114 On Biomimetics
3.3 Jacobian matrix
In order to evaluate the motion characteristics of the Arm, it is necessary to develop the
Jacobian matrix of the Arm structure.
As li’(i=1,···,6), , , h 1 ’, , and h 2 ’ can be taken as functions of time t, noticing l
．
i’,^ x．
, y．
, z．
arefunctions of
．
, ．
, h．
1 ’, ．
,．
, h．
2 ’, the Equation (9), (10), (12) can be differentiated with respect to
t, to get the following equations. The A, B, C are the matrixes with element aij, bij (i,j=1,2,3), cij
(i=1,2,3, j=1,···, 6), respectively.
,bbb 'h'hbbbbbb'l'l'l,aaa 'h'haaaaaa'l'l'l(^33323122)
232212
131211
6
5
4
(^33323111)
232212
131211
3
2
1
A B (13)
'h
'h
'h
'h
cccccc
cccccc
cccccc
z
y
x
2
1
2
1
363534333231
262524232221
161514131211
C (14)
Next, the orientation of B^1 PRE’ in OB 1 - XYZ can be expressed by Equation (15), where the
element of the matrix P^1 R 1 P^2 R 2 is presented by rij (i,j=1,2,3).
(^)
333231
232221
131211
rrr
rrr
rrr
coscossinsincos
cossincossinsin
cos sin
coscossinsincos
cossincossinsin
cos sin
0 0
2
P2
1
P1 RR
(15)
By using Equation (15), the Euler angles (, , ) can be acquired (Yoshikawa, 1988).
(^) 2 2 2 )(0
1323 132 232 33 r,ratan,r,rratan,r,ratan 3132 (16)
Equation (17) can be derived by differentiating Equation (16) with regard to time t
(Yoshikawa, 1988):
)(0
2
31
2
32
31323132
2
33
2
32
2
31
2
23
2
1333
2
23
2
13
2
13
2
23
13231323
rr
rrrr
rrr ,
rrrrrr
rr ,
rrrr^33 (17)
．
,
．
,
．
are functions of
．
,
．
,
．
,
．
.Therefore, by using a matrix D with element dij (i=1, 2, 3,
j=1, 2, 4, 5), (
．
,
．
,
．
)T can be expressed as: