reports in applied measurement ram reports in applied measurement A new transducer for directly measuring the forces and moments which a skier transfers to the skis F Z M Z A. Freddi, G. Olmi, D. Croccolo DIEM Institute for Machine Building University of Bologna Fig. 1: The forces and moments which a skier transfers to the skis z Introduction M x The aim of the work is the design and manufacture of a transducer with six degrees of freedom for measuring the forces and moments transferred from a skier s feet to the skis (see Fig. 1). F x x M y y Two possible configurations were analyzed and tested in order to arrive at an optimal solution. Both cases make use of sensors that are fitted with strain gages arranged between two aluminum plates. These sprung elements are incorporated into the ski binding at the point where vibration absorbing plates are normally found. The first solution is based on the design specified by P. Jung in Reports in Applied Measurement [1]. z 1 y x The second variant represents an independent development which is notable for having fewer components and greater sensitivity in measuring the six load components. These sensors were used in the laboratory and on the piste to test for the release of ski bindings in accordance with national and international standard specifications for insuring the safety of ski bindings (ISO 96) []. y 1 z x Initial observations With the aim of selecting the most suitable configuration for the problem in hand, the solutions known from published scientific works were analyzed first. Special attention was paid to monolithic sensors with three and six degrees of freedom. These sensors are noted for their compactness and for being very sensitive to forces and moments. One of the most commonly used designs is the Maltese cross sensor [8]. Figures and show Maltese cross configurations with three or six degrees of freedom. In figure the load acts upon an inner flange (1) linked by four prismshaped struts () to an outer, circular flange (). A strain gage () is installed on the two opposite sides of the x-y level of each strut. In each case these strain gages form part of a Wheatstone half bridge, which means there are four half bridges in total. F y Fig. : Maltese cross sensor with three degrees of freedom Fig. : Maltese cross sensor with six degrees of freedom ram 1/
The three axial forces (x, y, z) acting on the inner flange can be determined by measuring the tensile, compressive and bending deformations of the prism-shaped struts. This configuration makes it possible to compensate for temperature, although four measurement channels have to be used to analyze only three axial forces. The transfer matrix therefore has a x structure and can be inverted using the Moore algorithm. 1b-1-1a (15b-15-15a) 1 1 SEZ. R - R 1b-1-1a (16b-16-16a) To determine all six load components it is necessary to use this particular Maltese cross configuration (Fig. ). Here the ends of the struts (1) are connected to very thin membranes () that are notable for their high radial elasticity. Strain gages () are installed on all four sides of the struts, requiring 16 strain gages in all. y z x SEZ. K - K Here too, the strain gages installed on opposite sides of the struts are interconnected in half bridges. Thus there are eight half bridges, though only six load components occur. For calibration purposes the resulting transfer matrix is 8 x6 and this too can be inverted with the aid of the Moore algorithm. Another important configuration is the Junyich sensor introduced in []. Its shape is similar to the Maltese cross configuration, but different wiring is used for the strain gages in order to insure mechanical and electrical decoupling and obtain a quadratic form for the transfer/calibration matrix. The diagram in figure uses black points to show that 16, or 8 strain gages can be used. R R The individual elements b ij of transfer matrix B=C -1 are set out in the following equations. 1b-1-1a (5b-5-5a) b--a (6b-6-6a) b--a (8b-8-8a) b--a (b--a) 9b-9-9a (11b-11-11a) 1 1 1b-1-1a (1b-1-1a) Here ε expresses the measured strain in µm/m, where (ε SG-number ) Fi represents the strain induced by the i-th load component with F1=Fx, F=Fy, F=Fz, F=Mx, F5=My, F6=Mz (i =1-6). = b 1i (ε ) Fi + (ε 1 ) Fi (1) = b i (ε 1 ) Fi + (ε 1 ) Fi () Fig. : Junyich sensor, characterized by good decoupling and a quadratic calibration matrix = b i = b i (ε ) Fi + (ε ) Fi (ε 5 ) Fi (ε 8 ) Fi (ε ) Fi + (ε 6 ) Fi () () = b 5i (ε 1 ) Fi + (ε ) Fi (5) = b 6i (ε 11 ) Fi (ε 1 ) Fi + (ε 15 ) Fi (ε 16 ) Fi (6) Fig. 5: Configuration of a Junyich sensor ram 1/
reports in applied measurement Fig. 6: Single-axis sensor in D view... L Fig. :... and in cross-section F F F F Regarding decoupling it should be noted that when there is a combined torsion moment and bending moment, the third bridge output signal is zero (the SGs wired into the opposite branches of the bridge are subjected to the same strain). The same applies when a vertical force is acting. Here the output signal from the fourth and fifth Wheatstone bridge is theoretically zero, since the SGs on opposing struts are subjected to opposite strains. In order to increase measurement precision, the number of strain gages can be increased to. Then the b ij -equations - with the exception of (ε ka ) Fi + (ε kb ) Fi () instead of (ε k ) Fi (8) - have the same form. Figure 5 shows how the cross element of a Junyich sensor is designed with the inner load flange and outer flange. At this point it should be mentioned that roller bearings are built in to prevent torsion of the struts under the effect of torsion moments and bending moments or tensile and compressive forces in the event of longitudinal or transverse loading. The sensors just described can be used provided all forces and moments act upon the inner load flange. However, in this particular application the load cell cannot be fitted in a central position on the underside because the vibration absorbing plates are already located there. But since the underside is mainly stretched in the longitudinal direction the above precondition is not fulfilled, which would give rise to unacceptable measurement errors and inaccuracies. Using these sensors also leads to another problem, which is a reduction in the stability of the connection between the sensor and the upper plate to which the ski binding is fitted. International standards demand tests which demonstrate that all adverse effects on skiing and every risk of injury to the tester are prevented. Initial design and configuration for a transducer The first design for a load cell, published in [1], was fitted with seven separate sensors, each sensor being loaded and relieved of its load in a single direction. All sensors operate as beam spring elements and have a load imposed upon them by a force acting on two diametrically opposed points (Fig. 6 und ). The load is applied through a fork attached to opposite ends of the struts, producing decoupling in the other direction. The structure is isostatic. Four sensors measure the vertical loads only (two moments and one force). Two sensors measure the transverse horizontal forces only (another moment or force can be computed) and the seventh sensor measures the horizontal loading in the longitudinal direction only. Seven half bridges were installed so as to obtain good resolution and temperature compensation. Calibration was carried out in the laboratory, where each sensor was subjected to a series of defined loads. This process made it possible to determine the mathematical relationship between the measurement signals and the forces concerned. The precision of the regression lines to the measured values was found to be between 99.9 % and 1 %. ram 1/ 5
15 Bending moment M y Bending moment M x 1 Torsion moment M z 1 1 Moments [Nm] 5-5 Forces [N] Moments [Nm] -1-15 - 1 8 6 -,5 1 1,5,5,5 Time [s] Fig. 8: Measurement results during a test for binding release due to rotation of boot and leg - 6 65 5 8 85 9 95 1 15 11 Time [s] 1 1 8 6 - - -6 Vertical force F z Transverse force F y 5 Longitudinal force F x 5 Bending moment M y Bending moment M x 5 Torsion moment M z 5 6 65 5 8 85 9 95 1 15 11 Time [s] The diagram in figure 8 shows some typical laboratory test results. They represent the development of moments during a test for binding release due to rotation of boot and leg. The diagram illustrates the simultaneous occurrence of torsion moments and bending moments together with their peak values. Results of the skiing experiments Ski trials were carried out in the Dolomites (Trento, Italy). Typical measurement results from a test run are shown in figure 9. The new transducer The new transducer was intended to be noteworthy not only for its extreme compactness but also for its safety and sensitivity. It was important to have sufficient length in the torque arms in order to increase measurement precision, to avoid adverse effects on the skiing and to protect the tester from the risk of injury. By way of preparation the previously determined experimental results were first analyzed and processed, and the actual peaks for the longitudinal, transverse and vertical forces acting on the sensors were computed. The results shown in the diagrams were processed using a mathematical model to separate out the loads acting on the four sensors. In these diagrams the four color-coded lines represent the evolution of the vertical loads on the sensors. To make the most of the advantages of a modular configuration, each sensor was manufactured using the same electroerosion machine with identical settings.the new design placed the main emphasis on mechanical decoupling. Fig. 9: Test results for the giant slalom above: forces; below: moments 6 ram 1/
reports in applied measurement Fig. 11: Belleville springs placed under the heads of the screws that make the connection between the sensor struts and the load application device to prevent moments to be transferred when the load is applied 5 6 1 Fig. 1: Three-dimensional view of the new sensor with three degrees of freedom Fig. 1 shows a domed surface 1 and a tapped hole at the point where the loads act. The strain gages are installed in three areas to measure deformation. One measures chiefly the vertical deformation, one the longitudinal 5 and another the transverse 6. The four small tapped holes allow the sensor to be screwed to the under plate, while the fifth central hole is intended to save weight. Fig. 11 shows that Belleville springs are placed under the plate and under the heads of the screws that make the connection between the sensor struts and the load application device. This prevents a moment being transferred to the transducer when the load is applied. This degree of freedom is also achieved by means of a gap between the screw shank and the hole in the upper plate, and by means of an elongated drill-hole at the position of the Belleville springs. Wiring as four half bridges and four full bridges insures high resolution and provides temperature compensation. The configuration of the strain gages is illustrated in Fig. 1. Strain gages 1 and are sensitive in the longitudinal direction (longitudinal arrangement L) while SGs and are sensitive in the transverse direction (transverse arrangement T). These SGs are wired to a full bridge. In contrast strain gages 5 and 6, which mainly sense transverse forces, and strain gages and 8 for vertical forces, are wired to half bridges. In order to reduce the number of channels, the double-grid strain gages for the longitudinal forces (T rosettes) are installed on the two forward sensors only. The strain gages 8 (L) (T) 1(L) (T) V V V 1 Channel 1 6 5 for transverse forces are installed on the same side as the two front and two rear sensors, and are wired to two full bridges. The total number of channels then comes to eight (four for vertical forces, two for transverse forces and two for longitudinal forces). Channel Fig. 1: View of the new sensors showing the configuration and wiring of the SGs 6 5 8 Channel ram 1/
Fig. 1: Regression curves for vertical forces 15 Fig. 1b: Association of measured quantities Strain [µm/m] 1 5 Theory Sensor 1 Sensor Sensor Sensor Front section No. of the Wheatstone bridge 1 5 Output signal ε 1 ε ε ε ε 5 Rear section 6 ε 6 ε 5 1 15 5 8 ε 8 Vertical force [N] Initial calibration of the new transducer During initial calibration each sensor was loaded with predefined forces and the measurement signals related to the three bridges (or two for heel sensors) were analyzed. The measurement data was used in the appropriate transfer matrix, where good linearity and reproducibility were observed. By way of example, Fig. 1 shows the regression curves for vertical forces as determined by the calibration test.the calibration results were used to define and normalize the transfer matrices. Normalization was achieved by multiplying each column with the measured peak value for the force concerned. Computation of the coupling coefficients CC (by dividing each element of the normalized transfer matrix c ij by the sum of the elements in this line) shows good sensor decoupling. CC = c ij Σ j c ij The transfer matrices were inverted for computation of the calibration matrices. For the purpose of determining the definitive 6 x 8 matrices that define the relationship between the six degrees of freedom and the eight load signals, the matrices were linked as shown in Fig. 1a and b. Global calibration For global calibration of the load cell, the four sensors were interconnected with the aid of an upper plate as shown in Fig. 16 and then loaded with three forces and three moments. In this special case eight Wheatstone bridges were used, resulting in an 8x6 transfer matrix B and a 6 x 8 calibration matrix C. (9) The number of rows in a transfer matrix is equal to the number of measurement data channels, whereas the number of columns matches the degrees of freedom (load components). In the case of the calibration matrix the situation is reversed, since the mathematical relationship C=B + can be represented as C=B -1 when C and B are quadratic matrices. Otherwise C is treated as pseudoinverted matrix B and can be computed using the Moore algorithm. Calibration is carried out in the following sequence: The upper plate is loaded with a longitudinal force (e.g. 1 N) and the eight measurement data channels are analyzed. The force is then increased in steps until the peak intensity value is reached. This loading process must be repeated at least twice. The mean values then have to be computed in order to obtain a high degree of precision. The procedure enables eight functions to be Fx Fy Fz Mx My Mz =.1889688.51 -.5518611.655 -.55861.6515.58.116 -.598 -. -.5698.698 -.81.8.68.51E-5.115.191 -.5559 -.16561.181 -..858 -.15E-5 -.6699 -.951.169895.95.8995 -..9958 6.89E-5-9.6E-6-1.55E-5 -.111558 -.61.16568 -.9 -.15.18 -.11.16156.8 -.6.5 ε 1 ε ε ε ε 5 ε 6 ε ε 8 Fig. 1a: Computation of the load components from the measured strain 8 ram 1/
reports in applied measurement Fig. 15, right: Second generation sensor with HBM strain gages, e.g. LY11 /5 Measured quantity Longitudinal force on the front left sensor Longitudinal force on the front right sensor Transverse force at the toe Vertical force on the front left sensor Vertical force on the front right sensor Transverse force at the heel Vertical force on the rear left sensor Vertical force on the rear right sensor recorded. It also enables the computation of eight regression curves which describe the trend in the measurement data (measurement data compensation) and which are characterized by the fact that they show an increase. These coefficients are entered in the first column of the transfer matrix. They refer to the first load component and represent the signal for a 1 N loading. The above procedure is then repeated using a transverse force. The values determined in this way are entered in the second column of the transfer matrix. Finally this procedure is repeated for all the other load components. The last three columns of the transfer matrix then contain the results arising from the output signals from the Wheatstone bridges in the case of the three loads with a moment of 1 [N m]. In order to compute a calibration matrix the transfer matrix then has to be inverted. Determining the calibration matrix is the most important objective of this work, since computation of the six load components acting on the newly developed load cell is directly dependent upon it. The equations for computing the strain from the loading and for computing the loading from the strain are as follows: ε =B F (1) F=C ε (11) Multiplying by the B T matrix gives B T ε=b T B F (1) If the experimental or mathematical problem is given the right conditions, the resulting B is an 8 x 6 matrix of rank 6. Then B T B is a 6 x 6 matrix with full rank which can be easily inverted as follows: (B T B) -1 B T ε=f (1) On comparing (11) and (9) it is noticeable that the 6 x 8 matrix (B T B) -1 B T is identical to C and represents the calibration matrix B + which expresses the relationship between the strain signals and the loads concerned. Fig. 16: above: Individual sensors; below: The complete transducer, ready for global calibration ram 1/ 9
Summary This paper presented two transducers with six degrees of freedom, suitable for measuring the forces and moments that a skier transfers to the skis.the first transducer was fully tested in the laboratory, during calibration and during skiing trials under real load conditions.the second new and independently developed transducer was designed to increase compactness, safety, precision and sensitivity in relation to the six loads. At the same time satisfactory decoupling effects were achieved in three ways: - mechanically by means of the HBM strain gages which are optimally tuned to the geometry and stresses of the sprung elements. - electrically due to wiring the strain gages in a special way to compensate for unwanted signals. References [1] Peter Jung: Applying metrology to skis, Reports in Applied Measurement, 11 (195) Vol., pp. -, Vol., pp. 59-6 [] Alpine ski-bindings, Safety requirements and test methods, International Standard, ISO 96 [] Lu-Ping Chao, Ching-Yan Yin: The six-component force sensor for measuring the loading of the feet in locomotion, Materials and Design, (1999) [] Lu-Ping Chao, Kuen-Tzong Chen: Shape optimal design and force sensitivity evaluation of six-axis force sensors, Sensors and Actuators A Physical, 6 (199) 15 11 [5] Gab-Soon Kim, Dea-Im Kang, Se-Hun Ree: Design and fabrication of a six-component force/moment sensor, Sensors and Actuators A Physical, (1999) 9 [6] Tom Boyd, M.L.Hull, D.Wootten: An improved accuracy six-load component pedal Dynamometer for cycling, Journal of Biomechanics, 9 (1996) 115 111 [] D. Gorinevsky, A. Formalsky, A.Schneider: Force Control of Robotics Systems CRC Press, New York - mathematically by computing the calibration matrix. The individual sensors in the new load cell were fully calibrated. A global calibration then has to be carried out in the laboratory, during which forces and moments are simultaneously applied to the whole configuration along the x, y and z axes (Fig. 16). Skiing trials can then be carried out under real load conditions. 1 ram 1/