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

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

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AIAA Guidance, Navigation, and Control Conference

9 - 13 January 2017, Grapevine, Texas

AIAA 2017-1266

Copyright © 2017 by Radhakant Padhi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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

Adopting the nonlinear backstepping

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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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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