Analysis of the Impact of Sensor Noise on Formation
Flying Control1
Jonathan P. How2 and Michael Tillerson3
Space Systems Laboratory
Massachusetts Institute of Technology
Abstract
This paper analyzes the impact of sensing noise on formation flying control algorithms that have been developed for distributed spacecraft systems. The key
issue is that the sensing errors cause uncertainty in
the initial conditions of the trajectory planning process
which is at the core of the fuel-optimization algorithm.
For example, the relative velocity measurements using a carrier-phase differential GPS sensor are predicted to be accurate to approximately 2 mm/sec, but
a 2 mm/sec error in the intrack velocity corresponds
to approximately 30 m/orbit secular drift rate in the
relative positions between two spacecraft. This large
drift rate is comparable to the values expected from
differential J2 disturbances, so its impact on the fuel
used to perform formation flying is an important concern. To account for these sensing errors, modifications are presented to the station-keeping optimization
algorithms that have been developed using linear programming. The approach robustifies the design of the
control inputs to velocity errors, but this is achieved at
the expense of using much shorter design horizons. The
modified control approach has been demonstrated using a realistic nonlinear simulation environment. The
results from these simulations confirm that noise in the
relative velocity measurements will play a crucial role
in the fleet performance and/or the fuel cost.
1 Introduction
A large number of future planned space missions are
based on a new approach that will use coordinated
microsatellites to provide flexible, low-cost access to
space. In particular, many of these missions will use
clusters of relatively small spacecraft to form a “virtual
satellite bus” that replaces the standard monoliths used
today [1, 2]. However, to achieve these future mission
goals, several guidance, navigation, and control challenges must first be addressed. For example, very tight
coordination, control, and monitoring of the distributed vehicles in the cluster will be required to achieve
1
Funded under Air Force grant #F49620-99-1-0095 and
NASA GSFC grant #NAG5-6233-0005
2 Associate Professor, MIT, Dept. of Aeronautics and Astronautics, jhow@mit.edu
3 Research Assistant, MIT, Dept. of Aeronautics and Astronautics, mike t@mit.edu
the stringent payload pointing requirements for a synthetic aperture radar mission, such as TechSat 21 [3].
Much of the research for cluster dynamic modeling and
control has focused on the design of passive apertures,
which are (short baseline) periodic formation configurations that provide good, distributed, Earth imaging while reducing the tendency of the vehicles to drift
apart. These passive apertures can be designed using the closed-form solutions provided by Hill’s equations [4, 5] (also known as the Clohessy-Wiltshire equations), which assume a circular reference orbit. There
has been further analysis to develop apertures that are
insensitive to differential J2 disturbances [6] and reference orbit eccentricity [8]. These papers have also
analyzed various modeling errors and differential disturbances to determine their relative effects on the fuel
used to perform formation flying [4, 6, 7, 8, 9]. The
results indicate that differential J2 is the most important differential disturbance to be accounted for in the
control design, resulting in drift rates on the order of
10’s m/orbit in the relative separation between spacecraft (depending on the configuration) [8] that would
require a ∆V on the order of several (cm/sec)/orbit to
correct (the exact numbers are still under investigation,
see Ref. [9, 10]).
The purpose of this paper is to demonstrate that sensor
noise could also play a key role in the fuel consumption.
For example, designing a fuel-optimized control input
sequence will typically require a trajectory planning
technique, but these planning techniques will heavily
rely on the knowledge of the spacecraft’s initial conditions. But since these initial relative positions and velocities must be determined from measurements, they
will be corrupted by noise. For example, with filtered
Carrier-Phase Differential GPS (CDGPS) signals as the
relative navigation sensor, the position noise is currently predicted to be on the order of σp =2-5 cm and
velocity noise on the order of σv =2-3 mm/s [15, 16].
This paper will examine the effects of these measurement errors from three perspectives: satellite dynamics, ability to plan fuel-optimal trajectories, and expected fuel consumption.
2 Effects on Relative Motions
To analyze the effects of measurement errors, consider
the dynamics of the relative motion of two spacecraft
(Hill’s equations) with a circular reference orbit (orbital
frequency n) [11]
2
2nẏ + 3n2 x + fx
−2nẋ + fy
−n2 z + fz
10
(1)
The x-coordinate is in the radial direction, the ycoordinate is in the intrack direction and the zcomponent is in the cross-track direction. The closedform solution to these equations is well known to be of
the form
2ẏ0
ẋ0
sin nt −
+ 3x0 cos nt
x(t) =
n
n
2ẏ0
(2)
+ 4x0
+
n
4ẏ0
2ẋ0
y(t) =
cos nt +
+ 6x0 sin nt
n
n
2ẋ0
+ y0 −
(3)
− (3ẏ0 + 6nx0 )t
n
Taking x0 , y0 , ẋ0 , and ẏ0 to be nominally zero, each
initial condition can then be perturbed to values of
±0.02 m for the position and ±0.002 m/sec for the velocity to calculate the resulting relative motion. Initial
errors in position as well as radial velocity only result
in small errors in predicted motion, on the order of less
than a meter. However, a ±0.002 m/s intrack velocity
error results in approximately a 30 m intrack position
offset after only one orbit. For comparison, the relative
motion (as predicted by the FreeFlyer orbital simulation software [12]) due to differential J2 for a pair of
satellites in an orbit with a 35◦ inclination angle results in a drift rate of ≈5 m per orbit. The response to
each of these effects are shown in Figure 1.
Another way to analyze these errors is to note that the
last term in Eq. 3 shows that the relative velocity errors
(ẏ0 ) have an effect that is 1/(2n) ≈ 450 times larger
(in terms of the secular intrack drift) than the relative
position errors (x0 ). However, the filtered CDGPS is
only predicted to provide velocity knowledge that is a
factor of ten better (comparing 0.002 to 0.02). These
results indicate that obtaining better velocity estimates
is an important issue for future work in formation flying. From the control perspective, these results also
indicate that it is important for any control technique
that designs fuel-optimal trajectories to account for uncertainty in the spacecraft’s initial conditions. Modifications to account for these sensing errors in a recently
developed planning algorithm are outlined in the following sections.
3 Effects on Path Planning
As previously mentioned, fuel optimization techniques
generally require trajectory planning, and predicting
these trajectories depends heavily on the initial conditions that are corrupted by measurement noise. These
0
Radial/ Intrack Respponse
ẍ =
ÿ =
z̈ =
Response to velocity Perurbation and J
20
−10
−20
−30
radial
intrack
intrack vel error
J2
−40
−50
−60
0
0.5
1
1.5
2
Time (orbits)
2.5
3
3.5
4
Fig. 1: Comparison of the resulting relative motion
due to initial error in the intrack velocity (+0.002m/sec
intrack) (◦) and due to differential J2 effects (✷). The
results show that there is an intrack drift of ≈30 m per
orbit for intrack velocity error and ≈5 m per orbit due
to J2 .
points are analyzed in this section for the case of a
trajectory design technique that uses Linear Programming (LP) to determine the fuel optimal control inputs. Modifications to the LP optimization are then
presented to make the designed trajectories more robust to initial condition uncertainty.
3.1 Path Planning – Summary
A Linear Programming (LP) trajectory planning approach has recently been developed to design fueloptimized trajectories and station-keeping control inputs [13, 14, 21]. The basic form of the LP is
min u1 subject to Au ≤ b
(4)
where u is the vector of fuel inputs (∆V ) at each time
step and A, b are functions of the linearized spacecraft
dynamics, initial conditions, and final conditions. The
LP determines the control inputs for a specified time
interval that minimizes the fuel cost, the sum of the
inputs, while satisfying the constraints on the trajectory. Constraints to the problem can include: state
constraints such as remaining within some tolerance of
a specified point, maximum input values (actuator saturation), and terminal constraints. This approach can
include differential disturbances such as drag and linearized forms of the differential J2 effects (see Ref. [4, 9]
for details on these models). To complete the low-level
control design, the LP is also embedded within a realtime optimization control approach that monitors the
spacecraft relative positions and velocities, and then
redesigns the control input sequence if the vehicle approaches the edge of the error box [21].
3.2 Robust LP for Initial Condition Uncertainty
Any planned trajectory will rely heavily on the knowledge of the satellite’s initial conditions, but the initial
relative positions and velocities must be measured and
will be noisy. To examine this point further, consider
a velocity uncertainty of ±2 mm/s in the system described in the previous section with a plan horizon of
four orbits. Figure 2 shows the response to the nominal plan for the nominal case (solid line terminating
at the circle marked 0) as well as responses when the
velocity initial conditions are perturbed (±2 mm/s intrack, ±2 mm/s radial). As expected, for the nominal
case, the inputs keep the satellite within the error box
for four orbits and the path terminates near the center. However, when the velocity is perturbed, three of
the trajectories violate the constraint box and two of
the paths leave the box and never return. The alternatives at this point are to increase the size of the error
box, which will impact the payload performance, or to
modify the algorithm so that it is less susceptible to
measurement noise.
Uncertainty in the initial conditions can be addressed
in the LP by designing trajectories that are robust to
errors in ẋ(0) and ẏ(0). Based on the multiple-model
techniques successfully used for robust feedback control design [17, 18], one approach to robustify the trajectory planner is to design the input sequence to simultaneously satisfy the constraints for several (mic )
initial conditions. The initial conditions appear only
in the b column vector for the constraints in Eqn. 4.
For each possible initial condition, a constraint vector
b is formed. The actual b vector used in the LP optimization, bmin , is constructed as follows:
Form matrix of b vectors:
B = b1 . . . bm
(5)
Form new bmin vector:
bi = min Bij ; ∀ i = 1, . . . , N
(6)
j
Note that this modification does not lead to an increase in the size of the LP, which is a key point for
the numerical implementation. Because this approach
typically only considers several (mic =4–10) perturbed
initial conditions, it is not guaranteed to provide an input sequence that will not violate the design constraints
(e.g., keep the spacecraft within a specified error box
for the next 2 orbits). However, as was the case with
previous work on robust feedback control design, experience indicates that the results from this robustified
approach are much less sensitive to errors in the initial
conditions. Future work will compare this “multiplemodel” approach to robustification with the guaranteed techniques in Ref. [20].
By considering several different initial conditions, the
result of this LP design should be robustified to mea-
surement errors, but due to the large effect of the initial
velocity errors, the length of the planning horizon has
to be reduced in order to achieve a feasible solution.
For the examples considered in the remainder of this
section, the horizon was reduced to approximately one
quarter of an orbit and the terminal constraint on the
optimization (e.g., finish within 1 m of the origin) was
removed. Figure 3 shows the response to the perturbed
velocity initial conditions as well as two additional initial conditions for a quarter orbit plan designed using
only the nominal case. The
√ have
√ additional two cases
2
mm/s
intrack,
+
2 mm/s
initial velocity
errors
of
+
√
√
radial and + 2 mm/s intrack, - 2 mm/s radial. Note
that four of the paths exit the error box. This is not unexpected, as the LP was only designed for the nominal
case. The control inputs from the robustified LP were
applied to the same set of initial conditions to generate
the trajectories in Figure 4. Only four of these initial
conditions were included in the LP design (the labeled
ones). However, as shown, all six trajectories remain
within the box during this first quarter orbit.
With increasing noise levels and subsequent increasing
uncertainty in the initial conditions, the variation in
trajectories for a set of inputs can become quite large.
As a result, the control can only be successfully applied
over a reduced time horizon. The size of the error box
also determines the maximum allowable plan horizon.
For example, a 2 mm/s deviation in intrack velocity
results in a 30 m drift over one orbit. If the error
box is smaller than 30 m in the intrack direction, then
the plan horizon clearly must be less than one orbit.
However, a larger error box may be able to sustain this
drift from the planned destination. Figure 5 shows a
plot of feasible plan times versus the noise level for
increasing error box size. The LP optimization for the
plot only constrained the satellite to remain within the
specified error box for the duration of the plan time.
Note that as the noise level increases the plan time
drops very rapidly, but increasing the error box size
relieves the position constraint and allows for a longer
plan horizon.
4 Effects on Fuel Use
Several nonlinear simulations were performed using
FreeFlyer orbit simulator in order to determine the
effect of sensor noise on fuel use for satellite station
keeping. The simulations consists of two satellites, in
approximately 90 minute circular orbits, on a closed
form ellipse with 200 m semimajor axis. The differential drag is modeled as a constant ± 0.5 × 10−7 m/s2
acceleration. In order to clearly examine the effect of
sensor noise on fuel use for station keeping, only the
differential drag disturbances are implemented in the
FreeFlyer [12] simulation, other disturbances such as
J2 effects and solar radiation pressure could be implemented. The J2 and other disturbances are disabled
in the FreeFlyer simulations because the dynamics in
FORMATIONKEEPING USING INPUT FOR 4 ORBITS
FORMATIONKEEPING USING NOMINAL INPUT FOR 1/4 ORBITS
6
5
+2 I
4
4
3
+2 I
2
2
RADIAL (m)
RADIAL (m)
1
−2 R
0
0
+2 R
0
0
−1
+2 R
−2
−2
−2 R
−3
−2 I
−4
−4
−2 I
−20
−15
−10
−5
0
INTRACK (m)
5
10
15
20
−5
25
Fig. 2: Trajectory followed using a nominal plan designed for four orbits without considering the initial
condition uncertainties. The trajectories for ±2 mm/s
intrack error had final position errors of approximately
±130 m.
the LP do not account for these disturbances. With
these disturbances eliminated, the increase in fuel use
is caused by the sensor noise rather than a combination of noise and plant uncertainty. During maneuvers,
the satellite thrusters are restricted to provide a maximum acceleration of 0.003 m/s2 and a minimum of
5 × 10−6 mm/s2 . The maximum thrust is produced by
turning the thruster on for the full time step. The minimum thrust is determined from a minimum impulse bit
of 10 msec during the 5.4 second time step. The ±10 m
intrack × ±5 m radial error box is selected to meet the
requirements of the TechSat 21 mission. Simulations
were performed for position noise levels of 2 cm and
velocity noise levels ranging from 0.1 mm/s to 2 mm/s.
Only the velocity noise is varied because the LP is much
more sensitive to velocity errors than position errors,
as demonstrated by the simulations in Section 2. The
noise is modeled in the simulations as the true state
vector plus a white noise component. Multiple simulations are required for each noise level because of the
stochastic nature of the system response. Three, one
day simulations are performed for each noise level. The
fuel use, ∆V , is determined as an average fuel use per
orbit for each noise level.
For each simulation, the satellite begins in the center of
the error box and drifts to the edge due to a differential
drag. When the satellite nears the edge, a control input
is determined to keep the satellite within the box using
the robust LP. The constraints for the LP are the same
for each noise level, only the planning horizon is altered
in order to arrive at a feasible solution. The satellite
is constrained to remain inside the error box for the
duration of the plan.
Figures 6 and 7 display the typical relative error box
motion of a satellite for a low noise level of 0.1 mm/s
−10
−5
0
INTRACK (m)
5
10
Fig. 3: Trajectories for each of the five possible initial
conditions using plan for nominal case. The pentagram
and hexagram represent two additional cases within the
uncertainty ellipsoid but not considered in the plan.
FORMATIONKEEPING USING ROBUST INPUT FOR 1/4 ORBITS
5
4
+2 I
3
2
+2 R
1
RADIAL
−6
−25
0
0
−1
−2 R
−2
−3
−2 I
−4
−5
−10
−8
−6
−4
−2
0
INTRACK
2
4
6
8
10
Fig. 4: Trajectory followed using robust plan for each
initial condition using robustified plan. The pentagram
and hexagram represent two additional cases within the
uncertainty ellipsoid but not considered in the plan.
and high noise level of 2 mm/s, respectively. Notice
that the low noise case results in a smooth continuous motion, but the high level case is disjointed. The
abrupt changes in motion for the high noise case result
from the uncertainty in initial conditions. The robust
LP plans for the worst case possible and, as a result, often uses a large control input that completely reverses
the motion of the satellite.
To summarize these simulations, Figure 8 shows a
plot of fuel use versus noise level. The average fuel
used increases proportionally to the sensor noise, from
∆V = 1.15 mm/s per orbit for a noise level of 0.1 mm/s
to ∆V = 33.3 mm/s per orbit for 2 mm/s noise. Also
note that the variation in fuel use increases as the noise
level increases from 0.1 to 1.0 mm/s, and then decreases
Plan Time vs Noise Level− Increasing Error Box Size
Low Noise Error Box Motion
8
6
5 X 2.5
10 X 5
20 X 10
40 X 20
7
4
6
y−radial [m]
Plan Time (orbits)
2
5
4
0
3
−2
2
−4
1
0
0
0.2
0.4
0.6
0.8
1
1.2
Velocity Noise Level (mm/s)
1.4
1.6
1.8
−6
2
Fig. 5: Maximum plan time achievable versus noise
level for increasing error box size. Error box size are in
meters (± intrack×± radial).
−10
−8
−6
−4
−2
0
2
x−intrack [m]
4
6
8
10
Fig. 6: Typical errbox motion for 0.1 mm/s velocity
noise level.
High Noise Error Box Motion
6
Figure 9 compares the fuel cost for station keeping using LP with no uncertainty, the robust LP with 2 mm/s
velocity uncertainty, and a non-robust LP with 2 mm/s
velocity uncertainty. The simulation with no noise used
∆V ≈ 0.5 mm/s per orbit. Station-keeping using the
robust LP requires ∆V ≈ 30 mm/s per orbit for each
satellite, which is a significant increase. The non-robust
control works well at times, but not always and results
in a ∆V ≈ 90 mm/s per orbit.
5 Conclusions
The key observation from these preliminary results are
that sensing errors, and in particular the relative velocity error, could play a very important role in the performance and/or fuel cost associated with formation
flying. This appears to be particularly true for relative
navigation using differential carrier-phase GPS sensors
for which the velocity errors are predicted to be on the
order of 1-2 mm/sec. Note that these noise levels are
comparable to the expected relative velocities between
the formation flying spacecraft! A technique for robus-
4
2
y−radial [m]
again. The variation in the fuel used arises from the
uncertainty in the effectiveness of the LP generated inputs. The LP is designed to compensate for the worst
case effects of the initial condition errors, which turns
out to be more fuel efficient in some cases than in others. The decrease in the variability of fuel used for
higher noise levels is most likely due to the reduced
plan time. The shorter the plan time, the less the
satellite will diverge from a designed plan. The plot
suggests that a reduction in sensor noise by 50% (from
2 mm/s) will reduce fuel use by ≈50%. The results
of this simulation show that a ∆V ≈ 33.3 mm/s per
orbit is needed to account for a velocity noise level of
2 mm/s. The ∆V is on the same order as the ∆V
required by differential J2 .
0
−2
−4
−6
−10
−8
−6
−4
−2
0
2
x−intrack [m]
4
6
8
10
Fig. 7: Typical motion for 2 mm/s velocity noise level.
tifying the linear programming formation-keeping algorithm to this sensor noise was outlined. The robust approach works well, but this performance is achieved at
the cost of using much shorter design horizons and significantly more fuel. Simulations show a proportional
increase in fuel cost with sensor noise level. The variance of the fuel use also increases with increased noise
level, which complicates the predictions of the fuel requirements for a mission plan. The studies also show
that the increase in fuel cost for noise level is on the
same order as the increase in fuel cost due to neglecting
J2 effects in the LP model.
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45
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35
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