On Biomimetics by Lilyana Pramatarova

To Design a Small Pneumatic Actuator Driven Parallel
Link Mechanism forShoulder Prostheses for Daily Living Use 113



2 2

2

2

2

1

11 1 3 3

' cos sin sin cos

22 2 2 2

13

cos sin sin '

22

PB P P B

PP

rr r r r

rrh

l 

 



   











2 2

2

3

2

1

11 1 3 3

' cos sin sin cos

22 2 2 2

13 cos sin sin '

22

PB P P B

PP

rr r r r

rrh

l 

 



   









(9)

Equation (9) is a simultaneous equation with three unknown variables, , , h 1 ’. In the case
that the lengths l 1 ’, l 2 ’, l 3 ’ are given, it is possible to calculate , , h 1 ’ by using the Newton
method (Ku, 1999; Merlet, 1993; Press et al., 1992). Similarly, for the Segment 2 (i.e. in OP 1 -
xP 1 yP 1 zP 1 ), the following equation can be derived. In the case that the length l 4 ’, l 5 ’, l 6 ’, are
given, it is possible to calculate ,, h 2 ’.

     





2

P P 2

2

P P B

2

P B

2

6

2

P P 2

2

P P B

2

P B

5 2

2 2 2 2

4

'hsinrsincosr

l rcosr' rcosrsinsinr

sincosr 'hsinr

l rcosr' rcosrsinsinr

l' P B P P 2 'hsincosrsinsinrrcosr

















  

















   









 

















  

















   









 

    

 

  

 

  

  

2

3

2

1

2

3

2

3

2

1

2

1

2

1

2

3

2

1

2

3

2

3

2

1

2

1

2

1

(10)

The position of the Rod end PRE’ in OP 2 -xP 2 yP 2 zP 2 , i.e. P^2 PRE’ can be defined as:



T

RE R

2PP  00 l , ,'

(11)

Let x, y, z be the coordinate of B^1 PRE’, then end position of the Rod, x, y, z in OB 1 - XYZ can be
described as:





















































































 















 



















 

coscossinsincos 'h'hl

cossincossinsin

cos sin

coscossinsincos

cossincossinsin

cos sin

z

y

x

zyx '''

R 2 1

1P

1B

2P

1P

RE

2P

2

2P

1

T 1P

0

0

0

0

0

(^000)


 


 
OOPRR
(12)
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