# Three-axis optical force plate for studies in small animal locomotor mechanics

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4 S. Tonia Hsieh Rev. Sci. Instrum. 77, Signal-position calibration protocol Equidistant points were marked off in a grid on the force platform and a single known mass was placed at each point to obtain Z-axis channel responses. Since the total torque total acting on any static object is always zero, position calibration curves were derived by calculating the torques acting on each channel 1 4 for the Z axis, mass = =0, F L X = b 1 F 1 + b 2 F 2 + b 3 F 3 + b 4 F 4, F L Y = b 5 F 1 + b 6 F 2 + b 7 F 3 + b 8 F 4, 8 FIG. 2. Signal-force calibration results. a Schematic of the calibration setup. The thick, solid line is the tape and the dotted lines are string. The calibration mass is labeled as m. Individual channel calibration results were plotted against the actual calibration weight for the b X axis, c Y axis, and d Z axis. The dotted lines are the linear regressions. Slopes for all axes were 1.00, with R and solving for X and Y which are the positions on the force platform at which the mass is placed. Since cross talk between axes was low see Sec. III, the uncorrected force value for each channel, a i s i, was used as the value for F i. The constants b 1 8 are the perpendicular distances of the Z-axis sensors from the platform origin. If the weight is placed at p positions, the following system of matrices represents the calibration scheme and it is possible to solve for the vector of b constants using a least squares fit: X p = F1 /F L Y X F 1 /F L Y ] X F p p 1 /F L Y F 2 /F L F 3 /F L F 4 /F L F 1 /F L F 2 /F L F 3 /F L F b 4 /F 2 L F 2 /F L F 3 /F L F b 4 /F L F 1 /F L F 2 /F L F 3 /F L F b 4 4 /F L b 5 ] ] ] ] ] ] ] ] b F p p 2 /F L F p p 3 /F L F p p 6 4 /F L b F p p 1 /F L F p p 2 /F L F p p p b1 3 /F L F p 7 4 /F L b 8. 9 D. Data collection Fishes were induced to jump off the force plate in the X direction. Each jump was recorded with one high-resolution , high-speed camera Photron Ultima APX, San Diego, CA filming at 500 frames/s, zoomed in on the lateral view of the force platform and fish, and one camcorder Sony Handycam model DCR-TRV38 simultaneously filming the lateral and dorsal via a 45 mirror views of the entire length of the jump. A 0.5 cm grid was positioned in the background to permit video calibration and extraction of jump distance and height. The analog force plate response signals were electronically amplified, then converted to a digital signal using an ADInstruments Colorado Springs, CO analog-digital converter. Data were sampled at 1 10 khz and recorded on a laptop computer with POWERLAB CHART5.0 ADInstruments, Colorado Springs, CO software. Jump force during a trial was calculated using the a vector of constants determined during the signal-force calibration. E. Confirming force plate accuracy The blenny was modeled as a point-mass projectile initially subject to three forces in the X, Y, and Z directions. Time to peak height t p, maximum jump height z t p, and jump distance d jump were determined to be t p = ż 0 /g, 10 z t p = ż 2 0 t p /2g, and 11 d jump =2t p ẋ ẏ Calculated values for jump height and distance were then compared with those determined from the video images. The moment when the blenny s tail left the locomotor surface was set as t=0.

5 Optical force plate design Rev. Sci. Instrum. 77, FIG. 3. Signal-position calibration results for real vs. calculated a X position and b Y position. In both panels, the fitted linear regression dotted lines have a slope of 1.00, R 2 =0.999 X and Y, and p c Ball-rolling experiment result. X-Y position plots of the ball position digitized from video footage and the ball position calculated from individual force plate Z-axis channel output. To assess accuracy of center of pressure determination, a 1.06 g steel ball bearing was rolled across the force platform and filmed in dorsal view. The motion of the steel ball was then digitized and the calculated position data were then compared with the actual, digitized data for confirmation of center of pressure accuracy. III. RESULTS AND DISCUSSION A. Force plate calibration results Force calibration and position calibration curves were linear, all with R Figs. 2 b 2 d, 3 a, and 3 b. Force plate natural frequency was much lower than desired Hz because the moving elements of the force plate were heavy relative to the blades spring stiffness. Mean cross talk between channels were 4.3% from X to Y, 0.6% from X to Z, 0.6% from Y to X, 8.2% from Y to Z, 8.1% from Z to X, and 4.4% from Z to Y. B. Experimental results When rolling a ball across the surface of the force plate, calculation of the location of center of pressure matched up well Fig. 3 c, with maximum errors of 0.56 and 0.41 cm at the start and end of the roll when the ball was moving most rapidly. Mean errors were 0.07 cm along the X axis and 0.13 cm along the Y axis. The apparent mismatch between calibration accuracy along the Y axis as shown in Fig. 3 b with the calculated results in panel c may be caused by two factors. First, the system was not fully modeled and therefore some characteristics of force plate response, such as damping elements, were not accounted for. A more rigorous model of FIG. 4. Jumping blenny data. a A single high-speed video frame of a Pacific leaping blenny jumping off a horizontal surface on land. Representative unfiltered gray line and filtered black line force traces of a 0.6 g blenny jumping off the force plate for the b X axis, c Y axis, and d Z axis. The black triangles represent the location of peak positive and negative forces used to calculate signal/noise ratios presented in Table I. The dotted line in panel d is the fish s body weight in mn. The blenny s tail loses contact with the surface at t=0 s. the dynamic components of the system would increase its positional accuracy. Second, the positional calibration as described here assumes the cross talk between channels is negligible when determining the center of pressure location. Although cross talk was shown to be less than 10% among all axes, this assumption likely introduced some error into the final center of pressure localization calculations. The forces generated along each axis during an actual blenny jump are shown in Fig. 4. All data were filtered with a ninth-order Butterworth low-pass filter with a cut-off frequency of 20 Hz. A high order for the filter was chosen to increase the steepness of signal drop-off. For the trial shown

7 Optical force plate design Rev. Sci. Instrum. 77, I. K. Bartol, G. Morteza, D. Weihs, P. W. Webb, J. R. Hove, and M. S. Gordon, J. Exp. Biol. 206, E. G. Drucker and G. V. Lauder, J. Exp. Biol. 203, F. E. Fish, J. Hurley, and D. P. Costa, J. Exp. Biol. 206, J. E. Harris, J. Exp. Biol. 13, Y.-C. Pai and K. Iqbal, J. Biomech. 32, M. G. McCay, J. Exp. Biol. 204, D. L. Jindrich and R. J. Full, J. Exp. Biol. 205, J. Schmitt, M. Garcia, R. C. Razo, P. Holmes, and R. J. Full, Biol. Cybern. 86, M. Sun and Y. Xiong, J. Exp. Biol. 208, L. H. Ting, R. Blickhan, and R. J. Full, J. Exp. Biol. 197, G. A. Cavagna, J. Appl. Physiol. 39, A. A. Biewener and R. J. Full, in Biomechanics Structures and Systems: A Practical Approach, edited by A. A. Biewener Oxford University Press, Oxford, 1992, pp M. Bhikajee and J. M. Green, African Zoology 37, R. Blickhan and R. J. Full, in Biomechanics Structures and Systems: A Practical Approach, edited by A. A. Biewener Oxford University Press, Oxford, 1992, pp N. C. Heglund, J. Exp. Biol. 93, F. P. Beer and E. R. Johnston, Jr., Mechanics of Materials McGraw-Hill, New York, 1992, p. 740.

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