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Nonlinear least squares and SVD based algorithms for the estimation of a coordinate transformation
Gait & Posture 29S (2009) e1–e31
Contents lists available at ScienceDirect
Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Abstracts of the SIAMOC 2008 Congress
Nonlinear least squares and SVD based algorithms for the estimation of a coordinate transformation D. Sgattoni *, S. Fioretti Department of Electromagnetics and Bioengineering, Universita` Politecnica delle Marche, Ancona, Italy
Introduction: A typical problem in movement analysis is to determine, with the maximum accuracy, the coordinate transformation that allows to pass from one reference system to another, when the coordinates of N points embedded to the rigid body, and expressed both in a local reference system (SL) and in a global reference system (SG), are known. The aim of this work is the implementation of a recursive nonlinear least-squares algorithm (LSA) [1] to estimate the seven parameters of the coordinate transformation, assuming that the global coordinates are affected both by additive random noise and by systematic error. The results have been compared with those obtain through a well known algorithm, based on the Singular Value Decomposition (SVD) [2]. Methods: The coordinate transformation taken in account is yi ¼ sRxi þ k
(1)
where yixi are the ideal coordinates of a generic point, expressed in SG and in SL, respectively, s is the scale factor, R is the rotation matrix, and k is the translation vector. The SVD algorithm determines s, R and k minimizing, in the least squares sense, P T the functional F SVD ¼ N1 N i¼1 ð½sRxi þ k yi Þ ð½sRxi þ k yi Þ (with N 3). The LSA algorithm estimates s, R and k minimizing the T functional F LSA ¼ ð˜l lÞ Q 1 ð˜l lÞ, where l is the noisy observations vector (expressing the SG coordinates), ˜l is the estimate of l, and Q is the covariance matrix related to l. For this purpose, Eq. (1) can be re-written as sRxi + k yi = 0, or, in general as Fðl; pÞ ¼ 0
(2)
where p is the 7 independent parameters vector to be estimated (1 for s, 3 for R and 3 for k). Equation (2) is linearized by Taylor’s series expansion truncated at the first order. The Q matrix takes into
Table 1 RMSE of 7 LSA and SVD estimated parameters: s = scale factor, a, b, g = independent angular parameters related to R; k1, k2, k3 = k components. RMSE
s
a (8)
b (8)
g (8)
k1 (mm)
k2 (mm)
k3 (mm)
LSA SVD
0.01 0.02
0.28 0.92
0.28 1.06
0.22 0.30
10.0 12.2
0.7 2.3
2.0 7.8
account both the random and the systematic error. The LSA adjusts iteratively both the estimate of the observations and of the 7 parameters values. The algorithms were tested in simulation that considered 6 points embedded into a truncated-cone-shaped rigid body (whose dimension could be compared with those of a thigh). To the SG coordinates, we have added both random error (with standard deviation = 0.6 mm) and systematic error characterized by a sinusoidal trend with an amplitude ranging between 0 and 25 mm, dependent on the point position in the local system. Results: Table 1 shows the RMSE related to the estimate of the 7 parameters transformation calculated with both algorithms. The simulation has been done with 200 different realizations of random and systematic noise. Discussion: The LSA algorithm estimates all the 7 parameters with a precision greater than the SVD based algorithm. This is due to the fact that LSA considers both the random and the systematic noise into the structure of covariance matrix Q. Moreover the LSA algorithm allows to do this, taking into account point by point, and for each coordinate, how the superimposed error is distributed in the simulated data. The LSA application could be useful as compensation technique for soft tissue artefacts, provided that one knows approximately their spatial distribution. References [1] Mikhail EM, Ackermann F. Observations and least squares. New York: IEP; 1976. [2] Challis JH. J Biomech 1995;28(6):733–7.
doi: 10.1016/j.gaitpost.2008.10.003
Movement of the shoulder complex: A preliminary study L. Ladislao 1,*, A. Rocchetti 2, G. Sartini 2, S. Fioretti 1 1
Department of Electromagnetics and Bioengineering, Universita` Politecnica delle Marche, Ancona, Italy 2 Myolab, Sporting Rehabilitation Centre, Jesi, Ancona, Italy
Introduction: The shoulder is a complex joint that has a range of motion greater than any other joint in the body. Consequently the shoulder complex is susceptible to a wide range of pathologies and injuries, such as impingement and instability. This is particularly frequent in sporting activities. This paper introduces a protocol for the measurement of shoulder movement that uses a
Contents lists available at ScienceDirect
Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Abstracts of the SIAMOC 2008 Congress
Nonlinear least squares and SVD based algorithms for the estimation of a coordinate transformation D. Sgattoni *, S. Fioretti Department of Electromagnetics and Bioengineering, Universita` Politecnica delle Marche, Ancona, Italy
Introduction: A typical problem in movement analysis is to determine, with the maximum accuracy, the coordinate transformation that allows to pass from one reference system to another, when the coordinates of N points embedded to the rigid body, and expressed both in a local reference system (SL) and in a global reference system (SG), are known. The aim of this work is the implementation of a recursive nonlinear least-squares algorithm (LSA) [1] to estimate the seven parameters of the coordinate transformation, assuming that the global coordinates are affected both by additive random noise and by systematic error. The results have been compared with those obtain through a well known algorithm, based on the Singular Value Decomposition (SVD) [2]. Methods: The coordinate transformation taken in account is yi ¼ sRxi þ k
(1)
where yixi are the ideal coordinates of a generic point, expressed in SG and in SL, respectively, s is the scale factor, R is the rotation matrix, and k is the translation vector. The SVD algorithm determines s, R and k minimizing, in the least squares sense, P T the functional F SVD ¼ N1 N i¼1 ð½sRxi þ k yi Þ ð½sRxi þ k yi Þ (with N 3). The LSA algorithm estimates s, R and k minimizing the T functional F LSA ¼ ð˜l lÞ Q 1 ð˜l lÞ, where l is the noisy observations vector (expressing the SG coordinates), ˜l is the estimate of l, and Q is the covariance matrix related to l. For this purpose, Eq. (1) can be re-written as sRxi + k yi = 0, or, in general as Fðl; pÞ ¼ 0
(2)
where p is the 7 independent parameters vector to be estimated (1 for s, 3 for R and 3 for k). Equation (2) is linearized by Taylor’s series expansion truncated at the first order. The Q matrix takes into
Table 1 RMSE of 7 LSA and SVD estimated parameters: s = scale factor, a, b, g = independent angular parameters related to R; k1, k2, k3 = k components. RMSE
s
a (8)
b (8)
g (8)
k1 (mm)
k2 (mm)
k3 (mm)
LSA SVD
0.01 0.02
0.28 0.92
0.28 1.06
0.22 0.30
10.0 12.2
0.7 2.3
2.0 7.8
account both the random and the systematic error. The LSA adjusts iteratively both the estimate of the observations and of the 7 parameters values. The algorithms were tested in simulation that considered 6 points embedded into a truncated-cone-shaped rigid body (whose dimension could be compared with those of a thigh). To the SG coordinates, we have added both random error (with standard deviation = 0.6 mm) and systematic error characterized by a sinusoidal trend with an amplitude ranging between 0 and 25 mm, dependent on the point position in the local system. Results: Table 1 shows the RMSE related to the estimate of the 7 parameters transformation calculated with both algorithms. The simulation has been done with 200 different realizations of random and systematic noise. Discussion: The LSA algorithm estimates all the 7 parameters with a precision greater than the SVD based algorithm. This is due to the fact that LSA considers both the random and the systematic noise into the structure of covariance matrix Q. Moreover the LSA algorithm allows to do this, taking into account point by point, and for each coordinate, how the superimposed error is distributed in the simulated data. The LSA application could be useful as compensation technique for soft tissue artefacts, provided that one knows approximately their spatial distribution. References [1] Mikhail EM, Ackermann F. Observations and least squares. New York: IEP; 1976. [2] Challis JH. J Biomech 1995;28(6):733–7.
doi: 10.1016/j.gaitpost.2008.10.003
Movement of the shoulder complex: A preliminary study L. Ladislao 1,*, A. Rocchetti 2, G. Sartini 2, S. Fioretti 1 1
Department of Electromagnetics and Bioengineering, Universita` Politecnica delle Marche, Ancona, Italy 2 Myolab, Sporting Rehabilitation Centre, Jesi, Ancona, Italy
Introduction: The shoulder is a complex joint that has a range of motion greater than any other joint in the body. Consequently the shoulder complex is susceptible to a wide range of pathologies and injuries, such as impingement and instability. This is particularly frequent in sporting activities. This paper introduces a protocol for the measurement of shoulder movement that uses a