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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

t≥t0

. 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

Ci∈Rnq,i×nx

,

Di∈Rnq,i×nu

and

Ei∈Rnx×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,

p∈Rnp

and

q∈Rnq

, where

np

=

PNp

i=1 np,i

and

nq

=

PNp

i=1 nq,i

, and hence

C∈Rnq×nx

,

D∈Rnq×nu

and

E∈Rnx×np. By stacking each of the φito construct the function φ:Rnq→Rnp, we can write (4) as

˙η(t) = A(t)η(t) + B(t)ξ(t) + Ep(t)

q(t) = Cη(t) + Dξ(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×Rnx→Rdeﬁned 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>Q−1η≤

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=η∈Rnx|η>Q−1η≤1=nQ1/2w| kwk2≤1o.(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∈ EQ⇒u∈ 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

Q∈Snx

++

and a time-varying matrix

K∈Rnu×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

:

Rnq→Rnp

be a nonlinear map such that

q7→ φ

(

q

). A symmetric

matrix M∈S(nq+np)is a local multiplier matrix for φover the set Ωif and only if

µ(q):="q

φ(q)#>

M"q

φ(q)#≥0,for all q∈Ω.(14)

Denote the set of local multiplier matrices for φover the set Ωby

Mφ,Ω=nM∈S(nq+np)|µ(q)≥0for all q∈Ωo.(15)

Multiplier matrices as deﬁned in [

60

] account for the range space of

φ

, but require an inequality like

(14)

to

hold for all

q∈Rnq

. Therefore a multiplier matrix is a local multiplier matrix, but the converse is not necessarily

true. The set

Mφ,Ω

is a convex cone. Importantly, however, matrices

M∈ Mφ,Ω

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 Ωand 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

>Q−1+Q−1Acl −Q−1˙

QQ−1+αQ−1Q−1E

E>Q−10#"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 λ≥0such that

"Acl

>Q−1+Q−1Acl −Q−1˙

QQ−1+αQ−1+λCcl

>M11Ccl Q−1E+λCcl

>M12

E>Q−1+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,EGQ−1G>,(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

¯x∈int X

and

¯u∈int 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

Xf⊂Rnx

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,kxk2≤xmax},(22)

where each constraint function

hi

:

Rnx→R

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>

ix≤bi, i = 1, . . . , mx∩ {x| kxk2≤xmax }(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,kuk2≤umax },(26)

we form the approximate quasi-polytopic constraint set P¯uin the vicinity of the nominal trajectory as

P¯u=u|a>

ju≤bj, u = 1, . . . , mu∩ {u| kuk2≤umax }(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⇐⇒ "Q−1K>

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 (·),λ(·),M(·)log det Q(t0)(31a)

s.t. 0QQmax,0≤λ, M ∈ Mφ,EGQ−1G>, 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,GQ−1G>

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=Kη ∈ 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,GQ−1G>

; 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

0−I#(33)

is a valid local multiplier matrix if

γ

is a local Lipschitz constant for the nonlinear function

φ

over the set

EGQ−1G>

.

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>p≤γ2q>q⇐⇒ kpk2≤γkqk2.(34)

Introducing a matrix ∆

∈Rnp×nq

, we can then write

p

= ∆

q

and

k

∆

k2≤γ

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 +E∆Ccl)η, k∆k2≤γ. (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 k∆k2. That is, we need to solve

γ= max

ηΓ∗(η)

s.t. η∈ EQ⇐⇒ η>Q−1η≤1,

(36)

where,

Γ∗(η) = min

∆k∆k2

s.t. ˙η−Aclη=E∆Ccl 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 Γ∗inside 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=Kηs, us= ¯u+ξs

we can compute at each time the data:

Acl =A+BK, Ccl =C+DK, ˙ηs=fxs, us−f¯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 0−λ−1I

0.

Pre- and post-multiplying by diag I, λ−1I, I gives the equivalent condition that

F−˙

Q νE γG>

νE>−νI 0

γG 0−νI

0, ν :=λ−1.(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 α≥0are all given.

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Problem 2

(Q-Problem)

.

Given a local multiplier matrix

Mγ∈ Mφ,EGQ−1G>

, ﬁnd the matrix-valued functions

of time Q(t),Y(t)and scalar ν(t)that solve the following optimization problem.

max

Q(·),Y (·),ν(·)log det Q(t0)(40a)

s.t. 0QQmax,0≤ν, t ∈[t0, tf],(40b)

F−˙

Q νE γG>

νE>−νI 0

γG 0−νI

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

Q≈Qmax

. We measure how close a given solution

of the Q-problem is to achieving the upper bound of Q≈Qmax 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(0.5−κ)⇒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.

11

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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 Mγ←0

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(k−1), Ymax (k−1)}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(k−1) and Ymax(k)←ςYmax(k−1) 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

S∈Rnx×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

¯x∈int X

and

¯u∈int 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

kMγk2≤δ‡

. 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

QInx

are also feasible

solutions.

Suppose that the algorithm has not converged by iteration

k−

1, and that

Qmax(k−1) 1(k−1) Inx

for some

1(k−1) >

0. The contraction step implies that during the next iteration, we have

Qmax(k)1(k)Inx

, where

1(k)

=

ς1(k−1) < 1(k−1)

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)k2≤δ(k)

for some

δ(k)≤δ(k−1)

. 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(0.5−κtol)<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

J−1τB+r×

u,BFB−ω×

BJωB

,(44)

where

rI∈R3

is the inertial position vector,

vI∈R3

is the inertial velocity vector, Θ = (

ϕ, θ, ψ

)

∈R3

represents

the Euler angles, and

ωB∈R3

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 ≤ kFBk2≤Fmax and −Tmax ≤ kτBk∞≤Tmax (45)

for some Fmin, Fmax , Tmax ∈R++, and a gimbal angle constraint

kFBk2≤sec δmaxz>

BFB.(46)

The constraint sets X,Uand Xfare then taken to be

X={x|xlb ≤x≤xub, δxlb ≤x−¯x≤δxub ,kxk2≤ ∞} ,(47a)

U={u|ulb ≤u≤uub,(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 ×10−4s/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=(σ∈Rn+1

n+1

X

i=1

σi= 1, σi≥0, 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)∈ΣnM−1and ς(t) = ς1(t), . . . , ςnN(t)∈ΣnN−1.

We then have the following two results on the decomposition of the product M(t)N(t).

Lemma 5.

At any time

t∈R+

, 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

t∈R+

, 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, . . . , nM−1, 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

t∈R+

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+

nM−1

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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