SE(3)变换参数化教程：旋转表示与优化详解

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A tutorial on SE(3) transformation parameterizations and

on-manifold optimization

MAPIR Group

Technical report #012010

Dpto. de Ingenier´ıa de

Sistemas y Autom´atica

http://mapir.isa.uma.es/

1.2. Common parameterizations

1.2.1 3D translation plus yaw-pitch-roll (3D+YPR)

A 6D pose p

can be described as a displacement in 3D plus a rotation deﬁned by means of a speciﬁc

case of Euler angles: yaw (φ), pitch (χ) and roll (ψ), that is:

= [x y z φ χ ψ]

(1.4)

The geometrical meaning of the angles is represented in Figure 1.3. There are other alternative

conventions about triplets of angles to represent a rotation in 3D, but the one employed here is the

one most commonly used in robotics.

Yaw

(1st)

Pitch

(2nd)

Roll

(3rd)

Arrow indicates

positive direction

Figure 1.3: A common convention for the angles yaw, pitch and roll.

Note that the overall rotation is represented as a sequence of three individual rotations, each taking

a diﬀerent axis of rotation. In particular, the order is: yaw around the Z axis, then pitch around the

modiﬁed Y axis, then roll around the modiﬁed X axis. It is also common to ﬁnd in the literature the

roll-pitch-yaw (RPY) parameterization (versus YPR), where rotations apply over the same angles (e.g.

yaw around the Z axis) but in inverse order and around the unmodiﬁed axes instead of the successively

modiﬁed axes of the yaw-pitch-roll form. In any case, it can be shown that the numeric values of the

three rotations are identical for any given 3D rotation [6], thus both forms are completely equivalent.

This representation is the most compact since it only requires 6 real parameters to describe a

pose (the minimum number of parameters, since a pose has 6 degrees of freedom). However, in

some applications it may be more advantageous to employ other representations, even at the cost of

maintaining more parameters.

Degenerate cases: gimbal lock

One of the important disadvantages of the yaw-pitch-roll representation of rotations is the existence

of two degenerate cases, speciﬁcally, when pitch (χ) approaches ±90

◦

. In this case, it is easy to realize

that a change in roll becomes a change in yaw.

This means that, for χ = ±90

◦

, there is not a unique correspondence between any possible rotation

in 3D and a triplet of yaw-pitch-roll angles. The practical consecuences of this characteristic is the

A tutorial on SE(3) transformation parameterizations and

on-manifold optimization

MAPIR Group

Technical report #012010

Dpto. de Ingenier´ıa de

Sistemas y Autom´atica

http://mapir.isa.uma.es/

need for detecting and handling these special cases, as will be seen in some of the transformations

described later on.

Implementation in MRPT

Poses based on yaw-pitch-roll angles are implemented in the C++ class mrpt::poses::CPose3D:

# include < mrpt / p oses / CPos e3D . h >

usi ng n a m espace mrpt :: poses ;

usi ng mrpt :: utils :: DEG2RAD ;

CPo se3D p (1.0 /* x */ ,2.0 /* y */ ,3.0 /* z */ ,

DEG 2RAD (30.0) /* yaw */ , DEG2 RAD (20.0) /* pitc h */ , DEG2 RAD (90.0) /* rol l */ );

1.2.2 3D translation plus quaternion (3D+Quat)

A pose p

can be also described with a displacement in 3D plus a rotation deﬁned by a quaternion,

that is:

= [x y z q

]

(1.5)

where the unit quaternion elements are [q

, (q

, q

)]. A useful interpretation of quaternions is that

of a rotation of θ radians around the axis deﬁned by the vector ~v = (v

, v

) ∝ (q

, q

). The

relation between θ, ~v and the elements in the quaternion is:

= cos

= sin

This interpretation is also represented in Figure 1.4. The convention is q

(and thus θ) to be

non-negative. A quaternion has 3 degrees of freedom in spite of having four components due to the

unit length constraint, which can be interpreted as a unit hyper-sphere, hence its topology being that

of the special unitary group SU(2), diﬀeomorphic to S(3).

Implementation in MRPT

Poses based on quaternions are implemented in the class mrpt::poses::CPose3DQuat. The quaternion

part of the pose is always normalized (i.e. q

+ q

= 1).

# includ e < mrpt / poses / CPose 3 DQuat .h >

usi ng n a m espace mrpt :: poses ;

usi ng n a m espace mrpt :: math ;

CPos e3DQua t p ( 1.0 /* x */ ,2.0 /* y */ ,3.0 /* z */ ,

CQuat e r nionDou b l e (1.0 /* qr */ , 0.0 ,0.0 ,0.0 /* v e ctor pa rt */ ) );

Normalization of a quaternion

In many situations, the quaternion part of a 3D+Quat 7D representation of a pose may drift away of

being unitary. This is specially true if each component of the quaternion is estimated independently,

such as within a Kalman ﬁlter or any other Gauss-Newton iterative optimizer (for an alternative, see

§10).

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SE(3)变换参数化教程：旋转表示与优化详解

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