MPG - A Framework for Reasoning on 6 DOF Pose Uncertainty
Wendelin Feiten
Muriel Lang
Abstract— Reasoning about the pose, i.e. position and ori-
entation of objects is one of the cornerstones of robotic
manipulation under uncertainty. In a number of joint research
projects our group is developing a robotic perception system
that perceives and models an unprepared kitchen scenario with
many objects. Since no single sensor or measurement provides
sufficient information, a technique is needed to fuse a number
of uncertain estimates of the pose, i.e. estimates with a widely
stretched probability density function (pdf).
The most frequently used approaches to describe the pdfs are
sample based description and multivariate normal (Gaussian)
distributions. Sample based descriptions in 6D can describe
basically any type of pdfs, but they require a large number
of samples and there are no analytic formulae to fuse several
pdfs. For Gaussian distributions these formulae exist, but the
Gaussian distributions are unimodal and don’t model widely
spread distributions well.
In this paper we present a framework for probabilistic modeling
of 6D poses that combines the expressive power of the sample
based description with the conciseness and algorithmic power
of the Gaussian models. As parameterization of the 6D poses
we select the dual quaternions, i.e. any pose is represented by
two quaternions. The orientation part of a pose is described
by a unit quaternion. The translation part is described by
a purely imaginary quaternion. A basic probability density
function over the poses is constructed by selecting a tangent
point on the 3D sphere representing unit quaternions and taking
the Cartesian set product of the tangent space with the 3D
space of translations. In this 6D Euclidean space a 6D Gaussian
distribution is defined. Projecting this Gaussian back to the unit
sphere and renormalizing induces a distribution over 6D poses,
called a Projected Gaussian.
A convex combination of Projected Gaussians is called a
Mixture of Projected Gaussians (MPG). The set of MPG can
approximate the probability density functions that arise in our
application, is closed under the operations mentioned above
and allows for an efficient implementation.
I. INTRODUCTION
A framework for reasoning on the 6D pose should allow
for treating a 6D pose and a rigid motion in the same way.
This is important for the propagation of information, e.g. a
pose information taken by camera. The pose of this camera
is in itself uncertain w.r.t. the common reference frame for
several measurements.
The pose representation should use as few parameters as
possible. This reduces the required memory space, and if
more than the minimal number of parameters is used, it
makes the renormalization of the parameters easier. The
parameters of a composition of rigid motions should follow
from the parameters of the single motions in a simple
way. Also, the parameters should have no singularities or
Wendelin Feiten is with Corporate Technology, Intelligent Systems &
Control, Siemens AG, D-80200 Munich, Germany, Muriel Lang is with
Ludwig-Maximilians-University, Munich, Germany
discontinuities.
The representation of pdfs on the parameter space should
interface with the standard representations, i.e. on the
one hand it should be possible to find a correspondence
between a representation from the new framework and a
sample based description of a pdf, and on the other hand
a unimodal parametric description (like a Gaussian) should
be included in the new framework. Further on, we demand
that it is easy to calculate an estimate based on two pdfs
from the framework that describe the same pose. Last, but
not least, it should be straightforward to calculate the pdf
of a composition of rigid motions from the pdfs of the
individual rigid motions.
Since each position and orientation w.r.t a given coordinate
system is the result of a translation and a rotation. Position
and translation can be and will be used synonymously in
this paper, as well as orientation and rotation. Also, pose
and rigid motion are used synonymously.
In Section II we will recapitulate various approaches
to the parametrization of rigid motions and corresponding
probability density functions. None of them fulfills all re-
quirements listed above, but they provide ingredients to our
synthesis. In Section III we will present our approach to
probability density functions over rigid motions. The relation
between sample based descriptions and the MPG framework
is described in Section IV, and the convergence properties are
investigated in Section V. We describe the implementation
and experimental results in Section VI. In Section VII we
will recollect the presented system and indicate directions of
future work.
II. RELATED WORK
The Mixture of Projected Gaussians (MPG) was first
presented by Feiten et al. [1] and then expanded by Muriel
Lang in her Diploma Thesis [2]. See there for additional
references on previous work concerning the parameterization
of the rotation in 3D and the rigid motion.
The representation of rigid motions and especially of
orientation in three dimensions is a central issue in various
disciplines of arts, science and engineering. Rotation matrix,
Euler angles, Rodrigues vector and unit quaternions are
the most popular representations of a rotation in three
dimensions.
Rotation
matrices
have
many
parameters,
Euler angles are not invariant under transforms and have
singularities and Rodrigues vectors do not allow for an easy
composition algorithm. Stuelpnagel [3] points out that unit
quaternions are a suitable representation of rotations on the
hypersphere S3 with few parameters, but does not provide
arXiv:1707.01941v1  [cs.RO]  5 Jul 2017
probability distributions. Choe [4] represents the probability
distribution of rotations via a projected Gaussian on a
tangent space. He only deals with concentrated distributions
and does not take translations into account. Goddard and
Abidi [5], [6] use dual quaternions for motion tracking.
In their observations the correlation between rotation and
translation is captured also. The probability distribution
over the parameters of the state model is a unimodal normal
distribution. If the initial estimate is sufficiently certain
and if the information that shall be fused to the estimate
is sufficiently well focused this is an appropriate model.
As can be seen in [7] from Kavan et al. dual quaternions
provide a closed form for the composition of rigid motions,
similar to the transform matrix in homogeneous coordinates.
Antone [8] suggests to use the Bingham distribution in
order to represent weak information even though he does
not give a practical algorithm for fusion of information
or propagation of uncertain information. By now it is
known that propagated uncertain information only can be
approximated by Bingham distributions. Further Love [9]
states that the renormalization of the Bingham distribution
is computationally expensive. Glover [10] also works with
a mixture of Bingham distributions and recommends to
create a precomputed lookup table of approximations of
the normalizing constant using standard floating point
arithmetic. Mardia et al. [11] use a mixture of bivariate von
Mises distributions. They fit the mixture model to a data set
using the expectation maximization algorithm because this
allows for modeling widely spread distributions. Translations
are not treated by them. To propagate the covariance matrix
of a random variable through a nonlinear function, the
Jacobian matrix is used in general. Kraft et al. [12] use
therefore an unscented Kalman Filter [13]. This technique
would have to be extended to the mixture distributions.
From the analysis of the previous work, we synthesize our
approach as follows: We use unit quaternions to represent
rotations in 3D, and dual quaternions to obtain a concise
algebraic description of rigid motions and their composition.
The base element of a probability distribution over the rigid
motions is a Gaussian in the 6D tangent space, characterized
by the tangent point to the unit quaternions and the mean and
the covariance of the distribution. Such a base element is
called a Projected Gaussian. We use Mixtures of Projected
Gaussians to reach the necessary expressive power of the
framework.
III. POSE UNCERTAINTY BY MIXTURES OF PROJECTED
GAUSSIAN DISTRIBUTIONS
We assume that the quaternion as such is sufficiently well
known to the reader. In order to clarify our notation, at first
some basics are restated.
A. Rigid Motion and Dual Quaternions
Let H be the quaternions, i.e
H = {q|q = a + ib + jc + kd & a, b, c, d ∈R}
(1)
where a is the real part of the quaternion, and the vector
v = (b, c, d)⊤is the imaginary part. The quaternions can be
identified with R4 via the coefficients,
q = a + i · b + j · c + k · d ∼[a, b, c, d].
The norm and the conjugate are denoted by ∥q∥and q.
Further we denote the unit quaternions by H1 and the
imaginary quaternions by Hi.
Analogously to the way that unit complex numbers z =
cos(φ) + i sin(φ) = eiφ represent rotations in 2D via the
formula prot = zp for any point p ∈C, unit quaternions
represent rotations in 3D.
A point (u, v, w)⊤in 3D is represented as the purely imag-
inary quaternion p = i·u+j ·v +k ·w; a rotation around the
unit 3D axis v by the rotation angle θ is represented by the
quaternion q = cos(θ/2) + sin(θ/2) (i · v1 + j · v2 + k · v3).
The rotated point is obtained as prot = q ∗p ∗q . Clearly,
q and −q represent the same rotation, so the set U of unit
quaternions is a double coverage of the special orthogonal
group SO(3) of rotations in 3D. The composition of rotations
corresponds to the multiplication of the corresponding unit
quaternions.
The rigid motions in 3D can in a similar way be repre-
sented by dual quaternions. Let’s again clarify the notation.
The ring of the dual quaternions with the dual unit ϵ, which
has the properties ϵ · 1 = 1 · ϵ = ϵ and ϵ2 = 0, is defined as:
HD = {dq | dq = q1 + ϵ · q2 & q1, q2 ∈H}
(2)
As for the quaternions, the addition and the scalar mul-
tiplication are component wise. The multiplication follows
from the properties of the dual unit. The quaternion conjugate
(there is also a dual conjugate and a total conjugate) is given
by q1 + ϵ · q2 := q1 + ϵ · q2.
The rigid motion consists of a rotation part, represented
by a rotation unit quaternion qr, and a translation part,
represented by a purely imaginary translation quaternion qt.
A point p = (x, y, z)⊤is embedded in HD by
pd := [1, 0, 0, 0] + ϵ · [0, x, y, z].
(3)
With these definitions, the dual quaternion representing the
rigid motion is defined by
dq := qr + ϵ1
2 · qt ∗qr
(4)
A point transformed by a rigid motion is represented by
dq ∗∗pd ∗∗dq
(5)
As before, a composition of rigid motions is represented by
the product of the corresponding dual quaternions.
Note that H1 × Hi is a double coverage of SE(3).
B. Base Element
A mixture distribution generally consists of a convex
combination of base elements. In our case, base elements
are pdfs that are induced from a Gaussian distribution on
R6. This space can be interpreted as a linearization of
SE(3) w.r.t. the dual quaternion representation at a rotation
represented by a unit quaternion q0 = [a, b, c, d]. We take
the tangent space in R4 to the unit sphere S3 at the point
(a, b, c, d)⊤. We provide it with the basis that we derive
from the canonical basis {e1, e2, e3, e4} in R4 by applying
the quaternion formulation for rotations in R4 with unit
quaternions ql and qr:
rot(p)
=
ql ∗p ∗qr
(6)
bi
:=
q0 ∗ei ∗qid
for i = 1 . . . 4
(7)
Note that b1 ∼q0. The vectors {b2, b3, b4} are an orthonor-
mal basis of the tangent plane. The coefficients of elements
of the tangent plane, together with the coefficients of the
translation part, constitute the 6D tangent space TSq0 of
the rigid motion. This tangent space is mapped to the rigid
motions by a central projection for the rotation part and an
embedding for the translation part.
Let (u, v, w, x, y, z) be a point in this tangent space. Then
qr :=
1
√
1 + u2 + v2 + w2 · (b1 + u · b2 + v · b3 + w · b4)
(8)
corresponds to a unit quaternion representing rotation. The
imaginary quaternion
qt := [0, x, y, z]
(9)
represents the translation. Then the dual quaternion accord-
ing to (4) represents the corresponding rigid motion. Since
the projections depend on q0 we denote this mapping by
Πq0 : TSq0 −→S3 × R3 ∼SE(3)
(10)
We extend Πq0 to be two-valued by letting both qr and −qr
be in the image. Thus all points on S3 are reached except
for those orthogonal to q0.
With this mapping we can define the base element.
Definition: Let q0 be an arbitrary unit quaternion. Further,
let TSq0 be the tangent space defined above, and let N(µ, Σ)
be a Gaussian distribution on TSq0, calling the corresponding
probability density function pT . Then the function pSE(3) on
S3 × R3 ∼SE(3) given by
pSE(3)(m)
:=
1
C · pT S
Π−1
q0 (m)


(11)
C
:=
Z
S3×R3 pT S
Π−1
q0 (m)


dm
(12)
is a pdf on SE(3). For rigid motions m0 with a rotation
quaternion orthogonal to q0 we let pSE(3)(m0) := 0 - this
is a smooth completion. We call this type of distribution a
Projected Gaussian or PG and denote it by
pSE(3) ∼N (q0, µ, Σ)
(13)
See Figure 1 for examples of projected Gaussians. Note
that the peaks are lower due to renormalization.
The subset of pdfs for which µu = µv = µw = 0 in the
corresponding Gaussian on the tangent space is referred to
as PG0.
Before extending our framework to Mixtures of Projected
Gaussians, we would like to explain how to fuse and how to
compose two Projected Gaussians.
(a)
(b)
Fig. 1.
(a) A base element on the unit circle, obtained by projecting a
Gaussian on a tangent line (b) An illustration of a Gaussian distribution
projected to a 2D sphere
C. Fusion and Composition of Projected Gaussians
Two Gaussian distributions p1 ∼N (µ1, Σ1) and p2 ∼
N (µ2, Σ2) on Rn pertaining to the same phenomenon are
fused by
p3 ∼N (µ3, Σ3)
µ3 = (Σ1 + Σ2)−1 · (Σ1 · µ2 + Σ2 · µ1)
Σ3 =
Σ−1
1
+ Σ−1
2

−1
(14)
We can generalize this to PGs p1 ∼N (q1, µ1, Σ1)
and p2 ∼N (q2, µ2, Σ2) only if the tangents spaces are
reasonably close - the angle between the normals to the
tangent spaces should be less than 15◦, or, equivalently by
switching to the antipodal tangent point, larger than 165◦.
Then we define q3 by renormalizing q1+q2 to length 1 and
restate the original distributions approximately (we show p1,
p2 works the same way). With the Jacobian J at the mean
value µ1 of the mapping
f = Π−1
q3 ◦Πq1 : TSq1 −→TSq3
(15)
the parameters are estimated as
µ3,1 = f (µ1) and Σ3,1 = J · Σ1 · J⊤
(16)
Renormalization of the base elements involves an integral
that is hard to approximate with quadrature techniques.
Therefore we use Monte Carlo integration. Since the inte-
grands are bounded by exponential functions that are easy
to sample from, the integration is reasonably efficient.
The estimates resulting from both original distributions are
then fused on the common tangent space TSq3 using (14).
Generally, the rotation part of µ3 is not zero. Since it is
advantageous to refrain to base elements of type PG0, p3 is
restated as above with the new tangent point q4 = Πq3 (µ3).
The resulting probability density function on S3 × R3 needs
to be normalized according to Definition 1. The fused pdf
is denoted as p3 = φ (p1, p2).
In robotics we frequently need to estimate the pdf of
a composition of two subsequent rigid motions given the
pdfs of the individual rigid motions (e.g. from an uncertain
position in an uncertain camera frame to a position in world
coordinates).
Without loss of generality we refrain to PG0 to define the
composition, so p1 ∼N (q1, 0, Σ1) and p2 ∼N (q2, 0, Σ2).
From the composition in terms of dual quaternions dq3 =
dq2 ∗∗dq1 the natural choice for the tangent point q3 of the
composition is q3 = q2 ∗q1. This induces a mapping g
g : TSq2 × TSq1 −→TSq3
g (y2, y1) : = Π−1
q3 (Πq2 (y2) ∗∗Πq1 (y1))
(17)
Note that g (0, 0) = 0, thus µ3 = 0. With
Jγ =
∂g
∂(y2,y1)

(0,0) and Σγ =

Σ2
0
0
Σ1

the resulting
covariance matrix of the composition is Σ3 = Jγ · ΣC · J⊤
γ .
The composition p3 ∼N (q3, 0, Σ3) is denoted by
p3 = γ (p2, p1).
D. Mixture of Projected Gaussians MPG
As stated above, a precondition for the fusion of PG base
elements is that their tangent points are sufficiently close to
each other and that they are sufficiently well concentrated.
For this reason, widely spread probability density functions
should not be modeled in a single base element. Instead,
we use a mixture of PG base elements. Thus let pi ∈PG
be base elements, then a Mixture of Projected Gaussians is
defined as
pm = 1
n
n
X
i=1
πi · pi
with
0 ≤πi ≤1
and
n
X
i=1
πi = 1
(18)
Fusion and composition of PGs carry over to MPGS in
a similar way as this work for Mixtures of Gaussians [14].
Let pm,1 = 1
n
Pn
i=1 π1,ip1,i and pm,2 = 1
l
Pl
j=1 π2,jp2,j
be MPGS. The fused mixture pm,3 = Φ (pm,1, pm,2) is
obtained by fusing and weighting the base elements of the
original mixtures:
pm,3 = 1
C ·
n,l
X
i,j=1
λi,j · π1,i · π2,j · φ (p1,i, p2,j)
(19)
with a normalizing constant
C =
n,l
X
i,j=1
λi,j · π1,i · π2,j
(20)
The weights π1,i and π2,j are those of the prior mixture. The
plausibility is composed of two factors, λi,j = αi,j · δi,j.
The factor α says whether the mixture elements can share
a tangent space and thus probably pertain to the same cases
in the mixture (for detail see [2]). The factor δ is the
Mahalanobis distance of the mean values and covariances,
transported to the common tangent space.
δi,j = e−1/2·(µ3,1,i−µ3,2,j)·(Σ3,1,i+Σ3,2,j)−1·(µ3,1,i−µ3,2,j)⊤
It expresses that even if the mixture elements could share a
tangent space, they could still not be compatible.
The composition pm,3 = Γ (pm,1, pm,2) carries over in a
similar manner. In this case, there is no question of whether
two base elements could apply at the same time, since the
two probability distributions are assumed to be independent,
so the factor λi,j is omitted.
pm,3 = 1
C ·
n,l
X
i,j=1
π1,i · π2,j · γ (p1,i, p2,j)
(21)
with
C =


n,l
X
i,j=1
π1,i · π2, j

.
(22)
Note that in both cases the individually fused or combined
resulting base elements are assumed to be renormalized.
IV. MPG AND SAMPLE BASED DESCRIPTION OF pdfs
The MPGs try to fill the middle ground between sample
based descriptions and unimodal parametric descriptions of
pdfs. In our perception framework, we use sample based
descriptions a lot, so we need to restate pdfs available in
one description also in the other one.
Sampling from MPG is easy, we first sample from the
discrete pdf induced by the weights, and then draw a sample
for the Gaussian on the tangent space using the Box-Muller
algorithm.
For fitting a MPG to a sample set, we use a slight variant
of the Expectation Maximization Algorithm (see [15]):
1) Set the initial value for the means µi, covariance
matrices Σi and weighting coefficients λi and evaluate
the log likelihood with these values.
2) E step:
Evaluate the responsibilities γ(xn,i) using the current
parameter values
γ(zj,i) :=
λiN(xj|TSi, µi, Σi)
P
k λkN(xj|TSk, µk, Σk)
3) M step:
Reestimate the parameters using the current responsi-
bilities
µnew
i
= 1
Ni
N
X
j=1
γ(zj,i) · xj
Σnew
i
= 1
Ni
N
X
j=1
γ(zj,i)(xj −µnew
i
)(xj −µnew
i
)⊤
λnew
i
= Ni
N
where Ni = PN
j=1 γ(zj,i)
4) Evaluate the log likelihood:
ln P(X|TS, µ, Σ, λ)=
N
X
j=1
ln
 n
X
i=1
λiN(xj|TSi, µi, Σi)
!
and check for convergence of either the parameters or
the log likelihood. If the convergence criterion is not
satisfied return to the E step.
Note that we want to keep our base elements in PG0, so re-
estimating µi in the M-Step also means resetting the tangent
points qi.
V. CONVERGENCE PROPERTIES OF THE MPG
The good news of the MPG framework is that that we
get good approximations with few base elements, and that it
allows for fusing and composing pdfs without having to go
back to the original sensor readings.
The bad news is that the number of base elements tends
to increase quadratically with these operations. However, we
can drop base elements with small weights and we can merge
base elements with similar statistics without loosing much
information, while decreasing the number of base elements.
In order to use the framework for the task of grasping,
we defined a grasp criterion: Given a pdf p of an object
pose and a box B ⊂SE(3) that captures the tolerances
associated with the grasping task (e.g. width of the gripper,
possible grasp on the object), we try to find a rigid transform
m that maximizes the probability of successfully grasping:
m := arg
max
m∈SE(3)
Z
m(B)
p(x)dµm(B)
If pM(x) = Pn
i=1 λi · p(µi, Σi, x), the integral is not
very much affected by dropping a base element with a
small weight, let’s assume the last one. Renormalizing the
remaining weights
λ
′
i :=
λi
1 −λn
we get

Z
B
n
X
i=1
λipi dµB −
Z
B
n−1
X
i=1
λ′
ipi dµB

< 2λn
for any box B (for a proof see [2]).
The information loss due to combining two similar base
elements is investigated in terms of a symmetric version of
the Kullback-Leibler divergence, inspired by an investigation
of Runnalls for ordinary mixtures of Gaussians (see [16]).
Let’s assume the base elements p1 ∼N (TSq1, µ1, Σ1) and
p2 ∼N (TSq2, µ2, Σ2) from a MPG M have already been
restated to the same tangent space, i.e. q1 = q2. Then the
symmetric Kullback-Leibler divergence between them is
D =(Σ−1
1
+ Σ−1
2 )(µ1 −µ2)(µ1 −µ2)⊤
dsKL(p1, p2) =1
2tr
Σ−1
2 Σ1 + Σ−1
1 Σ2
+ D


−6
(23)
If we now replace both p1 and p2 with the combined base
element
p′ =
1
λ1 + λ2
· (λ1p1 + λ2 · p2)
to obtain the modified pdf M ′ with less base elements, then
we have:
dsKL(M, M ′) ≤
1
λ1 + λ2
·(λ1dsKL(p1, p′) + λ2dsKL(p′, p2))
(24)
For details and proofs see [2].
VI. IMPLEMENTATION AND EXPERIMENTAL RESULTS
The MPG framework is fully implemented in Python.
The probability density functions are visualized by drawing
random samples and for each pose painting a flag on the
screen. The foot of the flag represents position, the pole
represents the z-axis of the rotated coordinate system, and
the tip of the pennant marks the x-axis. As an application
example we demonstrate how to estimate the pose of an
object based on SIFT features, e.g. the salt box of figure 2.
Let’s assume that the robot detects the features ’B’ of the
word Bad (green) and ’l’ of Salz (blue). The mountain top
will be used as a third feature (purple).
(a)
(b)
Fig. 2.
(a) salt box (b) Stereo SIFT features on the salt box. The features
have an orientation around the sight line, but the visibility range is assumed
to be 15◦, the translation invariance is low. Each feature is represented by
a MPG with seven base elements.
(a)
(b)
Fig. 3.
(a) From the first two SIFT features, two distributions for the object
pose are derived by the composition of the rigid motions camera to feature
and feature to object coordinate system (b) The two distributions are fused
(red).
VII. CONCLUSION AND OUTLOOK
In this paper we present the framework of Mixtures of
Projected Gaussians that allows for modelling a large variety
of possible probability distribution functions of 6D poses. In
contrast to a sample based description, much fewer parame-
ters are needed to describe the distribution. The framework
interfaces well with the sample based descriptions, and
provides the inference mechanisms of fusing and composing
uncertain pose information.
The operations of fusion, propagation or multiplication
of MPG distributions generally result in a large number of
mixture elements. However, many of them have practically
(a)
(b)
Fig. 4.
(a) From originally 49 base elements, ten with low weights are
skipped (b) The base elements are merged until only ten are left.
(a)
(b)
Fig. 5.
(a) The object pose estimate from the next feature is added ... (b)
... and fused with the previous estimate (turquoise).
zero weight, while others are approximately identical. This is
used to drop some base elements, and to merge others, thus
reducing their number without losing much information.
The algorithms for probabilistic inference (fusion, propa-
gation, multiplication) are fully implemented in Python.
The covariance matrices are currently estimated using the
Jacobian of the non-linear transforms. These estimates could
be improved by using the unscented estimation technique
(see Julier and Uhlmann [13]).
In this paper we focus on the perception of static objects.
The MPG framework can be extended to the dynamic case
as well, following concepts by Goddard [6] and by Brox et
al. [17].
VIII. ACKNOWLEDGEMENT
This work was made possible by funding from the
ARTEMIS Joint Undertaking as part of the project R3-COP
and from the German Federal Ministry of Education and
Research (BMBF) under grant no. 01IS10004E.
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