Three-Dimensional Guidance Law Mimicking Realistic Ballistic Trajectories

Full text

Three-Dimensional Guidance Law Mimicking Realistic Ballistic
Trajectories
Robert Fonod ∗, Michael Proff †, and Spilios Theodoulis‡
French-German Research Institute of Saint-Louis, Saint-Louis, 68300, France
I. Introduction
Deve
lopment of guidance and control for precision-guided projectiles is an active field of research that has
emerged from increasing operational requirements of guided ammunition in range, accuracy, and cost [
1
,
2
].
Gun-launched projectiles follow a ballistic trajectory to reach their target. Impact accuracy can be significantly affected
by uncertainties in modeling aerodynamic characteristics and unknown atmospheric perturbations. Due to the severely
limited maneuverability of aerodynamically controlled projectiles, designing guidance laws that mimic realistic ballistic
trajectories is of great interest. Modern projectile guidance laws generally follow one of the three design paradigms:
trajectory following, trajectory shaping, or predictive guidance [3].
Trajectory tracking [
3
–
5
] aims at minimizing the in-flight error between the expected ballistic trajectory and the
actual position of the projectile. The tracked trajectory is uploaded into the projectile prior to or at launch. This type of
guidance logic is easy to implement, requires low computing power, and allows retaining the general ballistic tables [
3
].
On the other hand, it tends to be less robust and requires a higher level of maneuverability than predictive guidance [
6
].
Traditional trajectory shaping algorithms may force the projectile to follow non-ballistic trajectories and typically
require high control effort [
3
], often resulting in energy-inefficient actuation force distribution over the trajectory, which
can produce undesired actuator saturations [
3
]. The Proportional Navigation (PN) guidance law [
7
] is perhaps one of
the most popular trajectory shaping guidance laws designed to intercept stationary or low-speed targets. Despite its
implementation simplicity, is not well suited for gun-launched projectiles, unless it is employed during the terminal
phase. There have been several efforts to adapt the PN scheme to account for the curvature of the ballistic trajectory
[
8
–
11
]. Guo et al.
[8]
suggested a heuristic multi-stage guidance approach for dual-spin projectiles. The first stage is
employed at the ascending phase and consists of a parameter-varying lateral deviation correction scheme only, while the
second stage is engaged at the descending phase and employs a conventional constant-gain PN guidance scheme for
simultaneous lateral and longitudinal corrections. Zhang et al.
[9]
proposed an adaptive variant of the 2D PN guidance
law suitable for curved trajectories in the vertical plane. The variable guidance gain is computed off-line based on the
nominal ballistic trajectory and uploaded into the projectile, where it is later retrieved from a lookup table using the
projectile’s actual longitudinal position. Finally, Sharma and Ratnoo
[10]
derived an adaptive 2D PN-like guidance
∗Research Scientist, Department of Guidance, Navigation, and Control, robert.fonod@ieee.org (Corresponding Author).
†Research Engineer, Department of Guidance, Navigation, and Control, Michael.Proff@isl.eu, AIAA Member.
‡Senior Research Scientist, Department of Guidance, Navigation, and Control, Spilios.Theodoulis@isl.eu, AIAA Senior Member.

scheme to mimic short-range ballistic trajectories. In contrast to [
9
], the coefficients determining the time-varying
guidance gain were derived analytically and evaluated in-flight using bearings-only information. This method has been
later extended by the same authors to take into account look-angle constraints [11].
Predictive guidance uses a ballistic model to predict the impact point of the projectile and compute corrections
accordingly. Impact Point Prediction (IPP) based guidance laws were used to improve guidance performance for
gun-launched projectiles [
6
,
12
–
14
]. Alongside evaluating aerodynamic data, the EoM need to be solved in real-time for
each guidance update instant, leading to a high on-board computation burden. Several attempts have been made to
reduce the computational burden of IPP-based methods. Yang
[12]
proposed an IPP method based on solving algebraic
equations based on the estimated location, velocity, and acceleration of a spinning projectile. IPP based on perturbation
theory was proposed in [
13
] for projectiles with 2D trajectory correction fuse. Gagnon and Vachon
[6]
proposed an
improved IPP method for spin-stabilized projectiles. The method leverages the Modified Point Mass (MPM) model of
[
15
] with an additional term for spin rate. The additional term aims at taking into account the drift in the crossrange
direction due to the yaw of repose. Proff and Theodoulis
[14]
further improved the MPM model by assuming variable
spin rate, variable aerodynamic parameters, and variable angle of repose. The IPP based on the improved MPM model
of [
14
] was implemented using the so-called Zero-Effort-Miss (ZEM) guidance formulation and is referred to in this
paper as the state-of-the-art guidance method.
Despite the implementation simplicity of the existing trajectory shaping guidance laws, their main disadvantage is
the inability to adequately account for the drift in the crossrange direction. On the other hand, predictive guidance
methods are able to account for this issue, but they require high on-board computing capabilities. In this paper, we
aim at combining the benefits of these two guidance paradigms. We propose a new practical three-dimensional (3D)
guidance law suitable for a large class of gun-launched spin-stabilized or fin-stabilized projectiles. The proposed law
aims at mimicking a realistic ballistic trajectory by adapting the navigation gain of the conventional PN guidance law
to produce acceleration load factor commands similar to the ones experienced by an unguided projectile flying on a
nominal trajectory. Consequently, the resulting guidance strategy achieves the accuracy of state-of-the-art predictive
methods while maintaining a relatively simple implementation complexity, low on-board computational burden, and
reduced maneuverability requirements, ideal for low maneuverability canard-guided projectiles.
II. Background
A. Overview of Applicable Concepts
The guidance schemes discussed in this paper are applicable to a wide family of gun-launched fin-stabilized and
spin-stabilized guided projectiles. The presented concept potentially allows to consider canard or tail control approaches.
Fin-stabilized projectiles rely on aerodynamic surfaces for stability and control, and resemble more to missile systems,
2

albeit demonstrating less maneuverability. On the other hand, spin-stabilized guided projectiles are typically equipped
with a roll-decoupled course correction fuse (CCF), designed to provide range and/or lateral corrections, while at the
same time, they rely on the high spin rate of the aft part for gyroscopic and potential dynamic stability.
An example of a dual-spin canard-guided projectile concept is illustrated in Fig. 1, along with a dedicated reference
frame and some flight mechanics state variables, both of which will be described further down. The aft part typically
houses the payload and spins freely, while the front part accommodates a set of canards used for aerodynamic
trajectory correction. The fuze also contains all the necessary GNC software and hardware, such as g-hardened inertial
measurement units (IMUs), Global Navigation Satellite System (GNSS) receivers, and on-board computing units. The
main shortcoming of CCF projectiles is their very limited maneuverability. Since the maximum acceleration that a
typical CCF projectile can generate is way less than the amount needed to counteract gravity.
𝑝
𝑞
𝑟
b′
1
b′
3
b′
2
Fig. 1 ISL’s 155 mm course correction fuze concept. Adapted from [
16
]. ©2020 by French-German Research
Institute of Saint-Louis.
B. Frames and Coordinate Systems
Several right-handed orthonormal frames and their associated Cartesian coordinate systems will be used throughout
this paper. Neglecting the Coriolis and centrifugal forces, the Earth is chosen as an inertial reference frame Eby
unwrapping and centering the Earth’s surface into a plane tangential to the launch point E(flat-Earth approach). The
right-handed orthonormal triad of base vectors
(𝒆1,𝒆2,𝒆3)
and the base point Eof this frame is illustrated in Fig. 2. The
𝒆1
axis points northwards,
𝒆3
axis is parallel and in the direction of the gravity vector, and
𝒆2
completes the triad by
pointing eastwards. The body-fixed (rolling) frame Bis centered at the projectile’s Center of Mass (CoM) Band defined
by the triad
(𝒃1,𝒃2,𝒃3)
. The
𝒃1
axis points towards the projectile nose,
𝒃2
is normal to
𝒃1
and points rightwards, and
𝒃3
points downwards and is normal to
𝒃1−𝒃2
. The body non-rolling (BNR) frame
B′
is an intermediate frame also
centered at the projectile CoM Band defined by the triad
𝒃′
1,𝒃′
2,𝒃′
3
. Its second base vector
𝒃′
2
lies always on the
horizontal
𝒆1−𝒆2
plane and is normal to
𝒃1
,
𝒃′
1≡𝒃1
, and
𝒃′
3
completes the triad. Unlike B, the BNR frame
B′
will
only experience pitching and yawing angular motions, but not the rolling motion. The target-fixed frame Tis centered at
3

the target’s CoM T. The definition of the basis vectors for Tcan be arbitrary and depends on its type. The flight-path
frame Vis defined by the triad
(𝒗1,𝒗2,𝒗3)
. The
𝒗1
vector is parallel and in the direction of the velocity vector of the
projectile wrt the flat-Earth
𝒗E
B
, the
𝒗2
vector remains in the horizontal plane subtended by
𝒆1
and
𝒆2
,
𝒗3
completes the
triad. Finally, the line-of-sight (LoS) frame Ois defined by the triad
(𝒐1,𝒐2,𝒐3)
. The
𝒐1
-direction is parallel and in the
direction of the LoS vector
𝒔TB
, a vector connecting Bwith T; the
𝒐2
-direction is normal to
𝒐1
and lies in the
𝒆1−𝒆2
plane; and 𝒐3completes the triad. Both, Oand V, have their base point at the projectile CoM B.
{
Impact uncertainty
Fig. 2 Illustration of the employed reference frames.
The local-level Coordinate System (CS),
]L
, also known as the North-East-Down or NED, is the preferred CS for E.
Similarly, the preferred CS for B,
B′
,T,V, and O, respectively, is
]B
,
]B′
,
]T
,
]V
, and
]O
. The axes of all these coordinate
systems line up with the base vectors of the frames they are associated to. Figure 3 depicts the orientation between
selected coordinate systems, including the associated transformation angles as well as intermediate systems
]X
and
]Y
, where applicable. Specifically, Fig. 3a follows the aircraft Euler angle convention. Two rotations are needed in
order to pass from
]L
to
]B′
. The first by the yaw angle
𝜓
, and the second by the pitch angle
𝜃
. The third rotation by the
roll angle
𝜙
takes us to
]B
, the CS needed for control purposes (e.g., to compute the canards deflection angles). The
meaning and definition of angles shown in Fig. 3b and Fig. 3c will become evident in Section III.
C. Nonlinear Dynamics and Kinematics
A generic 6 degrees of freedom (DoF) nonlinear model of the projectile’s dynamics and kinematics is introduced
here. This model is suitable for simulating the 3D trajectory of a spin-stabilized or a fin-stabilized projectile and
will be used later for guidance design. The projectile EoM are expressed using the BNR frame
B′
rather than the
conventional body-fixed frame Bthat would roll along with the body. This is especially advantageous for roll-symmetric
spin-stabilized projectiles, where the high spin rate (typically between 170 and 300
Hz
) would require a very small
integration step to capture the true underlying roll dynamics. This would ultimately result in long simulation times and
may lead to large accumulated numerical errors [17].
The EoM of the translational and attitude airframe dynamics, coordinated to the BNR CS ]B′, are given by∗
∗
Capital subscripts denote points while capital superscripts signify frames. Two subscripts or superscripts are read from left to right, inserting the
4

B
1L
2L
3L, 3XB
1XB
2XB, 2YB
ψ
ψ
θ
θ
1YB, 1B
3YB
2B3B
φ
φ
(a) Body-fixed CS ]Bwrt local CS ]L
B
1L
2L
3L, 3XO
1XO
2XO, 2O
ψO
ψO
θO
θO
1O
3O
T
sTB
(b) LoS CS ]Owrt local CS ]L
B
1L
2L
3L, 3XV
1XV
2XV,2V
χ
χ
γ
γ
1V
3V
vE
B
(c) Flight-path CS ]Vwrt local CS ]L
Fig. 3 Orientation and associated transformation angles between selected coordinate systems.
𝑚¤
𝒗E
BB′+𝑚h𝛀B′EiB′𝒗E
BB′=𝒇a,gB′
,(1)
𝑰B
BB′¤
𝝎BEB′+h𝛀B′EiB′𝑰B
BB′𝝎BEB′=[𝒎a]B′.(2)
The components of linear velocity
(𝒗E
B)
, angular velocity
(𝝎BE)
, aerodynamic and gravity forces
(𝒇a,g)
, and aerodynamic
moments referred to the CoM (𝒎a), are defined, respectively, as†
𝒗E
BB′≜

𝑢
𝑣
𝑤

,𝝎BEB′≜

𝑝
𝑞
𝑟

,𝒇a,gB′≜

𝑋
𝑌
𝑍

,[𝒎a]B′≜

𝐿
𝑀
𝑁

.
The externally-applied airframe forces
(𝑋 , 𝑌, 𝑍 )
contain a sum of gravitational and aerodynamic components. The
aerodynamic components are due to drag-lift, damping, Magnus (for spinning projectiles only), and control forces.
The externally-applied airframe moments
(𝐿, 𝑀 , 𝑁 )
are a result of these aerodynamic forces and internal mechanical
friction, if applicable. In
(1)
,
𝑚
signifies the projectile mass. In
(2)
,
𝑰B
BB′
stands for the moment of inertia (MoI) of
the projectile body Breferred to the its CoM Band expressed in the CS
]B′
. Finally, the skew-symmetric matrix form of
word “relative to” [18]. For sake of clarity, the notation of time-dependency (𝑡) of some variables is omitted whenever the context is clear.
†By standard frame notation conventions, we should write 𝑢′,𝑣′,𝑤′,𝑝′, etc., but for sake of notational simplicity, we drop the ’′’ notation.
5

the angular velocity of B′wrt Eexpressed in ]B′is
h𝛀B′EiB′
=

0−𝑟 𝑞
𝑟0𝑟tan 𝜃
−𝑞−𝑟tan 𝜃0

.
The translational and attitude kinematics, coordinated into
]L
and expressed using Euler angles convention,
respectively, are given by
[¤
𝒔BE]L=¯
𝑻B′L𝒗E
BB′,(3)
¤
𝒆BE =𝑱𝑒𝜔 𝝎BE B′.(4)
The components of the linear position (𝒔BE ), angular position (𝒆BE ), and the matrix 𝑱𝑒𝜔 are
[𝒔BE]L≜

𝑥
𝑦
𝑧

,𝒆BE ≜

𝜙
𝜃
𝜓

,𝑱𝑒𝜔 =

1 0 tan 𝜃
0 1 0
0 0 sec 𝜃

.
Finally, the orthogonal time-dependent coordinate transformation matrix
[𝑻]B′L
in
(3)
is given as a function of the
projectile pitch 𝜃and yaw 𝜓Euler angles, i.e.,
¯
𝑻B′L
=

cos 𝜃cos 𝜓−sin 𝜓sin 𝜃cos 𝜓
cos 𝜃sin 𝜓cos 𝜓sin 𝜃sin 𝜓
−sin 𝜃0 cos 𝜃

,(5)
where the overbar indicates matrix transpose.
III. Guidance Law Design
A. Problem Formulation
The role of a guidance algorithm is to generate suitable reference pitch/yaw-channel load factor orders. Canard or
fin deflections are usually used as control surfaces. The normal
𝑛z,cmd
and lateral
𝑛y,cmd
load factor commands are used
6

as a tracking outputs for the autopilot and are defined in this paper as follows:

𝑛z,cmd
𝑛y,cmd

=
1
𝑔(ℎ)
001
010
𝒂E
BB′
cmd .(6)
Here, the two selected components of the 3D acceleration command
𝒂E
BB′
cmd
, which will be designed later, are converted
to units in g’s and sent to the autopilot in order to steer the projectile with accuracy to the target [
18
]. The third
component parallel to the body is normally unused, unless a so-called braking function is available to the projectile, see
Gagnon and Lauzon
[3]
and references therein. In
(6)
,
𝑔(ℎ)
stands for the altitude (
ℎ≜−𝑧
) dependent acceleration due
to the gravity. In this paper, a simplified first-order linear regression model is adopted,
𝑔(ℎ)=𝐺𝑀
𝑟2≈𝑔0+𝑔1ℎ, (7)
where
𝑀
is the mass of Earth,
𝑟
is the distance from Earth center, and
𝐺
is the universal gravitational constant. The
numerical values for 𝑔0and 𝑔1are considered as follows: 𝑔09.8084 [m/s2] and 𝑔1−3.0710 ×10−6[1/s2].
In order to avoid actuator saturation and/or autopilot instability, a flight envelope protection function
𝑓𝑝(·)
is often
applied to the load factor commands (6), i.e., for i∈ {y,z}, we have
𝑓𝑝(𝑛i,cmd)=









𝑛i,cmd if |𝑛i,cmd | ≤ ¯𝑛i(ℎ, 𝑉 ),
sign(𝑛i,cmd )¯𝑛i(ℎ, 𝑉 )else,
(8)
where ¯𝑛i(ℎ, 𝑉 )is, in general, the altitude ℎand speed 𝑉≜√𝑢2+𝑣2+𝑤2dependent load factor order limit in g’s.
B. Proportional Navigation
Adopted from physical intuition, the PN guidance law is perhaps one of the most common guidance laws in use,
particularly for air-to-air engagements. The acceleration command of the gravity-compensated 3D pure PN guidance
law, expressed in the CS ]B′, is given by [18]
𝒂E
BB′
cmd
=𝑁𝝎OEB′×𝒗E
BB′−[𝑻]B′L[𝒈]L,(9)
7

where
𝑁
is the so-called navigation gain and
[𝒈]L
is the gravitational acceleration vector in the local-level CS
]L
,
defined as
[𝒈]L≜

0
0
𝑔(ℎ)

,(10)
and transformed into the BNR CS using the transformation matrix
[𝑻]B′L
. The angular rate tensor of Owrt Eexpressed
in ]B′is given by 𝝎OEB′=[𝑻]B′L𝝎OE L
=[𝑻]B′L𝒔TBL×𝒗E
TBL
[𝒔TB]L2,(11)
where
∥·∥
stands for the Euclidean norm of a vector and
𝒔TB
is the relative displacement of the target CoM Twrt the
projectile CoM B. Coordinating 𝒔TB to CS ]L, we have
[𝒔TB]L≜[𝒔TE ]L−[𝒔BE]L.(12)
Here,
𝒔TB
is also referred to as the LoS vector, see Fig. 2. The target’s inertial position
𝒔TE
is considered to be perfectly
known and fixed through the engagement. The differential velocity vector,
𝒗E
TBL≜𝒗E
TL−𝒗E
BL
=−𝒗E
BL,(13)
is the differential velocity of the target Twrt the projectile Bas observed from E. In this paper, only stationary targets are
considered, hence
𝒗E
T
in
(13)
is considered zero. This is a deliberate choice of the authors since the class of projectiles
studied here does not have either the necessary maneuverability or the on-board sensors (for example seekers) to detect
and intercept moving targets.
C. Ballistic Proportional Navigation
In case of a stationary target, the PN guidance law is able to intercept the target with zero miss distance by any
𝑁 >
0, provided perfect navigation information, instantaneous execution of the requested load factor commands, and no
maneuverability limitations. Specifically, the last assumption poses a serious challenge for canard-guided projectiles,
which often have very limited maneuvering capabilities [19].
In missile guidance applications, the gain
𝑁
is normally assumed constant and typically takes values between 3 and 5
[
20
]. The higher the value, the closer the resulting trajectory to the straight-line [
21
]. Obviously, this is in contradiction
to the natural ballistic flight path curvature typical for gun-launched projectiles (cf. Fig. 2 for illustration).
During the terminal phase of a ballistic flight, the resulting trajectory resembles a quasi straight-line, approaching the
8

validity of the underlying assumptions used in the PN guidance law derivation (e.g., the engagement geometry staying
close to the collision course [
21
]). Therefore, the baseline PN guidance law is typically only engaged during the terminal
guidance phase of gun-launched projectiles. For low maneuverability projectiles, lower guidance gains, i.e.,
𝑁 <
3,
were suggested in order to limit the risk of canards saturation while maintaining acceptable impact accuracy [19].
In this paper, we introduce three alternatives to the conventional constant-gain 3D PN guidance law. Henceforth, we
will refer to these alternatives as: 1) Ideal Ballistic Proportional Navigation (IBPN); 2) IBPN with look-angle Constraints
(IBPN+C); and 3) Realistic Ballistic Proportional Navigation (RBPN). For each alternative, we aim to preserve the
relatively simple structure of the PN guidance law introduced in
(9)
. However, instead of using a constant navigation
gain
𝑁
, the goal is to design a time-varying alternative
𝑁(𝑡)
that shapes the 3D trajectory of a guided projectile to mimic
a ballistic one. Consequently, this scheme should be able to engage a closed-loop guidance strategy much earlier.
1. Ideal Ballistic Proportional Navigation
As the name suggests, IBPN aims at mimicking a ballistic trajectory under ideal ballistic trajectory assumptions, i.e.,
no lift, drag or side forces. Sharma and Ratnoo
[10]
derived in closed-form a time-varying navigation gain for a planar
2D engagement, which can be expressed as follows:
𝑁(𝑡)=𝑘1−𝑘2𝜃O(𝑡), 𝑡 ∈ [𝑡0, 𝑡f),(14)
where 𝜃Ois the LoS elevation angle (cf. Fig. 3b) defined as
𝜃O=arctan ©«
−𝑧TB
𝑥2
TB +𝑦2
TB ª®®¬
,(15)
and
𝑥TB
,
𝑦TB
,
𝑧TB
are the LoS vector components in
]L
, i.e.,
[¯
𝒔TB]L=[𝑥TB , 𝑦TB , 𝑧TB ]
. The guidance gains
𝑘1
and
𝑘2
aim at imposing the ballistic trajectory mimicking constraints.
Under the mild condition that the launch elevation angle
𝜃(𝑡0) ∈ (0, 𝜋/2)
, the gain
𝑘1
in
(14)
is determined by
matching the initial heading error rate of the guided trajectory to that of the ballistic trajectory [
10
]. For the notation
employed in this paper, 𝑘1takes the following form
𝑘1=
𝑔(𝑡0)𝑅go (𝑡0)
𝑉2(𝑡0)tan 𝛾(𝑡0),(16)
where the flight-path angle 𝛾(cf. Fig. 3c) is given by
𝛾=arctan −𝑤
√𝑢2+𝑣2,(17)
9

and the LoS separation 𝑅go , henceforth referred to as range-to-go, is given by
𝑅go =𝑥2
TB +𝑦2
TB +𝑧2
TB.(18)
The gain
𝑘2
is obtained by matching the final flight-path angle of the guided trajectory to that of the ballistic trajectory
[10], i.e.,
𝑘2=2𝑘1𝛾(𝑡f) − 𝛾(𝑡f) + 𝛾(𝑡0)
𝛾2(𝑡f).(19)
Note that
𝜃(𝑡0)=𝛾(𝑡0)
holds for a perfectly aligned projectile inside the canon tube and, for stationary targets and
perfect target hit, 𝛾(𝑡f)=𝜃O(𝑡f).
In this paper, we suggest using
(14)
directly in the 3D PN structure of
(9)
, while determining the guidance parameters
𝑘1
and
𝑘2
from a realistic 3D nonlinear simulation (cf. Section III.C.4). Compared to constant-gain PN, this strategy
may help to better account for the curvature of the ballistic path. However, no expectations are made in terms of the
ability to account for cross-range errors and deviations from the ideal ballistic path.
2. IBPN with Look-angle Constraints
For pursuers equipped with a strapdown seeker, Sharma and Ratnoo
[11]
proposed a modification to IBPN addressing
the seeker’s look-angle constraint. The resulting time-varying navigation gain can be expressed to have an identical
structure as the one given in (14), but the two guidance gains, 𝑘1and 𝑘2, take the following form
𝑘1=1+
𝛽+𝛽2+4𝛽𝛾(𝑡0)
𝛾∗
2,(20)
𝑘2=
𝛾∗𝛽−𝛽2+4𝛽𝛾(𝑡0)
𝛾∗+2𝛾(𝑡0)
(𝛾∗)2,(21)
where
𝛽=
4
(𝜎max −𝛾(𝑡0))/𝛾∗
. The design objective of
𝑘1
and
𝑘2
is to generate acceleration commands which, at the
final time 𝑡f, lead to [11]
lim
𝑡→𝑡f
𝑅go (𝑡)=lim
𝑡→𝑡f(𝛾(𝑡) − 𝛾∗)=0,
while satisfying
𝜎(𝑡) ∈ [−𝜎max, 𝜎max], 𝑡 ∈ [𝑡0, 𝑡f).
Here,
𝜎=𝛾−𝜃O
is the seeker’s look-angle,
𝜎max >
0represents the maximum look-angle constraint, and
𝛾∗∈ [−𝜋,
0
]
denotes the desired impact angle. According to Sharma and Ratnoo
[11]
, all impact angles
𝛾∗∈ [−𝜋,
0
]
can be achieved
10

with a bounded acceleration for any 𝜎max satisfying
𝜎max ≥









𝛾(𝑡0)if 𝛾∗∈ [−2𝛾(𝑡0),0],
−(𝛾∗)2
4(𝛾∗+𝛾(𝑡0)) if 𝛾∗∈ [−𝜋, −2𝛾(𝑡0)).
(22)
Similarly to the the IBPN case, the above guidance gains were derived for a planar engagement and under idealistic
assumptions. We will use the resulting time-varying navigation gain
𝑁(𝑡)
directly in the expression of the 3D PN
guidance law of
(9)
. As our goal is to mimic realistic ballistic trajectories, we will not impose any pre-defined impact
angle constraint. Instead, we will set
𝛾∗
to be equal to
𝛾(𝑡f)
obtained from a nonlinear simulation. Moreover, the value
of
𝜎max
has a direct influence on the resulting maneuverability requirements. Therefore, we will consider
𝜎max
as a
tuning parameter used to avoid actuator saturation and maximize projectile accuracy.
3. Realistic Ballistic Proportional Navigation
The proposed RBPN guidance law aims at addressing the shortcomings of PN, IBPN, and IBPN+C. Specifically, the
aim is to mimic a realistic 3D ballistic trajectory by designing the time-varying navigation gain
𝑁(𝑡)
such that the PN
guidance law of
(9)
produces acceleration commands only when the actual projectile path deviates from the predicted
nominal ballistic trajectory used for the computation of 𝑁(𝑡).
First, let us define
𝝎VE
as the rotational velocity vector of the velocity vector
𝒗E
B
wrt Earth and
𝝎OE
being the angular
velocity vector of the LoS frame Owrt the inertial frame E. For an unguided projectile, the experienced acceleration
normal to the projectile velocity can be expressed‡as a cross product of 𝝎VE and 𝒗E
B:
˜
𝒂E
B=𝝎VE ×𝒗E
B.(23)
Furthermore, the PN relation in 3D states that 𝝎VE must be made proportional to 𝝎OE,
𝝎VE =𝑁𝝎OE,(24)
where 𝑁serves as the proportionality constant. Substituting (24) into (23) yields§
˜
𝒂E
B=𝑁𝝎OE ×𝒗E
B.(25)
In order to mimic a realistic ballistic trajectory, we aim to find a proportionality constant
𝑁(𝑡)
that attempts to make
‡We will skip the use of coordinate systems, unless necessary.
§
Note that
(25)
resembles the PN equation of
(9)
, but without the gravity bias term
𝒈
, which was added to the PN law in order to avoid the
sagging tendency of the trajectory under seeker control [18].
11

the acceleration of the guided projectile
(25)
similar (in a L
2
-norm sense) to the acceleration of the unguided projectile
(23)
for all time instances
𝑡
during which the PN guidance law is actively engaged. This problem can be formulated as a
minimization problem of the following cost function:
𝐽=
1
2
𝑡f
∫
𝑡s𝝎VE (𝑡) − 𝑁(𝑡)𝝎OE (𝑡)2d𝑡, (26)
where
𝑡s≥𝑡0
denotes the start of the guidance law. It is obvious that in order to minimize
𝐽
, it is sufficient to minimize
the argument of the integral for all
𝑡∈ [𝑡s, 𝑡f]
. Thus, for a fixed
𝑡
, the first-order optimality condition can be expressed as
𝜕𝝎VE −𝑁𝝎OE2
𝜕𝑁
=−2
3

𝑖=1𝜔VE
𝑖−𝑁𝜔OE
𝑖𝜔OE
𝑖=0.(27)
Solving the above equality wrt 𝑁yields the least squares solution, i.e., for 𝑡∈ [𝑡s, 𝑡f]we have¶
𝑁(𝑡)=arg min
𝑁(𝑡)
𝐽=
3
Í
𝑖=1
𝜔VE
𝑖(𝑡)𝜔OE
𝑖(𝑡)
3
Í
𝑖=1𝜔OE
𝑖(𝑡)2
=
𝝎OE (𝑡) · 𝝎VE (𝑡)
∥𝝎OE (𝑡)∥2.(28)
To compute
(28)
, we need to obtain the off-line time-series for
𝝎OE
and
𝝎VE
in the same CS. To do this, we identify that
the angular rate tensor 𝝎OE can be expressed in ]Las follows (cf. Fig. 3b)
𝝎OEL
=¤
𝜓O[𝒆3]L+¤
𝜃O¯
𝑇XOL[𝒙O2]XO=¤
𝜓O

0
0
1

+¤
𝜃O

−sin 𝜓O
cos 𝜓O
0

=

−¤
𝜃Osin 𝜓O
¤
𝜃Ocos 𝜓O
¤
𝜓O

,(29)
where the LoS elevation angle 𝜃Owas defined in (15) and the LoS azimuth angle 𝜓Ois given by
𝜓O=arctan 𝑦TB
𝑥TB .(30)
Alternatively, part of the formula
(11)
can be used to obtain
𝝎OE
directly in
]L
. Similarly,
𝝎𝑉 𝐸
in
]L
can be expressed
as (cf. Fig. 3c)
𝝎𝑉 𝐸 L
=¤𝜒[𝒆3]L+ ¤𝛾[𝑇]LXV[𝒙V2]XV=¤𝜒

0
0
1

+ ¤𝛾

−sin 𝜒
cos 𝜒
0

=

−¤𝛾sin 𝜒
¤𝛾cos 𝜒
¤𝜒

,(31)
¶Here, "·" signifies the dot product of two vectors.
12

where the flight-path angle 𝛾was defined in (17) and the heading angle 𝜒is given by
𝜒=arctan 𝑣
𝑢.
The time derivatives of the heading angle
¤𝜒
and the flight-path angle
¤𝛾
can be either obtained by direct numerical
differentiation of 𝜒and 𝛾, respectively, or alternatively by evaluating the following expressions:
¤𝜒=
𝑢¤𝑣−𝑣¤𝑢
𝑢2+𝑣2,(32)
¤𝛾=
𝑤(𝑢¤𝑢+𝑣¤𝑣) − ¤𝑤(𝑢2+𝑣2)
𝑉2√𝑢2+𝑣2.(33)
Here, the components ¤𝑢,¤𝑣, and ¤𝑤are contained in ¤
𝒗E
BB′.
4. Implementation Details
The implementation of the IBPN or IBPN+C scheme is relatively straightforward. The two guidance gains 𝑘1and
𝑘2
need to be computed off-line. Knowing the target’s position, the projectile’s initial conditions, such as
𝑅go (𝑡0)
,
𝑉(𝑡0)
,
and
𝛾(𝑡0)=𝜃(𝑡0)
, are easily obtained using ballistic tables. To obtain
𝛾(𝑡f)
, one could simply consider the fact that for
an ideal ballistic trajectory, we have
𝛾(𝑡f)=−𝛾(𝑡0)
. Alternatively, as suggested in Section III.C.1, consider a more
realistic ballistic trajectory taking into account aerodynamic forces and moments and predict a more accurate estimate
of the terminal impact angle 𝛾(𝑡f)using an off-line trajectory simulation of an un-guided projectile.
To implement the RBPN scheme, first we need to pre-calculate, parametrize, and store the navigation gain of
(28)
in
a lookup table. Similarly as for IBPN(+C), the initial conditions for an unguided ballistic trajectory need to be found.
Then, the 6-DoF nonlinear differential equations of
(1)
–
(4)
can be numerically integrated to obtain a nominal trajectory
in terms of
𝝎OE
and
𝝎VE
. Alternatively, a reduced-order 3-DoF (for fin-stabilized) or 4-DoF (for spin-stabilized) model
can be considered. Once the trajectory estimate is obtained, the navigation gain can be parameterized using
𝑅go
, see
(18)
, or by time-to-go
𝑡go
defined as
𝑡go =𝑡f−𝑡
. Here,
𝑡
is the integration time and
𝑡f
is the impact time obtained from
the off-line simulation. Consequently, the discretized
𝑅go (𝑡𝑘)
-based or the
𝑡go (𝑡𝑘)
-based lookup table for
𝑁(𝑡𝑘)
can be
evaluated in-flight (e.g., by linear interpolation) using the estimates of 𝑅go or 𝑡go computed as:
ˆ
𝑅go =∥ˆ
𝒔TE −ˆ
𝒔BE ∥,or ˆ
𝑡go =
ˆ
𝑅go
ˆ
𝑉c,
where the hat signifies in-flight estimates and
𝑉c
is the projectile-target closing speed defined as
𝑉c=(𝒔TB ·𝒗E
TB)/∥𝒔TB ∥
.
Finally, Fig. 4 illustrates the integration of a 𝑅go-based lookup table in the PN guidance law of (9).
Remark 1
If the initial conditions used for the off-line trajectory simulation do not lead to a perfect target hit, the RBPN
13

Ini�al condi�ons Trajectory
simula�on
Interpola�on
x
PN guidance law
Ground computer
Flight computer
Lookup table crea�on
Fig. 4 Schematics of the RBPN guidance law with a 𝑅go look-up table.
navigation gain
𝑁(𝑡)
might have a spiky behavior as
𝑡→𝑡f
. A remedy is to either bound the range of attainable values,
i.e., 𝑁min ≤𝑁(𝑡) ≤ 𝑁max, or to freeze the value of 𝑁(𝑡),𝑡∈ [𝑡f−𝜀, 𝑡f]to 𝑁(𝑡f−𝜀)for some small constant 𝜀 > 0.
IV. Results
A. Simulation Overview
In this section, we will compare the effectiveness of the three guidance schemes presented in Section III.C, when
implemented for a spin-stabilized guided projectile equipped with a CCF [
22
]. For completeness, we also include
the benchmark guidance law of PN and the current state-of-the-art guidance method for low-maneuverability guided
projectiles, the ZEM guidance. The guidance laws are evaluated using a high-fidelity 7-DoF nonlinear simulator of a
ballistic dual-spin guided projectile [23].
If an unguided projectile is launched with a given muzzle velocity
𝑉0≜𝑉(𝑡0)
, initial spin rate
𝑝0≜𝑝(𝑡0)
, elevation
angle
𝜃0≜𝜃(𝑡0)
, and azimuth angle
𝜓0≜𝜓(𝑡0)
, it reaches a so-called ballistic impact point. However, due to uncertain
launch/wind conditions and model parameters, a complete GNC loop is necessary to ensure a stationary target hit with a
given CEP
‖
probability. In order to assess the guidance law performance solely, we assume that perfect navigation
information is available to the guidance and autopilot modules.
The decoupled roll and pitch/yaw channel autopilots [
22
] were designed using
H∞
control techniques to achieve
performance, robustness, and stability when the canard deflection angles
𝛿i
and their respective rates
¤
𝛿i
are within
nominal operation ranges. Beyond these ranges, even though the validity/linearity of the aerodynamic model may not be
guaranteed, anti-windup control laws were studied by the authors [
24
–
26
]. In this paper, the canard amplitude and rate
saturation levels are set to 𝛿c
sat =±30°and ¤
𝛿c
sat =±100°/s.
For all guidance laws, a static flight envelop protection function (cf. Section III.A) is applied with the following
values
¯𝑛y=¯𝑛z=
0
.
2g’s. The value of the PN navigation gain was tuned via Monte Carlo (MC) simulations and was
‖
The Circular Error Probable (CEP) is defined here as a circle, centered at the target’s center T, whose radius includes 50% of the projectile’s
landing points. Analogously, CEP1𝜎, CEP2𝜎, and CEP3𝜎includes 1𝜎=68.2%,2𝜎=95.4%, and 3𝜎=99.7% landing points, respectively.
14

set to
𝑁=
2
.
23. The value of
𝜎max
for the IBPN+C guidance law was set to
𝜎max =𝜃0(=𝛾0)=
42
°
. This value
satisfies
(22)
and leads to smaller, slower-varying and more homogeneous amplitudes for the canard actuator control
signals. Hence, the risk of actuator saturation is limited, while a good impact accuracy is maintained. In general, values
𝜎max > 𝜃0
result in higher amplitudes for the reference load factors and lead to more frequent canard saturations. For
both, IBPN and IBPN+C, the value of
𝛾(𝑡f)
was determined using the 6-DoF model presented in Section II.C. The same
model has been used to construct the
𝑅go
based lookup table for the RBPN guidance law (cf. Section III.C.4). The
implementation details and parameters of the ZEM guidance law can be found in [14].
B. Nominal Ballistic Trajectory
Before evaluating the guidance laws in a MC setting, we first simulate a nominal ballistic trajectory. A nominal
trajectory can be defined as an unguided ballistic flight without any uncertainties.
For the initial conditions presented in Table 1 and for a target located at 24
km
downrange from the projectile launch
point, the resulting ballistic trajectory shown in Fig. 5a represents a perfect target hit. To better appreciate the drift in the
crossrange direction due to the yaw of repose effect, the resulting trajectory was coordinated to the Canon’s
]C
CS. Here,
the
𝑥
-axis is referred to as downrange, the
𝑦
-axis as crossrange, and the
𝑧
-axis represents the altitude. The associated
coordinate transformation matrix from ]Lto ]Cis given by
[𝑻]CL =

cos 𝜓0sin 𝜓00
−sin 𝜓0cos 𝜓00
0 0 1

(34)
Figure 5b illustrates the various navigation gain variations as a function of range-to-go. These gains were computed
in an open-loop fashion from the available kinematic variables corresponding to the nominal ballistic trajectory and the
initial conditions presented in Table 1.
Table 1 Initial conditions of the nominal trajectory.
State Value Unit
Initial velocity (𝑉0) 940 m/s
Initial spin rate (𝑝0) 305 Hz
Launch elevation angle (𝜃0) 42 °
Launch azimuth angle (𝜓0) 1.69 °
15

Launch point
0
00.05
0
0.5
0.1
5
1
10 0.15
1.5
Impact point
15
2
0.2
20
2.5
(a) Nominal ballistic trajectory and impact point.
0510152025
0
0.5
1
1.5
2
2.5
3
Apogee
(b) Navigation gain variation for a perfect hit.
Fig. 5 Nominal ballistic trajectory simulation results.
C. Monte Carlo Results
To demonstrate the effectiveness of the proposed methods, we perform 2000 MC simulations for each guided case as
well as for the pure ballistic case. The initial conditions presented in Table 1, as well as some aerodynamic coefficients
(longitudinal force
𝐶X
, normal force slope
𝐶N𝛼
) and environmental parameters (pressure
𝑃
, temperature
𝑇
, longitudinal
wind
𝑢w
, lateral wind
𝑣w
) are altered according to a normally distributed random uncertainties summarized in Table 2.
Table 2 Parameter uncertainties.
Parameter 1𝜎value Unit
𝑉05m/s
𝜃01.5 mils (NATO)
𝜓02.5 mils (NATO)
𝐶X1 %
𝐶N𝛼3 %
𝑃,𝑇0.4 %
𝑢w,𝑣w3.5 m/s
For the guided cases, the entire flight is decomposed into two flight phases: a) ballistic open-loop phase and b)
closed-loop guided phase. At the start of the closed-loop phase at time
𝑡s
, the pitch-yaw guidance and control are
initiated. We will investigate two start times, namely
𝑡s=
23 s and
𝑡s=
60 s. The latter start time results in approximately
less than 21 sof guided phase. From this flight point onwards, the resulting trajectory resembles a quasi-straight line.
Even though the determination of an optimal starting time for the guided phase is an important topic per se, the authors
tend to believe that due to the particularity of the considered system (i.e. spin-stabilized projectile with very limited
control authority and hence maneuverability), this ought to start at the earliest time possible, for example immediately
after the GPS signal is available and any initial cannon-induced trajectory oscillations and transients have died out.
16

The simulation results in Fig. 6 and Fig. 7 represent the relative impact distributions for
𝑡s=
60 s and
𝑡s=
23 s,
respectively. These results are also summarized in Table 3. Fig. 8 shows the Cumulative Distribution Function (CDF) of
the maximal necessary canard deflection angle during the guided phase. As can be seen from Fig. 6a, the total ballistic
dispersion that needs to be corrected by the guidance is more than ±1000 m in downrange and ±500 m in crossrange.
-1000 -500 0 500 1000
Downrange [m]
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Crossrange [m]
Mission target
CEP = 282.277 m
1 = 369.266 m
2 = 665.017 m
3 = 922.844 m
(a) Ballistic case
-800 -600 -400 -200 0 200 400 600 800
Downrange [m]
-800
-600
-400
-200
0
200
400
600
800
Crossrange [m]
Mission target
CEP = 12.472 m
1 = 110.888 m
2 = 543.631 m
3 = 763.576 m
(b) PN
-800 -600 -400 -200 0 200 400 600 800
Downrange [m]
-800
-600
-400
-200
0
200
400
600
800
Crossrange [m]
Mission target
CEP = 47.275 m
1 = 236.436 m
2 = 546.459 m
3 = 766.157 m
(c) IBPN
-800 -600 -400 -200 0 200 400 600 800
Downrange [m]
-800
-600
-400
-200
0
200
400
600
800
Crossrange [m]
Mission target
CEP = 7.675 m
1 = 139.824 m
2 = 545.834 m
3 = 762.730 m
(d) IBPN+C
-800 -600 -400 -200 0 200 400 600 800
Downrange [m]
-800
-600
-400
-200
0
200
400
600
800
Crossrange [m]
Mission target
CEP = 1.678 m
1 = 131.963 m
2 = 550.905 m
3 = 868.054 m
(e) ZEM
-800 -600 -400 -200 0 200 400 600 800
Downrange [m]
-800
-600
-400
-200
0
200
400
600
800
Crossrange [m]
Mission target
CEP = 1.157 m
1 = 143.106 m
2 = 551.245 m
3 = 839.246 m
(f) RBPN
Fig. 6 Relative impact distributions for guidance starting at 𝑡s=60 s.
Table 3 Summary of the Monte Carlo run results in [m].
Evaluation metric CEP CEP1𝜎CEP2𝜎CEP3𝜎
Guidance starts 𝑡s=60 s 𝑡s=23 s 𝑡s=60 s 𝑡s=23 s 𝑡s=60 s 𝑡s=23 s 𝑡s=60 s 𝑡s=23 s
Ballistic case 282.277 369.266 665.017 922.844
PN [18] 12.472 1848.352 110.888 2011.929 543.631 2408.394 763.576 2718.922
IBPN [10] 47.275 400.581 236.436 504.082 546.459 880.248 766.157 1217.010
IBPN+C [11] 7.675 401.183 139.824 496.888 545.834 805.346 762.730 1257.186
ZEM [14] 1.678 0.008 131.963 0.013 550.905 0.028 868.054 0.109
RBPN 1.157 0.002 143.106 0.002 551.245 0.004 839.246 0.006
As it can be inferred from Table 3, all evaluated guidance laws are able to reduce the dispersion below 50 mof
CEP when engaging the guidance at
𝑡𝑠=
60 s. However, only RBPN and ZEM permit an earlier start of the guidance,
yielding to a significant performance gain, especially for the CEP
3𝜎
metric. This behavior is likely to be attributed to
the ability of these guidance laws to fully exploit the built-in (i.e., by design) information about the nominal ballistic
17

-1000 -500 0 500 1000
Downrange [m]
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Crossrange [m]
Mission target
CEP = 282.277 m
1 = 369.266 m
2 = 665.017 m
3 = 922.844 m
(a) Ballistic case
-2000 -1000 0 1000 2000
Downrange [m]
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
Crossrange [m]
Mission target
CEP = 1848.352 m
1 = 2011.929 m
2 = 2408.394 m
3 = 2718.922 m
(b) PN
-1000 -500 0 500 1000
Downrange [m]
-1000
-500
0
500
1000
Crossrange [m]
Mission target
CEP = 400.581 m
1 = 504.082 m
2 = 880.248 m
3 = 1217.010 m
(c) IBPN
-1000 -500 0 500 1000
Downrange [m]
-1000
-500
0
500
1000
Crossrange [m]
Mission target
CEP = 401.183 m
1 = 496.888 m
2 = 805.346 m
3 = 1257.186 m
(d) IBPN+C
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Downrange [m]
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Crossrange [m]
Mission target
CEP = 0.008 m
1 = 0.013 m
2 = 0.028 m
3 = 0.109 m
(e) ZEM
-0.01 -0.005 0 0.005 0.01
Downrange [m]
-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Crossrange [m]
Mission target
CEP = 0.002 m
1 = 0.002 m
2 = 0.004 m
3 = 0.006 m
(f) RBPN
Fig. 7 Relative impact distributions for guidance starting at 𝑡s=23 s.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
0
5
10
15
20
25
30
Max canard deflection angle [deg]
PN
IBPN
IBPN+C
ZEM
RBPN
(a) Guidance starts at 𝑡s=60 s.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Iteration
0
5
10
15
20
25
30
Max canard deflection angle [deg]
PN
IBPN
IBPN+C
ZEM
RBPN
(b) Guidance starts at 𝑡s=23 s.
Fig. 8 CDF of the maximal canard deflection angle.
trajectory and make the necessary trajectory corrections earlier in the guidance phase. Consequently, as can be observed
in Fig. 8b, earlier trajectory corrections help to avoid actuator saturations. Surprisingly, the same observation cannot be
made for IBPN and IBPN+C. Despite the fact that both of these schemes aim at taking into account the (ideal) ballistic
trajectory of the projectile, the performance significantly drops when compared to the deferred start of
𝑡𝑠=
60 s. On the
other hand, the earlier start of PN, IBPN, and IBPN+C considerably deteriorate the performance, even when compared
18

to the unguided ballistic case.
Figure 8a suggests that a deferred start of the guidance can lead to actuator saturations for all guidance strategies.
This is likely a consequence of not having enough time to make the necessary trajectory corrections. Nevertheless, the
guidance laws of PN, IBPN, and IBPN+C achieve a similar level of performance as RBPN and ZEM. This is likely to be
attributed to the fact that as the projectile approaches the target, the remaining trajectory is closer to the straight line,
hence the underlying design assumptions of these guidance laws become more and more valid.
V. Conclusions
The simulation results of a dual-spin projectile suggest that, when compared to the legacy PN guidance law, the
heuristic adaptations of IBPN and IBPN+C to the three-dimensional guidance case can provide an extra performance
boost at the expense of added complexity of having to measure the LoS elevation angle during the flight. On the other
hand, the newly proposed RBPN guidance law demonstrates, in general, better performance than the state-of-the-art
ZEM guidance law. Moreover, the on-board computational complexity of RBPN is considerably lower than the ZEM
which, at each discrete time step, performs an online trajectory simulation on-board. In contrast to this, the RBPN
scheme only consists of a simple PN structure augmented with a two-dimensional lookup table to select the appropriate
value of the PN navigation gain. The navigation gain is computed off-line and allows to fully take into account the true
curvature of the nominal ballistic trajectory. All these aspects additionally allow the RBPN to be engaged early enough
in the ballistic phase to command necessary trajectory corrections and avoid potential actuator limits violations.
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