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Discretization Performance and Accuracy Analysis for

the Powered Descent Guidance Problem

Danylo Malyuta∗

, Taylor P. Reynolds∗

, Michael Szmuk∗

,

Mehran Mesbahi†

, Beh¸cet A¸cıkme¸se†

,

Dept. of Aeronautics & Astronautics, University of Washington, Seattle, WA 98195, USA

and John M. Carson III‡

NASA Johnson Space Center, Houston, TX 77058, USA

In this paper we analyze the performance and accuracy properties of several diﬀerential

equation discretization methods in the context of powered descent guidance for pinpoint

planetary landing. The guidance problem is formulated as a continuous-time 6-DoF optimal

control problem with nonlinear dynamics and a multitude of state and control constraints.

This problem is to be solved via a direct method whereby it is temporally discretized and

solved iteratively as a sequence of parameter optimization problems. Proper discretization

thus becomes crucial if the resulting thrust commands are to be reproducible by the real

vehicle. Moreover, proper discretization can decrease the overall time required to obtain a

solution, and ultimately to satisfy a real-time computational requirement. We thus carry

out a Monte Carlo performance comparison of the piecewise constant, piecewise linear,

Runga-Kutta and three pseudospectral discretization methods. We study the method’s

performance and accuracy, and discuss how each method may impact the ability to achieve

a real-time solution. These empirical results are backed by a theoretical discussion of how

well each method preserves favorable properties about the discrete optimization problem,

such as sparsity. To the best of our knowledge, these are the ﬁrst results that provide a

back-to-back and fair test for a large number of discretization methods applied to a the

6-DoF powered descent guidance problem.

I. Introduction

This paper compares the accuracy and performance of several methods for converting a continuous-time

optimal control problem into a ﬁnite-dimensional optimization problem. The motivation for this study is to

assess which technique may be suitable for a real-time implementation of a 6 degree-of-freedom (DoF) rocket

powered descent guidance problem for pinpoint planetary landing.

The 6-DoF powered descent guidance problem is a continuous-time free-ﬁnal time nonlinear optimal control

problem subject to both state and control constraints.1,2 We refer to the process of converting this type of

problem into a ﬁnite-dimensional discrete optimization problem as discretization and the methods by which

this is achieved as discretization methods. Discretization consists of expressing the cost function, dynamical

equations and state and control constraints as functions of a ﬁnite number of parameters deﬁned on a set of

temporal nodes.

Numerical methods to solve continuous-time optimal control problems can be broadly classiﬁed intro indirect

and direct methods.3, 4 Direct methods parameterize either the state or control signal using a set of basis

functions whose coeﬃcients are found via parameter optimization. Indirect methods use techniques from

optimal control theory5–7 to determine the necessary conditions of optimality, which are then solved as a

two-point boundary-value-problem. While indirect methods have been used for powered descent and ascent

problems,8, 9 the necessary conditions for problems with multiple non-trivial state and control constraints

∗Doctoral Student, W.E. Boeing Department of Aeronautics & Astronautics, AIAA Student Member

{danylo,tpr6,szmuk}@uw.edu

†Professor, W.E. Boeing Department of Aeronautics & Astronautics, AIAA Associate Fellow, {mesbahi,behcet}@uw.edu

‡SPLICE Project Manager, AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

can be very diﬃcult to write down, even for problems with simple dynamics.10 As such, our approach is to

use a direct method.

Direct methods can be further classiﬁed into techniques that parameterize only the control signal or that

parameterize both the state and control signals. In the context of powered descent guidance, the former

group was the ﬁrst to appear in literature. For the 3-DoF translational guidance problem, several early works

adopted piecewise constant basis functions to approximate the control function.11–14 This design choice is

valid given that the structure of the optimal solution is known to be piecewise constant. A piecewise linear

approximation, however, can yield a more accurate representation of the optimal control signal over the same

(coarse) temporal grid. In eﬀect, if the density of the temporal discretization (which is ﬁxed) does not yield

a node suﬃciently close to an optimal switching time, then sub-optimal results may follow. The piecewise

linear approximation is better suited to handle such cases, and subsequent work on the 3-DoF guidance

problem used this approach.15–19 Recently, these same techniques have been applied to the 6-DoF powered

descent guidance problem, with both piecewise constant20 and piecewise linear approximations.1,2, 21, 22 The

latter group of direct methods that parameterize both the state and control signals are, for our purposes,

global pseudospectral methods.23, 24 These methods have recently been successfully applied to the 3-DoF

translational guidance problem,25 however their application to the 6-DoF powered descent problem has not

yet been reported in literature.

Beyond the speciﬁc example of powered descent guidance, there is a signiﬁcant body of work on both

types of direct methods for aerospace applications such attitude control, proximity operations and formation

reconﬁguration.26–32 Historically, it appears that parameterizing the control signal only is the more popular

approach, perhaps due to the fact that common discretization techniques (e.g., forward or backward Euler,

Runga-Kutta, etc.) fall under this category. There is, however, a growing body of work in these areas that

parameterize both the state and control via pseudospectral methods.33, 34

The common theme to all of these applications is that once a discretization method is selected for the dy-

namical equations, the entire continuous-time problem is converted into a discrete parameter optimization

problem. Depending on the intended solver, the optimization problem may be classiﬁed as a nonlinear

program (NLP) or as a particular class of convex optimization problems. Our approach is to convert the

non-convex 6-DoF powered descent guidance problem into a convex second order cone programming problem.

These convex problems are then solved iteratively in a process known as successive convexiﬁcation.1, 19, 21, 35

This choice arises primarily due to the nature of the constraints inherent to this problem, many of which are

expressed as second order cones, and necessitates the linearization of the (nonlinear) dynamical equations.

Convex optimization techniques are desirable for real-time safety-critical applications due to their determin-

istic time complexity and guaranteed convergence to the global optimum. Recent ﬂight heritage of convex

optimization-based powered descent guidance36 has increased the maturity of the technology.

A. Contributions

The primary contribution of this paper is the extensive and fair comparison of the computational performance

and solution accuracy of several popular discretization schemes. In particular, we examing the piecewise

constant and piecewise linear approximations of the control, Runga-Kutta discretization, the Chebyshev-

Gauss-Lobatto pseudospectral method and the diﬀerential and integral forms of the Legendre-Gauss-Radau

pseudospectral method. The example that we use to perform this analysis is the 6-DoF powered descent

guidance problem for planetary landing, which is a trajectory optimization problem. To the best of our

knowledge, no analysis as extensive as the one presented herein has been previously published for this

increasingly important spaceﬂight problem. Moreover, §II serves as a useful and succinct summary of each

discretization method used herein.

B. Outline

This paper is structured as follows. First, §II provides the mathematical details of the six discretization meth-

ods that we consider. Next, §III describes how each discretization method is used to convert a continuous-time

optimal control problem into a discrete ﬁnite-dimensional one. The same section describes our Monte Carlo

analysis approach to performance testing. In §IV, Monte Carlo simulation results are presented, and §V

provides a discussion of these results and highlights each discretization method’s merits and drawbacks.

Finally, §VI concludes the paper with an outlook for future research.

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C. Notation

We use the following notation and conventions. Let Rdenote the set of reals, Zthe set of integers and Z++

the set of positive integers. Scalars are lowercase italic, e.g. t∈R, vectors are lowercase bold, e.g. x∈Rn,

and matrices are uppercase italic, e.g. F∈Rm×n. Exceptions to this rule are Mand Nwhich are scalar

numbers of temporal collocation and discretization nodes respectively. Inline column vectors are written in

parentheses, e.g. v= (1,2,3) ∈R3. For a given matrix F,Fij shall denote the element in row iand column

j,Fishall denote column iand Fi:jshall denote columns ithrough jwhere all indexing is zero-based (i.e.

index 0 is the ﬁrst element, row or column). An exception to the element indexing rule is the diﬀerentiation

matrix D, where Dji is deﬁned Section II. Furthermore, when dealing with block vectors and matrices the

indexing refers to blocks rather than individual elements, e.g. if v= (a,b,c) then v1=b. The notation (k)

shall be shorthand for the evaluation of vector or matrix function at the k-th temporal node. Generally,

shall denote a “some object” placeholder. The symbol Ishall be reserved for the identity matrix, which

is square and whose size shall be clear from context. The number 0 shall denote at once a zero scalar, vector

or matrix of appropriate size depending on context. 1shall denote the vector of all ones whose length shall

be clear from context. We use three diﬀerent times: tfor real time, τfor scaled time and ηfor pseudo time.

Each of these shall be deﬁned later in the paper. We use the over-dot for diﬀerentation with respect to real

time, i.e. ˙

=d

dt , the circle for diﬀerentiation with respect to scaled time, i.e. ◦

=d

dτ , and the apostrophe

for diﬀerentiation with respect to pseudo time, i.e. 0=d

dη . Finally, the Kronecker delta function is deﬁned

by δii = 1 and δij = 0 if i6=j.

II. Discretization Methods

In this section we consider the problem of discretizing the following Linear Time Varying (LTV) system:

˙

x(t) = A(t)x(t) + B(t)u(t) + Σ(t)σ+z(t),x(0) = x0, t ∈[0, tf] (1)

where x:R→Rnis the state, u:R→Rmis the control input, σ∈Ris a parameter, A:R→Rn×nis the

state zero-input dynamics matrix, B:R→Rn×mis the control matrix, Σ : R→Rnis the parameter matrix

and z:R→Rnis a perturbation that arises if (1) is obtained by linearizing nonlinear dynamics about a

trajectory. Note that the system evolves in real time t, i.e., physical wall clock time. However, we shall see

in §III that by appropriately deﬁning σwe shall be able to equivalently write a scaled time version of (1).

We assume that A,B, Σ, uand zare piecewise continuous functions. Our purpose in the remainder of this

section is to outline several methods for temporally discretizing (1) on a grid of Nnodes tk∈[0, tf] known

as sampling times. These methods will each be applied to discretize a diﬀerential equation constraint of the

type (1) in the powered descent guidance optimal control problem.

Let Φ(·, t0) : R→Rn×ndenote the state transition matrix for (1). We will make use of the following theorem

from linear systems theory.

Theorem II.1. Φ(t, t0)has the fol lowing properties:

1. Φ(·, t0) : R→Rn×nis the unique solution of the linear matrix ordinary diﬀerential equation

d

dtΦ(t, t0) = A(t)Φ(t, t0),Φ(t0, t0) = I, (2)

and is continuous ∀t∈Rand diﬀerentiable everywhere except at the discontinuity points of A.

2. For all t, t0, t1∈R,Φ(t, t0) = Φ(t, t1)Φ(t1, t0).

3. For all t0, t1∈R,Φ(t1, t0)is invertible and its inverse is [Φ(t1, t0)]−1= Φ(t0, t1).

Proof. See a standard text on linear systems theory.37, 38

A. Zeroth Order Hold

We begin by considering discretization methods that parameterize the control directly. The ﬁrst and seem-

ingly most popular such method is to keep the input constant between the sampling times, and is known as

zeroth order hold (ZOH). Speciﬁcally, the control input is assumed to take the form

u(t) = u(tk)∀t∈[tk, tk+1).(3)

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For this purpose we consider a uniformly spaced temporal grid:

tk=k

N−1tf, k = 0, . . . , N −1.(4)

Note that (3) amounts to choosing piecewise constant basis functions that render the coeﬃcients of the

resulting control signal independent from each other. The vectors u(tk)∈Rmdenote the coeﬃcients of the

control input’s representation in this basis. We point out that for a temporal grid with Npoints, this choice

provides N−1 degrees of freedom.

Using (2), the state transition matrix can be found between each of the discrete sampling times. Using

an integration routine such as the classical Runge-Kutta method,39, 40 we can compute Φ(tk+1, tk) for each

k= 0, . . . , N −1. It is then a well-known result from linear systems theory that the state at time t∈[tk, tk+1)

is given by37,38

x(t) = Φ(t, tk)x(tk) + Zt

tk

Φ(t, ξ)B(ξ)u(ξ)dξ+Zt

tk

Φ(t, ξ)Σ(ξ)dξσ +Zt

tk

Φ(t, ξ)z(ξ)dξ. (5)

Using the state transition matrix properties from Theorem II.1 together with (5) and our ZOH assumption

(3) on the control input, we can write the discretized LTV system

x(k+1) =A|kx(k)+B|ku(k)+ Σ|kσ+z|k, k = 0, . . . , N −2,(6)

where we have

A|k= Φ(tk+1, tk),(7a)

B|k=A|kZtk+1

tk

Φ(ξ, tk)−1B(ξ)dξ, (7b)

Σ|k=A|kZtk+1

tk

Φ(ξ, tk)−1Σ(ξ)dξ, (7c)

z|k=A|kZtk+1

tk

Φ(ξ, tk)−1z(ξ)dξ. (7d)

The discretized dynamics (6) can be concatenated for ease of implementation. Deﬁne

X:=

x(0)

x(1)

.

.

.

x(N−1)

∈RnN ,U:=

0

u(0)

.

.

.

u(N−2)

∈RmN ,

so that (6) may be expressed as the single constraint

X=

I0 0 · · · 0

A|00 0 · · · 0

0A|10· · · 0

.

.

..

.

........

.

.

0 0 · · · A|N−20

X+

0 0 · · · 0

0B|0· · · 0

.

.

..

.

.....

.

.

0 0 · · · B|N−2

U+

0

Σ|0

Σ|1

.

.

.

Σ|N−2

σ+

0

z|0

z|1

.

.

.

z|N−2

.(8)

B. First Order Hold

We now consider a discretization method that paramterizes the control input as a linear function between

sampling times. This method is known as ﬁrst order hold (FOH), and assumes that the control input takes

the form

u(t) = λ−

k(t)u(tk) + λ+

k(t)u(tk+1),∀t∈[tk, tk+1 ),(9a)

λ−

k(t):=tk+1 −t

tk+1 −tk

, λ+

k(t):=t−tk

tk+1 −tk

.

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Note that we maintain the same uniformly spaced temporal grid as in (4). In this case, we are selecting

piecewise linear basis functions that render the coeﬃcients of the control input’s representation dependent on

one another. In particular, both u(tk) and u(tk+1) are used as vector coeﬃcients for the basis function at the

kth sampling time, eﬀectively linking the coeﬃcients of adjacent sampling times. Moreover, for a temporal

grid with Npoints, this choice provides Ndegrees of freedom, since the value of the control at the ﬁnal

time will now impact the control in the preceding sample interval. This change is reﬂected by rewriting (5)

and (6) using the new deﬁnition (9a) to arrive at

x(k+1) =A|kx(k)+B−

|ku(k)+B+

|ku(k+1) + Σ|kσ+z|k, k = 0, . . . , N −2,(10)

where A|k,Σ|kand z|kare the same as in (7) and

B−

|k=A|kZtk+1

tk

Φ(ξ, tk)−1λ−

k(ξ)B(ξ)dξ, (11a)

B+

|k=A|kZtk+1

tk

Φ(ξ, tk)−1λ+

k(ξ)B(ξ)dξ. (11b)

Like the ZOH case, the discretized dynamics (10) can be concatenated to a single constraint by deﬁning

X:=

x(0)

x(1)

.

.

.

x(N−1)

∈RnN ,U:=

u(0)

u(1)

.

.

.

u(N−1)

∈RmN ,

and expressing (10) as

X=

I0 0 · · · 0

A|00 0 · · · 0

0A|10· · · 0

.

.

..

.

........

.

.

0 0 · · · A|N−20

X+

0 0 0 · · · 0

B−

|0B+

|00· · · 0

0B−

|1B+

|1· · · 0

.

.

..

.

........

.

.

0 0 · · · B−

|N−2B+

|N−2

U+

0

Σ|0

Σ|1

.

.

.

Σ|N−2

σ+

0

z|0

z|1

.

.

.

z|N−2

.(12)

C. Classical Runge-Kutta Method

The third method that approximates the control input directly is the classical Runge-Kutta fourth order

integration method (RK41).40 In this case, the state transition over each sample interval is given by

x(k+1) =x(k)+∆t

6(k|1+ 2k|2+ 2k|3+k|4), k = 0, . . . , N −2,(13)

where,

k|1=A(k)x(k)+B(k)u(k)+ Σ(k)σ+z(k),(14a)

k|2=A(k+1/2) x(k)+∆t

2k|1+B(k+1/2)u(k+1/2) + Σ(k+1/2) σ+z(k+1/2),(14b)

k|3=A(k+1/2) x(k)+∆t

2k|2+B(k+1/2)u(k+1/2) + Σ(k+1/2) σ+z(k+1/2),(14c)

k|4=A(k+1)(x(k)+ ∆tk|3) + B(k+1) u(k+1) + Σ(k+1)σ+z(k+1) .(14d)

We use the notation (k+1/2) =(tk+ ∆t/2) and ∆t:=tf/(N−1) in accordance with (4).

Remark II.2. If the LTV system (1) arises from a linearization of a nonlinear system, then the quantities

(ξ)correspond to the ﬁrst-order terms in a Taylor series expansion about some reference {¯

x(ξ),¯

u(ξ),¯σ}. In

particular, they are Jacobian matrices with respect to the state, control or parameters evaluated at the time

ξ. The primary consequence of this is that the quantities in (14b) and (14c) become diﬀerent. For example,

A(k+1/2) in (14b) becomes the Jacobian with respect to the state evaluated at ¯

x(k)+∆t¯

k|1/2, whereas A(k+1/2)

in (14c) must be evaluated at ¯

x(k)+ ∆t¯

k|2/2. The same holds for each of B(k+1/2),Σ(k+1/2) and z(k+1/2).

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To avoid increasing the input vector dimension to include the intermediate values u(k+1/2), we make a

simplifying assumption that the control may be linearly interpolated to obtain

u(k+1/2) :=u(k)+u(k+1)

2.(15)

This choice retains the property that for a temporal grid of Npoints, this discretization method yields N

degrees of freedom. This simpliﬁcation of the input signal eﬀectively makes the RK41 method identical

to FOH, except that the discretization is coarsened by removing the integrations (7a), (7c), (7d), (11a)

and (11b). By substituting the k|i,i= 1,...,4, into (13) and simplifying, we obtain the dynamical system

update equation

x(k+1) =A|kx(k)+B0

|ku(k)+B1

|ku(k+1) + Σ|kσ+z|k, k = 0, . . . , N −2,(16)

where A|k,B0

|k,B1

|k, Σ|kand z|kare linear combinations of the elements in (14), and are distinct from

those obtained for ZOH and FOH. The discretized dynamics (16) can be concatenated for implementation

by deﬁning

X:=

x(0)

x(1)

.

.

.

x(N−1)

∈RnN ,U:=

u(0)

u(1)

.

.

.

u(N−1)

∈RmN ,

and expressing (16) as the single constraint

X=

I0 0 · · · 0

A|00 0 · · · 0

0A|10· · · 0

.

.

..

.

........

.

.

0 0 · · · A|N−20

X+

0 0 0 · · · 0

B0

|0B1

|00· · · 0

0B0

|1B1

|1· · · 0

.

.

..

.

........

.

.

0 0 · · · B0

|N−2B1

|N−2

U+

0

Σ|0

Σ|1

.

.

.

Σ|N−2

σ+

0

z|0

z|1

.

.

.

z|N−2

.(17)

D. Global Pseudospectral Methods

Pseudospectral methods oﬀer an alternative way of discretizing (1) by parameterizing both the state and

control functions. These methods are therefore inherently diﬀerent from those we have discussed previously.

One of their primary advantages is an exponential convergence rate to the exact solution as the temporal

resolution is increased, assuming that the solution is smooth.24 In this section we describe various pseu-

dospectral ﬂavors, all of which are uniﬁed by a common theme of approximating the state function xin a

basis of Lagrange polynomials. In particular, we deﬁne D:={ηi∈[−1,1]}N−1

i=0 to be the set of discretization

nodes and refer to ηas the pseudo time. Note that it is standard among pseudospectral methods for the

nodes ηi∈ D to be deﬁned on the [−1,1] interval.23 The transformation between pseudo time and real time

is given by

t(η) = tf(η+ 1)

2.(18)

Pseudospectral discretization methods use the following approximation for the state and input signals

x(η) =

N−1

X

i=0

x(i)φi(η),u(η) =

N−1

X

i=0

u(i)φi(η),where φi(η):=

N−1

Y

j=0

j6=i

η−ηj

ηi−ηj

.(19)

where we employ the shorthand notation x(η):=x(t(η)) using (18). The function φiis known as a Lagrange

interpolating polynomial of degree N−1.

In all pseudospectral methods discussed here, η0=−1 and ηN−1= 1. Note that |D| =Nwhich stems

from our desire to temporally discretize (1) on a grid of Nnodes to remain consistent with (4). Owing to

the property that φi(ηj) = δij , we see that x(j)=x(ηj) are the coeﬃcients for these basis functions for

j= 0, . . . , N −1. In other words, the coeﬃcients x(j)gives directly the signal approximation at the temporal

node ηj.

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American Institute of Aeronautics and Astronautics

Let C ⊆ D be the set of collocation nodes such that |C| =M≤N. The collocation nodes are the temporal

values at which the diﬀerential equation (1) gets imposed. The core diﬀerence between all pseudospectral

methods described below is C, which may or may not contain the terminal node ηN−1and which shall diﬀer

in the distribution of the nodes on the [−1,1] interval. In each case, Cis chosen so as to avoid the Runge

phenomenon.41

Associated with any given Cand Dis a diﬀerentiation matrix D∈RM×Nsuch that Dji :=φ0

i(ηj). The

matrix Dimposes the diﬀerential equation in the ﬁnite-dimensional basis of Lagrange polynomials. In other

words

x0(ηj) =

N−1

X

i=0

x(i)φ0

i(ηj) =

N−1

X

i=0

Dji x(i),∀ηj∈ C,(20)

where we draw attention to the fact that diﬀerentiation is with respect to pseudo time η. Note that when

M < N ,Dis not a square matrix and this has important ramiﬁcations for both a method’s formulation

and performance. For example, when Dis square we cannot write an equivalent “integral” version of the

dynamics (see §2). In order to use (20) to approximate (1), the temporal transformation (18) has to be

applied. First, note that

x0=dx

dη =dx

dt

dt

dη =tf

2

dx

dt =tf

2˙

x.(21)

Next, substituting (21) into (20) we obtain

˙

x(ηj) = 2

tf

N−1

X

i=0

Dji x(i),∀ηj∈ C,(22)

In the following summary of pseudospectral methods, our main focus is on deﬁning the collocation node

set C. Given C, the corresponding diﬀerentiation matrix Dcan be eﬃciently computed to within machine

rounding error via barycentric Lagrange interpolation.42 Without loss of generality, we assume that the

optimal control problem under consideration has no integral in its cost function – it is of Mayer form.6As

such, we do not discuss quadrature methods associated to each method.

1. Chebyshev-Gauss-Lobatto

Deﬁne the Chebyshev polynomial of degree Nas

ςN: [−1,1] →R

η7→ cos(Narccos(η)).

The Chebyshev-Gauss-Lobatto (CGL) collocation node distribution deﬁnes η0, . . . , ηN−1as the roots of

(1 −η2) ˙ςN−1, sorted in increasing order. This admits an explicit formula23

ηj=−cos πj

N−1, j = 0, . . . , N −1.(23)

The diﬀerential equation (1) can now be discretized by imposing that it holds at each collocation node.

Substituting (22) into (1), we obtain

2

tf

N−1

X

i=0

Dji x(i)=A(j)x(j)+B(j)u(j)+ Σ(j)σ+z(j), j = 0, . . . , N −1.(24)

The discretized dynamics (24) can be concatenated by deﬁning

X:=

x(0)

x(1)

.

.

.

x(N−1)

∈RnN ,U:=

u(0)

u(1)

.

.

.

u(N−1)

∈RmN ,

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so that (24) can be expressed as a single constraint

2

tf

D00I· · · D0(N−1)I

.

.

.....

.

.

D(N−1)0I· · · D(N−1)(N−1)I

X=

A(0) · · · 0

.

.

.....

.

.

0· · · A(N−1)

X+

B(0) · · · 0

.

.

.....

.

.

0· · · B(N−1)

U

+

Σ(0)

.

.

.

Σ(N−1)

σ+

z(0)

.

.

.

z(N−1)

.(25)

2. Legendre-Gauss-Radau

Deﬁne the Legendre polynomial of degree Nas

`N: [−1,1] →R

η7→ 1

2NN!

dN

dηN[(η2−1)N].

The Legendre-Gauss-Radau (LGR) collocation node distribution deﬁnes η0, . . . , ηN−2as the roots of `N−2+

`N−1, sorted in increasing order. Note that this deﬁnes a set Cwhich is non-symmetric about the origin

and which does not contain the terminal node ηN−1= 1. Nevertheless, the terminal node is included as

a discretization node such that D=C ∪ {ηN−1}. In this case, the diﬀerentiation matrix D∈R(N−1)×N.

Because ηN−1= 1 ∈ D, it is still possible to include terminal constraints via x(N−1) in (19). Previous work

on rocket powered descent guidance has explored the ﬂipped LGR distribution, where the negative of the

roots is taken and thus the initial node η0=−1 is omitted.43 Since it is more convenient to have u(0)

rather than u(N−1) as a primal variable in an optimal control problem, we prefer to use the regular LGR

distribution.44 In our approach we make a simplifying zeroth order hold assumption to recover the terminal

input, i.e., we assume u(N−1) =u(N−2). The diﬀerential equation (1) can now be discretized similarly to

the CGL case by using (22)

2

tf

N−1

X

i=0

Dji x(i)=A(j)x(j)+B(j)u(j)+ Σ(j)σ(j)+z(j), j = 0, . . . , N −2,(26)

and we highlight that (1) is not imposed at ηN−1, since it is not a collocation node. The discretized

dynamics (26) can be concatenated for implementation by deﬁning

X:=

x(0)

x(1)

.

.

.

x(N−1)

∈RnN ,U:=

u(0)

u(1)

.

.

.

u(N−2)

∈Rm(N−1),

so that (26) can be expressed as a single constraint in diﬀerential form

2

tf

D00I· · · D0(N−1) I

.

.

.....

.

.

D(N−2)0I· · · D(N−2)(N−1)I

X=

A(0) · · · 0

.

.

.....

.

.

0· · · A(N−2)

X0:N−2+

B(0) · · · 0

.

.

.....

.

.

0· · · B(N−2)

U

+

Σ(0)

.

.

.

Σ(N−2)

σ+

z(0)

.

.

.

z(N−2)

.(27)

Previous research has shown that there may be computational advantages, such as more consistent results,

to an alternative but mathematically equivalent integral form of (27).45 In fact, it has been shown that the

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submatrix D1:N−1obtained by removing the ﬁrst column of Dis full rank and −D−1

1:ND0=1.24 This allows

us to rewrite (27) in an integral form, which is henceforth refered to as LGR-integral (LGRI), and imposes

the discrete dynamics as

X1:N−1=

x(0)

.

.

.

x(0)

+tf

2

D01I· · · D0(N−1) I

.

.

.....

.

.

D(N−2)0I· · · D(N−2)(N−1)I

−1

A(0) · · · 0

.

.

.....

.

.

0· · · A(N−2)

X0:N−2+

B(0) · · · 0

.

.

.....

.

.

0· · · B(N−2)

U+

z(0)

.

.

.

z(N−2)

.

(28)

III. Monte Carlo Analysis

In this section we present the particular optimal control problem and method by which we will compare the

discretization methods presented in §II. This is a 6-DoF powered descent guidance problem and is outlined

in §A. In §B, we describe the Monte Carlo methodology by which the discretization methods are compared.

A. 6-DoF Powered Descent Guidance

The discretization schemes are tested on a 6-DoF powered descent guidance problem. The exact formulation

that we consider is presented as the “baseline problem” in [2, Problem 1]. By expressing the equations of

motion using dual quaternions, this problem models a rigid rocket-powered vehicle with variable mass and

inertia whose objective is to land at a target location using minimum fuel. The sole actuator is a bottom-

mounted rocket engine that can be gimbaled symmetrically about two axes (up to a maximum angle) and

can produce rate-limited thrust with non-zero lower and upper bounds. The following state constraints are

imposed on the vehicle’s trajectories: the vehicle’s vertical axis remains within a maximum tilt angle from

the inertial vertical direction; the position remains inside a glide-slope cone eminating from the landing site;

and the slew rate is bounded. This setup provides a rich and complex optimal control problem to serve as a

reasonable test of the eﬀectiveness of each discretization scheme. While the interested reader may consult2

for the precise problem formulation, we state the problem in a generic form:

min

x(·),u(·),tf

−J(x(tf)),(29a)

subject to ˙

x(t) = f(x(t),u(t)),(29b)

gx(x(t)) ≤0,(29c)

gu(x(t),u(t)) ≤0,(29d)

b0(x(0),u(0)) = 0,(29e)

bf(x(tf),u(tf)) = 0,(29f)

where J:Rn→Rreturns the ﬁnal spacecraft mass, gx:Rn→Rnxcaptures the pure state constraints,

gu:R→Rnucaptures the mixed state and input constraints and b0:R→Rn0,bf:R→Rnfhandle the

set of initial and ﬁnal boundary conditions. In order to handle the fact that [2, Problem 1] is a free ﬁnal

time problem, we introduce the following variable change

t=στ, (30)

where τ∈[0,1] is the scaled time and σis a time dilation factor deﬁned so that t∈[0, tf] under the

mapping (30). An application of the chain rule yields

◦

x=dx

dτ =dx

dt

dt

dτ=σ˙

x,(31)

and as a result the nonlinear dynamics (29b) may be written in scaled time as

◦

x(τ) = σf (x(τ),u(τ)).(32)

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In order to apply the discretization schemes of §II, (32) must be linearized in order to obtain the form of (1).

As explained in,1, 2 an iterative method is used to solve the optimal control problem. Each iteration generates

a state trajectory ¯

x(τ), control trajectory ¯

u(τ) and time dilation factor ¯σby solving an approximation of

(29). These quantities are then used to obtain the ﬁrst order Taylor series expansion of (32) for any τ∈[0,1]

as

◦

x(τ)≈¯σf(¯

x(τ),¯

u(τ)) + ¯σ∂f

∂x

¯

x,¯

u

(x(τ)−¯

x(τ)) + ¯σ∂f

∂u

¯

x,¯

u

(u(τ)−¯

u(τ)) + f(¯

x(τ),¯

u(τ))(σ−¯σ),

= ¯σ∂f

∂x

¯

x,¯

u

x(τ) + ¯σ∂f

∂u

¯

x,¯

u

u(τ) + f(¯

x(τ),¯

u(τ))σ−¯σ∂f

∂x

¯

x,¯

u

¯

x(τ) + ¯σ∂f

∂u

¯

x,¯

u

¯

u(τ),

=A(τ)x(τ) + B(τ)u(τ) + Σ(τ)σ+z(τ),(33)

where,

A(τ) = ¯σ∂f

∂x

¯

x,¯

u

, B(τ) = ¯σ∂f

∂u

¯

x,¯

u

,Σ(τ) = f(¯

x(τ),¯

u(τ)),

z(τ) = −¯σ∂f

∂x

¯

x,¯

u

¯

x(τ)−¯σ∂f

∂u

¯

x,¯

u

¯

u(τ).

We note that (33) now has the same form as (1) and can be used directly in the discretization schemes

outlined in §II by simply replacing twith τ.

The remaining state and input constraints of [2, Problem 1] are imposed point-wise at the temporal dis-

cretization nodes. The optimal control problem is thus approximated as a ﬁnite-dimensional parameter

optimization problem over x(k),u(k)and σthat can be solved using a numerical optimization algorithm.

For example if ZOH discretization from §II.A is used then (29) becomes:

min

x(k),u(k),σ −J(x(N−1)),(34a)

subject to: x(k+1) =A|kx(k)+B|ku(k)+ Σ|kσ+z|k, k = 0, .. . , N −2,(34b)

gx(x(k))≤0, k = 0, . . . , N −1,(34c)

gu(x(k),u(k))≤0, k = 0, . . . , N −2,(34d)

b0(x(0),u(0) )=0,(34e)

bf(x(N−1),u(N−2) )=0.(34f)

Other discretization methods yield similar ﬁnite-dimensional optimization problems, with the only diﬀerence

being that (34b) is replaced by the appropriate discretized dynamics formulation for the chosen method.

Problem (34) is then solved iteratively until convergence is achieved.1Each iteration solves a convex second

order cone program due to the linear equality constraints and (most generally) second order cone inequality

constraints. As mentioned previously, this strategy is a form of successive convexiﬁcation (SC), which we

now note uses soft quadratic trust regions and virtual control to guide the convergence process. We use

the SDPT3-4.0 solver46 to solve each of these optimization problems in our implementation. Each solution

produces the discrete state ¯

x(τk), input ¯

u(τk), and time dilation factor ¯σfor each scaled time grid node,

where tk= ¯στk. We can recover the continuous time input trajectory ¯

u(τ) over each sampling interval

τ∈[τk, τk+1) via (3), (9a) or (19), depending on the chosen discretization method. The continuous time

state trajectory on this interval can then be recovered by integrating the nonlinear dynamics

¯

x(τ) = ¯

x(τk) + Zτ

τk

¯σf(¯

x(ξ),¯

u(ξ))dξ, τ ∈[τk, τk+1 ), k = 0, . . . , N −1,(35)

which we do in practice via RK41 on an Nsub-step uniform grid of the [τk, τk+1] interval, where Nsub = 15.

The continuous time state and control and the time dilation factor are used to compute (33) at the next

iteration, as illustrated in Figure 1. We have found that resetting the initial condition of the integration

in (35) to ¯

x(τk) for each k= 0, . . . , N −1 yields superior convergence properties for the iterative algorithm

described in.1, 2 While this creates a discontinuous state trajectory with which the LTV system (33) is

computed, it can provide a more accurate, robust and faster solution than continuing from ﬁnal integrated

state of the previous interval. Per [1, Section III.A.3], this strategy is analogous to the well-known multiple

shooting technique. The improved convergence properties can thus be intuitively (though not rigorously)

compared to the improved performace obtained when using multiple shooting versus direct shooting.4

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Figure 1. Illustration of the integration procedure. The right-hand ﬁgure shows (35) and uses the states from

the previous successive convexiﬁcation iteration to initialize the integration over each time interval.

B. Monte Carlo Method

Using each of the discretization methods presented in §II, our goal is to obtain a set of statistics that

measure their relative performance. By controlling for the optimization solver, algorithm weights, constraint

boundaries, and terminal conditions, the performance of each discretization method depends solely on

•The resolution of the temporal grid, N;

•The initial condition, x(0).

Our approach to Monte Carlo simulation is thus to test each discretization method over a range of initial

conditions and values of N. To be consistent, the same set of initial conditions will be tested for each

discretization method and value of N. To compute the initial condition of the ith Monte Carlo trial, we ﬁrst

sample the inertial velocity from a uniform distribution according to

vi(0) ∼

U(−10,10)

U(−10,10)

U(−10,0)

m/s,(36)

where the ﬁrst and second components are in the horizontal plane and the third component is the vertical

direction (planetary curvature is neglected). The inertial position ri(0) is then generated via hit-and-run

sampling47 of the constrained controllability (CC) polytope48–50 generated for the simpliﬁed 3-DoF problem13

corresponding to [2, Problem 1]. This random initialization ensures that an initial guess is generated so that

we know the 3-DoF problem is feasible from this initial condition. The initial mass and angular velocity

are ﬁxed, while the initial attitude quaternion is an optimization variable. The optimal 3-DoF trajectory is

used as the initial guess for the algorithm, i.e., to generate the ﬁrst reference trajectory ¯

x(τ), ¯

u(τ) and ¯σfor

τ∈[0,1] that are used in (33) to generate the approximate LTV system.1The 6-DoF problem, however, is

not guaranteed to be feasible and the algorithm may fail either due to being fundamentally infeasible for this

initial condition or simply failing to converge. In such cases, the problem is marked as “failed” in our Monte

Carlo data. Note that failure can generally occur for one discretization method but not another for the same

initial condition. This Monte Carlo approach is illustrated in Figure 2 and is formalized as Algorithm 1.

IV. Results

This section analyzes and compares discretization method performance using the Monte Carlo data generated

by Algorithm 1. We use R= 200 initial conditions which are shown in Figure 3. One can see that while

increasing Nfrom 20 to 30 can have a large eﬀect on the trajectory using {ZOH, FOH, RK41}, pseudospectral

trajectories are nearly identically for all N. This echoes the conjecture that pseudospectral methods require

a smaller Nfor a given level of accuracy.25 We also note that FOH and the pseudospectral methods give

perfect success rates while ZOH fails sometimes and RK41 fails many times.

While Figure 3 visually conﬁrms that the trajectories generated by each discretization method appear to be

similar, we proceed to quantify this similarity. Let r(k)and v(k)be the position and velocity at time tkfor

a “baseline” discretization method. Similarly deﬁne ˜

r(k)and ˜

v(k)for a “comparison” discretization method.

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Figure 2. Illustration of our Monte Carlo procedure where each combination of discretization method and

temporal resolution Nis tested for a randomly generated initial condition sampled from a CC polytope for a

simpliﬁed 3-DoF problem. Two such initial conditions are shown.

Algorithm 1 Monte Carlo data collection for discretization method performance analysis.

1: for i= 1, . . . , R do

2: for discretization method = ZOH,FOH,RK4,CGL,LGR,LGRI do

3: for N= 20,30,40 do

4: Choose vi(0) by sampling (36)

5: Generate CC polytope50 for this v(0)

6: Choose ri(0) via hit-and-run sampling47 of the CC polytope

7: Solve problem 29 for this initial condition, discretization method and N

8: if failed to converge then

9: Save trial as failed

10: else

11: Save trial data

12: end if

13: end for

14: end for

15: end for

Deﬁne position and velocity discrepancy measures:

∆r:= max

k=0,...,N−1kr(k)−˜

r(k)k2,(37)

∆v:= max

k=0,...,N−1kv(k)−˜

v(k)k2.(38)

For a Monte Carlo simulation consisting of Rtrials, Rsuch measures are generated corresponding to a given

initial condition for each trial, excluding initial conditions where either of the discretization methods failed to

converge. Figure 4 plots the median, minimum and maximum ∆rand ∆vusing ZOH and CGL as baseline

methods (using more baseline methods is redundant since the discrepancy relationship is transitive). We

note two important characteristics:

1. The {ZOH, FOH, RK41}and {CGL, LGR, LGRI}method groups converge to distinct solutions of the

problem, since position and velocity discrepancies do not decrease between these groups for larger N;

2. The {ZOH, FOH, RK41}group appears to converge around the same solution as Nincreases, but has

much more variability in the discrepancy measures (up to 60 m and 9.5 m/s) than the {CGL, LGR,

LGRI}group (up to 16 m and 1.5 m/s) for N= 20.

These results suggest that pseudospectral methods provide more consistent trajectories that are less sensitive

to the discretization resolution N.

The next important quantity to consider is the inﬂuence of each discretization method on the solution time.

Because SDPT3-4.0 is a relatively slow solver, we look at the solution times as indicative of the relative

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(a) Success rate {99,99,99}%. (b) Success rate {100,100,100}%. (c) Success rate {73,98,95}%.

(d) Success rate {100,100,100}%. (e) Success rate {100,100,100}%. (f ) Success rate {100,100,100}%.

Figure 3. Initial condition cloud. Blue markers show initial conditions where successive convexiﬁcation failed to

converge. Light, medium and dark gray lines show the generated trajectories for N= 20,30 and 40 respectively

and subﬁgure captions give the corresponding success rates.

Figure 4. ∆rand ∆vstatistics. Bars show the median and error bars show the minimum and maximum

diﬀerence.

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(a) SC global iteration count. (b) Solver iteration count per SC iteration.

(c) Overall problem solution time. (d) Solution time per SC iteration.

Figure 5. Successive convexiﬁcation iteration and solution time statistics. Bars show the median and error

bars show the minimum and maximum values.

speed of each discretization method for a future real-time implementation. Figure 5 shows the SC iteration

and solution time statistics. From Figure 5a, we see that the group of pseudospectral methods generally

converges in fewer iterations and with less variance than the {ZOH, FOH, RK41}group. We set the SC

method to timeout at 20 iterations and indeed the RK41 method is the worst performing by this measure

as it sometimes requires all 20 iterations. As shown in Figure 5b, however, the median number of iterations

that SDPT3-4.0 takes to solve each SC sub-problem is relatively constant across all methods with a slight

increase for larger N. The most important quantity, however, is the overall time taken to solve the problem.

As shown in Figure 5c, for N= 20 all methods take a median of 12-16s to solve the problem. According to

Figure 5d, for N= 20, SDPT3-4.0 takes a median of 3 s to solve each SC subproblem using {ZOH, FOH,

RK41}and a median of 4-6 s for {CGL, LGR, LGRI}. However, this is oﬀset by the fewer SC iterations

required, yielding an overall similar SC solution time for all methods. Indeed, for N= 20 the LGR method

is the best performing as its SC total solution time’s median and maximum values are both small. The

situation changes for N= 30 and 40, however, as the SDPT3-4.0 solution time for the larger SC sub-

problems increases much more for {CGL, LGR, LGRI}than for {ZOH, FOH, RK41}. This indicates that

the overall solution time is more sensitive to Nfor pseudospectral methods.

The increase in solution time for pseudospectral methods may be explained by a loss of sparsity in the

resulting ﬁnite dimensional optimization problem. This is caused by the diﬀerentiation matrix Din (20)

which eﬀectively couples the states and controls across all temporal discretization nodes. This is not the case

for the {ZOH, FOH, RK41}methods, where only the neighboring states and controls are explicitly coupled.

Generally speaking, a loss of sparsity will slow down any optimization problem. In the particular case of

SDPT3-4.0 this is manifested through the sparsity of a Schur complement matrix Mwhich is involved in a key

computationally expensive step of ﬁnding the search direction in a primal-dual interior-point algorithm [51,

Section 2.1]. The authors note that factorization of Mbecomes very expensive when Mis dense. Figure 6

conﬁrms that the density of Mcan increase up to 3 times for pseudospectral discretization compared to

{ZOH, FOH, RK41}. Furthermore, Figure 7 shows that the Mmatrix can be several orders of magnitude

worse conditioned for pseudospectral methods. We believe that these poor numerical properties of the M

matrix are a key contributor to the observed slowdown in solver time.

Having considered the solution time, it is equally important to consider the accuracy and performance of

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(a) 4.89% non-zero elements. (b) 5.66% non-zero elements. (c) 5.66% non-zero elements.

(d) 14.62% non-zero elements. (e) 14.24% non-zero elements. (f ) 16.89% non-zero elements.

Figure 6. Sparsity patterns of the SDPT3-4.0 Schur complement matrix Musing N= 20. Similar patterns

hold for diﬀerent Nand initial conditions.

the trajectories generated by each discretization method. We measure accuracy by the ﬁnal position and

velocity errors obtained by propagating the nonlinear dynamics using the generated thrust proﬁle. That

is, we numerically integrate (29b) using the interpolated continuous thrust commands from the converged

solution. The integration is carried out over the interval [0, tf] in one shot. We stress that only the thrust

is used for this process, not the state vectors obtained as part of the optimization. The performance is

measured by fuel consumption. As illustrated in Figures 8a and 8b, FOH achieves approximately the same

level of accuracy as the pseudospectral methods. The RK41 method is the least accurate, since it mimics

the FOH input proﬁle, but does not include the ﬁne integration steps (11a) and (11b). As a result, RK41

does not suﬃciently capture the nonlinear dynamics between sample points. In terms of performance, all

methods achieve a similar fuel consumption.

An ideal discretization method yields both a small SC total solution time and small ﬁnal position and velocity

errors. The scatter plots in Figure 9 visualize how close each discretization method comes to this ideal, with

better methods located in the left hand corner. Because all methods generate less than about 1 m/s velocity

error (see Figure 8b), we consider only the position error in Figure 9. We see that for N= 20, {FOH, CGL,

LGR}are the best performing, with FOH matching the accuracy of CGL while LGR is the most consistently

faster method (it has less variance in the SC total solution time). The LGRI method is interestingly not

particularly accurate as was originally motivated in §II. Due to the poor Mmatrix conditioning in Figure 7,

this could be a result of solver numerical error (i.e. our implementation) rather than a fault of the LGRI

method itself. As we have already seen, ZOH and RK41 underperform. In particular, it is interesting to

note that the RK41 method is the worst in SC total solution time, this being due to the large number of

SC iterations that it sometimes requires to converge. For N= 30 and 40, the solution time increases for

pseudospectral methods much more than it does for {ZOH, FOH, RK41}. As a result, for higher Nthe

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Figure 7. Condition number of the SDPT3-4.0 Schur complement matrix M. Bars show the median and error

bars show the minimum and maximum values.

(a) Final position error. (b) Final velocity error. (c) Total fuel consumption.

Figure 8. Trajectory accuracy and performance statistics. Bars show the median and error bars show the

minimum and maximum values.

FOH method is the best performing.

An ideal discretization method also ensures that the state and input constraints are not violated between

temporal discretization nodes. In this context, one can check whether the thrust magnitude constraints are

satisﬁed and is illustrated in Figure 10. One can see that, by design, the {ZOH, FOH, RK41}methods

satisfy the thrust magnitude constraints at all times. The pseudospectral method, however, tends to violate

the constraints whenever the thrust magnitude is near one of the boundaries. This is due to the interpolated

polynomial nature of the input. Despite this fact, pseudospectral methods seem to produce input proﬁles

that are more similar across Nthan the {ZOH, FOH, RK41}methods, where it is seen that in some cases

the input changes drastically as Nincreases. The ZOH and FOH cases each produce input proﬁles that are

relatively consistent across N, but perhaps less so than each pseudospectral method. It must also be noted

that each pseudospectral method yields a small number of solutions where the burn time greatly exceeds

the average. Interestingly, since FOH achieved a 100% success rate, we know that it solved these same exact

problems but found a much shorter burn time (we cannot make the same deﬁnitive claim for ZOH and RK41

as they did fail some trials). The cause of this phenomenon is diﬃcult to pinpoint, and thus raises some

concern as the diﬀerence of roughly 20 s burn time is considerable.

V. Discussion

This section summarizes the ﬁndings of Section IV. Table 1 overviews the beneﬁts and drawbacks of each

group of methods (i.e., those that parameterize control only versus those that parameterize state and control).

FOH with N= 20 appears to be the best method that parameterizes the control as it has the lowest overall

solution time in that group, and is accurate for each N. Of the three methods that parameterize state and

control, LGR with N= 20 is preferred as it has the lowest overall solution time and is about as accurate as

CGL and LGRI. We recall that LGR and FOH appear to ﬁnd slightly diﬀerent solutions, since increasing N

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Figure 9. Final landing position error versus SC overall solution time. The spread generated by the random

initial conditions for each discretization method is covered with its convex hull polytope for visualization

purposes.

does not bring the trajectories produced by the two methods closer (see Figure 4). Despite LGR matching

the performance of FOH for N= 20, polynomial interpolation of the input induces inter-sample constraint

violation as illustrated in Figure 10. The result is that further constraints would need to be added to

the pseudospectrally-discretized optimal control problem in order to (conservatively) prevent inter-sample

constraint violation. This extra complexity would, however, likely increase the solution time. Our key

conclusion from the present analysis, therefore, is that:

•FOH yields an accurate and fast solution with no convex input constraint violation, but has the

disadvantage that the state and input trajectories can vary with N;

•LGR performance matches that of FOH for small N, produces state and input trajectories that are

more consistent across N, but has the disadvantage of inter-sample input constraint violation and less

sparse optimization problems.

VI. Conclusions

In this paper, we analyzed and compared the performance and accuracy of six discretization methods in the

context of solving a powered descent guidance problem via direct optimization. These six methods included

three that parameterized the control function only, and three that parameterized both the state and control

functions (using global pseudospectral methods). A Monte Carlo statistical analysis indicated that the

performance and accuracy of both groups matches for discretizations with a low temporal resolution, but the

pseudospectral methods become slower for higher temporal resolutions. It was concluded that the ﬁrst order

hold discretization and LGR pseudospectral method provide the fastest computational times and achieved

similar performance. While the ﬁrst order hold method guarantees inter-sample convex input constraint

satisfaction, it showed sensitivity to the discretization’s temporal resolution. Conversely, the LGR method

produces trajectories that are less sensitive to the temporal resolution, however, inter-sample constraint

violation was consistently observed and the optimization problems were noted to be much less sparse. Each

characteristic is important for a real-time implementable algorithm designed to solve 6-DoF powered descent

guidance problems. Our future work will provide discussion of the implementation of these methods in

concert with a real-time capable optimization solver.

Acknowledgements

This research has been supported by NASA grant NNX17AH02A. Government sponsorship acknowledged.

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Figure 10. Thrust proﬁles with upper and lower bounds shown as dashed black lines. Top to bottom: ZOH,

FOH, RK41, CGL, LGR and LGRI. Left to right: N= 20,30 and 40.

{ZOH, FOH, RK41} {CGL, LGR, LGRI}

+ Higher sparsity in the discretized optimization prob-

lem (Figure 6);

+ Faster solver solution times, in particular for larger

N;

+ FOH provides a terminal position and velocity accu-

racy on par with pseudospectral (Figures 8a and 8b);

−Potentially large changes in trajectory across N(Fig-

ure 3);

−Potentially large changes in input trajectory across N

(Figure 10);

−Generally more SC iterations required (Figure 5a).

+ More consistent trajectories across N(Figure 3);

+ More consistent optimal input trajectories across N

(Figure 10);

+ Fewer SC iterations (Figure 5a);

+ Similar performance to FOH for N= 20;

−Longer solver solution time per SC iteration (Fig-

ure 5d);

−Higher sensitivity of SC solution time to N(Figures 5c

and 5d);

−Lower sparsity in the discretized optimization problem

(Figure 6);

−Input constraint violation due to polynomial intepro-

lation (Figure 10).

Table 1. Beneﬁts and drawbacks of the discretization methods.

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