I recently spent a bit of time wondering how the position of the bow planes
and stern planes, on submarines so equipped, are properly co-ordinated for
dynamic manoeuvers. An internet search didn't turn up much information, so
I did the next best thing - I sat down and reasoned it out. Assuming a
vessel is equipped with dive planes that can be independently controlled, I
came up with what I think is a good automated control scheme, for a sub
with a computer of some sort to manage the automatic control. I post it
here for comment or criticism - not on the practicality of automated dive
plane control (expensive for a psub, I know), but rather on the suitability
and robustness of the algorithm.
-Sean
The following description of hydroplane positioning logic is applicable to
a submarine vessel with control surfaces consisting of independently
operated port side bow plane, starboard side bow plane, port side stern
plane and starboard side stern plane. The vessel is also assumed to have
one or more rudders for steering control, but the rudder position will
always correspond to the commanded rudder position according to the
desired rate of turn, and is therefore not addressed herein.
NOMENCLATURE
alpha
the commanded angle of the bow planes
beta
the commanded angle of the stern planes
gamma
the maximum angle that a control plane can assume before turbulence
occurs (velocity dependent), subject to mechanical limits (in the case of
near-zero velocities)
theta
the current angle of roll of the vessel
theta-c
the commanded angle of roll of the vessel (special purposes -
ordinarily zero)
phi
the current angle of pitch of the vessel
phi-c
the commanded angle of pitch of the vessel
delta
pitch leveling threshold corresponding to the minimum distance from
target depth at which a leveling manoeuvre must be initiated in order to
reach the target depth without overshoot. This value is a function of
both ascent / descent rate and of phi, and shall be determined
experimentally with the vessel in a neutral buoyancy condition. (i.e.
dynamic manoeuvering only)
ANTI-ROLL STABILIZATION
Except where specifically disabled by command, all control planes act
in conjunction to stabilize the vessel dynamically. This behaviour occurs
as follows: each plane will act in opposition to vessel roll, with a
commanded angle proportional to the difference between theta and theta-c.
The zero position (commanded position when theta is equal to theta-c),
will be equal to alpha and beta for the bow and stern planes,
respectively. In this fashion, the dynamic anti-roll stabilization
behaviour superimposes itself on commanded plane positions for vessel
manoeuvres. When the vessel is in the surfaced condition, the maximum
correction will occur when theta equals (or exceeds) the maximum standard
banking angle during normal operation (maximum commandable angle of roll
when submerged). When the vessel is submerged, the maximum correction
will occur when theta equals (or exceeds) the maximum permissible angle of
roll. In either case, at maximum correction, the control planes will be
oriented at an angle of +/- gamma with respect to the vessel.
HYDROPLANE BEHAVIOUR - SURFACED CONDITION
When the vessel is in the surfaced condition, any commanded roll angle
(theta-c) is disregarded - trim tank and control surface systems all act
to maintain theta equal to zero with respect to gravity. Trim in pitch
(phi-c) may still be commanded, subject to limits determined by
permissible propellor depth and/or available freeboard and hatch coaming
height. Ordinarily, the vessel will operate in the surfaced condition
with a phi-c of zero; however, it may be desireable to command a different
pitch angle under certain circumstances, such as when towing or being
towed, or when facilitating the launch or recovery of personnel or
materials between the sea and the weather deck. Regardless of commanded
pitch angle (phi-c), when the vessel is in the surfaced condition alpha
and beta will correspond to horizontal positions with respect to gravity.
HYDROPLANE BEHAVIOUR - DIVE MANOEUVER FROM SURFACE
When a dive manoeuvre from surface is initiated, alpha is initially set
hard down at an angle of -gamma, and assumes subsequent values
proportional to the difference between phi and phi-c. The stern plane
angle beta acts similarly in the opposite direction; however, the initial
value of beta can not be set to gamma immediately, since this would act to
raise the propellor depth above the normal surfaced propellor depth at the
beginning of the dive manoeuvre. Accordingly, the initial value of beta,
(the value which controls the magnitude of the proportional response), is
not gamma, but rather a value which starts at zero (horizontal with
respect to gravity) and rapidly increases to gamma in a manner which keeps
the propellor below the normal surfaced propellor depth.
HYDROPLANE BEHAVIOUR - DIVE MANOEUVER IN MIDWATER
When a dive manoeuver is initiated while submerged, the bow plane angle
alpha is initially set hard down at an angle of -gamma, and assumes
subsequent values proportional to the difference between phi and phi-c.
The stern plane angle beta is initially set hard up at an angle of gamma,
and assumes subsequent values proportional to the difference between phi
and phi-c.
HYDROPLANE BEHAVIOUR - ASCENT MANOEUVER
When an ascent manoeuver is initiated, the bow plane angle alpha is
initially set hard up at an angle of gamma, and assumes subsequent values
proportional to the difference between phi and phi-c. The stern plane
angle beta is initially set hard down at an angle of -gamma, and assumes
subsequent values proportional to the difference between phi and phi-c.
HYDROPLANE BEHAVIOUR - BANK CONTROL
Commanded roll angles (theta-c) for specific purposes aside, the vessel
will normally operate in an upright condition, with theta-c equal to zero.
Under this condition, at high speeds, a turn will result in a normal
component acceleration which moves the apparent gravity normal away from
perpendicular to the vessel normal. To correct this, a roll correction
will be applied which increases theta-c to bring the apparent gravity
normal back to perpendicular to the vessel, subject to the limits of +/-
gamma angle on the control surfaces, and maintaining theta within the
maximum permissible theta. Thus, in a sustained turn at a high rate of
speed, vessel trim will be as comfortable as possible for the occupants.
HYDROPLANE BEHAVIOUR - RATE CONTROL
The above description of hydroplane behaviour details how the dive
control surfaces function with respect to the commanded pitch angle,
phi-c. Response time of the control planes must be rapid in order for the
dynamic stabilization system to function effectively; however, rapid
rotation of control surfaces to extreme positions should not be confused
with extreme manoeuvers. Such extreme control surface positions are
generally short lived under the control scheme. Extreme manoeuvers, on
the other hand, are generally associated with a large magnitude rate of
change of phi-c or theta-c.
A comprehensive description of dive control surface behaviour must also
include the behaviour necessary to level off the vessel as it approaches
the target dive depth, to prevent the vessel from overshooting the desired
depth during dynamic manoeuvering. This is accomplished by varying phi-c
as follows: If the difference between the vessel's current depth and the
target depth is greater than delta (a function of ascent / descent rate),
then phi-c is maintained at the setpoint. If this difference is less than
delta, then phi-c is reduced in a manner proportional to this difference,
becoming zero when the target depth is reached.
HYDROPLANE BEHAVIOUR - MAINTENANCE OF DEPTH
When levelled off at the desired depth setpoint, alpha and beta will be
adjusted by a common angle (i.e. in the same direction) to compensate for
a difference between the current depth and the setpoint depth. The amount
of correction shall be proportional to this depth difference, subject to
the limits of +/- gamma on the dive plane positions. In the event of a
persistent difference (indicating a buoyancy condition other than
neutral), the ballast system should be automatically compensating,
reducing the required dynamic correction to zero.
MANUAL OPERATION
When control of the vessel is not by the autopilot, but rather by the
operator, provided power is available, all functions of the automatic
system described above will continue to operate, with the exception of
rate control. In this case, phi-c will correspond directly to operator
input.
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