Background
==========
A spacecraft (S/C e.g. New Horizons) is flying on a linear
trajectory past a target (e.g. Pluto), with a 99%+ probability that
the target is inside a volume shaped like a cigar with its long axis
parallel to the flyby velocity vector. The observation scans an
instrument across the target, which instrument has a Field Of View
(FOV) far greater than the smaller axes of the cigar. So the target
will be visible in the instrument FOV at some point as long as the
instrument boresight is scanned along that long axis during the
observation.
However, there is a much more stringent requirement determining
exactly *how* the boresight moves along that axis. Specifically,
the instrument boresight must be scanned along the long axis of the
cigar such that, when the boresight passes over the target, the
angular rate of the target with respect to the instrument frame is a
precise fixed value. That value is typically set, via a S/C command
sequence, as the scan rate of a Time Delay Integration (TDI) CCD.
DPDTRUN purpose
===============
For a given flyby of a nominal body position Pnom, calculate the
ephemeris of a pseudo-body (CB3 a.k.a. P) along-track wrt Pnom,
such that if any S/C instrument points its boresight at CB3 throughout the flyby, the scan rate (radian/second) of the actual body, at
an unknown position but assumed to be fixed somewhere along-track
wrt Pnom, across that boresight will remain at the constant
sequenced value.
Initial conditions
-----------------xEll_km
Extent of along-track to be covered wrt Pnom, km
- If one value specified: -|xEll_km| to +|xEll_km|
- If two values specified: min(xEll_km) to max(xEll_km)
- positive implies +X i.e. downtrack
- This procedure integrates the CB3 position between
these two values at a minimum
xEll_s
IFF xEll_km not specified, extent of along-track covered
wrt nominal target, s past CA (t - TCA) at vFb
- default: 150s
- same scheme as xEll_km for one or two values
- positive implies +X i.e. downtrack
xCb3_0_km
xCb3_0_s
xSc0_km
xSc0_s
Initial CB3 position wrt nominal target, km
- positive implies +X => downtrack
IFF xCb3_0_km not specified, initial CB3 position
wrt nominal target, s past CA (CB3t0 - TCA) at vFb
- default: 0s
- positive implies +X => downtrack
Initial S/C position wrt nominal target, km
- positive implies +X => downtrack
IFF xSc0_km not specified, initial S/C position
wrt nominal target, s past CA (SCt0 - TCA) at vFb
- default: 0s
- positive implies +X => downtrack
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FUNCTION dydx
=============
Derivative function to support DPDTRUN: used to calculate dy/dx
(derivative) for fourth-order Runge-Kutta solution; see derivation below.
Arguments
--------t
Time at which to calculate derivative (t=0 is start of integration)
P
CB3(t), current value of P
////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////*/
public double dpdt(double t, double P) {
deltaX = P - (vFb * t);
return dThDtTDI * (deltaY + (deltaX * deltaX / deltaY));
}
/*//////////////////////////////////////////////////////////////////////
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Derivation of dPdT function
===========================
Refer to the Background and diagram above.
The instantaneous angular (scan) rate of a target across such a S/C
instrument's boresight, with said target fixed at position
[Pnom + CB3(t)] at any time t, is
dThDtTDI = dThDt + dThDtFb
(radian/second; rad/s)
[1]
where
Pnom
= Nominal target position = inertial frame origin, constant
CB3(t)
= Time-dependent pseudo-body* offset from Pnom along X, km
dThDtTDI = Desired (sequenced) scan rate, constant, converted to rad/s
Theta(t) = Boresight angle wrt inertial +Y, positive toward +X, radian
dThDt
dThDtFb
= dTheta / dt, inertial angular rate of Theta(t), rad/s
= Instantaneous angular rotation rate of a line from [the
fixed target at CB3(t)] to [the spacecraft] due to
S/C-target relative translational motion, radian/second
* The to-be-solved-for pseudo-body position model, CB3(t), does not
represent a real target; it is instead a synthetic trajectory loaded
into the S/C GN&C** subsystem, and to which GN&C will point the
instrument boresight during the observation scan. The CB3(t) pseudobody trajectory will at some point during the scan pass through the
the real target position, which position is at an unknown but fixed
offset from Pnom. The goal of this exercise is to solve for (derive)
that CB3(t) trajectory model.
** Guidance, Navigation and Control
Since Theta(t) is by definition the instantaneous angle of the
boresight pointing at any actual fixed target at CB3(t) from S/C(t),
the scan rate contribution of that fixed target across the boresight
due to relative translational motion is the ratio of [the motion of
the fixed target perpendicular to the boresight] to [the range
between the S/C and the fixed target]:
vFb * cos(Theta)
2
dThDtFb = ---------------------- = vFb * cos (Theta) / deltaY
deltaY / cos(Theta)
[2]
Substituting into [1] and solving for dThDt,
dThDt = dThDtTDI
-
2
vFb * cos (Theta) / deltaY
[3]
The distance along the flyby direction (+X) between the spacecraft
and the target can be derived from two formulae:
deltaX = P(t) - vFb * t
deltaX = deltaY * tan(Theta)
[4]
[5]
where t is time and is zero at TCA to Pnom.
and solving for P(t) yields
P(t) = deltaY * tan(Theta)
+
Setting [4] equal to [5]
vFb * t
[6]
Solving for dP/dt yields
2
dP/dt = (deltaY/cos (Theta)) * dTh/dt
+
vFb
[7]
Substituting [3] for dTh/dt into [7]:
deltaY
dP/dt = ----------2
cos (Theta)
/
* ( dThDtTDI
\
-
2
vFb * cos (Theta) \
----------------- )
deltaY
/
+
vFb
2
dP/dt = deltaY * dThDtTDI / cos (Theta)
-
vFb
+
vFb
2
dP/dt = deltaY * dThDtTDI / cos (Theta)
[8]
Substituting the trigonomtric identity
2
cos (Theta(t)) =
2
2
2
deltaY
/ (deltaY
+ deltaX
)
yields
2
2
deltaY + deltaX
dP/dt = deltaY * dThDtTDI * ------------------2
deltaY
and, after rearranging,
2
/
dP/dt = dThDtTDI * (
deltaY
\
+
deltaX
-----deltaY
\
)
[9]
/
Finally, substituting equation [4] for deltaX yields the equation
[10], which is used in method dpdt above as part of the 4th-order
Runge-Kutta solution:
2
(P(t) - vFb * t) \
dP/dt = dThDtTDI * ( deltaY + ----------------- )
\
deltaY
/
/
[10]
////////////////////////////////////////////////////////////////////////
Final note:
When the range to the target is large and/or the flyby distance
(deltaY) is large compared to the distance the S/C moves during the
scan, dThDtFb is essentially constant, and this derivation is not
relevant.
////////////////////////////////////////////////////////////////////////
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/*//////////////////////////////////////////////////////////////////////
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Analytical solution (developed after the RK4 solution)
======================================================
dThDt = dThDtTDI Substituting
2
vFb * cos (Theta) / deltaY
[3]
b^2
= vFb / (dThDtTDI * deltaY)
- see Note 1 below for case where this quantity is negative
yields
dTheta
--------------------- = dThDtTDI * dt
2
1 - b^2 * cos (Theta)
[11]
2
N.B. for b^2=1, the left side of [11] becomes dTheta/sin (Theta)
Substituting
broot = SQRT(b^2 - 1)
if b^2 > 1
broot = SQRT(1 - b^2)
if b^2 < 1
and integrating both sides of [11] (see Notes 1) depending on the
value of b^2:
1
/ tan(Theta) - broot \
--------- ln( -----------------) = dThDtTDI * t + C [12a]
2 * broot
\ tan(Theta) + broot /
(See also Note 2 below)
b^2 > 1:
b^2 < 1:
1
-1 / tan(Theta) \
----- tan ( ---------- )
broot
\
broot
/
= dThDtTDI * t + C
[12b]
b^2 = 1:
- cot(Theta)
= dThDtTDI * t + C
[12c]
=
- 1 / tan(Theta)
where C is the constant of integration and may be evaluated at t = zero
as the left hand sides of [12*]; any other time tReal may be converted
to 't' by means of a constant offset (t = tReal - tReal0), and Theta(t=0)
defined as ThetaReal(tReal=tReal0), the initial condition for the ODE.
Note 1:
Integrals 376, 375, & 308 in section A of "CRC Handbook of
Chemistry and Physics," Weast, Robert C. (ed), 56th edition,
CRC Press, pp. A-136, A-138, 1974. The first form of integral
376 and the is essentially identical to 375 where 375 covers the
case where dThDtTDI is negative.
Note 2:
In equation [12a], it is possible that broot is greater
than tan(Theta), which results in a negative numerator
and a negative value passed to the natural logarithm
function. In that case this approach fails, because the
constant of integration cannot be calculated.
Substituting
deltaX/deltaY = tan(Theta)
T
= dThDtTDI * t + C
into [12*] and solving for deltaX:
b^2 > 1:
deltaY * broot * ( exp(2*broot*T) + 1 )
deltaX = --------------------------------------1 + exp(2*broot*T)
[13a]
b^2 < 1:
deltaX = deltaY * broot * tan(broot*T) + 1 )
[13b]
b^2 < 1:
deltaX = - deltaY / T
[13c]
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//////////////////////////////////////////////////////////////////////*/
Acknowledgement: this problem was brought to me by Ann Harch while
we were working together on the New Horizons mission. The
spacecraft had an on-board algorithm to compensate for translational
motion during a scan of the RALPH/MVIC instrument, but that
algorithm did not account for significant changes in the range from
spacecraft to target. Others, such as Gabe Rogers and the late,
great Gene Heyler also solved this problem by coding an Euler-style
ODE solver.