Proceedings ArticleDOI

# An Optimal Explicit Guidance Algorithm for Terminal Descent Phase of Lunar Soft Landing

09 Jan 2017-
TL;DR: In this paper, an explicit guidance algorithm for multi-constrained terminal descent phase of lunar soft landing is presented, where a minimum jerk guidance is designed and extended for this purpose to achieve the terminal control and state constraints.
Abstract: An explicit guidance algorithm for multi-constrained terminal descent phase of lunar soft landing is presented in this paper. A minimum jerk guidance is designed and extended for this purpose to achieve the terminal control and state constraints. The closed form jerk expression, obtained using the minimum jerk guidance is analyzed to obtain an explicit expression for acceleration command, which is the physical control variable for the guidance loop. The guidance formulation ensures the minimum rate of change of acceleration and vertical touchdown of the spacecraft towards a designated landing site with high terminal accuracy. The design features of the proposed guidance law are demonstrated using simulation results.

### An Optimal Explicit Guidance Algorithm for Terminal

• Descent Phase of Lunar Soft Landing Avijit Banerjee∗ and Radhakant Padhi† Indian Institute of Science, Bangalore, Karnataka, 560012, India.
• The guidance formulation ensures the minimum rate of change of acceleration and vertical touchdown of the spacecraft towards a designated landing site with high terminal accuracy.
• Many researches have been carried out to formulate a terminal guidance with autonomous re-targeting feature.
• It is inadequate to ensure the terminal vertical orientation of the spacecraft and also the trajectory generated using CTVG may results into subsurface travel.
• The jerk commands need to be integrated numerically to obtain equivalent accelerations as physical guidance command, in order to implement the same in actual guidance-loop.

### II. Problem Objective

• At the end of the braking phase, the lunar module is supposed to be placed at an altitude of 100m from the local lunar surface with near zero velocity and vertical orientation.
• The autonomous re-targeting guidance objective is to drive the spacecraft safely towards the determined landing site and ensure the terminal velocity and vertical orientation requirements.
• Where, (x, z) represents the position of the spacecraft described in local inertial frame of reference, (vx, vz) and (ax, az) describe the velocity and acceleration components respectively of the spacecraft motion in respective directions.
• To satisfy all the terminal constraints accurately and also to reduces the possibility of subsurface travel, the terminal descent phase is divided into two intermediate sub phases.
• After reaching 2m the propulsion system needs to shut down to prevent the contamination of lunar dust over the solar panel and spacecraft is allowed to free fall with negligible impact.

### III. Generic Formulation of Minimum Jerk Guidance

• 5 derived based on acceleration minimization provides the closed form feedback expression for acceleration, the guidance law is inadequate to ensure the terminal control constrain (vertical look angle).
• As the thrust engine is attached with the lunar module (along the roll axis), direction of thrust vector essentially represents the spacecraft orientation.
• Moreover, the resulting guidance law comes with an additional freedom of choice for the initial acceleration.
• The control continuity essentially leads to a smooth spacecraft trajectory.
• Solving the above system of equations for λxf , λzf , λvxf , λvzf , λaxf and λazf in terms of system state and putting that solution into the control expressions Eq.(7)and Eq.(8) the closed form solution for minimum jerk in respective directions are obtained.

### IV. Analytic Expression of Acceleration as Physical Guidance Command

• A typical guidance loop operates on acceleration command as the physical control variable.
• Solution of such time varying differential equations is not very trivial.
• To obtain the closed form acceleration expression an alternate approach has been adapted in the present work.
• The following terms are defined for brevity.

### V. Simulation Results

• To perform a precise maneuver, the spacecraft trajectory for the terminal descent phase is segmented into two intermediate sub phases, namely the re-targeting phase and vertical descent phase.
• At the end of re-targeting phase, the guidance algorithm needs to ensure terminal positional accuracy and nullify the horizontal velocity component.
• Look angle profile distance and smoothly placed above the desired landing site at an altitude of 20m without any subsurface maneuver as shown in Fig.(2).
• The result obtained in Fig.(3) also ensures the spacecraft motion is restricted in horizontal direction during vertical descent phase.

### VI. Conclusion

• Minimum jerk based explicit guidance algorithm for terminal descent phase is presented for the lunar soft landing.
• Based on the minimum jerk solution, a closed form expression for acceleration command as physical control variable for guidance loop has been derived.
• The formulation inherently poses the terminal state and control requirements as hard constraints and leads to high terminal accuracy of position, velocity and ensure the vertical orientation of the lunar module.
• The minimum jerk guidance along with explicit acceleration expression as physical guidance command formulated in this paper is demonstrated for the lunar soft landing mission, the formulation is quite generic and it can address similar class of problems with desired terminal state and control objectives.

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An Optimal Explicit Guidance Algorithm for Terminal
Descent Phase of Lunar Soft Landing
Avijit Banerjee
Indian Institute of Science, Bangalore, Karnataka, 560012, India
An explicit guidance algorithm for multi-constrained terminal descent phase of lunar
soft landing is presented in this paper. A minimum jerk guidance is designed and extended
for this purpose to achieve the terminal control and state constraints. The closed form jerk
expression, obtained using the minimum jerk guidance is analyzed to obtain an explicit
expression for acceleration command, which is the physical control variable for the guid-
ance loop. The guidance formulation ensures the minimum rate of change of acceleration
and vertical touchdown of the spacecraft towards a designated landing site with high ter-
minal accuracy. The design features of the proposed guidance law are demonstrated using
simulation results.
Nomenclature
x Downrange, m
z Altitude, m
v
x
, v
z
Horizontal and vertical components of velocity, m/s
a
x
, a
z
Horizontal and vertical components of acceleration, m/s
2
u
x
, u
z
Horizontal and vertical components of jerk, m/s
3
J Cost function for jerk minimization, m
2
/s
5
λ Co-state vector
λ
x
, λ
z
, λ
v
x
, λ
v
z
, λ
a
x
, λ
a
z
Components of co-state vector
λ
xf
, λ
zf
, λ
v
xf
, λ
v
zf
, λ
a
xf
, λ
a
zf
Terminal components of co-state vector
t Time, s
t
f
Final ﬂight time, s
t
g o
Available ﬂight time, s
t
g o
0
Initial available ﬂight time, s
Z
x
, Z
z
Zero eﬀort miss along downrange and altitude, m
V
x
, V
z
Zero eﬀort velocity miss along downrange and altitude, m/s
A
x
Zero eﬀort acceleration miss along downrange, m/s
2
c
1x
, c
2x
, c
3x
, c
1z
, c
2z
, c
3z
Integration Constants
J
a
Cost function for acceleration minimization, m
2
/s
3
J
ax
Cost function for acceleration minimization along downrange, m
2
/s
3
J
az
Cost function for acceleration minimization along altitude, m
2
/s
3
β Look angle, Deg
x
0
, z
0
Initial position components, m
v
x
0
, v
z
0
Initial velocity components, m/s
a
x
0
, a
z
0
Initial acceleration components, m/s
2
x
f
, z
f
Terminal position components, m
v
x
f
, v
z
f
Terminal velocity components, m/s
a
x
f
, a
z
f
Terminal acceleration components, m/s
2
g gravitational acceleration, , m/s
2
PhD Student, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India.
Professor, Dept. of Aerospace Engineering, IISc, Bangalore, India and Associate Fellow, AIAA
1 of 12
American Institute of Aeronautics and Astronautics
AIAA Guidance, Navigation, and Control Conference
9 - 13 January 2017, Grapevine, Texas
AIAA 2017-1266
AIAA SciTech Forum

H Hamiltonian
X R
6×1
State Vector
P R
6×6
System matrices
Q R
6×1
Input matrices
I. Introduction
Over the years, space exploration has become a potential platform for demonstrating technological
prowess of a nation. One key objective of space exploration is precise and safe landing of a space vehi-
cle over the unknown territory of a celestial body. An autonomous descent guidance for landing on moon,
which has to be essentially a soft landing, starts from the perilune, the nearest point of the elliptical orbit
(about 18km) from the lunar surface. The descent trajectory typically involves three intermediate phases,
viz., rough braking, ﬁne braking and terminal descent phase. The phase-I (rough braking) incorporates a
reverse thrust braking mechanism to decelerate the spacecraft from initial high orbital velocity to relatively
a lower velocity. The objective of the ﬁne braking phase is to guide the spacecraft towards a designated
landing site with a residual altitude (100m). At the end of the braking phase, the spacecraft is placed
just above the predetermined landing site with near zero velocity. At this point, a simple vertical terminal
descent may not be adequate as the presence of process noise during ﬁne braking may cause a signiﬁcant
deviation of the downrange which leads to deviation of the spacecraft from desired location. Following the
entire successful journey from earth, the possibility of landing over a small boulder or large pebble may
tumble down the spacecraft. Hence a vision based autonomous re-targeting is necessary. Onboard image
processing unit captures the local image of lunar surface and provide a safe landing site. The objective of the
autonomous re-targeting guidance is to drive the spacecraft towards the selected landing site with zero veloc-
ity. Also, the terminal orientation of the spacecraft needs to be vertical with respect to the local lunar surface.
Many researches have been carried out to formulate a terminal guidance with autonomous re-targeting
feature. A path shaping guidance
1
for terminal soft landing is presented by Colin et al. The guidance formu-
lation provides an analytical solution based on gravity turn approach.
2
control approach Feng et al.
3
addressed the terminal descent problem in Intergrated Guidance and Con-
trol (IGC) framework. A hybrid thrust-tether based terminal guidance strategy in presented in.
4
D’souza
5
framed a fuel optimal feedback guidance law based on acceleration minimization approach for planetary
landing. The resulting guidance law is commonly known as constrained terminal velocity guidance (CTVG).
The inherent similarities of the CTVG with the generalized explicit guidance (GENEX) are demonstrated by
Guo et al.
6
Although the close form acceleration expression obtained from CTVG is applicable for onboard
implementation, it is inadequate to ensure the terminal vertical orientation of the spacecraft and also the
trajectory generated using CTVG may results into subsurface travel. Also, Intercept Angle Control Guid-
ance (IACG), as a combination of CTVG and Free Terminal Velocity Guidance (FTVG), is proposed to
address the look-angle constraints. However, the IACG may not satisﬁe the soft landing requirement as the
formulation inherently poses a directionally constrained terminal velocity vector with velocity magnitude
being free parameter. To ensure the terminal orientation of lunar module, the desired terminal acceleration
is incorporated with the cost function of CTVG as a soft constraint.
7
Depending on the choice of the sen-
sitivity parameter of the additional soft constraint, the formulation incorporates tradeoﬀ between the fuel
optimality and terminal control constraint. The augmentation of the additional soft constraint
7
eventually
leads to a modiﬁed expression of the ﬁnal time, derived based on transversality conditions. An improved
CTVG,
8
prevents the spacecraft trajectory from sub surface travel. But the modiﬁed acceleration command
may results in an unacceptable terminal position and velocity error.
Zhang et al.
8
introduced the concept of pseudo control as time derivative of acceleration to attain the
terminal state and control constraints simultaneously . In order to eliminate the unacceptable terminal error,
the Model predictive Static Programming (MPSP) is proposed for generating the guidance command. The
MPSP based algorithm provides an onboard computable fast numerical guidance with high precision terminal
accuracy. A similar concept like pseudo control as the time derivative of acceleration has been adopted by
Uchiyama et al.
9
A uniﬁed feedback guidance law, based on jerk minimization, in constrast to the earlier
acceleration minimization approach is introduced. The closed form jerk expression obtained using calculus
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American Institute of Aeronautics and Astronautics

of variation,
9
addresses all the terminal constraints (state and control) for the retargeting phase of the lunar
landing. The ﬁnal ﬂight time is computed based on the transversality conditions and results into a sixth
order polynomial of available ﬂight time. A minimum jerk based missile guidance law was given by Grinfeld
et al.
10
Here a diﬀerent solution approach is considered based on state transition matrix. Although the
closed form solution of the optimal jerk is presented, the jerk commands need to be integrated numerically
to obtain equivalent accelerations as physical guidance command, in order to implement the same in actual
guidance-loop. The numerical integration may involve sensors biases and is prone to integration errors.
In this paper, the minimum jerk guidance has been designed for the re-targeting phase of the lunar soft
Figure 1. Schematic spacecraft trajectory and reference frame
landing. Based on the state feedback jerk expression, a closed form solution of the acceleration command has
been derived. That eliminates the requirement of numerical integration. The minimum jerk based guidance
command provides a family of acceleration proﬁles based on initial choice of available ﬂight time (t
g o
0
),
such that all of them ensure the desired terminal constraints (position, velocity and look angle) for the soft
lunar landing with high terminal accuracy. Along with aforementioned characteristics, the minimum jerk
based guidance formulation assures minimum rate of change of acceleration which is much desirable from
propulsion perspective as the thrust engine has a physical limitation to respond to the change in acceleration
demand.
II. Problem Objective
At the end of the braking phase, the lunar module is supposed to be placed at an altitude of 100m from
the local lunar surface with near zero velocity and vertical orientation. The velocity and vertical look angle
requirement is necessary to capture good quality local terrain images (free from motion blur). The image
processing unit determines a safe landing site based on the local terrain images captured by onboard camera.
The autonomous re-targeting guidance objective is to drive the spacecraft safely towards the determined
landing site and ensure the terminal velocity and vertical orientation requirements.
The lunar terminal descent initiate at about an altitude of 100m, where the lunar surface can be reason-
ably considered as ﬂat and gravitational acceleration is fairly constant. spacecraft dynamics in local inertial
reference frame is the same adopted by D’souza
5
which is described by the following equation of motion
Eq.(1) The origin of the local reference frame is considered to be located on the local terrain, vertically
below the position of the spacecraft at the end of braking phase(100 m altitude as shown in Fig.(1)). The
origin and the initial position of spacecraft together determine the ‘z’ axis. The vector, from the origin
towards the re-targeted landing site construct the ‘x’ axis. The re-targeting maneuver of the spacecraft is
considered to be conﬁned in planar motion (‘x-z’ plane) as shown in Fig.(1). where, (x, z) represents the
position of the spacecraft described in local inertial frame of reference, (v
x
, v
z
) and (a
x
, a
z
) describe the
velocity and acceleration components respectively of the spacecraft motion in respective directions. The
lunar gravitational acceleration is denoted by g. The time derivative of acceleration components (u
x
, u
y
)
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essentially represent the jerks, in respective directions, which are considered to be the control variables. The
spacecraft initial position (x
0
, z
0
), velocity (v
x
0
, v
z
0
), and acceleration (a
x
0
, a
z
0
) of the re-targeting phase
are essentially obtained from the terminal states of the braking phase. Similarly the desired terminal states
are given by, (x
f
, z
f
), (v
x
f
, v
z
f
) and (a
x
f
, a
z
f
) respectively.
˙x = v
x
˙z = v
z
˙v
x
= a
x
˙v
z
= a
z
+ g (1)
˙a
x
= u
x
˙a
z
= u
z
In order to accomplish a successful soft landing mission, very precise spacecraft motion must be carried out
during the terminal descent phase. To satisfy all the terminal constraints accurately and also to reduces
the possibility of subsurface travel, the terminal descent phase is divided into two intermediate sub phases.
The phase-I (re-targeting phase) is designed to drive the spacecraft from 100m altitude and place it just
above the re-targeted landing site of about 20m altitude. At the end of phase-I, only the horizontal velocity
component v
x
f
= 0 must be nulliﬁed. During the phase-II(vertical descent phase) the spacecraft is only
allowed to perform an unidirectional motion and from 20m it descends down vertically to 2m altitude with
zero velocity. The introduction of the additional sub-phase in guidance design provides an auxiliary safe
guard (of 20m)to subsurface travel. Also, it reduces the possibility of the terminal residual errors. After
reaching 2m the propulsion system needs to shut down to prevent the contamination of lunar dust over the
solar panel and spacecraft is allowed to free fall with negligible impact.
III. Generic Formulation of Minimum Jerk Guidance
Although the constrained terminal velocity guidance,
5
derived based on acceleration minimization pro-
vides the closed form feedback expression for acceleration, the guidance law is inadequate to ensure the
terminal control constrain (vertical look angle). As the thrust engine is attached with the lunar module
(along the roll axis), direction of thrust vector essentially represents the spacecraft orientation. The thrust
vector orientation (look angle) is expressed in terms of acceleration components as
β = tan
1
a
z
a
x
(2)
To ensure the the terminal vertical orientation (β = 90
0
), terminal horizontal acceleration component needs
to be nulliﬁed (a
xf
= 0). The explicit guidance law based on jerk minimization
9
is presented in this section
for the shake of completeness. The derivation of minimum jerk guidance considers the acceleration as the
state variable. Hence the formulation inherently poses the desired terminal look angle constraint. Moreover,
the resulting guidance law comes with an additional freedom of choice for the initial acceleration. This
additional freedom can be used to assure control continuity between the braking and the re-targeting phases.
The control continuity essentially leads to a smooth spacecraft trajectory. The minimum jerk guidance is
formulated as the following optimal control problem
min
˙a
x
, ˙a
z
J =
1
2
t
f
Z
0
˙a
2
x
+ ˙a
2
z
dt (3)
= min
u
x
,u
z
1
2
t
f
Z
0
u
2
x
+ u
2
z
dt
The objective is to obtain a closed form jerk expression that minimizes the cost function of Eq.(3) subjected
to the system dynamics Egn.(1) along with the initial conditions x(0) = x
0
, z(0) = z
0
, v
x
(0) = v
x0
, v
z
(0) =
v
z0
, a
x
(0) = a
x0
, a
z
(0) = a
z0
and the following terminal conditions needs to be ensured.
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x(t
f
) = x
f
, z(t
f
) = z
f
, v
x
(t
f
) = v
xf
, v
z
(t
f
) = v
zf
, a
x
(t
f
) = a
xf
, a
z
(t
f
) = a
zf
. Considering the co-state
vector
λ
T
=
h
λ
x
λ
z
λ
v
x
λ
v
z
λ
a
x
λ
a
z
i
The Hamiltonian for the above problem is formulated as,
H =
1
2
u
2
x
+ u
2
z
+ λ
x
v
x
+ λ
z
v
z
+ λ
v
x
a
x
+ λ
v
z
(a
z
+ g) + λ
a
x
u
x
+ λ
a
z
u
z
(4)
Using the calculus of variation the co-state dynamics is obtained as
˙
λ
x
= 0
˙
λ
z
= 0
˙
λ
v
x
= λ
x
˙
λ
v
z
= λ
z
(5)
˙
λ
a
x
= λ
v
x
˙
λ
a
z
= λ
v
z
The optimal control equation in terms of co-state variables is presented as
u
x
+ λ
a
x
= 0
u
z
+ λ
a
z
= 0
considering the terminal co-state variables as follows,
λ
x
(t
f
) = λ
xf
λ
z
(t
f
) = λ
zf
λ
v
x
(t
f
) = λ
v
xf
λ
v
z
(t
f
) = λ
v
zf
λ
a
x
(t
f
) = λ
a
xf
λ
a
z
(t
f
) = λ
a
zf
Denoting t
g o
= t
f
t, the solution of the co-state equation is represented as
λ
x
= λ
xf
λ
z
= λ
zf
λ
v
x
= λ
xf
t
g o
+ λ
v
xf
λ
v
z
= λ
zf
t
g o
+ λ
v
zf
λ
a
x
=
λ
xf
2
t
2
g o
+ λ
v
xf
t
g o
+ λ
a
xf
(6)
λ
a
z
=
λ
zf
2
t
2
g o
+ λ
v
zf
t
g o
+ λ
a
zf
Using the solution of the co-state equations, the optimal control equations are rewritten as
u
x
=
λ
xf
2
t
2
g o
+ λ
v
xf
t
g o
+ λ
a
xf
(7)
u
z
=
λ
zf
2
t
2
g o
+ λ
v
zf
t
g o
+ λ
a
zf
(8)
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##### References
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Proceedings ArticleDOI
11 Aug 1997
TL;DR: In this article, a guidance law which minimizes the commanded acceleration along with the (weighted) final time is developed, which is a linear function of the states relative to the landing point.
Abstract: A guidance law which minimizes the commanded acceleration along with the (weighted) final time is developed. This guidance law is a linear function of the states (relative to the landing point) and a nonlinear function of the time-to-go. The time-togo is obtained as a solution to a quartic equation which is solved analytically. The advantage of this guidance law is that it does not involve any iterations whatsoever. It is the exact solution to the two-point boundary-value problem associated with the first variation necessary conditions. It also satisfies the second variation necessary conditions for a minimum. An example of a lunar landing is given to demonstrate the optimality of this guidance law.

140 citations

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TL;DR: In this article, the authors proposed to perturb a nominal gravity-turn trajectory by offsetting the thrust vector for some period of time from the nominal attitude, anti-parallel to the instantaneous velocity vector The offset is removed when the predicted and required landing points coincide.
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TL;DR: In this paper, a filter-based backstepping control strategy is proposed to overcome the explosion of terms problem existing in standard back-stepping, and the stability of the closed-loop system is guaranteed based on Lyapunov stability theory and singular perturbation theory.

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