Conference Paper

Funnel Synthesis for the 6-DOF Powered Descent Guidance Problem

Abstract and Figures

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.
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Funnel Synthesis for the 6-DOF
Powered Descent Guidance Problem
Taylor P. Reynolds, Danylo Malyuta, Mehran Mesbahi, Behçet Açıkmeşe
Dept. of Aeronautics & Astronautics, University of Washington, Seattle, WA 98195, USA
John M. Carson III
NASA Johnson Space Center, Houston, TX 77058, USA
This paper presents a new implicit trajectory generation technique called quadratic
funnel synthesis. In contrast to more standard explicit trajectory generation meth-
ods that compute a single trajectory that connects two single-vector boundary con-
ditions, implicit trajectory generation uses a group of functions to define a set of
trajectories that connect two sets of boundary conditions. Explicit tra jectory gener-
ation 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 integra-
tion. 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.
I. Introduction
Powered descent guidance refers to the problem of transferring a vehicle from an estimated initial state to a
target state using rocket-powered engines and/or reaction control systems. A guidance trajectory, in this context,
is understood as a trajectory that will autonomously achieve this objective using the available actuation methods.
The inherent difficulty of trajectory generation for powered descent is compounded by the addition of precision
landing requirements, atmospheric interactions and constraints that arise due to navigation, communication,
and/or thermal system requirements. While each celestial body targeted for landing presents unique design
considerations, all powered descent trajectories are subject to some set of such constraints. The payoffs for
overcoming these challenges are numerous: increased scientific return from a given mission, lower cost-per-kilogram
to launch spacecraft from Earth, and the possibility of establishing permanent human outposts elsewhere in our
solar system [1].
The need for autonomy is clear, as it is not feasible for a rocket booster returning to the Earth to be piloted
and landed by humans, and (robotic) missions to other celestial bodies suffer time delays that prohibit direct
control by humans on Earth. Despite the historic success of piloted lunar landings during the Apollo program,
autonomous powered descent and precision landing have been identified as key enabling technologies for future
robotic and human missions [
2
4
]. Apollo-derived methods still form the core of nearly all guidance systems that
have been flown on flagship missions [
5
7
]. Apollo guidance is a 3-degree-of-freedom (3-DOF) guidance law that
considers the translational motion of the vehicle only, and assumes that the attitude is controlled by a faster
inner-loop to achieve the required thrust vectors. A significant body of literature exists that studies the optimality
and real-time computation of 3-DOF powered descent trajectories [8–15].
Modern landing systems that use vision-based navigation are, however, inherently 6-DOF [
2
,
16
]. The field of
view requirements for vision-based sensors couple the rotational and translational motion. For future missions
where navigation and guidance may be tightly coupled, the feasibility of using 3-DOF guidance solutions is called
into question, and may require limiting the trajectory design space and/or considerable backstage hand-tuning and
edge-case handling efforts to certify that the computed trajectories will be feasible in a 6-DOF sense. To support
hazard avoidance and real-time intelligent landing site re-targeting, there is therefore a desire for algorithms that
Ph.D. Candidate, AIAA Student Member {tpr6,danylo}@uw.edu
Professor, AIAA Associate Fellow, {mesbahi,behcet}@uw.edu
SPLICE Principal Investigator, AIAA Associate Fellow, john.m.carson@nasa.gov
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11–15 & 19–21 January 2021, VIRTUAL EVENT 10.2514/6.2021-0504
Copyright © 2021 by Taylor Patrick Reynolds. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA SciTech Forum
are capable of solving constrained 6-DOF landing problems. These algorithms must have predictable convergence
behavior and be real-time capable on computationally constrained hardware. Moreover, the algorithms must
be capable of handling a large suite of input and state constraints, and in particular may need to account for
pointing requirements imposed by new sensing and navigation architectures.
In previous work, we have developed explicit trajectory generation techniques that are capable of solving
general nonconvex optimal control problems [
17
23
]. These techniques are part of a broader increase in explicit
trajectory optimization methods reported in the literature, such as successive convexification (SCvx) [
24
26
],
DESCENDO [
27
], GuSTO [
28
,
29
], ALTRO [
30
], and more [
31
37
]. Explicit trajectory optimization methods,
when implemented numerically, return a complete trajectory from one state to another state by solving a general
nonconvex optimal control problem. A subset of these works have specifically studied the 6-DOF landing problem
and developed real-time capable solution strategies for problem scenarios with numerous state and control
constraints [
18
,
23
,
38
]. A recent flight test on Blue Origin’s NS-13 mission demonstrated the ability to integrate
and run this type of algorithm—specifically [18]—on representative space flight hardware.
Alas, explicit trajectory generation methods suffer from two fundamental drawbacks. First, if a trajectory is
computed by an explicit method and any problem data is subsequently changed (e.g., an initial condition), then
this trajectory is no longer strictly feasible with respect to either the dynamics or the constraints. An attempt to
follow the originally computed trajectory by executing the control commands would place an undue burden on
the downstream tracking controllers that would be responsible for cleaning up the dispersions. Explicit trajectory
generation is inherently specific to the given problem data, and any changes necessitate a full re-solve of the
optimal control problem. Second, there is a lack of convergence guarantees for the solution of nonconvex optimal
control problems. No known algorithm can be formally guaranteed to solve a nonconvex optimal control problem
from an arbitrary initial guess. There is therefore no theoretical reason that we should expect to always be able
to solve a nonconvex tra jectory optimization problem in real-time – though certainly there is ample empirical
data to support expectations of reliable convergence. It is natural, in response, to study alternative methods for
which the ability to provide a feasible tra jectory in real-time can be theoretically established.
To this end, we have investigated the use of implicit trajectory optimization methods. Implicit trajectory
optimization provides one (or more) functions that implicitly define a set of trajectories in both state and control
space. Instead of a single trajectory that connects two boundary conditions, we obtain a group of functions that
connect two sets of initial and terminal boundary conditions. Funnel synthesis is the class of implicit trajectory
optimization methods that we explore in this work. The benefit of funnel synthesis is that we are able to derive a
theoretical guarantee that a feasible trajectory can be computed for a constrained nonconvex optimal control
problem under certain conditions. Moreover, the design procedure transfers the majority of the computational
load to offline processes, and the primary computation required onboard the vehicle is numerical integration.
Because there are well-established methods for numerical integration in space flight software, there is reason to
believe that funnel synthesis can be a real-time capable trajectory generation technique.
A. Funnel Synthesis
Consider the following guiding question: if problem data changes after an explicit trajectory optimization
method has been executed, can the resulting solution be used to obtain a new trajectory without re-solving the
original problem, and can we guarantee that any new tra jectory will be both dynamically feasible and feasible
with respect to all state and control constraints? Funnel synthesis provides one avenue towards an affirmative
answer to this question.
This guiding question is intimately related to neighboring optimal control, a concept that was initially developed
by the trajectory optimization community [
39
,
40
]. The main idea is to use a first-order model of a nonlinear system
and a second-order model of the cost function around some trajectory, and study the necessary conditions for
optimality given by the maximum principle. A state feedback law that is obtained by a solution of these necessary
conditions can provide a near-optimal controller and one that is rather robust to parameter variations [
41
43
].
The key difference between neighboring optimal control and the funnel synthesis methods presented in this paper
are the use of a Lyapunov function (as opposed to the maximum principle-based techniques) and by extension the
degree to which state and control constraints can be handled. In fact, we can do more than separate the topics
based on their technical approaches; neighboring optimal control is designed to seek out nearby optimal solutions,
whereas funnel synthesis is designed to seek out nearby feasible trajectories.
Implicit trajectory optimization and funnel synthesis also have roots in the field of robotics. Burridge et al. appear
to be the first to have made the explicit connection between Lyapunov functions and a “funnel” [
44
]. The authors
discuss the sequential composition of atomic funnels that are each designed for a single control objective; with
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the net result of funnel composition being that a robot can achieve a broader control objective (called preimage
backchaining [
45
]). These ideas have been advanced significantly in the recent decade [
46
49
]. These works,
for the most part, use Sums-of-Squares (SOS) to compute polynomial Lyapunov functions for systems whose
dynamics can be presented as polynominals in the state and control vectors. SOS optimization connects the
search for a polynomial Lyapunov function with semidefinite programming [
50
], providing a practical connection
between funnel synthesis and convex optimization.
Prior to reviewing additional relevant literature, it will be useful to provide a formal definition of a funnel.
Definition 1
(Funnel)
.
A funnel, denoted by
F
(
t
), is a time-varying set in state and control space that is both
invariant and contained inside a feasible region.
The term funnel synthesis refers to the algorithmic procedure designed to compute a funnel. The invariance
property of a funnel means that if a particular initial condition is inside the entry of the funnel (at some initial
time
t0
), then the entire subsequent trajectory remains inside the funnel as well. Stated mathematically, if
x
(
t0
)
, u
(
t0
)
∈ F
(
t0
)then
x
(
t
)
, u
(
t
)
∈ F
(
t
)for all
tt0
. Relative to existing definitions of a funnel, e.g., [
47
],
Definition 1 adds the second clause that requires the funnel to lie inside a feasible region.
Based on the guiding question posed earlier, it should be no surprise that we seek the largest possible funnel.
This allows us to implicitly define a large family of trajectories by using the functions that define the funnel –
thereby providing the ability to guarantee the availability of a feasible trajectory over a larger region of parameter
variations.
A connection between funnel synthesis and convex optimization is made in this work by defining a sub-class of
funnels, called quadratic funnels. Instead of using SOS and polynomial dynamics, we use a first-order approximation
of the dynamics and a quadratic Lyapunov function to derive a differential matrix inequality (DMI). The resulting
DMI is similar to the differential Riccati equations that are pervasive in the robust control literature. In fact, the
algorithm that we propose is reminiscent of the Kleinman iteration [
51
], or of the D-K iteration [
52
], but with
additional requirements to ensure constraint satisfaction and to maximize the funnel’s size.
Indeed, there are many interesting connections between quadratic funnel synthesis and the
H
and robust MPC
literature [
53
57
]. Philosophically, a portion of what differentiates this work from others is the intent to maximize
a controlled-invariant set for a nonlinear function, as opposed to minimizing an invariant set in the presence of
disturbances or uncertainty. We are aiming to approximate the largest controlled-invariant set around a trajectory
of a nonlinear system while simultaneously choosing a control policy that maximizes this same invariant set.
Quadratic funnel synthesis, as it is defined in this work, can be viewed as a new combination of old techniques in
order to compute feasible solutions for nonconvex optimal control problems in real-time.
This paper is organized as follows. Section II outlines a representation of the nonlinear equations of motion by
using so-called structured nonlinearity, before introducing the definition of a quadratic funnel and culminating in
the framing of the quadratic funnel synthesis problem. Section III then derives an iterative algorithm to solve the
quadratic funnel synthesis problem that is shown to converge in a finite number of iterations. Next, Section IV
provides a case study of a 6-DOF powered descent guidance scenario. Lastly, Section V offers some concluding
remarks.
II. Quadratic Funnel Synthesis
A. Structured Nonlinear System Model
Consider the nonlinear dynamics
˙x(t) = fx(t), u(t), t [t0, tf],(1)
where
x
(
t
)
Rnx
and
u
(
t
)
Rnu
represent the state and control vectors. We assume that
f
is at least once
differentiable, and that
tf
is fixed. Let
{¯x
(
t
)
,¯u
(
t
)
}tf
t=t0
be a nominal trajectory that satisfies the dynamics
(1)
for
some initial condition, and define
η(t):=x(t)¯x(t)and ξ(t):=u(t)¯u(t).(2)
Because
f
is differentiable, we can write
(1)
in terms of the difference variables
(2)
by using a first-order Taylor
series expansion around the nominal trajectory:
˙η(t) = A(t)η(t) + B(t)ξ(t) + gx(t), u(t),(3)
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where
A
(
t
)and
B
(
t
)are the partial derivatives of
f
evaluated along the nominal trajectory. The dynamical
system (3) can then be equivalently expressed using structured nonlinearities according to
˙η(t) = A(t)η(t) + B(t)ξ(t) + PNp
i=1Eipi(t)
qi(t) = Ciη(t) + Diξ(t), i = 1, . . . , Np
pi(t) = φiqi(t), i = 1, . . . , Np
(4)
where the input/output pairs (
qi, pi
)
Rnq,i ×Rnp,i
are related through the nonlinear functions
φi
. The details of
this transformation are laid out in [
58
], and the constant matrices
CiRnq,i×nx
,
DiRnq,i×nu
and
EiRnx×np,i
serve as nonlinear input and output channel selectors. For example, a nonlinear gravity model would represent
a position-to-acceleration nonlinear channel, and so
Ci
would be constructed by placing an identity matrix in
the columns corresponding to the position vector (all other entries zero),
Di
would be zero, and
Ei
would be
constructed by placing an identity matrix in the rows corresponding to the linear acceleration vector in the
state derivative (all other entries zero). We can rewrite the system
(4)
in a more compact form by making the
definitions
p=
p1
.
.
.
pNp
, q =
q1
.
.
.
qNp
, C =
C1
.
.
.
CNp
, D =
D1
.
.
.
DNp
, E =hE1· · · ENpi.(5)
Here,
pRnp
and
qRnq
, where
np
=
PNp
i=1 np,i
and
nq
=
PNp
i=1 nq,i
, and hence
CRnq×nx
,
DRnq×nu
and
ERnx×np. By stacking each of the φito construct the function φ:RnqRnp, we can write (4) as
˙η(t) = A(t)η(t) + B(t)ξ(t) + Ep(t)
q(t) = (t) + (t)
p(t) = φq(t)
(6)
B. Quadratic Funnels
We now introduce the definition of a quadratic funnel – the specific class of funnels that is obtained by using
quadratic stability. Let
u
(
t
) =
K
(
t
)
x
(
t
)for some matrix-valued function of time
K
(
t
)
Rnu×nx
so that the
closed-loop system becomes
˙η(t) = Acl(t)η(t) + Ep(t)
q(t) = Ccl(t)η(t)
p(t) = φq(t)
(7)
where
Acl
(
t
)
:
=
A
(
t
) +
B
(
t
)
K
(
t
)and
Ccl
(
t
)
:
=
C
+
DK
(
t
). By Definition 1, a funnel
F
(
t
)must be both invariant
and feasible, and so we require that any computed
K
(
t
)renders the closed-loop system
(7)
stable. The funnel
synthesis techniques that we develop are based on the notion of quadratic stability as defined in [
59
] and [
57
,
60
,
61
],
the latter of which offer necessary and sufficient conditions for stability based on quadratic Lyapunov functions.
To this end, consider the scalar-valued function V:R×RnxRdefined by
Vt, η(t)=η(t)>Q(t)1η(t),(8)
where
Q
(
t
)
Snx
++
is a matrix-valued function of time whose range space lies in the set of positive definite matrices.
As a result, we have Vt, η (t)>0for all t[t0, tf]whenever η(t)6= 0.
Having introduced each of the time-varying terms, we henceforth omit the argument of time,
t
, whenever possible.
The 1-level set of
V
(
t, η
)is the set of states that satisfy the quadratic inequality
η>Q1η
1, which is also the
equation of a non-degenerate
nx
-dimensional ellipsoid. We denote the ellipsoid defined by the positive definite
matrix Qand centered at the origin as
EQ=ηRnx|η>Q1η1=nQ1/2w| kwk21o.(9)
If
x∈ EQ
, then
Cx ∈ EC QC>
, a fact that can be proven easily via Schur complements when
C
is full row-rank
.
The assumption that
u
=
Kx
used to form the closed-loop system
(7)
thus results in the following implication:
x∈ EQu∈ EKQK >.(10)
When Cis not full row-rank, the ellipsoid ECQC>is a degenerate ellipsoid.
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State-Space: EQnominal
state, ¯x(t)
new
state,
x
(
t
)
obstacle
Control-Space: EKQK>
nominal
control, ¯u(t)
new
control,
u
(
t
)
constraint
boundary
Fig. 1 A depiction of a quadratic funnel in both state- and control-spaces. At each instant of
time, the quadratic funnel is an ellipsoid (green) centered around the nominal trajectory (blue).
Any new trajectory (red) that starts in the funnel will remain in the funnel, and the entire funnel
is contained in the feasible region.
Suppose that
X Rnx
and
U Rnu
are the (possibly nonconvex) sets of feasible state and control vectors.
Using these feasible sets, we formally define a quadratic funnel in Definition 2.
Definition 2
(Quadratic Funnel)
.
A quadratic funnel,
F
, is a set in state and control space that is parameterized
by a time-varying positive definite matrix
QSnx
++
and a time-varying matrix
KRnu×nx
. Specifically, we have
F=EQ× EKQK >and EQ X ,EKQK >⊆ U.(11)
We call Kthe correction law associated with the quadratic funnel.
Figure 1 provides an illustration of the quadratic funnel concept. Definition 1 (funnels) and Definition 2
(quadratic funnels) are intended to mirror the definitions of stability and quadratic stability. As mentioned before,
part of what differentiates this work from others is the intent to maximize the (quadratic) funnel, as opposed to
minimize an invariant set in the presence of disturbances or uncertainty.
1. Invariance of a Quadratic Funnel
For the closed-loop system in (7), the condition that must be met to achieve quadratic stability is
˙
V(t, η)≤ −αV (t, η),t[t0, tf],q∈ ECcl QCcl >, p =φ(q).(12)
The decay rate
α >
0ensures that the value of
kηk2
decreases at some non-zero minimum rate, implying that
a new trajectory will eventually approach and follow the nominal trajectory [
59
]. When the condition
(12)
is
met, the level sets of the Lyapunov function
V
(
t, η
)are invariant sets. The invariance of a quadratic funnel thus
follows from the invariance of the 1-level set of V(t, η)under a quadratically-stabilizing correction law K.
The quadratic funnel’s invariance property means that
η∈ EQ
for all times
t
[
t0, tf
]. This implies that the
vector q, which is the input to the nonlinear terms in system (7), satisfies the set inclusion
q∈ ECclQCcl >.(13)
To use the condition
(12)
to synthesize a quadratic funnel, we need to express the condition “
q∈ ECclQCcl >, p
=
φ
(
q
) in such a way that it is consistent with the quadratic form of the function
V
(
t, η
). To this end, we use
multiplier matrices and the
S
-procedure [
57
,
59
,
60
]. Because the funnel synthesis procedure results in a bounded
set of possible inputs to the nonlinear terms, due to
(13)
, we are able to refine the definition of multiplier matrices
to a new “local” version.
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Definition 3
(Local Multiplier Matrix)
.
Let
φ
:
RnqRnp
be a nonlinear map such that
q7→ φ
(
q
). A symmetric
matrix MS(nq+np)is a local multiplier matrix for φover the set if and only if
µ(q):="q
φ(q)#>
M"q
φ(q)#0,for all q.(14)
Denote the set of local multiplier matrices for φover the set by
Mφ,=nMS(nq+np)|µ(q)0for all qo.(15)
Multiplier matrices as defined in [
60
] account for the range space of
φ
, but require an inequality like
(14)
to
hold for all
qRnq
. Therefore a multiplier matrix is a local multiplier matrix, but the converse is not necessarily
true. The set
Mφ,
is a convex cone. Importantly, however, matrices
M∈ Mφ,
are not necessarily positive
semidefinite (the set of which is also a convex cone). This is because we only require that
(14)
holds over the
subset of
R(nq+np)
defined by the set and the range space of the function
φ
. A positive semidefinite matrix
M
would of course immediately satisfy the definition
(14)
, but would enforce the inequality over all of
R(nq+np)
and
provide no characterization of the nonlinearity. As a result, using a positive semidefinite matrix would be overly
conservative (and as we will see, would lead to an infeasible funnel synthesis problem).
Using local multiplier matrices, we can rewrite the quadratic stability condition (12) as
˙
V(t, x)≤ −αV (t, x),t[t0, tf],"q
p#>
M"q
p#0,(16)
for some M∈ Mφ,ECclQCcl >. Expanding this condition in terms of the closed-loop system (7) leads to
"η
p#>"Acl
>Q1+Q1Acl Q1˙
QQ1+αQ1Q1E
E>Q10#"η
p#0,
for all "η
p#>"Ccl
>0
0I#M"Ccl 0
0I#"η
p#0(17)
By using the S-procedure, the condition (17) holds if and only if there exists a scalar λ0such that
"Acl
>Q1+Q1Acl Q1˙
QQ1+αQ1+λCcl
>M11Ccl Q1E+λCcl
>M12
E>Q1+λM>
12Ccl λM22 #0,(18)
where we have used the block-decomposition
M="M11 M12
M>
12 M22#∈ Mφ,ECcl QCcl >(19)
Remark 1.
In the case of multiple nonlinearities,
Np>
1, the block decomposition shown in
(19)
has a banded
structure. Each matrix
M11, M12
and
M22
is itself a block-diagonal matrix, with the size of each block determined
by the dimension of the respective nonlinear channel, denoted by
nq,i
and
np,i
in §II.A. This follows by constructing
an independent characterization of each nonlinear channel by using a suitable multiplier matrix and the definitions
in (5).
For synthesis using convex optimization, we must pose the matrix inequality
(18)
as an LMI in the variable
Q
,
not its inverse [59]. Pre- and post-multiplying (18) by diag {Q, I}yields the equivalent condition
"QAcl
>+AclQ˙
Q+αQ +λQCcl
>M11Ccl Q E +λQCcl
>M12
E>+λM>
12Ccl Q λM22 #0, Q 0, λ 0,(20)
and M∈ Mφ,ECclQCcl >. Immediately we can see that to satisfy (20) requires M22 0.
The matrix inequality
(20)
is a nonlinear DMI in four variables: the matrices
Q, K, M
and the scalar
λ
. For
later use, we reformulate
(20)
into another nonlinear DMI by expanding the closed-loop system matrices and
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using the change of variables
Y
=
KQ
. In particular,
(20)
can be rewritten equivalently in terms of the variables
Q, Y, M and λas
"F˙
Q+λG>M11G E +λG>M12
E>+λM>
12G λM22 #0, Q 0, λ 0, M ∈ Mφ,EGQ1G>,(21a)
where,
F=QA>+AQ +BY +Y>B>+αQ, (21b)
G=CQ +DY. (21c)
Satisfaction of the nonlinear DMI
(20)
or
(21)
is the condition that provides the invariance property of a quadratic
funnel.
2. Feasibility of a Quadratic Funnel
We now discuss the conditions that must be met in order for a quadratic funnel to be contained in a given
feasible region. Suppose again that
X Rnx
and
U Rnu
are the (possibly nonconvex) sets of feasible state
and control vectors. We assume that the nominal trajectory satisfies
¯xint X
and
¯uint U
, or, in words, the
nominal trajectory lies in the strict interior of the two respective sets. This assumption is necessary to avoid
degenerate solutions during funnel synthesis. Note that this assumption does not preclude nominal trajectories
that activate a constraint; it merely suggests that the constraint sets used for funnel synthesis are slightly more
relaxed than those imposed during nominal trajectory synthesis, if necessary.
We also assume that
XfRnx
is a set that determines the maximum funnel size at the final time
t
=
tf
(i.e.,
the acceptable set of state deviations at the final time).
Because quadratic funnels are constructed using ellipsoids, it is natural to compute the largest ellipsoids that
are able to fit in the feasible space defined by
X,U
and
Xf
. This is equivalent to a “maximum quadratic funnel”
denoted by Fmax that satisfies
Fmax =EQmax × ERmax and EQmax X ,ERmax ⊆ U
where
Qmax Snx
++
and
Rmax Snu
++
. Inclusion in the terminal set
Xf
can be guaranteed by ensuring that
EQmax ⊆ Xfwhen t=tf.
To compute the matrix Qmax at any time t<tf, we assume that the set Xcan be expressed as
X={x|hi(x)0, i = 1, . . . , mx,kxk2xmax},(22)
where each constraint function
hi
:
RnxR
is a scalar-valued function of the state. Note that equality constraints
must be excluded from the definition
(22)
because of the assumption that the nominal trajectory lies in the strict
interior of the set X. The best quasi-polytopic approximation of Xin the vicinity of the nominal trajectory is
P¯x=x|a>
ixbi, i = 1, . . . , mx∩ {x| kxk2xmax }(23)
where the data
{ai, bi}mx
i=1
are first-order approximations to the functions
hi
computed using either direct
linearization along the nominal trajectory, or by using a project-and-linearize technique if any
hi
is a concave
function [25]. Note that if the state constraints hiare all affine then X=P¯x.
The matrix Qmax is computed as the maximum volume ellipsoid so that
¯x⊕ EQmax ⊂ P¯x X ,(24)
where
denotes the Minkowski sum of a vector and set, and so the term
¯x⊕ EQmax
is equivalent to the ellipsoid
defined by
Qmax
centered at the vector
¯x
. The assumption that
¯x
lies in the strict interior of
X
ensures that
Qmax
describes a full nx-dimensional ellipsoid.
Using techniques described in [
59
], we can write the condition
(24)
for any time
t
[
t0, tf
]as the following
optimization problem:
Q1/2
max(t) = arg max
Zlog det Z
s.t. kZai(t)k2+ai(t)>¯x(t)bi(t), i = 1, . . . , mx
0ZxmaxInx
(25)
Common constraints that fit this description include ellipsoidal collision avoidance constraints, or in the control space, a minimum
two-norm constraint. The project-and-linearize technique enlarges the feasible domain defined by the approximation and can lead to
larger funnels.
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When
t
=
tf
, a set of constraints can be added to constrain
EQmax ⊆ Xf
. By assuming that the set
Xf
can be
described in the same manner as
X
, this set of constraints is identical to those that are shown in
(25)
, albeit
calculated using the data corresponding to Xf.
The matrices
Rmax
can be computed in precisely the same manner. Starting with the analogous description of
Uas
U={u|hj(u)0, j = 1, . . . , mu,kuk2umax },(26)
we form the approximate quasi-polytopic constraint set P¯uin the vicinity of the nominal trajectory as
P¯u=u|a>
jubj, u = 1, . . . , mu∩ {u| kuk2umax }(27)
and then solve the following optimization problem for any time t[t0, tf]:
R1/2
max(t) = arg max
Zlog det Z
s.t. kZaj(t)k2+aj(t)>¯u(t)bj(t), j = 1, . . . , mu
0ZumaxInu
(28)
Problems
(25)
and
(28)
can be solved to obtain matrices that approximate the maximum ellipsoidal regions fully
contained within the feasible state and control domains. Note that if
X
=
P¯x
, then this is not an approximation
(similarly for U=P¯u).
Recall from the definition that a quadratic funnel is defined by the ellipsoids
EQ
and
EKQK >
. A sufficient
condition to ensure that a quadratic funnel is fully contained within the original feasible domain defined by
X
and Uis
EQ⊆ EQmax and EKQK >⊆ ERmax .
By using the definition of an ellipsoid (9), one can show that these conditions are equivalent to
QQmax and KQK >Rmax .(29)
The former is affine in the variable
Q
, whereas the latter is not jointly convex in the variables
Q
and
K
. However,
a reformulation provides the necessary remedy. Using a Schur complement, we can rewrite the second matrix
inequality in (29) to be
Rmax KQK >0"Q1K>
K Rmax#0.
Pre- and post-multiplying by diag {Q, Inu}then yields the equivalent condition that
"Q Y >
Y Rmax#0,(30)
where we’ve reused the variable substitution
Y
=
KQ
. Together, the constraints
QQmax
and
(30)
ensure that
a quadratic funnel is contained in the feasible region defined by
X
and
U
. Both are linear matrix inequalities and
convex in the variables Qand Y.
C. The Quadratic Funnel Synthesis Problem
We have now formulated the two properties of a quadratic funnel separately as a set of three matrix inequalities.
In this section, we combine them and pose a single optimization problem that can be solved in order to obtain
a quadratic funnel. The continuous-time quadratic funnel synthesis problem for the nonlinear system
(6)
is
summarized in Problem 1.
Problem 1
(Quadratic Funnel Synthesis)
.
Given a nominal trajectory
{¯x(t),¯u(t)}tf
t=t0
that satisfies the nonlinear
dynamics
(1)
, an appropriate definition of the system
(6)
and
α
0, find the matrix-valued functions of time
Q(t),Y(t)and M(t)and scalar λ(t)that solve the following optimization problem.
max
Q(·),Y (·)(·),M(·)log det Q(t0)(31a)
s.t. 0QQmax,0λ, M ∈ Mφ,EGQ1G>, t [t0, tf],(31b)
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"F˙
Q+λG>M11G E +λG>M12
E>+λM>
12G λM22 #0, t [t0, tf],(31c)
"Q Y >
Y Rmax#0, t [t0, tf].(31d)
The matrices Fand Gare given by (21b) and (21c).
Problem 1 is a nonlinear, nonconvex optimization problem with matrix variables. The main difficulty in solving
Problem 1 lies in the fact that it is not possible (to the best of our knowledge) to express
(31c)
as a linear DMI in
the problem’s variables. Moreover, the inclusion
M∈ Mφ,GQ1G>
can at best be approximated – a point we will
discuss in more detail shortly in §III.A.
A quadratic funnel obtained by solving Problem 1 can be used to generate a feasible trajectory in the following
manner. Let
¯x
(
t0
)be the initial condition of the reference state trajectory. For any initial condition
x
(
t0
)such
that x(t0)¯x(t0)∈ EQ(t0), the state and control vectors
u(t) = ¯u(t) + K(t)x(t)¯x(t),(32a)
x(t) = x(t0) + Zt
t0
fx(τ), u(τ)dτ, (32b)
satisfy:
η=x¯x∈ EQ,and ξ=u¯u=∈ EK QK>
for all
t
[
t0, tf
]. Feasibility with respect to the constraint sets
X
and
U
follows. Notice that dynamic feasibility
follows from
(32)
by construction, and the control
u
is guaranteed to be stabilizing for the original nonlinear
dynamics.
III. Solution Strategy: The γ-Iteration
Because a direct solution of Problem 1 is in general not possible, we now present an iterative method that uses
the following observation: if we fix the triple
{Q, Y, λ}
, then Problem 1 is convex in
M
provided we have an
appropriate description of the convex cone
Mφ,GQ1G>
; conversely, for a fixed
M
, Problem 1 is convex in the
variables
{Q, Y, λ}
. A natural and common technique is to alternate between fixing one set of variables and solving
for the other. This has been done in similar contexts for funnel generation [
47
], as well as the Newton-Kleinman
iteration [51] and the D-K iteration [52], and is the premise of the γ-iteration.
The nomenclature “γ-iteration” comes from the fact that the matrix
Mγ="γ2I0
0I#(33)
is a valid local multiplier matrix if
γ
is a local Lipschitz constant for the nonlinear function
φ
over the set
EGQ1G>
.
By solving for
γ
, we may easily construct a local multiplier matrix that can be used to obtain a new iterate for
the variables {Q, Y, λ}. We now discuss a strategy to compute γfor an arbitrary nonlinear function and set.
A. The M-Problem
The M-problem is the subproblem that is obtained by fixing the triple
{Q, Y, λ}
in Problem 1. The quadratic
inequality (14) under the assumption that M=Mγbecomes
p>pγ2q>q⇒ kpk2γkqk2.(34)
Introducing a matrix
Rnp×nq
, we can then write
p
= ∆
q
and
k
k2γ
in place of the nonlinear function
φ
in (6). By using the closed-loop matrices Acl and Ccl, we can rewrite the system (6) as
˙η= (Acl +ECcl)η, kk2γ. (35)
The question becomes: How to compute, or estimate, the value of
γ
? There are several procedures for doing so
(see, e.g., [62]); and for the sake of brevity, we present only one method that is based on sampling [58, 63].
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To compute the local multiplier matrix
Mγ
, we would like to find the point
η∈ EQ
that corresponds to the
largest value of kk2. That is, we need to solve
γ= max
ηΓ(η)
s.t. η∈ EQη>Q1η1,
(36)
where,
Γ(η) = min
kk2
s.t. ˙ηAclη=ECcl η. (37)
The inner optimization problem
(37)
finds the smallest matrix (in the spectral norm) that satisfies the nonlinear
equations of the dynamic model at a particular point
η
. This inner problem is the key component that allows
us to be as non-conservative as possible in our search for a local Lipschitz constant. The outer optimization
problem (36) then finds the point ηthat maximizes the value of Γinside the funnel.
By sampling a set of states {ηs}Ns
s=1 from the state-funnel EQ,at different temporal points, and using
xs= ¯x+ηs, ξs=s, us= ¯u+ξs
we can compute at each time the data:
Acl =A+BK, Ccl =C+DK, ˙ηs=fxs, usf¯x, ¯u.
These are collected and the following optimization problem is solved to obtain a lower bound of the true local
Lipschitz constant:
γ= max
s=1,...,Ns
Γ(ηs)(38)
where Γ
is given by the solution of
(37)
. The outer optimization problem
(36)
is effectively replaced by selecting
the sample that produces the maximum value across the Nssamples.
This spatiotemporal sampling procedure has been observed to be accuracte enough for practical problems with a
reasonable number of points
Ns
(typically on the order of 10
3
or 10
4
for an entire trajectory). Note that because
it is possible to solve
Ns
independent optimization problems, each of which is quite small, this sampling procedure
can be done very quickly and is parallelizable.
B. The Q-Problem
We now describe the Q-problem, which is the subproblem that is obtained by fixing the local multiplier matrix
M
in Problem 1 and solving for the variables
{Q, Y, λ}
. Consider the differential matrix inequality
(31c)
. Due to
the form of the local multiplier matrix
Mγ
in
(33)
, we know that
M11
0. In this case, the matrix inequality
(31c)
can be written
"F˙
Q E
E>λI#+"γG>
0#(λI)hγG 0i0
If E6= 0 then we must have λ > 0. Hence we can use a Schur complement to arrive at the equivalent condition
F˙
Q E γG>
E>λI 0
γG 0λ1I
0.
Pre- and post-multiplying by diag I, λ1I, I gives the equivalent condition that
F˙
Q νE γG>
νE>νI 0
γG 0νI
0, ν :=λ1.(39)
The matrix inequality (39) is now a linear differential matrix inequality in the variables Q,Yand ν.
The continuous-time Q-problem is summarized below in Problem 2. It is assumed that a nominal trajectory,
system definition, and decay rate α0are all given.
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Problem 2
(Q-Problem)
.
Given a local multiplier matrix
Mγ∈ Mφ,EGQ1G>
, find the matrix-valued functions
of time Q(t),Y(t)and scalar ν(t)that solve the following optimization problem.
max
Q(·),Y (·)(·)log det Q(t0)(40a)
s.t. 0QQmax,0ν, t [t0, tf],(40b)
F˙
Q νE γG>
νE>νI 0
γG 0νI
0, t [t0, tf],(40c)
"Q Y >
Y Rmax#0, t [t0, tf].(40d)
The matrices Fand Gare given by (21b) and (21c).
In order to solve the Q-problem numerically, we make the following assumption regarding the continuous-time
problem data.
Assumption 2.
Let
Z
(
t
)denote any of the matrices
Q
(
t
)
, Qmax
(
t
)
, Y
(
t
)
, Rmax
(
t
)
, A
(
t
)or
B
(
t
). We assume
that Z(t)can be expressed or approximated as the convex combination
Z(t) =
nM
X
i=1
σi(t)Zi,(41)
for some integer nM>1, constant matrices Zi, and interpolating functions σi(t)0such that PnM
i=1 σi(t)=1.
By using the temporal matrix decomposition methods discussed in the Appendix, the DMI in
(40c)
can be
written as a set of
1
2nM
(
nM
+ 1) LMIs. As a result of Lemma 6 in the Appendix, these LMIs constitute a set of
sufficient conditions for the satisfaction of the DMI
(40c)
. The constraints
(40b)
and
(40d)
can be written as a set
of LMIs by using
(41)
and constraining each summand in the resulting expression individually. For example,
(40d)
is written "Q Y >
Y Rmax#=
nM
X
i=1
σi(t)"QiY>
i
YiRmax,i#0.
By constraining each matrix in the right-hand sum to be positive semidefinite, we can ensure that the continuous-
time expression holds.
C. Algorithm Summary and Convergence
Having defined the Q- and M-problems independently, we now collect them together in order to design a
convergent algorithm that is capable of solving Problem 1. The premise of the
γ
-iteration is that we can decrease
the upper bound,
Qmax
, until we are able to synthesize a funnel
QQmax
. We measure how close a given solution
of the Q-problem is to achieving the upper bound of QQmax by using the fill ratio, defined by
κ= min
i=1,...,nxprojiEQ
projiEQmax 1/2
.(42)
The notation
projiEQ
denotes the projection of
EQ
onto the dimension
i
and is equivalent to the maximum
distance that
EQ
extends along the
ith
axis in
nx
-dimensional space. The initial value of
Qmax
provides the largest
feasible ellipsoidal set in the state-space. After a solution of the Q-problem, if the fill ratio exceeds a certain
threshold, then the algorithm is terminated. If the fill ratio is below the threshold, then we use a contraction step
that shrinks
Qmax
by an amount proportional to the fill ratio. The contraction step is simple, we maintain the
shape of the ellipsoid
EQmax
and shrink its size by multiplying by a contraction factor
ς
[
ςmin,
1). The fill ratio
determines the value of the contraction factor in this interval through the use of a sigmoid function
ς=ςmin + (1 ςmin)1
1 + eh(0.5κ)Qmax ςQmax,(43)
where
h
is a width parameter and
κ
is given by
(42)
. Equation
(43)
is designed so that when the fill ratio
κ
is 0
.
5,
the contraction factor is chosen as the midpoint of the interval [
ςmin,
1). An example set of contraction factor
curves versus the fill ratio is provided in Figure 2 for an array of width parameters h.
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Fig. 2 Example range of contraction factors as a function of the width parameter husing ςmin
= 0
.
2
.
Algorithm 1
The
γ
-iteration designed to solve the quadratic funnel synthesis problem. A variable
z
at the
kth
iteration is denoted using z(k).
Input:
A nominal trajectory
{¯x
(
t
)
,¯u
(
t
)
}tf
t=t0
, the system matrices for
(6)
, a decay rate
α
0, a tolerance
κtol (0,1).
1solve Problems (25) and (28) for Qmax and Rmax
2set Mγ0
3solve the Q-problem to get {Q(0), Y (0)}Problem 2
4update Qmax(0) Q(0) and Ymax(0) Y(0)
5for k= 1, . . . do
6solve the M-problem to get Mγ(k)using {Qmax(k1), Ymax (k1)}see (38)
7solve the Q-problem to get {Q(k), Y (k)}using Mγ(k)Problem 2
8if κ(k)(t0)κtol then see (42)
9Converged.
10 break
11 else
12 Qmax(k)ς Qmax(k1) and Ymax(k)ςYmax(k1) contraction step, see (43)
13 end if
14 end for
Output: Time-varying matrices Q(t)and K(t)that define a quadratic funnel in the sense of Definition 2.
Remark 3.
It is also possible to use a spatiotemporal contraction factor, as opposed to the solely temporal one
given by
(43)
. For this method, one would compute a diagonal matrix
SRnx×nx
where the
(i, i)th
entry is
the minimand of
(42)
. One then would notionally compute
Qmax SQmax S
, but there is no guarantee that
ESQmax S⊆ EQmax
, because the scaling can rotate the ellipsoid. To remedy this, one simply solves for the maximum
volume ellipsoid that can be inscribed in the intersection of EQmax and ESQmaxS– see [59].
The key idea that leads to a convergent algorithm for which every iteration’s funnel is a valid quadratic funnel,
is to use
Qmax
and its associated correction law to compute the local multiplier matrix
Mγ
at each iteration. We
therefore discard the intermediate solutions of the Q-problem, and only make use of the final iteration’s value.
The
γ
-iteration is summarized in Algorithm 1. Note that no user-specified initial guess is needed, as the algorithm
can self-start by solving the Q-problem with
Mγ
= 0. Theorem 4 establishes the
γ
-iteration as a convergent
algorithm, where we denote the value of a variable zat the kth iteration of the algorithm by z(k).
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Theorem 4.
Suppose that there exists a real number
 >
0such that
InxQ(1)
. Then for any tolerance
0
< κtol <
1, the
γ
-iteration in Algorithm 1 converges to a solution of Problem 1 in a finite number of iterations.
Proof.
Note that for each instance of Problem 2, the solution
Q
= 0 and
Y
= 0 is always feasible. Because the
inclusions
¯xint X
and
¯uint U
hold, the initial solution to Problem 2 is feasible and produces a non-zero and
positive definite solution Qmax(0) 1(0) Inxfor some 1(0) >0.
For any
Qmax
0, the local multiplier matrix obtained by solving
(38)
will be non-zero. Because
Qmax
is
bounded from above and the dynamics
(1)
are assumed to be differentiable, we know that at any iteration, there
must exist some
δ >
0such that
kMγk2δ
. Let
δ(1)
be this bound at the first iteration. Due to the assumption
in the statement of the theorem, there exists some
 >
0such that there is a feasible solution to the Q-problem at
the first iteration that satisfies
InxQ(1)
. This implies that matrices that satisfy
QInx
are also feasible
solutions.
Suppose that the algorithm has not converged by iteration
k
1, and that
Qmax(k1) 1(k1) Inx
for some
1(k1) >
0. The contraction step implies that during the next iteration, we have
Qmax(k)1(k)Inx
, where
1(k)
=
ς1(k1) < 1(k1)
due to the contraction factor 0
< ς <
1. Because we hold
Kmax
constant through the
iterations via the update rules in step 12, we must have
kMγ(k)k2δ(k)
for some
δ(k)δ(k1)
. Therefore, there
must still exist a feasible solution to the Q-problem at the
kth
iteration for which
InxQ(k)
. By induction on
k
,
each instance of the Q-problem has a feasible solution that satisfies InxQ(k).
Because the contraction factor satisfies
ς < ςmin + (1 ςmin)1
1 + eh(0.5κtol)<1,
at each iteration prior to convergence, there must exist some finite integer
K
for which the value of
1(K)< 
. In
this case, we have
Qmax(K)Inx
, at hence
Qmax(K)
is a feasible solution and results in a fill ratio of exactly one.
Because the cost function
(40a)
maximizes the volume of the fill ratio at the initial time, the optimal solution at
the Kth iteration will be Qmax(K)and the algorithm will terminate.
IV. Case Study: 6-DOF Powered Descent
This section provides a numerical example of the
γ
-iteration applied to a 6-DOF powered descent guidance
problem. This problem has nonlinear dynamics, state and control constraints, and large state and control
dimensions that collectively make funnel synthesis challenging.
If we use solely the thrust vector from a single (gimbaled) main engine as the control input, then we will find
that the linearized system is not stabilizable (in the linear systems sense of the term) at any fixed instant in time.
As a result, the Lyapunov-type inequalities that must be solved during the Q-problem will be infeasible. This is a
fundamental limitation of funnel synthesis as we have presented it. To alleviate this issue, we must add another
actuation mechanism to the problem formulation; differential thrust from multiple rocket engines or a reaction
control system (RCS) will suffice. We shall use the latter for this example.
It is assumed that the maneuver occurs close enough to the landing site and over a short enough duration
that gravity can be approximated as constant vector. It is further assumed that planetary rotation may be
neglected and that atmospheric effects can be neglected for nominal trajectory design. If necessary, each of these
assumptions can be removed with relative ease by suitable modification of the dynamics. 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.
Consider a surface-fixed landing frame
FI
with origin at the intended landing site and constructed from the
orthonormal vectors
{xI, yI, zI}
. These basis vectors are oriented such that
xI
represents the downrange
direction,
yI
represents the crossrange direction, and
zI
points locally up. Similarly, a body frame
FB
has its
origin at the vehicle’s center of mass and is constructed from the orthonormal vectors
{xB, yB, zB}
, where
zB
is
chosen to point along the vehicle’s vertical axis. Because the definition of
η
in
(2)
requires that the difference
between the nominal state trajectory,
¯x
, and the actual state trajectory,
x
, be additive (and not multiplicative),
If γ < 1, then this bound is δ= 1. Otherwise, it can be γ2.
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we cannot use either unit quaternions or dual quaternions to represent the state of the vehicle. We therefore
parameterize the orientation of FBwith respect to FIwith a 3-2-1 Euler angle sequence.
The state, control, and nonlinear dynamics for this problem are taken to be
x=
m
rI
vI
Θ
ωB
, u ="FB
τB#,˙x=f(x, u) =
α˙mkFBk2
vI
1
m(CIB (Θ)FB) + gI
T(Θ)ωB
J1τB+r×
u,BFBω×
BJωB
,(44)
where
rIR3
is the inertial position vector,
vIR3
is the inertial velocity vector, Θ = (
ϕ, θ, ψ
)
R3
represents
the Euler angles, and
ωBR3
is the angular velocity in the body frame. The controlled inputs are the engine
thrust vector
FB
and an RCS torque vector
τB
. Note that
nx
= 13 and
nu
= 6 for this problem. The nominal
trajectory was computed by using the PTR algorithm, see [
18
,
23
], with the following boundary conditions and a
final time of tf= 29.7 s:
t0: ¯m= 3250 kg,¯rI= (250,0,433) m,¯vI= (35.7,0,11.8) m/s,Θ = (0,59.8,0) deg, ωB= 03×1,
tf: ¯m= 3130.3 kg,¯rI= (0,0,30) m,¯vI= (0,0,1) m/s,Θ=03×1, ωB= 03×1.
For this example, we enforce two nonlinear control constraints that serve to demonstrate a case where
U 6
=
P¯u
.
In particular, the nominal trajectory is computed using both upper and lower thrust bounds
Fmin ≤ kFBk2Fmax and Tmax ≤ kτBkTmax (45)
for some Fmin, Fmax , Tmax R++, and a gimbal angle constraint
kFBk2sec δmaxz>
BFB.(46)
The constraint sets X,Uand Xfare then taken to be
X={x|xlb xxub, δxlb x¯xδxub ,kxk2 ∞} ,(47a)
U={u|ulb uuub,(45),(46)},(47b)
Xf=EQmax,f , Q1/2
max,f =diag 1637,1
2,1
2,1
2,1
4,1
4,1
4,π
60 ,π
60 ,π
60 ,π
60 ,π
60 ,π
60 (47c)
Note that
X
enforces both an absolute bound on the state vector, as well as a bound on the deviation from the
nominal trajectory. The sets Xand Uare described by
xlb =2100,150,150,0,40,40,30, π, π
2, π, 0.5,0.5,0.5,
xub = + 3737.7,350,300,500,30,30,5, π, π
2, π, 0.5,0.5,0.5,
δxlb =,100,100,100,,,,2
9π, 2
9π, 2
9π, 2
9π, 2
9π, 2
9π,
δxub = + ,100,100,100,,,,2
9π, 2
9π, 2
9π, 2
9π, 2
9π, 2
9π,
ulb =(7695.5,7695.5,5400,150,150,150) ,
uub = + (7695.5,7695.5,24750,150,150,150) .
The terminal constraints imposed by
Xf
ensure that any trajectory does not deviate from the nominal path at
the final time by more than 0
.
5 m in position, 0
.
25
m/s
in velocity, or 3
deg
and 3
deg/s
in attitude and angular
rate in any direction. The remaining data for this problem are provided in Table 1, and are (loosely) based on an
Apollo-class lander.
The matrices
A
and
B
are the partial derivatives of
f
along the nominal trajectory, and the parameters
C, D
and Eare constructed using a total of six (Np= 6) nonlinear channels to be
C=
I101×301×301×301×3
03×103×303×3I303×3
03×103×303×3I303×3
03×103×303×303×3I3
03×103×303×303×3I3
03×303×303×303×303×3
, D =
01×301×3
03×303×3
03×303×3
03×303×3
03×303×3
I303×3
14
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Table 1 The parameters for the 6-DOF powered descent case study.
Parameter Value Parameter Value
Jdiag {13600,13600,19150}kg m2α˙m4.5324 ×104s/m
rF,B0,0,0.25mgI0,0,1.62m/s2
Fmin 5400 N Fmax 24750 N
δmax 25 deg Tmax 150 Nm
nM25 Ns100nM
κtol 0.5α0.1 s
E=
01×301×301×301×301×3I1
03×303×303×303×303×301×1
I3I303×303×303×301×1
03×303×3I3I303×301×1
03×303×303×303×3I301×1
and therefore we have np= 16 and nq= 16 for this problem.
We employ Assumption 2 in order to temporally discretize each problem variable for the Q-problem using the
value of nMin Table 1, with the exception of νand γ, which are taken to be piecewise constant§.
The
γ
-iteration converges after 6iterations for this problem setup. In the state space, the synthesized quadratic
funnel has the following projections onto each of the nxstate dimensions at the initial time t0:
m: 3.0 kg rI:
20.6
26.5
17.5
m, vI:
1.2
4.3
2.7
m/s,Θ :
10.6
8.0
10.6
deg, ωB:
6.0
4.4
1.1
deg/s
Figure 3 shows the computed quadratic funnel. In Figure 3a, the ellipsoid
EQ
is projected onto each state
dimension (mass is omitted) and depicted as the shaded grey area. The red tra jectories correspond to test cases
for which an initial condition was randomly (uniformly) selected from the ellipsoid
EQ(t0)
, and the nominal control
and correction law were used to numerically integrate the equations of motion according to
(32)
. Figure 3b shows
the ellipsoid
ERmax
projected into each control dimension along with the corresponding control trajectories from
each test case. Figure 4 further displays the three-dimensional thrust space over the entire maneuver. The portion
of the feasible set
U
that corresponds to the thrust constraints, constructed from the (convex) thrust upper bound,
(nonconvex) thrust lower bound and (convex) gimbal angle constraints, is shown in green in order to visually
confirm the feasibility of the quadratic funnel in the thrust-space.
Figure 5a provides the fill ratio versus the iteration number. To give a better sense of how the convergence process
looks, the fill ratios shown are computed by using the three-dimensional ellipsoids that result from projecting
both
EQ
and
EQmax
into each of the position, velocity, attitude, and angular rate dimensions. It can be seen that
by the last iteration, each of these projections has surpassed the desired fill ratio, with the attitude and angular
rate ratio’s being nearly at the theoretical limit of one.
These results are quite promising – the powered descent problem is a challenging problem with nonlinear
dynamics, some nonlinear and nonconvex constraints, and relatively large state and control dimensions. What
these results show is that we are able to generate feasible trajectories (both dynamically and with respect to
the constraints that were considered) for any initial condition in a set that stretches more than 35 m in every
position direction, 2
m/s
in each velocity direction, 16
deg
in each Euler angle, and 2
deg/s
in each angular
velocity direction, all by using a single quadratic funnel.
V. Conclusions
This paper presents a new implicit trajectory generation algorithm, namely quadratic funnel synthesis. We
developed a specific algorithm, called the
γ
-iteration, that computes a quadratic funnel subject to certain
§
Only the assumption that
ν
is piecewise constant is without loss of generality. Taking
γ
to be piecewise constant can introduce
some additional conservatism.
15
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(a) In state space, EQ.
(b) In control space, ERmax .
Fig. 3 The quadratic funnel computed by the γ-iteration for the powered descent problem. The
initial condition of each test case was randomly sampled from the funnel entry.
assumptions on the nonlinear equations of motion for a given dynamical system. Funnel synthesis addresses the
two fundamental drawbacks of explicit trajectory generation – namely the stringent association of tra jectories to
problem data and lack of formal guarantees of real-time solvability. Ultimately, funnel synthesis depends on the
availability of a feasible nominal trajectory, and is therefore intimately connected and complementary to explicit
trajectory optimization.
For future safety-critical aerospace missions, the ability to theoretically guarantee the availability of a feasible
guidance trajectory (with respect to nonlinear dynamics and nonconvex constraints) in real-time is a mission-
enabling technology. The results presented in this paper have shown that with a single quadratic funnel computed
offline, such feasible trajectories can be computed using only numerical integration for any initial state in a set
that spans a relatively large distance in each of the position, velocity, attitude, and angular rate dimensions.
16
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Fig. 4 The ellipsoid ERmax projected into thrust space (gray) compared to the thrust portions of
the original feasible set, U, (green). All thrust curves must lie inside the gray set.
Fig. 5 The fill ratio of EQ(t0)projected into four states across each iteration of the γ-iteration,
and the value of the Lyapunov function V(t)for each test case for the powered descent problem.
Acknowledgements
This research has been supported by NASA grant NNX17AH02A and National Sciences and Engineering
Research Council of Canada (NSERC) grant PGSD3-502758-2017, government sponsorship is acknowledged.
Appendix: Temporal Matrix Decompositions
Denote the standard n-dimensional simplex by
Σn=(σRn+1
n+1
X
i=1
σi= 1, σi0, i = 1, . . . , n).(48)
Let M(t)Rn×mand N(t)Rm×nbe matrix valued functions of time. Suppose that
M(t) =
nM
X
i=1
σi(t)Miand N(t) =
nN
X
j=1
ςj(t)Nj,(49)
17
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where σ(t) = σ1(t), . . . , σnM(t)ΣnM1and ς(t) = ς1(t), . . . , ςnN(t)ΣnN1.
We then have the following two results on the decomposition of the product M(t)N(t).
Lemma 5.
At any time
tR+
, the product
M
(
t
)
N
(
t
)
Rn×n
can be written as a convex combination of the
nMnNmatrices
MiNj, i = 1, . . . , nM, j = 1, . . . , nN.(50)
Proof.
Let
M
(
t
)and
N
(
t
)be as in
(49)
. Then
M
(
t
)
N
(
t
) =
PnM
i=1 PnN
j=1 σi
(
t
)
ςj
(
t
)
MiNj
. It is straightforward to
show that PnM
i=1 PnN
j=1 σi(t)ςj(t) = 1, and that σi(t)ςj(t)0for any i, j.
The second result is a special case of Lemma 5 for which the dimensions of
M
(
t
)and
N
(
t
)are equal and the
time-varying coefficients in the expansions (49) are the same.
Lemma 6.
If
nM
=
nN
and
σ
(
t
) =
ς
(
t
)for any
tR+
, then the product
M
(
t
)
N
(
t
)
Rn×n
can be written as a
convex combination of the 1
2nM(nM+ 1) matrices:
MiNi, i = 1, . . . , nM(51a)
MiNj+MjNi, i = 1, . . . , nM1, j =i+ 1, . . . , nM(51b)
Proof.
The proof is by construction. Let
M
(
t
)and
N
(
t
)be as in
(49)
with
nM
=
nN
. From the proof
of Lemma 5 we know that the product
M
(
t
)
N
(
t
)can be written as the convex combination
M
(
t
)
N
(
t
) =
PnM
i=1 PnM
j=1 σi
(
t
)
ςj
(
t
)
MiNj
, where
PnM
i=1 PnM
j=1 σi
(
t
)
ςj
(
t
)=1. If
σ
(
t
) =
ς
(
t
)for any
tR+
then we have
σi(t)ςj(t) = σj(t)ςi(t), and so:
M(t)N(t) =
nM
X
i=1
nM
X
j=1
σi(t)ςj(t)MiNj
=
nM
X
i=1
σi(t)ςj(t)MiNi+
nM
X
i=1
nM
X
j=1,j6=i
σi(t)ςj(t)MiNj
=
nM
X
i=1
σi(t)ςj(t)MiNi+
nM1
X
i=1
nM
X
j=i+1
σi(t)ςj(t) (MiNj+MjNi)
The last expression establishes that the product M(t)N(t)is a convex combination of the matrices in (51).
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21
Downloaded by 184.147.114.9 on January 5, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2021-0504
... One way to resolve this issue is to replace the trajectory with a robust forward invariant set that can be viewed as a controlled invariant funnel (CIF) around the nominal trajectory. The process of computing this invariant set and an associated feedback controller is commonly referred to as funnel synthesis [2]. ...
... Prior research has attempted to develop efficient algorithms for CIF generation for nonlinear systems [2], [3], [4], [5]. Most proposed methods generate a nominal trajectory by using a trajectory optimization algorithm and then compute the CIF around the nominal trajectory, e.g., by using techniques from robust control. ...
... In contrast, the proposed work computes the CIF for nonlinear systems that are locally Lipschitz with an LMI-based approach. The research most similar to this paper is given in [2]. We extend this work to the cases where the systems are subject to disturbances. ...
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This paper presents a joint synthesis algorithm of trajectory and controlled invariant funnel (CIF) for locally Lipschitz nonlinear systems subject to bounded disturbances. The CIF synthesis refers to a procedure of computing controlled invariance sets and corresponding feedback gains. In contrast to existing CIF synthesis methods that compute the CIF with a pre-defined nominal trajectory, our work aims to optimize the nominal trajectory and the CIF jointly to satisfy feasibility conditions without relaxation of constraints and obtain a more cost-optimal nominal trajectory. The proposed work has a recursive scheme that mainly optimizes two key components: i) trajectory update; ii) funnel update. The trajectory update step optimizes the nominal trajectory while ensuring the feasibility of the CIF. Then, the funnel update step computes the funnel around the nominal trajectory so that the CIF guarantees an invariance property. As a result, with the optimized trajectory and CIF, any resulting trajectory propagated from an initial set by the control law with the computed feedback gain remains within the feasible region around the nominal trajectory under the presence of bounded disturbances. We validate the proposed method via a three dimensional path planning problem with obstacle avoidance.
... No known algorithm can be formally guaranteed to solve a general nonconvex optimal control problem from an arbitrary initial guess. There is, therefore, no theoretical reason that could enable one to solve a nonconvex trajectory optimization problem in real-time -although certainly there is ample empirical data to support expectations of reliable local convergence [16]. It is natural, in response, to study alternative methods for which, the ability to provide a feasible trajectory in real-time can be theoretically established. ...
... The funnel synthesis techniques are based on the notion of quadratic stability as defined in [21] and [22][23][24], the latter of which offers necessary and sufficient conditions for stability based on quadratic Lyapunov functions. As described in [16], funnel synthesis makes use of a Lyapunov function (as opposed to the maximum principle-based techniques) to seek out nearby feasible trajectories. Following is a formal definition of a funnel. ...
... The assumption that = used to form the closed-loop system (16) thus results in the following implication: ...
Conference Paper
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This work develops a Robust Controller synthesis method for Vision-based Spacecraft(RCVS) guidance and control, integral to the robust autonomy framework for multi-spacecraft formation control and reconfiguration applications. The method is built around the use of a photo-realistic simulator, where a camera is deployed on a tracking spacecraft (ego) in order to observe an uncontrolled spacecraft (target) in a Low Earth Orbit (LEO). In this direction, the proposed approach performs the relative state (attitude and position) estimation of the target spacecraft using a Convolutional Neural Network (CNN). The state estimation error is then modeled and the corresponding error bounds are obtained around a nominal trajectory of the ego and target spacecraft. Next, this work proposes a linear matrix inequality (LMIs) based approach to controller synthesis, guaranteed to be robust against both model uncertainties and measurement errors, resulting from vision-based estimation. This controller is comprised of two distinct components, one synthesized based on the nominal trajectory, while the “robust” component corrects for deviations from the nominal trajectory. Finally, a tracking scenario that directly utilizes the image data for spacecraft guidance and control, is presented to showcase the performance of the proposed robust autonomy framework.
... To mitigate this issue, we gradually decrease the magnitude of variance σ 2 i in each iteration to minimize the constraint violation. The promising future direction to handle the constraint violation is to employ funnel synthesis approaches in which the feedback gain is calculated with considering the constraints [18]. ...
... Subsequently, supervised learning is used to independently train a neural network on the samples. In every epoch in training, we evaluate the policy with the objective given in (18) using the state and input pairs in the trajectories starting The trajectory comparison between the proposed approach and the imitation learning method. The figures in the first column represent the trajectories starting from the initial states in the training set. ...
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Nonlinear trajectory optimization algorithms have been developed to handle optimal control problems with nonlinear dynamics and nonconvex constraints in trajectory planning. The performance and computational efficiency of many trajectory optimization methods are sensitive to the initial guess, i.e., the trajectory guess needed by the recursive trajectory optimization algorithm. Motivated by this observation, we tackle the initialization problem for trajectory optimization via policy optimization. To optimize a policy, we propose a guided policy search method that has two key components: i) Trajectory update; ii) Policy update. The trajectory update involves offline solutions of a large number of trajectory optimization problems from different initial states via Sequential Convex Programming (SCP). Here we take a single SCP step to generate the trajectory iterate for each problem. In conjunction with these iterates, we also generate additional trajectories around each iterate via a feedback control law. Then all these trajectories are used by a stochastic gradient descent algorithm to update the neural network policy, i.e., the policy update step. As a result, the trained policy makes it possible to generate trajectory candidates that are close to the optimality and feasibility and that provide excellent initial guesses for the trajectory optimization methods. We validate the proposed method via a real-world 6-degree-of-freedom powered descent guidance problem for a reusable rocket.
... Therefore, an extra search for the optimal cutoff time is usually adopted, and thus, the realtime calculation of the optimal steering law could not be achieved 1 online. Recently, some studies focusing on the robust and convergence guarantees for the trajectory optimization methods emerged (Afshari et al. 2011;Reynolds et al. 2021). And some are concerned about the safety of the lander for the crewed lunar landing mission (Lu and Sandoval 2021). ...
... 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.
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Autonomous vehicles and robots promise many exciting new applications that will transform society. For example, autonomous aerial vehicles (AAVs) operating in urban environments could deliver commercial goods and emergency medical supplies, monitor traffic, and provide threat alerts for national security [1] . At the same time, these applications present significant engineering challenges related to performance, trustworthiness, and safety. For instance, AAVs can be a catastrophic safety hazard should they lose control or situational awareness over a populated area. Space missions, self-driving cars, and applications of autonomous underwater vehicles share similar concerns.
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Reliable and efficient trajectory generation methods are a fundamental need for autonomous dynamical systems of tomorrow. The goal of this article is to provide a comprehensive tutorial of three major convex optimization-based trajectory generation methods: lossless convexification (LCvx), and two sequential convex programming algorithms known as SCvx and GuSTO. In this article, trajectory generation is the computation of a dynamically feasible state and control signal that satisfies a set of constraints while optimizing key mission objectives. The trajectory generation problem is almost always nonconvex, which typically means that it is not readily amenable to efficient and reliable solution onboard an autonomous vehicle. The three algorithms that we discuss use problem reformulation and a systematic algorithmic strategy to nonetheless solve nonconvex trajectory generation tasks through the use of a convex optimizer. The theoretical guarantees and computational speed offered by convex optimization have made the algorithms popular in both research and industry circles. To date, the list of applications includes rocket landing, spacecraft hypersonic reentry, spacecraft rendezvous and docking, aerial motion planning for fixed-wing and quadrotor vehicles, robot motion planning, and more. Among these applications are high-profile rocket flights conducted by organizations like NASA, Masten Space Systems, SpaceX, and Blue Origin. This article aims to give the reader the tools and understanding necessary to work with each algorithm, and to know what each method can and cannot do. A publicly available source code repository supports the provided numerical examples. By the end of the article, the reader should be ready to use the methods, to extend them, and to contribute to their many exciting modern applications.
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This paper presents a numerical algorithm for computing six-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics and several standard path constraints. Our formulation also includes a special line-of-sight constraint that is enforced only within a specified band of slant ranges relative to the landing site: a novel feature that is especially relevant to terrain and hazard relative navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting nonconvex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the nonconvex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process toward dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line-of-sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm’s robustness to initial conditions and real-time performance.
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The sizing and capability definitions of reusable launchers during high-speed recovery are very challenging problems. In this paper, a convex optimization guidance algorithm for this type of system is proposed, based on performance improvements arising from the study of the coupled flight mechanics, guidance, and control problem. To appreciate the obtained improvements, tradeoff analyses of powered descent and landing scenarios are presented first using traditional guidance techniques. Subsequently, these results are refined by using the proposed online successive convex optimization-based guidance strategy. The descending over extended envelopes using successive convexification-based optimization (DESCENDO) algorithm has been designed as a middle ground between efficiency and optimality. This approach contrasts with previous convexification algorithms that either aimed at increasing computational efficiency (by typically disregarding aerodynamic deceleration) or reaching trajectory design optimality (by using exhaustive convex approximations). More critically, the algorithm is not confined to the mild coverage conditions assumed by previous approaches and can successfully handle the incorporation of the operational dynamics of reusable launchers. Insights provided by DESCENDO operating in a closed-loop fashion over full recovery scenarios enable a computationally efficient mission performance assessment.