Comparison of TRIAD+EKF and TRIAD+UKF Algorithms for

Nanosatellite Attitude Estimation

MEHMET ASIM GOKCAY

Hezârfen Aeronautics and Astronautics Technologies Institute for Space Sciences

Turkish National Defense University

Yeşilyurt, 34149, Bakırköy, Istanbul

TURKEY

CHINGIZ HAJIYEV

Faculty of Aeronautics and Astronautics

Istanbul Technical University

Ayazağa, 34469, Maslak, Istanbul

TURKEY

Abstract: - Two most commonly used sensors on nanosatellites are magnetometer and sun sensor. In this paper,

magnetometer and sun sensor measurements are combined gyro measurements to produce enhanced attitude

estimation. Tri-Axial Attitude Determination (TRIAD) algorithm is used with Kalman filter to form a complete

attitude filter. Sun sensor and magnetometer measurements are selected as inputs to TRIAD algorithm and

output is fed to Kalman filter as a measurement. Two different Kalman filters, extended and unscented, are

used with TRIAD algorithm. A comparison is given between performances of both Kalman filter.

Key-Words: Nanosatellite, Magnetometer, Sun Sensor, Kalman Filter, Attitude Estimation.

Received: May 15, 2021. Revised: March 4, 2022. Accepted: April 8, 2022. Published: May 4, 2022.

1 Introduction

Increasing demand for the space operations,

space industry turns its face to cost effective

solutions. Small satellites, due to their size and cost,

are receiving interest from many organizations. The

amount of attitude determination and control

equipment that can be placed in small satellites are

considerably lower than the regular size satellite.

Kalman filter has been the backbone of the

space industry since the publication of the Dr.

Kalman’s work [1]. Many different versions of the

Kalman filter have been derived. Harold Black

published an algorithm called Tri-Axial Attitude

Determination (TRIAD) in 1964 [2]. It is the earliest

algorithm that was published to find satellite attitude

with two measurements. In 1965, Grace Wahba

suggested her famous problem [3]. Solution

methods such as q-method [4] and Quest algorithm

have been widely used [5]. A computationally

expensive method, SVD, is also published [6].

Many of the mentioned methods are coupled with

Kalman filters for higher accuracy [7], [8]. In this

work algebraic method is used with sun sensor and

magnetometer measurements.

EKF is probably the most used version of

the Kalman filter [9]. One of the earliest work of

Kalman filter for attitude determination used Euler

angle rotations [10]. It is known that all three-

parameter representations of the special orthogonal

group suffer from singularity and discontinuity

problems. To overcome this challenge new

representation methods have been studied [11].

Quaternions have become the most used form of the

attitude representations. Euler angles, Rodrigues and

modified Rodrigues parameters are avoided for most

of the agile missions due to their singularity

problems. In the last two decades new approaches

have been suggested for replacing EKF. Unscented

Kalman filter is one of them [12]. UKF uses the

unscented transformation to achieve high-order

approximations of the nonlinear functions in order

to estimate mean and covariance of the state vector.

Filter uses predefined number of sigma points to

approximate Gaussian distribution. Each of the

sigma points are propagated through the propagation

functions [12, 13].

In this work, using common sensors, couple

of filters are design to overcome to attitude

determination problem. Two of most common

sensors that are being used in nanosatellites are

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magnetometers and sun sensors. Magnetic dipole

model is selected for magnetic field model. VSOP87

theory is used for sun direction vector [14]. Using

these two models, sensor measurement models have

been established. For attitude representation of the

spacecraft Euler angles are selected. TRIAD

algorithm is used for combining sun sensor and

magnetometer measurements. Body angles that are

produced by TRIAD are used as linear

measurements to the Kalman filter. Algorithm is

constructed for both Extended and Unscented

Kalman filters for comparison.

2 Mathematical Model

Satellite kinematic, dynamic and sensor

models are derived in this section.

2.1 Satellite Kinematic and Dynamic Model

Firstly, for kinematic model, rate equations

are given below

cos sin     

(1)

( sin cos )sec       

(2)

u v w

tan ( sin cos )         

(3)

where



is the roll angle about

axis,



is the

pitch angle about

axis,



is the yaw angle about

axis,



and



are the components of

vector in body frame with respect to the reference

frame.

Rotations need to be defined with respect to

inertial frame.

v y 0



   



     

     

     

(4)

where

and

are the components of vector

in body frame with respect to the inertial frame,

A is the attitude transformation matrix composed of

Euler angles,





defines the orbital angular

velocity with respect to the inertial frame,



is the

gravitational constant of the Earth and r is the

distance between center of mass of the satellite and

the Earth,

rr

Attitude dynamics is related with time

derivative of the angular momentum vector. If the

origin of the body frame is selected as the center of

mass, then

h Jw

(5)

where h is the angular momentum vector, J is the

moment of inertia matrix.

Relation between time derivative of the

angular momentum and the angular velocity is

c b c

h h w (h )  

(6)

where hb is the time derivative of angular

momentum in body-fixed frame.

Rewriting derivative equations for discrete

time we have,

 

 

k 1 k

x x z y y z m

w w w w J J T





   

(7)

 

 

k 1 k

y y x z z x m

w w w w J J T





   

(8)

 

 

k 1 k

z z x y x y m

w w w w J J T





   

(9)

where Tm is the external torque and

t

is the

sampling time interval.

Equations above describe the satellite attitude

motion.

2.2 Sensor Models

Magnetometer is the most commonly used

sensor particularly in nanosatellite applications. For

magnetic field vector, dipole model is used. Sensor

model is given below,

y 2 m m

B ( , , , t) B (t)

B ( , , , t) A B (t) b

B ( , , , t) B (t)



      



   

   

   

(10)

where B1(t), B2(t) and B3(t) indicates Earth magnetic

field vector components in orbit frame,

Bx(

, , , t

), By(

, , , t

) and Bz(

, , , t

) show

the Earth magnetic field vector components in body

frame as a function of body angles and time. The

magnetometer bias vector is given as

 

m x y z

b b b b

. Bias vector is modeled via the

expression below:

b0

. (11)

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The noise term,



is added linearly to the model

and modeled as zero mean Gaussian white noise

with the characteristic of

 

1 k 1 j 3 x 3 m kj

    



(12)

where I3x3 is the identity matrix,



is the standard

deviation of magnetometer errors and



is the

Kronecker symbol.

In order to construct a sun sensor model,

sun direction vector is used. Using VSOP87 theory,

a direction cosine matrix is calculated which shows

the sun’s position relative to Earth in ECI frame

[14].

B E s

s A s





 

 



(13)

Construction of sun direction vector, sE, requires

two assumptions. Comparing the distance between

Sun-Earth, 1 AU, and Earth-satellite, satellite

altitude is negligible. Therefore, satellite’s sun

direction vector is always parallel to Earth’s sun

direction vector. The other assumption is taking the

right ascension node of Sun’s orbit as zero. The sun

direction vector,

ecliptic

E ecliptic

ecliptic

cos

s sin cos

sin sin













(14)

The noise term,



is added linearly to the model

and modeled as zero mean Gaussian white noise

with the characteristic of

1k 1 j 3 x 3 s kj

    







(15)

where I3x3 is the identity matrix,



is the standard

deviation of sun sensor errors.

The gyro model is constructed with satellite

dynamic equations. A commonly used model for

gyro measurements given by

BI BI g g

  

  

(16)

where

is the gyro bias vector and the



modeled as zero mean Gaussian white noise with

the characteristic of

1k 1 j 3 x 3 g kj

    







(17)

where I3x3 is the identity matrix,



is the standard

deviation of gyro errors. Gyro bias vector is

modeled as,

b0

. (18)

3 Filter Design

3.1 TRIAD

A rotation matrix describes the attitude of a

spacecraft with respect to a known reference frame.

It takes at least two measured vectors to determine

the orientation of the vehicle. The algebraic method

constructs two triads of orthonormal vectors, two

triads are expressed by sun sensor and

magnetometer unit vectors in body and reference

frames. Let magnetometer measurement vector is

denoted by B and sun sensor vector is denoted by S.

For initial base vector, sun sensor is selected.

v S B / S B

r u v



  



(19)

It is important to note that two vectors, S and B

cannot be parallel,

S.B 1

. Constructing the

direction cosine matrix,

  

r r r b b brb

A u v w u v w

(20)

Final step of the TRIAD is to find error covariance

of the algorithm

2 T 2 T

m b b s b b

s b b

S S B B

P v v

 







(21)

Body angles and error covariances that are obtained

from TRIAD algorithm are used as measurement

inputs to the Kalman filter. These results are

combined with gyro measurements in order to

achieve better accuracy.

3.2 Extended Kalman Filter

In 1960, Dr. Kalman published his famous

paper “A new Approach to Linear Filtering and

Prediction”. It represented a sequential solution to

the time-varying filtering problem. The filter is

particularly good for dynamical systems and also

removed the non-dynamical requirements of Weiner

filter [1]. The extended Kalman filter uses the first

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order Taylor series linearization for non-linear

systems. This process requires two assumptions.

Propagation and measurement functions must be

differentiable. Filter algorithm is given below.

Propagation of state vector is conducted with

equation,

 

k k 1

ˆˆ

x f x







(22)

Equation of state estimation,

  

k k k k

ˆˆ

x x K z h x

  

  

(23)

Here, h is the measurement function and K is the

Kalman gain,

K P H HP H R 









(24)

where R is the covariance matrix of measurement

error, H is the measurement matrix of system,

P

the covariance matrix of extrapolation error.

Covariances of both extrapolation and filtering

errors are given below. For error equations,

Jacobian matrix

is needed. Jacobian matrix can

be constructed by taking partial derivatives of

propagation function with respect to state variables

k k 1

FP F QP





(25)

 

P I KH P





(26)

where F is the Jacobian matrix and Q is the

covariance matrix of process noise.

3.3 Unscented Kalman Filter

EKF, due to nature of the first-degree

linearization, is not as accurate for highly linear

systems. Jacobians are hard to derive and the

linearization needs very short time intervals

otherwise filter becomes unstable. But this comes

with the computational power gets higher. The main

idea behind the UKF is distributions are easier to

approximate from nonlinear functions [13].

Therefore, it introduces “sigma points”. With these

points, filter removes need for derivation of

Jacobian matrix and it is more robust to the initial

estimation errors [15]. Sigma point can be obtained

by [16]





(27)

 

x L P i 1,....L



   

(28)

 

x L P i L 1,....2L





    

(29)



is denotation for the sigma vector. In order to

capture the true nature of the reflection, sigma

points are weighted

 

( m )





(30)

 

( c ) 2





   



(31)

(m ) (c)

WW

(32)

where

 

2LL

  

  

is the scaling parameter,



is the secondary scaling parameter,



determines how spread the sigma points are and is

used to incorporate prior knowledge for x. Using the

weights, propagated state matrix can be determined

by,

(m)

k i i

x W Y







(33)

And propagated covariance

2L T

(c )

k i i k i k

ˆˆ

P W Y x Y x Q

  



   

   

   



(34)

where Q is the process noise. Obtaining the Kalman

gain is similar to EKF. Hence,

xy yy

K P P



(35)

where Pxy and Pyy are,

2L T

(c)

yy i i k i k

ˆˆ

P W Z z Z z R





   

   

   



(36)

2L T

(c)

xy i i k i k

ˆˆ

P W Y z Z z





  

   

   



(37)

The last part is to update both state and covariance

matrix with measurements

 

k k k k

ˆ ˆ ˆ

x x K z z



  

(38)

k k yy

P P KP K





(39)

where zk is the measurement vector.

4 Simulation

Simulations are performed with a

hypothetical nanosatellite with principal moment of

inertia J = diag(2.1x10e-3 2x10e-3 1.9x10e-3).

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Simulations are conducted with a sampling time of 1

second. Orbit of the satellite is a circular orbit with

an altitude of 400 km. Argument of perigee of the

orbit is 207.4 degrees and inclination of the orbit is

97.65 degrees.

Figures below give attitude estimations in

terms of Euler angles. First figure shows estimation

with only TRIAD algorithm. Figure 1 and 2 show

attitude estimations combining TRIAD and two

different Kalman filters

Fig. 1: TRIAD-EKF attitude estimation

Fig. 2: TRIAD - UKF attitude estimation

Mean square errors of each method are

given in Table 1.

Table 1: RMSE values

Roll (deg)

Pitch (deg)

Yaw (deg)

TRIAD

6.2395

7.2175

11.4611

EKF

4.5339

5.4990

10.5363

UKF

3.4718

6.1306

10.7370

RMSE results clearly show that Kalman

filters increase quality of estimation compared to

TRIAD-only algorithm. Both extended and

unscented Kalman filters estimated similarly. The

only significant difference in RMSE values is at roll

estimation.

5 Conclusion

Algebraic method, even though it is an aging

algorithm, can estimate satellite attitude well. The

sun sensor and magnetometer are selected for inputs

to algebraic method because of their wide usage in

space industry. Many different filtering algorithms

are presented to this day but proven algorithms are

still getting attention from engineers. Extended

Kalman filter proves itself on many missions.

Therefore, selecting the EKF for this work was a

must. As expected, it performs really well on

simulations especially in low initial angular velocity

cases. UKF is known for its superiority to EKF

when dealing with nonlinear functions. In many

simulations attempts, UKF performed relatively

same as EKF.

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