Part B | 11.2
288 Part B Autonomous Ocean Vehicles, Subsystems and Control
" Na control the oscillation time scale; I Ca and I Na drive
the depolarization levels; and k sets a relative time
scale between the uv- and zw-subsystems. The nonlinear functions are
p iu .u i / D u i .u i a/.1 u i / ;
p iz .u i / D z i .z i a/.1 z i / :
The function I exti .t/ is the extracellular stimulus, which
can be used for the purpose of control, such as phase
synchronization. Note that in a UUV, we do not need to
be concerned with the ionic interpretations of the variables and constants; we are only concerned with the
generic nonlinear dynamics [11.28].
The (u i ; v i ) oscillator can generate spikes (analogous to neural action potential), but this feature has
not been used in the context of UUV maneuvering.
The (z i ; w i ) oscillator, however, has been used successfully to synchronize the phase of a six-finned vehicle [11.33].
The salient features of this approach to control compared with conventional engineering controllers (i. e.,
proportional, integral, derivative (PID) controllers) are
that no sensors are required, no reference is required,
the controller has built-in disturbance rejection ability,
and, in principle, the controller has no size or time scale
limitation.
The lower oscillator in (11.2) can then be written as
R
z i C F.z i /P z i C kz i C "I D 0 ;
(11.3)
where F is a cubic polynomial function and k is a constant if I exti .t/ D 0. Equation (11.3) resembles Lienard’s
oscillator (in contrast, the function F is a well-defined
quadratic in the more familiar van der Pol oscillator [11.34, p. 13]). The oscillator exhibits a closed
orbit i in the state space .z i P z i /, that is, (z i w i ), which
is also known as LCO, with the constant parameters determining the form of i .
Below, we consider the notions of integrated design
and nonlinear sensing because they are theoretically attractive.
Principles of Integrated Design
Swimming animals offer clue to homing, autonomy,
and station keeping. The mechanisms are at a theoretical stage of understanding and we touch upon them
notionally to spur future work. The principles of integration of the controller, actuator, and sensor are
based on the hypothesis of persistent synchrony with
the environment which after all should enhance mission
effectiveness (pursuit of food, mating, and evasion in
the case of animals). According to this hypothesis, not
only the controller, but also the actuator and the sensor must follow similar dynamics as given essentially
in (11.1)–(11.3). We have shown that the actuator does
indeed follow laws similar to olivo-cerebellar dynamics [11.15]. It has been shown that it is beneficial for
a UUV if olivo-cerebellar dynamics is a common principle of the three elements of the platform. In [11.28],
it is shown that an integrated design accelerates homing
if we build-in 1015% of handedness.
Nonlinear Sensing as a Theoretical
Requirement of Autonomy
Nonlinear acoustic sensing is another piece of the autonomy puzzle which however is the least developed
of the three components, namely sensor, actuator, and
controller. Most hydrophones are based on linear vibration of transducers which require lines of them for
improved aperture. In [11.35], an alternative theoretical foundation is given based on nonlinear LCO of
tiny three-dimensional transducer elements. (These are
similar to cilium sensors – widely used by animals.)
This transducer has a built-in disturbance rejection
(of vibration, e.g.) property. Furthermore, by the use
of the property of metachronism in a cluster of such
nonlinearly oscillating transducers, acoustic sensing
methodologies can be miniaturized and aperture can be
increased.
The above rationale for the integration of actuators, controllers, and sensors, together with the need to
develop nonlinear oscillatory versions of these three individual systems, gives a mathematically founded basis
of autonomy.
288 Part B Autonomous Ocean Vehicles, Subsystems and Control
" Na control the oscillation time scale; I Ca and I Na drive
the depolarization levels; and k sets a relative time
scale between the uv- and zw-subsystems. The nonlinear functions are
p iu .u i / D u i .u i a/.1 u i / ;
p iz .u i / D z i .z i a/.1 z i / :
The function I exti .t/ is the extracellular stimulus, which
can be used for the purpose of control, such as phase
synchronization. Note that in a UUV, we do not need to
be concerned with the ionic interpretations of the variables and constants; we are only concerned with the
generic nonlinear dynamics [11.28].
The (u i ; v i ) oscillator can generate spikes (analogous to neural action potential), but this feature has
not been used in the context of UUV maneuvering.
The (z i ; w i ) oscillator, however, has been used successfully to synchronize the phase of a six-finned vehicle [11.33].
The salient features of this approach to control compared with conventional engineering controllers (i. e.,
proportional, integral, derivative (PID) controllers) are
that no sensors are required, no reference is required,
the controller has built-in disturbance rejection ability,
and, in principle, the controller has no size or time scale
limitation.
The lower oscillator in (11.2) can then be written as
R
z i C F.z i /P z i C kz i C "I D 0 ;
(11.3)
where F is a cubic polynomial function and k is a constant if I exti .t/ D 0. Equation (11.3) resembles Lienard’s
oscillator (in contrast, the function F is a well-defined
quadratic in the more familiar van der Pol oscillator [11.34, p. 13]). The oscillator exhibits a closed
orbit i in the state space .z i P z i /, that is, (z i w i ), which
is also known as LCO, with the constant parameters determining the form of i .
Below, we consider the notions of integrated design
and nonlinear sensing because they are theoretically attractive.
Principles of Integrated Design
Swimming animals offer clue to homing, autonomy,
and station keeping. The mechanisms are at a theoretical stage of understanding and we touch upon them
notionally to spur future work. The principles of integration of the controller, actuator, and sensor are
based on the hypothesis of persistent synchrony with
the environment which after all should enhance mission
effectiveness (pursuit of food, mating, and evasion in
the case of animals). According to this hypothesis, not
only the controller, but also the actuator and the sensor must follow similar dynamics as given essentially
in (11.1)–(11.3). We have shown that the actuator does
indeed follow laws similar to olivo-cerebellar dynamics [11.15]. It has been shown that it is beneficial for
a UUV if olivo-cerebellar dynamics is a common principle of the three elements of the platform. In [11.28],
it is shown that an integrated design accelerates homing
if we build-in 1015% of handedness.
Nonlinear Sensing as a Theoretical
Requirement of Autonomy
Nonlinear acoustic sensing is another piece of the autonomy puzzle which however is the least developed
of the three components, namely sensor, actuator, and
controller. Most hydrophones are based on linear vibration of transducers which require lines of them for
improved aperture. In [11.35], an alternative theoretical foundation is given based on nonlinear LCO of
tiny three-dimensional transducer elements. (These are
similar to cilium sensors – widely used by animals.)
This transducer has a built-in disturbance rejection
(of vibration, e.g.) property. Furthermore, by the use
of the property of metachronism in a cluster of such
nonlinearly oscillating transducers, acoustic sensing
methodologies can be miniaturized and aperture can be
increased.
The above rationale for the integration of actuators, controllers, and sensors, together with the need to
develop nonlinear oscillatory versions of these three individual systems, gives a mathematically founded basis
of autonomy.