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Quaternion kinematics for the error-state KF

Joan Sol`a

July 24, 2016

Contents

1 Quaternions and rotation operations 2

1.1 Deﬁnition of quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Alternative representations of the quaternion . . . . . . . . . . . . . . . . . 3

1.3 Some quaternion properties . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Rotations and cross-relations . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Quaternion conventions. My choice. . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Frame composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.7 Perturbations and time-derivatives . . . . . . . . . . . . . . . . . . . . . . 24

1.8 Time-integration of rotation rates . . . . . . . . . . . . . . . . . . . . . . . 26

1.9 Useful, and very useful, Jacobians of the rotation . . . . . . . . . . . . . . 29

2 Error-state kinematics for IMU-driven systems 31

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 The error-state Kalman ﬁlter explained . . . . . . . . . . . . . . . . . . . . 31

2.3 System kinematics in continuous time . . . . . . . . . . . . . . . . . . . . . 32

2.4 System kinematics in discrete time . . . . . . . . . . . . . . . . . . . . . . 38

3 Fusing IMU with complementary sensory data 40

3.1 Observation of the error state via ﬁlter correction . . . . . . . . . . . . . . 41

3.2 Injection of the observed error into the nominal state . . . . . . . . . . . . 43

3.3 ESKF reset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 The ESKF using global angular errors 45

4.1 System kinematics in continuous time . . . . . . . . . . . . . . . . . . . . . 45

4.2 System kinematics in discrete time . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Fusing with complementary sensory data . . . . . . . . . . . . . . . . . . . 48

A Runge-Kutta numerical integration methods 50

A.1 The Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.2 The midpoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.3 The RK4 method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.4 General Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . 53

B Closed-form integration methods 54

B.1 Integration of the angular error . . . . . . . . . . . . . . . . . . . . . . . . 54

B.2 Simpliﬁed IMU example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.3 Full IMU example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

C Approximate methods using truncated series 61

C.1 System-wise truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

C.2 Block-wise truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

D The transition matrix via Runge-Kutta integration 64

D.1 Error-state example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

E Integration of random noise and perturbations 67

E.1 Noise and perturbation impulses . . . . . . . . . . . . . . . . . . . . . . . . 69

E.2 Full IMU example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

1 Quaternions and rotation operations

1.1 Deﬁnition of quaternion

One introduction to the quaternion that I ﬁnd particularly attractive is given by the Cayley-

Dickson construction: If we have two complex numbers A = a + bi and C = c + di, then

constructing Q = A + Cj and deﬁning k , ij yields a number in the space of quaternions

Q = a + bi + cj + dk ∈ H , (1)

where {a, b, c, d} ∈ R, and {i, j, k} are three imaginary unit numbers deﬁned so that

= j

= k

= ijk = −1 , (2a)

from which we can derive

ij = −ji = k , jk = −kj = i , ki = −ik = j . (2b)

From (1) we see that we can embed complex numbers, and thus real and imaginary num-

bers, in the quaternion deﬁnition, in the sense that real, imaginary and complex numbers

are indeed quaternions,

Q = a ∈ R ⊂ H , Q = bi ∈ I ⊂ H , Q = a + bi ∈ Z ⊂ H . (3)

Likewise, and for the sake of completeness, we may deﬁne numbers in the tri-dimensional

imaginary subspace of H (we refer to them as pure quaternions),

Q = bi + cj + dk ∈ I

⊂ H . (4)

It is noticeable that, while regular complex numbers of unit length z = e

iθ

can encode

rotations in the 2D plane (with one complex product, x

= z ·x), “extended complex

numbers” or quaternions of unit length q = e

i+u

j+u

k)θ/2

encode rotations in the 3D

space (with a double quaternion product, x

= q ⊗ x ⊗ q

∗

, as we explain later in this

document).

CAUTION: Not all quaternion deﬁnitions are the same. Some authors write the

products as ib instead of bi, and therefore they get the property k = ji = −ij, which results

in ijk = 1 and a left-handed quaternion. Also, many authors place the real part at the end

position, yielding Q = ia + jb + kc + d. These choices have no fundamental implications

but make the whole formulation diﬀerent in the details. Please refer to Section 1.5 for

further explanations and disambiguation.

CAUTION: There are additional conventions that also make the formulation diﬀerent

in details. They concern the “meaning” or “interpretation” we give to the rotation op-

erators, either rotating vectors or rotating reference frames –which, essentially, constitute

opposite operations. Refer also to Section 1.5 for further explanations and disambiguation.

1.2 Alternative representations of the quaternion

The real + imaginary notation {1, i, j, k} is not always convenient for our purposes. Pro-

vided that the algebra (2) is used, a quaternion can be posed as a sum scalar + vector,

Q = q

+ q

i + q

j + q

k ⇔ Q = q

+ q

, (5)

where q

is referred to as the real or scalar part, and q

= q

i + q

j + q

k = (q

, q

) as

the imaginary or vector part.

It can be also deﬁned as an ordered pair scalar-vector

Q = hq

, q

i . (6)

We mostly represent a quaternion Q as a 4-vector q ,

q ,

















, (7)

Our choice for the (w, x, y, z) subscripts notation comes from the fact that we are interested in the

geometric properties of the quaternion in the 3D Cartesian space. Other texts often use alternative

subscripts such as (0, 1, 2, 3) (more suited for algebraic interpretations) or (1, i, j, k) (for mathematical

interpretations).

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