ABSTRACT
This paper presents a terminal guidance method for a reusable launch vehicle dur-
ing approach and landing phase. This guidance technology do not require any predeter-
mined reference trajectories, instead would be capable of obtaining feasible trajectories
online. In scenarios in which the re-entry vehicle is significantly deviated from its nor-
mal trajectories upon entry into the landing phase, the usefulness of such an online
method can be clearly realised. To solve the approach and landing guidance problem, a
concept called the sliding mode terminal guidance is presented. This approach takes the
advantage of the finite time reaching phase of a sliding mode technique to ensure that
any desired state constraints can be fulfilled within a finite time. Through a set of sim-
ulations, it was made clear that some robustness to variations in the initial downrange
and velocity is possible.

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TABLE OF CONTENTS
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Sliding mode control 3
3 System Model and Equations of Motions 4
4 Reentry SMTG Design 6
4.1 Sliding Mode Control for Systems with Relative Degree of 2 . . . . 6
4.2 Application of SMTG to Reentry Terminal Guidance . . . . . . . . 8
5 Results and Analysis 12
6 Conclusions 18
i
LIST OF FIGURES
3.1 . . . . . . . . . . . . . . . . 4
Diagram of the Approach and Landing Phase
5.1 . . . . . . . 12
Attitude vs downrange for varying value of final vertical velocity
5.2 . . . . . . . . . . 13
Vertical velocity vs time for varying final vertical velocity
5.3 . . . . . . . . . . . 13
Lift coefficient vs time for varying final vertical velocity
5.4 . . . . . . . . . . . . . . 14
Attitude vs downrange for varying lift coefficient
5.5 . . . . . . . . . . . . . 14
Vertical velocity vs time for varying lift coefficients
5.6 . . . . . . . . . . . . . . 15
Lift coefficient vs time for varying lift coefficients.
5.7 . . . . . . . . . . . . . . . . . . . . 15
Attitude vs downrange for varying n
5.8 . . . . . . . . . . . . . . . . . . . 16
Vertical velocity vs time for varying n
5.9 . . . . . . . . . . . . . . . . . . . . 16
Lift coefficient vs time for varying n
ii
CHAPTER 1
Introduction
1.1 Overview
In recent years, it has become apparent that there is a need for developing
advanced reentry guidance technologies that can improve the safety and reliability of
reusable launch vehicles (RLVs). In particular, guidance technologies are desired that
can accommodate for aerosurface failures, poor vehicle performance, or dispersions
from the desired trajectory . These types of technologies are particularly critical during
the approach and landing (A and L) reentry phase. Of these sources of uncertainty, this
method focuses on accounting for dispersions from the desired trajectory. However, the
method presented here is not intended to compensate for the aerosurface failures.
For the space shuttle, the A and L phase is the final phase of the reentry pro-
cess, beginning at the end of the terminal area energy management phase at an altitude
of roughly 10,000 ft and ending with touchdown on the runway. The goal of this flight
phase is for the RLV to land at a desired runway with a near-zero vertical velocity. The
vertical velocity at touchdown is desired to be below 5ft/s , but velocities up to 9 ft/s
are generally still considered acceptable for the space shuttle. One of the concerns with
guidance during the A and L phase is that there is the possibility that crucial control sur-
faces on the RLV may be damaged during the previous reentry phases, leading to a loss
of controllability. Another concern is that upon entering the A and L phase there may
be a significant deviation of the RLV from its desired trajectory. This concern is very
significant because current space shuttle guidance methods during the A and L phase
rely on the shuttle to follow a predetermined trajectory. There is thus a need for A and
L guidance technologies that do not require predetermined reference trajectories to be
implemented, but instead would be capable of obtaining feasible trajectories online.
There have been many previous papers that solve the A and L guidance prob-
lem while attempting to minimize the amount of offline information required. Schier-
man [2] presented an approach for obtaining feasible A and L trajectories in the pres-
ence of major control surface errors. He solved the problem using an optimum-path-
to-go (OPTG) approach. In this approach, a large database of neighboring optimal
trajectories is first generated offline. Then, when integrating the trajectory online, the
states are observed at each point and the trajectory is reconfigured to follow the partic-
ular offline trajectory that leads to the best performance.
This method presents a novel method for generating online A and L trajectories
by taking advantage of the finite-time-reaching phase of the sliding mode control tech-
nique. The only information the control law requires in addition to the instantaneous
state of the system is the desired runway location and the A and L initial conditions.
Thus, a certain level of robustness to initial conditions is obtained. To solve this prob-
lem, a concept of sliding mode terminal guidance (SMTG) is used. In SMTG the sliding
surface is chosen as a terminal constraint, which is the altitude at the landing site in this
case. Then, by making use of the finite-time-reaching phase of sliding mode, it can be
guaranteed that the terminal constraint will be reached in a finite time. A key principle
in these types of problems is that once the sliding surface is reached, the problem is
complete, and so no movement along the sliding surface is required. It should be noted
that while there have been previous approaches to reentry guidance with the sliding
mode technique, these approaches involved tracking a trajectory generated offline and
not creation of new trajectories to adapt to current conditions.
2
CHAPTER 2
Sliding mode control
In control theory, sliding mode control, or SMC, is a form of variable structure control
(VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system
by application of a high-frequency switching control. The state-feedback control law is
not a continuous function of time. Instead, it switches from one continuous structure to
another based on the current position in the state space. Hence, sliding mode control is
a variable structure control method. The multiple control structures are designed so that
trajectories always move toward a switching condition, and so the ultimate trajectory
will not exist entirely within one control structure. Instead, the ultimate trajectory will
slide along the boundaries of the control structures. The motion of the system as it slides
along these boundaries is called a sliding mode and the geometrical locus consisting of
the boundaries is called the sliding (hyper)surface.
CHAPTER 3
System Model and Equations of Motions
Figure 3.1: Diagram of the Approach and Landing Phase
This section details the system model of an RLV during the approach and landing
phase. The equations of motion of an unpowered RLV during the A and L phase are,
D
̇
V = − − gsinγ (3.1)
m
L gcosγ
−
γ=
̇ (3.2)
mV V
̇
h = V sinγ (3.3)
x = V cosγ
̇ (3.4)
In Eqs. (3.1) and (3.2), the lift and drag forces are defined as,
L = q SCL
̄ (3.5)
D = q SCD
̄ (3.6)
q is the dynamic pressure. q = 1 ρV 2
2
ρ = ρ0 e( − h/H) (3.7)
CL = CL0 sin2 αcosα (3.8)
2
CD = CD0 + KCL (3.9)
H = Scale height, 8.5 km
CD = Drag Coeffcient
CD0 = Zero lift drag coeffcient
CL = Lift Coeffcient
CL0 = Zero angle of attack lift coeffcient
D = Drag force
g = Earth’s gravitational acceleration
h = Altitude
K = Lift-induced drag coefficient parameter
L = Lift force
m = Reusable launch vehicle mass
S = Surface area
V = Velocity magnitude
α = Angle of attack
γ = Flight-path angle
ρ = Air density
5
CHAPTER 4
Reentry SMTG Design
This section covers the development of SMTG and its application to the A and L
reentry phase and consists of two parts. First, a general approach for solving sliding
mode problems with a relative degree of 2 is derived. Afterwards, the steps in the use
of this approach for solving the A and L guidance problem are given.
4.1 Sliding Mode Control for Systems with Relative Degree of 2
To handle problems with larger relative degrees, a higher-order sliding mode
(HOSM) approach must be used. In HOSM, both the sliding surface and its succes-
sive derivatives are driven to zero. Most HOSM algorithms are of the second order and
can be used to either handle problems with a relative degree of 2 or to reduce chattering
in problems with a relative degree of 1. One of the more popular second-order sliding
mode approaches is the twisting algorithm, in which the sliding surface and its deriva-
tive make an infinite number of rotations (twists) about the origin and reach zero in a
finite time. Although the twisting algorithm is a very useful approach that has been
used for a variety of applications, since it forces the sliding surface to oscillate between
positive and negative values before converging to zero, there are certain applications for
which it cannot be used. In particular, the twisting algorithm cannot be used for cer-
tain terminal sliding mode problems, in which once the sliding surface crosses zero, the
problem is generally considered to be over. Thus, there is a need for a new second-order
sliding mode approach that can be used to solve these types of problems. In this section,
a novel second-order sliding mode method based on an adaptive backstepping approach
is derived. The usefulness of this method is that it guarantees the sliding surface and
its derivative will go to zero in a finite time, while also ensuring that the sliding surface
will not cross zero until the final time.
First assume that is desired to reach a sliding surface of the form,
s1 = f (x) = 0 (4.1)
s ̇1 = h(x) (4.2)
s1 = l(x) + g(x)u
̈ (4.3)
functions h(x),l(x) and g(x) are assumed to be known functions of x, the state vector.
It can be seen that the control input u appears in the second derivative of the sliding
surface, so the surface s1 has a relative degree of 2. The goal is to find an expression
for u such that the sliding surface s1 and its first derivative will both go to zero at some
desired finite time tr. This goal is achieved through the use of a backstepping approach,
where first derivative of s1 is taken as a virtual control input. To find the value of
first derivative of s1 that would be needed to drive s1 to zero, a candidate Lyapunov
functionVl1 is chosen as,
1
Vl1 = s2 (4.4)
2 1
̇
Vl1 = s1 s ̇1 (4.5)
To guarantee that the surface s1 will go to zero in a finite time, first derivative of
s1 must be chosen such that Vl1 is negative-definite. This negative-definite property can
be achieved by choosing first derivative of s1 as,
ns2 1
s ̇1 = − ,n > 1 (4.6)
tr − t
so that
ns2
Vl = − tr −t < 0
1
One way to guarantee that the desired trajectory will be reached in a finite time is
to consider Eq. (4.6) as a new sliding surface:
ns1
=0
s2 = s ̇1 + (4.7)
tr − t
7
ns ̇1 (tr − t) + ns1
s ̇2 = l(x) + + g(x)u (4.8)
(tr − t)2
It can be seen from Eq. (4.8) that the new sliding surface s2 has a relative degree of
one with respect to the control input u. To find an expression for u that will drive s2 to
zero in some finitetime t∗ , a candidate Lyapunov function is chosen as,
r
1
Vl2 = s2 (4.9)
2 2
ns ̇1 (tr − t) + ns1
V ̇l2 = s2 s ̇2 = s2 l(x) + + g(x)u (4.10)
tr − t)2
make Eq. (4.10) negative-definite, u is selected as,
ns ̇1 (tr − t) + ns1 ns1
1
u=− + ηsgn s ̇1 +
l(x) + (4.11)
2
(tr − t) tr − t)
g(x)
s2 (0)
η= (4.12)
t∗r
necessary condition for the convergence,
t∗ < tr (4.13)
r
4.2 Application of SMTG to Reentry Terminal Guidance
This section details the steps in the application of the SMTG method to the reen-
try terminal guidance problem. For the A and L reentry phase, the two primary terminal
constraints are as follows:
1) The RLV must land on a desired runway located at some distance downrange of the
entry point of the A and L phase.
2) The RLV must land with a sufficiently small vertical velocity. The SMTG-based
guidance law developed in this section will lead to both of the above objectives being
8
satisfied for a desired runway location relative to the A and L entry point.
It should be noted that only longitudinal two-dimensional motion is analyzed in this
work. Thus, the vehicle attitude/orientation and any crossrange errors are not consid-
ered.
A sliding surface is first chosen to make the final altitude zero as,
s=h=0 (4.14)
̇ ̇
s = h = V sinγ (4.15)
s=A
̈ (4.16)
̇
s = B + C CL
̈ (4.17)
terms A, B, and C are highly nonlinear functions with the expressions,
1 S 1 S
A = −g + ρV 2 cosγCL − KsinγCL − ρV 2
2
sinγCD0 (4.18)
2 m 2 m
S ̇
2
cosγCL − KsinγCL − sinγCD0 V
B = ρV
m
(4.19)
1 S
+ ρV 2 2
−sinγCL − − cosγCD0 γ
KcosγCL ̇
2 m
S
1
C = ρV 2 [cosγ − 2KsinγCL ] (4.20)
2 m
By inspecting Eqs. (4.14) and (4.15), it can be seen that the RLV will be at a zero
altitude with a zero vertical velocity. However, a zero vertical velocity at touchdown is
unnecessarily restrictive, as the maximum vertical velocity at touchdown for the space
shuttle is typically between 7 and 10 ft/s. A more reasonable approach would thus
involve constraining the vertical velocity at touchdown to be some specific finite value
9
rather than zero. To allow for this constraint, a new variable is defined as,
z = s − sdes = 0
̇ ̇ ̇ (4.21)
where,
̇
sdes = hdes
̇ (4.22)
Integrating Eq.(4.21) leads to the expression,
z = s − sdes t + c
̇ (4.23)
where c is a constant. Now define a new term, called the time-to-go, as,
r
tgo = (4.24)
V
where r is the instantaneous range from the RLV to the runway given by,
(xr − x)2 + h2
r= (4.25)
The time-to-go gives an approximation of the time it will take the RLV to reach the
runway if it follows its present course. Using this time-to-go expression, the term c in
Eq. (4.21) is now chosen as ,
c = sdes (tgo − t)
̇ (4.26)
Substituting Eq. (4.26) into Eq. (4.21) leads to,
z = s − sdes t − sdes (tgo − t)
̇ ̇ (4.27)
Finally, simplifying Eq.(4.27) and substituting Eqs. (4.14) and (4.22), expression
for z
̇
z = h + hdes tgo (4.28)
Now, suppose that z is driven to zero at the runway. Since the time-togo, tg o, be-
comes, goes to zero as the distance to the runway (i.e. the range r) goes to zero, the
10
second term in Eq. (4.28) will go to zero at the runway. Thus, in order for Eq. (4.28)
to be zero at the runway, the first term in the equation (h) must also be zero. This im-
plies that the altitude will be zero at the runway, satisfying the touchdown requirement.
Equation (4.28) can be considered as a new sliding surface, which satisfies the same
property as the original sliding surface in Eq. (4.14). As was done before, Eq. (4.28) is
now differentiated until the control input appears, leading to,
z = y − ydes
̇ ̇ ̇ (4.29)
z=A
̈ (4.30)
̇
z = B + C CL
̈ (4.31)
where the terms A, B, and C were defined previously. Note that the condition
z =0 and first derivative of z= 0 implies that the RLV will be at a zero altitude with the
desired amount of vertical velocity. Thus, the reentry guidance problem can be solved
by finding an expression for the derivative of the lift coefficient that will guarantee
that the sliding surface z and its derivative will go to zero at some desired downrange
position xr corresponding to the runway.
11
CHAPTER 5
Results and Analysis
In this section, simulations of the proposed reentry guidance approach is presented.
For each of these simulations, the initial values for the altitude, velocity, and flight-
path angle are taken as 10,000 ft, 570 ft/s, and -25 degree respectively. First, it was
desired to see the effect of varying the desired vertical velocity at touchdown, initial
lift coefficient, and value for n. To analyze these effects, a set of simulations were
performed for an initial downrange of 28,000 ft. Following figures contain the results
of a simulation of varying the desired vertical velocity at touchdown for a specific initial
lift coefficient and n of 0.6 and 2, respectively.
Figure 5.1: Attitude vs downrange for varying value of final vertical velocity
Following are the results of the simulation of varying the initial lift coefficient for
a set desired vertical velocity at touchdown and n of 5 ft/s and 2 respectively .
Following are the results of a simulation of varying n for specific initial lift coeffi-
cient and desired vertical velocity at touchdown velocities of 0.6 and 5 ft/s. Inspect-
ing the results of varying the desired vertical velocity at touchdown , it is interesting
to observe that the final vertical velocity has a fairly negligible effect on the resulting
trajectories. However, the values of the initial lift coefficient and n have a large im-
Figure 5.2: Vertical velocity vs time for varying final vertical velocity
Figure 5.3: Lift coefficient vs time for varying final vertical velocity
13
Figure 5.4: Attitude vs downrange for varying lift coefficient
Figure 5.5: Vertical velocity vs time for varying lift coefficients
14
Figure 5.6: Lift coefficient vs time for varying lift coefficients.
Figure 5.7: Attitude vs downrange for varying n
15
Figure 5.8: Vertical velocity vs time for varying n
Figure 5.9: Lift coefficient vs time for varying n
16
pact on the transient performance and nature of the trajectories. By increasing the lift
coefficient, the RLV begins to perform an increasingly large pull-up maneuver at the
beginning of the trajectory. Also, each initial lift coefficient satisfies the desired vertical
velocity condition of 5 ft/s at touchdown, the horizontal velocity at touchdown increases
as the initial lift coefficient increases.
17
CHAPTER 6
Conclusions
A method for A and L reentry phase guidance has been introduced through a concept
of sliding mode terminal guidance (SMTG). A control law is obtained that allows for
the rapid construction of feasible trajectories that depend only on the instantaneous
states of the system and the initial and final conditions of the A and L phase. Thus,
the availability of a predetermined reference trajectory is not required. Through a set
of simulations, it was shown that the approach shows some robustness with respect to
variations in the initial downrange and velocity. From various simulations, it was found
that for a downrange of 28000 ft, a high initial lift coefficient and a low value of n lead
to acceptable transient performance.
REFERENCES
[1] Nathan Harl and S.N.Balakrishnan,“Reentry Terminal Guidance Through Sliding
Mode Control,” Journal Of Guidance, Control, And Dynamics,Vol. 33, no. 1,
January-February pp.186-199.2010.
[2] Schierman, J., Hull, J., and Ward, D, “Online Trajectory Command Reshaping for
Reusable Launch Vehicles,”AIAA Paper 2003-5439, August. 2003.
[3] Shtessel, Y., and Krupp, D,“Reusable Launch Vehicle Trajectory Control in Slid-
ing Modes,”Proceedings of the American Control ConferenceInst. of Electrical and
Electronics Engineers, Piscataway,NJ, June 1997, pp. 2557-2561.
[4] Schierman, J., Hull, J., and Ward, D,“Adaptive Guidance with Trajectory Reshaping
for Reusable Launch Vehicles,”AIAA Paper 2002-4458, August. 2002.
[5] Bollino, K., Ross, M., and Doman, D, “Optimal Nonlinear Feedback Guidance for
Reentry Vehicles,” AIAA Paper 2006-6074, August. 2006, pp.376-387.
[6] Schierman, J., Ward, D., Hull, J., Gahndi, N., Oppenheimer, M., and Doman,
D,“Integrated Adaptive Guidance and Control for Re-Entry Vehicles with Flight-
Test Results,”Journal Of Guidance, Control, And Dynamics, Vol. 27, No. 6,
November-December. 2004, pp. 975-988.