112 On Biomimetics
0
0
010
0
0
0
001
0) ,
2
3
- ,
2
1
-( : 0), ,
2
3
2
1
-( : 0), 0, ,( : 0), 0, (0, :
coscossinsincos
cossincossinsin
cos sin
sin cos
cos sin
cossin
sincos
r r,r rr
1
1P
13 P P
1P
12 P P
1P
P11
1P
1P
1P
R
O P P P
And,
1,2,3)1P i1' 1P 1 1PPRP i1 (i (3)
Therefore, the coordinate of each element of P^1 P 1 i’ is:
)
2
3
2
,^1
2
3
2
,^1
2
( :^1
)
2
3
2
1
,
2
3
2
1
,
2
1
( :
,( : , )
sinsinrcosr' sincosrcosr sinr
sinsinrcosr' sincosrcosr sinr
sincosrsinsinrcosr'
13 P P P P P
1P
12 P P P P P
1P
P11 P P
1P
P
P
P
(4)
Next, P 1 i’, B 1 i’in OB 1 - XYZ, i.e. B^1 P 1 i’ and B^1 B 1 i’ can be defined as:
1,2,3)( 0, 0,
0) ,
2
,^3
2
( : 0), ,^1
2
,^3
2
( : 0), 0, ,( :^1
i'h''
r rr rr
T
i1 1
1P
i1
1B
13 B B
1B
12 B B
1B
B11
1B
PP
B B B
So, each element of B^1 P 1 i’ can be expressed as:
)
2
3
2
,^1
2
3
2
,^1
2
( :^1
)
2
3
2
,^1
2
3
2
,^1
2
( :^1
,( : - , )
sinsinrcosr' 'hsinrsincosrcosr
sinsinrcosr' sincosrcosr 'hsinr
'hsincosrsinsinrcosr'
13 P P P P P 1
1B
12 P P P P P 1
1B
P11 P P 1
1B
P
P
P
(7)
Accordingly, the length of each actuator can be defined:
i1 i''l 1,2,3)(
1B
i1
1B
i^ BP^ (8)
By Equation (5), (7) and (8), the equations describing the relation between the lengths of
pneumatic actuators, wires attached in Segment 1, Backbone 1, and the angles of two revolute
joints can be defined as follows:
(^) 1 2
22 2
l' rrrPBPcos- sin sin -rPcos sin h 1 '^
(1)
(2)
(5)
(6)