ROBOTICS : A DVANCED C ONCEPTS

&
A NALYSIS
M ODULE 10 – A DVANCED T OPICS

Ashitava Ghosal1
1 Department of Mechanical Engineering
&
Centre for Product Design and Manufacture
Indian Institute of Science
Bangalore 560 012, India
Email: asitava@mecheng.iisc.ernet.in

NPTEL, 2010

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 .
. .1 C ONTENTS

. ∗
. .2 L ECTURE 1
Chaos and Non-linear Dynamics in Robots

.
. .3 L ECTURE 2
Gough-Stewart Platform based Force-torque Sensors

. ∗
. .4 L ECTURE 3
Modeling and Analysis of Deployable Structures

.
. .5 A DDITIONAL M ATERIAL
References and Suggested Reading

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 O UTLINE

.
. .1 C ONTENTS
. ∗
. .2 L ECTURE 1
Chaos and Non-linear Dynamics in Robots

.
. .3 L ECTURE 2
Gough-Stewart Platform based Force-torque Sensors

. ∗
. .4 L ECTURE 3
Modeling and Analysis of Deployable Structures

.
. .5 A DDITIONAL M ATERIAL
References and Suggested Reading

. . . . . .
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 C ONTENTS OF L ECTURE

Introduction to non-linear dynamics and chaos.

Chaos in robot control equations1 .
Simulation results
Analytical criteria
Summary

1 Majorportions of this Lecture are from Shrinivas & Ghosal (1996 &
1997) and Ravishankar & Ghosal (1999). More details are available in
references listed at the end of the module. . . . . . .
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 I NTRODUCTION
Deterministic physical and mathematical systems whose
time history has sensitive dependence on initial conditions.
Occurs in many non-linear dynamical systems2 .
Observed in a wide range of systems – Electric circuits,
fluid dynamics, double pendulum (Levien and Tan, 1993),
large deformation in plates, mechanical and
electro-mechanical systems with friction and
hysteresis(Sekar and Narayanan, 1992), and mathematical
equations modeling complex phenomenon such as weather.
Controversially related to fields such as economics and
medical phenomenon such as epilepsy.
Origin: classical gravitational 3− body problem (Poincaré
late 19th century).
Duffing’s equation (see Moon (1987) & Dowell and
Pezeshki (1986)) is a common and well-studied example
Ẍ + C Ẋ + X 3 = B cos t – Non-linear ‘hardening’ spring.
Different time histories for different values of C and B.
2 To see more details, copy and paste link in a New Window/Tab by

right click. Close New Window/Tab after viewing... . . . . .

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 C HAOTIC S YSTEMS
Non−Chaotic Non−Chaotic
4 8

3 6

2 4

1 2

XDOT
0 0
X

−1 −2

−2 −4

−3 −6

−4 −8
0 10 20 30 40 50 60 70 80 90 100 −4 −3 −2 −1 0 1 2 3 4
Time(seconds) X

Figure 1: Non-chaotic behaviour of a Figure 2: Phase plot – Ẋ (t) Vs. X (t) –

Duffing’s oscillator in non-chaotic case

C = 0.08, B = 0.2, and two “close-by” initial conditions —

(3.0, 4.0) and (3.01, 4.01).
Two trajectories do not deviate much as time increases and
phase plot settle to a limit cycle!
. . . . . .
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 C HAOTIC S YSTEMS Chaotic
Chaotic
4 15

3
10
2

1 5

XDOT
0
X

−1

−2 −5

−3
−10
−4

−5 −15
0 10 20 30 40 50 60 70 80 90 100 −5 −4 −3 −2 −1 0 1 2 3 4
Time(seconds) X

Figure 3: Chaotic behaviour of a Figure 4: Phase plot – Ẋ (t) Vs. X (t) –

Duffing’s oscillator in chaotic case

C = 0.05, B = 7.5 and two “close-by” initial conditions —

(3.0, 4.0) and (3.01, 4.01).
Large deviation in trajectories3 with t and dense phase plot.
3A diagnostic tool for chaos, called the Lyapunov exponent, is based on
this deviation of trajectories (see Parker and Chua (1989) for computation
of Lyapunov exponent and other characteristic of chaos.)
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 K EY F EATURES OF C HAOTIC
S YSTEMS
Main features of a chaotic system:
Sensitive dependence on initial conditions (see Duffing’s
equation) for certain parameter values.
Every point in phase space is eventually visited – Periodic
system (single or finitely many periods), phase plot will
settle in a region (see limit cycle in Duffing’s equation).
Chaotic system, the attractor is not a fixed point or a limit
cycle → Strange attractor has fractal dimension!
Only in non-linear dynamical systems – Finite dimensional
linear systems can never exhibit chaos.
In continuous non-linear dynamical system (described by
differential equation), the dimension must be 3 or more.
Integrable non-linear differential equations → No chaos!
1D discrete systems (logistic map) can exhibit chaos.
Well known book on mathematical aspects of chaos theory
– Guckenheimer and Holmes (1983).
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 ROBOT DYNAMICS & C ONTROL

Equations of motion of a robot are non-linear (see Module

6, Lecture 1)

[M(q)]q̈ + C(q, q̇) + G(q) + F(q, q̇) = τ

[M(q)] — Non-linear mass matrix – Trigonometric terms

C(q, q̇) — Nonlinear Coriolis/centripetal – Trigonometric
and quadratic products q̇i q̇j .
G(q) — Nonlinear gravity term – Trigonometric terms.
F(q, q̇) — Nonlinear friction and other terms.
τ — Control torque/force at joints
Control schemes are sometimes nonlinear.

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 ROBOT DYNAMICS & C ONTROL
To track a desired trajectory, various control schemes are
used (see Module 7, Lecture 3).
Proportional + Derivative(PD) or a PID control scheme
∫
τi = q̈di + Kpi (qdi − qi ) + Kvi (q̇di − q̇i ) + KIi (qdi − qi ) d τ

Model-based control schemes

\
\ τ ′ + C(q,
τ = [M(q)] [ + F(q,
q̇) + G(q) \ q̇)

τ ′ is same as in PD control scheme and (.)c are estimates

used in model-based terms.
PD and model-based control is asymptotically stable for a
regulator problem under certain conditions (see Module 7,
Lecture 6).
Stability not proved for trajectory following with arbitrary
q̇d (t).
. . . . . .
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 ROBOT DYNAMICS & C ONTROL
Equations of motion are same as DOF in number and
non-autonomous in nature.
Simulation of 2 DOF robots – RR, RP, PR, and PP — 4
dimensional system!
Assumptions
No gravity, friction and other non-linear terms.
Equation of motion contain only inertia and
Coriolis/Centripetal Terms
[M(q)]q̈ + C(q, q̇) = τ
Desired repetitive joint space trajectory –
qd i = Ai sin(ωi t), i = 1, 2
Model estimates obtained by perturbing model parameters
— Multiply by (1 + ε )
Numerical integration of equations of motion with control
laws.
Observe evolution of state variables for various controller
gains and estimates of model parameters.
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 D IAGNOSTIC T OOLS FOR C HAOS
Phase plots – Plot of position Vs. velocity
Non-chaotic – Closed with one or more but finite number
of loops.
Chaotic – Not closed and fills up a region in phase space.
Lyapunov exponents – Measures divergence of adjacent
trajectories as t → ∞ (see Parker and Chua (1989), Wolf et
al. (1985)).
n exponents for n dimensional system – One exponent is
zero always.
At least one positive for chaotic system.
Poincaré maps — Stroboscopic sampling of phase plots
One or finite number of points in non-chaotic case
Points tend to fill up a region — Strange Attractor.
Bifurcation diagrams
Plot of a state variables as a parameter is varied.
Pre- and post-chaotic behavior.
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 T HE RP MANIPULATOR
Equations of motion
(m1 l1 2 +I1 +I2 +m2 d2 2 )θ¨1 +2m2 d2 θ˙1 d˙2 = τ1
2
link 2 m2 d¨2 − m2 d2 θ˙1 = F2
m2 Non-dimensional
p(x,y)
parameters
Y
ρ1 = (1 + Im1 +Il 22 ), ρ2 = m2
m1
11
m l2
d2 ρ3 = 11.01 , ρ4 = m11.0l1
X = dl12 , τ1∗ = ρτ13 , F2∗ = F2
ρ4

link 1
Kp∗ = Kp /ω 2 , Kv∗ =
m 1, l1 , r1 Kv /ω , t ∗ = ω t
O θ 1
Non-dimensional parameters are
X
fewer! → Easier to search parameter
Figure 5: The RP
space for obtaining chaotic
Manipulator behaviour.
For model based control
ρ̂i = (1 + ε )ρi
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 RP MANIPULATOR UNDER PD
C ONTROL PD control Kp*=1.2 Kv*=0.7 Chaotic
Pd control Kp*=10.0 Kv*=5.0 Nonchaotic
2 4

1.5 3

1 2

0.5 1

XDOT
XDOT

0 0

−0.5 −1

−1 −2

−1.5 −3

−2 −4
−1.5 −1 −0.5 0 0.5 1 1.5 −6 −4 −2 0 2 4 6
X
X

Figure 6: Phase plot in non-chaotic Figure 7: Phase plot in chaotic

case case(Ravishankar and Ghosal, 1999

Aθ = π , AX = 1.0 and ω = 1.0

Non-dimensional parameters – ρ1 = 2.5, ρ2 = 0.5, ρ3 = 0.4,
and ρ4 = 2.0.
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 RP MANIPULATOR UNDER PD
C ONTROL largest Lyapunov exponent −chaotic
Largest lyapunov exponent(non chaotic) 0.7
0

0.6 Kp*=1.2
−0.2

0.5 Kv*=0.7
−0.4 Kp*=10.0,Kv*=5.0

largest lyapunov exponent

largest lyapunov exponent

0.4
−0.6

−0.8 0.3

−1 0.2

−1.2 0.1

−1.4 0
0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
normalized time

Figure 8: Lyapunov exponent in Figure 9: Lyapunov exponent in chaotic

non-chaotic case case

Largest Lyapunov exponent for chaotic and non-chaotic

cases.
Aθ = π , AX = 1.0, ω = 1.0, ρ1 = 2.5, ρ2 = 0.5, and
ρ3 = 0.4, ρ4 = 2.0 . . . . . .
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 RP MANIPULATOR UNDER PD
C ONTROL
3.5

2.5
Poincaré map
2
(θ˙1 , θ1 )
projection.
Theta1dot

1.5
Aθ = π ,
1
AX = 1.0,
0.5 ω = 1.0,
0 Kp*=1.2 ρ1 = 2.5,
−0.5 Kv*=0.7
ρ2 = 0.5, and
ρ3 = 0.4,
−1
−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2
Theta1

Figure 10: Poincaré map for RP manipulator under

PD control
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 RP MANIPULATOR UNDER PD AND
M ODEL - BASED C ONTROL
8
1.2

1
6

0.8 5

Kv*
4
Kv*

0.6

0.4
2

0.2 1

0
0 0 5 10 15 20 25 30 35 40 45
0 2 4 6 8 10 12 Kp*
Kp*

Figure 11: Chaos maps for RP Figure 12: Chaos maps for RP
manipulator under PD control manipulator under model-based control

Chaos maps – Values of gains for chaotic behavior.

Kv∗ in steps of 0.1, Kp∗ in steps of 1.0.
Initial conditions – (0, 0, π , 1.0)
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 RP MANIPULATOR UNDER PD AND
M ODEL - BASED C ONTROL
Kv*=0.5

−3.12 −0.5

−3.14
−1
−3.16

−3.18
−1.5
THETA

X
−3.2

−2
−3.22

−3.24
−2.5

−3.26

−3.28
9.2 9.3 9.4 9.5 9.6 9.7 9.8
9.6 9.65 9.7 9.75 9.8 9.85 9.9 Kp*
KP*

Figure 13: Bifurcation diagram for RP Figure 14: Bifurcation diagram for RP
manipulator under PD control manipulator under model-based control

Bifurcation diagrams – Plot of state-variable as Kp∗ is

changed at a fixed Kv∗ .
Period doubling route to chaos! . . . . . .
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 T HE RR MANIPULATOR
Y
p(x,y)

link 2

m 2 , l 2 , r2 , I2
θ2

link 1

m 1, l1 , r 1 , I 1

θ1

Figure 15: The RR Manipulator

Equations of motion (see Module 6, Lecture 2)

(m1 r12 + I1 + m2 r22 + I2 + m2 l12 + 2m2 l1 r2 cos θ2 )θ̈1 +
(m2 r22 + I2 + m2 l1 r2 cos θ2 )θ̈2 − m2 l1 r2 sin θ2 (2θ̇1 + θ̇2 )θ̇2 = τ1
(m2 r22 + I2 + m2 l1 r2 cos θ2 )θ̈1 + (m2 r22 + I2 )θ̈2 + m2 l1 r2 sin θ2 θ̇12 = τ2
The RR manipulator (or a double pendulum) is known to
be chaotic (see Mahout et al. (1993)).
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 RR MANIPULATOR UNDER PD AND
MODEL - BASED CONTROL

VARIATION OF THE LARGEST LYAPUNOV EXPONENTS

0.8

MODEL BASED KP=15, KV=1.5, EPS=−0.7

0.6

PD KP=50, KV=2.0
LARGEST LYAPUNOV EXPONENT

0.4
Aθ1 = π /2, Aθ2 = π /4
0.2 and ω = 2.0
0 Mass and DH parameters –
−0.2
Correspond to the first two
links of the CMU DD Arm
−0.4
II (see Khosla (1986))
−0.6
0 100 200 300 400 500 600 700 800 900 1000
TIME in seconds

Figure 16: Largest Lyapunov exponent

for the RR manipulator under PD and
model-based control

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 RR MANIPULATOR UNDER PD AND
MODEL - BASED CONTROL
PD Controller EPS = −0.9
3 8

7
2.5

2
5

KV
KV

1.5 4

3
1

0.5
1

0 0
10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
KP KP

Figure 17: Chaos map for PD control Figure 18: Chaos map for model-based
of the RR Manipulator control of the RR Manipulator

Kv∗ in steps of 0.1, Kp∗ in steps of 1.0

Initial conditions – (0, π , 0, π /2), ε = −0.9
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 RR MANIPULATOR UNDER MODEL -
BASED CONTROL

2.2

2.15 eps=−0.9

2.1 Kp=49

2.05 ε = −0.9 and

theta2

2
Kp = 49
Period doubling
1.95
route to chaos
1.9

1.85

6.6 6.65 6.7 6.75 6.8 6.85 6.9 6.95

Kv

Figure 19: Bifurcation diagram for RR

manipulator
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 S UMMARY OF S IMULATION R ESULTS

Numerical simulation of 2 DOF planar robots under PD

and model-based controller.
Both the RR and RP robot can exhibit chaotic motions
Chaotic motions for low controller gains
Chaotic motions for large mismatch between model and
plant
Chaotic motions seen more easily for underestimations
Route to chaos appear to be through period doubling.
PR and PP robot do not show chaotic motions even after
extensive simulations!

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 A NALYTICAL C RITERIA

The manipulator mass matrix, [M(q)], is positive definite.

[M(q)] defines a Riemannian metric in the configuration
space (q)
From [M(q)] one can compute Riemannian curvature
tensor
n
Rijkl = ∑ Mih Riklh
h=1

Equations of motion in absence of potential energy

∂H ∂H
q̇ = , ṗ = − +τ
∂p ∂q

p is the momentum and H(p, q) is the Hamiltonian of the

system.

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 A NALYTICAL C RITERIA (C ONTD .)
If Rijkl = 0, then the mass matrix can be factorized (Stoker
1969, Spong 1992)

[M(q)] = [N(q)]T [N(q)]

Equations of motion can be written as

q̇ = P, Ṗ = [N(q)]−T τ

For τ = 0 ⇒ Equations of motion can be integrated in

closed-form ⇒ Cannot exhibit chaos!
Can obtain Rijkl easily for 2 DOF robots since [M(q)] is
known.
If Rijkl = 0 then chaotic motion not possible.
Not required to compute full tensor Rijkl – Gaussian
curvature of 2D subspace enough!
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 A NALYTICAL C RITERIA (C ONTD .)
ε (t)

ε (t)
ε0
ε0

G<0 G>0

Figure 20: Gaussian curvature and trajectories

τ = 0 – Trajectories along geodesics of manifold (Arnold
1989).
Rijkl ̸= 0 → Gaussian curvature of 2D subspace –
G = (R1212 / det[M]) √
In figures aboves, ε (t) = ε0 e −G t
G < 0, nearby trajectories diverge exponentially — Chaos!
Analytical criteria — G < 0 in any 2D subspace → Chaotic
(see also Zak (1985a & b)).
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 G AUSSIAN C URVATURE OF ROBOTS :
PP ROBOT ˙2 1 2
1
KE = 2 m1 d1 + 2 m2 d˙2
Elements of the mass matrix
X
M11 = m1 , M12 = 0, M22 = m2
link 2
m2

p(x,y)

d2 F2
Mass matrix is constant.

link 1 m1

O Y
d1

F1 G = 0 → Not chaotic – Expected

as it is linear system!
Figure 21: The PP Robot
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 G AUSSIAN C URVATURE OF ROBOTS :
RP ROBOT

link 2

m2 p(x,y)

Elements of mass matrix

d2
M11 = I +m2 d22 , M12 = 0, M22 = m2

link 1 G < 0 for I > 0 → Always

m 1, l1 , r1
O θ 1
chaotic!
X

Figure 22: The RP Robot

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 G AUSSIAN C URVATURE OF ROBOTS :
PR ROBOT
Y
Elements of mass matrix

M11 = m1 +m2 , M12 = −m2 r2 sin θ2

p(x,y)
link 2 M22 = m2 r22 + I2
m2 , l 2 , r 2

F 1, d 1 θ2

O X
link 1 Γ2 G = 0 → Not chaotic!

m1

Figure 23: The PR Robot

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 G AUSSIAN C URVATURE OF ROBOTS :
RR ROBOT
Y
p(x,y)

link 2
Elements of mass matrix
m 2 , l 2 , r2 , I2
θ2
M11 = c1 + c2 cos θ2
link 1

m 1, l1 , r 1 , I 1
M12 = 2(c3 + c4 cos θ2 ), M22 = c3

θ1 ci , i = 1, 2, 3, 4 are constants.

Figure 24: The RR Robot

a2 c1 − b1 + 2a3 c3 − a4 c1 + 2a5 c3
G < 0 if cos θ2 < − ,
a1 + a2 c2 + 2a3 c4 − a4 c2 + 2a5 c4
ai , i = 1, 2, 3, 4 constants.
Conditionally chaotic. . . . . . .
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 A NALYTICAL C RITERIA

Gaussian curvature is zero for PP and PR robots → PP

and PR manipulators do not show chaotic behaviour.
Gaussian curvature is less than zero for RR and RP robots
→ Shows chaotic behaviour in numerical simulation.
Gaussian curvature of a 2D subspace less than zero for
RRR and RRP robots.
Negative Gaussian curvature criteria is for unforced motion.
Numerical simulation and Lyapunov exponent or other
diagnostic criteria needs to be used for forced and/or
controlled motion.

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 S UMMARY

Robot dynamic and control equations are non-linear ODEs.

Nonlinearity different from nonlinear spring or other
commonly studied chaotic systems.
Equations are higher dimensional and more complicated
than commonly studied ones.
Feedback control equations for robots can exhibit chaos.
Suggest a re-look at some of the robustness results in
robot control (see Craig (1989)).
Lower bounds on controller gains can be obtained by
numerical simulations.
For unforced motion negative Gaussian curvature criteria
may be useful.

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 O UTLINE

.
. .1 C ONTENTS
. ∗
. .2 L ECTURE 1
Chaos and Non-linear Dynamics in Robots

.
. .3 L ECTURE 2
Gough-Stewart Platform based Force-torque Sensors

. ∗
. .4 L ECTURE 3
Modeling and Analysis of Deployable Structures

.
. .5 A DDITIONAL M ATERIAL
References and Suggested Reading

. . . . . .
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 C ONTENTS OF L ECTURE

Introduction4
Kinematics and statics of Gough-Stewart platform.
Isotropic and singular configurations
Six component force-torque sensors based on a
Gough-Stewart platform at a near singular configuration.
Modeling, analysis and design of Gough-Stewart platform
based sensors.
Hardware and experimental results.
Summary

4 Majorportions of this Lecture are from Bandyopadhyay & Ghosal

(2006, 2008 & 2009) and Ranganath et. al (2004). Please see these and
references listed at the end for more details. . . . . . .
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 I NTRODUCTION
First used as tyre testing machine
in UK.
Now known as Gough-Stewart
platform.

A moving platform connected to

fixed ground by six actuated
extendable legs — 6 DOF
(Fichter, 1986)
Linear motion of platform along
X , Y and Z axes & Rotational
motion about X , Y and Z axes.
Known also as Heave, Surge,
Stewart 1965 Sway & Roll, Pitch and Yaw.
Extendable `legs’
Best known parallel robot
Figure 25: The Stewart (Module 4, Lecture 5).
platform . . . . . .
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 I NTRODUCTION

Industrial manufacturing
Modern tyre testing Micro-positioning
machine

Physik Instrumetente
http://www.physikinstrumente.com

Precise alignment of
Robotic surgery mirror
Figure 26: Some modern uses of Gough-Stewart platform
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 I NTRODUCTION
P2 Top Platform
P3

{P0 }
P1
P4 Spherical Joint

Prismatic
Moving top platform
Joint
P6
P5 Fixed base

6 extendable legs actuated

B6 by prismatic joints.
B5 {B0 } B1
Coordinated motion of 6
prismatic joints →
B4 Arbitrary 6 DOF motion of
U Joint top platform

B2
Extensible Leg
B3
Fixed Base

Figure 27: The Gough-Stewart platform . . . . . .

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 M OTION S IMULATION
Motion simulations
done using
ADAMS⃝ R
.
Click herea for a
video showing
motion of a
Gough-Stewart
platform due to
combined motion of
all actuated joints.
a Copy & paste link in a
New Window/Tab by right
click. Close New
Figure 28: Motion simulation of Gough-Stewart Window/Tab after viewing.
platform

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 G OUGH -S TEWART PLATFORM AS A
S ENSOR

With actuators (P joints) locked → 0 degrees of freedom.

Instead of actuators, strain gauge based sensors at actuator
location.
External force-moment applied at top platform can be
related to axial forces along legs at P joint locations.
Axial forces in legs related to strains.
Measured strains can be related to external force-torque at
top platform.
All six components of externally applied force-torque can
be measured.

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 G OUGH -S TEWART PLATFORM
Kinematics and statics (see
Module 5, Lecture 5 for more
details).
{P0 }
Pi Direct kinematics involve solution
P0
pi
of a 40 degree polynomial.
S Joint
Leg vector
P Joint

B0
Si = B
P0 [R] pi + t − bi
0 P0 B0 B0
B0
t Ẑ
li
Ŷ Unit vector along leg
B0
Bi si =B0 Si /li
B0
bi Relation between external
X̂
force-moment at top platform
{Tool } and leg forces fi
{B0 }
U Joint    
B0 F
Tool ∑6i =1 B0 si fi
Figure 29: A leg of the  −−− = −−− 
Gough-Stewart platform B0 M 6 (B0 b ×B0 s )f
Tool ∑i=1 i i i
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 K INEMATICS AND S TATICS OF
G OUGH -S TEWART PLATFORM
Statics equation in matrix form
 B 
0F
Tool
FTool =  − − −  = B
B0 ∆ 0
Tool [ H ]f
B0 M
Tool

The force transformation matrix B 0

Tool [ H ] is given by
 B0 s B0 s B0 s

1 2 ... 6
B0  −−− −−− −−− −−− 
Tool [ H ] =
( 0 b1 × 0 s1 ) ( 0 b2 × 0 s2 )
B B B B ... ( 0 b 6 × 0 s6 )
B B

where f is the vector of forces at the prismatic joints

(f1 , f2 , ..., f6 )T .
The force transformation matrix is related to equivalent
Jacobian (Module 5, Lecture 5).
−1 B
Leg forces can be obtained as f = B 0
Tool [ H ]
0
FTool
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 I SOTROPIC C ONFIGURATIONS
Isotropic configuration
det(BTool [ H ]) ̸= 0
0

Eigenvalues of top left 3 × 3

and bottom right 3 × 3 matrix
are equal (not necessary equal
to each other) (see Klein and
Milkos 1991, Fattah and
Ghasemi 2001 and
Dwarakanath et al. 2001).
All directions are equivalent in
terms of force (or moment)
components.
Figure 30: Gough-Stewart platform Isotropic configuration can be
in an isotropic configuration
obtained in closed-form (see
Bandyopadhyay & Ghosal,
2008)
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 S INGULAR C ONFIGURATIONS
Singular configuration
det(B 0
Tool [ H ]) = 0
One or more eigenvalue of
B0
Tool [ H ] is zero!
Gain singularity → Platform
cannot resist one or more
component of force/moment
applied at the top platform.
Figure 31: Singularity manifold Singularity manifold(s) can be
of Gough-Stewart platform at a obtained in closed-form (see
given orientation
Bandyopadhyay & Ghosal, 2006)
Position singularity manifold shown for a semi-regular
Stewart platform manipulator (SRSPM) – Cubic in z and a
quadratic curve, but not an ellipse, in x and y .
Orientation singularity manifold, at a given position, can
also be obtained (see Bandyopadhyay & Ghosal, 2006).
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 S INGULAR C ONFIGURATION
Planar truss with hinges.
Force F is applied at the hinge C at
an angle ϕ .
R1 R2
Axial forces in links AC and BC are
( )( ) ( )
cos θ − cos θ R1 cos ϕ
=F
sin θ sin θ R2 sin ϕ
Figure 32: A planar two
link hinged truss
LHS matrix is [H] and for θ ̸= 0
( ) ( ) ( )
R1 F cos ϕ F cos ϕ / cos θ + sin ϕ / sin θ
= [H]−1 =
R2 F sin ϕ 2 − cos ϕ / cos θ + sin ϕ / sin θ
For θ → 0 and ϕ ̸= 0, R1 , R2 → ∞ — F cannot be resisted.
For θ = 0 and ϕ ̸= 0, the eigenvalues are 1 and 0.
0 eigenvector is Y axis – Fy not resisted at θ = 0.
A small Fy will give large output R1 → Enhanced sensitivity
or mechanical amplification for certain components!
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 N EAR S INGULAR C ONFIGURATION IN
A H INGED P LANAR T RUSS
300 For ϕ = π /2,
|R1 | = |R2 | = F /(2 sin θ )
250
— θ small, |R1 | and |R2 |
200
large!
At θ = 1◦ , magnification
Magnification

150
Rigid
|R1 |/F is approximately
100
28.6.
Elastic If AC and BC are elastic
50
θnew = arctan(δ + δ1 )
0
(Srinath 1983) where,
0 0.5 1 1.5 2 2.5 3
THETA in deg
δ = l sin θ and
Figure 33: Force amplification Vs. θ δ1 = l cos θ × (F /EA)1/3
and Poisson’s ratio is 0.3.
For elastic links R1 = −R2 = F /(2 sin θnew ) —
Amplification/enhanced sensitivity is present but lower!
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 S INGULAR C ONFIGURATIONS
Force transformation [H] matrix in 6 × 6 Gough-Stewart
 B0 s B0 s B0 s

1 2 ... 6
B0
Tool [ H ]= −−− −−− −−− −−− 
(B0 b1 ×B0 s1 ) (B0 b2 ×B0 s2 ) ... (B0 b6 ×B0 s6 )

Singular configuration det[H] = 0 (Merlet 1989, St-Onge

and Gosselin 2000).
Example — All legs parallel along (0 0 1)T (parallel to Z
axis) and base connection points on a plane
 
0 0 0 0 0 0
 0 0 0 0 0 0 
 
 1 1 1 1 1 1 
[H] = 



 b1y b2y b3y b4y b5 y b6y 
 −b1x −b2x −b3x −b4x −b5x −b6x 
0 0 0 0 0 0

Singular directions: (1, 0, 0; 0, 0, 0)T , (0, 1, 0; 0, 0, 0)T , and

(0, 0, 0; 0, 0, 1)T .
Cannot resist or enhanced sensitivity for Fx , Fy and Mz .
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 A LGORITHM TO OBTAIN SINGULAR
DIRECTIONS
External force and leg forces are related by
[ ]
F = [Hf ]f = s1 s2 s3 s4 s5 s6 f
Maximum, minimum and intermediate values of FT F
subject to a constraint f T f = 1 are the eigenvalues of
[gf ] = [Hf ]T [Hf ]5 .
Rank of [gf ] is at most 3 ⇒ 3 eigenvalues are 0 & 3
non-zero eigenvalues obtained from solution of a cubic and
in closed-form.
The tip of F lies on an ellipsoid and the axes of ellipsoid are
obtained from eigenvectors corresponding to non-zero
eigenvalues.
Principal axis of ellipsoid are along principal forces.
Directions corresponding to zero eigenvalues of [gf ] are
principal moments at origin.
5 See Module 5, Lecture 2 for a similar treatment for .velocities.
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 S INGULAR C ONFIGURATIONS IN 6 ×
6 G OUGH -S TEWART PLATFORM
Eigenvector of [gf ] mapped by [H] are
 
[0] [F ]∗
[H][X ] =  − − − −−− 
[M]O ∗ [M]p ∗

[F ]∗ is a 3 × 3 matrix of principal forces, [M]O ∗ is a 3 × 3

matrix of principal moments at the origin, and [M]p ∗ is a
3 × 3 matrix of principal moments at centre of platform.
Rank of [gf ] less than 3 ⇒ Singularity in force domain.
Eigenvectors of [gf ], corresponding to zero eigenvalue,
mapped by [H] give direction(s) where force cannot be
resisted — Same as null space [F ]∗ .
Singular directions of moment — Null space of [M]O ∗ .
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 A LGORITHM TO OBTAIN SINGULAR
DIRECTIONS
Enumerate all possible 6 − 6 Gough-Stewart platforms by
choosing pairs of base and platform points. For each of the
configurations,
Compute the number of zero eigenvalues of [H]. This will
give the total number of singular directions including force
and moments.
Obtain all eigenvalues and corresponding eigenvectors
symbolically for [gf ] using a symbolic manipulation
package.
Obtain the matrix [H][X ] and sub-matrices [F ]∗ and [M]∗O
(see previous slide).
Obtain null space vectors of [F ]∗ and [M]∗O to obtain the
singular force and moment directions (if any).
Eigenvalues and eigenvectors can be obtained symbolically
→ Singular directions can be obtained symbolically.
Singular directions obtained symbolically using
Mathematica⃝ R
(Wolfram 2004).
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 E XAMPLES OF S INGULAR
D IRECTIONS
Table 1: Examples of Singular Directions in 6 × 6 Gough-Stewart platform
configurations

Leg Connections ∗ Singular

Leg 1 Leg 2 Leg 3 Leg 4 Leg 5 Leg 6 Directions
1 B1 − P1 B2 − P2 B 3 − P3 B 4 − P4 B 5 − P5 B 6 − P6 3 Fx , Fy , Mz
2 B1 − P2 B2 − P1 B 3 − P3 B 4 − P4 B 5 − P5 B 6 − P6 2 Fx , Mz
3 B1 − P2 B2 − P1 B 3 − P4 B 4 − P3 B 5 − P5 B 6 − P6 1 Mz
4 B1 − P2 B2 − P1 B 3 − P4 B 4 − P3 B 5 − P6 B 6 − P5 0 none
5 B1 − P1 B2 − P3 B 3 − P2 B 4 − P5 B 5 − P4 B 6 − P6 1 Mz
6 B1 − P1 B2 − P6 B 3 − P5 B 4 − P4 B 5 − P3 B 6 − P2 2 Fx , Mz
7 B1 − P1 B2 − P3 B 3 − P2 B 4 − P4 B 5 − P6 B 6 − P5 1 Fy
8 B1 − P2 B2 − P3 B 3 − P4 B 4 − P5 B 5 − P6 B 6 − P1 3 Mx , My , Mz
∗ – Column indicate number of zero eigenvalues of [H].

Bi , Pi , i = 1, ...6 are Base and Platform connection points.

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 N EAR S INGULAR C ONFIGURATION IN
6 × 6 G OUGH -S TEWART PLATFORM
Configuration 1 chosen for sensor developement —
Enhanced sensitivity for Fx , Fy and Mz .
Both top and bottom platform are regular hexagons of
equal sides.
At exactly singular configuration, legs are exactly vertical
and amplification is infinite — Not desirable!
Gough-Stewart platform, Configuration # 1, at a near
singularity
The legs are not exactly vertical.
Top and bottom platform not aligned and included
half-angle changed from 30◦ to 33◦ → Top platform
rotated by 3◦ !
det [H] ̸= 0 → Near singular with condition number of [H]
about 1900.
Amplification of about 10 (and not infinity)!
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 N OMINAL G EOMETRY OF S ENSOR

Table 2: Nominal geometry of 6-6 Stewart Platform with γ = 33◦

Base coordinates Platform coordinates

Point x y z Point X Y Z
No. mm mm mm No mm mm mm
b1 43.30 25.0 0.0 p1 41.93 27.23 100
b2 0 50.0 0.0 p2 2.616 49.93 100
b3 -43.30 25.0 0.0 p3 -44.55 22.70 100
b4 -43.30 -25.0 0.0 p4 -44.55 -22.70 100
b5 0 -50 0.0 p5 2.616 -49.93 100
b6 43.3 -25.0 0.0 p6 41.93 -27.23 100

Expected to give enhanced sensitivity to Fx , Fy and Mz .

Near singular configuration – Can invert [H] if and when
required.
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 F LEXIBLE H INGES
Kinematic joints (S or U) give rise to unpredictable friction!
Flexible hinges (Paros and Weisboard 1965, Zhang and
Fasse 2001) much better – No friction! (see also McInroy
and Hamann 2000).
FY
FY M
Y
b M
Y

h
t
M Z
d
MZ
FX D
R FZ FX
FZ R

Figure 34: Flexure hinges with Figure 35: Flexure hinges with circular
rectangular cross-section cross-section
Geometry (t, R, θ ) or (d , D, θ ) can be designed to give
required lateral and longitudinal stiffness (or compliance).
For small motion good approximation .to kinematic
. .
joints.
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 F LEXIBLE H INGES

Figure 36: Detailed view of flexure hinges

Hinges (also leg and ring): Titanium alloy of yield strength

880 N /mm2 .
No rotation permitted beyond 3.8◦ to prevent failure!
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 S ENSING E LEMENT

Figure 37: Schematic of ring shaped sensing element

Ring shaped sensing element from Titanium alloy rod.

Ring mid-plane has largest stress (and strain) when axial
load applied.
For 30 N axial compressive load, 145 µ -strains
(compressive) at inside surface, 110 µ -strains (tensile) at
outside surface – 510 µ -strains in full bridge configuration.
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 F INITE E LEMENT A NALYSIS STRESS CONTOURS
DISPLACED-SHAPE
MX DEF= 5.84E-01 VON-MISES (VONM )
NODE NO.= 247 VIEW : 0.0149899
SCALE = 1.0 RANGE: 298.1128
(MAPPED SCALING)
298.1
276.8
255.5
234.2
212.9
191.6
170.4
149.1
127.8
106.5
85.19
63.89
42.60
21.31
1E-02

Z ROTX
ROTX Y -45.0
Z -45.0
Y ROTY MIDDLE LAYER ROTY
0.0 0.0
ROTZ X
-45.0 ROTZ
X -45.0

Figure 38: Deflection (mm) of sensor Figure 39: Stress (N/mm2 ) in sensor
FE model (in NISA) of top and bottom platform, six legs
with hinges and sensing element created.
Applied Fx = Fy = Fz = 0.98 N, Mx = My = Mz = 49.05
Maximum deflection 0.5 mm and maximum stress about
294 N/mm2 at the flexible hinges — Safe design!
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 G OUGH -S TEWART P LATFORM BASED
S ENSOR

Figure 40: Prototype Gough-Stewart platform based force-torque sensor

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 P ROTOTYPE S ENSOR : E XPERIMENTS
Figure. a Figure. b
10 10

Force in leg 1, N

Force in leg 2, N
Fx
Fz
0 0

Fy
−10 −10
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
Figure. c Figure. d
10 10
Force in leg 3, N

Force in leg 4, N
Fx
Fz
0 0

−10 −10 Fy

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

Figure. e Figure. f
10 10
Force in leg 5, N

Force in leg 6, N
0 0

−10 −10
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
Applied external force, N Applied external force, N

Figure 41: Experimental data for external applied force (∗: Fx , +: Fy and ♢: Fz
in all plots)
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 P ROTOTYPE S ENSOR : E XPERIMENTS
Figure. a Figure. b

Force in leg 2, N
Force in leg 1, N
5 5
Mz
My (+)
0 0
Mx ( * )

−5 −5

−50 0 50 −50 0 50
Figure. c Figure. d

Force in leg 4, N
Force in leg 3, N

5 5

0 0

−5 −5

−50 0 50 −50 0 50
Figure. e Figure. f

Force in leg 6, N
Force in leg 5, N

5 5

0 0

−5 −5

−50 0 50 −50 0 50
Applied external moment, N−mm Applied external moment, N−mm

Figure 42: Experimental data for external applied moment (∗: Mx , +: My and
♢: Mz in all plots)
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 C ALIBRATION
Calibration of leg – Measure strain in legs for known
loading.
Obtain calibration constant µ −strain /N for each leg.
Leg 1 – 13.786, Leg 2 – 13.958, Leg 3 – 14.102
Leg 4 – 13.921, Leg 5 – 13.994, leg 6 – 14.046
Convert measured strains to leg forces fi , i = 1, ..., 6 for
applied loads.
Obtain elements of [H] matrix from experimental data
From (F; M)T = [H]f, write
Fx = f1 H11 + f2 H12 + f3 H13 + f4 H14 + f5 H15 + f6 H16
fi measured leg forces, H1j unknown first row of [H].
From n sets of measurements fi form n × 6 matrix [f ].
H1j ’s are
(H1j , H2j , H3j , H4j , H5j , H6j )T = [f ]# (F1x , F2x , ..., Fnx )T
where [f ]# is the pseudo-inverse of [f ].
Find other rows of [H] in similar manner.
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 C ALIBRATION (C ONTD .)

Calibration [Hc ] matrix is obtained as

 
−0.0195 0.0279 −0.0266 −0.0223 0.0369 −0.0117
 0.0287 −0.0076 −0.0368 0.0280 0.0036 −0.0272 
 
 0.8890 0.8294 0.8321 0.8845 0.9704 0.9712 
[Hc ] = 



 22.7237 44.3631 21.0266 −18.6015 −45.1386 −26.4990 
 −6.7289 −5.5169 −5.0906 −4.8826 −5.1129 −6.4894 
1.3319 −1.5084 1.8969 −1.4110 1.2823 −1.9917

Condition No. is 1351 compared to a computed 1910.

Obtain unknown (F; M)T from [Hc ]f, where f is measured
leg forces.

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 E XAMPLES OF FORCE - TORQUE
MEASUREMENTS

For a combined 3D external loading of (0.9123, 0.9123, 0)T

N force and (−10.0356, 10.03560, 0)T N-mm moment, the
measured values of forces and moments are
(0.9270, 0.8819, 0.0265)T N and
(−13.0081, 10.1789, −1.4352)T N-mm respectively. The
FEA computed values are (0.9241, 0.8809, 0.0932)T N of
force and (−19.2041, 12.3772, −0.5258)T N-mm.
For a combined 3D loading of (0.9123, 0.9123, 0)T N force
and (−10.0356, 10.0356, −45.6165)T N-mm moment, the
measured values are (0.8937, 0.9153, 0.1462)T N and
(−12.2085, 8.9987, −45.9569)T N-mm respectively. The
computed FEA values are (0.8780, 0.9261, 0.2688)T N and
(−21.8783, 18.0896, −43.7448)T N-mm.

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 G OUGH -S TEWART P LATFORM BASED
S ENSOR

The performance of the prototype sensor is very good for

sensing forces and moments in the chosen sensitive
directions and errors are around 3%.
A magnification of about 10 is observed in the sensitive
directions.
The performance of the prototype sensor in the
non-sensitive directions is less accurate — More electronic
amplification is required.
The computed FEA values are in general larger. This is
expected since FE based models are known to be stiffer.

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 M OMENT S ENSITIVE
C ONFIGURATION
Configuration # 8 is sensitive to moments.
The connection sequence is B1 − P2 , B2 − P3 ... B6 − P1

CAD model of a leg

with sensing ring

Flexible hinge – 2 DOF

Figure 43: CAD model of sensor sensitive to moment components

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 M OMENT S ENSITIVE
C ONFIGURATION

Figure 44: Prototype sensor sensitive to moment components

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 S UMMARY

Gough-Stewart as a six component force-torque sensor.

Isotropic and singular configurations.
Algorithm to obtain singular directions – Can be done
symbolically!
Design of a 6 component force-torque sensor sensitive to
Fx , Fy and Mz .
Kinematic design – Choice of configuration and geometry.
Design of flexible hinges and sensing element.
Finite element analysis of full sensor.
Prototyping, calibration and testing.
Sensor sensitive to moments.
Can design a class of Gough-Stewart platform based
sensors with desired (enhanced) sensitivities for chosen
force/moment components!
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 O UTLINE

.
. .1 C ONTENTS
. ∗
. .2 L ECTURE 1
Chaos and Non-linear Dynamics in Robots

.
. .3 L ECTURE 2
Gough-Stewart Platform based Force-torque Sensors

. ∗
. .4 L ECTURE 3
Modeling and Analysis of Deployable Structures

.
. .5 A DDITIONAL M ATERIAL
References and Suggested Reading

. . . . . .
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 C ONTENTS OF L ECTURE

Introduction
Over-constrained mechanisms and deployable structures6
Constraint Jacobian and obtaining redundant links and
joints.
Kinematics of SLE based deployable structures.
Statics of SLE based deployable structures.
Summary

6 This
Lecture is based on material from Nagaraj (2009) and Nagaraj et
al. (2009, 2010). Please see these and reference listed at the end for more
details. . . . . . .
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 I NTRODUCTION
Large deployable structures
Space applications — Small payload bay.
Modern communication and other satellites in orbit have
large appendages.
Compact folded state in payload bay→ Large deployed
state in orbit.
Large number of links and joints present.
In stowed state — Locked/strapped one DOF mechanism.
During deployment, behaves as a one degree of freedom
mechanism.
At the end of deployment, actuated joint is locked.
In deployed state — Structure capable of taking load.
Main ones: coilable and pantograph masts, antennae and
solar panels.
This lecture deals with pantograph based deployable
structures.
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 E XAMPLES OF DEPLOYABLE
STRUCTURES

Figure 45: Folded articulated square Figure 46: Deployment of FAST (see
mast (FAST) Warden 1987)

Eight FAST masts are used in the International Space

Station to support solar arrays.
Source: AEC-Able Engineering Company, Inc.
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 E XAMPLES OF DEPLOYABLE
STRUCTURES
Revolute joint in middle connects
two links of equal length.
Passive cable: connects two
points such that it is slack when
fully or partially folded and
becomes taught when fully
deployed.
Passive cable(s) terminate
deployment and increase stiffness
of structure – Sometimes more
Figure 47: Planar
than one passive cable.
scissor-like-element (SLE) or a Active cable: length decreases
pantograph
continuously and control
deployment.
Typically only one active cable — To avoid multiple
mechanisms and actuators. . . . . . .
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 E XAMPLES OF DEPLOYABLE
STRUCTURES

Figure 48: Stacked planar SLE masts (a) Fully deployed, (b) Partially deployed
Four SLE’s stacked on top of each other.
Deployment angle varies from fully folded (β = 0◦ ) to fully
deployed (β = 45◦ ).
8 passive cables and one active cable. . . . . . .
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 E XAMPLES OF DEPLOYABLE
STRUCTURES

Figure 49: Deployment sequence of a cable stiffened pantograph deployable

antennae (You & Pellegrino (1997))
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 E XAMPLES OF DEPLOYABLE
STRUCTURES

Circular pantograph ring and

radial tensioned membrane rib
connected to a central hub.
5.6 m by 6.4 m elliptical version
tested in MIR space station.
Made by Energia-GPI Space
(EGS), Russia. Visit website for
more information.

Figure 50: Schematic of a 5.6

m EGS antennae
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 OVER - CONSTRAINED M ECHANISMS

(a) 3 sliders (b) RPPR (c) Parallelogram (d) Kempes-Burmester

(Mallik et al. mechanism linkage focal mechanism
1994) (Wunderlich (1968)

Figure 51: Over-constrained Mechanisms

Most well known DOF or mobility equation:
Grübler-Kutzbach
j
M = λ (n − j − 1) + ∑ fi , λ = 3 or 6
i =1
M ̸= 1 in all example, although all can move!
Case (a): Special geometry, Case (b): Passive DOF along
PP line af , Case (c): Redundant link pq, and Case (d):
Redundant R joint at d . . . . . . .
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 D EGREE OF F REEDOM & M OBILITY
Many other well-known mechanisms such as Bennett
mechanism (Bennett, 1903), deployable pantograph masts
etc. gives M ̸= 1 by Grübler-Kutzbach formula.
Grübler-Kutzbach fails since special geometry is not taken
in to account → Formula based on counting alone!
Many attempts to derive a “more universal” DOF/mobility
formula (see Gogu, 2005)
Passive DOF fp subtracted by Tsai (2001): S − S pair or
P − P pair cases.
Equivalent screw system to choose λ (Waldron, 1966).
Null space of Jacobian matrix (Freudenstein, 1962):
M = Nullity([J]) – Used in this Lecture!
Including state of self-stress s and number of internal
mechanisms m (Guest and Fowler, 2005).
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 D IFFERENT TYPES OF C OORDINATES

(a) Relative or joint coordinates

(b) Reference point coordinates

(c) Cartesian coordinates

Figure 52: Three kinds of coordinates in RRPR mechanism

Relative coordinates are with respect to previous link
(Denavit and Hartenberg, 1965).
Reference point (or absolute) coordinates – Planar 3
coordinates and 6 coordinates in 3D (Nikravesh, 1988).
Cartesian (or natural coordinates) – Reference point moved
to joint (Garcia de Jalon and Bayo, 1994).
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 C OORDINATES AND C ONSTRAINTS
For relative coordinates loop-closure constraints (see
Module 4, Lecture 1) for RRPR mechanism
l1 cos ϕ1 + d cos(ϕ1 + ϕ2 ) + l3 cos(ϕ1 + ϕ2 − π /2) = l4
l1 sin ϕ1 + d sin(ϕ1 + ϕ2 ) + l3 sin(ϕ1 + ϕ2 − π /2) = 0
q = (ϕ1 , ϕ2 , d ) are the coordinates (see Figure 52).
For reference point coordinates, the constraints are
xa + l1 /2 cos ϕ1 = x1 , ya + l1 /2 sin ϕ1 = y1
x1 + l1 /2 cos ϕ1 + l2 /2 cos ϕ2 = x2 , y1 + l1 /2 sin ϕ1 + l2 /2 sin ϕ2 = y2
ϕ2 − ϕ3 = π /2, (y2 − y3 ) cos ϕ2 + (x3 − x2 ) sin ϕ2 = l3 /2
x3 + l3 /2 cos ϕ3 = xd , y3 + l3 /2 sin ϕ3 = yd
q = (x1 , y1 , ϕ1 , x2 , y2 , ϕ2 , x3 , y3 , ϕ3 ) are coordinates.
For Cartesian coordinates
(x1 − xa )2 + (y1 − ya )2 = l1 2 , (x2 − x1 )2 + (y2 − y1 )2 = l2 2
(x3 − xb )2 + (y3 − yb )2 = l3 2 , (x2 − x1 )(x3 − xb ) + (y2 − y1 )(y3 − yb ) = l2 l3 cos ϕ
(x3 − x1 )/(x2 − x1 ) − (y3 − y1 )/(y2 − y1 ) = 0
q = (x1 , y1 , x2 , y2 , x3 , y3 ) are the coordinates.
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 C ONSTRAINTS WITH NATURAL
C OORDINATES
Distance between two points remain
k
constant: rij · rij = Lij 2
Link with three points: distance
between i, j and k remain constant.
i j
Link with 3 co-linear points:
rij · rij = Lij 2 and rij − krik = 0.
i k j
Link with three points and included
angle.
k
rij · rij = Lij 2
α

rik · rik = Lik 2

i j
rij · rik = Lij Lik cos(α )
Figure 53: Constraints
associated with rigid link
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 C ONSTRAINTS WITH NATURAL
C OORDINATES
Spherical joint – Two adjacent
links share a point.
Rotary joint constraints

rij · um − Lij cos(αi ) = 0

rij · un − Lij cos(αj ) = 0
rij · rij = L2ij , un · um = cos(γ )
un · un = um · um = 1

γ is the angle shown in figure.

Cylindrical joint constraint

rik × rij = 0
rij × uc = 0
Figure 54: Constraints
associated with joints
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 SLE & B OUNDARY C ONSTRAINTS
Two length constraint equations

rij · rij = L2ij , rkl · rkl = L2kl

Two co-linearity constraints

rij − λ1 rip = 0
rkl − λ2 rkp = 0

λ1 = a+b
a and λ2 = c+d
c .
Figure 55: Constraints
associated with SLE
Simplifying, SLE constraints are
b a c d
P + P − P − P =0
a+b i a+b j c +d l c +d k
Pm (m = i, j, k, l ) are the position vectors of 4 points.
Boundary constraints: If point P is fixed, its coordinates
are 0. . . . . . .
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 S YSTEM C ONSTRAINTS AND
C ONSTRAINT JACOBIAN

Rigid, joint and boundary constraints together can be

written as

fj (X1 , Y1 , Z1 , X2 , · · · , Yn , Zn ) = 0 for j = 1 to nc

nc is the total number of constraint equations and 3n is the

number of Cartesian coordinates of the system.
Derivative of all constraint equations in symbolic form

[J]δ X = 0

Homogeneous equation ⇒ Non-trivial δ X if dimension of

null space of [J] is at least one.
Dimension of null-space of [J] same as DOF of mechanism!
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 A LGORITHM TO OBTAIN DOF

Add the derivative of the constraint equations one at a

time in the following order
arising out of length constraints
arising out of joint constraints
At each step evaluate dimension of null-space of [J].
Nullity([J]) does not decrease when a constraint is added
→ Constraint is redundant.
Boundary constraints are added last: Nullity([J]) does not
decrease → Boundary constraint is redundant.
Final dimension of the null-space of [J] is the
mobility/degree of freedom of the system.

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 K INEMATIC A NALYSIS OF OVER -
CONSTRAINED M ECHANISMS

Figure 56: Constraint Jacobian analysis of three slider mechanism

Constraint Jacobian analysis correctly predicts DOF as 1.

Also determines redundant constraints which resulted in
M ̸= 1.

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 K INEMATIC A NALYSIS OF OVER -
CONSTRAINED M ECHANISMS

Joint d is seen to
be redundant

Link cd rotates
about d without
a joint at d !!

Figure 57: Constraint Jacobian analysis of Kempes -Burmester mechanism

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 K INEMATIC A NALYSIS OF SLE
BASED M ASTS

Spherical joint replaced with

Revolute joint shown above

SLE - 2 and 3 are redundant

DOF is 1 without SLE - 2
and SLE - 3

Figure 58: Constraint Jacobian analysis of triangular SLE mast with revolute
. . . . . .
joints
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 K INEMATIC A NALYSIS OF SLE
BASED M ASTS

Spherical joint replaced with

Revolute joint shown above

SLE - 3 and 4 are redundant

DOF is 1 without SLE - 3
and SLE - 4

Figure 59: Constraint Jacobian analysis of box SLE mast with revolute joints
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 K INEMATIC A NALYSIS OF SLE
BASED M ASTS

SLE – 6 and R joints on

FACE 5 & 6 are redundant
DOF is 1 without Cable
DOF is 0 with Cable
(modeled as rigid rod)

Figure 60: Constraint Jacobian analysis of hexagonal SLE mast with cables
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 S IMULATION OF SLE BASED M ASTS
Once redundancy identified → Can solve kinematics!
L = 30.0, Joint 2 moves horizontally and height decreases!

coordinate of joints 2 and 3 along Y axis

30
Joint 3
20 Joint 2
Joint 2
10

−10

−20

−30
0 5 10 15 20 25 30
coordinate of joints 2 and 3 along X axis
coordinate of joints 4/5/6 along Z axis

30

25

20

15

10

0
0 5 10 15 20 25 30
coordinate of joints 2 along X axis

Figure 61: Trajectory of joint coordinates for a triangular mast

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 S UMMARY

Over-constrained mechanisms do not give correct DOF

from Grübler-Kutzbach criterion.
Grübler-Kutzbach criterion does not take into account
geometry!
Null space dimension of the constraint Jacobian
Correctly determines degrees of freedom.
Can identify redundant links, joints and boundary
conditions.
Constraint Jacobian approach is local – Results valid at a
chosen configuration & does not account for singularities.
Global analysis possible for pantograph masts and simple
mechanisms.
Constraint Jacobian approach applied to SLE based masts
– Redundant SLEs can be identified!
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 S TATIC A NALYSIS OF SLE BASED
M ASTS

At the end of deployment, the actuator is locked &

Mechanism becomes a structure.
Various approaches to analyse structures (see Kwan &
Pellegrino (1994), Shan (1992) and Gantes et al. (1994))
Constraint Jacobian matrix extended for static analysis.
Stiffness matrix obtained from each type of constraints and
then assembled.
Rank of system stiffness matrix gives redundant links and
joints.
Deflection analysis from stiffness matrix.

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 L INKS SEGMENTS – A XIAL LOAD
For elastic members, [Sm ]δ L = δ T; Elongation is δ L for a
load δ T
The member stiffness matrix is
 AE 
1 1
l 0 0 0
 01 A2 E2 
 0 0 
[Sm ] =  l2
A3 E3 
 0 0 l3 0 
A4 E4
0 0 0 l4

where l , A and E are length, cross-sectional area and

elastic modulus, respectively.
External force is related to δ T by the Jacobian matrix:
[Jm ]T δ T = δ F
Hence, [Jm ]T [Sm ][Jm ]δ X = δ F.
Elastic stiffness matrix is [Km ] = [Jm ]T [Sm ][Jm ].
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 L INKS SEGMENTS – B ENDING
For rotations δ ϕ ′′ and member moments δ M′′

[Sn ]δ ϕ ′′ = δ M′′

The member stiffness matrix [Sn ] for the SLE is given by

 3E1 Iz 
l1 +l2 0 0 0
 3E1 Iy 
 0 0 0 
[Sn ] =  l1 +l2
3E2 Iz 
 0 0 l3 +l4 0 
3E2 Iy
0 0 0 l3 +l4

E is the Young’s modulus, Iz and Iy are moments of inertia.

Elastic stiffness matrix – [Kn ] = [Jn ]T [Sn ][Jn ]
Combined stiffness matrix

[Ks ] = [Km ] + [Kn ]

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 R ANK OF S TIFFNESS M ATRIX
Stiffness matrix is given by [Ks ] = [Js ]T [Ss ][Js ] where
 A 1 E1 
l1 0 0 0 0 0 0 0
 A2 E2 
 0 0 0 0 0 0 0 
 l2

 0 0 A3 E3
0 0 0 0 0 
 l3 
 0 0 0 A4 E4
0 0 0 0 
 l4 
[Ss ] =  3E1 Iz 
 0 0 0 0 0 0 0 
 l1 +l2 
 0 0 0 0 0
3E1 Iy
0 0 
 l1 +l2 
 3E2 Iz 
 0 0 0 0 0 0 l3 +l4 0 
3E2 Iy
0 0 0 0 0 0 0 l3 +l4

Rank of stiffness matrix [Ks ] same as rank of Jacobian

matrix

rank([Ks ]) = rank(([Js ][Ss ])T ([Js ][Ss ])) = rank([Js ][Ss ]) = rank([Js ])

Cable modeled as bar capable of taking tension only.

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 E XAMPLES
70
Transverse to mast
Along the mast

60

50

Stiffness (N/mm)
40

30

20

10

0
0 5 10 15 20 25 30 35 40 45
Angle of deployment (deg)

Figure 63: Axial and lateral stiffness during

Figure 62: Stacked SLE units deployment
Deployment from β = 0 to β = 45◦ , 0.5 N applied along X
and Y .
AE = 1.5 × 105 N, L = 1m, EIz = 9.6 × 107 Nmm2 .
Results match those by Kwan and Pellegrino (1994).
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 E XAMPLES

Figure 64: Nested hexagonal SLE mast with cables

X stiffness Y stiffness in Z stiffness
(N/mm) (N/mm) (N/mm)
Top or bottom cables 32.01 104.31 17.56
Only vertical cables 40.46 81.17 10.28
Top and bottom cables 65.44 175.42 27.25
All cables 114.23 326.64 39.26
Table 3: Variation of stiffness with addition of cables for assembled hexagonal
mast . . . . . .
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 S UMMARY

Over-constrained mechanisms and many deployable

structures do not give correct DOF using Grübler-Kutzbach
criterion.
Deployable structures are very important for space and
other applications.
A constraint Jacobian based approach is useful to
Determine correct DOF of over-constrained mechanisms
and deployable structures.
Determine redundant links and joints which make such
mechanisms violate Grübler-Kutzbach criterion.
Kinematic and static analysis of several pantograph based
structures performed.

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 O UTLINE

.
. .1 C ONTENTS
. ∗
. .2 L ECTURE 1
Chaos and Non-linear Dynamics in Robots

.
. .3 L ECTURE 2
Gough-Stewart Platform based Force-torque Sensors

. ∗
. .4 L ECTURE 3
Modeling and Analysis of Deployable Structures

.
. .5 A DDITIONAL M ATERIAL
References and Suggested Reading

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 A DDITIONAL M ATERIAL

References & Suggested Reading

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