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Comment on the paper “A technique for obtaining spatial kinematic parameters of segments of biomechanical systems from cinematographic data”
LETTER
TO THE EDITOR
COMMENT ON THE PAPER “A TECHNIQUE FOR OBTAINING SPATIAL KINEMATIC PARAMETERS OF SEGMENTS OF BIOMECHANICAL SYSTEMS FROM CINEMATOGRAPHIC DATA”*
Miller er al. (1980) have presented an interesting treatment of the methodological and experimental aspects of 3-D kinematics of rigid bodies. However, their paper would seem to warrant some comments of a fundamental and computational nature. First, the position and attitude of a rigid body is estimable from knowIedge of 3 ~o~-co~li~ear points. With observations from only onecamera, stable but not always unique sohttions of a comparatively low precision may be evaluated; in this situation, precision is worst along the camera axis and about the camera’s image axes, in a similar fashion as discussed in Woltring (1980). Secondly, the authors seem to confound the technique of least squares with the properties of statistical error modelling and propagation (see Mikhail, 1976, Chap. 5.3). Admittedly, their equation (10) yields a least squares solution, but this fact does not in itself render their equation (11) a covariance estimate. Such is only the case after defining an additive, stochastic error term in their equation (9), Z&A< I- F, + v
(9’)
where the noise vector v has certain known (or assumed) statistical properties. Typically, E(v) = 0, and E(vvr) = tr2 V, with V known and positive definite, and D possibly unknown. For such a model, the minimum oariance adjustment is identrcal to the weighted least squares adjustment At=(Z~V-‘Z,)-‘ZTV-‘F,
(lo’)
Upon convergence. the covariance matrix of the last adjustment is equal to the covariance matrix of the final estimate. If 0 is unknown, Miller et al’s residual sum of squares S’ in their equation (11) provides an unbiased estimate of ci’; see Mikhail (1976). In many cases, V may be taken to be the identity matrix, i.e. the noise terms in Y are uncorrelated and have the same standard deviation o. In that case (IO’) reduces to the least squares adjustment (10) in Miller vr ul. Generally. if V has a (block) diagonal structure, the admstment (ill’) may be more efficiently evaluated without explicit inversion of the matrix product (2,’ V-i Z,). Also, if only the diagonal elements of the a posteriori covariance matrix are required. complete inversion of this matrix product may be avoided (see Mikhail, 1976, Chap. 11.6.5). It is worthwhile to note that these diagonal elements may have large values by virtue of large correlations between estimated parameters. For example, the Euler angles a and y become perfectly correlated for b = 0” or 180”, since the first rotation axes is mapped onto the third, with only (n&y) being estimable. This will result in numerical problems with assessing (IO’) and with convergence attainment on the basis
of the criterium proposed by Miller et al. (p. 538 ). smce Aa and Ay may assume arbitrary values, with only A(a i. ;‘I becoming vanishingly small. In this case, one should either adopt a different rotation sequence (Ohkami. 1976). or adjoin a constraint equation of the type Aa = 0 or A;, = 0 to the original linearized system (9’): see Woltring (1980). Furthermore, the estimated standard deviations may serve as guidelines to establish convergence levels for parameter adjustments in (10’): if the criteria are chosen too small, numerical problems may result in convergence failure due to. for example, oscillations between two neighbourmg final values, especially in the less well-condittoned, single-camera case. Thirdly, Miller er al. suggest the use of more powerful optimization algorithms if the initial value when based on the final estimate at the previous measurement instant. is too far away (p. 538). In this writer’s opinion, extrapolation from the previous two estimates is a simpler and more efficient alternative. A generalization of these approaches is to define a srare-space model of rigid body positions, attitudes and their (first and second) derivatives, and to apply optimal prediction, filtering, and smoothing techniques (see, e.g. Gelb, 1974). This has the additional advantages of simultaneous derivative estimation and of automatic interpolation m the case of partial or complete data loss durmg finite time intervals, for example, due to loss of sight by one camera Unfortunately, such more umversal procedures are numerically expensive, and they require familiarity with contemporary developments in the realm of optimal control, system identification, and parameter estimation.
HFRV~L J Wru ~KIU~; Laboratory for Medical Unic;ersiry of Nijrneyen. P.O. Box 9101,
NL-6500
Phy.sics and B~oph\ ,$I(‘,
HB N~}nege?z,
The Netherlands
REFERENCES Gelb, A. (Ed.) (1974) Applied Optmu/ Est~mur~on M.I.T. Press, Cambridge. Mikhail, E. M. (1976) Observations and Leusr .Sqwr~. I E.P ~ New York. Ohkami, Y. (1976) Computer algorithms for computation of kinematical relations for three attitude angle systems, Al.44 J1. 14, 1135-1137. Woltring, H. J. (1980) Planar control m multt-camera calibration for 3-D gait studies. J. Eioniechanirc 13, 39 48.
* By Norman R. Miller, Robert Shapiro and Thomas M. McLaughlin, Journ& of Biomechanics 13, 535-547 (1980). ‘77
TO THE EDITOR
COMMENT ON THE PAPER “A TECHNIQUE FOR OBTAINING SPATIAL KINEMATIC PARAMETERS OF SEGMENTS OF BIOMECHANICAL SYSTEMS FROM CINEMATOGRAPHIC DATA”*
Miller er al. (1980) have presented an interesting treatment of the methodological and experimental aspects of 3-D kinematics of rigid bodies. However, their paper would seem to warrant some comments of a fundamental and computational nature. First, the position and attitude of a rigid body is estimable from knowIedge of 3 ~o~-co~li~ear points. With observations from only onecamera, stable but not always unique sohttions of a comparatively low precision may be evaluated; in this situation, precision is worst along the camera axis and about the camera’s image axes, in a similar fashion as discussed in Woltring (1980). Secondly, the authors seem to confound the technique of least squares with the properties of statistical error modelling and propagation (see Mikhail, 1976, Chap. 5.3). Admittedly, their equation (10) yields a least squares solution, but this fact does not in itself render their equation (11) a covariance estimate. Such is only the case after defining an additive, stochastic error term in their equation (9), Z&A< I- F, + v
(9’)
where the noise vector v has certain known (or assumed) statistical properties. Typically, E(v) = 0, and E(vvr) = tr2 V, with V known and positive definite, and D possibly unknown. For such a model, the minimum oariance adjustment is identrcal to the weighted least squares adjustment At=(Z~V-‘Z,)-‘ZTV-‘F,
(lo’)
Upon convergence. the covariance matrix of the last adjustment is equal to the covariance matrix of the final estimate. If 0 is unknown, Miller et al’s residual sum of squares S’ in their equation (11) provides an unbiased estimate of ci’; see Mikhail (1976). In many cases, V may be taken to be the identity matrix, i.e. the noise terms in Y are uncorrelated and have the same standard deviation o. In that case (IO’) reduces to the least squares adjustment (10) in Miller vr ul. Generally. if V has a (block) diagonal structure, the admstment (ill’) may be more efficiently evaluated without explicit inversion of the matrix product (2,’ V-i Z,). Also, if only the diagonal elements of the a posteriori covariance matrix are required. complete inversion of this matrix product may be avoided (see Mikhail, 1976, Chap. 11.6.5). It is worthwhile to note that these diagonal elements may have large values by virtue of large correlations between estimated parameters. For example, the Euler angles a and y become perfectly correlated for b = 0” or 180”, since the first rotation axes is mapped onto the third, with only (n&y) being estimable. This will result in numerical problems with assessing (IO’) and with convergence attainment on the basis
of the criterium proposed by Miller et al. (p. 538 ). smce Aa and Ay may assume arbitrary values, with only A(a i. ;‘I becoming vanishingly small. In this case, one should either adopt a different rotation sequence (Ohkami. 1976). or adjoin a constraint equation of the type Aa = 0 or A;, = 0 to the original linearized system (9’): see Woltring (1980). Furthermore, the estimated standard deviations may serve as guidelines to establish convergence levels for parameter adjustments in (10’): if the criteria are chosen too small, numerical problems may result in convergence failure due to. for example, oscillations between two neighbourmg final values, especially in the less well-condittoned, single-camera case. Thirdly, Miller er al. suggest the use of more powerful optimization algorithms if the initial value when based on the final estimate at the previous measurement instant. is too far away (p. 538). In this writer’s opinion, extrapolation from the previous two estimates is a simpler and more efficient alternative. A generalization of these approaches is to define a srare-space model of rigid body positions, attitudes and their (first and second) derivatives, and to apply optimal prediction, filtering, and smoothing techniques (see, e.g. Gelb, 1974). This has the additional advantages of simultaneous derivative estimation and of automatic interpolation m the case of partial or complete data loss durmg finite time intervals, for example, due to loss of sight by one camera Unfortunately, such more umversal procedures are numerically expensive, and they require familiarity with contemporary developments in the realm of optimal control, system identification, and parameter estimation.
HFRV~L J Wru ~KIU~; Laboratory for Medical Unic;ersiry of Nijrneyen. P.O. Box 9101,
NL-6500
Phy.sics and B~oph\ ,$I(‘,
HB N~}nege?z,
The Netherlands
REFERENCES Gelb, A. (Ed.) (1974) Applied Optmu/ Est~mur~on M.I.T. Press, Cambridge. Mikhail, E. M. (1976) Observations and Leusr .Sqwr~. I E.P ~ New York. Ohkami, Y. (1976) Computer algorithms for computation of kinematical relations for three attitude angle systems, Al.44 J1. 14, 1135-1137. Woltring, H. J. (1980) Planar control m multt-camera calibration for 3-D gait studies. J. Eioniechanirc 13, 39 48.
* By Norman R. Miller, Robert Shapiro and Thomas M. McLaughlin, Journ& of Biomechanics 13, 535-547 (1980). ‘77