Mathematical model establishment and validation of irregular characteristic morphology for reindeer foot bottom

Abstract: In recent years, the traffic accidents occur frequently in winter and how to prevent frequent traffic accidents has become an urgent problem to be solved. Reindeer lives in frigid areas and possesses the superior walking ability on ice. Reindeer feet are the exclusive parts which contact with the ice directly. The sole characteristic morphology of foot bottom is the key factor to the excellent anti-skid performance of reindeer. The research of the reindeer feet bottom characteristics will help to apply the superior characteristics to snow tire tread and improve the trafficability property of tire on ice ground. Therefore, the mathematical models of the reindeer feet were studied. The feet were taken from 4 eighteen-year-old adult homebred reindeers, which were bought from the Ewenki nationality located in Genhe City, Inner Mongolia, China. In order to avoid impurity interference, the feet were cleared up before the experiments. The geometric point clouds of the reindeer feet were obtained by using a three-dimensional (3D) hand-held non-contact laser scanner. After that, the data were imported to Geomagic Studio software and we reconstructed the analytical model. With the cutting function of the software, the reconstructed model of reindeer foot was divided into 4 typical characteristic areas: The edge curves (inside edge and outside edge), the ridge groove surface and spherical cap surface. In addition, these point clouds were imported to CATIA (computer aided three-dimensional interactive application) to be filtered, which reduced the calculation amount of the fitting curves and surfaces. We filtered the dense point clouds through adopting the reasonable method and the retained characteristic points that reflected the edge and sole morphology of reindeer foot with the digitized shape editor module of CATIA. Through the above processes, the 3D coordinate data points of these characteristic areas were exported. The edge curves were fitted using curve fitting software 1stOpt (First Optimization), and then fitting the equation and the fitted parameters were acquired, respectively. We also used the surface fitting function of Matlab software to fit 2 surfaces: Ridge groove surface and spherical cap surface. Finally, 2 surface fitting models were achieved. The fitting results revealed that the R2 (coefficient of determination) values of 2 curves and 2 surfaces were 0.994, 0.992, 0.96, and 0.98, respectively, which were all close to 1. It showed that the characteristic areas of reindeer feet were successfully transformed from the biological model to the mathematical model. In order to verify the model's scientificity, the specimen of other reindeer foot was used. We acquired the 3D coordinate data of the same characteristic areas of other reindeer foot by adopting the same treatment. The values of x and y axis for 4 areas were imported into relevant mathematical models and the relevant values that were dependent variables were acquired. By comparing the differences between fitting values and actual values, the residual error range, relative error range and mean relative error were analyzed. The relative error range of 2 curves models was 0.16%-2.73% and 0.12%-1.66% respectively, and the mean relative error was 1.5% and 0.68%, respectively. The relative error range of surfaces models was 0.43%-10.28% and 1.25%-9.74%, respectively. The maximum was about 10%, due to the influence of air impurities. However, the mean relative errors of surfaces models were 4.83% and 4.27%, respectively. The mean relative errors of 4 models were within 5%, which proved the mathematical models' effectiveness in this paper. In addition, the elements of reindeer feet were examined with EDS (energy dispersive spectroscopy). It included such elements as S, Si, Fe, Al, K and Ca, besides C, O and N. Different chemical elements meant different compounds, and different surfaces with different skid-resistance were composed of different elements. This paper will provide research direction for studying tire on ice ground with engineering bionic technology.