Estimation of rigid body parameters using the matrix logarithm
Institute for Statics and Dynamics of Aerospace Structures
University of Stuttgart, 70569 Stuttgart, Germany
Summary Recently, several techniques , ,  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
m I3 −m ˜BSr ¨
uB T uB f(B)
+ ki gB Ei gB Ei = , uB = uy , (1)
r J(B) ϕ¨ ϕ m(B)
ϕx 0 −(zS − zB ) y S − yB
ϕ = ϕy , ˜BS = zS − zB
r 0 −(xS − xB ) , gB Ei = .
˜B Ei n
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
−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 , where Z+ is the Moore-Penrose pseudoinverse of Z0 .
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.
uP1 I3 ˜BP1
. . . uB
. = .
. . .
uPn I3 ˜BPN
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
sum of matrices
M = M0 + Mi pi , M0 = , (3)
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.
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