# Estimation of rigid body parameters using the matrix logarithm

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```					       Estimation of rigid body parameters using the matrix logarithm

Nils Wagner
Institute for Statics and Dynamics of Aerospace Structures
University of Stuttgart, 70569 Stuttgart, Germany
e–mail: nwagner@isd.uni-stuttgart.de

ABSTRACT

Summary Recently, several techniques [1], [2], [3] have been developed for identifying rigid body properties of
a given system. A new method of estimating the location of the center of gravity (S), and the moments of inertia
of a rigid body is presented.
The linearized time domain equations of motion of an elastically supported rigid body for an arbitrary
reference point (B) can be expressed as
   
n                                                      ux
m I3 −m ˜BSr       ¨
uB                     T     uB         f(B)
+       ki gB Ei gB Ei        =                             , uB =  uy  , (1)
m ˜BS
r      J(B)      ϕ¨                            ϕ         m(B)
i=1                                                              uz
                                                    
ϕx                  0        −(zS − zB )  y S − yB
n
ϕ =  ϕy  , ˜BS =  zS − zB
r                         0      −(xS − xB )  , gB Ei =                                .
˜B Ei n
r
ϕz              −(yS − yB )   xS − xB         0

Index (Ei ) denotes support locations of spring elements ki on the rigid body; n is a unit vector de-
scribing the orientation of a spring, respectively. The whole system can be represented by the state-
˙
space system z = Az, where the state vector z is z = uB ϕ            ˙
uB ϕ  ˙      and A is deﬁned as
O I6
A=                       .
−M−1 K O
After sampling with constant period ∆t and transformation of the ﬁrst order differential equations into
a discrete time equation, one obtains the discrete time state-space model zk+1 = exp [A(p)∆t]zk .
The ﬁrst goal is to obtain an estimate of the system matrix A from the observations. One way to do
this is to form the matrices Z0 = z0 z1 . . . zn−1 , Z1 = z1 z2 . . . zn . Z1 and Z0
are combined by the exponential, Z1 = exp [A(p)∆t]Z0 and thus A can be found by a logarithmic
realization A(p) = ∆t log [Z1 Z+ ] =: G [4], where Z+ is the Moore-Penrose pseudoinverse of Z0 .
1
0                    0

The approach assumes the availability of the responses measurements at the full vector of degrees of
freedom. However, the angular quantities ϕ are more difﬁcult to measure than the translational ones.
The transformation
                       
uP1          I3 ˜BP1
r
 .   .              .  uB
 . = .
.           .     . 
.                                          (2)
ϕ
uPn          I3 ˜BPN
r

provides the relation between the motion at the reference point (B) and the pure translational measure-
ments at given points on the surface (Pi ). Therefore the six-degrees of freedom of the reference point (B)
can be obtained from this relationship by the least squares method. If the stiffness matrix K is known,
M can be obtained by the submatrix A21 = −M−1 K.
The parameterisation of the mass matrix M = −KA−1 =: G is then done by decomposing it into a
21
sum of matrices
9
mI3 O
M = M0 +            Mi pi ,   M0 =               ,                          (3)
O O
i=1

xx   yy   zz   xy   xz   yz
where 9 quantities can be deﬁned pT =      xBS yBS zBS J(B) J(B) J(B) J(B) J(B) J(B) .
Manipulating this relationship leads to a matrix equation Bp = f , where B = b1 . . .        b9    is a real
36 × 9 matrix with bi = vec(Mi ) and f = vec(G − M0 ) is a real m−vector.

REFERENCES

[1] M. Niebergall, H. Hahn, Identiﬁcation of the ten inertia parameters of a rigid body. Nonlinear Dy-
namics, 13: 361-372, (1997).

[2] H. Lee, Y.-B. Lee, Y.-S. Park, Response and excitation points selection for accurate rigid-body inertia
properties identiﬁcation.Mechanical Systems and Signal Processing, 13: 571-592, (1999).

[3] S. M. Pandit, Z.-Q. Hu, Determination of rigid body characteristics from time domain modal test
data. Journal of Sound and Vibration, 177: 31-41, (1994).

[4] N. Wagner, The matrix logarithm and its application in system identiﬁcation. Proc. Int. Conf. on
Structural System Identiﬁcation, Kassel 2001.

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