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406 Biomimetics: Biologically Inspired Technologies

Joint space Work space
J

R (J)
N (J)
O

Figure 16.3 Nonlinear and redundant mapping.

T
T
½
½
where u ¼ u 1 , u 2 , ... , u m Š , x ¼ x 1 , x 2 , ... , x n Š , m > n, and
dx ¼ J du (16:2)
where J is a n m matrix. As shown in Figure 16.3, the range and null spaces of J are

n
_
_
_
_
m
u
u
u
R(J) ¼ ˙ x 2 R : ˙ x ¼ J(u)u for 8u 2 R m , N(J) ¼ u 2 R : J(u)u ¼ 0 (16:3)
u
and dim R(J) þ dim N(J) ¼ m.
Assuming the Jacobian J is known, we summarize five typical inverse kinematics approaches:
1. By using the transpose of the matrix J, we calculate
_
d
T
u u ¼ J (x x) (16:4)
d
where x is the desired end-effector position (Chiacchio et al., 1991).
þ
2. For the case when rank (J) ¼ n, we use J , the pseudo-inverse of J, to obtain
þ
_
þ
_
þ
u
u u ¼ J ˙ x or u ¼ J ˙ x þ (I J J)h (16:5)
þ
þ
where JJ ¼ I and vector (I J J)h 2 N(J). When rank(J(u)) < n, then J is singular, the joint u is
the singular configuration (Klein and Huang, 1983).
3. By specifying additional task constraints to extend J as a full rank square matrix J e , we have
(Baillieul, 1985)
_
1
u u ¼ J x ˙ (16:6)
e
4. The regularization method to minimize the cost function k dx Jdu kþl k du k.
5. Based on compliance control, by using the relations:
T
t ¼ K u du , F ¼ K x dx; and t ¼ J F; dx ¼ J du (16:7)
T
1 T
T
then we have K u ¼ J K x J; and therefore du ¼ (J K x J) J K x dx.
In approach 3, the specification of the additional task constraints may be closely related to the
Bernstein’s concept of synergy. However, from the biological point of view, the main problem
inherent in all the above approaches is the assumption that the system’s Jacobian is known
a priori, which seems unlikely in biological system. In addition, the cost functions and task