Identification of Fully Physical Consistent Inertial Parameters using
Optimization on Manifolds
Silvio Traversaro1, Stanislas Brossette2, Adrien Escande3, and Francesco Nori1
Abstract— This paper presents a new condition, the fully
physical consistency for a set of inertial parameters to determine
if they can be generated by a physical rigid body. The
proposed condition ensure both the positive definiteness and
the triangular inequality of 3D inertia matrices as opposed to
existing techniques in which the triangular inequality constraint
is ignored. This paper presents also a new parametrization
that naturally ensures that the inertial parameters are fully
physical consistency. The proposed parametrization is exploited
to reformulate the inertial identification problem as a manifold
optimization problem, that ensures that the identified parameters
can always be generated by a physical body. The proposed opti-
mization problem has been validated with a set of experiments
on the iCub humanoid robot.
I. INTRODUCTION
A large part of existing robotic systems are modeled as
a system of multiple rigid bodies. The knowledge of the
dynamical characteristics of these rigid bodies is a key as-
sumption of model-based control and estimation techniques.
The dynamics of a rigid body, i.e. how the acceleration of a
rigid body is related to the forces applied on it, is completely
described by the mass distribution of the body in the 3D
space. The mass distribution itself is completely described
by 10 inertial parameters [1]. These parameters may be
available if a good Computer-Aided Design (CAD) model
of the robot is available, but often such models are either
not available, or the mass distribution of the rigid bodies in
the robot changes during operation, as in the case of an end
effector that grabs and heavy object.
Inverse robot dynamics models can be written linearly
with respect to the inertial parameters of the rigid bodies
composing the robot. Classical identification techniques [1],
[2] consider the parameters of each body to be an element
of the Euclidean space R10. Exploiting this fact, the inertial
parameters identification problem has been classically posed
as a Linear Least Square optimization problem [1]. The
resulting problem is convenient from a computational point
of view, but it neglects the fact that not all vectors in R10 can
*This paper was supported by the FP7 EU projects CoDyCo (No. 600716
ICT-2011.2.1 - Cognitive Systems and Robotics), Koroibot (No. 611909
ICT-2013.2.1 - Cognitive Systems and Robotics) and the H2020 European
project COMANOID.
1Silvio Traversaro and Francesco Nori are with iCub Facility, Is-
tituto Italiano di Tecnologia, Via Morego 30, Genoa 16163, Italy
name.surname@iit.it
2Stanislas
Brossette
is
with
CNRS-UM2
LIRMM
Interactive
Digital
Humans,
UMR5600,
Montpellier,
France
and
CNRS-AIST
Joint
Robotics
Laboratory
(JRL),
UMI3218/RL,
Tsukuba,
Japan.
name.surname@gmail.com
3Adrien Escande is with CNRS-AIST Joint Robotics Laboratory (JRL),
UMI3218/RL, Tsukuba, Japan. name.surname@gmail.com
be generated by a physical rigid body, i.e. it is possible that
some inertial parameters are identified even if no physical
rigid body could generate them.
A necessary condition for the inertial parameters to be
generated by a physical rigid body was first proposed in
[3]: the physical consistency condition. This condition is
important for control purposes because it ensures, if it is
valid for all the links of a robot, the positive definiteness and
the invertibility of the joint mass matrix [4]. This property
is a key assumption in proving the stability of model-based
control laws. The physical consistency has been enforced in
identification of inertial parameters using several techniques:
[4], [5], [6], [7], [8], [9]. However this condition is not
sufficient: it is possible that some inertial parameters that
respect this condition do not correspond to any physical
body: in particular this condition does not encode the triangle
inequalities of the 3D inertia matrix [10, Chapter 3], as it
will be explained in the remainder of the paper.
The main contribution of this paper is a new necessary
and sufficient condition for the inertial parameters to be gen-
erated by rigid body: the full physical consistency condition.
We show that this condition implies the already proposed
physical consistency condition and that the triangle inequal-
ities are respected. Furthermore, we propose a nonlinear
optimization formulation that takes into consideration this
constraint by using state of the art optimization techniques
on non-Euclidean manifolds [11]. The proposed optimization
technique is validated with a rigid body inertial identification
experiment on the arm of the iCub humanoid robot.
For the sake of simplicity, in this paper, we only consider
the problem of identifying the inertial parameters of a single
rigid body. However, the full physical consistency condition
and the optimization on manifolds are general contributions,
that could be applied to the case of the identification of
inertial parameters in generic multibody structures.
The paper is organized as follows. Section II presents the
notations used in the paper and the background on rigid body
dynamics. Section III details the proposed full physical con-
sistency condition, the proposed nonlinear parametrization
of the inertial parameters that ensures that this condition is
always satisfied and the optimization technique on the mani-
fold of the proposed parametrization. Section IV describe the
experiments used for validation. Remarks and perspectives
conclude the paper.
II. BACKGROUND
A. Notation
The following notation is used throughout the paper.
arXiv:1610.08703v1  [cs.RO]  27 Oct 2016
• The set of real numbers is denoted by R, while the set
of nonnegative real numbers is denoted by R≥0.
• Let u and v be two n-dimensional column vectors of
real numbers, i.e. u, v ∈Rn, their inner product is
denoted as u⊤v, with “⊤” the transpose operator.
• SO(3) denotes the set of R3×3 orthogonal matrices with
determinant equal to one, namely
SO(3) := { R ∈R3×3 | RT R = I3, det(R) = 1 }.
• Given u, v ∈R3, S(u) ∈R3×3 denotes the skew-
symmetric matrix-valued operator associated with the
cross product in R3, such that S(u)v = u × v
• The Euclidean norm of a vector of real numbers is
denoted by ∥·∥.
• 1n ∈Rn×n denotes the identity matrix of dimension n;
0n ∈Rn denotes the zero column vector of dimen-
sion n; 0n×m ∈Rn×m denotes the zero matrix of
dimension n × m.
• A denotes an inertial frame and B a body-fixed frame.
• pB ∈R3 denotes the origin of the B frame expressed in
the inertial frame, while ARB ∈SO(3) is the rotation
matrix that transforms a 3D vector expressed with the
orientation of the B frame in a 3D vector expressed in
the A frame. ω ∈R3 denotes the body angular velocity
of body B, defined as S(ω) = AR⊤
B
A ˙RB.
• v ∈R6 indicates the body twist [12, Chapter 3], i.e.
v =
AR⊤
B ˙pB
ω

• f ∈R6 indicates an external wrench exerted on the body
[12, Chapter 3], i.e.
f =
f
µ

where f is the 3D external force and µ the 3D external
moment, that are expressed in frame B.
• Given v =

v⊤
ω⊤⊤∈R6, v ¯×∗is the 6D force
cross product operator [13], defined as:
v ¯×∗=
S(ω)
03×3
S(v)
S(ω)

• Given a symmetric matrix I ∈R3×3 the vech operator
denotes the serialization operation on symmetric matri-
ces:
vech
 Ixx Ixy Ixz
Ixy Iyy Iyz
Ixz Iyz Izz

=


Ixx
Ixy
Ixz
Iyy
Iyz
Izz


• Given a symmetric matrix I ∈Rn×n, I ⪰0 denotes
that the matrix is positive semidefinite, i.e. that all its
eigenvalues are nonnegative.
• g ∈R3 is the constant vector of gravity acceleration
expressed with the orientation of the inertial frame A.
• ag ∈R6 denotes the proper body acceleration, i.e. the
difference between the body acceleration and the gravity
acceleration:
ag = ˙v −
 AR⊤
Bg
03

B. Rigid Body Dynamics
The well known Newton-Euler equations for a rigid body
[12, Chapter 3] are given by:
Mag + v ¯×∗Mv = f
(1)
where M ∈R6×6 is the 6D inertia matrix (also known as
spatial inertia):
M =
 m13
−mS(c)
mS(c)
IB

.
(2)
Where:
• m ∈R is the mass of the rigid body,
• c ∈R3 is the center of mass of the rigid body, expressed
in the frame B,
• IB ∈R3×3 is the 3D inertia matrix of the rigid body,
expressed with the orientation of frame B and with
respect to the frame B origin.
The 6D inertia matrix is parametrized by 10 parameters,
usually called the inertial parameters of the rigid body [1],
that are defined as π ∈R10:
π =


m
mc
vech(IB)

.
(3)
The Newton-Euler equations (1) can be written linearly
[14], [1] in the inertial parameters :
Y (ag, v)π = Mag + v ¯×∗Mv = f.
(4)
C. Relationship between the inertial parameters and the
density function
The mass distribution of a rigid body in space is described
by its density function:
ρ(·) : R3 7→R≥0.
(5)
The domain of this function is the points of body expressed
in the body-fixed frame B. We consider the density equal to
zero for the points outside the volume of the rigid body, so
we can define the domain of ρ(·) to be all the points in the
3D space R3.
The inertial parameters are obviously a functional of the
density ρ(·), in particular, we can define the functional πd(·) :
(R3 7→R≥0) 7→R10 that maps the density function to the
corresponding inertial parameters:
πd(ρ(·)) =


m(ρ(·))
mc(ρ(·))
vech(IB(ρ(·)))

=
=


RRR
R3 ρ(r)dr
RRR
R3 rρ(r)dr
vech
 
RRR
R3 S⊤(r)S(r)ρ(r)dr
!


.
(6)
D. 3D Inertia at the Center of Mass and Principal Axes
The 3D inertia at the center of mass is defined as
IC =
ZZZ
R3
S⊤(r −c)S(r −c)ρ(r)dr.
(7)
Exploiting the fact that S(·) is linear, the inertia matrix
with respect to the center of mass can be written as:
IC = IB + mS(c)S(c).
(8)
This result is known as parallel axis theorem.
As IC is symmetric, it can be diagonalized with an
orthogonal matrix Q ∈SO(3):
IC = Q diag (J)Q⊤.
(9)
Using (7) the diagonal matrix diag (J) (with J ∈R3) can
be written as:
diag (J)
=
ZZZ
R3
Q⊤S⊤(r −c)S(r −c)Qρ(r)dr =
ZZZ
R3
S⊤(Q⊤(r −c))S(Q⊤(r −c))ρ(r)dr. (10)
The operation mapping the body point r to QT (r −c) can
be interpreted as a change of reference frame, from the body
frame B to a frame C (a principal axes frame) whose origin
is the center of mass of the body and whose orientation is
one in which the IC matrix is diagonal.
By expressing with ˜r = Q⊤(r −c) the generic point of
the body expressed in the C frame, and with ˜ρ(˜r) the density
with respect to the C frame, we can write diag J as:
diag J =
ZZZ
R3
S⊤(˜r)S(˜r)ρ(˜r)d˜r.
(11)
The elements of the J vector are:
Jx =
ZZZ
R3
(˜y2 + ˜z2)˜ρ(˜r)dr′,
(12a)
Jy =
ZZZ
R3
(˜x2 + ˜z2)˜ρ(˜r)d˜r,
(12b)
Jz =
ZZZ
R3
(˜x2 + ˜y2)˜ρ(˜r)d˜r.
(12c)
We can write them as:
Jx = Ly + Lz,
Jy=Lx + Lz,
Jz=Lx + Ly. (13)
Lx =
ZZZ
R3
˜x2˜ρ(˜r)d˜r,
Ly=
ZZZ
R3
˜y2˜ρ(˜r)d˜r,
(14)
Lz =
ZZZ
R3
˜z2˜ρ(˜r)d˜r.
(15)
Where Lx, Ly, Lz are the central second moments of mass
of the density ˜ρ(˜r). It is clear that the non-negativity of ˜ρ(˜r)
constraints L =
Lx
Ly
Lz
⊤to be non-negative as well.
Furthermore, it is possible to see that the non-negativity of
L induces the triangular inequalities on diag (J):
Jx ≤Jy + Jz,
Jy ≤Jx + Jz,
Jz ≤Jx + Jy. (16)
E. Inertial Parameters Identification
Assuming that N values for F,Ag and V are measured,
equation (4) can be used to estimate π solving the following
optimization problem:
ˆπ = arg min.
π∈R10
N
X
i=1
∥Y (ag
i , vi)π −fi∥2 .
(17)
However, this optimization does not take into account the
physical properties of the inertial parameters π. For this
reason, the following definition was introduced.
Definition 1: A vector of inertial parameters π is called
physical consistent [3], [4] if:
m(π) ≥0,
IC(π)⪰0.
(18)
This condition has nice properties (it ensures that the matrix
M is always invertible), but is still possible to find some
physical consistent inertial parameters that can’t be generated
by a physical density.
III. CONTRIBUTION
A. Full physical consistency
In this subsection, we propose a new condition for assess-
ing if a vector of inertial parameters can be generated from
a physical rigid body. We will show that all the constraints
that emerge due to this full physical consistency condition
are due to the non-negativity on the density function.
Definition 2: A vector of inertial parameters π∗∈R10 is
called fully physical consistent if:
∃ρ(·) : R3 7→R≥0 s.t. π∗= πd(ρ(·)).
(19)
This definition extends the concept of physical consistent
inertial parameters to include also all possible constraints of
inertial parameters, such as the triangular inequalities (16)
of the diagonal elements of the inertia matrix.
Lemma 1: If a vector of inertial parameters π ∈R10
is fully physical consistent if follows that it is physical
consistent, according to Definition 1.
Proof: If π is fully physical consistent, then it follows
that there exists ρ(·) such that the corresponding 3D inertia
at the center of mass IC can be written as a function of ρ(·).
The positive semi-definiteness of m and IC then follows
from the classical properties of mass and the inertia matrix
of a rigid body, see for example subsection 3.3.3 of [10].
Lemma 2: If a vector of inertial parameters π ∈R10 is
fully physical consistent, the associated inertia matrices at the
body origin IB(π) and at the center of mass IC(π) respect
the triangular inequalities (16).
Proof: This lemma can be proved by writing IB or IC
as a functional of the density function ρ(·), as in the proof of
Lemma 1. Once IB or IC are written as a functional of ρ(·),
the demonstration that they respect the triangle inequality
can be found in any rigid body mechanics textbook, see for
example subsection 3.3.4 of [10].
To get a hint of the demonstration of Lemma 2, consider
that the diagonal elements of the 3D inertia matrix with
respect to an arbitrary frame can still be written as the sum
of two non-negative second moments of mass. The triangle
inequality then arises in a way similar to the case of the
inertia expressed in the principal axes.
B. Full physical consistent parametrization of inertia param-
eters
In this subsection, we introduce a novel nonlinear
parametrization of inertial parameters that ensures the full
physical consistency condition.
We choose to parametrize the mass as an element of the
spaces of non-negative numbers m ∈R≥0.
The center of mass do not have any constraints on its
location, so we choose to parametrize it as an element of the
3D space c ∈R3.
For parametrize the 3D inertia matrix ensuring the prop-
erties described in subsection II-D we choose the second
moments of mass L ∈R3
≥0 to be components of our
parametrization. In the following, we will show how this
choice ensures the full physical consistency of the resulting
inertial parameters.
The inertial parameters of a rigid body can then be
parametrized by an element θ ∈P = R≥0 × R3 × SO(3) ×
R3
≥0. In particular the components of θ are:
• m ∈R≥0 the mass of the body
• c ∈R3 the center of mass of the body
• Q ∈SO(3) the rotation matrix between the body frame
and the frame of principal axis at the center of mass
• L ∈R3
≥0 the second central moment of mass along the
principal axes
In other terms, there is a function πp(θ) : P 7→R10 that
maps this new parametrization to the corresponding inertial
parameters:
πp(θ) =

m(θ)
mc(θ)
vech(IB(θ))

=

m
mc
vech(Q diag (P L)Q⊤−mS(c)S(c))

Where P =
h 0 1 1
1 0 1
1 1 0
i
is a matrix that maps L to J.
Theorem 1: For each θ
∈P, there exists a density
function ρ(·) : R3 7→R≥0 such that πd(ρ(·)) = πp(θ),
i.e. every θ ∈P generates fully physical consistent inertial
parameters.
Proof: The proof is given in Appendix V-A.
Using this parametrization, it is possible to recast the
identification optimization problem (17) as:
ˆπ = π(ˆθ)
(20)
ˆθ = arg min.
θ∈P
N
X
i=1
∥Y (ag
i , vi)π(θ) −fi∥2
(21)
The main advantage of (20) with respect to (17) is that
thanks to Theorem 1 the identified inertial parameters ˆπ
are ensured to be fully physically consistent. However, the
optimization variable θ does not live anymore in a Euclidean
space, because P includes SO(3), so to solve this optimiza-
tion problem we exploit specific techniques related to the
optimization on manifolds.
C. Optimization on manifolds
In a nutshell, an n-dimensional manifold M is a space
that locally looks like Rn, i.e., for any point x of M there
exists a smooth map ϕx between an open set of Rn and a
neighborhood of x, with ϕx(0) = x.
For this paper, we focus on SO(3), a 3-dimensional
manifold. As such, it can be parametrized locally by 3
variables, for example, a choice of Euler angles, but any
such parametrization necessarily exhibits singularities when
taken as a global map (e.g. gimbal lock for Euler angles),
which can be detrimental to our optimization process.
For this reason, when addressing SO(3) with classical
optimization algorithms, it is often preferred to use one of
the two following parametrizations:
• unit quaternion, i.e. an element q of R4 with the
additional constraint ∥q∥= 1,
• rotation matrix, i.e. an element R of R3×3 with the
additional constraints RT R = I and det R ≥0.
The alternative is to use optimization software working
natively with manifolds [11][15] and solve
arg min.
θ∈R×R3×SO(3)×R3
N
X
i=1
∥Y (ag
i , vi)π(θ) −fi∥2
(22)
subj. to m ≥0, Lx ≥0, Ly ≥0, Lz ≥0 (23)
This alternative has an immediate advantage: we can
write directly the problem (20) without the need to add
any parametrization-related constraints. Because there are
fewer variables and fewer constraints, it is also faster to
solve. To check this, we compared the resolution of (20)
formulated with each of the three parametrizations (native
SO(3), unit quaternion, rotation matrix). We solved the
three formulations with the solver presented in [11], and the
two last with an off-the-shelf solver (CFSQP [16]), using
the dataset presented in section IV. The formulation with
native SO(3) was consistently solved faster. We observed
timings around 0.5s for it, and over 1s for non-manifold
formulations with CFSQP. The mean time for an iteration
was also the lowest with the native formulation (at least 30%
when compared to all other possibilities).
Working directly with manifolds has also an advantage
that we do not leverage here, but could be useful for future
work: at each iteration, the variables of the problem represent
a fully physical consistent set of inertial parameters. This
is not the case with the other formulations we discussed,
as the (additional) constraints are guaranteed to be satis-
fied only at the end of the optimization process. Having
physically meaningful intermediate values can be useful to
evaluate additional functions that presuppose it (additional
constraints, external monitoring . . .). It can also be leveraged
for real-time applications where only a short time is allocated
repeatedly to the inertial identification, so that when the
optimization process is stopped after a few iterations, the
output is physically valid. With non-manifold formulations,
at any given iteration, the parametrization-related constraints
can be violated, thus, the variables might not lie in the
manifold. It is then needed to project them on it. Denoting π
the projection (for example π =
q
∥q∥in the unit quaternion
formulation), to evaluate a function f on a manifold, we
need to compute f ◦π. If further the gradient is needed,
that projection must also be accounted for ([17] explains this
issue in great details for free-floating robots).
In this study, we use the same solver and approach
as presented in [11] which was inspired from [15]. The
driving idea of the optimization on manifold is to change the
parametrization at each iteration. The problem at iteration k
becomes:
min
zk∈Rn f ◦ϕxk(z)
s.t.
c ◦ϕxk(z) = 0.
(24)
Then xk+1 = ϕxk(zk) is guaranteed to belong to M. The
next iteration uses the same formulation around xk+1.
The smooth maps ϕx are built-in and are used automati-
cally by the solver while the user only has to implement the
functions of the optimization problem without the burden of
worrying about the parametrization.
IV. EXPERIMENTS
The iCub is a full-body humanoid with 53 degrees of
freedom [18]. For validating the presented approach, we
used the six-axis force/torque (F/T) sensor embedded in
iCub’s right arm to collect experimental F/T measurements.
We locked the elbow, wrist and hands joints of the arm,
simulating the presence of a rigid body directly attached to
the F/T sensor, a scenario similar to the one in which an
unknown payload needs to be identified [19].
FT sensor

Upper arm
?
Forearm







Fig. 1.
CAD drawing of the iCub arm used in the experiments. The used
six-axis F/T sensor is visible in the middle of the upper arm link.
We generated five 60 seconds joint positions paths in
which the three shoulder joints were reaching random joint
position using point to point minimum-jerk like trajectories.
The point to point trajectory completion times were 10s, 5s,
2s, 1s and 0.5s for the different paths. We played these joint
paths on the robot and we sampled at 100Hz the F/T sensors
and joint encoders output. We filtered the joint positions and
obtained joint velocities and accelerations using a Savitzky-
Golay filtering of order 2 and with a windows size of 499,
41, 21, 9, 7 samples. We used joint positions, velocities
and accelerations with the kinematic model of the robot
to compute ag and v of the F/T sensor for each time
sample. We removed the unknown offset from the F/T
measurements using the offset removal technique described
in [20]. We then solved the inertial identification problem
using the classical linear algorithm (17) and the one using
the proposed fully physical consistent parametrization (22).
We report the identified inertial parameters in Table I. It
is interesting to highlight that for slow datasets (trajectory
time of 10s or 5s) the unconstrained optimization problem
(17) results in inertial parameters that are not fully physical
consistency. In particular, this is due to the low values of
angular velocities and acceleration, that do not properly
excites the inertial parameters, which are then numerically
not identifiable. The proposed optimization problem clearly
cannot identify these parameters anyway, as the identified
parameters are an order of magnitude larger than the ones
estimated for faster datasets, nevertheless, it always estimates
inertial parameters that are fully physical consistent. For
faster datasets (trajectory time of 1s or 0.5s) the results
of the two optimization problems are the same because the
high values of angular velocities and accelerations permit to
identify all the parameters perfectly. While this is possible
to identify all the inertial parameters of a single rigid body,
this is not the case when identifying the inertial parameters
of a complex structure such as a humanoid robot, for
which both structural [2] and numerical [21] not identifiable
parameters exists. In this later application, the enforcement
of full physical consistency will always be necessary to get
meaningful results.
V. CONCLUSIONS
The condition for the full physical consistency of the
rigid body inertial parameters was introduced, to extend the
existing concept of the physical consistency of the inertial
parameters. A nonlinear parametrization of the inertial pa-
rameters that ensures full physical consistency was also intro-
duced and a nonlinear optimization on manifolds technique
was adapted to solve the resulting nonlinear identification
problem. The results have been validated with experiments
on the identification of the inertial parameters of the right
arm of the iCub humanoid robot.
For the sake of simplicity and for space constraints, in
this work, only the case of the identification of a single rigid
body was discussed. However, the full physical consistency
concept and the optimization on manifolds are general con-
cepts, that we plan to integrate with existing techniques for
whole body inertial parameters identification in humanoids
[2], [9] and for adaptive control of underactuated robots [22].
APPENDIX
A. Proof of Theorem 1
Proof: We prove the statement in a constructive way:
given an arbitrary element θ = (m, c, Q, L) ∈P we build
TABLE I
INERTIAL PARAMETERS IDENTIFIED WITH THE DIFFERENT DATASETS AND THE DIFFERENT OPTIMIZATION PROBLEMS.
Trajectory Time
Optimization Manifold
m
mcx
mcy
mcz
Ixx
Ixy
Ixz
Iyy
Iyz
Izz
R10
1.836
0.062
0.001
0.208
0.580
0.593
-0.541
1.022
0.190
-0.129
10s
P
1.836
0.062
0.001
0.208
0.215
0.012
-0.064
0.227
0.038
0.028
R10
1.842
0.061
0.000
0.206
0.128
-0.018
-0.125
0.125
0.026
-0.001
5s
P
1.842
0.060
0.000
0.206
0.166
0.001
-0.089
0.216
0.001
0.050
R10
1.852
0.060
0.001
0.206
0.065
0.001
-0.035
0.066
0.006
0.007
2s
P
1.852
0.060
0.001
0.206
0.067
0.001
-0.030
0.086
0.003
0.014
R10
1.820
0.060
0.002
0.205
0.032
0.0014
-0.017
0.036
0.002
0.008
1s
P
1.820
0.060
0.002
0.205
0.034
0.001
-0.015
0.042
0.001
0.009
R10
1.843
0.060
0.005
0.204
0.033
0.003
-0.014
0.035
0.000
0.008
0.5s
P
1.844
0.059
0.004
0.204
0.037
0.001
-0.013
0.039
0.000
0.008
Inertial parameters identified on R10 optimization manifold that are not fully physical consistent are highlighted.
Masses are expressed in kg, first moment of masses in kg.m, inertia matrix elements in kg.m2.
a density function ρ(·) : R3 7→R≥0 such that πp(θ) =
πd(ρ(·)). For example we can think of a cuboid of uniform
unit density, with the center of the cuboid coincident with
the center of mass of the inertial parameters (given by c),
with the orientation of its symmetry axis aligned with the C
principal axes frame defined by the Q rotation matrix and
the cuboid sides of lengths 2dx , 2dy and 2dz, with:
d =

dx
dy
dz
⊤=
hq
3 Lx
m
q
3 Ly
m
q
3 Lz
m
i⊤
(25)
Its density function in the C frame is given as:
˜ρ(˜r) =
(
1
if −d ≥˜r ≥d
0
otherwise
while the density function in the B frame is given by:
ρ(r) =
(
1
if −Qd + c ≥r ≥Qd + c
0
otherwise
.
(26)
The density defined in (26) and (25) can be seen as a
function γ(·) : P 7→(R3 7→R≥0). The theorem is then
demonstrated by using (6) and (20) to verify that:
πd(γ(θ)) = πp(θ)
is true ∀θ ∈P.
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