Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 112 (2015) 332 – 337
7th Asia-Pacific Congress on Sports Technology, APCST 2015
An estimation method of ski friction coefficients in ski running on actual snow fields by sensor system Akiko Kondo*, Kiyoshi Hirose, Ayuko Saito, Hitoshi Doki a
Akita University, Tegata Gakuen-machi, Akita-shi, Akita, 010-8502, Japan
Abstract This paper proposes an estimation method of ski friction coefficients in ski running by using sensor system. It is important for development of skies, ski waxes to obtain ski friction coefficient in ski running on actual snow filed. The ski friction coefficients are assumed as the sum of the Coulomb friction coefficient and the viscous friction coefficient of a ski. The ski friction coefficients are estimated using the Kalman filter by the information of reaction force from snow surface and gliding velocity in ski’s coordinate. The reaction force from snow surface was measured by 6-axis force sensor installed to between ski and binding, and the direction and vertical components of reaction force from snow surface are used for estimation of ski friction coefficient. The gliding velocity was estimated by the 3D posture and the acceleration of ski boot, and the 3D position. The 3D posture was estimated by using the sensor fusion from the angular velocity of gyro sensor and the acceleration of accelerometer attaching the ski boot. The 3D position was obtained by GPS receiver attaching to top of head. We used the estimation method of ski friction coefficients to measurement information obtained by the skier conducting carving turns on an actual snow field, and we obtained the ski friction coefficients varying from moment to moment on an actual snow field. The estimation results of ski friction coefficients fitted for the previous reference. Therefore, the proposed method can use to evaluation of skies, ski waxes. © 2015 The Authors. Published by Elsevier Ltd. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
Keywords: Skiing; Snow friction; Inertial sensor; Force sensor; Culomn friction; Viscous friction;
* Corresponding author. Tel.: +81-18-889-2341; fax: +81-18-837-0405. E-mail address:
[email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
doi:10.1016/j.proeng.2015.07.258
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1. Introduction Skiing has progressed with improvement of equipment and technique. There are several studies on skiing such as development of ski robot [1], simulation of skiing turn [2], characteristics analysis of ski [3] and the motion analysis of skier [4]. It is important to measure ski states such as ski characteristics, ski deformation and ski friction on actual snow filed. Ski friction coefficients are small. It is important to obtain the ski friction coefficients for the evaluations of skies and waxes. In the previous studies, the estimation of snow glide resistance in cross-country ski running [5], the ski friction coefficients were measured in the static and kinetic conditions [6], and the motion measurement and analysis of skier were conducted using these values [7]. However, ski friction has complexity mechanism, and ski friction is widely changed by skies and waxes, and the estimation method of snow friction coefficients on actual snow fields has not been developed. Therefore, it is necessary to develop the estimation method of ski friction coefficients using the gliding information of skiers on actual snow fields. We have proposed the measurement method of reaction force from snow surface using 6-axis force sensor [8], and we have developed the estimation method of 3D gliding velocity and 3D posture of skier by using wearable sensor system and GPS receiver [9, 10]. Therefore, the component of ski friction coefficients such as the frictional force, the normal force and the direction component of gliding velocity can be obtained by using these methods. In this study, we propose the estimation method of ski friction coefficients on actual snow fields by using the sensor system. 2. Method The ski friction coefficients are estimated by the gliding velocities and the reaction force from surface. The inertial and magnetic field measurement system installing gyro sensor (Analog Devices, ADIS16110), accelerometer (Hitachi metals, H30CD) and magnetic field sensor (Aichi steel, AMI 304), GPS receiver (FV-M8, San Jose Technology) and 6-axis force sensor are shown in Fig. 1(a), Fig. 1(b) and Fig. 2, respectively. The setting positions of measurement system and the coordinate system are shown in Fig .3. The GPS receiver was attached to top of head, the inertial and magnetic field sensors were attached to ski boots, and the 6-axis force sensors for measurement of reaction force from snow surface were installed to between skies and bindings. In this study, we assumed the ski friction coefficients as the sum of the Coulomb friction and the viscous friction. The assumed ski friction is represented by Eq. (1), where Fx is the friction force, μ is the Coulomb friction coefficient, Fz is the normal force, γ is the viscous friction coefficient, and vx is the direction component of gliding velocity.
Fx = μ Fz + γ vx
μ > 0, γ > 0
(1)
The friction force equals the direction component of reaction force from snow surface, and the normal force equals the vertical components of reaction force from snow surface. The information can be obtained by the 6-axis force sensor. In this study, we used the information of inertial and magnetic field sensors attached to the right boot and the information of 6-axis force sensor installed to between the right ski boot and the right binding. The direction component of gliding velocity is estimated by the previous method [8]. a
b
Fig. 1 (a) Inertial and magnetic field measurement system; (b) GPS receiver
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Fig. 2 Measurement system of reaction force from snow surface installing 6-axis force sensor
Fig. 3 Setting positions of measurement system and definition of coordinate system
The Kalman filter was used to the estimation of ski friction coefficients. This filter estimates the optimal values from the established nonlinear state and measurement equations. The nonlinear and measurement equations are represented by Eq. (2) and Eq. (3), where xt is the state value, yt is the measurement value, At and Ct are system matrix, wt and vt are white noise. The friction force of measurement value, the nominal force and the directional component of gliding velocity were used
xt +1 = At xt + wt
(2)
yt = Ct xt + vt
(3)
⎡μ ⎤ xt = ⎢ ⎥, yt = [Fx ] ⎣γ ⎦ ⎡1 0⎤ At = ⎢ ⎥, Ct = [ Fz ⎣0 1⎦
vx ]
The Kalman filter algorithm was used for the estimation of constant values (the Coulomb friction coefficient and the viscous coefficient). Therefore, the state equation is unit matrix. The measurement equation is consisted by Eq.
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(1). The friction coefficients are positive values. We used the absolute values of the nominal force and the directional component of gliding velocity for estimating the friction coefficient of positive values to the measurement equation. The Kalman filter algorithm was conducted repeatedly until the estimation values converged at the every constant time (0.1sec). The Kalman filter algorithm is represented by Eqs. (4-8), where Pt is the error covariance matrix, Kt is the Kalman gain, Q is the covariance matrix about process noise wt and R is the covariance matrix about measurement noise vt.
xt−+1 = At xt
(4)
− Pt +1 = At Pt At + Q
(5)
(
(
xt = xt− + K t yt − Ct xt−
−1
) )
Kt = Pt −CtT Ct Pt −CtT + R
Pt +1 = (I − K t +1Ct )Pt +−1
(6) (7) (8)
The covariance matrix of process noise Q was zero matrixes, and the covariance matrix of measurement noise R was configured by considering the sensor noise experimentally. The configured matrixes are represented by Eq. (9).
⎡0 0⎤ Q=⎢ ⎥, R = [0.1] ⎣0 0⎦
(9)
The execution method of Kalman filter algorithm in the estimation of ski friction coefficients is shown in Fig. 4, and the flow chart for estimation of ski friction coefficients is shown in Fig. 5. 3. Experiment We conducted the measurement experiment by a skier gliding on the actual snow slope. The measurement systems were attached to the skier, and the skier conducted the carving and skidding turns in the experiment. We used the measurement information of carving turns. The sampling frequency of inertial and magnetic field sensors is 100Hz, and the sampling frequency of GPS receiver is 5Hz. The measurement time is 30sec.
Fig.4 Execution method of Kalman filter algorithm in the estimation of ski friction coefficients
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Fig.5 Flow chart for estimation of ski friction coefficients
4. Results Result for the 3D gliding trajectory obtained by the GPS receiver is shown in Fig. 6. The 3D gliding trajectory was translated from latitude, longitude and altitude measured by the GPS receiver to meter. The result showed that the skier conducted the right turns and left turns repeatedly after running start. Results for the ski friction coefficients are shown in Fig .7. The coulomb friction coefficient increased during beginning phase, and the value decreased during final phase. The viscous friction coefficient changed in beginning phase, and the value slowly increased until the final phase. These results showed the effect by the difference of snow condition, and the results indicated the difference of snow condition in same ski course.
Fig. 6 Results for 3D gliding trajectory
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Fig. 7 Results for estimation of ski friction coefficients (Coulomb friction coefficient and viscous friction coefficient)
5. Discussion In the previous studies [5, 6], the coulomb friction coefficients were indicated about 0.05 on soft snow and 0.03 on hard snow. In this study, the coulomb friction coefficient indicated the values between 0.03 and 0.04 during intermediate phase and final phase. The estimation ski friction coefficients were used the measurement information of the experiment on the ski course of hard snow. Therefore, this result corresponded to the reference for the previous studies, and we indicated the effectiveness of this method. This method doesn’t consider aerodynamic drag against skier’s body due to the experiment in the low velocity and the low slope. The method considering aerodynamic drag will be developed to use in high speed skiing. 6. Conclusion In this study, we propose an estimation method of ski friction coefficients using sensor system. This method can estimate ski friction coefficients (coulomb coefficient and viscous coefficient) changing moment to moment by using the motion information (gliding velocity, trajectory and reaction force from snow surface). We estimated ski friction coefficients using the measurement information of carving turns by skier attaching the inertial and magnetic field sensors, GPS receiver and measurement system of reaction from snow surface. The effectiveness of this method can be indicated by the results. The proposed method can use for evaluation of skies and ski waxes. References [1] T. Yoneyama, H. Kagawa, M. Unemoto, T. Iizuka, N. W. Scott, A Ski robot system for qualitative modeling of the carved turn, Sports Engineering, 2009, 11, 131-141. [2] P. Federolf, M. Roos, A. Luthi and J. Dual, Finite element simulation of the ski-snow interaction of an alpine ski in a carved turn, Sports Engineering, 2010, 12, 123-133. [3] A. Subic, P. Clifton, J. Beneyto-Ferre, A. Leflohic, Y. Sato, V. Pichon, Investigation of snowboard stiffness and camber characteristics for different riding styles, Sports Enginnering, 2009, 11, 93-101. [4] T. Yoneyama, M. Kitabe and K. Osada, Investigation on the ski-snow interaction in a carved turn based on the actual measurement, Procedia Enginnering 2, 2010, 2901-2906. [5] A. Koptyug, L. Kuzmin, Experimental field studies of the cross-country ski running surface interaction with snow. Procedia Engineering, 13, 23-29. [6] M. Shimbo, Friction on snow of ski soles, unwaxed and waxed, Scientific study of skiing in Japan. 1971, pp. 99-112. [7] M. Brodie, A. Walmsley, W. Page, Fusion motion capture: a prototype system using inertial measurement units and GPS for the biomechanical analysis of ski racing, Sports Technology, 2008, 1-1, 17-28. [8] K. Hirose and H. Doki, A. Kondo, Dynamic analysis and motion measurement of ski turns using inertial and force sensors, The impact of sports technology on sport V, 2013, pp. 365–360. [9] A. Kondo, H. Doki, K. Hirose, An attempt of a new motion measurement method for alpine ski turns using inertial sensors, The Engineering of Sport 9, 2012, pp. 421–426. [10] K. Hirose and H. Doki, A proposal for the motion analysis method of skiing turn by the measurement of orientation and gliding trajectory, The impact of sports technology on sport IV, 2011, pp. 17–22.
