CN113124878A - Lunar surface large-range road topology network construction method, system and device - Google Patents

Lunar surface large-range road topology network construction method, system and device Download PDF

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CN113124878A
CN113124878A CN202110431195.5A CN202110431195A CN113124878A CN 113124878 A CN113124878 A CN 113124878A CN 202110431195 A CN202110431195 A CN 202110431195A CN 113124878 A CN113124878 A CN 113124878A
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白成超
郭继峰
王平
于晓强
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Harbin Institute of Technology
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Abstract

A method, a system and a device for constructing a lunar surface large-range road topology network belong to the technical field of lunar exploration path planning. The method aims to solve the problem that the task area coverage is poor in the current lunar path planning method or road topology network. The method firstly analyzes the lunar reachable area based on a lunar terrain feature calculation method, then improves a Poisson disc sampling mode to enable the sampled network nodes to have the optimality of the coverage rate of the flat area, improves a heuristic function of an A-algorithm and utilizes a path generated by the A-algorithm to realize the construction of the road topology network. The method is used for lunar exploration path planning.

Description

Lunar surface large-range road topology network construction method, system and device
Technical Field
The invention relates to a method, a system and a device for constructing a lunar road topology network, and belongs to the technical field of lunar exploration path planning.
Background
The construction of the existing road topology network is mainly realized based on a Poisson disc sampling algorithm and an A-x algorithm.
The Poisson Disc Sampling (Poisson Disc Sampling) algorithm is a plane random Sampling algorithm, generated Sampling points meet the characteristic that the Sampling points are randomly and uniformly distributed on a plane as much as possible, and the distances between all the points are not less than the specified minimum distance. Firstly, setting the minimum distance between sampling points as r, then randomly generating an active sampling point in a sampling range, and randomly generating k candidate sampling points in an annular region around the sampling point, wherein the annular region takes the active sampling point as a center of a circle and the radius of the annular region extends from r to 2 r. And eliminating points with the distance less than r from the selected sampling points from the k random candidate sampling points, and taking the rest points as new active sampling points. If all the k sampling points are rejected and no usable points are left, the selected active sampling point at the center of the annular area is marked as inactive and is not used for generating a candidate. When candidate sampling points are eliminated and screened, a unit grid with the length of a diagonal line r is used for accelerating distance check. Each unit grid can only contain one sampling point at most, and only a fixed number of adjacent unit grids around the candidate sampling point need to be checked. And when all the sampling points are in the inactive state, the iteration of the algorithm is finished. The main drawback of the poisson distributed sampling method is that the number and quality of the sampling points cannot be accurately controlled.
The a-Star algorithm is a direct search method most effective for solving the shortest path, and is a common heuristic algorithm, which expands nodes of the search according to a heuristic function f (n), where f (n) is g (n) + h (n), where f (n) is a cost estimate from an initial state to a target state via a state n, g (n) is an actual cost from the initial state to the state n in a state space, and h (n) is an estimated cost of an optimal path from the state n to the target state. When the heuristic function f (n) satisfies the consistency condition, a x can find the optimal solution. Meanwhile, the algorithm A stores nodes to be detected in the path planning process in an Open table, stores the detected grids in a Close table, then expands all neighbor nodes of the current node to be added into the Open table, and then selects the node with the minimum value of f (n) to expand, and the iteration is carried out until the target node is searched. The final path is then traced back from the target node using the parent node (parent) to store the traceback node. The heuristic function is a core function of the A-star algorithm and directly determines the characteristics and quality of the algorithm generation path.
Disclosure of Invention
The invention aims to solve the problem that the task area coverage is poor in the current lunar path planning method or road topology network.
A lunar surface large-range road topological network construction method comprises the steps of firstly, carrying out lunar surface reachable area analysis based on lunar surface topographic features to obtain a lunar surface reachable area map; then, analyzing and searching a flat area based on a lunar accessible area map, and then carrying out optimal Poisson disc sampling on the alternative flat area to form a network node set for constructing a road topology network; determining a road topological network structure according to a nearest neighbor network structure based on the network node set; finally, according to the network node set and the road topology network structure, connecting each network node by using an A-star algorithm to realize the construction of the road topology network;
the process of analyzing and searching the flat area, then carrying out optimal Poisson disc sampling on the alternative flat area, and forming the network node set for constructing the road topology network comprises the following steps:
selecting a flat area by adopting a sliding window method, namely setting a second sliding window with a fixed size on the lunar surface accessibility map, calculating the coverage rate of an accessible area in the second sliding window, and selecting all second sliding window areas with the accessible coverage rate exceeding a coverage rate threshold value as alternative flat areas;
based on a Poisson disc sampling mode, selecting the center position of a flat area with the highest reachable area coverage rate of a second sliding window as a new sampling point when generating the sampling point: firstly, calculating the reachable area coverage rate based on a second sliding window to obtain a position set P of a candidate flat areaplainAnd its corresponding set of area coverage SplainThen in the current area coverage rate set SplainSelecting the node with the highest coverage rate as the current node, and then selecting the current node from the SplainRemoving; then judging whether the distances between the current node and the existing sampling nodes are all larger than the sampling radius RsIf yes, adding the sampling node as a new sampling node into the sampling point position set PNodeOtherwise, the node is discarded, at SplainUntil all the alternative flat regions are traversed.
Further, the process of determining the road topology network structure according to the nearest neighbor network structure based on the network node set includes the following steps:
according to the nearest neighbor network structure, connectivity among nodes is set through the distance among the nodes, and the distance is smaller than 2RsThe nodes are arranged to be communicated and the distance is more than 2RsThe nodes are set to be not communicated, so that the selection of the road topology network nodes and the setting of the connectivity can be completed.
Further, an a-algorithm is used for connecting each network node to form a heuristic function of the a-algorithm in the construction of the road topology network, which is a safety heuristic function, and the method comprises the following specific steps:
fsafe(n)＝g(n)+ωsafe(n)+Diagonal_heuristic(n)
in the formula: g (n) is the actual cost from the initial state to state n in the state space, defined as the actual distance from the starting point to the current node; omegasafe(n) a safety cost for node n, defined as a total number of obstacles within a second sliding window centered on node n; diagnonal _ itself () is a Diagonal heuristic distance function, Euclidean distance function, or a heuristic function that satisfies a consistency condition.
Further, the size of the second sliding window is determined according to the size of the actual space, and the actual space is 20km × 20 km.
Further, the lunar reachable area analysis process based on the lunar terrain features obtains the terrain features of the DEM map in a sliding window mode.
Further, the process of obtaining the topographic characteristics of the DEM map in the form of the sliding window comprises the following steps:
setting a sliding window with the size of a 3 x 3 pixel grid, namely a first sliding window, and taking terrain information obtained by calculation in the first sliding window as terrain features of a central grid, wherein the terrain features comprise a terrain slope theta, a terrain relief degree R and a terrain roughness delta;
then setting threshold limits of maximum gradient, undulation degree and roughness according to the ability of the patrol device;
and regarding the grid area exceeding the threshold limit of the ability of the rover as an obstacle area, and converting the DEM map into a reachable area map.
Further, the terrain slope θ, the terrain undulation degree R, and the terrain roughness δ are respectively as follows:
Figure BDA0003031413750000031
R＝Hmax-Hmin
Figure BDA0003031413750000032
in the formula: f. ofxThe rate of change of the center grid in the east-west direction elevation is taken as the rate of change of the center grid in the east-west direction elevation; f. ofyCalculating the elevation change rate of the central grid in the north-south direction according to the following formula; hmaxIs the maximum elevation value in the first sliding window; hminIs the minimum height value in the first sliding window; hiThe elevation value corresponding to the ith grid in the first sliding window is obtained;
Figure BDA0003031413750000033
is the average of all elevation values within the first window.
Further, the center grid has an east-west elevation change rate fxCentral grid north-south direction elevation change rate fyRespectively as follows:
Figure BDA0003031413750000034
Figure BDA0003031413750000035
in the formula: h1-H9Elevation values corresponding to 9 grids in a 3 x 3 window respectively; and g is the resolution size of DEM data.
Further, the threshold limits for setting the maximum gradient, the waviness and the roughness according to the ability of the patrol instrument are as follows:
θ≤20°
R≤2*g*tan(20°)
Figure BDA0003031413750000036
in the formula: and g is the resolution size of DEM data.
A lunar surface large-range road topological network construction system is used for executing a lunar surface large-range road topological network construction method.
A lunar surface large-range road topological network construction device is used for storing and/or operating a lunar surface large-range road topological network construction system.
Has the advantages that:
the lunar large-range road topology network refers to a large-range transfer channel network which is interconnected and communicated in the full-moon range, and the large-range transfer path of the detector in the full-moon reachable area can be planned by constructing the full-moon traffic road topology network. Compared with the traditional planning algorithm which directly searches a large-range transfer path, the method for constructing the road topology network plan has the following advantages: 1) after the road topology network is constructed, the lunar terrain environment is basically unchanged, and the method can be used for planning large-range detection paths for a long time. 2) The planning speed of planning the large-scale transfer path through the road topology network is obviously higher than that of directly searching the transfer path on a large-scale map. 3) The completeness of the large-range transfer planning probability can be improved, when a certain path cannot pass, other alternative paths can be rapidly provided, and Rome of strip large-path passing is realized. 4) By constructing road topology networks with different scales and densities, autonomous exploration path planning with different ranges and different granularities can be realized.
The reachable area analysis method provided by the invention can better extract the lunar surface topographic characteristics and provide the lunar surface reachable area meeting the patroller capacity constraint. Meanwhile, the optimal Poisson disc sampling result provided by the invention is moderate in density and good in coverage of the whole task area, and the coverage efficiency of the network can be improved. The improved A-algorithm provided by the invention can make the generated path far away from the barrier area as far as possible, thereby improving the safety and the passing probability of the network.
Drawings
Fig. 1 is an Apollo mission area DEM map.
Fig. 2 is a result of analysis of reachable areas of the Apollo task area.
Fig. 3 shows the selection result of the nodes of the road topology network in the Apollo task area.
Fig. 4 is a construction result of an Apollo task area road topology network.
Detailed Description
The first embodiment is as follows:
in the method for constructing a lunar wide-range road topology network according to the embodiment, analysis of a lunar reachable area is performed based on a lunar terrain feature calculation method, a poisson disc sampling mode is improved to enable a sampled network node to have the optimal coverage rate of a flat area, a heuristic function of an algorithm A is improved to enable a generated path to be far away from an obstacle area, and the method specifically comprises the following steps:
the method comprises the following steps: the lunar reachable area analysis method specifically comprises the following steps:
the method adopts a sliding window form to calculate the topographic features of the DEM map, namely, a sliding window with the size of 3 multiplied by 3 pixel grid, namely a first sliding window is arranged, and the topographic information calculated in the first sliding window is used as the topographic features of the central grid. The lunar terrain feature is mainly calculated from three aspects of terrain gradient theta, terrain undulation degree R and terrain roughness delta, and the calculation method is shown as the following formula.
Figure BDA0003031413750000051
R＝Hmax-Hmin
Figure BDA0003031413750000052
In the formula: f. ofxAs a central grid(ii) east-west direction elevation change rate; f. ofyCalculating the elevation change rate of the central grid in the north-south direction according to the following formula; hmaxIs the maximum elevation value in the first sliding window; hminIs the minimum height value in the first sliding window; hiThe elevation value corresponding to the ith grid in the first sliding window is obtained;
Figure BDA0003031413750000053
is the average of all elevation values within the first window.
Figure BDA0003031413750000054
Figure BDA0003031413750000055
In the formula: h1-H9Elevation values corresponding to 9 grids in a 3 x 3 window respectively; and g is the resolution size of DEM data.
Then, threshold limits of maximum gradient, undulation and roughness are set according to the capacity of the patrol instrument, the maximum gradient threshold value which can normally run is set to be 20 degrees according to the off-road performance of the patrol instrument, and the threshold values of the undulation and the roughness are determined and converted into the gradient limit, as shown in the following formula.
R≤2*g*tan(20°)
Figure BDA0003031413750000056
The DEM map can be converted to a reachable area map by treating grid areas that exceed the threshold limit of the rover's capability as obstacle areas.
Step two: analyzing and searching a flat area based on the lunar surface reachable area map generated in the step one, and then carrying out optimal Poisson disc sampling on the alternative flat area to form a network node set for constructing a road topology network, wherein the method specifically comprises the following steps:
the method adopts a sliding window method to select a flat area, namely a second sliding window with a fixed size is arranged on a lunar accessibility map, the size of the second sliding window is determined according to the actual space size, preferably the actual space is 20km multiplied by 20km, the coverage rate of an accessible area in the second sliding window is calculated, and all areas of the second sliding window with the accessible coverage rate exceeding 95 percent are selected as alternative flat areas.
Because the quality of the sampling point cannot be guaranteed by the Poisson disc sampling, the optimal Poisson disc sampling is provided by the method, namely a random sampling strategy is not used when the sampling point is generated, and the central position of a flat area with the highest area coverage rate which can be reached by a second sliding window is selected as a new sampling point, so that the road topology network node obtained by final sampling can be guaranteed to meet the optimality. Firstly, calculating the reachable area coverage rate based on a second sliding window to obtain a position set P of a candidate flat areaplainAnd its corresponding set of area coverage SplainThen in the current area coverage rate set SplainSelecting the node with the highest coverage rate as the current node, and then selecting the current node from the SplainIs removed. Then judging whether the distances between the current node and the existing sampling nodes are all larger than the sampling radius RsIf yes, adding the sampling node as a new sampling node into the sampling point position set PNodeOtherwise, the node is discarded, at SplainAnd re-sampling until all the alternative flat areas are traversed, wherein the specific process of the algorithm is shown in table 1.
TABLE 1
Figure BDA0003031413750000061
Step three: determining a road topological network structure according to the nearest neighbor network structure based on the network node set generated in the second step, and specifically comprising the following steps:
the nearest neighbor network is a most common topological network structure, and the specific form of the nearest neighbor network is that each network node is only connected with neighbor nodes within a certain distance of the nearest neighbor network, and the nearest neighbor network is commonly used for network structure design of a road network, a logistics network and the like. The invention is based onA neighbor network structure, which sets connectivity among nodes by the distance among the nodes, and sets the distance to be less than 2RsThe nodes are arranged to be communicated and the distance is more than 2RsThe nodes are set to be not communicated, so that the selection of the road topology network nodes and the setting of the connectivity can be completed. By adjusting the sampling radius RsThe density of the road network can be adjusted, and the proper density of the road network can ensure good accessibility of the lunar surface area and the coverage area of the road network, thereby improving the coverage efficiency of the road network.
Step four: according to the network node set in the second step and the network structure in the third step, an improved A-star algorithm is used for connecting all the network nodes to form the construction of the road topology network, and the method specifically comprises the following steps:
the invention also provides a path planning method for connecting each network node to form a large-range road topology network, which comprises the following steps: due to the grid map characteristics of the DEM map and the optimality requirement of the lunar patrol route planning, the invention mainly researches a heuristic graph searching algorithm (A-algorithm) with optimality guarantee, improves the heuristic function of the A-algorithm, and leads the generated route to be far away from the obstacle area as far as possible, thereby improving the safety and the passing probability of the road topology network.
Using a security heuristic function fsafe(n) to guide the algorithm to perform the safety path search, wherein the safety heuristic function is as follows:
fsafe(n)＝g(n)+ωsafe(n)+Diagonal_heuristic(n)
in the formula: g (n) is the actual cost from the initial state to state n in the state space, defined as the actual distance from the starting point to the current node. Omegasafe(n) is the safety cost of the node n, defined as the total number of obstacles in a second sliding window taking the node n as the center, wherein the larger the safety cost is, the more obstacles around the node n are, and the lower the safety is; the Diagram _ itself () is a Diagonal heuristic distance function, and can be replaced by an Euclidean distance function or other heuristic functions meeting consistency conditions;
the safety path among the connectable nodes is planned by using the improved A-star algorithm, so that the connection among the nodes of the road topology network can be realized, and the construction of the full-month large-range road topology network is completed.
The second embodiment is as follows:
the embodiment is a lunar large-range road topological network construction system, and the system is used for executing the lunar large-range road topological network construction method. The system according to this embodiment may be a computer application program or software, which is used to execute a coded program corresponding to the method for constructing a lunar wide-range road topology network.
The second embodiment is as follows:
the embodiment is a lunar large-range road topological network construction device, and the device is used for storage and/or a lunar large-range road topological network construction system. The device described in this embodiment includes, but is not limited to, a storage device, a PC, a server, a workstation, a mobile device, and the like, and may also be a specially developed single chip microcomputer and the like.
Example (b):
the following examples are employed to demonstrate the beneficial effects of the present invention
The experimental environment is as follows: the section selects the landing points of Apollo11 and Apollo12 as the starting and stopping points of the large-range movement detection task. Firstly, DEM data of a task area is obtained, and since a CE-2 lunar terrain data product has obvious advantages in the aspects of spatial resolution, lunar coverage, positioning accuracy, topographic structure detail expression and the like compared with other lunar terrain data, the DEM-50m data set in the Chang' e II CE2TMap2015 data product is adopted for lunar reachable area analysis, the elevation information of the task area is extracted, and the elevation information of the task area is shown in figure 1, and the area of the task area reaches 2207.2km multiplied by 871.3 km.
The simulation test software environment of all algorithms of the invention is Windows 10+ MATLAB 2016, and the hardware environment is Intel (R) core (TM) i5-7200U CPU +12.0GB RAM.
Experimental results and analysis: according to the invention, firstly, the terrain and reachable area analysis method is used for analyzing the passability of the area, and the result shows that as shown in fig. 2, the reachable area calculation method provided by the section can better restore the lunar terrain characteristics, and finally the maximum reachable area of the task area is obtained, and the reachable area coverage rate of the area is 94.42%.
Then, the optimal Poisson disc sampling is carried out on the task area in an experiment, network nodes required for constructing a road topology network are selected, the size of a sampling sliding window is set to be 20km multiplied by 20km, the sampling radius Rs is set to be 100km, the simulation result of midway point selection of the Apollo task area is shown in figure 3, 111 network nodes are sampled in total, and the highest 100% coverage rate and the lowest 95.2% coverage rate of the reachable area in the sampling node window are achieved. As can be seen from FIG. 3, the network node is basically located in a safe flat area within the range of 20km, the optimal Poisson disc sampling result density is moderate, and the coverage of the whole task area is good, so that the coverage efficiency of the road topology network is improved.
Finally, the invention carries out the connection construction of the road topology network by the experiment, and plans the path among all the nodes by using the improved A-star algorithm provided by the step four, so that the generated path can be far away from the obstacle area as far as possible, thereby improving the safety and the passing probability of the road topology network. The window size of the safety cost function of the improved A-x algorithm is set to be 1km x 1km, namely the safety cost weight of the algorithm considers the total number of obstacles in the range, and the construction result of the road topology network in the Apollo task area is shown in FIG. 4. As can be seen from fig. 4, the road topology network constructed by the method can completely cover the Apollo task area, has moderate network density, and can ensure good accessibility and coverage efficiency of the lunar surface area. Meanwhile, the transfer path based on the road topology network is far away from a multi-obstacle area as far as possible, so that the safety and the passing probability of the network are improved.
According to the method, the lunar large-range road topology network construction method can be realized, and a new thought is provided for lunar exploration planning problem research.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope of the present invention.

Claims (11)

1. A lunar surface large-range road topological network construction method is characterized in that lunar surface reachable area analysis is carried out based on lunar surface topographic features to obtain a lunar surface reachable area map; then, analyzing and searching a flat area based on a lunar accessible area map, and then carrying out optimal Poisson disc sampling on the alternative flat area to form a network node set for constructing a road topology network; determining a road topological network structure according to a nearest neighbor network structure based on the network node set; finally, according to the network node set and the road topology network structure, connecting each network node by using an A-star algorithm to realize the construction of the road topology network;
the process of analyzing and searching the flat area, then carrying out optimal Poisson disc sampling on the alternative flat area, and forming the network node set for constructing the road topology network comprises the following steps:
selecting a flat area by adopting a sliding window method, namely setting a second sliding window with a fixed size on the lunar surface accessibility map, calculating the coverage rate of an accessible area in the second sliding window, and selecting all second sliding window areas with the accessible coverage rate exceeding a coverage rate threshold value as alternative flat areas;
based on a Poisson disc sampling mode, selecting the center position of a flat area with the highest reachable area coverage rate of a second sliding window as a new sampling point when generating the sampling point: firstly, calculating the reachable area coverage rate based on a second sliding window to obtain a position set P of a candidate flat areaplainAnd its corresponding set of area coverage SplainThen in the current area coverage rate set SplainSelecting the node with the highest coverage rate as the current node, and then selecting the current node from the SplainRemoving; then judging whether the distances between the current node and the existing sampling nodes are all larger than the sampling radius RsIf yes, adding the sampling node as a new sampling node into the sampling point position set PNodeOtherwise, the node is discarded, at SplainUntil all the alternative flat regions are traversed.
2. The method for constructing the lunar surface wide-range road topology network according to claim 1, wherein the process of determining the road topology network structure according to the nearest neighbor network structure based on the network node set comprises the following steps:
according to the nearest neighbor network structure, connectivity among nodes is set through the distance among the nodes, and the distance is smaller than 2RsThe nodes are arranged to be communicated and the distance is more than 2RsThe nodes are set to be not communicated, so that the selection of the road topology network nodes and the setting of the connectivity can be completed.
3. The method for constructing the lunar surface wide-range road topology network according to claim 2, wherein an A-algorithm is used for connecting each network node, and the heuristic function of the A-algorithm in the construction of the road topology network is a safety heuristic function, and specifically, the method comprises the following steps:
fsafe(n)＝g(n)+ωsafe(n)+Diagonal_heuristic(n)
in the formula: g (n) is the actual cost from the initial state to state n in the state space, defined as the actual distance from the starting point to the current node; omegasafe(n) a safety cost for node n, defined as a total number of obstacles within a second sliding window centered on node n; diagnonal _ itself () is a Diagonal heuristic distance function, Euclidean distance function, or a heuristic function that satisfies a consistency condition.
4. The method for constructing a lunar wide-range road topology network according to claim 1, 2 or 3, wherein the size of the second sliding window is determined according to the size of the actual space, which is 20km x 20 km.
5. The method for constructing the lunar surface large-range road topology network as claimed in claim 1, 2 or 3, wherein the lunar surface reachable area analysis process based on lunar surface topographic features obtains the topographic features of the DEM map in a sliding window mode.
6. The method for constructing the lunar surface wide-range road topology network as claimed in claim 5, wherein the process of obtaining the topographic features of the DEM map in the form of a sliding window comprises the following steps:
setting a sliding window with the size of a 3 x 3 pixel grid, namely a first sliding window, and taking terrain information obtained by calculation in the first sliding window as terrain features of a central grid, wherein the terrain features comprise a terrain slope theta, a terrain relief degree R and a terrain roughness delta;
then setting threshold limits of maximum gradient, undulation degree and roughness according to the ability of the patrol device;
and regarding the grid area exceeding the threshold limit of the ability of the rover as an obstacle area, and converting the DEM map into a reachable area map.
7. The method for constructing the topological network of the lunar surface and large-scale road according to claim 6, wherein the terrain slope θ, the terrain undulation degree R and the terrain roughness δ are respectively as follows:
Figure FDA0003031413740000021
R＝Hmax-Hmin
Figure FDA0003031413740000022
in the formula: f. ofxThe rate of change of the center grid in the east-west direction elevation is taken as the rate of change of the center grid in the east-west direction elevation; f. ofyCalculating the elevation change rate of the central grid in the north-south direction according to the following formula; hmaxIs the maximum elevation value in the first sliding window; hminIs the minimum height value in the first sliding window; hiThe elevation value corresponding to the ith grid in the first sliding window is obtained;
Figure FDA0003031413740000023
is the average of all elevation values within the first window.
8. The method for constructing the topological net of the lunar surface and large-scale road according to claim 7, wherein the center grid has an east-west elevation change rate fxCentral grid north-south direction elevation change rate fyRespectively as follows:
Figure FDA0003031413740000024
Figure FDA0003031413740000025
in the formula: h1-H9Elevation values corresponding to 9 grids in a 3 x 3 window respectively; and g is the resolution size of DEM data.
9. The method for constructing the topological net of the lunar surface and large-scale road according to claim 7, wherein the threshold limits for setting the maximum gradient, the waviness and the roughness according to the capacity of the patrol instrument are as follows:
θ≤20°
R≤2*g*tan(20°)
Figure FDA0003031413740000031
in the formula: and g is the resolution size of DEM data.
10. A lunar road topology network construction system, characterized in that said system is used to execute a lunar road topology network construction method according to one of claims 1 to 9.
11. A lunar road topology network construction device, characterized in that said device is used to store and/or operate a lunar road topology network construction system according to claim 10.
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