FYS3240- 4240 Data acquisition & control

FYS3240- 4240

Data acquisition & control

Navigation and attitude (orientation)

Spring 2021– Lecture #6 and #7

Bekkeng 10.02.2021

What we will cover

• An introduction to navigation

– Inertial navigation

– Satellite navigation

• Reference frames / Coordinate systems

• Attitude representation & determination

– Why do we need to know the attitude/orientation of an object?

– The attitude matrix (DCM) for 3D orientation

– Heading (2D orientation)

– The Earth’s gravity acceleration

– The Earth’s magnetic field

Support literature

Chapter (1), 2, 4 and 7.

For this lecture read chapter 2:

Page 21 - page 32 (but not 2.2.4 or 2.2.7)

2.3 – 2.3.2

Equation (2.101)

2.3.9 and 2.3.10

Introduction

Vector notation – column vectors

Right-hand-rule:

A vector must be decomposed in a

coordinate system for implementation

on a computer.

Notation

• ABR

: transformation from frame R to frame B.

– NB: read superscripts from right to left.

• 𝐴𝑅𝐵 : transformation from frame R to frame B.

– NB: read from subscript to superscript

Navigation

• Estimate the position, orientation and velocity of a vehicle

Need to select a suitable

coordinate frame!

Need for navigation solution

• Accelerometer

• GPS receiver

• Pressure sensor (for altitude)

• Camera– If known points can be identified in the picture

• Indoor positioning technology using beacons (based on infrastructure)

– UWB - Ultra-Wideband (GHz frequency)

– Ultrasound (20 kHz – 40 kHz)

– WiFi

– Bluetooth

– Bluetooth Low Energy (BLE)

Examples of position sensors

Received signal strength Indication (RSSI) and

fingerprinting (and angle of arrival – AoA).

Bluetooth 5.1 Puts Bluetooth In Its Place (nordicsemi.com)

TOF (time-of-flight) /TDOA (time-

difference-of-arrival)

Forkbeard - ultrasound indoor GPS

Orientation sensors I

• Gyroscopes

• Magnetometer

2F m v

A

Orientation sensors II

• GPS receiver

– two/three or more antennas (two for heading only)

• Sun sensor

• Star sensor

• Accelerometer

– when static

Common in space applications

Inertial Navigation

Inertial Navigation System (INS)

• Uses inertial sensors – gyroscopes

and accelerometers.

• The first INS systems were used in

the V1 and V2 rockets during the

second world war.

• In the 1960s the Apollo program used

inertial navigation in space.

Relative positioning – inertial navigation

• Relative measurements, dead reckoning (DR).

• Need to know initial position and orientation.

• Based on mathematical integration of accelerometer and

gyroscope measurements.

• The positioning solution drift (error increase) with time due to

the integration.

No external signals or infrastructure is required!

For comparison - Absolute positioning

• Active beacons (indoor)

• GPS / GNSS (outdoor)

• Landmark navigation

– for instance camera-based navigation

Requires external signals and “infrastructure” !

GNSS satellites, beacons (with known position)

or known landmarks

Satellite Navigation

Satellite navigation

• A satellite navigation system with global coverage is called a global navigation satellite system, or GNSS.

• Limitations of satellite navigation

– Need line of sight

– Can be jammed• Noise jamming

• Spofing

– Update rate 1 – 50 Hz

Figure from VectorNav

Triangulation in 3D

One satellite

Two satellites

Three satellites

Need a minimum of four satellites to determine position (latitude,

longitude, altitude), since clock error in the receiver also need to be

estimated.

r = c * t

Height is not so easy ….

• The geoid approximates mean sea level (MSL) .

• A mathematical model can only approximate the real shape of the

Earth.

• Different methods/definitions give different height measurements!

Extra

Space weather affect GNSS!

South Atlantic Anomali

Figure from

Kartverket

solar wind.

FYS4640 – Space Weather and Navigation Satellite

Systems

Extra

Space weather research at UIO

UIO raketterSatellites

Sounding rockets

Andøya

Svalbard

100 – 1000 km

Extra

• Every sensor have errors and noise!

• Need multi-sensor data fusion to build accurate

and robust systems!

Data fusion

Sensor data fusion

Data processing

algorithm

(Estimator)

Computer

Sensor 1

Sensor 2

Sensor n

.

.

• Position

• Velocity

• Acceleration

• Attitude/orientation

• Sensor errors, such as

bias (offset)

Sensor fusion gives reduced uncertainty and makes

the system more robust!

Estimated values

Data fusion example: INS + GNSS (GPS)

• The INS error increases with time because of integration

– angle error ∝ 𝑡, position error ∝ 𝑡3

• INS is not dependent on external signals such as GPS.

Combine INS and GNSS sensor data fusion

INS and GNSS are complementary!

INS/GPS integration example

• A very common INS/GPS integration

• Only INS solution when GPS not available

GPS

INS

Estimator+

-+

v, p + GPS error

v, p + INS error

INS error estimate

-

GPS error – INS error

update rate: 1 – 50 Hz

update rate: ~1 kHz

Gyroscopes &

accelerometers

Position (p) and

velocity (v)

estimates

Reference frames / Coordinate systems

What is an inertia reference frame?

• A reference frame (coordinate system) where Newton’s laws

are valid!

– No linear (translational) acceleration.

– No angular acceleration.

• If we are to use Newton's laws in a non-inertial frame we need

to add fictive forces, such as Coriolis force and

centripetal/centrifugal forces.

ECEF (Earth-Centered Earth-Fixed)

• A global reference frame with origin in the Earth’s center of mass.

• The Ez axis is normal to the equator (points along the rotation axis

• of the Earth), and is positive in the direction of the north pole.

• The Ex axis is in the direction of the Greenwich prime meridian.

Often a positon on/close to the earth is

described with polar coordinates (geodetic

coordinate system), in terms of latitude (φ),

longitude (λ) and altitude (h).

ECI (Earth-Centered Inertial)

• A global reference frame with origin in the Earth’s center of mass.

• The Ez axis is normal to the equator (points along the rotation axis of

the Earth), and is positive in the direction of the north pole.

• The Ex axis is in the direction of the vernal equinox. (The vernal

equinox is defined by the intersection of the Earth equator plane and

the ecliptic plane, where the ecliptic is the plane of the Earth’s mean

orbit about the Sun).

The frame does not rotate with Earth and serves as an inertial

reference frame for objects flying high and fast, for instance

• satellites orbiting the Earth,

• sounding rockets,

• long range ballistic missiles, and

• hypersonic vehicles

Local level frame

• The North-East-Down (NED) frame is

the most common local reference

frame.

• Often this frame is fixed to the vehicle

and moves with the body frame.

• But, it can also be fixed to the

ground, for instance at the launch

point of a rocket/missile.

• Similar to the NED frame, there is

also an East-North-Up (ENU) frame.

• A flat Earth approximation is used

– assumed inertial (not rotating). NED-frame

When can we use the local level frame?

• Examples:

– NED (North East Down)

– ENU (East North Up)

• Note: the book usually refers to ENU.

Can be used as an inertial reference frame for objects flying

low and slow (short term navigation), including tactical missiles.

E

NU

When do we have to take into account

the rotation of the Earth ?

New York

Chicago

• Assume a vehicle launched from the

North pole with the intention of flying to

New York.

• Assumed the vehicle travels at a speed

of 5795 km/h.

• During the flight time, of approximately

one hour, the Earth will have rotated by

about 15 degrees.

• If no Coriolis correction is made to the

onboard INS during the flight the vehicle

will arrive in the Chicago area rather

than New York.

What is the effect of ignoring the

rotation of the Earth ?

• We can calculate this!

• Depends on how big the contribution from the Coriolis and

centripetal acceleration are, relative to ae.

• See lecture 7 for more details!

2i ie e ei e ei ei eR a a v r

Coriolis

accelerationcentripetal

acceleration

0

0

0

z

y

z y

BA

x

x

Body frame

• A platform has its own reference frame known as the body

frame

• Origin typically placed at the platform’s center of gravity.

• Three orthogonal axes that comprise a right-handed system.

• Common definition:

– x-axis is pointing forward (“out the nose”),

– y-axis is pointing to the right, and

– z-axis is defined by the right hand system.

• Note: Some authors do not

follow this definition ….

• See next slide

Usually the x-axis!

Standard axis definitions!

x

y

z

Sensor frame (S)

• Fixed to the sensor

• Typically not (perfectly) aligned

with the body frame!

Note: The measurement (sensitive) axes can

be misaligned with the indicated sensor frame!

Typically many frames in use …..

Need to know the transformation between frames/coordinate systems.

Decompose measurements from each sensor in the body frame

Attitude representation &

determination

Example: Spacecraft attitude (orientation)

• Definition: The angular orientation

of a body-fixed coordinate frame

with respect to an external

reference frame.

• Spacecraft, aircrafts and many

other vehicles must know their

attitude.

Satellites Sounding rocketss

Pointing accuracy

• Sensors need to be pointing towards the point of interest

– Higher accuracy better results

Sensor not

looking at the

correct scene

Correct pointing

Low pointing

error (jitter) High pointing jitter

Example from http://www.s3l.be/usr/files/di/fi/2/Lecture13_ADCS_TjorvenDelabie_20181111202.pdf

Why is attitude determination

important?

• To control the pointing direction of an object, we need to first

determine where it is pointing.

• In strapdown inertial navigation, we need to calculate the attitude

(orientation) before we can determine position and velocity of an

object.

• Orientation has three degrees of freedom (DOF)

– Need to determine three or more parameters

• Simple example: What is up and what is down?

– Orientation in the Earth's gravitational field.

– Control the orientation of our phone/tablet

display (using accelerometer data).

Examples (from aerospace)

• Attitude determination & control

– Point solar panels on a spacecraft/ satellite

towards the Sun.

– Point communication antennas on a

spacecraft/satellite towards the Earth.

– Point the payload (sensors) towards the point

of interest.

Pointing accuracy:

1/36000°~ 2.78*10-5 °

Examples (from aerospace)

• Attitude determination

– Transform a vector (for instance an electric

field vector) measured on board a sounding

rocket to a non-rotating Earth-fixed frame.

– Use a common coordinate system for multi-

sensor data fusion.

• For instance to fuse data from an IMU and

a magnetometer.

IMU Magnetometer

Arduino Nano 33 BLE sense

Measured What we want

Eamb

The direction cosine matrix (DCM)

• The direction cosine matrix (DCM) is one of

the many ways to mathematically represent

an object’s orientation (attitude)

• It utilizes nine parameters. Each of these

parameters are referred to as the direction

cosine values between a reference frame (n)

and a frame (b).

DCM (attitude matrix) A

Unit vectors

The direction cosine matrix (DCM)

• This matrix is orthogonal:

• Given a vector vB

in the body frame (B). The

representation of this vector in the reference frame (R) is

given by

A rotation from frame R to frame B. Can also be labelled ABR

where

Euler Angles

• A transformation from one coordinate frame to another can be

carried out as three successive rotations about different axes.

• It is common to define the Euler roll angle φ as a rotation in the

positive sense (according to the right hand rule) about the x-axis,

the pitch angle θ about the y-axis and the yaw angle ψ about

the z-axis.

– Note that the book deviates from this.

Euler Angles

Euler Angles

• The three principal rotation matrices for rotations about the

three axes is defined as:

Note: DCMs from reference frame {R}

to body frame {B}, 𝐴𝑅𝐵

Note: axis (1, 2, 3) = (x, y, z) = (roll, pitch, yaw)

Coordinate transformation example

• Transform a vector represented in the ECEF-frame to a vector

represented in the ECI-frame.

• Use the principal rotation matrix R3 for rotations about the z-

axis

The Greenwich Mean Sidereal Time (the hour angle)

Figure: Wiley

Calculation of the principal rotation

matrices

• Given by the projection of the body vectors xB, y

Band z

Bonto

the vectors xR, y

Rand z

Rof the reference frame.

R2

R1

𝜑

𝜑

cos𝜑 cos𝜑

sin𝜑

sin𝜑

R3, B3

Extra

Rotation about axis 3 (yaw)

Euler Angles to/from DCM

• The DCM for any Euler angle sequence can be constructed

from the individual axis rotations.

• There are many different combinations of Euler angles,

however, the (3-2-1) set of Euler angles corresponding to yaw-

pitch-roll (ψ-θ-φ) is considered to be the standard

The order of these rotations is important! A (3-2-1) set of Euler angles

typically result in a different orientation than applying those same

angles in a (1-2-3) sequence!

Attitude determination - sensors

• Inertial sensors

• Attitude sensors

Attitude determination

Problem: At each time point, find the orientation of the spacecraft

body frame {b} with respect to the reference frame {r}

Attitude determination

• The fundamental problem is to calculate the

attitude matrix 𝐴𝑅𝐵 (DCM)

• Can be calculated based on two or more

vector observations from a single point in

time. Deterministic algorithms require at

least two vector measurements

• With only a single vector measurement, the

rotation about this vector can not be

resolved.

ESA

Attitude determination

• For spacecraft the Earth magnetic field vector B and the pointing

vector S to the Sun (and other celestial objects) are common to

use.

• If we ignore measurement errors we can write:

𝑺𝑏 = 𝐴𝑒𝑐𝑖𝑏 𝑺𝑒𝑐𝑖

𝑩𝑏 = 𝐴𝑒𝑐𝑖𝑏 𝑩𝑒𝑐𝑖

the attitude matrix

From modelMeasuredProblem: determine the matrix 𝐴𝑒𝑐𝑖

𝑏 that

satisfies the two (or more) measurement

equations.

Solutions: Many standard techniques

• TRIAD

• q-method

• QUEST

• Extended Kalman filter

• ++

For info only!

Heading

• Heading (= azimuth angle or yaw angle) is an important 2D

measurement.

– Navigation on land and sea can be constrained to 2D.

• Definition: Heading is the direction in which a vehicle (x-axis) is

pointing at any given moment. It is expressed as the angular distance

relative to north (true or magnetic).

• The heading may not necessarily be the direction that the vehicle

actually travels (the velocity vector v).

N

E

𝜑

x

v

y

The_Seven_Ways_to_Find_Heading

The Earth’s gravity acceleration

• The Earth gravitational vector 𝒈𝒗 is given by

• Standing on the Earth we are also subject to the centrifugal

acceleration. The total gravity acceleration on the Earth is

therefore

• Since the Earth have an ellipsoid shape (not circular) the

gravity vector magnitude varies with latitude

– However, for many applications it is sufficient to only assume

variation with altitude.

– In addition comes the inhomogeneous mass distribution of the Earth.

𝒈𝒗 = −GM𝒔𝑩𝑬

|𝑺𝑩𝑬 |𝟑

where |𝒈𝒗| = 9.82023 m/s2 on average on the Earth’s surface

B

E

𝑺𝑩𝑬

𝒈

𝒈 = 𝒈v − 𝝎𝑬𝑰x 𝝎𝑬𝑰𝒔𝑩𝑬

where the standard average value is 𝑔 = 9.8066 ~9.81 m/s2

Use of the Earth gravity acceleration

vector

• Since this vector is almost vertical we can not use

it alone to determine heading.

• Using an accelerometer we can measure roll and

pitch angles, but not heading/yaw (rotation about

the gravity vector).

• When standing still we can determine what is up

and what is down

– Think of how your phone/tablet flip the image ….

g

Earth’s magnetic field

• Magntic field vector

– Horizontal component and a vertical

component

• Models:

– World Magnetic Model (WMM).

– International Geomagnetic Reference

Field (IGRF)

Earth’s magnetic field II

• The IGRF and WMM models represent only the main

geomagnetic field, generated in the Earth’s outer core (which

resembles the field generated by a dipole magnet). This is the

dominating component of the field, accounting for over 95% of

the field strength at the Earth’s surface.

• The slow change in time of this field is also included in the

models.

• The field contributions from the Earth’s crust (arising from

magnetized crustal rocks), from currents flowing in the

ionosphere and magnetosphere, and induced currents in the

sea and the ground are not included in the model, as they are

difficult to predict.

Earth’s magnetic field II

• Magnetic inclination

– the angle between the Earth’s magnetic field

lines and a horizontal plane

• Magnetic declination

– the angle between the magnetic North Pole and

the True North (geographic North Pole)

• Varies with position on the Earth (and

slowly with time)

NCEI Geomagnetic Calculators (noaa.gov)

The South Atlantic Anomaly

• A dip in the Earth’s magnetic field over South America and the South

Atlantic ocean (dark blue = weakest field).

• This local weakness in Earth's magnetic field leads to an enhanced level

of charged particles which can cause damage to onboard electronic

systems in satellites and spacecraft.

Earth’s magnetic field

South Atlantic Anomaly

Extra