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A State-Triggered Line of Sight Constraint for 6-DoF Powered Descent Guidance Problems

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A State-Triggered Line of Sight Constraint for 6-DoF
Powered Descent Guidance Problems
Taylor P. Reynolds
, Michael Szmuk
, Danylo Malyuta
,
Mehran Mesbahi
, Beh¸cet A¸cıkme¸se
,
Dept. of Aeronautics & Astronautics, University of Washington, Seattle, WA 98195, USA
and
John M. Carson III
NASA Johnson Space Center, Houston, TX 77058, USA
This paper presents the formulation of a constrained 6-degree-of-freedom (6-DoF) pow-
ered descent guidance problem. The goal of this work is to design algorithms that obtain
locally optimal solutions to such problems, and that are amenable to real-time implemen-
tation. Using unit dual quaternions to parameterize the equations of motion, we devise
a free final time continuous optimal control problem that is subject to state and control
constraints. A novel feature of this formulation is the use of state-triggered constraints,
which are constraints enforced only when a certain state-dependent criterion is met. We
use these constraints to model a line of sight pointing constraint that is enforced condition-
ally based on the distance from the landing site. A numerical example highlights how the
inclusion of this constraint alters control commands and the resulting descent trajectory.
I. Introduction
Powered descent guidance refers the problem of transferring a vehicle from an estimated initial state to a
desired final state by using rocket-powered engines and/or reaction control systems. The problem of optimal
powered descent has been well studied since the Apollo program sought to achieve soft-landings on the
moon.1–3 The first tractable solutions adequate for flight operations, however, were limited to one-degree-
of-freedom (1-DoF) systems for purely vertical descent trajectories.1The structure of optimal solutions for
the more general 3-DoF translational guidance problem has been understood since the work of Lawden,2,4
but numerical solutions were difficult to obtain at the time. While optimal guidance was not incorporated
in the Apollo flight code, there is evidence that the system designers were aware of how their polynomial
guidance scheme stacked up relative to the optimal solution.5
Near the turn of the century, there was renewed interest in optimal powered descent guidance problems
due primarily to robotic Martian landing missions. These works expanded on previous theory in seek of
analytical solutions to the 3-DoF problem. D’Souza studied the free final time, minimum energy solutions
and obtained an analytical feedback control law as a function of the time-to-go.6Topcu, Casoliva and Mease
presented further results on the minimum fuel problem and compared theoretical predictions against rapidly
maturing numerical solvers.7,8 Several other authors continued to study the necessary conditions of the
minimum fuel 3-DoF guidance problem using optimal control theory.9–11
At the same time, A¸cıkme¸se and Ploen published work with an alternative viewpoint on the 3-DoF prob-
lem.12–14 They took a convex programming approach, and showed that a non-convex (non-zero) lower bound
on thrust magnitude can be relaxed by introducing a slack variable.14 Using Pontryagin’s maximum princi-
ple, they showed that in fact this relaxation is lossless, and yields the same optimal solution as the original
problem. A subsequent change of variables and relaxation led to a fully convex problem formulation that
can be solved efficiently. This lossless convexification result began a fruitful series of work that expanded the
Doctoral Student, AIAA Student Member {tpr6,szmuk,danylo}@uw.edu
Professor, AIAA Associate Fellow, {mesbahi,behcet}@uw.edu
SPLICE Project Manager, AIAA Associate Fellow, john.m.carson@nasa.gov
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theory to non-convex thrust pointing constraints,15–17 minimum-landing error problems18 and more general
optimal control problems.19–21 Interior time state and control constraints can render the solution of the nec-
essary conditions of optimal control theory a difficult proposition; but convex optimization does not suffer
from such issues provided the constraints are convex. Fortunately, a majority of the constraints of interest
in the powered descent guidance problem come in the form of second-order cones, a class of convex sets that
we exploit in this work.
More recent work has explored extensions to the 6-DoF problem that considers both translational and
rotational motion. These include the use of Lyapunov techniques,22 model predictive control23, 24 and feed-
forward trajectory generation techniques.25–28 The latter techniques devise iterative strategies that are used
to obtain feasible solutions to nonlinear and nonconvex optimal control problems that approximate local
optimality. A complete characterization of fuel optimal solution(s) for the 6-DoF problem is an active area
of research. However, numerical techniques offer promising results that locally optimal solutions can be
found by sequentially solving convex optimization problems with guaranteed convergence properties.29–31
The state variables selected for the 6-DoF problem formulation can be used to classify various methods.
For example, one may use standard Cartesian variables in conjunction with unit quaternions25–27 or dual
quaternions.22–24, 32, 33 This distinction should be viewed as a design choice during guidance system design.
We find that when there are constraints that naturally couple rotation and translation (such as line of sight
constraints), parameterizations using dual quaternions provide an efficient alternative. In this work, we elect
to parameterize the equations of motion and state constraints using dual quaternions.
Dual quaternions are a generalization of Hamilton’s quaternions that encode both relative orientation and
position information in a single parameter.34 An attractive feature of this parameterization is that the
equations of motion can be expressed in a form similar to the standard quaternion kinematic and dynamic
equations. Moreover, the formulation of several key constraints – including line of sight – are convex over a
given set of dual quaternions.24, 35
A. Contributions of This Work
The Apollo guidance system designated three powered descent phases: the braking phase, the approach phase
and the terminal-descent phase.5The braking phase slowed the vehicle from orbital speeds by thrusting
primarily in the anti-velocity direction. Prior to the approach phase, a pitch-up maneuver to a desired
attitude was executed to serve as an initial attitude for the approach phase. During the approach phase,
the lunar module maintained continuous visibility of the landing site until roughly 5 seconds before the
terminal-descent phase began. The attitude guidance system was designed to ensure a line of sight to the
landing site only when the geometry permitted.5There was no a-priori guarantee that this would occur.
We consider a similar scenario whereby a landing vehicle must maintain a line of sight until a certain distance
from the landing site, and use the newly introduced state-triggered constraints to model this constraint.28
We note that time-based criteria can equivalently be used in our framework. This allows the vehicle to
be free of the line of sight constraint once it is sufficiently close to the landing site, while maintaining
a continuous optimization framework (i.e., we do not resort to binary or integer variables). As a result,
guidance trajectories can be designed with the line of sight explicitly enforced when it is required, and not
enforced when it is not required.
State-triggered constraints (STCs) model constraints that are enforced only if a criterion conditioned on the
state vector is met. While this work focuses in particular on a distance-triggered line of sight constraints,
previous work has shown their applicability to speed-versus-angle-of-attack constraints for aerodynamic
descent maneuvers.28 Though not addressed here, state-triggered constraints can also be used to model
state-based keep-out zone constraints for autonomous collision avoidance maneuvers, and minimum (or
maximum) time-based constraints, among others. The continuous formulation of STCs offers a novel and
elegant way to incorporate such constraints without resorting to engineering heuristics or mixed-integer
programming. The resulting continuous optimal control problems are nonconvex; however we find that they
can be readily solved using a successive convexification procedure. In this work, we use such a procedure to
obtain feed-forward guidance solutions.
This paper is organized as follows. First, §II introduces our notation and description of 6-DoF motion using
dual quaternions. Next, §III details the state and control constraints as a function of the dual quaternion
and concludes with a statement of the problem that is solved. The solution method used in this work is
briefly discussed in §IV. Lastly, §V provides a numerical example that highlights the distance-triggered line
of sight constraint for a lunar landing scenario.
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B. Notation
We denote the set of real numbers using R, and use R+and R++ to denote non-negative and positive real
numbers, respectively. We use 0n×mto denote a matrix of size n×mwhose entries are all zero, and Into
denote the n×nidentity matrix. We use sp ec {·} to denote the set of eigenvalues of a matrix and denote
a symmetric positive semi-definite matrix Musing MSn
+or M0. The skew symmetric cross product
operator is defined as ·×:R3R3×3so that for any two vectors a,bR3we have a×b=a×b.
Unit quaternions are used to parameterize the set of isometric transformations (rotation matrices) of the
Special Orthogonal group in 3-dimensions, SO(3). We use Qto denote the quaternion manifold, and a
general element qQis composed of a vector part, qv, and a scalar part q0. We will write the set of
unit quaternions as Qu={q|q·q= 1}, where ·denotes the Euclidean scalar product. The set of unit
quaternions QuQis said to form a three-dimensional hypersphere within the quaternion manifold. We
refer to quaternions that have a zero scalar part as pure quaternions. Dual quaternions are denoted with a
˜
·to distinguish them from their quaternion counterparts.
II. Dual Quaternions and Rigid Body Motion
The Special Euclidean group, SE(3), contains all possible configurations of a rigid body relative to a fixed
inertial frame. Elements of SE(3) can be described using 4 ×4 homogeneous transformation matrices,
SE(3) = (TR4×4
T="Cr
03×11#, C SO(3),rR3).(1)
Unit dual quaternions parameterize this set in a similar way that unit quaternions parameterize the Special
Orthogonal group in three dimensions. Dual quaternions may be elegantly derived using the theory of
Clifford algebras as in,36 or by using geometric construction as in the original work of Clifford.34 We adopt
the notation that a dual quaternion is represented as
˜
q=q1+q2Q2(2)
where q1,q2Qare quaternions and 6= 0 is termed the dual unit that satisfies the property 2= 0. We
call q1the real part and q2the dual part of the dual quaternion ˜
q. Dual quaternions are elements of the
manifold Q2, an inherently different algebraic construct than the usual Euclidean vector space R8. Unit dual
quaternions form a subset Q2
uQ2within the dual quaternion manifold. Under the usual scalar product, a
dual quaternion of unit norm should satisfy
˜
q·˜
q= (q1+q2)·(q1+q2) = q1·q1+(q1·q2+q2·q1) = 1 + 0.(3)
It can then be observed that the real and dual parts of ˜
qmust satisfy
q1·q1= 1 and q1·q2= 0,(4)
in order for ˜
qto be a unit dual quaternion. Consequently, we define the set of unit dual quaternions as
Q2
u:={˜
q=q1+q2|q1·q1= 1,q1·q2= 0}.(5)
Note that the first constraint forces the real part of a unit dual quaternion to be a regular unit quaternion,
i.e. an element of the three dimensional hypersphere Qu. The second constraint dictates that the dual part
of a unit dual quaternion must be an element of the (three dimensional) tangent plane of the hypersphere
at the point q1. As such, we may view the set of unit dual quaternions in (5) as the three dimensional
hypersphere plus all of its tangent planes. Since attitude is encoded in the three dimensional hypersphere
via unit quaternions, the additional three dimensional tangent planes shall provide a natural environment in
which to encode position states.
A. Dual Quaternion Operations
Let ˜
q,˜
pQ2
ube two unit dual quaternions, and let a,bQube two unit quaternions. Recall that
quaternions are composed of a vector part and a scalar part, which we shall denote by a= (av, a0) and
b= (bv, b0). We first define quaternion multiplication as
ab=a0bv+b0av+a×
vb, a0b0av·bv,(6)
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from which dual quaternion multiplication is defined as
˜
q˜
p=q1p1+(q1p2+q2p1).(7)
Next, the quaternion cross product is defined as
ab=a0bv+b0av+a×
vb,0,(8)
which is used in turn to define the dual quaternion cross product23, 24
˜
q˜
p=q1p1+(q1p2+q2p1).(9)
We define the quaternion conjugate as a= (av, a0), which permits the definition of the dual quaternion
conjugate as
˜
q=q
1+q
2.(10)
For our purposes, we shall embed unit dual quaternions in the eight-dimensional Euclidean space R8so
that we may use more familiar matrix-vector analysis to manipulate them. To be clear, the elements of Q2
u
defined in (5) form a submanifold of Q2; we merely view them as elements of R8for convenience. Using the
natural isomorphism
˜
q=q1+q2Q2
u7→ ˜
q="q1
q2#R8
u:={˜
qR8|qT
1q1= 1 and qT
1q2= 0},(11)
we henceforth view unit dual quaternions as the subset R8
uR8defined by the constraints qT
1q1= 1 and
qT
1q2= 0. By virtue of the first four elements of ˜
qR8
u, it follows that we view unit quaternions as four
dimensional unit vectors embedded in R4. As such, we may now define special matrices to represent the
operations in (7) and (9). Specifically, when viewed as an element of R4we may rewrite (6) as
qp= [q]p= [p]
q(12)
where,
[q]:="q0I3+q×
vqv
qT
vq0#and [p]
:="p0I3p×
vpv
pT
vp0#.
Using these definitions, we can then rewrite (7) as
˜
q˜
p= [ ˜
q]˜
p= [ ˜
p]
˜
q,(13)
where,
[˜
q]:="[q1]04×4
[q2][q1]#and [ ˜
p]
:="[p1]
04×4
[p2]
[p1]
#.
The quaternion and dual quaternion cross products can be rewritten using matrices via the same methods,
see23, 24 for more details. It is important to note that due to (11) the dual unit is no longer present in these
expressions. The matrices in (13) are structured so that the matrix-vector multiplication gives the same
result as the definition in (7)a.
In deriving constraints as a function of a dual quaternion, we will make heavy use of the following two results.
For a,b,cR4we have
aT(bc) = bT(ac) = cT(ba),(14)
and if qis a unit quaternion, then
aTb= (aq)T(bq)=(qa)T(qb),(15)
which is referred to as the quaternion triple identity.23
aThe columns of these matrices can also be interpreted as the projection of the dual quaternion onto the basis vectors of
the Clifford sub-algebra used to derive them.36
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B. Rigid Body Motion
Let rdenote the origin of a body-fixed coordinate frame FBwith respect to an inertial coordinate frame
FI, and let q:=qB←I R4be the unit quaternion representing the orientation of FBwith respect to FI.
The composition of a rotation and translation is represented using dual quaternion multiplication. As such,
the unit dual quaternion that represents a translation by rfollowed by rotation qis given by (taking (11)
into account)
˜
q="q
1
2qrI#="q
1
2rBq#R8
u,(16)
where rIand rBdenote the coordinates of the vector rin the inertial and body frame respectively. The first
expression describes a translation by rIfollowed by a rotation q, whereas the second expression describes a
rotation by qfollowed by a translation rB. Figure 1 depicts each of these cases, and shows how they result
in the same geometric definition of relative position and orientation.
The equivalence of the two expressions in (16) leads to the observation that
rI=qrBqand rB=qrIq.(17)
FI
FB
q
rB
˜
q
FI
FB
q
rI
Figure 1: Rotation followed by translation is geometrically equivalent to a translation followed by a rotation.
This leads to the two equivalent definitions of the dual quaternion in (16).
Similarly, we can represent the velocity states using dual quaternions as follows. Let ωB,vBR3denote the
angular and linear velocity of a rigid body, respectively, expressed in body frame coordinates. By appending
a zero to each vector, we represent these as pure quaternions and define the dual velocity to be
˜
ω="ωB
vB#R8.(18)
Note that there is no requirement that this is unit dual quaternion.
1. Kinematics & Dynamics
We can take a time derivative of (16) to arrive at the dual quaternion kinematic equation.
d˜
q
dt =d
dt "q
1
2rIq#="˙
q
1
2(˙
rIq+rI˙
q)#="1
2qωB
1
2(vIq+1
2rIqωB)#
=1
2"q
1
2rIq#"ωB
vB#=1
2˜
q˜
ω(19)
where we’ve used (17) and (13) to write vIq=qvBand obtain the second to last equality.
The dynamics are obtained using the Newton-Euler equations in a rotating frame. We assume that neither
the mass nor the inertia are constant. Rather, the mass, mR+, is assumed to vary as a linear function of
the thrust magnitude according to
˙m=αkuBk2, α :=1
Ispge
(20)
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where uBR3is the thrust vector in body frame coordinates, Isp is the specific impulse of the rocket
engine in vacuum and ge= 9.806 m/s2is the acceleration due to gravity at sea level on Earth. The inertia
is assumed to be a function of the mass,
J:=J(m)S3
++,(21)
where the specific form of J(m) may vary with applications. We shall define the form assumed for our lunar
landing application in §V. The dynamical equations of motion are then obtained by taking a derivative of
the linear and angular momenta in the rotating body frame according to
d
dt(mvB) = m˙
vB+ω×
B(mvB) = XFB+uB,(22a)
d
dt(JωB) = ˙
JωB+J˙
ωB+ω×
B(JωB) = XTB+r×
uuB,(22b)
where FBR3and TBR3represent the externally applied forces and torques, respectively, and rudenotes
the constant vector from the vehicle’s center of mass to the point where the thrust is applied. We assume
in this work that PFB=gBand PTB= 0, where gBis the force due to gravity in the body frame. Note
that our definition of the mass depletion dynamics in (20) allows us to capture momentum changes due to
mass variability in (22a) using the term uB; see37 for details.
Combining the expressions in (22) with the definition of the dual velocity in (18) leads us to express the
equations of motion in terms of the dual velocity as
J˙
˜
ω+˜
ωJ+˙
JEr˜
ω= ΦuB+˜
gB,(23)
where Er=diag {I4,04×4}and,
J=
04×4mI30
0 1
J0
0 1 04×4
8×8
Φ =
I30
0 1
r×
u03×1
01×30
8×4
˜
gB="gB
0#8×1
.
We encourage the reader to refer to23, 24, 32, 33 for more details on rigid body kinematics and dynamics using
dual quaternions.
III. Problem Statement
This section details the ingredients necessary to state the continuous-time powered descent guidance problem
that we consider in this paper. Having already stated the equations of motion, this section focuses specifically
on the state and control constraints imposed during powered descent maneuvers. We now assume that the
inertial frame can be described by a set of three orthonormal vectors {xI,yI,zI}such that zIpoints locally
up, xIpoints south and yIpoints east. The body frame can similarly be described by the orthonormal
vectors {xB,yB,zB}that are assumed to describe the vehicle’s principal axes of inertia. We assume that
zBis the vector closest to the vehicle’s vertical axis.
This section is organized as follows. First, §A introduces the state and control constraints that we consider to
form a baseline problem. Next, §B introduces the concept of state-triggered constraints, and §C details their
application to distance-triggered line of sight constraints. We conclude in §D with a complete statement of
the optimal control problem that is to be solved.
A. Baseline Problem
Powered descent guidance problems are subject to several state and control constraints to ensure that
trajectories adhere to both safety considerations and vehicle limitations. We begin our formulation of the
baseline problem by discussing the control constraints.
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1. Control Constraints
The control authority of a rocket-powered vehicle is limited by the rocket engine it is equipped with. These
complex machines necessitate, at times, operation in restricted regimes of thrust and maneuverability. For
example, the main engines of Apollo-era landers could operate either at 93% thrust, or in the permitted
thrust interval of 11% to 65% of the rated thrust value.5The intervening thrust regions were forbidden to
avoid oxidizer and fuel cavitation, which could lead to the deterioration of propulsion system components.
We have assumed (in (23)) that main engine thrust is the only source of actuation, and thus do not consider
the additional constraints imposed by reaction control systems.
To model permitted thrust regions, we place restrictions on the norm of the thrust vector as
0< umin ≤ kuBk ≤ umax,(24)
where [umin, umax ]R++ denotes the permitted thrust interval. This constraint implies that once ignited,
the engines are not turned off until touchdown.
Next, limited maneuverability of the engine is modeled with a gimbal angle constraint. The gimbal angle,
δ[0,90), is defined as the total angular deflection of the thrust vector from its nominal position. We
express a gimbal angle constraint as the following second order cone constraint,
zT
BuB+kuBk2cos δmax 0⇒ kuBk2¯
zT
BuB(25)
where δmax [0,90) is the maximum allowable gimbal angle and ¯
zB= (1/cos δmax)zB.
The final constraint imposed on the control is a rate constraint that ensures commanded thrust vectors do
not change too rapidly for the engine to follow. This is formulated as
uB
tumax (26)
where ∆umax R3
++ denotes the vector of maximum allowable thrust deviations over a specified time
interval ∆tR++.
2. State Constraints
We now proceed to describe the state constraints enforced in the baseline problem formulation. Powered
descent maneuvers must not use more fuel than is stored on board, a constraint enforced on the mass of the
spacecraft using
mmdry (27)
where mdry R++ is the dry mass of the vehicle. Next, we use a glide slope cone to ensure the vehicle’s
altitude lies above the surface of the planet, while also ensuring sufficient elevation at large distances from
the landing site. If rIdenotes the inertial position of the vehicle, then we define the glide slope angle to be
the angle formed between rIand zIand denote it by γ[0,90]. A glide slope constraint enforces that
the glide slope angle must be less than some prescribed maximum value, γmax. Formally, the constraint is
expressed as
rT
IzI+krIk2cos γmax 0.(28)
We can express this with dual quaternions by using (15) to write
krIk=rT
IrI1/2=(rIq)T(rIq)1/2=
"04×1
rIq#
2
=
2"04×1
1
2rIq#
2
=k2Ed˜
qk(29)
where Ed=diag {04×4, I4}. Another application of (15) yields
rT
IzI= (rIq)T(zIq) = (rIq)T[zI]q=1
2(rIq)T[zI]q+1
2(rIq)T[zI]q,
="q
1
2rIq#T"04×4[zI]T
[zI]04×4#" q
1
2rIq#
=˜
qTMg˜
q.(30)
Using (29) and (30), we arrive at the following proposition.
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Proposition 1. The constraint (28) is expressed in terms of the dual quaternion as
cg(˜
q):=˜
qTMg˜
q+k2Ed˜
qk2cos γmax 0.(31)
Moreover, the function cg:R8
uRis convex over the domain dom cg=R8
uT˜
q|˜
qT˜
q1 + 1
42, where
krIk2is an upper bound on the distance from the landing site.
Proof. See23, 24 for the proof.
The next constraint we consider is a tilt angle constraint. The vehicle’s tilt angle is the angle formed between
zIand zB, denoted by θ[0,90), and constrained to be less than some prescribed value. By expressing
zBin inertial coordinates using (17), a tilt constraint is written in terms of the attitude quaternion as
zT
I(qzBq) + cos θmax 0,(32)
where θmax [0,90) is the maximum allowable tilt angle. Note that zIand zBare being treated as pure
quaternions in (32). To write this constraint as a function of the dual quaternion, we first use (15) to write
zT
I(qzBq)=(zIq)T(qzBqq) = (zIq)T(qzB) = qT(zIqz
B) = qT[zI][z
B]
q,
with which we can rewrite (32) to be
˜
qTMt˜
q+ cos θmax 0, Mt="[zI][zB]
04×4
04×404×4#(33)
by noting that [z
B]
= [zB]
. Since both zIand zBare unit vectors, we find that the eigenvalues of Mt
lie in the set {−1,0,1}, and thus it is a symmetric but indefinite matrix.
Proposition 2. The tilt constraint (32) is equivalently expressed in terms of the dual quaternion as
ct(˜
q):=˜
qT˜
Mt˜
q+ cos θmax ζ0,˜
Mt=Mt+ζEr(34)
where Er=diag {I4,04×4}and ζis a constant. Moreover, the function ctis convex for all ˜
qR8
uwhen
θ(0,90]and ζ1.
Proof. The proof follows by noting that ˜
qTEr˜
q1 = 0, due to the fact that the real part of the dual
quaternion is a unit quaternion. Hence for any ζ > 0 we have ζ˜
qTEr˜
q1= 0. Adding this term to (33)
yields
˜
qTMt˜
q+ cos θmax +ζ˜
qTEr˜
qζ0
˜
qT(Mt+ζEr)˜
q+ cos θmax ζ0
˜
qT˜
Mt˜
q+ cos θmax ζ0.
Now the eigenvalues of ˜
Mtcan be obtained as a function of the parameter ζ. Specifically, we find that
spec n˜
Mto={−1 + ζ, 0,1 + ζ}.
It is clear then that for ζ1, we have ˜
Mt0. When this is the case, the Hessian of ctis positive semidefinite
for all ˜
qR8
uand θ[0,90). Hence ctis convex over this domain.
The final state constraint considered part of our baseline problem is a bound on the angular rate of the
vehicle. We note that due to our assumption that the main engine is the only actuator producing torque,
angular accelerations can be limited by suitable choice of ∆umax in (26). We impose an addition angular
rate constraint of the form
kωBkωmax (35)
where ωmax R++ is the maximum allowable angular velocity about any axis.
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B. State-Triggered Constraints
We now detail the state-triggered constraints and their continuous formulation. State-triggered constraints
(STCs) were introduced in28 and represent constraints that are enforced only when a state-dependent crite-
rion is met. When formulated using the continuous variables of an optimization problem, these constraints
model an if-statement conditioned on the solution variables that they are constraining. In a trajectory gener-
ation problem, optimal solutions are thus obtained with a simultaneous understanding of how the constraint
affects the solution, and of how the solution enables or disables the constraint.
An STC is composed of two functions, the trigger function and the constraint function. Using zRnzto
denote an arbitrary solution vector, we denote these by g:RnzRand c:RnzR, respectively. Vector-
valued trigger and constraint functions are addressed in.38 The constraint function is to be conditionally
enforced based on the value of the trigger function, and thus we formally define the STC to be
g(z)<0c(z)0.(36)
We refer to g(z)<0 as the trigger condition and c(z)0 as the constraint condition. If the trigger function
is non-negative, then the optimization variable zis not subject to the constraint condition. If however, the
trigger function becomes strictly negative (i.e., becomes active), then the state is subject to the constraint
condition. The feasible region in the g(z)-c(z) space is depicted in the bottom two axes of Figure 2.
1. Continuous Formulation
In order to incorporate constraints of the form (36) into a continuous optimization problem, they must be
represented using continuous variables. To do so, we introduce the auxiliary variable σR++ and the
following system of equations
σ0,(37a)
g(z) + σ0,(37b)
σc(z)0.(37c)
The idea is to represent the (binary) logical implication in (36) as the outcome of this system of equations
in continuous variables. The equations in (37) are such that when the trigger condition is satisfied, (37b)
and (37a) imply that σ > 0. It follows then that (37c) holds if and only if constraint condition is met.
Therefore (37) is logically equivalent to (36). We refer to this continuous formulation as continuous state-
triggered constraints (cSTCs).
The formulation in (37) does not, however, admit a unique solution for σ. As illustrated in Figure 2, when
g(z)<0, the auxiliary variable satisfies σ[g(z),). Moreover, when the trigger condition is not met, σ
is free to be any non-negative number, including a non-zero value that enforces the constraint condition. We
have found that this ambiguity can both inadvertently constrain the solution variables and cause numerical
issues during solutions due to the unboundedness of the slack variable.
To alleviate these problems, we introduce an altered set of constraints motivated by the linear complemen-
tarity problem (LCP).39 The new set of constraints form a complementarity condition between the left-hand
sides of (37b) and (37a), and results in
0σg(z) + σ0,(38a)
σc(z)0,(38b)
where the notation in (38a) represents the trio of constraints σ0, g(z) + σ0 and s·g(z) + σ= 0.
For a given z, (38a) defines an LCP in σ, and we refer to this formulation as the projected continuous
state-triggered constraints (Projected cSTCs). This problem has a unique solution that varies continuously
as a function of z,39 and can be solved for analytically as
σ:=min(g(z),0).(39)
Substituting σinto (38) guarantees the satisfaction of (38a), and thus we may remove it from the for-
mulation. The result is that (38b) becomes a single, logically equivalent, constraint to (36) that uses only
continuous variables. We define the constraint (38b) with σto be
h(z):=min(g(z),0) ·c(z)0.(40)
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Feasible Set of cSTC Feasible Set of Projected cSTC
σ0, g(z) + σ0, σc(z)00σg(z) + σ0, σc(z)0
g(z)
c(z)
σ
g(z)
c(z)
c(z)
σ
g(z)
c(z)
σ
g(z)
c(z)
Feasible sets of corresponding STC
Figure 2: Geometric interpretation of cSTCs with an inequality constraint condition. The blue axes on the
left represent the cSTCs in (37), while the green axes on the right represent the Projected cSTCs in (38).
The red regions depict the feasible space of the auxiliary variable σ, with the volume observed in the central
axes portraying the ambiguity noted for (37).
The geometry resulting from this analytical solution for σis illustrated in the upper-rightmost axes of
Figure 2. The removal of the ambiguity from (37) was accomplished by the additional complementarity
constraint by effectively removing the volume of the red region seen in the central blue set of axes that
denote the feasible space of (37).
C. Distance-Triggered Line of Sight Constraints
We now consider the application of STCs to line of sight constraints. Line of sight (LOS) constraints restrict
the permissible attitude and position of the vehicle such that the vector to the landing site is aligned with
a particular boresight direction within some maximum angle. We denote the LOS angle by ξ[0,180],
and similarly denote its maximum allowable angle by ξmax. We highlight that this type of constraint has
been considered in powered descent problems in past work,23,24 however it was applied at all times during
a trajectory. The current methodology is required for powered descent problems with tight LOS bounds
imposed by vision-based sensors for two reasons. First, the LOS angle is a constant offset from the sum of
the glide slope and tilt angles. As a result, a small LOS angle can severely limit the maneuverability of the
vehicle, thus limiting the set of feasible initial conditions. Second, long trajectories may require pointing to
different regions of a planets surface to acquire navigation data, something that would typically require the
solution of two problems that are properly joined. The use of STCs can handle these scenarios in a single
optimization framework.
To formulate a distance-triggered LOS constraint, we propose a trigger function gl:R8
uRof the form
gl(˜
q):=d− k2Ed˜
qk2(41)
where from (29) the second term is equivalent to the normed distance from the landing site. This function
is seen to satisfy the trigger condition for distances strictly greater than dR. We note that d= 0 recovers
the previous LOS constraints considered in the literature.23, 24
To construct the constraint function, consider a unit vector in body coordinates, pBR3, that defines the
boresight of an optical sensor or window. The line of sight angle ξis defined as the angle between pBand
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FB
FI
rB
Body-fixed
ξmax-cone
centered
around pB
ξ
Landing site
Figure 3: A line of sight constraint to be conditionally imposed based on the distance from the landing site.
rB, and the line of sight constraint can be expressed as
rT
BpB+krBk2cos ξmax 0,(42)
and is visualized in Figure 3. To write this constraint with dual quaternions, we note that the same trick
from (30) can be used on the first term in (42) to yield
rT
BpB= (qrB)T(qpB) = ˜
qTMl˜
q, Ml="04×4[pB]
T
[pB]
04×4#
which in conjunction with (29) leads us to the following proposition.
Proposition 3. The line of sight constraint (42) is expressed in terms of the dual quaternion as
cl(˜
q):=˜
qTMl˜
q+k2Ed˜
qk2cos ξmax 0 (43)
Moreover, the function cl:R8
uRdefined above is convex over the domain dom cl=R8
uT˜
q|˜
qT˜
q
1 + 1
42, where krIk2is an upper bound on the distance from the landing site.
Proof. See23, 24 for the proof.
The distance triggered line of sight constraint can thus be expressed using (40) as
hl(˜
q):=min(gl(˜
q),0) ·cl(˜
q)0.(44)
Remark III.1. The formulation presented here is equally applicable to time-triggered constraints for prob-
lems where either the final or ignition time is a variable. In this case, it is a simple matter to construct the
trigger function to model an arbitrary interval of time as a function of the ignition time t0, the final time tf
and the current time t.
Remark III.2. The state-triggered constraints are formulated in a way that avoids using iterative schemes
that update the constrained temporal intervals based on the value of the trigger function from a previous
iteration. The advantage lies primarily in al lowing the optimization process to understand how adjusting the
state will enable or disable the constraint, a feature not enjoyed by these heuristic methods.
D. Problem Statement
We conclude this section with a full statement of the problem to be solved. We state the problem as a
free-final time continuous optimal control problem subject to nonlinear dynamics and both state and control
constraints. We wish to find the burn time, tfR++ and the piecewise continuous thrust commands uB(t)
for t[0, tf] that solve the following optimal control problem.
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min
tf,uB(·)m(tf)
subject to: ˙m=αkuBk2(20)
˙
˜
q=1
2˜
q˜
ω(19)
J˙
˜
ω= ΦuB+˜
gB˜
ωJ+˙
JEr˜
ω(23)
umin ≤ kuBk2umax (24)
kuBk2¯
zT
BuB(25)
uB
tumax (26)
mdry m(27)
cg(˜
q)0 (31)
ct(˜
q)0 (34)
kωBkωmax (35)
hl(˜
q)0 (44)
(Problem 1)
Finally, we note that the baseline problem is Problem 1 with constraint (44) removed.
IV. Solution Method
This section will briefly describe the solution method for Problem 1. In,28 a method was presented to
convert a general free-final time nonlinear continuous-time optimal control problem into a sequence of fixed-
final time convex discrete-time parameter optimization subproblems. The general formulation was specified
for a powered descent guidance problem using Cartesian variables, however the steps remain the same when
using dual quaternions. The algorithm works by iteratively solving these subproblems until a converged
solution is attained. Each iteration can be decomposed into two main steps; a propagation step and a solve
step. The propagation step is responsible for obtaining a convex approximation to Problem 1, while the solve
step solves this subproblem to full optimality using well-studied algorithms.40–43 The solve step’s solution
is then used during the next iteration’s propagation step to obtain an improved approximation of Problem
1. Upon convergence, the algorithm is designed to obtain solutions that adhere exactly to the nonlinear
dynamics of the problem, while approximating constraint feasibility and local optimality. For brevity, we do
not repeat the steps to obtain a convex approximation of Problem 1 here. We refer the reader to28 for these
details, but provide a summary of the key steps and assumptions made for the current work.
In,44 several methods for the propagation step are compared in a Monte Carlo campaign, and the methods
that appear suitable for real-time implementations were discussed. In particular, the trade study of direct
transcription methods in44 compared methods that parameterize the control trajectory to global pseudospec-
tral methods (that parameterize the state and control trajectories). It was found that the former group of
methods result in a more sparse optimization, leading to faster solution times. When the accuracy of the
solutions was taken into account, a piecewise linear approximation of the control signal was seen to perform
the best over the scenarios tested. As a result of this analysis, we proceed in this work with this assumption
in the propagation step.
We discretize the time interval [0, tf]R+into N1 evenly spaced temporal intervals such that 0 = t0<
t1< . . . < tk< . . . < tN1=tf. The control trajectory can then be expressed by a set of Nvectors
uB,k R3such that uB,k =uB(tk). The continuous-time signal is reconstructed using the piecewise linear
interpolation scheme between these discrete values. Each solve step therefore obtains the value of the control
signal at discrete temporal points along the trajectory.
During the propagation step, the nonlinear dynamics in (19), (20) and (23) are approximated by a first
order Taylor series centered about the previous iteration’s solve step solution. We point out that due to
the chosen discretization method,28, 44 the behaviour of the original nonlinear dynamics during inter-node
temporal intervals is captured even in the discrete solution. All constraints in Problem 1 are enforced only
at the discrete temporal points, and non-convex state and control constraints are approximated by a first
order Taylor series expansion centered about the previous iteration’s solve step solution.
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V. Numerical Examples
This section provides example solutions to both the baseline problem and Problem 1 discussed in §III. These
examples are intended to demonstrate the main contribution of this paper, namely the distance-triggered
line of sight constraint. The scenario is modeled as a lunar landing whereby the vehicle must provide a line of
sight to the nominal landing site until it has reached a distance of 200m from the landing site. The problem
parameters are given in Table 1, and we assume that the inertia matrix is computed as an affine function of
mass according to
J(m) =
αxm+βx0 0
0αym+βy0
0 0 αzm+βz
,(45)
where α=αxαyαzT,β=βxβyβzTR3
++ are vehicle specific parameters. For an appropriate choice
of αand β, this model amounts to a linear interpolation between the wet and dry inertia matrices. It is
important to note that the use of this model in conjunction with the solution strategy discussed in §IV means
that all thrust commands are computed with an understanding of how mass depletion will alter the inertia
of the vehicle and affect rotation induced by gimbaling the engine.
Table 1: Problem parameters for the solution of Problem 1.
Parameter Value Units Parameter Value Units
α[1.85 1.85 1.83] m2β[7605 7605 13395] kg m2
mwet 3250 kg mdry 2100 kg
rI(0) [250 150 433] m vI(0) [30 0 15] m/s
rI(tf) [0 0 30] m vI(tf) [0 0 1] m/s
ru0.25 ·zBmωB(0),ωB(tf) [0 0 0] /s
Isp 225.0 s q(tf) [0 0 0 1] -
θmax 80.0N20 -
ωmax 28.6/sξmax 30
γmax 75.0pB[0.906 0 0.423] -
δmax 20.0d200 m
Tmin 6000 N Tmax 22,500 N
All problems are solved using SDPT343 and CVX.45 We solve both the baseline problem (Problem 1 with
constraint (44) removed) and Problem 1 for comparison. The solution is initialized with the straight-line
interpolation method detailed in,28 in which each state is linearly interpolated between its boundary values,
and the thrust is chosen to oppose the force of gravity.
Converged trajectories are shown in Figure 4, while corresponding state and control trajectories are given
in Figure 5a and Figure 5b, respectively. In all figures, the black dots represent the discrete solution values
(both states and controls) from the optimization process. The solid curves are the trajectories obtained by
integrating these control signals through the nonlinear dynamics in (19), (20) and (23).
In Figure 4, the light blue lines represent the boresight vector pBat the discrete temporal points used for the
optimization procedure. In particular, the upper-right South-Zenith projection shows how both the attitude
and position of the vehicle are changed from the baseline scenario to account for the pointing constraint
enforced at this distance from the landing site. Figure 5a illustrates the distance from the landing site versus
the line of sight angle for each trajectory. This figure demonstrates that the line of sight angle is maintained
below its desired bound while the vehicle is far enough from the landing site (i.e., when the trigger condition
is active). Since the baseline problem does not enforce the line of sight constraint, the corresponding angles
are seen to violate the desired limit during the initial portion of the descent. Figure 5b indicates, at times,
large deviations in the thrust commands between the two problem instances as the solution to Problem 1
must accommodate for the additional line of sight constraint.
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-50 0 50 100 150 200 250 300
-50
0
50
100
150
200
0 100 200 300
0
100
200
300
400
500
0 100 200
0
100
200
300
400
500
0
300
200
100
200
200
100 100
300
00
400
500
Figure 4: Converged trajectories for both the baseline problem (brown curve) and Problem 1 (blue curve).
For Problem 1, a line of sight constraint is enforced conditionally on the distance from the landing site. The
light blue line represents the boresight vector pBin both cases.
VI. Conclusions
In this paper, the 6-degree-of-freedom powered descent guidance problem with state-triggered constraints was
formulated. Using dual quaternions to represent the equations of motion, the kinematics, dynamics, and state
and control constraints for a variable mass and variable inertia rigid body with a single engine configuration
were presented. This work introduced the inequality form of state-triggered constraints with an application
to distance-triggered line of sight constraints. The formulation enables the use of continuous optimization
tools to generate and study feasible trajectories that are subject to pointing constraints only during certain
portions of the descent. The solution method leverages recent work in developing iterative methods to solve
nonlinear and nonconvex problems in a way that is amenable to real-time and on-board computation. A
numerical example demonstrates these methods for a representative lunar landing scenario, and highlights
the utility and impact of state-triggered constraints on the resulting state and control trajectories.
Acknowledgements
This research has been supported by NASA grant NNX17AH02A and was partially carried out at the Johnson
Space Center. Government sponsorship acknowledged.
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0 10 20 30 40 50 60 70 80
0
50
100
150
200
250
300
350
400
450
500 Discrete Solution
Baseline
Problem 1
STC Trigger Distance
(a) Altitude versus line of sight angle.
0 5 10 15 20 25
0
1
2
104
0 5 10 15 20 25
0
5
10
15
20
(b) Thrust magnitude and gimbal angle commands.
Figure 5: State-triggered constraint phase plot and thrust commands for the converged trajectories. Con-
straint boundaries are represented by dashed red lines.
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... In previous work, we have developed explicit trajectory generation techniques that are capable of solving general nonconvex optimal control problems [17][18][19][20][21][22][23]. These techniques are part of a broader increase in explicit trajectory optimization methods reported in the literature, such as successive convexification (SCvx) [24][25][26], DESCENDO [27], GuSTO [28,29], ALTRO [30], and more [31][32][33][34][35][36][37]. ...
... The inertia matrix, center of mass and center of pressure of the vehicle are all assumed to be constant. In previous work, the effects of variable inertia on the trajectories generated were investigated [17]. In the scenarios studied in [17], trajectories did not deviate significantly from those obtained with a constant inertia matrix, and so these variations are ignored for trajectory design. ...
... In previous work, the effects of variable inertia on the trajectories generated were investigated [17]. In the scenarios studied in [17], trajectories did not deviate significantly from those obtained with a constant inertia matrix, and so these variations are ignored for trajectory design. ...
Conference Paper
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This paper presents a new implicit trajectory generation technique called quadratic funnel synthesis. In contrast to more standard explicit trajectory generation methods that compute a single trajectory that connects two single-vector boundary conditions, implicit trajectory generation uses a group of functions to define a set of trajectories that connect two sets of boundary conditions. Explicit trajectory generation for nonconvex optimal control problems has inherent limitations that do not permit theoretical guarantees that a feasible trajectory can be computed in real-time. These limitations motivate the study of new methods that are able to provide such guarantees. This paper introduces quadratic funnel synthesis and establishes a provably-convergent offline algorithm that is able to provide a group of functions that permit the computation of feasible trajectories using only numerical integration. A case study using the nonconvex 6-DOF powered descent guidance problem shows that a single quadratic funnel can provide feasible trajectories for any initial condition in a relatively large set in the state-space.
... Improved formulations of the state-triggers are presented in [39] and extended to compound STCs formulation in [21] where logical conjunctions are used to formulate multiple trigger and constraint conditions. Given a logical (binary) trigger function g(z) of an arbitrary solution vector z and the constraint equation c(z) to be satisfied, the formal definition of a continuous STC for the logical constraint: ...
... is given by [39]: ...
... However, there may arise ambiguity in the formulation (21) which does not yield a unique solution for η. This set of equations are therefore modified in [39] with a new set of constraint conditions as given in (22). ...
... Lossless convexification transforms fuel-optimal PDG into a convex optimization problem, which allows for the global optimal descent trajectory to be computed with guaranteed convergence [Acikmese and Ploen, 2007, Blackmore et al., 2010, Açıkmeşe et al., 2013. Solving the problem by convex optimization further facilitates enforcing convex path constraints such as minimum glide slope and maximum off-vertical thrust direction [Acikmese and Ploen, 2007, Blackmore et al., 2010, Açıkmeşe et al., 2013, Reynolds et al., 2019, Lee and Mesbahi, 2017. The guidance for fuel-optimal large diverts (G-FOLD) algorithm , solving fix-time 3D fuel-optimal PDG through lossless convexification, has been demonstrated by test flights. ...
Preprint
Atmospheric powered descent guidance can be solved by successive convexification; however, its onboard application is impeded by the sharp increase in computation caused by nonlinear aerodynamic forces. The problem has to be converted into a sequence of convex subproblems instead of a single convex problem when aerodynamic forces are ignored. Besides, each subproblem is significantly more complicated, which increases computation. A fast real-time interior point method was presented to solve the correlated convex subproblems onboard in the work. The main contributions are as follows: Firstly, an algorithm was proposed to accelerate the solution of linear systems that cost most of the computation in each iterative step by exploiting the specific problem structure. Secondly, a warm-starting scheme was introduced to refine the initial value of a subproblem with a rough approximate solution of the former subproblem, which lessened the iterative steps required for each subproblem. The method proposed reduced the run time by a factor of 9 compared with the fastest publicly available solver tested in Monte Carlo simulations to evaluate the efficiency of solvers. Runtimes on the order of 0.6 s are achieved on a radiation-hardened flight processor, which demonstrated the potential of the real-time onboard application.
... SPLICE is developing a descent and landing guidance algorithm capable of incorporating 6-DOF constraints into the guidance solution [16][17][18]. The algorithm development began as a research effort through a Co-operative Agreement with the University of Washington. ...
Conference Paper
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The Safe and Precise Landing—Integrated Capability Evolution (SPLICE) Project’s suite of technologies provides a spacecraft with Precision Landing and Hazard Avoidance (PL&HA) capabilities for conducting precise and safe landing. SPLICE has been a focal PL&HA project since 2017 within the Space Technology Mission Directorate (STMD) Game Changing Development (GCD) Program and has funding planned through 2024. STMD/GCD has pursued SPLICE as a technology push to enable PL&HA capabilities for human and robotic lander missions to the Moon, with extensibility to Mars, icy moons, ocean worlds, and other solid-surface solar system destinations. PL&HA technologies are prioritized within NASA Technology Roadmaps, the Artemis roadmap, and the Entry, Descent and Landing (EDL) Systems Capability Leadership Team (SCLT) technology development plan. SPLICE has multiple active partnerships, including funded efforts to demonstrate PL&HA technologies on terrestrial suborbital rocket flights and planned infusion to lunar spaceflight missions. This paper describes the SPLICE technologies in development, maturation progress, and recent suborbital rocket flight testing.
Article
In this paper, a guidance algorithm is presented based on successive convexification for the generation of fuel-optimal 6-DOF spacecraft close proximity trajectory subject to multiple path constraints on sensor field-of-view, collision avoidance, and plume impingement. The whole maneuver process is divided into fly-around and docking phases, where the collision avoidance constraints are only considered in the first stage, while the plume impingement restrictions are implemented when the chaser spacecraft lies within the docking corridor. These discrete decisions are embedded into continuous optimization framework by using the scheme of compound state-triggered constraints. Besides, in order to make full use of the control ability of actuator, the translational and rotational dynamics are reformulated with the thruster configuration. The original optimal guidance problem is converted into a series of convex subproblems by the linearization and discretization of nonlinear system dynamics and constraint functions around the reference trajectory, and iteratively solved by convex programming. Particularly, the algorithm is initialized with an imprecise trajectory obtained from pseudospectral method, and the reference states in the subsequent iterations are obtained by the propagation of the system dynamics in each discrete time interval rather than directly inheriting the latest solution. The convergence analysis is presented to prove that the proposed guidance algorithm converges to the stationary point of original problem under mild conditions. Finally, the effectiveness of the algorithm is demonstrated by numerical simulations.
Conference Paper
View Video Presentation: https://doi.org/10.2514/6.2022-0950.vid This paper presents a convex optimization based trajectory planning algorithm that solves a generalized fixed-final-time, 2-degree-of-freedom hypersonic entry vehicle guidance problem. This planar Cartesian formulation is a prototype problem that introduces an abstracted control input of aerodynamic accelerations in order to simplify the highly nonlinear dynamics, which are instead dealt with as control constraints. These abstracted controls are made to be dynamically feasible via state-dependent constraints, which allow the optimizer to sweep a convex subspace of aerodynamic accelerations corresponding to a feasible range of bank angles and angles-of-attack. First, the full nonconvex optimal control problem for hypersonic entry is formulated. Then a sequence of convex optimization sub-problems are generated whose solutions will converge to a solution of the trajectory planning problem. This convex optimization based method is referred to as successive convexification. After an optimal solution to this problem is determined, a set of feasible corresponding low-level control inputs such as bank angle and angle-of-attack are extracted in post processing. This Cartesian formulation aims to improve upon the historical numerical difficulties in prior literature associated with solving the 6-degree-of-freedom hypersonic entry formulation in polar form as a standalone optimization problem by instead applying convex optimization for abstracted control inputs and deferring the computation of low-level control to a secondary step. Moving the primary nonlinearities of the aerodynamics into the control constraints opens the door to exploring future extensions for lossless convexification techniques for the 3-degree-of-freedom formulation, where the feasible control set for a given velocity becomes a shell rather than a convex region as in the planar formulation.
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Space mission design places a premium on cost and operational efficiency. The search for new science and life beyond Earth calls for spacecraft that can deliver scientific payloads to geologically rich yet hazardous landing sites. At the same time, the last four decades of optimization research have put a suite of powerful optimization tools at the fingertips of the controls engineer. As we enter the new decade, optimization theory, algorithms, and software tooling have reached a critical mass to start seeing serious application in space vehicle guidance and control systems. This survey paper provides a detailed overview of recent advances, successes, and promising directions for optimization-based space vehicle control. The considered applications include planetary landing, rendezvous and proximity operations, small body landing, constrained attitude reorientation, endo-atmospheric flight including ascent and reentry, and orbit transfer and injection. The primary focus is on the last ten years of progress, which have seen a veritable rise in the number of applications using three core technologies: lossless convexification, sequential convex programming, and model predictive control. The reader will come away with a well-rounded understanding of the state-of-the-art in each space vehicle control application, and will be well positioned to tackle important current open problems using convex optimization as a core technology.
Preprint
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Space mission design places a premium on cost and operational efficiency. The search for new science and life beyond Earth calls for spacecraft that can deliver scientific payloads to geologically rich yet hazardous landing sites. At the same time, the last four decades of optimization research have put a suite of powerful optimization tools at the fingertips of the controls engineer. As we enter the new decade, optimization theory, algorithms, and software tooling have reached a critical mass to start seeing serious application in space vehicle guidance and control systems. This survey paper provides a detailed overview of recent advances, successes, and promising directions for optimization-based space vehicle control. The considered applications include planetary landing, rendezvous and proximity operations, small body landing, constrained reorientation, endo-atmospheric flight including ascent and re-entry, and orbit transfer and injection. The primary focus is on the last ten years of progress, which have seen a veritable rise in the number of applications using three core technologies: lossless convexification, sequential convex programming, and model predictive control. The reader will come away with a well-rounded understanding of the state-of-the-art in each space vehicle control application, and will be well positioned to tackle important current open problems using convex optimization as a core technology.
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This paper presents the SCvx algorithm, a successive convexification algorithm designed to solve non-convex optimal control problems with global convergence and superlinear convergence-rate guarantees. The proposed algorithm handles nonlinear dynamics and non-convex state and control constraints by linearizing them about the solution of the previous iterate, and solving the resulting convex subproblem to obtain a solution for the current iterate. Additionally, the algorithm incorporates several safe-guarding techniques into each convex subproblem, employing virtual controls and virtual buffer zones to avoid artificial infeasibility, and a trust region to avoid artificial unboundedness. The procedure is repeated in succession, thus turning a difficult non-convex optimal control problem into a sequence of numerically tractable convex subproblems. Using fast and reliable Interior Point Method (IPM) solvers, the convex subproblems can be computed quickly, making the SCvx algorithm well suited for real-time applications. Analysis is presented to show that the algorithm converges both globally and superlinearly, guaranteeing the local optimality of the original problem. The superlinear convergence is obtained by exploiting the structure of optimal control problems, showcasing the superior convergence rate that can be obtained by leveraging specific problem properties when compared to generic nonlinear programming methods. Numerical simulations are performed for an illustrative non-convex quad-rotor motion planning example problem, and corresponding results obtained using Sequential Quadratic Programming (SQP) solver are provided for comparison. Our results show that the convergence rate of the SCvx algorithm is indeed superlinear, and surpasses that of the SQP-based method by converging in less than half the number of iterations.
Conference Paper
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This paper presents a successive convexification ($ \texttt{SCvx} $) algorithm to solve a class of non-convex optimal control problems with certain types of state constraints. Sources of non-convexity may include nonlinear dynamics and non-convex state/control constraints. To tackle the challenge posed by non-convexity, first we utilize exact penalty function to handle the nonlinear dynamics. Then the proposed algorithm successively convexifies the problem via a $ \textit{project-and-linearize} $ procedure. Thus a finite dimensional convex programming subproblem is solved at each succession, which can be done efficiently with fast Interior Point Method (IPM) solvers. Global convergence to a local optimum is demonstrated with certain convexity assumptions, which are satisfied in a broad range of optimal control problems. As an example, the proposed algorithm is particularly suitable for solving trajectory planning problems with collision avoidance constraints. Through numerical simulations, we demonstrate that the algorithm converges reliably after only a few successions. Thus with powerful IPM based custom solvers, the algorithm can be implemented for real-time autonomous control applications.
Conference Paper
In this paper, we employ successive convexification to solve the minimum-time 6-DoF rocket powered landing problem. The contribution of this paper is the development and demonstration of a free-final-time problem formulation that can be solved iteratively using a successive convexification framework. This paper is an extension of our previous work on the 3-DoF free-final-time and the 6-DoF fixed-final-time minimum-fuel problems. Herein, the vehicle is modeled as a 6-DoF rigid-body controlled by a single gimbaled rocket engine. The trajectory is subject to a variety of convex and non-convex state- and control-constraints, and aerodynamic effects are assumed negligible. The objective of the problem is to determine the optimal thrust commands that will minimize the time-of-flight while satisfying the aforementioned constraints. Solving this problem quickly and reliably is challenging because (a) it is nonlinear and non-convex, (b) the validity of the solution is heavily dependent on the accuracy of the discretization scheme, and (c) it can be difficult to select a suitable reference trajectory to initialize an iterative solution process. To deal with these issues, our algorithm (a) uses successive convexification to eliminate non-convexities, (b) computes the discrete linear-time-variant system matrices to ensure that the converged solution perfectly satisfies the original nonlinear dynamics, and (c) can be initialized with a simple, dynamically inconsistent reference trajectory. Using the proposed convex formulation and successive convexification framework, we are able to convert the original non-convex problem into a sequence of convex second-order cone programming (SOCP) sub-problems. Through the use of Interior Point Method (IPM) solvers, this sequence can be solved quickly and reliably, thus enabling higher fidelity real-time guidance for rocket powered landings on Mars.
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The problem of powered descent guidance and control for autonomous precision landing for next-generation planetary missions is addressed. The precision landing algorithm aims to trace a fuel-optimal trajectory while keeping geometrical constraints such as the line of sight to the target site. The design of an autonomous control algorithm managing such mission scenarios is challenging due to fact that critical geometrical constraints are coupled with the translational and rotational motions of the lander spacecraft, leading to a complex motion-planning problem. This problem is approached within the model predictive control framework by representing the dynamics of the rigid body in a uniform gravity field via a piecewise affine system taking advantage of the unit dual-quaternion parameterization. Such a parameterization in turn enables a six-degree-of-freedom motion planning in a unified framework while also admitting a quadratic cost on the required control commands to minimize propellant consumption. A novel feature of the proposed approach is the development of convex representable subsets in the configuration space in terms of unit dual quaternions. The stability and reachability issues of the corresponding piecewise affine model predictive control approach are then discussed. Numerical simulations are presented to demonstrate the effectiveness of the proposed methodology for autonomous precision landing.