Conventional Mechanisms
M = A (= F)
⇒ KM = Kψ + KI + KE ≈ Kψ = Kφ
Kφ = diag (∞)
donc T = 1A ⇒ JC = J

Conclusion 1 - Examples of application

Simple 2-DOF Mechanism

Coeﬃcient ∂θ1a /∂φ
xp = [xp , yp ]T
1.2

ρb 1.1

θ1a 1.0

φ
K
1 0 1 2 3 4



Simple 2-DOF Mechanism

Coeﬃcient ∂θ1a /∂φ

1.2

1.1

1.0

K
1 0 1 2 3 4



Compliant Tripteron

2 3
0, 85 −0, 04 −0, 04
6
6 −0, 05 0, 88 −0, 02 7
7

1 0 0
 6 0, 00 −0, 00 0, 99 7
Jc = 6 7
 0 1 0 
6 −0, 00 0, 01 −0, 00 7
J= 6
4 −0, 03 −0, 01 −0, 01
7
5
0 0 1 0, 03 0, 01 0, 00

(For φ = [0.5; 0; 0]T and f = 0)



Sub-centimeter Underactuated Compliant Gripper
fq

θq
lq
y φ
b
x
θ3
θ2
h
fp
a
r (θ1) c
lp
θ1
g
e θp
d1 d2 θ4
ly
lx ρ

dθp
 
 dρ 
 dθ1  = CM (fp , fq ) · Kρ
T= 

dρ

Conclusion 2 - Conclusion

Contribution and future works

Contribution of the thesis

The formulations of the presented models are general
The complementarity of kinematics and statics has been used
Improvement of the precision by taking into account more parameters


Conclusion 2 - Conclusion

Contribution and future works

Future works

Application to underactuated mechanisms
Properties of the stiﬀness matrix
Prototype
Dynamic modelling and more


Brief presentation of the thesis

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (20)

Recently uploaded

Recently uploaded (20)

Brief presentation of the thesis