Grey Box Modeling and Dynamic of Human Forearm

8/18/2019 Grey Box Modeling and Dynamic of Human Forearm

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Grey Box Modeling and dynamic of Human Forearm

Introduction

The study of dynamics of human arm has been developed by many researchers. Within

the available research most of them use robot arm as a system that resembles the actual

system of human arm. This can be done by assuming that for both robot and human arm

mechanical ob!ectives of movement and manipulation are identical and by applying the

same la" of physics. Ho"ever this system cannot be used to represent the #real$ human

arm system as its model involves many constraints that cannot be applied to human

model.

Many of analyses are carried out for different activities of human arm. Most of the

current research uses a t"o lin% mechanical structure "ith parameters in the hori&ontal

motion. This can simplify the analysis since the gravity in their models can be neglected.

T.'ode% et.al()**+, studied shoulder elbo" and "rist dynamics and static tor-ue in the

elbo" flexion extension movement. /no et.al (0121, study the tra!ectory planning and

control in voluntary human arm movement. They proposed a mathematical model by

ta%ing account for formation of arm tra!ectory on the basis of optimi&ation theory. They

computed the optimal tra!ectory of human arm by "hat they called #minimum tor-ue

change model$. Ho"ever it is still difficult to define the cost function and ob!ectives of the model.

In many physical systems include human movement system there are some parameters

"hich could not be measured directly. These parameters have to be estimated by using

measurement data. 3ome measurement devices from the conventional one "hich using

electromyography (4MG, video based and sensors. Ho"ever the measurement data can

be corrupted by noise either from the environment or the device itself.

Modeling

4ssentially there are t"o "ays of obtaining mathematical model namely physical model

and system identification. 5hysical model is analytic approach "here basic la"s from

physics are used to describe the dynamic behaviour of the system hence this techni-ue

re-uires physical data of the system. 3ystem identification on the other hand does not


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re-uire any physical data. 3ystem identification is a "ay of building mathematical

models based on operational data of the system.

Grey Box is the common method used in 3ystem Identification if the physics of the

system are %no"n and can be represented using ordinary differential or differencee-uations (674s, "ith un%no"n parameters either it is linear or nonlinear model. Grey-

box model 674s specify the mathematical structure of the model explicitly including

couplings bet"een parameters and %no"n parameter values. Grey box modeling is useful

"hen the relationships bet"een variables constraints on model behavior or explicit

e-uations representing system dynamics are %no"n.

The concept of Grey Box modelling has not been applied much in the study of human

arm movement. Burdet et.al employed this concept to examine the concept of stability in

human arm movement. Firstly they derived a computational model of human arm

movements "hich interact "ith the environment by using a !oint space model. They

computed the !oint tor-ue produced by muscles according to an inverse dynamic model

of the planned movements and considers motor output variability and tor-ue dependent

impedance. They then compared the computational model "ith the available data on

human arm movement from previous research.

In our system human forearm is modeled based on the concept of spherical pendulum

"ith t"o degree of freedom. The result geometry model is then determined as model

structure of the system "ith un%no"n parameter.

5hysical Model

The part of human arm "hich "ill be considered in my research is only forearm "ith the

elbo" lin%. To simplify the study I use the concept spherical pendulum "ith t"o degree

of freedom to represent human forearm "ith elbo" as single !oint. Wrist is fixed and

considered as an integral part of forearm. For the mathematical modelling all the lin%s

are considered rigid and friction is neglected at !oints.

For the present "or% 8agrangian 4uler Formulation is used in this "or% "hich is given

by 9


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iii q

Lq L

dt d

τ =∂∂−

∂∂

:::::::::::::..(0,

Where 8agrange function is defined as 9

V T L −= ::::::::::::::::::(),

T 9 Total 'inetic energy

; 9 Total 5otential energy

iq are the generali&ed coordinates "hich in my model are the position of the arm

denoted as ,( φ θ . The length of arm is denoted by 8. 2l θ φGenerali&ed coordinates -> ( φθ,

sin cos( , ) sin sin

cos

l r l

l

φ θ θ φ φ θ

θ

=

To derive 8agrangian "e must first determine %inetic and potential energy of pendulum 9T > %inetic energy;> potential energy

First derivative of r 9cos cos sin sin

sin cos cos sin

sin

x

y

z

φ θ φ θ θ φ

φ θ φ θ θ φ

φ φ

= −= −=−

& &&

& &&

&&


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cos cos sin sin

sin cos cos sin

sin

l l

r l l

l

φ θ φ θ θ φ

φ θ φ θ θ φ

φ φ

− ÷= − ÷ ÷−

& &

& &&

&

The total %inetic energy of the system becomes 92 21 1 1

2 2 21 2T T r m r J J θ φ = + +& && &

The first term 92 2 2 21 1

2 2

2 2 2 212

2 2 2 2 2 2 212

2 2 2 2 2 2

( )

(( cos cos sin sin ) ( sin cos cos sin ) ( sin ) )

( cos cos sin sin 2 sin cos sin cos

sin cos cos sin 2 sin cos sin cos

T r m r m l x y z

m l

m l

φ θ φ θ θ φ φ θ φ θ θ φ φ φ

φ θ φ θ θ φ φθ θ θ φ φ

φ θ φ θ θ φ φθ θ θ φ φ

= + += − + + + −= + −

+ + +

& & & &

& & & & &

& & &&

& & &&

2 2

2 2 2 2 2 2 2 2 2 2 212

2 2 2 2 2 2 212

2 2 2 212

sin )

( cos (sin cos ) sin (sin cos ) sin )

( cos sin sin )

( sin )

m l

m l

m l

φ φ

φ φ θ θ θ φ θ θ φ φ

φ φ θ φ φ φ

φ θ φ

+= + + + += + += +

&

& & &

& & &

& &

Then the total %inetic energy 92 2 2 2 2 21 1 1

2 2 21 2( sin )T ml J J φ θ φ θ φ = + + +& & & &

The potential energy 9cosV m g l φ =

To bring the viscous damping into account no" the virtual "or% "ill be implemented.

Ta%e the "or% from the virtual displacement 9a,. damping coefficient.

Then 8agrangian e-uation 9


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2 2 2 2 2 21 1 12 2 21 2( sin ) cos

L T V

L mL J J mgLφ θ φ θ φ φ

= −= + + + −& & & &

7eriving 8agrange?s e-uation of motion 9

i i

d L LQ

dt q q ∂ ∂− = ÷∂ ∂ &

( )

2 2 2 21 12 2 1

2 21

2 21

2 2 2

1

( sin )

sin

( sin sin 2 )

( sin ) sin2

0

d L d mL J

dt dt d

mL J dt mL J

mL J mL

L

θ φ θ θ θ

θ φ θ

θ φ θ φ φ θ

φ θ θ φ φ

θ

∂ ∂ = + ÷ ÷∂ ∂

= +

= + +

= + +∂ =∂

& && &

& &

&& && &&

&& &&

( )

( )

( )

2 2 21 12 2 2

22

22

22

2 2 212

2 2

( )

sin cos

cos sin

d L d mL J

dt dt

d mL J

dt mL J

mL J

LmL mgL

mL mgL

φ φ φ φ

φ φ

φ φ

φ

θ φ φ φ φ

θ φ φ

∂ ∂= + ÷ ÷∂ ∂

= +

= += +

∂ ∂= −∂ ∂= +

& && &

& &

&& &&

&&

&

&

Then the e-uation of motion becomes 9

2 2 21

222

0( sin ) sin 2 0sincos( )

mL J mL c mgLmLmL J c

φ θ θ φ φ θ τ

φ θ φ φ φ

+ + − + + + = ÷ ÷ ÷ ÷ ÷ ÷ −−+ −

&& && &

&&& &

22 21

2 22

sin2 0sin 0sin 00 cos

c mLmL J mgLmL J mL c

θ θ φ θ τ φ φ φ θ φ φ

+ + + = ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ −+ −

&& & &

&& & &


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3imulation result 9

For τ > sin ) π f t (f > ) H&,m> *.0 %g

g > *.12 m@s)

l > )

A0>A)> 0

0 2 4 6 8 10 12 14 16 18 2010

10.01

10.02

10.03

10.04

10.05

10.06

10.07

10.08

10.09

time

p e n d u l u m

p o s i t i o n ( t h e t a )


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0 2 4 6 8 10 12 14 16 18 20-7

-6

-5

-4

-3

-2

-1

0x 10

-3

time

p e n d u l u m

p o s i t i o n ( p h i )

0 2 4 6 8 10 12 14 16 18 20-0 .1

-0.05

0

0.05

0.1

0.15

time

t h e t a

d o t


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0 2 4 6 8 10 12 14 16 18 20-14

-12

-10

-8

-6

-4

-2

0

2x 10

-4

time

p h i d o t


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(5arameter derived from agarsheth et.al .)**2. $Modeling and 7ynamics of Human

) H&,m> 0.0C0 %g

g > *.12 m@s)

l > 0*.D0 cm

A0>A)> 0

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

time

p e n d u l u m



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0 2 4 6 8 10 12 14 16 18 20-2

-1 .5

-1

-0 .5

0

0.5

1

1.5

2

time

p e n d u l u m


0 2 4 6 8 10 12 14 16 18 20-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time

v e l o c i t y I ( t h e t a

d o t )


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0 2 4 6 8 10 12 14 16 18 20-0 .5

-0 .4

-0 .3

-0 .2

-0 .1

0

0.1

0.2

0.3

0.4

0.5

time

v e l o c i t y I I ( p h i d o t )


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For τ > sin ) π f t (f > ) H&,m> 0.0C0 %g

g > *.12 m@s)

l > 0*.D0 cm

A0>A)> 0

Initial state 9

00.785

0

rad rad

θ φ

θ φ

=== =& &

0 2 4 6 8 10 12 14 16 18 20-4

-3

-2

-1

0

1

2x 10

-4

time

p e n d u l u m



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0 2 4 6 8 10 12 14 16 18 20-0 .8

-0 .6

-0 .4

-0 .2

0

0.2

0.4

0.6

0.8

time

p e n d u l u m


0 2 4 6 8 10 12 14 16 18 20-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10

-4

time

v e l o c i t y I ( t h e t a

d o t )


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0 2 4 6 8 10 12 14 16 18 20-0.25

-0 .2

-0.15

-0 .1

-0.05

0

0.05

0.1

0.15

0.2

0.25

time

v e l o c i t y I I ( p h i d o t )


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Grey Box Modeling and Dynamic of Human Forearm

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