Continuous ambulatory hand force monitoring during manual materials handling using instrumented force shoes and an inertial motion capture suit

Hand forces (HFs) are commonly measured during biomechanical assessment of manual materials handling; however, it is often a challenge to directly measure HFs in field studies. Therefore, in a previous study we proposed a HF estimation method based on ground reaction forces (GRFs) and body segment accelerations and tested it with laboratory equipment: GFRs were measured with force plates (FPs) and segment accelerations were measured using optical motion capture (OMC). In the current study, we evaluated the HF estimation method based on an ambulatory measurement system, consisting of inertial motion capture (IMC) and instrumented force shoes (FSs).Sixteen participants lifted and carried a 10-kg crate from ground level while 3D full-body kinematics were measured using OMC and IMC, and 3D GRFs were measured using FPs and FSs. We estimated 3D hand force vectors based on: (1) FP+OMC, (2) FP+IMC and (3) FS+IMC. We calculated the root-mean-square differences (RMSDs) between the estimated HFs to reference HFs calculated based on crate kinematics and the GRFs of a FP that the crate was lifted from.Averaged over subjects and across 3D force directions, the HF RMSD ranged between 10-15N when using the laboratory equipment (FP + OMC), 11-18N when using the IMC instead of OMC data (FP+IMC), and 17-21N when using the FSs in combination with IMC (FS + IMC). This error is regarded acceptable for the assessment of spinal loading during manual lifting, as it would results in less than 5% error in peak moment estimates.     

Determination of joint moments with instrumented force shoes in a variety of tasks

Ground reaction forces (GRFs) are often used in inverse dynamics analyses to determine joint loading. These GRFs are usually measured using force plates (FPs). As an alternative, instrumented force shoes (FSs) can be used, which have the advantage over FPs that they do not constrain foot placement. This study tested the FS system in one normal weight subject (77 kg) performing 19 different lifting, pushing and pulling and walking tasks. Kinematics were measured with an optoelectronic system and the GRFs and the positions of the centre of pressure (CoP) were synchronously measured with FPs and FSs. Differences between the outcomes of the two measurement systems (i.e. CoP and GRFs) and the resulting ankle and L5/S1 joint moments were determined at the instant of the peak GRF (DaPF). For most lifting and pushing and pulling tasks, the difference between the FP and FS measurements remained small: GRF DaPF remained below 3% body weight, CoP DaPF remained below 10 mm, ankle moment DaPF remained below 7% of the peak total ankle moment that occurred during normal walking and L5/S1 moment DaPF remained below 7% of the peak total L5/S1 moment that occurred during normal symmetric lifting. More substantial differences were only found in the maximal pushing tasks. For the walking tasks, peak vertical GRFs were somewhat underestimated. However, differences in ankle and L5/S1 moments remained small, i.e. DaPF below 7% of the peak total moment that occurred during normal walking.     

Oblique abdominal muscle activity in response to external perturbations when pushing a cart

Cyclic activation of the external and internal oblique muscles contributes to twisting moments during normal gait. During pushing while walking, it is not well understood how these muscles respond to presence of predictable (cyclic push-off forces) and unpredictable (external) perturbations that occur in pushing tasks. We hypothesized that the predictable perturbations due to the cyclic push-off forces would be associated with cyclic muscle activity, while external perturbations would be counteracted by cocontraction of the oblique abdominal muscles. Eight healthy male subjects pushed at two target forces and two handle heights in a static condition and while walking without and with external perturbations. For all pushing tasks, the median, the static (10th percentile) and the peak levels (90th percentile) of the electromyographic amplitudes were determined. Linear models with oblique abdominal EMGs and trunk angles as input were fit to the twisting moments, to estimate trunk stiffness. There was no significant difference between the static EMG levels in pushing while walking compared to the peak levels in pushing while standing. When pushing while walking, the additional dynamic activity was associated with the twisting moments, which were actively modulated by the pairs of oblique muscles as in normal gait. The median and static levels of trunk muscle activity and estimated trunk stiffness were significantly higher when perturbations occurred than without perturbations. The increase baseline of muscle activity indicated cocontraction of the antagonistic muscle pairs. Furthermore, this cocontraction resulted in an increased trunk stiffness around the longitudinal axis.     

Calibration and application of an intra-articular force transducer for the measurement of patellar tendon graft forces: an in situ evaluation.

The objective of this study was to evaluate two calibration methods for the "Arthroscopically Implantable Force Probe" (AIFP) that are potentially suitable for in vivo use: (1) a direct, experimentally based method performed by applying a tensile load directly to the graft after it is harvested but prior to implantation (the "pre-implantation" technique), and (2) an indirect method that utilizes cadaver-based analytical expressions to transform the AIFP output versus anterior shear load relationship, which may be established in vivo, to resultant graft load (the "post-implantation" technique). The AIFP outputs during anterior shear loading of the knee joint using these two calibration methods were compared directly to graft force measurements using a ligament cutting protocol and a 6 DOF load cell. The mean percent error (actual-measured)/(actual)* 100) associated with the pre-implantation calibration ranged between 85 and 175 percent, and was dependent on the knee flexion angle tested. The percent error associated with the post-implantation technique was evaluated in two load ranges: loads less than 40 N, and loads greater than 40 N. For graft force values greater than 40 N, the mean percent errors inherent to the post-implantation calibration method ranged between 20 and 29 percent, depending on the knee flexion angle tested. Below 40 N, these errors were substantially greater. Of the two calibration methods evaluated, the post-implantation approach provided a better estimate of the ACL graft force than the pre-implantation technique. However, the errors for the post-implantation approach were still high and suggested that caution should be employed when using implantable force probes for in vivo measurement of ACL graft forces.     

Validation of a biodynamic model of pushing and pulling.

Pushing and pulling during manual material handling can increase the compressive forces on the lumbar disc region while creating high shear forces at the shoe-floor interface. A sagittal plane dynamic model derived from previous biomechanical models was developed to predict L5/S1 compressive force and required coefficients of friction during dynamic cart pushing and pulling. Before these predictions could be interpreted, however, it was necessary to validate model predictions against independently measured values of comparable quantities. This experiment used subjects of disparate stature and body mass, while task factors such as cart resistance and walking speed were varied. Predicted ground reaction forces were compared with those measured by a force platform, with correlations up to 0.67. Predicted erector spinae and rectus abdominus muscle forces were compared with muscle forces derived from RMS-EMGs of the respective muscle groups, using a static force build-up regression relationship to transform the dynamic RMS-EMGs to trunk muscle forces. Although correlations were low, this was attributed in part to the use of surface EMG on subjects of widely varied body mass. The biodynamic model holds promise as a tool for analysis of actual industrial pushing and pulling tasks, when carefully applied.     

A technique for conditioning and calibrating force-sensing resistors for repeatable and reliable measurement of compressive force

Miniature sensors that could measure forces applied by the fingers and hand without interfering with manual dexterity or range of motion would have considerable practical value in ergonomics and rehabilitation. In this study, techniques have been developed to use inexpensive pressure-sensing resistors (FSRs) to accurately measure compression force. The FSRs are converted from pressure-sensing to force-sensing devices. The effects of nonlinear response properties and dependence on loading history are compensated by signal conditioning and calibration. A fourth-order polynomial relating the applied force to the current voltage output and a linearly weighted sum of prior outputs corrects for sensor hysteresis and drift. It was found that prolonged (>20 h) shear force loading caused sensor gain to change by approximately 100%. Shear loading also had the effect of eliminating shear force effects on sensor output, albeit only in the direction of shear loading. By applying prolonged shear loading in two orthogonal directions, the sensors were converted into pure compression sensors. Such preloading of the sensor is, therefore, required prior to calibration. The error in compression force after prolonged shear loading and calibration was consistently <5% from 0 to 30 N and <10% from 30 to 40 N. This novel method of calibrating FSRs for measuring compression force provides an inexpensive tool for biomedical and industrial design applications where measurements of finger and hand force are needed.     

Modelling torque generation by the mero-carpopodite joint of the american lobster and the snow crab

The torque generated by a rotating joint comprises the useful force exerted by the joint on the external environment, and both the magnitude and distribution of torque through the step cycle during walking are important variables in understanding the mechanics of walking. The mechanics of the American lobster (Homarus americanus) and snow crab (Chionoecetes opilio) during walking were modelled to examine the relative roles of flexor versus extensor apodeme-muscle complexes, investigate which legs of these decapods likely contribute the greatest to locomotion, determine scaling effects of torque generation, and assess the relative roles of various model variables on torque production. Force generated along the length of the apodeme by the muscle was modelled based on apodeme surface area, muscle stress, and muscle fibre pinnation angle. Torque was then calculated from this estimated force and the corresponding moment arm. The flexor apodeme-muscle complex is calculated to generate consistently greater forces than the extensor, and generally this results in flexor torque being larger than extensor, though the snow crab does illustrate the opposite in two of its legs. This greater torque generation in flexion suggests that, in addition to the pushing of the trailing legs, the pulling action of the leading legs may play a significant role, at least during lateral walking. Leg 4 of both species appears to generate greater torques and thus provide the greatest forces for locomotion. Torque generation as a function of body size shows a second order response due to the increase in apodeme surface area. The pinnation angle of the muscle fibre is found to be insignificant in force generation, apodeme surface area (representing muscle cross sectional area) likely plays the most influential role in total force production, and moment arm controls the distribution of this force through the step cycle. Muscle stress remain a largely unknown quantity however, and may significantly affect both magnitude and distribution through step cycle of forces, and thus torque. Despite the uncertainty associated with the muscle stress parameter, the modelled results fit well with previously published force measurements.     

Balance assessment during squatting exercise: A comparison between laboratory grade force plate and a commercial,low-cost device

Testing balance through squatting exercise is a central part of many rehabilitation programs and sports and plays also an important role in clinical evaluation of residual motor ability. The assessment of center of pressure (CoP) displacement and its parametrization is commonly used to describe and analyze squat movement and the laboratory-grade force plates (FP) are the gold standard for measuring balance performances from a dynamic view-point. However, the Nintendo Wii Balance Board (NWBB) has been recently proposed as an inexpensive and easily available device for measuring ground reaction force and CoP displacement in standing balance tasks. Thus, this study aimed to compare the NWBB-CoP data with those obtained from a laboratory FP during a dynamic motor task, such as the squat task. CoP data of forty-eight subjects were acquired simultaneously from a NWBB and a FP and the analyses were performed over the descending squatting phase. Outcomes showed a very high correlation (r) and limited root-mean-square differences between CoP trajectories in anterior-posterior (r > 0.99, 1.63 ± 1.27 mm) and medial-lateral (r > 0.98, 1.01 ± 0.75 mm) direction. Spatial parameters computed from CoP displacement and ground reaction force peak presented fixed biases between NWBB and FP. Errors showed a high consistency (standard deviation < 2.4% of the FP outcomes) and a random spread distribution around the mean difference. Mean velocity is the only parameter which exhibited a tendency towards proportional values. Findings of this study suggested the NWBB as a valid device for the assessment and parametrization of CoP displacement during squatting movement.     

The coordination of force oscillations and of leg movement in a walking insect (Carausius morosus)

As in the preceding paper stick insects walk on a treadwheel and different legs are put on platforms fixed relative to the insect's body. The movement of the walking legs is recorded in addition to the force oscillations of the standing legs. The coordination between the different legs depends upon the number and arrangement of the walking legs and the legs standing on platforms. In most experimental situations one finds a coordination which is different from that of a normal walking animal.Supported by DFG (Cr 58/1)     

Oscillations of force in the standing legs of a walking insect (Carausius morosus)

In the experiments presented here adult stick insects (Carausius morosus) walk on a treadwheel with various legs standing on platforms fixed relative to the body of the insect. These standing legs produce large forees directed towards the rear which are modulated in the rhythm of the walking legs. Neighbouring legs which both stand on a platform often oscillate in phase. Possible reasons for the occurrence of the force oscillations are discussed.Supported by DFG (Cr 58/1)     

Hip contact forces and gait patterns from routine activities.

In vivo loads acting at the hip joint have so far only been measured in few patients and without detailed documentation of gait data. Such information is required to test and improve wear, strength and fixation stability of hip implants. Measurements of hip contact forces with instrumented implants and synchronous analyses of gait patterns and ground reaction forces were performed in four patients during the most frequent activities of daily living. From the individual data sets an average was calculated. The paper focuses on the loading of the femoral implant component but complete data are additionally stored on an associated compact disc. It contains complete gait and hip contact force data as well as calculated muscle activities during walking and stair climbing and the frequencies of daily activities observed in hip patients. The mechanical loading and function of the hip joint and proximal femur is thereby completely documented. The average patient loaded his hip joint with 238% BW (percent of body weight) when walking at about 4 km/h and with slightly less when standing on one leg. This is below the levels previously reported for two other patients (Bergmann et al., Clinical Biomechanics 26 (1993) 969-990). When climbing upstairs the joint contact force is 251% BW which is less than 260% BW when going downstairs. Inwards torsion of the implant is probably critical for the stem fixation. On average it is 23% larger when going upstairs than during normal level walking. The inter- and intra-individual variations during stair climbing are large and the highest torque values are 83% larger than during normal walking. Because the hip joint loading during all other common activities of most hip patients are comparably small (except during stumbling), implants should mainly be tested with loading conditions that mimic walking and stair climbing.     

Amino acid similarity matrices based on force fields

MOTIVATION: We propose a general method for deriving amino acid substitution matrices from low resolution force fields. Unlike current popular methods, the approach does not rely on evolutionary arguments or alignment of sequences or structures. Instead, residues are computationally mutated and their contribution to the total energy/score is collected. The average of these values over each position within a set of proteins results in a substitution matrix. RESULTS: Example substitution matrices have been calculated from force fields based on different philosophies and their performance compared with conventional substitution matrices. Although this can produce useful substitution matrices, the methodology highlights the virtues, deficiencies and biases of the source force fields. It also allows a rather direct comparison of sequence alignment methods with the score functions underlying protein sequence to structure threading. AVAILABILITY: Example substitution matrices are available from http://www.rsc.anu.edu.au/~zsuzsa/suppl/matrices.html. SUPPLEMENTARY INFORMATION: The list of proteins used for data collection and the optimized parameters for the alignment are given as supplementary material at http://www.rsc.anu.edu.au/~zsuzsa/suppl/matrices.html.     

Finger force vectors in multi-finger prehension

In a majority of studies on grasp, only normal forces were measured and only when a zero torque was exerted on a hand-held object. This study concerns finger force vectors during the torque production tasks. Subjects (n=8) stabilized a handle with an attachment that allowed for change of external torque from -1.5 to 1.5 Nm. Forces and moments exerted by the digit tips on the object were recorded. At the large (>-0.375 Nm) supination torques the index/middle and ring/little pairs of fingers generated oppositely directed tangential forces. The index and middle finger produced forces in a downward direction and therefore did not support the load. At a zero torque and pronation torques, the middle, ring and little fingers produced forces along nearly the same direction. The vector of the index finger force was always directed differently from the vectors of other finger forces, the angles ranged from 19 degrees 30' to 47 degrees 40'. The points of force application were systematically displaced with the torque, with the exception of the little finger. Tangential finger forces contributed substantially to the total torque exerted on the hand-held object.     

An investigation of rugby scrimmaging posture and individual maximum pushing force

Although rugby is a popular contact sport and the isokinetic muscle torque assessment has recently found widespread application in the field of sports medicine, little research has examined the factors associated with the performance of game-specific skills directly by using the isokinetic-type rugby scrimmaging machine. This study is designed to (a) measure and observe the differences in the maximum individual pushing forward force produced by scrimmaging in different body postures (3 body heights x 2 foot positions) with a self-developed rugby scrimmaging machine and (b) observe the variations in hip, knee, and ankle angles at different body postures and explore the relationship between these angle values and the individual maximum pushing force. Ten national rugby players were invited to participate in the examination. The experimental equipment included a self-developed rugby scrimmaging machine and a 3-dimensional motion analysis system. Our results showed that the foot positions (parallel and nonparallel foot positions) do not affect the maximum pushing force; however, the maximum pushing force was significantly lower in posture I (36% body height) than in posture II (38%) and posture III (40%). The maximum forward force in posture III (40% body height) was also slightly greater than for the scrum in posture II (38% body height). In addition, it was determined that hip, knee, and ankle angles under parallel feet positioning are factors that are closely negatively related in terms of affecting maximum pushing force in scrimmaging. In cross-feet postures, there was a positive correlation between individual forward force and hip angle of the rear leg. From our results, we can conclude that if the player stands in an appropriate starting position at the early stage of scrimmaging, it will benefit the forward force production.     

Inter-joint coupling effects on muscle contributions to endpoint force and acceleration in a musculoskeletal model of the cat hindlimb

The biomechanical principles underlying the organization of muscle activation patterns during standing balance are poorly understood. The goal of this study was to understand the influence of biomechanical inter-joint coupling on endpoint forces and accelerations induced by the activation of individual muscles during postural tasks. We calculated induced endpoint forces and accelerations of 31 muscles in a 7 degree-of-freedom, three-dimensional model of the cat hindlimb. To test the effects of inter-joint coupling, we systematically immobilized the joints (excluded kinematic degrees of freedom) and evaluated how the endpoint force and acceleration directions changed for each muscle in 7 different conditions. We hypothesized that altered inter-joint coupling due to joint immobilization of remote joints would substantially change the induced directions of endpoint force and acceleration of individual muscles. Our results show that for most muscles crossing the knee or the hip, joint immobilization altered the endpoint force or acceleration direction by more than 90° in the dorsal and sagittal planes. Induced endpoint forces were typically consistent with behaviorally observed forces only when the ankle was immobilized. We then activated a proximal muscle simultaneous with an ankle torque of varying magnitude, which demonstrated that the resulting endpoint force or acceleration direction is modulated by the magnitude of the ankle torque. We argue that this simple manipulation can lend insight into the functional effects of co-activating muscles. We conclude that inter-joint coupling may be an essential biomechanical principle underlying the coordination of proximal and distal muscles to produce functional endpoint actions during motor tasks.     

A probabilistic method to estimate gait kinetics in the absence of ground reaction force measurements

Human joint torques during gait are usually computed using inverse dynamics. This method requires a skeletal model, kinematics and measured ground reaction forces and moments (GRFM). Measuring GRFM is however only possible in a controlled environment. This paper introduces a probabilistic method based on probabilistic principal component analysis to estimate the joint torques for healthy gait without measured GRFM. A gait dataset of 23 subjects was obtained containing kinematics, measured GRFM and joint torques from inverse dynamics in order to obtain a probabilistic model. This model was then used to estimate the joint torques of other subjects without measured GRFM. Only kinematics, a skeletal model and timing of gait events are needed. Estimation only takes 0.28 ms per time instant. Using cross-validation, the resulting root mean square estimation errors for the lower-limb joint torques are found to be approximately 0.1 Nm/kg, which is 6–18% of the range of the ground truth joint torques. Estimated joint torque and GRFM errors are up to two times smaller than model-based state-of-the-art methods. Model-free artificial neural networks can achieve lower errors than our method, but are less repeatable, do not contain uncertainty information on the estimates and are difficult to use in situations which are not in the learning set. In contrast, our method performs well in a new situation where the walking speed is higher than in the learning dataset. The method can for example be used to estimate the kinetics during overground walking without force plates, during treadmill walking without (separate) force plates and during ambulatory measurements.     

Influence of loading rate and cable migration on fiberoptic measurement of tendon force

Several investigators have recently used fiberoptic cables to measure tendon forces in situ. The technique may be subject to significant error due to cable migration and differences in the loading rates used for calibration and those experienced during measurement. This in vitro study examined the impact of these potential sources of error on transducer accuracy. A fiberoptic cable was passed perpendicular to the fibers of four Achilles tendons in the mediolateral direction and each specimen was cyclically loaded to 1000 N. The influence of loading rate on transducer output was investigated by comparing results from tests conducted at 20, 200 and 1000 N/s. The effect of cable migration was examined by comparing the outputs obtained after displacing the cable one tendon width medially and laterally along its path in the tendon and then repeating the 200 N/s testing protocol. It was possible to obtain nonlinear specimen-specific relationships between the fiberoptic output and tendon force. Differences in loading rate resulted in root-mean-square (RMS) errors not larger than 17% maximum load. Hysteresis effects caused RMS errors smaller than 5% maximum load. Cable migration errors were less than 27%. The total RMS error due to the combined effects of loading rate difference and cable movement was less than 32%. Fiberoptic measurement of tendon force is attractive due to its low cost, easy implementation and comparable accuracy relative to other implantable force transducers. Although additional factors such as cable placement, edge artifacts due where the transducer exits the skin and non-uniform loading may also influence fiberoptic output, careful control of loading rate and transducer movement during calibration is imperative if maximum accuracy is to be achieved.     

Individual muscle contributions to the axial knee joint contact force during normal walking

Muscles are significant contributors to the high joint forces developed in the knee during human walking. Not only do muscles contribute to the knee joint forces by acting to compress the joint, but they also develop joint forces indirectly through their contributions to the ground reaction forces via dynamic coupling. Thus, muscles can have significant contributions to forces at joints they do not span. However, few studies have investigated how the major lower-limb muscles contribute to the knee joint contact forces during walking. The goal of this study was to use a muscle-actuated forward dynamics simulation of walking to identify how individual muscles contribute to the axial tibio-femoral joint force. The simulation results showed that the vastii muscles are the primary contributors to the axial joint force in early stance while the gastrocnemius is the primary contributor in late stance. The tibio-femoral joint force generated by these muscles was at times greater than the muscle forces themselves. Muscles that do not cross the knee joint (e.g., the gluteus maximus and soleus) also have significant contributions to the tibio-femoral joint force through their contributions to the ground reaction forces. Further, small changes in walking kinematics (e.g., knee flexion angle) can have a significant effect on the magnitude of the knee joint forces. Thus, altering walking mechanics and muscle coordination patterns to utilize muscle groups that perform the same biomechanical function, yet contribute less to the knee joint forces may be an effective way to reduce knee joint loading during walking.     

Rate-independent characteristics of an arthroscopically implantable force probe in the human achilles tendon

The effect of loading rate on specimen calibration was investigated for an implantable force sensor of the two-point loading variety. This variety of sensor incorporates a strain gage to measure the compressive load applied to the sensor due to tensile loading in a soft tissue specimen. The Achilles tendon in each of four human cadaveric lower extremities was instrumented with a force sensor and then loaded in tension using a materials testing machine. Each specimen was tensile tested at three different displacement rates, 0.25, 2.5 and 12.7 cm s(-1), corresponding with mean loading rates of 33.8, 513.2, and 2838.6 N s(-1), respectively. A calibration curve relating the force sensor signal and applied tendon tension was generated for each specimen/ displacement rate combination. For each specimen, calibration curves were compared by calculating an RMS error for the entire data set (eRMS = 1.6% of the full load value) and a coefficient of determination, R2, of a curve fit through all of the data (R2 = 99.6%). Over the range of rates tested, no measurable change in sensor sensitivity due to loading rate was observed. Hysteresis for all displacement rates was on the order of 2.4%.     

Matrix analyses of interaction among fingers in static force production tasks

The fingers on a hand show interactions in force production tasks. The interfinger connection matrices (IFMs) quantify these interactions (Li et al. 2002; Zatsiorsky et al. 2002b; Danion et al. 2003). The goal of the present study was to explore the differences in the IFMs of individual subjects and, in particular, to establish a procedure that may be used in the future for diagnostic purposes. Subjects (n=20) pressed downward maximally with ten different combinations of the four fingers, index (I), middle (M), ring (R), and little (L): I, M, R, L, IM, MR, RL, IMR, MRL, and IMRL. Voluntary activation of a subset of the four fingers was accompanied by an involuntary force production by fingers that were not intentionally activated (enslaving). Interfinger connection matrices were computed for each subject by the artificial neural network. The similarities/dissimilarities (proximities) between the individual matrices were determined. This procedure was performed twice: (a) for nonnormalized IFMs whose elements represented the amount of force (in newtons) exerted by a finger i in response to a unit command to a finger j; and (b) for normalized IFMs, after dividing the elements of each IFM by the total force produced by the four fingers acting together (the elements of the matrix are in percents). The 20×20 matrix of the proximities was subjected to multidimensional scaling (MDS) to reduce the number of dimensions and identify the major ones. To interpret the meaning of the computed dimensions, they were regressed on a set of finger force parameters described in the text. For the nonnormalized IFMs an interpretable dimension was the strength of the subjects. For the normalized IFMs two dimensions were interpreted: (a) the location of the point of resultant force application along the mediolateral axis that is defined by the pattern of force sharing among the fingers and (b) the total contribution of the enslaved forces into the total finger force. We speculate that the similarity of typical everyday tasks across the population promotes the similarity of the IMFs reflecting optimal hand functioning over these tasks. AcknowledgementsThis study was partly supported by NIH grants NS-35032, AR-048563 and AG-18751. The support from the Whittaker Foundation to Dr. Z.M. Li is also acknowledged.