Aircraft Dynamic Navigation
for
Unmanned Aerial Vehicles

A dissertation submitted in partial fulfilment
for the degree of Doctor of Philosophy

Lennon Cork
B.Eng Aerospace Avionics
Queensland University of Technology

Australian Research Centre for Aerospace Automation
School of Robotics and Aerospace Systems
Science and Engineering Faculty
Queensland University of Technology
Brisbane, Australia

May 5, 2014


Aircraft Dynamic Navigation for Unmanned Aerial Vehicles

Keywords: Unmanned Aircraft Systems (UAS), Unscented Kalman Filtering
(UKF), Inertial Navigation Systems (INS), Global Positioning System (GPS),
GPS/INS Integration, Aircraft Dynamic Modelling (ADM), Aircraft Dynamic
Navigation (ADN)

c Copyright 2014 by Lennon R. Cork
All Rights Reserved

To Leonard and Shirley Stevens



QUT Verified Signature



Abstract

Aircraft navigation is a well established field which has seen considerable research and development over the last 50 years. Of particular significance is the
evolution of the low-cost, strapdown, Inertial Navigation System (INS), and their
integration with the Global Positioning System (GPS). Combined, the two systems provide both an accurate and consistent estimate of the aircraft’s position,
attitude and velocity; and for this reason GPS/INS navigation play a significant
role in the development of Unmanned Aerial Vehicles (UAVs).
This thesis describes the investigation of an Aircraft Dynamic Navigation
(ADN) approach, which incorporates an Aircraft Dynamic Model (ADM) directly
into the navigation filter of a fixed-wing aircraft of UAV. The ADM provides the
filter with information on the forces and moments acting on the aircraft as a
result of the control surface actuation, and enables the filter to directly predict
the system’s inertial sensor measurements. The result is a dynamic, model aiding
approach, that offers performance improvements over the standard GPS/INS
solution.
ADMs have been applied to the navigation problem in the past, however
each example presented in the literature has used simplified models and limited
the scope to specific regions of the aircraft’s dynamics. This is acceptable when
the application is a runway approach or autonomous take-off, but neglects the
complex interactions that occur in-between these stable modes and over the
longer term. This research investigates the ADN approach in a broader context


vii


viii

Abstract

than has been done in the past, in particular it covers multiple phases of flight
and presents a robust analysis of filtering performance.
The primary contribution of this research is the formulation of a directly
aided, loosely coupled, Unscented Kalman Filter (UKF) which incorporates a
complex, non-linear, laterally and longitudinally coupled, ADM. The use of the
UKF provides an opportunity to integrate the ADM without the need for model
linearisation that would otherwise be required by the Extended Kalman Filter
(EKF). A significant constraint applied was the use of a sensor suite representing
a typical UAV, consisting of a Global Positioning System (GPS) receiver and
Inertial Measurement Unit (IMU), with the addition of an Electronic Compass
(EC), and Air Data (AD) Pitot Static System.
In order to investigate the proposed approach, a detailed Monte-Carlo simulation environment was implemented. This allowed for a large spectrum of the
aircraft’s dynamics to be reviewed in the context of the navigation problem. This
is important as the dynamic model introduces additional sources of error, which
may only become evident during specific modes of flight. 100 random missions
were simulated using this environment each with a one hour duration, totalling
36 × 106 data-points against which the ADN approach has been assessed.
The results demonstrated an 80% improvement, i.e. reduction in estimate
error, in the system’s attitude estimate, a 75% improvement in the velocity estimate, and a 50% improvement in the angular rate estimate compared to the
standard GPS/INS approach. The bias estimates of the accelerometers and rate
gyros are improved by 30%, and the navigation filter was used to provide direct
estimates of the aircraft’s rotational rates, accelerations and angular accelerations. In addition, the filter was able to maintain an estimate of the background
wind within 1m/s of the true value. This was demonstrated to be 80% better than using the aircraft’s cross-track error, and in addition did not need to

perform intentional wind-finding manoeuvres.


Acknowledgments

I would like to thank my principle supervisor, the late Professor Rodney Walker,
who passed in 2011 and was unable to see the completion of this research. Rod
was continually patient and encouraging throughout the course of this extended
candidate, and this work owes a lot to his enthusiasm and dedication. As the
founding Director of the Australian Research Centre for Aerospace Automation
(ARCAA), Rod was a driving force in the Australian UAS industry, he made a
large number friends, and will be missed.
This research also owes thanks to Professor Duncan Campbell for taking over
Rod’s supervisory role and for the years where he acted as associate supervisor,
and to Dr Luis Mejas for taking on the role of associate supervisor. Special
thanks to Daniel Fitzgerald, Damien Dusha, Duncan Greer, Reece Clothier and
the rest of my colleagues at the ARCAA; to Torsten Merz and the Autonomous
Systems Laboratory (ASL) at the Commonwealth Science and Industry Research
Organisation (CSIRO); and the Engineering team at Insitu Pacific. To my friends
and family, and most importantly to Christina who provided me with more
support an encouragement than all others combined.
This research was supported by a QUT Postgraduate Research Award and
BEE top-up scholarships, and was originally established under the Cooperative
Research Centre for Satellite Systems (CRCSS). Computational (and/or data
visualisation) resources and services used in this work were provided by the
QUT, High Performance Computing (HPC) and Research Support Group.

ix




Contents

Statement of Authorship

v

Abstract

vii

Acknowledgments

ix

Table of Contents

xv

List of Figures

xix

List of Tables

xxii

Glossary

xxiii


Nomenclature

xxv

1. Introduction

1

1.1

Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Aircraft Dynamic Navigation

. . . . . . . . . . . . . . . . . . . .

4

1.3

Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4


Contributions and Publications . . . . . . . . . . . . . . . . . . .

9

1.5

Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . 12

2. Literature Review
2.1

15

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

xi


xii

Contents
2.2

Inertial Navigation System (INS) . . . . . . . . . . . . . . . . . . 16

2.3

Global Positioning System (GPS) . . . . . . . . . . . . . . . . . . 20


2.4

GPS/INS Integration Architectures . . . . . . . . . . . . . . . . . 23

2.5

GPS/INS Filtering Algorithms . . . . . . . . . . . . . . . . . . . . 25

2.6

Additional Aiding Sensors . . . . . . . . . . . . . . . . . . . . . . 29

2.7

Aircraft Dynamic Navigation
2.7.1

2.8

. . . . . . . . . . . . . . . . . . . . 31

Model Selection . . . . . . . . . . . . . . . . . . . . . . . . 36

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3. Aircraft Dynamic Modelling

39

3.1


Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2

Reference Frames and Coordinate Systems . . . . . . . . . . . . . 41

3.3

3.4

3.5

3.2.1

Reference Ellipsoid (WGS-84) . . . . . . . . . . . . . . . . 41

3.2.2

Earth-Centred Inertial (ECI) Frame . . . . . . . . . . . . . 42

3.2.3

Earth-Centred Earth-Fixed (ECEF) Frame . . . . . . . . . 43

3.2.4

Body-Fixed Navigation (NED) Frame . . . . . . . . . . . . 45

3.2.5


Body-Fixed (Body) Frame . . . . . . . . . . . . . . . . . . 47

3.2.6

Sensor-Fixed (Sensor) Reference Frame . . . . . . . . . . . 50

Dynamic and Kinematic Motion . . . . . . . . . . . . . . . . . . . 50
3.3.1

Rotational Motion . . . . . . . . . . . . . . . . . . . . . . 50

3.3.2

Translational Motion . . . . . . . . . . . . . . . . . . . . . 54

Atmosphere and Gravitation . . . . . . . . . . . . . . . . . . . . . 56
3.4.1

Standard Atmosphere (ISA-75) . . . . . . . . . . . . . . . 56

3.4.2

Dynamic Atmosphere (Wind and Turbulance) . . . . . . . 59

3.4.3

Gravitation (WGS-84) . . . . . . . . . . . . . . . . . . . . 61

Aerodynamics and Propulsion . . . . . . . . . . . . . . . . . . . . 62

3.5.1

Mass and Inertia . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.2

Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.3

Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.4

Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


Contents
3.5.5

xiii
Combined Forces and Moments . . . . . . . . . . . . . . . 72

3.6

Aircraft Dynamic Model (ADM) . . . . . . . . . . . . . . . . . . . 73

3.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


4. Aircraft Dynamic Navigation

77

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2

Sensor Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3

4.4

4.5

4.6

4.7

4.2.1

Global Positioning System (GPS) . . . . . . . . . . . . . . 78

4.2.2

Inertial Measurement Unit (IMU) . . . . . . . . . . . . . . 79


4.2.3

Electronic Compass (EC)

4.2.4

Air-Data (AD) . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.5

Measurement Uncertainty . . . . . . . . . . . . . . . . . . 83

4.2.6

Calibration and Alignment . . . . . . . . . . . . . . . . . . 84

. . . . . . . . . . . . . . . . . . 81

The State Estimation Problem . . . . . . . . . . . . . . . . . . . . 87
4.3.1

Bayesian Filter (BF) . . . . . . . . . . . . . . . . . . . . . 89

4.3.2

Unscented Kalman Filter (UKF) . . . . . . . . . . . . . . 90

4.3.3


Consistency and Tuning . . . . . . . . . . . . . . . . . . . 94

Inertial Navigation (IN) . . . . . . . . . . . . . . . . . . . . . . . 95
4.4.1

Process Model . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.2

Observation Model . . . . . . . . . . . . . . . . . . . . . . 98

Air-Data Inertial Navigation (AIN) . . . . . . . . . . . . . . . . . 99
4.5.1

Process Model . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.5.2

Observation Model . . . . . . . . . . . . . . . . . . . . . . 102

4.5.3

Wind Tracking . . . . . . . . . . . . . . . . . . . . . . . . 103

Aircraft Dynamic Navigation (ADN) . . . . . . . . . . . . . . . . 104
4.6.1

Process Model . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6.2


Observation Model . . . . . . . . . . . . . . . . . . . . . . 108

4.6.3

Sensor Update . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6.4

Processing Overhead . . . . . . . . . . . . . . . . . . . . . 110

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111


xiv

Contents

5. Results and Analysis

113

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2

Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . 114


5.3

5.2.1

Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2.2

Wind and Atmosphere . . . . . . . . . . . . . . . . . . . . 117

5.2.3

Monte-Carlo Simulations . . . . . . . . . . . . . . . . . . . 120

Filtering Performance . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.1

Navigation Estimates . . . . . . . . . . . . . . . . . . . . . 122

5.3.2

Bias Estimates . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.3

Wind Estimates . . . . . . . . . . . . . . . . . . . . . . . . 131

5.4

Dynamic Performance . . . . . . . . . . . . . . . . . . . . . . . . 133


5.5

Coasting Performance . . . . . . . . . . . . . . . . . . . . . . . . 137

5.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.6.1

Inertial Navigation (IN) . . . . . . . . . . . . . . . . . . . 139

5.6.2

Air-Data Inertial Navigation (AIN) . . . . . . . . . . . . . 139

5.6.3

Aircraft Dynamic Navigation (ADN) . . . . . . . . . . . . 140

6. Conclusion

143

6.1

Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2


Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.3

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4

Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.5

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

A. Sensitivity Analysis

153

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.2 Coefficient Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.3 Coefficient Variation . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B. Extended Results

159


Contents

xv


B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2 Inertial Navigation (IN) . . . . . . . . . . . . . . . . . . . . . . . 160
B.3 Air-Data Inertial Navigation (AIN) . . . . . . . . . . . . . . . . . 167
B.4 Aircraft Dynamic Navigation (ADN) . . . . . . . . . . . . . . . . 176
B.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Bibliography

193



List of Figures

1.1-1

Smart Skies manned and unmanned aircraft . . . . . . . . . . . .

2.2-1

Inertial Navigation System (INS) architecture . . . . . . . . . . . 17

2.3-2

Global Position System (GPS) receiver architecture . . . . . . . . 21

2.4-3

Loosely coupled, direct aiding GPS/INS integration . . . . . . . . 24


2.4-4

Loosely coupled, indirect aiding GPS/INS integration . . . . . . . 25

3.2-1

Earth-Fixed and Navigation coordinate systems . . . . . . . . . . 43

3.2-2

Body-Fixed aircraft (fixed-wing) coordinate systems . . . . . . . . 48

3.5-3

F16 yaw moment coefficient variation . . . . . . . . . . . . . . . . 66

4.4-1

Inertial Navigation (IN) Filter Implementation . . . . . . . . . . . 96

4.5-2

Air-Data Inertial Navigation (AIN) Filter Implementation . . . . 100

4.6-3

Aircraft Dynamic Navigation (ADN) filter implementation . . . . 105

5.2-1


Flight test scenario (Simulation 1 of 100) . . . . . . . . . . . . . . 114

5.2-2

Flight test wind rose (Simulation 1 of 100) . . . . . . . . . . . . . 118

5.2-3

Pressure altitude error (Simulation 1 of 100) . . . . . . . . . . . . 119

5.3-4

Position error comparison (Simulation 1 of 100) . . . . . . . . . . 124

5.3-5

Position RMS error comparison (All Simulations) . . . . . . . . . 124

5.3-6

Attitude error comparison (Simulation 1 of 100) . . . . . . . . . . 125

5.3-7

Attitude RMS error comparison (All Simulations) . . . . . . . . . 125

5.3-8

Velocity error comparison (Simulation 1 of 100) . . . . . . . . . . 126


5.3-9

Velocity RMS error comparison (All Simulations) . . . . . . . . . 126

5.3-10

Rotation error comparison (Simulation 1 of 100) . . . . . . . . . . 127

5.3-11

Rotation RMS error comparison (All Simulations) . . . . . . . . . 127

xvii

2


xviii

List of Figures

5.3-12

Accelerometer bias error comparison (Simulation 1 of 100) . . . . 129

5.3-13

Accelerometer bias RMS error comparison (All Simulations) . . . 129

5.3-14


Gyroscope bias error comparison (Simulation 1 of 100) . . . . . . 130

5.3-15

Gyroscope bias RMS error comparison (All Simulations) . . . . . 130

5.3-16

Wind error comparison (Simulation 1 of 100) . . . . . . . . . . . . 132

5.3-17

Wind RMS error comparison (All Simulations) . . . . . . . . . . . 132

5.4-18

Roll dynamic performance (Simulation 1 of 100) . . . . . . . . . . 134

5.4-19

Pitch dynamic performance (Simulation 1 of 100) . . . . . . . . . 135

5.4-20

Yaw Dynamic Performance (Simulation 1 of 100) . . . . . . . . . 136

6.1-1

AIN/ADN results compared to IN . . . . . . . . . . . . . . . . . . 144


B.2-1

IN position estimate errors (Simulation 1 of 100) . . . . . . . . . . 162

B.2-2

IN attitude estimate errors (Simulation 1 of 100) . . . . . . . . . . 163

B.2-3

IN velocity estimate errors (Simulation 1 of 100) . . . . . . . . . . 164

B.2-4

IN accelerometer bias estimate errors (Simulation 1 of 100) . . . . 165

B.2-5

IN gyroscope bias errors (Simulation 1 of 100) . . . . . . . . . . . 166

B.3-6

AIN position estimate errors (Simulation 1 of 100) . . . . . . . . . 169

B.3-7

AIN attitude estimate error (Simulation 1 of 100) . . . . . . . . . 170

B.3-8


AIN Velocity estimate errors (Simulation 1 of 100) . . . . . . . . . 171

B.3-9

AIN wind velocity estimate errors (Simulation 1 of 100) . . . . . . 172

B.3-10 AIN atmosphere estimate errors (Simulation 1 of 100) . . . . . . . 173
B.3-11 AIN accelerometer bias estimate errors (Simulation 1 of 100) . . . 174
B.3-12 AIN Gyroscope bias estimate errors (Simulation 1 of 100) . . . . . 175
B.4-13 ADN position estimate errors (Simulation 1 of 100) . . . . . . . . 179
B.4-14 ADN attitude estimate errors (Simulation 1 of 100) . . . . . . . . 180
B.4-15 ADN velocity estimate errors (Simulation 1 of 100) . . . . . . . . 181
B.4-16 ADN rotation estimate errors (Simulation 1 of 100) . . . . . . . . 182
B.4-17 ADN acceleration estimate errors (Simulation 1 of 100) . . . . . . 183
B.4-18 ADN angular acceleration estimate errors (Simulation 1 of 100) . 184
B.4-19 ADN power and actuator estimate errors (Simulation 1 of 100) . . 185
B.4-20 ADN actuator estimate errors (Simulation 1 of 100) . . . . . . . . 186


List of Figures

xix

B.4-21 ADN wind velocity estimate errors (Simulation 1 of 100) . . . . . 187
B.4-22 ADN atmosphere estimate errors (Simulation 1 of 100) . . . . . . 188
B.4-23 ADN accelerometer bias estimate errors (Simulation 1 of 100) . . 189
B.4-24 ADN gyroscope bias estimate errors (Simulation 1 of 100) . . . . 190




List of Tables

2.2-1

Inertial Navigation System (INS) performance . . . . . . . . . . . 19

2.3-2

Global Positioning System (GPS) performance . . . . . . . . . . . 23

3.2-1

World Geodetic System 1984 (WGS-84) ellipsoid parameters . . . 42

3.4-2

International Standard Atmosphere 1975 (ISA-75) parameters . . 57

3.4-3

Beaufort wind speed scale . . . . . . . . . . . . . . . . . . . . . . 60

3.4-4

Earth Gravitational Model 1996) (EGM-84) parameters . . . . . . 61

3.5-5

F16 mass and inertia reference data . . . . . . . . . . . . . . . . . 63


3.5-6

F16 control surface lag, saturation and rate limits . . . . . . . . . 64

3.5-7

F16 wing geometry reference data . . . . . . . . . . . . . . . . . . 66

3.5-8

F16 dimensionless force and moment coefficients . . . . . . . . . . 67

4.2-1

Sensor measurement noise properties . . . . . . . . . . . . . . . . 84

5.2-1

Flight test dynamic range (All Simulations) . . . . . . . . . . . . 116

5.2-2

Simulated mission uncertainties. . . . . . . . . . . . . . . . . . . . 120

5.3-3

Navigation estimate error comparison (All Simulations) . . . . . . 122

5.3-4


Inertial sensor bias error comparison (All Simulations)

5.3-5

Wind and atmosphere error comparison (All Simulations) . . . . . 131

5.5-6

Coasting error comparison (All Simulations) . . . . . . . . . . . . 138

A.2-1

Acceleration sensitivity to ±10% variations in force coefficients. . 155

A.2-2

Sensitivity to ±10% variations in moment coefficients. . . . . . . . 155

A.3-3

Variation in force coefficients compared to their maximum. . . . . 157

A.3-4

Variation in moment coefficients compared to their maximum. . . 157

B.2-1

IN tuned process and observation uncertainties . . . . . . . . . . . 160


xxi

. . . . . . 128


xxii

List of Tables

B.2-2

IN estimate error statistics (All Simulations) . . . . . . . . . . . . 161

B.3-3

AIN tuned process and observation uncertainties . . . . . . . . . . 167

B.3-4

AIN estimate error statistics (All Simulations) . . . . . . . . . . . 168

B.4-5

ADN tuned process and observation uncertainties . . . . . . . . . 176

B.4-6

ADN estimate error statistics (All Simulations) . . . . . . . . . . 177



Glossary

AD

Air-Data

ADC

Analogue to digital conversion

AIN

Air-Data Inertial Navigation

ADM

Aircraft Dynamic Model

ADN

Aircraft Dynamic Navigation

AGL

Above Ground Level (altitude above)

ARCAA

Australian Research Centre for Aerospace Automation


BF

Bayesian Filter

CFD

Computational Fluid Dynamics

COTS

Commercial of the shelf components.

CSIRO

Commonwealth Scientific and Research Organization

DAC

Digital to analogue conversion

DCM

Direction Cosine Matrix

DGPS

Differential GPS

DR


Dead Reckoning

EC

Electronic Compass

EKF

Extended Kalman Filter

EKM

Euler Kinematic Matrix

GCS

Ground Control Station

GNSS

Global Navigation Satellite Systems

GPS

Global Positioning System

IMU

Inertial Measurement Unit

xxiii


xxiv

Glossary

INS

Inertial Navigation System

ISA

International Standard Atmosphere

KF

Kalman Filter

L1

Link 1

L2

Link 2

MEMS

Micro-Electro-Mechanical Systems


MMSE

Minimum Mean Square Error

MSL

Mean Sea Level (altitude above)

NAS

National Airspace System

NED

North-East-Down

NEES

Normalise Estimate Error Squared

PF

Particle Filter

QUT

Queensland University of Technology

RTK


Real-Time Kinematic

SBAS

Satellite Based Augmentation System

SMC

Sequential Monte Carlo

SP

Single Point

SUT

Scaled Unscented Transformation

UAS

Unmanned Aircraft System

UAV

Unmanned Aerial Vehicle

UKF

Unscented Kalman Filter


UT

Unscented Transformation

WAAS

Wide Area Augmentation System


Nomenclature

Conventions
1. Vectors are indicated by lower-case bold font symbol a, and Matrices are
indicated by upper-case bold font symbol B.
2. Reference frames are indicated by a lower-case, right subscript Fi , and
Coordinate systems are indicated by a lower case right superscript C b .
3. Coordinate transforms, and Direction Cosine Matrices (DCM), are indicated by Cbi and Cib respectively. The order of the lower-case subscripts
indicate the direction of the transform (from i to b).
4. Position is given by peC/O where the upper-case, right subscripts denotes a
point (C with respect to O) and the lower-case, right superscripts denote
the coordinate system.
5. Velocities are given by vbC/i (translational) and ω bb/i (rotational), where
the right subscripts denotes the motion frames and the lower-case right
superscripts denote the coordinate system.
b
6. Accelerations are given by aC/i
(translational) and αbb/i (rotational), where

the subscripts and superscripts follow the same convention as velocities.

7. Derivatives of a vector i v˙ bC/i are defined with a left superscript which specifies the frame in which the derivative is taken.

xxv