A.3.2.4 Spin Stabilization
1
A.3.2.4 Spin Stabilization of Launch Vehicle Third Stage
A.3.2.4.1 Introduction
We leave the third stage without any active control, placing the majority of the avionics
in the second stage and subsequently causing a large reduction in cost for the launch
system. Thrust misalignment within the solid rocket motor and center of mass offset still
present instability problems that we must address. Thus we decide to spin stabilize the
third stage, much like a football quarterback spinning the ball during a pass to ensure that
it reaches the receiver just in time for the game winning touchdown. We introduce a
specialized adapter called a spin table between the second and third stages, allowing the
third stage to spin freely. At the appropriate time, transversely mounted thrusters are
activated to achieve the required spin rate. The technique of spin stabilization nearly
eliminates thrust offset instability and greatly reduces final pointing error during the
burning of solid rocket motors.
Through communication with A. Guzik in the Trajectory group, we determine that an
acceptable final pointing error for the third stage will be no more than 1o from the
nominal pointing value for each payload.
Using analysis methods developed by
Longuski, we develop MATLAB codes that numerically predict pointing errors
associated with both the spinning up of the third stage and the thrusting of the third
stage.1 These codes are named AAE450_DNC_spin_offset and Spin_Rate respectively.
We create the codes based on a further simplification of the equations of motion as
detailed in section A.3.2.1, namely the aerodynamic forces have been eliminated and the
spinning-up and the thrusting maneuvers are separate so that the spin moment and the
thrust force are not applied at the same time. While we can calculate the velocity
pointing error directly for certain cases, we can also calculate the pointing error from the
pointing error of the angular momentum vector. Both methods are used to analyze the
thrusting maneuver, while only the angular momentum method is used to analyze the
spin-up maneuver.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
2
In Figure A.3.2.4.1.1, we sketch the free body diagram of the third stage, with associated
body fixed unit vector basis r̂ . Let us note that the weight force causes a negligible
moment about the body center of mass, while the thrust force causes a much larger scale
moment. Any pointing error associated with the gravitational moment will be very small
in comparison to the error from thruster offset, especially over the short duration of the
thruster burn. Thus we neglect the gravity force from the derived equations of motion.
Figure A.3.2.4.1.1: Simplified free body diagram of third stage with spin moment and thrust vector.
(Author: Jeffrey Stuart)
A.3.2.4.2 Thrusting Maneuver
Analysis of the thrusting maneuver is complicated by the changing position of the center
of mass along the rocket axis of symmetry and the time dependence of the moments of
inertia. We use Euler’s equations of motion detailed in Eq. A.3.2.4.2.1-A.3.2.4.2.3 and
the definition of angular momentum (Eq. A.3.2.4.2.4) to numerically calculate time
histories of the angular velocity and momentum in the body-fixed frame.
M 1  I 1 1  I11  ( I 3  I 2 ) 2 3
A.3.2.4.2.1
M 2  I 2 2  I2 2  ( I 1  I 3 )31
A.3.2.4.2.2
M 3  I 3 3  I3 3  ( I 2  I 1 )1 2
A.3.2.4.2.3
where M is the moment about one of the body axes, I is the moment of inertia, and ω is
the angular velocity.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
3
H i  I i i
A.3.2.4.2.4
where H is the angular moment in the body fixed frame, I is the moment of inertia, and ω
is the angular velocity.
Additionally, we derive equations for the change in velocity from Newton’s Second Law,
which will be used to calculate the velocity pointing error.
Equation A.3.2.4.2.5
summarizes these relations for the inertial accelerations and an Euler 3-1-2 rotation. Note
for simplicity that we assume the thrust is offset in only one of the transverse body axes.
a X  c3 c2  s1 s 2 s3
 a   s c  s s c
 Y  3 2 1 2 3
 a Z    s 2 c1
 s3 c1
c3 c1
s1
c3 s 2  s1c2 s3  T sin(  ) / m

s3 s 2  s1c 2 c3  
0

  T / m 
c1c2
A.3.2.4.2.5
where T is the thrust, χ is the angular offset to the thrust vector, m is the instantaneous
mass and a is the components of the inertial acceleration.
Using Eq. A.3.2.4.2.4 and the Euler 3-1-2 rotation, we arrive at the relation between
inertial and body-fixed angular momentum in Eq. A.3.2.4.2.6.
 H X  c3 c2  s1 s 2 s3
 H   s c  s s c
 Y  3 2 1 2 3
 H Z    s 2 c1
 s3 c1
c3 c1
s1
c3 s 2  s1c2 s3   I 11 
s3 s 2  s1c 2 c3   I 2 2 
  I 33 
c1c2
A.3.2.4.2.6
where s and c are sine and cosine of the associated Euler angles, and the X,Y,Z subscripts
denote inertial unit vectors initially aligned with the 1,2,3 body-fixed unit vectors
respectively.
Discovering these quantities is useful in that it now allows us to calculate the pointing
error from relations as noted by Longuski, namely those presented in Eq. A.3.2.4.2.7A.3.2.4.2.8.1
X  H X / HZ
A.3.2.4.2.7
Y  H Y / H Z
A.3.2.4.2.8
where ρ is the inertial pointing error and H is the inertial angular velocity.
For this analysis, we make the simplifying assumption that there is only one moment
caused by thruster offset, given by Eq. A.3.2.4.2.9.
M 1  T ( L  d cm ) sin(  )
Author: Jeffrey Stuart
A.3.2.4.2.9
A.3.2.4 Spin Stabilization
4
where M is the moment, T is the thrust, L is the total length of the third stage from the
forward tip, dcm is the displacement of the center of mass from the forward tip, and χ is
the angular offset to the thrust vector.
From research conducted by N. Wilcox, the 3σ thrust offset χ is a constant 0.5871o for all
solid motors. Because the plots for each case are so similar, only the results from the 5kg
payload are discussed visually, while the plots for the 200g and 1kg payloads are placed
at the end of this section. Unless explicitly stated otherwise, any observation made about
the 5kg plots also applies to the 1kg and 200g payload cases.
Figure A.3.2.4.2.1 shows the angular momentum thrusting error plot for the full burn
time. The large band we see is in fact a spiral starting at the origin and moving inward
toward the point (0.05,0.75). This spiraling behavior is in large part a consequence of the
non-constant center of mass with additional influence from the changing moments of
inertia.
1.5
Hy/Hz (deg)
1
0.5
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Hx/Hz (deg)
Figure A.3.2.4.2.1: Third stage pointing error for 5kg payload over full burn time
(Author: Albert Chaney)
Though this plot provides information for the full burn time, the true area of interest is
the behavior during the final few seconds of the burn, which we have plotted in Fig.
A.3.2.4.2.2. We can find the final pointing error much easier from this plot than from
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
5
Fig. A.3.2.4.2.1. The final pointing error is found by simply finding the magnitude of the
vector from the origin (marked by an asterisk) to the center of the circle formed by the
pointing error curve.
While visual methods of calculating this displacement are
somewhat subjective, we can mitigate these effects by ensuring that the pointing error is
well below the limit of 1o. As we can see in Fig. 3.2.4.2.2, the final pointing error is
approximately 0.75o for a spin rate of 200rpm.
0.8
0.7
Hy/Hz (deg)
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Hx/Hz (deg)
Figure A.3.2.4.2.3: Third stage pointing error for 5kg payload during last few seconds of burn time
(Author: Albert Chaney)
To complete the analysis, we also compare the results from the angular momentum
method to the results for the direct calculation method. We plot the results for the direct
calculation in Fig. A.3.2.4.2.4. The pointing error traces out a large scale spiral from the
beginning of the burn to the end of the burn. The smaller oscillations about the spiral are
caused by the non-constant inertias and center of mass location. As we can see, the final
velocity pointing error is approximately 0.77o for the same spin rate of 200rpm.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
6
0.9
0.8
0.7
vY/vZ (deg)
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.2
0
0.2
0.4
0.6
0.8
vX/vZ (deg)
Figure A.3.2.4.2.4: Third stage pointing error for 5kg payload calculated directly from numerically
integrated velocities.
(Author: Jeffrey Stuart)
3.2.4.3 Spin-Up Maneuver
While the spinning-up maneuver is not complicated by non-constant moments of inertia,
we still developed the code under the assumption that such analysis would at some point
be useful, which is why all equations presented here include terms for varying moments
of inertia. Additionally, rather than calculate Euler angles and use them to calculate the
elements of a direction cosine matrix, the elements of the direction cosine matrix were
directly calculated from differential relations. We also simplify Euler’s equations to
arrive at the differential equations A.3.2.4.3.1-A.3.2.4.3.3.
P  I 1 1  I11  ( I 3  I 2 ) 2 3
A.3.2.4.3.1
0  I 2 2  I2 2  ( I 1  I 3 )1 3
A.3.2.4.3.2
S  I 3 3  I3 3
A.3.2.4.3.3
where S is the applied spin moment, P is the moment caused by thruster misalignment
(again assumed to be 0.5871o).
Using the same techniques as for the thrusting maneuver angular momentum method, we
determine the pointing error associated with the spinning-up maneuver.
Author: Jeffrey Stuart
In Fig.
A.3.2.4 Spin Stabilization
7
A.3.2.4.3.1, we plot the pointing error for the entire spin-up maneuver. From visual
inspection of this figure and Fig. A.3.2.4.2.3, we can see that the pointing error associated
with the thrusting maneuver is much larger than the pointing error from the spinning-up
maneuver. This is as we expect, for two reasons: first, the thrusting maneuver is of a
much longer duration than the spinning-up maneuver, and second, the thrusting maneuver
also has higher destabilizing moments than the spinning up maneuver.
0.3
0.25
Hy/Hz (deg)
0.2
0.15
0.1
0.05
0
-0.05
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Hx/Hz (deg)
Figure A.3.2.4.3.1: Pointing error from spinning-up maneuver of third stage, 5kg
(Author: Jeffrey Stuart)
There are several other constraints on the spin-up maneuver besides the goal of attaining
the needed spin rate for the thrusting maneuver. One major constraint that we face is the
structural limitations of the launch vehicle itself. From data provided by the S. Izzo in
the Structures group, we know that the maximum acceptable angular acceleration is
170rpm per second. Another constraint is that we must keep the mass of the spin up
system as low as possible to remain within the constant mass and inertia assumption from
the analysis. This constraint on the size will also necessitate that the spin up thrusters be
relatively low power.
Within AAE450_DNC_spin_offset, we can select the spin
moment and the duration of the spin up maneuver to satisfy these constraints. We
calculate the needed thrust from the spin moment and the diameter of the third stage,
assuming two small solid motors are used for the spin-up.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
8
Figure A.3.2.4.3.2 contains plots for the angular velocities about the body-fixed axes r̂ .
We note that the scale for ω3 is in revolutions per minute while the scales for ω1 and ω2
are in radians per second. The dotted line in the plot of ω3 is the desired angular velocity
for the thrusting maneuver. The plots of ω1 and ω2 indicate that they are sinusoidal in
nature of the form given by Eqs. A.3.2.4.3.4 and A.3.2.4.3.5.
Angular Velocity, rpm
Omega 3
300
200
Burn
200 rpm
100
0
0
0.5
1
1.5
2
-3
1
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
Omega 1
x 10
0.5
Angular Velocity, rad/sec
0
-0.5
-1
0
0.5
1
1.5
2
-3
1
2.5
Omega 2
x 10
0.5
0
-0.5
-1
0
0.5
1
1.5
2
2.5
Time, sec
Figure A.3.2.4.3.2: Body-fixed angular velocities during spin-up maneuver, 5kg
(Author: Jeffrey Stuart)
1  A sin( Kt 2 )  B cos( Kt 2 )
A.3.2.4.3.4
 2  C sin( Kt 2 )  D cos( Kt 2 )
A.3.2.4.3.5
where ω is the angular velocity, K is a constant determined by the inertia properties, and
A, B, C, and D are constants determined by the initial conditions.
A.3.2.4.4 Results
In Table A.3.2.4.4.1, we have summarized the required spin rate, spin-up thrust, and
pointing errors for all three payload cases. As is expected from the similar size of the
third stages, the required thrust and spin-up times are quite close across all three designs.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
9
Additionally, we satisfy the constraints by noting that the needed thrust from the spin-up
motors is less than 10 Newtons, while the required burn time is typically about 4.5
seconds. For all cases, the total pointing error is less than 0.9o, less than the required 1o.
We also note that the angular acceleration is well below the structural limit.
Table A.3.2.4.4.1 Third Stage Spin Results
Payload
200g
1kg
5kg
Spin rate
180
200
200
Spin moment
2.00
2.25
2.15
Thruster average force
7.35
7.76
7.82
Spin-up time
4.13
4.53
4.49
Spin-up pointing error
0.05
0.05
0.05
Thrusting pointing error
0.82
0.82
0.75
Total pointing error
0.87
0.87
0.80
Pointing error limit
1.00
1.00
1.00
Angular acceleration
43.62
44.15 44.55
Max angular accel
170.00 170.00 170.00
Units
rpm
N-m
N
sec
deg
deg
deg
deg
rpm/sec
rpm/sec
When we inspect the results, we come to the startling and rather amusing realization that
the results for the spin thrusters are in fact rather close to the performance characteristics
of low powered solid grain model rocket motors. Provided in Table A.3.2.4.4.2, we have
the performance characteristics of one such motor, the Estes E9.2 Though the idea seems
almost ludicrous at first, when we stop to consider the actual sizes of the third stages we
quickly realize that these motors are at least on the right scale of what we will need for
the physical rocket. We will have to design and build a system that is much more reliable
than the average model rocket motor, but we expect the resulting design to be very
similar to what we see in the low powered rocket community.
Table A.3.2.4.4.2 Estes E9 Performance Data
Total Impulse:
27.87 N-sec (σ = 0.42)
Peak Thrust:
19.47 N-sec (σ = 1.25)
Burn Time:
3.09 sec
(σ = 0.06)
Average Thrust:
9.02 N-sec
Propellant Mass:
35.80 g
Inert Mass:
22.10 g
Price
$15.19 Per 3 pack
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
10
A.3.2.4.5 Conclusions
The analysis presented here shows the characteristics of the spin-stabilized third stage of
our three launch vehicles. As designed, the third stage will have a velocity pointing error
of no more than 1o from the nominal and will undergo angular accelerations well below
structural failure values. The launch vehicle now has a viable scheme for removing the
avionics package from the third stage while still maintaining the required trajectory and
achieving the required orbit.
Though the work presented here is complete in and of itself, we can identify several
aspects that can be developed further:
1. The effect of a non-constant center of mass and moments of inertia can be more
fully investigated, in particular for the spin-up maneuver.
2. Longuski and Javorsek present methods for reducing pointing error using
specialized burning schemes.1,3 These techniques can be investigated further for
application to the launch vehicle design.
3. We have only simplified designs for the spin-up thrusters and spin table, designs
which must be more fleshed out to be truly viable.
References
1.
Longuski, J.M., Kia, T., and Breckenridge, W.G. “Annihilation of Angular Momentum Bias
During Thrusting and Spinning-up Maneuvers,” The Journal of the Astronautical Sciences, Vol.
37, No. 4, Oct-Dec 1989, pp. 433-450.
2.
“Estes E9,” National Association of Rocketry Standards and Testing [online], 2001, URL:
http://www.nar.org/pdf/Estes/E9.pdf
3.
Javorsek, D., and Longuski, J.M. “Velocity Pointing Errors Associated with Spinning Thrusting
Spacecraft,” Journal of Spacecraft and Rockets, Vol. 37, No. 3, May-June 2000, pp. 359-365.
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
11
A.3.2.4.6 Accessory Plots for 200g Launch Vehicle
1.6
1.4
Hy/Hz (deg)
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Hx/Hz (deg)
Figure A.3.2.4.6.1: Third stage pointing error for 200g payload over full burn time
(Author: Albert Chaney)
0.9
0.8
Hy/Hz (deg)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.4
-0.2
0
0.2
0.4
0.6
Hx/Hz (deg)
Figure A.3.2.4.6.2: Third stage pointing error for 200g payload during last few seconds of burn time
(Author: Albert Chaney)
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
12
1
0.9
0.8
vY/vZ (deg)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.2
0
0.2
0.4
0.6
0.8
vX/vZ (deg)
Figure A.3.2.4.6.3: Third stage pointing error for 200g payload calculated directly from numerically
integrated velocities.
(Author: Jeffrey Stuart)
0.35
0.3
Hy/Hz (deg)
0.25
0.2
0.15
0.1
0.05
0
-0.3
-0.2
-0.1
0
0.1
0.2
Hx/Hz (deg)
0.3
0.4
0.5
0.6
Figure A.3.2.4.6.4: Pointing error from spinning-up maneuver of third stage, 200g
(Author: Jeffrey Stuart)
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
13
Angular Velocity, rpm
Omega 3
200
150
100
Burn
180 rpm
50
0
0
0.5
1
1.5
2
Angular Velocity, rad/sec
-3
1
2.5
3
3.5
4
4.5
3
3.5
4
4.5
3
3.5
4
4.5
Omega 1
x 10
0.5
0
-0.5
-1
0
0.5
1
1.5
2
-3
1
2.5
Omega 2
x 10
0.5
0
-0.5
-1
0
0.5
1
1.5
2
2.5
Time, sec
Figure A.3.2.4.6.5: Body-fixed angular velocities during spin-up maneuver, 200g
(Author: Jeffrey Stuart)
A.3.2.4.7 Accessory Plots for 1kg Launch Vehicle
1.6
1.4
Hy/Hz (deg)
1.2
1
0.8
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Hx/Hz (deg)
Figure A.3.2.4.7.1: Third stage pointing error for 1kg payload over full burn time
(Author: Albert Chaney)
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
14
0.9
0.8
Hy/Hz (deg)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.4
-0.2
0
0.2
0.4
0.6
Hx/Hz (deg)
Figure A.3.2.4.7.2: Third stage pointing error for 1kg payload during last few seconds of burn time
(Author: Albert Chaney)
1
0.9
0.8
vY/vZ (deg)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.2
0
0.2
0.4
0.6
0.8
vX/vZ (deg)
Figure A.3.2.4.7.3: Third stage pointing error for 1kg payload calculated directly from numerically
integrated velocities.
(Author: Jeffrey Stuart)
Author: Jeffrey Stuart
A.3.2.4 Spin Stabilization
15
0.3
0.25
Hy/Hz (deg)
0.2
0.15
0.1
0.05
0
-0.05
-0.3
-0.2
-0.1
0
0.1
0.2
Hx/Hz (deg)
0.3
0.4
0.5
0.6
Figure A.3.2.4.7.4: Pointing error from spinning-up maneuver of third stage, 1kg
(Author: Jeffrey Stuart)
Angular Velocity, rpm
Omega 3
300
200
Burn
200 rpm
100
0
0
0.5
1
1.5
2
Angular Velocity, rad/sec
-3
1
2.5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
3
3.5
4
4.5
5
Omega 1
x 10
0.5
0
-0.5
-1
0
0.5
1
1.5
2
-3
1
2.5
Omega 2
x 10
0.5
0
-0.5
-1
0
0.5
1
1.5
2
2.5
Time, sec
Figure A.3.2.4.7.5: Body-fixed angular velocities during spin-up maneuver, 1kg
(Author: Jeffrey Stuart)
Author: Jeffrey Stuart