Constrained least-squares optimization for robust estimation of center of rotation

This paper presents a new direct method for estimating the average center of rotation (CoR). An existing least-squares (LS) solution has been shown by previous works to have reduced accuracy for data with small range of motion (RoM). Alternative methods proposed to improve the CoR estimation use iterative algorithms. However, in this paper we show that with a carefully chosen normalization scheme, constrained least-squares solutions can perform as well as iterative approaches, even for challenging problems with significant noise and small RoM. In particular, enforcing the normalization constraint avoids poor fits near plane singularities that can affect the existing LS method. Our formulation has an exact solution, accounts for multiple markers simultaneously, and does not depend on manually-adjusted parameters. Simulation tests compare the method to four published CoR estimation techniques. The results show that the new approach has the accuracy of the iterative methods as well as the short computation time and repeatability of a least-squares solution. In addition, application of the new method to experimental motion capture data of the thumb carpometacarpal (CMC) joint yielded a more plausible CoR location compared to the previously reported LS solution and required less time than all four alternative techniques.     

A technique for joint center analysis using a stored program calculator

For a long time in the study of joint kinematics, the instant center of rotation in plane motion was obtained through graphic drawings. Since then the study of joint kinematics has become three-dimensional involving the use of computers. For this paper a stored-program calculator has been used as it is a precise instrument and several films can be used even if their positions are very close to one another.

A movement is never perfectly plane, it was important to define a coefficient (in percentage) to qualify the more or less plane character of a movement.

We believe that an analytical location is a better way than using graphic drawings of I.C.R.:

1. (1) to smooth the raw coordinates;

2. (2) to calculate the plane motion coefficient in order to eliminate an X-Ray picture of a whole series of pictures for lack of plane character;

3. (3) to define in the results an error rectangle whose dimensions are linked to errors in the observation and then to pick out among the points of a body those with the smallest risk for error.

To probe this method the two radio-ulnaris joints have been studied. At present studies are being carried on to compare the I.C.R.'s behaviour of the lumbar spine during a motion of lateral inflexion both in the case of normal people and people with scoliosis.     

A new method for estimating the axis of rotation and the center of rotation.

A new method is presented for estimating the parameters of two different models of a joint. The two models are: (1) A rotational joint with a fixed axis of rotation, also referred to as a hinge joint and (2) a ball and socket model, corresponding to a spherical joint. Given the motion of a set of markers, it is shown how the parameters can be estimated, utilizing the whole data set. The parameters are estimated from motion data by minimizing two objective functions. The method does not assume a rigid body motion, but only that each marker rotates around the same fixed axis of rotation or center of rotation. Simulation results indicate that in situations where the rigid body assumption is valid and when measurement noise is present, the proposed method is inferior to methods that utilize the rigid body assumption. However, when there are large skin movement artefacts, simulation results show the proposed method to be more robust.     

Tests of a linear model of visual-vestibular interaction using the technique of parameter estimation

The goal of this study was to test whether a superposition model of smooth-pursuit and vestibulo-ocular reflex (VOR) eye movements could account for the stability of gaze that subjects show as they view a stationary target, during head rotations at frequencies that correspond to natural movements. Horizontal smooth-pursuit and the VOR were tested using sinusoidal stimuli with frequencies in the range 1.0–3.5 Hz. During head rotation, subjects viewed a stationary target either directly or through an optical device that required eye movements to be approximately twice the amplitude of head movements in order to maintain foveal vision of the target. The gain of compensatory eye movements during viewing through the optical device was generally greater than during direct viewing or during attempted fixation of the remembered target location in darkness. This suggests that visual factors influence the response, even at high frequencies of head rotation. During viewing through the optical device, the gain of compensatory eye movements declined as a function of the frequency of head rotation (P < 0.001) but, at any particular frequency, there was no correlation with peak head velocity (P > 0.23), peak head acceleration (P > 0.22) or retinal slip speed (P > 0.22). The optimal values of parameters of smooth-pursuit and VOR components of a simple superposition model were estimated in the frequency domain, using the measured responses during head rotation, as each subject viewed the stationary target through the optical device. We then compared the model's prediction of smooth-pursuit gain and phase, at each frequency, with values obtained experimentally. Each subject's pursuit showed lower gain and greater phase lag than the model predicted. Smooth-pursuit performance did not improve significantly if the moving target was a 10 deg × 10 deg Amsler grid, or if sinusoidal oscillation of the target was superimposed on ramp motion. Further, subjects were still able to modulate the gain of compensatory eye movements during pseudo-random head perturbations, making improved predictor performance during visual-vestibular interactions unlikely. We conclude that the increase in gain of eye movements that compensate for head rotations when subjects view, rather than imagine, a stationary target cannot be adequately explained by superposition of VOR and smooth-pursuit signals. Instead, vision may affect VOR performance by determining the context of the behavior. Received: 16 June 1997 / Accepted: 5 December 1997     

The colorimetric estimation of protein by the Lowry technique using a liquid scintillation counter

A comparison has been made of the use of the spectrophotometer and liquid scintillation counter for the colorimetric quantification of protein. The attenuation of photon detection from sealed miniature ¹⁴C radioactive standards enabled protein quantification over a concentration range far in excess of that achievable by the use of a spectrophotometer. Accuracy of quantification was high over the entire range of protein concentration. The ability of the technique to provide quick and accurate protein estimations is discussed.     

Movement distances and density estimation of small mammals using the spool-and-line technique

Edge effect is an inherent problem when using trapping grids to estimate density of small mammals, resulting in a sampling area larger than the area of the grid. Distances between captures of individuals are used to estimateA(W), the effective sampling area of a trapping grid, but grid size sets a limit for the largest detectable distance. The spool-and-line technique is proposed here as a new method to estimateA(W). Movement distances based on the spool-and-line technique were compared to similar movement distances based on capture-recapture of three species of marsupials of the Atlantic Forest of Brazil. Distances based on the two methods were uncorrelated, and only ln-transformed distances based on the spool-and-line were normally distributed. The maximum distance moved (MaxD) estimated by the spool-and-line was chosen as the more accurate and practical distance to estimate edge effect. Estimates of the effective sampling area and densities for the common opossumDidelphis aurita (Wied-Neuwied, 1826), were compared using MaxD based on spool-and-line (MaxDs p o o l ) , capture-recapture (MaxD_cap), and also the distance between traps (DT). MaxD_spool reflected more accurately density variation between seasons. Movement distances of small mammals based on the spool-and-line technique permit more accurate estimates of density and its dynamics.     

Lateral inhibition as a mechanism for stimulus intensity estimation

Results of imitational modeling of neural network are presented. Results show that lateral inhibitory connections between neurons with certain parameters are necessary for estimation of stimulus intensity in the central nervous system. An essential characteristic of suggested mechanism of intensity detection is known from physiology fact of negative dependence of response latency from the stimulus intensity.     

The position of the rotation center of the glenohumeral joint

To validate the assumption that the center of rotation in the glenohumeral (GH) joint can be described based on the geometry of the joint, two methods for calculation of the GH rotation center were compared. These are a kinematic estimation based on the calculation of instantaneous helical axes, and a geometric estimation based on a spherical fit through the surface of the glenoid. Four fresh cadaver arms were fixed at the scapula and fitted with electromagnetic sensors. Each arm was moved in different directions while at the same time the orientation of the humerus was recorded. Subsequently, each specimen was dissected and its glenoid and humeral head surfaces were digitized. Results indicate no differences between the methods. It is concluded that the method to estimate the GH center of rotation as the center of a sphere through the glenoid surface, with the radius of the humeral head, appears to be valid.     

Bias compensated least squares estimate of the center of rotation

It is shown that a recently published least squares method for the estimation of the average center of rotation is biased. Consequently, a correction term is proposed, and an iterative algorithm is derived for finding a bias compensated solution to the least squares problem.The accuracy of the proposed bias compensated least squares method is compared to the previously proposed least squares method by Monte-Carlo simulations. The tests show that the new method gives a substantial improvement in accuracy.     

Robust estimation of dominant axis of rotation

A simple method is developed for robustly estimating a fixed dominant axis of rotation (AoR) of anatomical joints from surface marker data. Previous approaches which assume a model of circular marker trajectories use plane fitting to estimate the direction of the AoR. However, when there is limited joint range of motion and rotation due to a second degree of freedom, minimizing only the planar error can give poor estimates of the AoR direction. Optimizing a cost function which includes the error component within a plane, instead of only the component orthogonal to a plane, leads to improved estimates of the AoR direction for joints which exhibit additional rotational motion from a second degree of freedom. Results from synthetic data validation show the ranges of motion where the new method has lower estimation error compared to plane-fitting techniques. Estimates of the flexion-extension AoR from empirical motion capture data of the knee and index finger joints were also more anatomically plausible.     

Instantaneous center of rotation by least square method

In the design of the total replacement of the human joint, the joint center of rotation is often important. This paper describes an analytical technique to study joint kinematics, especially the instantaneous center of rotation. The empirical data for the analysis were obtained by the sonic digitizer system which works acoustically and electronically. As biomechanical applications of the technique, the elbow, the metacarpophalangeal, and wrist joints were studied.     

Comparison of two methods for in vivo estimation of the glenohumeral joint rotation center (GH-JRC) of the patients with shoulder hemiarthroplasty

Determination of an accurate glenohumeral-joint rotation center (GH-JRC) from marker data is essential for kinematic and dynamic analysis of shoulder motions. Previous studies have focused on the evaluation of the different functional methods for the estimation of the GH-JRC for healthy subjects. The goal of this paper is to compare two widely used functional methods, namely the instantaneous helical axis (IHA) and symmetrical center of rotation (SCoRE) methods, for estimating the GH-JRC in vivo for patients with implanted shoulder hemiarthroplasty. The motion data of five patients were recorded while performing three different dynamic motions (circumduction, abduction, and forward flexion). The GH-JRC was determined using the CT-images of the subjects (geometric GH-JRC) and was also estimated using the two IHA and SCoRE methods. The rotation centers determined using the IHA and SCoRE methods were on average 1.47±0.62 cm and 2.07±0.55 cm away from geometric GH-JRC, respectively. The two methods differed significantly (two-tailed p-value from paired t-Test ～0.02, post-hoc power ～0.30). The SCoRE method showed a significant lower (two-tailed p-value from paired t-Test ～0.03, post-hoc power ～0.68) repeatability error calculated between the different trials of each motion and each subject and averaged across all measured subjects (0.62±0.10 cm for IHA vs. 0.43±0.12 cm for SCoRE). It is concluded that the SCoRE appeared to be a more repeatable method whereas the IHA method resulted in a more accurate estimation of the GH-JRC for patients with endoprostheses.     

Muscle fiber conduction velocity estimation by using normalized peak-averaging technique

In order to compute the muscle fiber conduction velocity (MFCV) and to clarify how action potentials are conducted, the normalized peak-averaging technique (NPAT) was newly employed together with computer softwares. Twelve pairs of surface electromyograms were selected from biceps brachii muscles during contraction at a level of 50% of the maximum voluntary isometric contraction in seven healthy volunteers. The techniques to compute MFCV from the time delay of the peaks (P-NPAT) and from the cross correlation (CC-NPAT) of averaged pulses were compared to the cross-correlation technique (CCT). The spread rate of averaged pulses was computed to estimate the spread of MFCVs in different motor units. Tri-phasic averaged pulses were obtained clearly by averaging more than 500 detected pulses. The P-NPAT and CC-NPAT highly correlated with the CCT in the computed MFCVs. The MFCVs obtained by P-NPAT were generally larger than those obtained by CCT, and the spread rates had in the definite values. These results suggest that the MFCV could be computed and the spread of MFCVs would be estimated from averaged pulses. The MFCV of a patient with myotonic dystrophy was also studied, and it was suggested that the NPAT would be clinically useful.     

Mobile location estimation using density-based clustering technique for NLoS environments

Mobile location technologies have drawn much attention to cope with the mass demands of wireless communication services. Although clustering spatial data is viewed as an effective way to access the objects located in a physical space, little has been done in estimating mobile location. In wireless communication, one of the main problems with accurate location is nonline of sight (NLoS) propagation. To solve the problem, we present a new location algorithm with clustering technique by utilizing the geometrical feature of cell layout, time of arrival range measurements, and three base stations. The mobile location is estimated by solving the optimal solution of the objective function based on the high density cluster. Furthermore, our proposed algorithm only needs three range measurements and does not distinguish between NLoS and LoS environments. Simulations study was conducted to evaluate the performance of the algorithm for different NLoS error distributions and various upper bound of NLoS error. The results of our experiments demonstrate that the proposed algorithm is significantly more effective in location accuracy than range scaling algorithm, linear lines of position algorithm, and Taylor series algorithm, and also satisfies the location accuracy demand of E-911.

A numerical simulation of trichromatic equations in chlorophyll estimation using the spectrophotometric technique

A numerical simulation of trichromatic pigment equations is made with the aid of a computer utility program. Significant quantitative differences in the estimates of pigment concentration result from using different sets of trichromatic equations. Estimates of chlorophylls a, b, and c were found highly correlated with the application of the equations, even though the absorbance values used as input for the stimulation are not correlated.Contribution No. 284 of the Great Lakes Research DivisionContribution No. 284 of the Great Lakes Research Division