Biomimetic and Biologically Inspired Control - Biomimetics: Biologically Inspired Technologies - page 419

h

T , d 0 ¼ d 1 d n T which is parameterized by c 0 ¼ c 1 , and k 0 .

With respect to the unknown system of (16.24)-(16.26), the feedforward controller Q ( u )is
described by

½ c 2 c n d 2

d j 1 ( t )
dt

¼ F j 1 ( t ) þ gf d ( t )

(16 : 28)

d j 2 ( t )
dt

¼ F j 2 ( t ) þ gu ( t ) (16 : 29)

u ff ( t ) ¼ c T j 1 ( t ) þ d T j 2 ( t ) þ kf d ( t ) ¼ u T j ( t )

(16 : 30)

so as to generate the feedforward input u ff from the desired force signal f d ( t ). Here

T , T

u ¼ c T d T j ( t ) ¼ j 1 ( t ) T j 2 ( t ) T k f d ( t ) (16 : 31)

and the robot's control input u ( t ) is then given by

u ( t ) ¼ u ff þ u fb ¼ u T ( t ) j ( t ) þ K ( s ) e ( t ) (16 : 32)

Next, introducing a new state equation for the vector j e ( t ) with respect to the control error input
e ( t ) ¼ f d ( t ) f ( t )as

d j e ( t )
dt

¼ F j e ( t ) þ ge ( t ) (16 : 33)

and defining

T

j ( t ): ¼ j 1 ( t ) T j e ( t ) T j 2 ( t ) T f d ( t ) e ( t ) (16 : 34)

we can express the total control input u ( t )as

u ( t ) ¼ u 0 j ( t ) (16 : 35)

which is linearly parameterized by u 0 , see Muramatsu and Watanabe (2004) for the details of the
derivation.
Finally, to derive an adaptive rule, let us define

u ( t ): ¼ u T ( t )

j ( t ) (16 : 36)

by replacing u 0 in Equation (16.35) with u ( t ), and defining an error signal « ( t )as

g T j ( t ) ¼c ( t ) T

« ( t ): ¼ u ( t ) u ( t ) ¼ u 0 u ( t ) f j ( t ) (16 : 37)

where

c ( t ): ¼ u ( t ) u 0 (16 : 38)