�
114
Biologically Inspired Robotics
where β is the flexural rigidity of the spring under compression, and it is
inversely proportional to the number of coils per unit length of the spring
(Timoshenko 1936); hence,
β =
H
h
β
where H 0 is the original length of the spring, and β 0 can be calculated based
on physical parameters of the compressive spring.
The initial conditions at the two ends of the spring are
( )
p
= 0, p 0 = 0;( ) h= = p b ,( ) h = θ
h=0
( )
′ h=
p h 0
p′ h= 0
p
(6.5)
By neglecting the corresponding vertical displacement h b of the spring under
small bending, as shown in Figure 6.4b, the spring length under compression
is calculated as
F 2 = K H
0 )
(6.6)
( 0 − h
where K is the spring stiffness and L 0 is the initial length of the spring.
Based on Equations (6.1)–(6.6), for a configured cable-driven, spring-based
parallel mechanism as shown in Figures 6.4a and 6.4b, both the inverse kinematics and statics can be solved analytically.
6.3.2 Cable and Housing
The drive cable used in this application is braided extra-strength polyester
twine. Because the length of the drive cable is longer than 2 m, the extension
of the cable has to be considered. While a cable is pulled in a curved housing,
friction force is unavoidable. Here it is assumed that the shape of the cable
and housing is unchanged for each robot motion control, and we can depict
a cable and housing with n turns as shown in Figure 6.5.
The coefficient of static friction and kinetic friction between the cable and
the inner wall of its hose is taken as the same μ. Each turn in the curving
path can define a plane and we define the hose bends counterclockwise away
from the velocity of the cable as its turn angle Λ j . As in this application, the
∈[ , ]
π π j
turn angles are defined as Λ j 0 2 \ , = 1, , n . We use F j and l j to denote
the cable force and hose length of each segment. Note that the j here is in
[ ,
1 Q − 1] . The F 0 and F n are the input driving force and output driving force of
the cable-hose drive mechanism. The l 0 denotes the cable length measured
114
Biologically Inspired Robotics
where β is the flexural rigidity of the spring under compression, and it is
inversely proportional to the number of coils per unit length of the spring
(Timoshenko 1936); hence,
β =
H
h
β
where H 0 is the original length of the spring, and β 0 can be calculated based
on physical parameters of the compressive spring.
The initial conditions at the two ends of the spring are
( )
p
= 0, p 0 = 0;( ) h= = p b ,( ) h = θ
h=0
( )
′ h=
p h 0
p′ h= 0
p
(6.5)
By neglecting the corresponding vertical displacement h b of the spring under
small bending, as shown in Figure 6.4b, the spring length under compression
is calculated as
F 2 = K H
0 )
(6.6)
( 0 − h
where K is the spring stiffness and L 0 is the initial length of the spring.
Based on Equations (6.1)–(6.6), for a configured cable-driven, spring-based
parallel mechanism as shown in Figures 6.4a and 6.4b, both the inverse kinematics and statics can be solved analytically.
6.3.2 Cable and Housing
The drive cable used in this application is braided extra-strength polyester
twine. Because the length of the drive cable is longer than 2 m, the extension
of the cable has to be considered. While a cable is pulled in a curved housing,
friction force is unavoidable. Here it is assumed that the shape of the cable
and housing is unchanged for each robot motion control, and we can depict
a cable and housing with n turns as shown in Figure 6.5.
The coefficient of static friction and kinetic friction between the cable and
the inner wall of its hose is taken as the same μ. Each turn in the curving
path can define a plane and we define the hose bends counterclockwise away
from the velocity of the cable as its turn angle Λ j . As in this application, the
∈[ , ]
π π j
turn angles are defined as Λ j 0 2 \ , = 1, , n . We use F j and l j to denote
the cable force and hose length of each segment. Note that the j here is in
[ ,
1 Q − 1] . The F 0 and F n are the input driving force and output driving force of
the cable-hose drive mechanism. The l 0 denotes the cable length measured