Optimal Path Calculation in the Countryside A Raster-attribute-based Model Approach Philipp Le1, Gerald Eichler2 1 Otto-von-Guericke-University Magdeburg Faculty of Electrical Engineering and Information Technology Universitätsplatz 2, 39106 Magdeburg, Germany [email protected], [email protected] 2 Deutsche Telekom AG Technology Innovation: Portfolio, Strategy and Data Deutsche-Telekom-Allee 7, 64295 Darmstadt, Germany [email protected], [email protected] Abstract. Using a landscape raster-based model for optimal route navigation in orienteering sports, velocity related attributes are mapped onto single squares. Both, accelerators and dampeners as well as barriers are taken into consideration. The optimization calculation between points is based on A* while the entire route is calculated by asymmetric traveller salesman problem solutions. Keywords: Asymmetric Traveller Salesman Problem (ATSP), Orienteering Map, Digital Elevation Model (DEM), Amateur Radio Direction Finding (ARDF), Navigation, Route Optimization, Raster Graph Transformation. 1 Foot Orienteering and Amateur Radio Direction Finding Orienteering is an example, where routing is not limited to predefines paths. Foot ori- enteering is considered the mother of all derivatives in orienteering sports. Primarily, the competitor aims to find the shortest path between two control points (CP) in terms of time consumption. This leads into the question: Where is the right place to leave a given foot path network into vegetation covered countryside? A detailed map (Fig. 1) and compass assist the competitor. Except for restricted ar- eas, any way through rural terrain is valid. Amateur Radio Direction Finding (ARDF), where CPs are hidden transmitters called foxes, extends the competition goal by two dimensions, using a portable direction selective radio receiver: 1. Fox location: The location of CPs within the countryside is not known from the very beginning. However, after five minute running, a competitor has a rough idea about their locations by means of his radio. 2. Fox order: The order in which competitors search for and discover the foxes is en- tirely at their discretion except for the finish beacon, which shall be registered as the last one of the transmitters. 62 LBAS 2017 This contribution focuses on (2) and therefore, it is better applicable to Foxoring. The artificial term is constructed out of “FOXhunting” and “OrienteeRING”. To discover the optimal path, beside the nature of countryside, there is also a set of secondary cri- teria like personal maximum speed and continuation or abilities in navigation in com- plex environments. Furthermore, weather and time of the year influence the competi- tion terrain. Differences between the three sports are listed in Table . While Orienteer- ing is essentially targeting navigation and object discovery capabilities, the key of Fox- oring is the discovery of the optimal order of control points. ARDF requires even more strategic decisions during the entire competition, due to limited air time of each fox with one out of five minutes. Table 1. Comparison of Orienteering, Foxoring and classical ARDF. Orienteering Foxoring ARDF CP sequence Fix Flexible Flexible CP position Exactly known Roughly known Not known CP indication Kite 30 by 30 cm Log station only Kite 30 by 30 cm Radio transmitter n/a Constantly 1 min interval Transmission pwr. n/a Low power (mW) High power (W) Number of CPs 10 to 30 7 to 20 + beacon 5 + beacon For all three disciplines, the sportsman is given an orienteering map (Fig. 1). In the approach of this contribution, the map will be overlaid by a regular raster (Section 2). Incorporated information is based on public geo-data. Section 3 will elaborate on maps and a Digital Elevation Model (DEM) as data sources. Fig. 1. Section of orienteering map of “Magdeburg Herrenkrug”, Germany [1]. The model approach (Section 4) will overlay a raster with equal regular shapes of a defined granularity [2]. To apply a classical Traveller Salesman Problem (TSP) solu- tion, after A* reduction, the final routing is based on vectored graphs (Section 5). Ortsbezogene Anwendungen und Dienste 2017 63 2 Regular Raster Shapes Natively, vectored paths are perfect for route optimization problems. However, orient- eering offers the entire landscape as competition area. Therefore, a “navigation on ar- eas” approach [2] will be targeted. As ARDF competitions mostly not exceed a race length of 12 km bee line, the relevant part of the map is typically not larger than 3 km by 3 km. At a scale of 1:10.000 this will fit an A3 sheet of paper. 2.1 Options for Regular Shape Selection The types of convex polygons with equal edge length which are sufficient to raster the projection of a surface into regular shapes is limited to triangle, square and hexagon (Fig. 2). For reasons of given data sources, the rectangle will be evaluated as an addi- tional candidate right here. (a) (b) (c) (d) Fig. 2. Regular shape raster decomposition (a) triangle, (b) rectangle, (c) square, (d) hexagon. Having path mapping in mind, the neighbourhood relations of raster fields are im- portant. Starting from a single shape, two types of direct neighbours can be identified: 1. Line neighbour (L): Two adjacent shapes have exactly two corners in common. 2. Corner neighbour (C): Two adjacent shapes have exactly one corner in common. By design, each raster field has the same number of neighbours. Neighbours with the distance one, sharing either one or two corners, should be called first (1st) order neigh- bours. In other words, L neighbours share one edge. Table 2. Neighbourhood relationship of shapes of 1st order. Triangle Rectangle Square Hexagon Number of L-neighbours 3 4 4 6 Number of C-neighbours 9 4 4 0 Different L-distances 1 2 1 1 Different C-distances 2 1 1 n/a Different L-directions 3 4 4 6 Different C-directions 9 4 4 6 Table 2 summarizes the absolute number and the number of different distances and directions of first order neighbours. As the comparison shows, triangles have a rather complex neighbourhood, which can be simplified by using hexagons as a super-shape 64 LBAS 2017 of triangles, i.e. each hexagon can be split into six triangles. In small map scale, a rec- tangle would perfectly fit to geo-coordinates, as 1 arc sec in Central Europe equals about a size of 20 m by 30 m. However, a square better represents the symmetric reso- lution for the given problem in latitude and longitude. Therefore, squares are taken for this raster approach model. 2.2 Multi-order Neighbourhood of Squares More general the order of neighbourhood is defined as the number of circle-free and outwards first order steps between two selected raster fields. Outwards indicates an increasing distance by each step. Distances are always measured between the centres of squares; see white arrows in Fig. 3. Fig. 3. Multi-order neighbourhood specification of squares. As distance reference the edge length of a single square is to be considered as a. Table 3. Neighbourhood relationship of shapes of 1st order. Neighbour type Order Neighbours Distance New directions L 1 4 a 4 8 C 1 4 √2*a 4 LL 2 4 √4*a = 2*a 0 LC/CL 2 8 √5*a 8 1*8 CC 2 4 √8*a = √2*a 0 LLC/LCL/CLL 3 8 √10*a 8 2*8 LCC/CLC/CCL 3 8 √13*a 8 Two findings characterize the multi-order model for squares: 1. Line (L) and corner (C) direction extension towards higher order (O) is always com- mutative regarding distance and direction. 2. Only 8*(O-1) new directions in higher order are potential of interest, as others can be modelled iteratively by lower order without any routing gain. Ortsbezogene Anwendungen und Dienste 2017 65 To simplify the methodology for the upcoming sections, only the first order relation- ship will be applied within this contribution. 3 Data Sources for Attribution Orienteering typically takes place in difficult rural areas. There are two factors that heavily influence the speed of the competitor: height differences and the surface cov- erage e.g., plants or swamp. This information is given to the competitor by a detailed “orienteering map”. For the envisaged approach the information needs to be transferred into the raster model (Section 4). 3.1 SRTM Height Data Although height information is given by contour lines within the orienteering map, there is a better source to meet the raster approach. The Shuttle Radar Topography Mission (SRTM) has collected earth surface height data by Spacebone Imaging Radar measurements (SIR-C). The data is public available all over the world between 60 de- gree latitude south and 60 degree latitude north. There are data sets of two resolutions: 1 arc sec and 3 arc sec. The latter is generated by calculating the average of nine SRTMGL1 values1. The access to the SRTMGL1 resolution requires a personal regis- tration. The other sets of Table 4 are subject to instant download. The sources claim an accuracy of about 6 (DLR) to 12 meters (NASA) [3]. However, height differences of neighbour fields are more important, which imply better results. The data is provided as file of 16-bit-integer numbers in big endian (BE) notation with- out a header as a bulk for one degree latitude by one degree longitude. Positive values represent heights in meter above mean sea level. Values are arranged from West to East and then North to South. The value “-32768” is reserved for not known height. SRTM Void Filled altitude data are the result of additional processing to address areas of miss- ing data or voids in the SRTM Non-Void Filled set. Table 4. Available SRTM data sets [4]. File set Resolution Number of DTED Rectangle in values accuracy Central Europe SRTMGL1 1 arc sec 3601*3601 Level 2 20 m * 30 m SRTMGL3 3 arc sec 1201*1201 Level 1 60 m * 90 m SRTMGL30 30 arc sec 121*121 Level 0 600 m * 900 m There is a unique file naming convention.