Highly Maneuverable Biorobotic Underwater Vehicles 11.2 Theoretical Foundation of Animal-Inspired Hydrodynamics and Control 287
Part B | 11.2
where
.t/ D .t/ C Â.t/ ;
D 0 sin.2ft/ ;
Â.t/ D Â 0 sin.!t C / C Â Bias ;
! s D 2!!
StU
A
;
A D 2 o R avg :
F
0
x is the fluctuation in force, and F
0
xo is the amplitude
of F
0
x ; G.D0:02/ and T.D0:5/ are constants (determined by trial) that affect the shape and the blur of
the oscillatory solution (the limit cycle), respectively.
In other words, the flapping (rolling and pitching) fin is
a nonlinear oscillator and has a limit cycle oscillation
(LCO) [11.15]. (The roll, pitch, and twist oscillation of
the hinged fin may be seen to act as the I exti .t/ term
(see olivo-cerebellar control Sect. 11.2.2 below) to the
nonlinear oscillator arising out of the natural vortex
shedding of the nonflapping fin.)
Flapping Fin Parameters for Cruising
and Maneuvering
Table 11.1 summarizes the fin kinematics that a controller (in the BAUV, SPLINE, and Razor vehicles,
Sect. 11.3) uses for optimization of thrust and efficiency
during cruising, hovering, and yawing. The total forward velocity is much higher than the wing velocity
(which is cross-stream) during cruising, and these velocities are nearly equal during hovering (the forward
speed is zero during hovering). Biasing the fin pitch at
half the required yawing angle is the key fin setting required for yawing. For optimization of efficiency, the
controller makes use of the observed direct coupling
between Strouhal number St, whose definition includes
roll oscillation amplitude 0 and pitch amplitude  0 ;
a higher pitch amplitude of about 40
ı is used when St
is near 0:40, and a lower pitch amplitude of 20
ı is used
when St is near 0:25 (Fig. 11.2f) [11.15]. Also, note that
in a single flapping fin, thrust is a function of the square
of the flapping fin frequency f for given pitch amplitude
and forward speed [11.15, 19].
The flapping fin mechanism can be understood by
observing that the forces produced are describable by
van der Pol (11.3) [11.15] or Stuart Landau nonlinear
oscillators [11.28]. Using the equation for the latter oscillator, it has been shown theoretically that a coupling
exists between the fin and the wake [11.28]. Table 11.1
can be viewed as tuning of the oscillator under various performance requirements. The rightmost column
in Table 11.1 summarizes the wall shear stress on the
fin for the fin kinematics given in the other columns.
Characteristic patterns can be observed:
1. The point of stagnation oscillates, shifting from one
pressure side of the fin to another pressure side as
the pressure and suction roles of the fin surfaces alternate.
2. Downstream of the stagnation line and on the suction side, a generally spanwise low-wall-shear region develops, indicating the presence of a LEV.
3. Chordwise low-wall-shear bands appear in the suction side in hovering and yawing fins, indicating
a complex and less efficient surface flow.
11.2.2 Animal-Like Motion Control Laws
and the Principles
of Integrated Design
We describe the unmanned underwater vehicle (UUV)
design to be integrated when the actuators (fins), controllers, and sensors have similar laws of dynamics. In
the case of animal-inspired swimming, these will be
equations of nonlinear oscillators.
We assume that the principles of integrated design
are derived from the olivo-cerebellar motion control
laws [11.24–27, 33], as evidenced by the dynamics of
the inferior-olive (IO) neurons that are responsible for
motion and balance in all mammals, from rats to human
beings [11.24, 25]. We first consider the motion control
laws and then how the controller, actuator, and sensors
are integrated in a common framework of dynamic systems mathematics.
The motion control laws are self-regulating. In such
control, a sensor is not needed for error estimation with
respect to a reference because a conventional closed
control loop is not present. In other words, these controllers reject disturbances, which make them robust.
(See [11.3, 28] for discussion of self-regulation.) Owing to this robustness, the controllers have remained
unchanged in many animals.
The olivo-cerebellar dynamics of an IO neuron (i D
1; 2; : : :) can be represented by two coupled nonlinear
oscillators (u i ; v i ) and (z i ; w i ) as given below [11.26,
27],
2
6
6
6
4
P
u i
P
v i
P
z i
P
w i
3
7
7
7
5
D
2
6
6
6
4
k." Na /
1
.p iu .u i / v i /
k.u i z i C I Ca I Na /
p iz .z i / w i
" Ca .z i I Ca /
3
7
7
7
5
C
2
6
6
6
4
0
0
0
" Ca
3
7
7
7
5
I exti .t/ ;
(11.2)
where the variables (z i ; w i ) are associated with the
subthreshold oscillations and low-threshold (Ca-dependent) spiking, and (u i ; v i ) describe the higher threshold
(Na
C -dependent) spiking (this is similar to FitzHugh–
Nagumo equation). The constant parameters " Ca and
Part B | 11.2
where
.t/ D .t/ C Â.t/ ;
D 0 sin.2ft/ ;
Â.t/ D Â 0 sin.!t C / C Â Bias ;
! s D 2!!
StU
A
;
A D 2 o R avg :
F
0
x is the fluctuation in force, and F
0
xo is the amplitude
of F
0
x ; G.D0:02/ and T.D0:5/ are constants (determined by trial) that affect the shape and the blur of
the oscillatory solution (the limit cycle), respectively.
In other words, the flapping (rolling and pitching) fin is
a nonlinear oscillator and has a limit cycle oscillation
(LCO) [11.15]. (The roll, pitch, and twist oscillation of
the hinged fin may be seen to act as the I exti .t/ term
(see olivo-cerebellar control Sect. 11.2.2 below) to the
nonlinear oscillator arising out of the natural vortex
shedding of the nonflapping fin.)
Flapping Fin Parameters for Cruising
and Maneuvering
Table 11.1 summarizes the fin kinematics that a controller (in the BAUV, SPLINE, and Razor vehicles,
Sect. 11.3) uses for optimization of thrust and efficiency
during cruising, hovering, and yawing. The total forward velocity is much higher than the wing velocity
(which is cross-stream) during cruising, and these velocities are nearly equal during hovering (the forward
speed is zero during hovering). Biasing the fin pitch at
half the required yawing angle is the key fin setting required for yawing. For optimization of efficiency, the
controller makes use of the observed direct coupling
between Strouhal number St, whose definition includes
roll oscillation amplitude 0 and pitch amplitude  0 ;
a higher pitch amplitude of about 40
ı is used when St
is near 0:40, and a lower pitch amplitude of 20
ı is used
when St is near 0:25 (Fig. 11.2f) [11.15]. Also, note that
in a single flapping fin, thrust is a function of the square
of the flapping fin frequency f for given pitch amplitude
and forward speed [11.15, 19].
The flapping fin mechanism can be understood by
observing that the forces produced are describable by
van der Pol (11.3) [11.15] or Stuart Landau nonlinear
oscillators [11.28]. Using the equation for the latter oscillator, it has been shown theoretically that a coupling
exists between the fin and the wake [11.28]. Table 11.1
can be viewed as tuning of the oscillator under various performance requirements. The rightmost column
in Table 11.1 summarizes the wall shear stress on the
fin for the fin kinematics given in the other columns.
Characteristic patterns can be observed:
1. The point of stagnation oscillates, shifting from one
pressure side of the fin to another pressure side as
the pressure and suction roles of the fin surfaces alternate.
2. Downstream of the stagnation line and on the suction side, a generally spanwise low-wall-shear region develops, indicating the presence of a LEV.
3. Chordwise low-wall-shear bands appear in the suction side in hovering and yawing fins, indicating
a complex and less efficient surface flow.
11.2.2 Animal-Like Motion Control Laws
and the Principles
of Integrated Design
We describe the unmanned underwater vehicle (UUV)
design to be integrated when the actuators (fins), controllers, and sensors have similar laws of dynamics. In
the case of animal-inspired swimming, these will be
equations of nonlinear oscillators.
We assume that the principles of integrated design
are derived from the olivo-cerebellar motion control
laws [11.24–27, 33], as evidenced by the dynamics of
the inferior-olive (IO) neurons that are responsible for
motion and balance in all mammals, from rats to human
beings [11.24, 25]. We first consider the motion control
laws and then how the controller, actuator, and sensors
are integrated in a common framework of dynamic systems mathematics.
The motion control laws are self-regulating. In such
control, a sensor is not needed for error estimation with
respect to a reference because a conventional closed
control loop is not present. In other words, these controllers reject disturbances, which make them robust.
(See [11.3, 28] for discussion of self-regulation.) Owing to this robustness, the controllers have remained
unchanged in many animals.
The olivo-cerebellar dynamics of an IO neuron (i D
1; 2; : : :) can be represented by two coupled nonlinear
oscillators (u i ; v i ) and (z i ; w i ) as given below [11.26,
27],
2
6
6
6
4
P
u i
P
v i
P
z i
P
w i
3
7
7
7
5
D
2
6
6
6
4
k." Na /
1
.p iu .u i / v i /
k.u i z i C I Ca I Na /
p iz .z i / w i
" Ca .z i I Ca /
3
7
7
7
5
C
2
6
6
6
4
0
0
0
" Ca
3
7
7
7
5
I exti .t/ ;
(11.2)
where the variables (z i ; w i ) are associated with the
subthreshold oscillations and low-threshold (Ca-dependent) spiking, and (u i ; v i ) describe the higher threshold
(Na
C -dependent) spiking (this is similar to FitzHugh–
Nagumo equation). The constant parameters " Ca and