# Design Methodology and Performance Evaluation of New Generation Sounding Rockets

Authors: Marco Pallone; Mauro Pontani; Paolo Teofilatto; Angelo Minotti
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

Sounding rockets are currently deployed for the purpose of providing experimental data of the upper atmosphere, as well as for microgravity experiments. This work provides a methodology in order to design, model, and evaluate the performance of new sounding rockets. A general configuration composed of a rocket with four canards and four tail wings is sized and optimized, assuming different payload masses and microgravity durations. The aerodynamic forces are modeled with high fidelity using the interpolation of available data. Three different guidance algorithms are used for the trajectory integration: constant attitude, near radial, and sun-pointing. The sun-pointing guidance is used to obtain the best microgravity performance while maintaining a specified attitude with respect to the sun, allowing for experiments which are temperature sensitive. Near radial guidance has instead the main purpose of reaching high altitudes, thus maximizing the microgravity duration. The results prove that the methodology at hand is straightforward to implement and capable of providing satisfactory performance in term of microgravity duration.


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## Body

## 1. Introduction

Sounding rockets are specialized missiles generally used to investigate the region of space between 50 and 700 km where it is difficult to enter with the traditional atmospheric balloon or low orbit satellites. They can be used in several missions, such as detection of the solar activity and anomalies, analysis of the constituents of the upper atmosphere, thermal analysis on new materials, and generally, measurements of the space surrounding the Earth [1–3]. The most important characteristic of a sounding rocket is the capability of achieving microgravity conditions for the payload, without the need of a manned mission in the ISS to perform experiments and then reducing the costs of the mission. The quality of microgravity depends on the absence of drag and other gyroscopic forces, so the guidance of the rocket assumes the utmost importance. Another good characteristic of the sounding rocket in contrast to the satellite missions is the sounding rocket’s ability to retrieve the payload with the help of a heat shield and a parachute. For all of these reasons, a vast number of sounding rocket families were developed, such as the American “Black Brant” and the European “Maxus.” The Black Brant family includes sounding rockets that range from one through three stages. The Black Brant VC [4] (Figure 1) can be launched from a steerable launch tower with different kick angles to achieve different apogee altitudes. It is controlled via spin-up motors and canard wings, which provide a fast response at the cost of small instability. Four rear tail surfaces guarantee an adequate static margin. The Maxus [4, 5] is a very powerful European single-stage sounding rocket launched in Kiruna (Sweden) and can reach a maximum altitude of 700 km, while the range does not exceed 100 km. It has a microgravity duration of about 12 min, and the microgravity level Δg is near 10−4m/sec2. TVC control system is highly sophisticated in order to guarantee strict margins for the range and to avoid damages to both people and goods. Table 1 portrays some useful data to compare the performance of the two rockets.Figure 1
Black Brant VC.Table 1
“Maxus” and “Black Brant VC” data.
BBVC
Maxus
Length (m)
8.3
16.2
Max diameter (m)
0.44
1
Mass liftoff (kg)
1544
12298
Propellant mass (kg)
1017
10042
Av. thrust (N)
69374
372500
Burnout time (sec)
32
63Both of these sounding rockets are propelled with a solid state motor, due to their simplicity and light weights with respect to liquid propellant. A crucial point is that a sounding rocket has to be cheaper than a high-tech rocket, and in order to achieve this objective, new design methodologies are needed. Many authors (e.g., Chowdhury et al. [6, 7] and Casalino and Pastrone [8, 9]) are interested in designing new sounding rockets to explore the potentiality of new hybrid motors for hypersonic flight. Nonaka et al. [10] and Woo et al. [11] investigated the reusability of the sounding rocket, and Cremaschi et al. [12] investigated the optimization of a trajectory with the use of a commercial trajectory simulation tool (ASTOS). However, a very limited number of studies are concerned with fast sizing and aerodynamic losses during the ascent path. The purpose of this work is to find a novel and simple methodology to size and design for future sounding rockets, considering accurate aerodynamics and three guidance algorithms. In particular, the technique explained in this work is applied to a single-stage rocket whose three-dimensional trajectory is simulated using the following guidance schemes:
(i)
Constant attitude guidance(ii)
Near radial guidance(iii)
Sun-pointing guidanceThis work purpose is to find a procedure that can provide the minimal volume rocket that can guarantee a fixed time of microgravity with different scientific payloads inside. Definitely, the methodology at hand yields configurations that can provide a well-developed preliminary design for future sounding rockets, with an accurate aerodynamic model and using three distinct guidance schemes.


## 2. Rocket Geometry and Sizing

### 2.1. Basic Geometry

With the purpose of maintaining the configuration as simple as possible, the sounding rocket is composed of one stage and a module hosting the avionics, experiments, and recovery system (parachute and the heat shield).The basic geometry of the rocket is composed of the following:(i)
A nose cone of parabolic shape(ii)
A fuselage of cylindrical shape(iii)
Four canard wings in cruciform configuration(iv)
Four tail surfaces of trapezoidal shapeThis configuration is chosen with the purpose of creating an algorithm as general as possible. Incidentally, this geometry is adopted by a lot of sounding rockets in the market, like the “Black Brant VC” in Figure1.Every rocket configuration in this work is defined by the total lengthl and the maximum diameter d, while all the other parameters like the propulsive ones are obtained through scaling factors. The grid used for the computation is
(i)
l from 3 m to 15 m with step of 1 m(ii)
d from 0.2 m to 1.5 m with step of 0.05 m.Thus, a total of 243 different configurations were analyzed. The design methodology that is being presented relates geometry, weight, aerodynamics, and propulsive performances.


### 2.2. Main Body

For each configuration, the length of the ogive is proportional to the total length of the sounding rocket(1)logive=Cogivel,where Cogive=0.17 is a dimensionless coefficient. The dimensional ogive point distribution is then obtained by multiplying the nondimensional points in the (ξ, η) frame in Figure 2 with logive and d. The main body, which contains the propulsive system and the avionics, is cylindrical with diameter d, while its length is found through
(2)lbody=l−logive.Figure 2
Main surfaces and reference quantities.
(a)
Canard and tail wing geometry. The profile adopted is a symmetric double-wedge profile
(b)
Canard and tail wing disposition along the longitudinal axis
(c)
Nondimensional ogive


### 2.3. Aerodynamic Surfaces

The aerodynamic surfaces are sized according to the following ratios:(3)AP=l−logived,AC=145AP,AT=112AP.Then the following quantities are computed as follows:(4)hT=43AC,hC=2AT,while the distances with respect to the nose tip are as follows:
(5)dC=logive+0.4m,dT=l−hT.The main concept under this procedure is to maintain the shape of the wings, while selecting their size in relation to the overall size of the main body. This is in order to avoid wings with nonconventional shapes that would result in undesired aerodynamic effects. Figure2 portrays the wing geometry with the profile adopted for both wing systems. The nonvariable distance in (5) 0.4 m is meant to provide a fixed volume needed for the canard avionic and the recovery system. Figure 2 portrays an example of a configuration with the disposition of the leading edges along the longitudinal axis of the rocket.


### 2.4. Mass Scaling

In order to describe properly a real sounding rocket mission, a fixed mass including parachute, heat shield, and metal case is used for every configuration. Table2 portrays in detail these auxiliary masses, which form maux.Table 2
Auxiliary massmaux of the ogive.
Parachute mass (kg)
10
Heat shield mass (kg)
35
Metal case mass (kg)
5
Totalmaux (kg)
50With regard to the initial mass at launch, a common initial density is used in order to find a reasonable and scaled mass distribution for every rocket configuration. This initial density isCm0=1092.3kg/m3, which is taken from the Black Brant VC and is considered to have a reasonable value. This value includes propellant, structural, and auxiliary masses. So the initial mass for every configuration without the ogive mass is
(6)m0body=Cm0Vbody.For every configuration. 50, 100, and 200 kg of scientific payload are used, so the total initial mass at launch is(7)m0=m0body+mU+maux.The propulsive technology of this single-stage rocket is a solid propellant one, allowing for the following structural coefficient:(8)ϵ=mSmP+mS=0.11.The structural and propellant masses are obtained through the following expressions:(9)u=mU+ϵm0−ϵmUm0,mP=m01−u,mS=ϵm01−u1−ϵ.In the end, (6), (7), (8), and (9) yield the overall rocket mass at launch, portrayed in Figure3 as a function of l and d, together with the corresponding propellant mass.Figure 3
Example of scaledm0 and mP with mU=50kg.
(a)
Initial mass
(b)
Propellant mass


### 2.5. Propulsion System Scaling

To scale the thrust for every sounding rocket size, the initial launch acceleration and the specific impulse are maintained constant:(10)T=n0gm0,where n0=3g, while the propellant mass consumption is
(11)ṁ=−TIspg.The value forn0 is chosen in order to constrain the rocket dynamical stress at launch. Equation (11) leads to the burnout time:
(12)tB=mPṁ.Figure4 portrays the thrust and the burnout time scaling with the rocket size.Figure 4
Example of scaledT and tB with mU = 50 kg.
(a)
Thrust
(b)
Burnout time


## 2.1. Basic Geometry

With the purpose of maintaining the configuration as simple as possible, the sounding rocket is composed of one stage and a module hosting the avionics, experiments, and recovery system (parachute and the heat shield).The basic geometry of the rocket is composed of the following:(i)
A nose cone of parabolic shape(ii)
A fuselage of cylindrical shape(iii)
Four canard wings in cruciform configuration(iv)
Four tail surfaces of trapezoidal shapeThis configuration is chosen with the purpose of creating an algorithm as general as possible. Incidentally, this geometry is adopted by a lot of sounding rockets in the market, like the “Black Brant VC” in Figure1.Every rocket configuration in this work is defined by the total lengthl and the maximum diameter d, while all the other parameters like the propulsive ones are obtained through scaling factors. The grid used for the computation is
(i)
l from 3 m to 15 m with step of 1 m(ii)
d from 0.2 m to 1.5 m with step of 0.05 m.Thus, a total of 243 different configurations were analyzed. The design methodology that is being presented relates geometry, weight, aerodynamics, and propulsive performances.


## 2.2. Main Body

For each configuration, the length of the ogive is proportional to the total length of the sounding rocket(1)logive=Cogivel,where Cogive=0.17 is a dimensionless coefficient. The dimensional ogive point distribution is then obtained by multiplying the nondimensional points in the (ξ, η) frame in Figure 2 with logive and d. The main body, which contains the propulsive system and the avionics, is cylindrical with diameter d, while its length is found through
(2)lbody=l−logive.Figure 2
Main surfaces and reference quantities.
(a)
Canard and tail wing geometry. The profile adopted is a symmetric double-wedge profile
(b)
Canard and tail wing disposition along the longitudinal axis
(c)
Nondimensional ogive


## 2.3. Aerodynamic Surfaces

The aerodynamic surfaces are sized according to the following ratios:(3)AP=l−logived,AC=145AP,AT=112AP.Then the following quantities are computed as follows:(4)hT=43AC,hC=2AT,while the distances with respect to the nose tip are as follows:
(5)dC=logive+0.4m,dT=l−hT.The main concept under this procedure is to maintain the shape of the wings, while selecting their size in relation to the overall size of the main body. This is in order to avoid wings with nonconventional shapes that would result in undesired aerodynamic effects. Figure2 portrays the wing geometry with the profile adopted for both wing systems. The nonvariable distance in (5) 0.4 m is meant to provide a fixed volume needed for the canard avionic and the recovery system. Figure 2 portrays an example of a configuration with the disposition of the leading edges along the longitudinal axis of the rocket.


## 2.4. Mass Scaling

In order to describe properly a real sounding rocket mission, a fixed mass including parachute, heat shield, and metal case is used for every configuration. Table2 portrays in detail these auxiliary masses, which form maux.Table 2
Auxiliary massmaux of the ogive.
Parachute mass (kg)
10
Heat shield mass (kg)
35
Metal case mass (kg)
5
Totalmaux (kg)
50With regard to the initial mass at launch, a common initial density is used in order to find a reasonable and scaled mass distribution for every rocket configuration. This initial density isCm0=1092.3kg/m3, which is taken from the Black Brant VC and is considered to have a reasonable value. This value includes propellant, structural, and auxiliary masses. So the initial mass for every configuration without the ogive mass is
(6)m0body=Cm0Vbody.For every configuration. 50, 100, and 200 kg of scientific payload are used, so the total initial mass at launch is(7)m0=m0body+mU+maux.The propulsive technology of this single-stage rocket is a solid propellant one, allowing for the following structural coefficient:(8)ϵ=mSmP+mS=0.11.The structural and propellant masses are obtained through the following expressions:(9)u=mU+ϵm0−ϵmUm0,mP=m01−u,mS=ϵm01−u1−ϵ.In the end, (6), (7), (8), and (9) yield the overall rocket mass at launch, portrayed in Figure3 as a function of l and d, together with the corresponding propellant mass.Figure 3
Example of scaledm0 and mP with mU=50kg.
(a)
Initial mass
(b)
Propellant mass


## 2.5. Propulsion System Scaling

To scale the thrust for every sounding rocket size, the initial launch acceleration and the specific impulse are maintained constant:(10)T=n0gm0,where n0=3g, while the propellant mass consumption is
(11)ṁ=−TIspg.The value forn0 is chosen in order to constrain the rocket dynamical stress at launch. Equation (11) leads to the burnout time:
(12)tB=mPṁ.Figure4 portrays the thrust and the burnout time scaling with the rocket size.Figure 4
Example of scaledT and tB with mU = 50 kg.
(a)
Thrust
(b)
Burnout time


## 3. Rocket Aerodynamic Modeling

Aerodynamic forces affect the rocket motion, which partially occurs in the atmosphere. Thus, the rocket aerodynamics is to be modeled appropriately. In this work, four quantities are assumed to determine the aerodynamic forces: Mach numberM, Reynolds number Re, angle of attack α, and sideslip angle β. To provide an accurate model for each configuration, two fundamental steps are needed:
(1)
Derivation of the aerodynamic coefficientsCD, CL, and CQ from DATCOM [13] at a relevant number of Mach, Reynolds, angles of attack, and sideslip angles(2)
Cubic interpolation of the aerodynamic coefficients during the integration of the dynamics equationsFollowing the approach presented by Mangiacasale [14], Pallone et al. [15], and Pontani [16], the aerodynamics of the rocket was modeled through the Missile DATCOM software [13]. This software provides an accurate esteem of the aerodynamic forces without using a lot of computational resources; thus, it can provide a complete aerodynamic database for each configuration in a short time. The following discrete values of M, Re, α, and β are used:
(i)
M = [0.2, 0.5, 0.6, 0.8, 0.9, 1.1, 1.2, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 10](ii)
Re = [0.5, 1, 1.5, 2]∗105(iii)
α = [0, 1, 2, 3, 4, 5] deg(iv)
β = [0, 1, 2, 3, 4, 5] degThis means that DATCOM was run 3024 times for eachl,d combination. The aerodynamic surfaces used in the computation are the cross surfaces of each configuration and correspond to the circular section, that is,
(13)Si=πd22.Figure5 portrays an example of the aerodynamic coefficients for a single configuration.Figure 5
Example of the drag force coefficients provided by the DATCOM software for the configurationl=6 m and d=0.8 m.
(a)
Drag coefficient forβ=0 deg and Re=4∗107
(b)
Drag coefficient forM=3 and Re=4∗107


## 4. Rocket Dynamics

The problem investigated in this work consists in finding the sounding rocket with the minimum total volume that can achieve microgravity durations not less than 10 or 15 minutes. In the absence of significant attitude angular velocities, a rocket is considered in microgravity conditions when it is flying at an altitude above 100 km [5].This approximation is based on real evidence and calculations, allowing for it to be used as a practical rule to analyze the rocket performance. Thus, the problem at hand reduces to finding the minimal rocket volumel,d that can guarantee ΔTmg of 10 and 15 min, fixing the propulsive technology (Isp,ϵ) and the geometrical ratios between its parts (Cm0,Cogive,AC/AP,AT/AP).


### 4.1. Reference Frames

A description of the reference frames used in this work is useful to understand the guidance implementation. In the following, the notationRj∓ξ (j = 1, 2, 3) denotes the elementary rotation about axis j by angle ξ while the ∓ sign means, respectively, clockwise and counterclockwise.
(1)
TheEarth-centered frame (ECI) c^1,c^2,c^3 is a Cartesian inertial reference frame defined as follows. Its origin O is the center of the Earth. The unit vector c^3 is aligned with the Earth axis of rotation and is positive northward, whereas c^1 is aligned with the vernal axis, which corresponds to the intersection of the Earth equatorial plane with the ecliptic plane.(2)
TheEarth-centered Earth-fixed frame (ECEF) i^,j^,k^ represents a reference system that is rigidly attached to the Earth. Under the assumption that the planet rotates with a constant angular rate ωE, also the ECEF frame rotates with angular rate ωE. The origin of the coordinate system is the center of the Earth, while the unit vector i^ intersects the Greenwich reference meridian at all times, and k^ is aligned with the planet axis of rotation and is positive northward. ωEk^ is the vector rotation rate of the ECEF frame with respect to the ECI frame. The reference meridian is the Greenwich side real-time θG, given by θGt=θGt¯+ωEt−t¯ where t¯ refers to a generic time instant. The ECEF frame is obtained through a counterclockwise rotation by angle θG around the third axis, that is, i^,j^,k^T=R3θGc^1,c^2,c^3T.(3)
Thelocal horizontal frame (LH) (r^,E^,N^) is a reference system with the origin in the rocket center of mass. Specifically, r^ is the unit vector aligned with the instantaneous position r^, E^ is directed along the local east direction, and N^ is aligned with the local north direction. LH frame is obtained from the ECEF frame through the following rotations: r^,E^,N^T=R2−LaR3λgi^,j^,k^T.(4)
Theauxiliary orbital frame (AO) r^,θ^r,h^r is defined making reference to the rocket relative velocity vR=vI−ωE×r. The unit vector θ^R is aligned with the projection of the velocity vr onto the horizontal plane (E^,N^). The AO frame is obtained from the LH frame with the simple rotation r^,θ^R,h^RT=R1ζRr^,E^,N^T.(5)
Therelative velocity frame (RV) (n^R,v^R,h^R) is obtained from the AO frame through the following clockwise rotation: n^R,v^R,h^RT=R3−γRr^,θ^R,h^RT.(6)
Thewind axis frame (WA) (n^W,v^W,h^W) is defined with reference to the rocket velocity with respect to the local velocity of the atmosphere va=ωE×r so va=vR and just a counterclockwise rotation about axis 2 by the bank angle σ: n^W,v^W,h^WT=R2σn^R,v^R,h^RT.(7)
Theauxiliary body axis frame (ABA) (i^B,j^B,k^B) is a reference system aligned with the rocket inertia axes. The unit vector k^B is orthogonal to the rocket plane of symmetry, while the two remaining unit vectors lie on it. The ABA frame is obtained from the WA frame through a sequence of two rotations: σi^B,j^B,k^BT=R3−αR1βn^W,v^W,h^WT.(8)
Thebody axis (BA) (x^B,y^B,z^B) is, like the ABA frame, also attached to the rocket inertial axes only with a different orientation. Specifically, x^B is aligned with the rocket longitudinal axis (x^B=j^B); z^B is directed downward so z^B=−i^B and as consequence y^B=−k^B. These relations lead to the following elementary rotation:
(14)x^By^Bz^B=01000−1−100i^Bj^Bk^B=RAi^Bj^Bk^B.The knowledge of the instantaneous orientation of (x^B,y^B,z^B) is equivalent to identifying the instantaneous attitude of the rocket.(9)
Thelaunch inertial frame (LI) (i^0,j^0,k^0) is an inertial system centered in the launch location at the launch instant. The unit vector i^0 is aligned with the local north direction, j^0 is directed toward the local east direction, and k^0 is aligned with the downward direction. The LI frame is obtained from the ECI frame with a sequence of three rotations: the first two of them take the ECI frame to an intermediary reference frame, named auxiliary launch inertial frame (ALI) (r^0,E^0,n^0) using the geographical latitude and longitude of the launch base at the launch time, while the third one relates the axes of the ALI frame to the LI frame. The complete sequence is as follows:
(15)i^0j^0k^0=R2−LLR3λaL001010−100c^1c^2c^3=R2−LLR3λaLRBc^1c^2c^3.The BA frame is also related to the LI frame through a sequence of three rotations using the attitude angles (ϕ,θ,ψ). The complete sequence is x^B,y^B,z^BT=R1ϕR2θR3ψi^0,j^0,k^0T.In order to avoid ambiguities in the definition of the vectors of interest, the angles introduced in this section have to be constrained. Table3 shows the limitation for the angles used in this work.Table 3
Angle constraints.
−
π
<
θ
G
≤
π
−
π
<
λ
G
≤
π
−
π
2
<
L
a
≤
π
2
−
π
<
ζ
R
≤
π
−
π
2
<
γ
R
≤
π
2
−
π
<
σ
≤
π
−
π
2
<
β
≤
π
2
−
π
<
α
≤
π
−
π
<
ψ
≤
π
−
π
2
<
θ
≤
π
2
−
π
<
ϕ
≤
π
—Any vector can be obtained with respect to any reference frame using the rotations defined in this section. These rotations can be used to define the rocket kinematics and dynamics through the motion, and Figures6 and 7 portray all the reference frames used in this work, together with the related rotations.Figure 6
Reference frames and related rotation matrices.Figure 7
Main reference frames and related angles.
(a)
ECI-ECEF-LH frames and their respective angles
(b)
LH frame
(c)
WA-ABA frames and their relative angles
(d)
Thrust vector direction with respect to the RV frame


### 4.2. Forces

The main term of the gravitational forceG is given in the RV frame simply by
(16)Gm=−μEr3=−μEr2cosγRsinγR0Tn^Rv^Rh^R.The components of the aerodynamic force can be referred to the WA frame and then projected in the RV frame with a simple rotation. The three components are the lift force, the drag force, and the side force.(17)Am=L+D+Qm=Ln^W−Dv^W+Qh^Wm,where
(18)L=Ln^W=12ρvW2CLα,β,Re,Mn^W,D=−Dn^W=−12ρvW2CDα,β,Re,Mn^W,Q=Qn^W=12ρvW2CQα,β,Re,Mn^W.The aerodynamic forces have to be projected in the RV frame with a rotation about the bank angle around the second axis:(19)A=AnAvAhn^Rv^Rh^R=L−DQR2σn^Rv^Rh^R.The aerodynamic forces are taken into account up to 100 km of altitude where the atmospheric density becomes negligible.The thrust can be projected along the RV frame with the in-planeαT and the out-of-plane βT:
(20)Tm=TmcosβTsinαTTmcosβTcosαTTmsinβTn^Rv^Rh^RT,and to avoid ambiguities, the following constraints must hold:
(21)−π<αT≤π,−π2≤βT≤π2.The thrust vectorT is considered aligned with the rocket longitudinal axis so two angles, αT and βT suffices to describe its direction.


### 4.3. Equations of Motion

The equations of motion that govern the three-dimensional rocket dynamics can be conveniently written in terms ofr,λG,La,γR,vR,ζR. These variables refer to the relative motion in the RV frame. They form the state vector xR of the launch vehicle (in rotating coordinates).
(22)ṙ=vRsinγR,λ̇G=vRcosγRcosζRrcosLa,L̇a=vRcosγRsinζRr,γ̇R=−μEvRr2+vRrcosγR+T/mcosβTsinαTvR+AnvR+ωE2rvRcosLacosLacosγR+sinLasinγRsinζR+2ωEcosLacosζR,v̇R=−μEr2sinγR+TmcosβTcosαT+Av+ωE2rcosLacosLasinγR−sinLacosγRsinζR,ζ̇R=−vRrtanLacosγRcosζR+T/mvRcosγRsinβT+2ωEcosLatanγRsinζR+AhvRcosγR−ωE2rvRcosγRsinLacosLacosζR−2ωEsinLa.The initial values for (22) are reported in Table 4.Table 4
Initial condition for (22).
r (km)
R
E+0.001
v
R
k
m
/
s
0.008
γ
R (deg)
86
L
a (deg)
−2.938
ζ
R (deg)
0
λ
G (deg)
40.213Equation (22) is valid for both the propelled arc and the ballistic flight of the ogive. The values for La and λG in Table 4 are referred to the Malindi launch base, while the launch is in the east direction (ζR). The control laws for (αT,βT), are determined using three distinct guidance scheme, namely, the constant attitude guidance, the near radial guidance, and the sun-pointing guidance, which are described in the following section. The state xR is continuos across the first stage separation, which occurs at time tB. After that, the payload continues its motion in a ballistic trajectory.


## 4.1. Reference Frames

A description of the reference frames used in this work is useful to understand the guidance implementation. In the following, the notationRj∓ξ (j = 1, 2, 3) denotes the elementary rotation about axis j by angle ξ while the ∓ sign means, respectively, clockwise and counterclockwise.
(1)
TheEarth-centered frame (ECI) c^1,c^2,c^3 is a Cartesian inertial reference frame defined as follows. Its origin O is the center of the Earth. The unit vector c^3 is aligned with the Earth axis of rotation and is positive northward, whereas c^1 is aligned with the vernal axis, which corresponds to the intersection of the Earth equatorial plane with the ecliptic plane.(2)
TheEarth-centered Earth-fixed frame (ECEF) i^,j^,k^ represents a reference system that is rigidly attached to the Earth. Under the assumption that the planet rotates with a constant angular rate ωE, also the ECEF frame rotates with angular rate ωE. The origin of the coordinate system is the center of the Earth, while the unit vector i^ intersects the Greenwich reference meridian at all times, and k^ is aligned with the planet axis of rotation and is positive northward. ωEk^ is the vector rotation rate of the ECEF frame with respect to the ECI frame. The reference meridian is the Greenwich side real-time θG, given by θGt=θGt¯+ωEt−t¯ where t¯ refers to a generic time instant. The ECEF frame is obtained through a counterclockwise rotation by angle θG around the third axis, that is, i^,j^,k^T=R3θGc^1,c^2,c^3T.(3)
Thelocal horizontal frame (LH) (r^,E^,N^) is a reference system with the origin in the rocket center of mass. Specifically, r^ is the unit vector aligned with the instantaneous position r^, E^ is directed along the local east direction, and N^ is aligned with the local north direction. LH frame is obtained from the ECEF frame through the following rotations: r^,E^,N^T=R2−LaR3λgi^,j^,k^T.(4)
Theauxiliary orbital frame (AO) r^,θ^r,h^r is defined making reference to the rocket relative velocity vR=vI−ωE×r. The unit vector θ^R is aligned with the projection of the velocity vr onto the horizontal plane (E^,N^). The AO frame is obtained from the LH frame with the simple rotation r^,θ^R,h^RT=R1ζRr^,E^,N^T.(5)
Therelative velocity frame (RV) (n^R,v^R,h^R) is obtained from the AO frame through the following clockwise rotation: n^R,v^R,h^RT=R3−γRr^,θ^R,h^RT.(6)
Thewind axis frame (WA) (n^W,v^W,h^W) is defined with reference to the rocket velocity with respect to the local velocity of the atmosphere va=ωE×r so va=vR and just a counterclockwise rotation about axis 2 by the bank angle σ: n^W,v^W,h^WT=R2σn^R,v^R,h^RT.(7)
Theauxiliary body axis frame (ABA) (i^B,j^B,k^B) is a reference system aligned with the rocket inertia axes. The unit vector k^B is orthogonal to the rocket plane of symmetry, while the two remaining unit vectors lie on it. The ABA frame is obtained from the WA frame through a sequence of two rotations: σi^B,j^B,k^BT=R3−αR1βn^W,v^W,h^WT.(8)
Thebody axis (BA) (x^B,y^B,z^B) is, like the ABA frame, also attached to the rocket inertial axes only with a different orientation. Specifically, x^B is aligned with the rocket longitudinal axis (x^B=j^B); z^B is directed downward so z^B=−i^B and as consequence y^B=−k^B. These relations lead to the following elementary rotation:
(14)x^By^Bz^B=01000−1−100i^Bj^Bk^B=RAi^Bj^Bk^B.The knowledge of the instantaneous orientation of (x^B,y^B,z^B) is equivalent to identifying the instantaneous attitude of the rocket.(9)
Thelaunch inertial frame (LI) (i^0,j^0,k^0) is an inertial system centered in the launch location at the launch instant. The unit vector i^0 is aligned with the local north direction, j^0 is directed toward the local east direction, and k^0 is aligned with the downward direction. The LI frame is obtained from the ECI frame with a sequence of three rotations: the first two of them take the ECI frame to an intermediary reference frame, named auxiliary launch inertial frame (ALI) (r^0,E^0,n^0) using the geographical latitude and longitude of the launch base at the launch time, while the third one relates the axes of the ALI frame to the LI frame. The complete sequence is as follows:
(15)i^0j^0k^0=R2−LLR3λaL001010−100c^1c^2c^3=R2−LLR3λaLRBc^1c^2c^3.The BA frame is also related to the LI frame through a sequence of three rotations using the attitude angles (ϕ,θ,ψ). The complete sequence is x^B,y^B,z^BT=R1ϕR2θR3ψi^0,j^0,k^0T.In order to avoid ambiguities in the definition of the vectors of interest, the angles introduced in this section have to be constrained. Table3 shows the limitation for the angles used in this work.Table 3
Angle constraints.
−
π
<
θ
G
≤
π
−
π
<
λ
G
≤
π
−
π
2
<
L
a
≤
π
2
−
π
<
ζ
R
≤
π
−
π
2
<
γ
R
≤
π
2
−
π
<
σ
≤
π
−
π
2
<
β
≤
π
2
−
π
<
α
≤
π
−
π
<
ψ
≤
π
−
π
2
<
θ
≤
π
2
−
π
<
ϕ
≤
π
—Any vector can be obtained with respect to any reference frame using the rotations defined in this section. These rotations can be used to define the rocket kinematics and dynamics through the motion, and Figures6 and 7 portray all the reference frames used in this work, together with the related rotations.Figure 6
Reference frames and related rotation matrices.Figure 7
Main reference frames and related angles.
(a)
ECI-ECEF-LH frames and their respective angles
(b)
LH frame
(c)
WA-ABA frames and their relative angles
(d)
Thrust vector direction with respect to the RV frame


## 4.2. Forces

The main term of the gravitational forceG is given in the RV frame simply by
(16)Gm=−μEr3=−μEr2cosγRsinγR0Tn^Rv^Rh^R.The components of the aerodynamic force can be referred to the WA frame and then projected in the RV frame with a simple rotation. The three components are the lift force, the drag force, and the side force.(17)Am=L+D+Qm=Ln^W−Dv^W+Qh^Wm,where
(18)L=Ln^W=12ρvW2CLα,β,Re,Mn^W,D=−Dn^W=−12ρvW2CDα,β,Re,Mn^W,Q=Qn^W=12ρvW2CQα,β,Re,Mn^W.The aerodynamic forces have to be projected in the RV frame with a rotation about the bank angle around the second axis:(19)A=AnAvAhn^Rv^Rh^R=L−DQR2σn^Rv^Rh^R.The aerodynamic forces are taken into account up to 100 km of altitude where the atmospheric density becomes negligible.The thrust can be projected along the RV frame with the in-planeαT and the out-of-plane βT:
(20)Tm=TmcosβTsinαTTmcosβTcosαTTmsinβTn^Rv^Rh^RT,and to avoid ambiguities, the following constraints must hold:
(21)−π<αT≤π,−π2≤βT≤π2.The thrust vectorT is considered aligned with the rocket longitudinal axis so two angles, αT and βT suffices to describe its direction.


## 4.3. Equations of Motion

The equations of motion that govern the three-dimensional rocket dynamics can be conveniently written in terms ofr,λG,La,γR,vR,ζR. These variables refer to the relative motion in the RV frame. They form the state vector xR of the launch vehicle (in rotating coordinates).
(22)ṙ=vRsinγR,λ̇G=vRcosγRcosζRrcosLa,L̇a=vRcosγRsinζRr,γ̇R=−μEvRr2+vRrcosγR+T/mcosβTsinαTvR+AnvR+ωE2rvRcosLacosLacosγR+sinLasinγRsinζR+2ωEcosLacosζR,v̇R=−μEr2sinγR+TmcosβTcosαT+Av+ωE2rcosLacosLasinγR−sinLacosγRsinζR,ζ̇R=−vRrtanLacosγRcosζR+T/mvRcosγRsinβT+2ωEcosLatanγRsinζR+AhvRcosγR−ωE2rvRcosγRsinLacosLacosζR−2ωEsinLa.The initial values for (22) are reported in Table 4.Table 4
Initial condition for (22).
r (km)
R
E+0.001
v
R
k
m
/
s
0.008
γ
R (deg)
86
L
a (deg)
−2.938
ζ
R (deg)
0
λ
G (deg)
40.213Equation (22) is valid for both the propelled arc and the ballistic flight of the ogive. The values for La and λG in Table 4 are referred to the Malindi launch base, while the launch is in the east direction (ζR). The control laws for (αT,βT), are determined using three distinct guidance scheme, namely, the constant attitude guidance, the near radial guidance, and the sun-pointing guidance, which are described in the following section. The state xR is continuos across the first stage separation, which occurs at time tB. After that, the payload continues its motion in a ballistic trajectory.


## 5. Guidance Algorithms

### 5.1. Constant Attitude Guidance

With this guidance, the rocket orientation does not change during the flight and was actually implemented in the Black Brant X (cf. [17]). This is ideal to achieve nominal (perfect) microgravity conditions (Δg=0), due to rotation rates equal to 0. The attitude angles ϕ,θ,ψ remain constant; therefore, in this type of guidance, the following expressions can be assumed as follows:
(23)ϕ=ϕ0θ=θ0ψ=ψ0.The remaining anglesαT,βT,α,β,σ can be found using analytical formulas at any times using the rotation matrices portrayed in Figure 6.


### 5.2. Near Radial Guidance

This guidance is called near radial because it is defined in the AO reference frame and the thrust direction is defined by(24)αT=γR0+ttBπ2−γR0−γR,βT=0whereas γR0=θ0 is the pitch angle at the initial time and is equal to the initial flight path angle. With βT=0, the angle that the rocket forms with the local horizontal axis is given by (αT+γR). At the initial time, this angle corresponds to the initial flight path angle, while at the burnout time, it corresponds to αT+γR=π/2. The roll angle is can be chosen arbitrarily and is set to 0 deg. For the remaining angles θ,ψ,α,β,σ, they are obtained through analytical expressions through the rotation matrices in Figure 6. The thrust direction tends to be aligned with the radial direction. Thus, in principle, this guidance leads to obtain the mean maximal altitude for the rocket at hand. Moreover, with this guidance, the usual values of Δg are in the order of 10−4m/sec2.


### 5.3. Sun-Pointing Guidance

The sun-pointing guidance is based on maintaining constant attitude of the rocket with respect to the sun position. The attitude anglesϕ,θ,ψ are computed in an analytical fashion using the rotations in Figure 6. The first step to obtain the attitude of the sounding rocket is to locate the sun position with respect to the ECI frame:
(25)r^S=cosλScosξScosλSsinξSsinλSc^1c^2c^3T,where λS and ξS are the sun declination and the right ascension, respectively. Then, projection of this unit vector into the ALI frame yields as follows:
(26)r^⊙=r^SR3−λaLR2LLr^0E^0N^0=cosλ⊙cosξ⊙cosλ⊙sinξ⊙sinλ⊙r^0E^0N^0=X⊙Y⊙Z⊙r^0E^0N^0.So the thrust direction is obtained by aligning the body axis of the launcher with the sun direction in the ALI frame, as shown in Figure8. To avoid that the rocket will fly with a γr near to zero while trying to catch the sun slightly above the local horizon, the thrust direction is constrained to have a displacement from r^⊙ not exceeding the maximal value η=10deg (cf. Figure 8). So the thrust direction is defined imposing ξT=ξ⊙ and λT=maxλ⊙,π/2−η. The angles αT and βT are then obtained through rotation of T^ from the ALI frame to the RV frame using the rotations reported in Figure 6 and considering that the thrust is aligned with the longitudinal axis of the rocket x^B. The pitch and yaw angles are then obtained with the following expressions:
(27)sinθ=cosλTcosξT→θ,cosψ=sinλT,sinψ=cosλTsinξT→ψ.Figure 8
Sun-pointing guidance principal angles and axes.
(a)
Sun positionr^S in the ECI frame
(b)
Sun positionr^⊙ in the ALI frame. This case is an example of the sun being outside the spherical section allowed by η
(c)
Sun positionr^⊙ in the BA frame. χS and LS are obtained projecting ιS on the BA frameAs roll angle is arbitrary, it can be selected after choosing the angleιS related to the sun position in the body frame. The roll angle ϕ is obtained by comparing r^⊙ in the ALI frame with its projection in the RV frame.From Figure8(c), it is straightforward to obtain the following angles:
(28)sinLSsinιS=sinλT→LS,tanLStanιS=sinξS,cosλTcosLS=cosξS→ξS.They univocally define the vectorr^⊙=XB⊙YB⊙ZB⊙,x^By^Bz^BT in the body frame. Comparing the coordinates of r^⊙ in the ALI frame to the ones in the BA frame leads the following expression:
(29)X⊙Y⊙Z⊙RBTR3TψR2TθR1Tϕ=XB⊙YB⊙ZB⊙.Equation (29) leads to the roll angle ϕ.Then, the aerodynamic anglesα,β,σ can be obtained through the rotation matrices in Figure 6, which are not reported for the sake of brevity. In short, the sun-pointing guidance requires specifying the following parameters:
(i)
θS which is the sun position in the ECI frame(ii)
η which constraints the possible thrust direction(iii)
θG0, defined by the date and hour of the launch(iv)
ιS, needed to obtain the roll angle ϕAs the attitude is also in this case constant, the sun-pointing guidance is a particular case of the constant attitude guidance. Hence, it maintains its main advantages like the ideally perfect microgravity condition (Δg=0) and can provide the perfect conditions for thermal experiments in the space.


## 5.1. Constant Attitude Guidance

With this guidance, the rocket orientation does not change during the flight and was actually implemented in the Black Brant X (cf. [17]). This is ideal to achieve nominal (perfect) microgravity conditions (Δg=0), due to rotation rates equal to 0. The attitude angles ϕ,θ,ψ remain constant; therefore, in this type of guidance, the following expressions can be assumed as follows:
(23)ϕ=ϕ0θ=θ0ψ=ψ0.The remaining anglesαT,βT,α,β,σ can be found using analytical formulas at any times using the rotation matrices portrayed in Figure 6.


## 5.2. Near Radial Guidance

This guidance is called near radial because it is defined in the AO reference frame and the thrust direction is defined by(24)αT=γR0+ttBπ2−γR0−γR,βT=0whereas γR0=θ0 is the pitch angle at the initial time and is equal to the initial flight path angle. With βT=0, the angle that the rocket forms with the local horizontal axis is given by (αT+γR). At the initial time, this angle corresponds to the initial flight path angle, while at the burnout time, it corresponds to αT+γR=π/2. The roll angle is can be chosen arbitrarily and is set to 0 deg. For the remaining angles θ,ψ,α,β,σ, they are obtained through analytical expressions through the rotation matrices in Figure 6. The thrust direction tends to be aligned with the radial direction. Thus, in principle, this guidance leads to obtain the mean maximal altitude for the rocket at hand. Moreover, with this guidance, the usual values of Δg are in the order of 10−4m/sec2.


## 5.3. Sun-Pointing Guidance

The sun-pointing guidance is based on maintaining constant attitude of the rocket with respect to the sun position. The attitude anglesϕ,θ,ψ are computed in an analytical fashion using the rotations in Figure 6. The first step to obtain the attitude of the sounding rocket is to locate the sun position with respect to the ECI frame:
(25)r^S=cosλScosξScosλSsinξSsinλSc^1c^2c^3T,where λS and ξS are the sun declination and the right ascension, respectively. Then, projection of this unit vector into the ALI frame yields as follows:
(26)r^⊙=r^SR3−λaLR2LLr^0E^0N^0=cosλ⊙cosξ⊙cosλ⊙sinξ⊙sinλ⊙r^0E^0N^0=X⊙Y⊙Z⊙r^0E^0N^0.So the thrust direction is obtained by aligning the body axis of the launcher with the sun direction in the ALI frame, as shown in Figure8. To avoid that the rocket will fly with a γr near to zero while trying to catch the sun slightly above the local horizon, the thrust direction is constrained to have a displacement from r^⊙ not exceeding the maximal value η=10deg (cf. Figure 8). So the thrust direction is defined imposing ξT=ξ⊙ and λT=maxλ⊙,π/2−η. The angles αT and βT are then obtained through rotation of T^ from the ALI frame to the RV frame using the rotations reported in Figure 6 and considering that the thrust is aligned with the longitudinal axis of the rocket x^B. The pitch and yaw angles are then obtained with the following expressions:
(27)sinθ=cosλTcosξT→θ,cosψ=sinλT,sinψ=cosλTsinξT→ψ.Figure 8
Sun-pointing guidance principal angles and axes.
(a)
Sun positionr^S in the ECI frame
(b)
Sun positionr^⊙ in the ALI frame. This case is an example of the sun being outside the spherical section allowed by η
(c)
Sun positionr^⊙ in the BA frame. χS and LS are obtained projecting ιS on the BA frameAs roll angle is arbitrary, it can be selected after choosing the angleιS related to the sun position in the body frame. The roll angle ϕ is obtained by comparing r^⊙ in the ALI frame with its projection in the RV frame.From Figure8(c), it is straightforward to obtain the following angles:
(28)sinLSsinιS=sinλT→LS,tanLStanιS=sinξS,cosλTcosLS=cosξS→ξS.They univocally define the vectorr^⊙=XB⊙YB⊙ZB⊙,x^By^Bz^BT in the body frame. Comparing the coordinates of r^⊙ in the ALI frame to the ones in the BA frame leads the following expression:
(29)X⊙Y⊙Z⊙RBTR3TψR2TθR1Tϕ=XB⊙YB⊙ZB⊙.Equation (29) leads to the roll angle ϕ.Then, the aerodynamic anglesα,β,σ can be obtained through the rotation matrices in Figure 6, which are not reported for the sake of brevity. In short, the sun-pointing guidance requires specifying the following parameters:
(i)
θS which is the sun position in the ECI frame(ii)
η which constraints the possible thrust direction(iii)
θG0, defined by the date and hour of the launch(iv)
ιS, needed to obtain the roll angle ϕAs the attitude is also in this case constant, the sun-pointing guidance is a particular case of the constant attitude guidance. Hence, it maintains its main advantages like the ideally perfect microgravity condition (Δg=0) and can provide the perfect conditions for thermal experiments in the space.


## 6. Numerical Results

Figures9–11 portray the results of the computation, which are, respectively, for the constant attitude guidance (Figure 9), the near radial guidance (Figure 10), and the sun-pointing guidance (Figure 11). From Figure 9 to Figures 11(b), 11(d), and 11(f)), if the desired ΔTmg is not chosen, it is possible to obtain a maximum ΔTmg for each length. This result is particularly visible for the shorter lengths while is less evident for the bigger configurations. Panels (a), (c), and (e) of Figures 9–11 show every l,d combination which guarantee 600 and 900 sec of ΔTmg. The optimal configuration corresponds to the point in which each curve is tangent to an equilateral isovolume hyperbola in the (l,d2) graph. As the tangence point always occurs in the lowest right part of the graph, two constraints were inserted, for the sake of avoiding too slender or bulky configurations. In general, the constraint for the l/d is
(30)5≤ld≤20.Figure 9
Optimal results for the constant attitude guidance.
(a)
50 kg payload
(b)
50 kg payload
(c)
100 kg payload
(d)
100 kg payload
(e)
200 kg payload
(f)
200 kg payloadFigure 10
Optimal results for the near radial guidance.
(a)
50 kg payload
(b)
50 kg payload
(c)
100 kg payload
(d)
100 kg payload
(e)
200 kg payload
(f)
200 kg payloadFigure 11
Optimal results for the sun-pointing guidance.
(a)
50 kg payload
(b)
50 kg payload
(c)
100 kg payload
(d)
100 kg payload
(e)
200 kg payload
(f)
200 kg payloadIt should be noticed that that the optimal configurations for all the three guidance are nearly the same, and this means that the preliminary design of the sounding rocket can be done regardless of the guidance type. Tables5 and 6 show the optimal rocket configuration for 600 and 900 sec of ΔTmg and for all the payload masses. The configuration with l=7.29m and d=0.36m in Table 5 is particularly similar to the Black Brant VC, demonstrating that with similar mU and ΔTmg, the algorithm can be effective to optimize the size of the rocket.Table 5
Optimal configurations for each payload mass,ΔTmg=10 min.
m
U (kg)
50
100
200
l (m)
7.29
8.25
9.69
d (m)
0.36
0.41
0.48Table 6
Optimal configurations for each payload mass,ΔTmg=15 min.
m
U (kg)
50
100
200
l (m)
10.98
12.42
14.53
d (m)
0.55
0.62
0.73Figures12–14 portray the complete state vector elements and other variables for the optimal configuration with mU= 50 kg and ΔTmg= 10 min, using the near radial guidance.Figure 12
State variables for the configuration withl = 7.29 m and d = 0.36 m (near radial guidance).
(a)
Altitude
(b)
Speed
(c)
Flight path angle
(d)
Heading angle
(e)
Latitude
(f)
LongitudeFigure 13
Control law, attitude angles, and dynamic pressure in the thrust arc for the configuration withl = 7.29 m and d = 0.36 m (near radial guidance).
(a)
In-plane control angle
(b)
Out-of-plane control angle
(c)
Roll angle
(d)
Pitch angle
(e)
Yaw angle
(f)
Dynamic pressure constraintFigure 14
Aerodynamic angles and coefficients in the thrust arc for the configuration withl = 7.29 m and d = 0.36 m (near radial guidance).
(a)
Lift coefficient
(b)
Drag coefficient
(c)
Side force coefficient
(d)
Angle of attack
(e)
Sideslip angle
(f)
Bank angleFor the rocket structural integrity, the constraintQαtot<150000Pa∗deg must be satisfied, where
(31)Q=12ρvR2and
(32)αtot=arccoscosαcosβ.From Figure13(f), it is apparent that the constraint is satisfied, and the same occurs for all the cases.


## 7. Conclusions

Designing a sounding rocket is a difficult task, and this work proposes a method to simplify the early stage modeling process and performance evaluation. The new methodology at hand has four principal phases: (i) mass and geometry sizing based on reasonable assumptions, (ii) scaled propulsion modeling, (iii) accurate aerodynamic modeling, and (iv) the performance evaluation using different guidance schemes. With regard to the guidance algorithms, three schemes where presented which are the constant attitude, near radial, and sun-pointing guidances. While the constant attitude guidance is a well-known scheme, the other two are novel guidances. In particular, the benefits of the sun-pointing guidance are presented, which are the nearly perfect microgravity condition and the possibility to orient the rocket with a defined attitude with respect to the sun. The results clearly show that the algorithm is capable of providing a fast and reasonable sizing in accordance with the mission requirements. The results obtained can be used as a reference point for a montecarlo campaign to analyze the impact points in the presence of local winds or as a first guess solution for more sophisticated optimization algorithms.



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*Source: 1678709-2018-02-06.xml*