Optimised procedure for the calibration of the force platform location

Gait and Posture 17 (2003) 75
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Optimised procedure for the calibration of the force platform location M. Rabuffetti a,*, M. Ferrarin a, P. Mazzoleni a, F. Benvenuti b, A. Pedotti a a

Centro di Bioingegneria, Fondazione Don Carlo Gnocchi ONLUS IRCCS, Politecnico di Milano, Via Capecelatro 66, I-20148 Milan, Italy b Laboratorio di Fisiopatologia e Riabilitazione del Movimento, Dipartimento di Geriatria ‘I Fraticini’, INRCA, Florence, Italy Accepted 29 April 2002

Abstract An innovative optimised method, including an experiment and a mathematical model, for the calibration of the force platform location in the optoelectronic reference frame is proposed. The calibration experiment adopts a bearing-marker testing object contacting the platform and does not directly measure the platform location. The experiment is designed in order to avoid the main drawbacks possibly occurring in commonly adopted methods. The mathematical model of the experiment estimates the expected ground reaction. An optimisation algorithm identifies the optimal platform location as the one that best matches the measured outcome of the calibration experiment with the corresponding model estimate. The innovative calibration procedure has been assessed in terms of inter-tester reliability and compared with commonly used calibration procedures of platform location. These results evidenced how the introduction of such optimised procedure could improve the reliability of the calibrated platform location and, consequently, of the kinetic variables considered in posture and gait analysis. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Motion analysis; Force platform; Calibration; Optimisation; Laboratory management; Rehabilitation outcome assessment

1. Introduction The accuracy and precision of the outcomes, mostly joint kinematics and dynamics, of any motion analysis are primarily determined by the accuracy and precision of any involved measurement device, and by the correct integration of all the measurements into one absolute time spatial reference system. Particularly the measured ground reaction, quantified in the platform local coordinate system, must be correctly transformed into the optoelectronic reference frame, assumed as absolute reference frame, for the solution of the inverse dynamic problem and the estimation of joint dynamics. The calibration procedure of the platform location usually implies the placement of at least three markers, or eventually a rigid cluster of markers, onto the platform plate: these points have a known relation with the platform reference frame, thus allowing to define the transformation from the platform to the

* Corresponding author. Tel.: /39-0240-308-305; fax: /39-024048919 E-mail address: [email protected] (M. Rabuffetti).

absolute reference frame. Several errors may affect this procedure: the operator may misplace the markers; non linearity distortion may affect the optoelectronic system [1]; still markers may produce a distorted image due to the pixel quantisation of the camera sensors, thus causing an offset to be added to the true position. These possible occurrences negatively affect the identification of the platform location and of the transformation between relative and absolute reference frames. An inaccurate calibration of the platform location distorts the computed joint moments [2], causing errors that may be compared with the errors due to inaccurate joint centre position identification [3,4]. These errors may alter the interpretation of a gait analysis [5] and may completely disrupt the computed joint dynamics in posture analysis [6]: joint moments may definitively swap from being extensor to flexor, from being adductor to abductor and vice versa. The assessment of the calibration of the platform location has been addressed by spot-check procedures [7
/10], whose implicit indications are to repeat the platform calibration with the current methods until the quality indices are inside an acceptable range.

0966-6362/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 6 6 - 6 3 6 2 ( 0 2 ) 0 0 0 6 1 - 9

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In the present paper an innovative method is proposed for the calibration of the force platform location that derives from a spot check approach [8]. The method includes an experimental protocol, a testing object of known characteristics and a mathematical model. The indices derived from the spot check model are combined in a target function, whose independent variables are the transformation parameters (or alternatively the platform location coordinates). The current value of the target function quantifies the discrepancy between two different estimates of the same quantities: one estimate does not use the platform location, while the alternative estimate does. The optimisation process identifies the optimal platform location, as the one that minimise the target function. The performance of the proposed method has been quantified in terms of inter and intra test-retest reliability, and a comparison has been made with currently adopted method for platform location calibration.

2. Methods The testing object consists of a rigid pointed rod (distance between extremities: 1065 mm; mass: 1.31 kg) bearing a set of eight reflective markers. The object design/construction complies with a multi-planar symmetry in geometrical features and mass distribution. Particularly, the object, crafted similarly to calibration objects for stereophotogrammetric systems, has the following features: the markers uniquely identify a plane; the markers are placed, two by two, in symmetrical positions relatively to a longitudinal axis, which passes through the rod extremities; the object centre of mass coincide with the barycentral point of the markers’ cluster; the centre of mass coincides with the midpoint between the two rod extremities. The previous features allow, when the markers’ positions are measured by an optoelectronic system, to estimate the positions of the rod extremities and of the centre of mass. Two metal plates, one placed on the ground and one handheld by an operator, are frictionlessly connected to both rod extremities and allow to keep the object in equilibrium by means of a compressive force produced by the operator. According with previous works [8,10], we assume that only the operator dynamic action, the rod weight and the ground reaction (evidenced in Fig. 1, respectively, as A, W and F) are applied to the rod, while no moment is transmitted across the two extremities. After the calibration of the optoelectronic system, the experimental protocol requires the operator, who is standing outside the platform, to move the rod, which is pushed against the platform, in a circular trajectory, thus describing a cone. No strict requirements are held for the exerted force and for the rod inclination: the

Fig. 1. The testing object, a rigid rod bearing a set of reflective markers, is handheld by the operator. The forces that keep the rod in equilibrium are superimposed on the photograph (A is the force exerted by the operator, F is the ground reaction, W is the rod weight). The four platform vertices are evidenced with squares, and the four force sensors, where markers have been placed for the standard calibration, by circles.

general guidelines suggest that the load must be in the magnitude order of the subsequent analysis. The movement should be slow, a circular trajectory performed in about 2 s, in order to minimise the inertial components. The experiment is recorded by the optoelectronic system and by the force platform. Particularly, the optoelectronic system provides the positions of the rod extremities, P1 and P2, and their midpoint, the rod centre of mass. The force platform provides the ground reaction force FM, along with its direction vector fM, and its application point PM expressed in the platform reference frame (subscript M). The platform location and, consequently, the six degrees-of-freedom transformation T (actually the quaternion matrix) between the

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platform and the absolute reference frames are unknown. The mathematical model allows estimating the position P of the application point and the direction f of the ground reaction force F in the absolute reference frame, without reference to the platform location. The inputs of the model are the measurements of the optoelectronic system (P1 and P2) and of the force platform (FM), and geometrical and inertial features (l, w and h ) of the testing object (see Fig. 2 for a schematic representation of the model showing all the variables and parameters involved in the computations). With reference to a single sample, the static equilibrium of the rod with respect to P2, is described by the following equation (where F /jFMj): Fl sin(a)

1 2

w sin(b)0

(1)

and the angle q between the reaction force and the gravity line is defined by: 
w sin(b) (2) q bsin1 2F The unit vector f is f [fx

fy

fz ]

[sin(q )sin(g);

cos(q );

sin(q )cos(g)]

(3)

Having positioned the inferior plate on the platform, the application point P of the ground reaction force is

77

determined simply by moving the point P1, the rod extremity, along the direction f to the platform surface. The vector equation follows P P1

h cos(q )

f

(4)

The quantities, already estimated by the model and expressed in the absolute reference frame, are also directly measured by the force platform and expressed in the local coordinate system: PM and fM. The components of PM and fM in the absolute reference frame (subscript T) are function of the unknown variable T, fT(T)/TfM and PT(T)/TPM, and are expected to coincide with the corresponding model derived quantities. The mean vector error, referred to the whole data set composed by N samples, between the alternative estimates of f is defined as follows: N X

Ef (T)

(f i  f T (T)i )

i1

N

(5)

Two scalar variables can be derived from the previous mean vector error according to the Carnot theorem as follows: 
½Ef (T)½2 1 mf (T)cos (6) 1 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX u (cos1 (1  ½(f  f (T) )  E (T)½2 =2))2 i T i f u t sf (T) i1 N (7) Additionally, with reference to the alternative estimates of P, a mean vector error is defined as follows N X (Pi  PT (T)i )

EP (T) i1

N

(8)

and similarly two scalar variables are defined:

Fig. 2. The model of the experimental setup includes the rod (P1 and P2 are the rod extremities and W is the weight); the plate (h is the plate height); the expected ground reaction (F vector applied in the point P; direction vector f is implicitly described by vector F); the force exerted by the operator (refer to vector A in Fig. 1) is not represented because it is not included in the model. The graphical representation shows the plane identified by the rod and by the reaction force, including also the Y -axis gravity line. The angular variables a , b and u describe the orientations of the rod and of the reaction force. The angular variable g , spatial angle between the represented plane and the absolute XY plane is not represented.

mP (T)½EP (T)½ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX u ½((Pi  PT (T)i ))  EP (T)2 ½ u t i1 sP (T) N

(9)

(10)

The previous four scalar variables can be combined in one scalar function J /J (T) that quantifies a global discrepancy between the two alternative estimates: J(T)(mP (T)sP (T))(mf (T)sf (T))

(11)

Goal of the non-linear optimisation algorithm is to

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identify the transformation Tˆ that minimises the target ˆ function J, which means J(T)5J(T) for  T" Tˆ/ The computational program was implemented in MATLAB (The Mathworks, Natick, MA) adopting the available simplex optimisation algorithm [11].

3. Results The presented method has been applied in the clinical gait analysis laboratory of the Centro di Bioingegneria in Milano, Italy. The laboratory setup included a motion capture optoelectronic system (ELITE system, BTS, Milan, Italy) and a force platform (model 9218B, Kistler Instrumente AG, Winterthur, Switzerland). The two instruments were calibrated according to the manufacturers’ specifications in order to capture gait in a working volume of 1.8 /2.2 /3.0 m; the force platform was positioned almost centrally in this volume. In the considered conditions, the accuracy of the calibration of the optoelectronic system was assessed by moving a rigid object bearing two markers, the difference between the true inter-marker distance and the average measured inter-marker distance was 0.8 mm, which is approximately one on three thousandth of the diagonal of the stereophotogrammetric calibrated volume. Three operators repeated ten times each the above described platform calibration protocol and the optimised platform locations were recorded as positions of the four vertices. Additionally, the platform was calibrated ten times using a classical technique adopting direct marking of the platform: four spherical markers were placed onto the screw holes of the four force transducers (markers are represented as circles in Fig. 1), being this choice, according to the last techniques suggested by the manufacturers, much more reliable than putting hemispherical markers on the platform vertices. No filtering was applied to the measured quantities. The platform vertices (represented as squares in Fig. 1) were derived from the markers’ positions by the following procedure: the platform centre was obtained as the mean of the acquired position of the four markers; the longitudinal and the transversal platform axes were identified as the axes connecting the midpoints of each side of the four markers’ rectangular shape; the Gram
/Schmidt orthogonalisation procedure was applied to the above axes and defined the platform main axes parallel to the platform brims; finally, the four vertices were extrapolated from the platform centre along the main axes considering the platform dimension. The results are reported in Table 1 as target function values and coordinates of each vertex (the mean and standard deviation (S.D.) values about ten repetitions are reported), for each operator performing the opti-

mised calibration and for the direct marking reference experiments. Additionally a graphical presentation of the results, in the XZ horizontal plane, is provided in Fig. 3. In order to provide a reference for comparison among results, each mean position is attached with a circular shape representing a marker with a diameter of 15 mm, the one adopted in direct marking. The S.D. values are represented in terms of rectangles (actually included in the circular shapes because of their small values). The differences among the alternative vertices position, already reported in Table 1, are even more evidenced in Fig. 3. Such differences are quantified by the distances between the mean positions as determined by the optimised method applied by different operators, and by the distances between the direct marking and the optimised results. Such distances are expected to be null when comparing accurate calibrations. In the present study, the mean value of all the distances between optimised results is 3.0 mm, ranging from 1.4 to 5.0 mm. When considering the direct marking compared with the optimised calibration, the mean values of all the distance is 15.2 mm, ranging from 4.3 to 24.2 mm. The reported values of the S.D. show an intraoperator test-retest reliability comparable to the one of directly marking the transducers holes and not dependent from the operator. The computed distances among the positions of vertices, as obtained by the optimised method, show also a clear inter-operator reliability of the optimised method. Moreover, in the presented results, the positions of vertices as obtained by direct marking are characterised by a clear offset from the positions of vertices as obtained by the optimised method.

4. Discussion and conclusion The proposed method for an optimised platform location calibration consists of an experimental protocol, which measures some mechanical quantities in the platform reference frame, and a mathematical model, which estimates the same quantities in the absolute reference frame. The minimisation of a residual scalar, a target function of the unknown transformation between the two reference frames, allows identifying the platform location, thus performing the platform location calibration. The operative aspects of the proposed method overcome the drawbacks of commonly applied methods: no constraints are imposed on the rod markers position, thus completely avoiding the negative effects of markers misplacement; the rod markers are positioned in the middle of the stereophotogrammetric calibrated volume, where accuracy and precision are expected to be maximised (while markers positioned on the platform

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Table 1 Comparison of optimised and direct marking platform calibrations Operator A

Target function J X 1 (mm) Y 1 (mm) Z 1 (mm) X 2 (mm) Y 2 (mm) Z 2 (mm) X 3 (mm) Y 3 (mm) Z 3 (mm) X 4 (mm) Y 4 (mm) Z 4 (mm)

Operator B

Operator C

Direct marking

Mean

S.D.

Mean

S.D.

Mean

S.D.

Mean

S.D.

0.96 /647.7 /0.1 175.5 /48.1 1.1 197.1 /62.5 /4.3 596.8 /662.1 /5.5 575.2

0.12 1.4 0.8 1.6 1.3 0.6 1.9 1.1 0.5 1.8 0.9 0.4 1.6

1.22 /646.6 /0.4 174.3 /47.1 0.6 198.4 /63.2 /4.2 598.0 /662.7 /5.2 573.9

0.32 0.7 0.4 1.2 0.6 0.3 1.1 0.8 0.3 1.0 0.7 0.2 1.2

1.06 /644.6 /0.2 171.6 /45.3 0.7 200.5 /64.5 /4.5 599.9 /663.8 /5.4 571.1

0.11 2.1 0.3 2.7 1.9 0.3 2.4 1.6 0.3 2.2 1.4 0.3 2.9

8.38 /649.4 1.8 179.0 /49.4 1.2 180.4 /50.4 /5.3 580.4 /650.4 /4.7 578.9

1.83 1.7 1.0 1.6 1.7 0.5 1.2 0.6 1.1 1.2 0.6 0.9 1.6

The results of the optimised calibration procedure as performed by three operators (A, B, C), compared with the results of a standard procedure based on platform direct marking. The first row reports the statistics of the target function value resulting after the optimisation and for the direct marking. The following rows report the statistics of the coordinate (X , Y , Z ) of the positions of the four vertices (1, 2, 3, 4).

‘S.D.’ indices to the mean indices granted robustness to the minimisation process, because a simple product would be minimised by only one null factor, thus having infinite solutions. The optimised target function value was practically obtained by the product of the two ‘deviation’ indices, while the ‘mean’ indices were null. The test-retest reliability of the present method has been assessed and compared with a current calibration procedure implying the direct marking of some points belonging to the platform. The reported results showed that:

Fig. 3. The horizontal absolute coordinates (the scale and the axes’ directions are reported in the bottom left plot) of the four vertices are presented, one for each subplot. Data are from the three sessions of optimised calibration, performed by three operators, as reported in Table 1. Moreover, the direct marking session data are reported. The mean positions are represented by crosses, the S.D. by the attached rectangles. Additionally, circles are centred on the mean positions (dotted lines distinguish the direct marking result), whose diameters are equal to the diameters of the markers (15 mm) adopted in the experiments.

are probably located outside the stereophotogrammetric calibrated volume); rod markers are moved during the experiment, thus pixel quantisation adds white noise to the markers’ position time series. The target function includes the ‘mean’ residuals of the application point and of the direction of the ground reaction force vector (they are multiplied being, respectively, linear and angular quantities). The sum of the

. the indices values obtained with the optimised method are about one order of magnitude less than the values obtained with the standard procedure; . both the direct-marking procedure and the optimised procedure allow a high intra-operator test-retest reliability (the largest S.D. of the platform vertices coordinates was 2.7 mm); . no relevant inter-operator effect was noted for the optimised method (the largest distance between any couple of alternative positions of one vertex was 5.0 mm, while the mean distance was 3.0 mm); . the direct marking identifies a platform location that had a relevant offset displacement relatively to the platform location identified by the optimised method (the distance between any couple of positions of one vertex, one position obtained by direct marking and one by optimised method, ranged from 4.3 to 24.2 mm, while the mean distance was 15.2 mm). The last point does not necessarily imply that direct marking technique produces inaccurate platform location calibration. Nonetheless, larger target function values show that the standard calibration is less

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coherent with the expected results, and, indirectly, it can be assumed to be less accurate. Some possible causes of inaccuracies, in direct marking procedures, are that: . markers are positioned on the platform that is placed in the ground, thus resulting outside the stereophotogrammetric calibrated volume; . markers are still, with possible bias effect due to the pixel quantisation of the marker image on the camera sensor; . the number of markers usually adopted does not support a more robust redundant approach; . markers can be possibly misplaced. On the other hand, the accuracy of the optimised calibration procedure here proposed relies on the accuracy of the testing object, which could be certified. An interesting feature is that the proposed procedure optimises the current instruments setup, despite their individual levels of accuracy [12], by practically reducing to null the ‘mean’ indices. Nonetheless, excessively larger ‘S.D.’ indices resulting from the optimised platform calibration may warn the operator about the possible occurrence of a miscalibration affecting one of, or both, the measuring systems. Possible enhancements of the present method include the development of an experimental protocol that is completely independent from human operators. A recent publication [13] proposes a mechanical structure, a pendulum, which may be possibly adopted to exert a varying dynamic load on the platform still being completely identifiable by a mathematical model.

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