二维三边测量 [英] 2d trilateration

问题描述

我正在编写一些代码来参加 AI 挑战.AI 挑战的主要目标是让模拟机器人在迷宫中导航到目的地区域.可选的次要目标是找到放置在迷宫中未知位置的充电器.这一切都在 2D 网格中完成.

I am writing some code to participate in an AI challenge. The main objective for the AI challenge is to take a simulated robot and navigate it through a maze to a destination zone. The secondary objective which is optional is to find a recharger placed in the maze at an unknown location. This is all done in a 2D grid.

我的程序可以调用一个方法来从充电器获取距离测量值.所以使用三边测量，我应该能够通过调用这个方法来定位充电器，记录我的 ai 当前位置以及充电器与该点的距离 3 次以上.

My program can call a method to get a distance measurement from the recharger. So using trilateration I should be able to locate the recharger by calling this method, recording my ai's current position and the distance the recharger is away from that point 3 times over.

我在维基百科上找到了这个三边测量的例子http://en.wikipedia.org/wiki/Trilateration 但这适用于 3d 空间.我只处理二维空间.此外，我不明白如何使用维基百科中显示的公式，在网络上搜索插入数字并归结到最终坐标的工作示例在 Google 搜索中很少见.

I found this example of trilateration on wikipedia http://en.wikipedia.org/wiki/Trilateration but this applies to a 3d space. I'm only dealing with a 2D space. Also I don't understand how to use the formula shown in Wikipedia, searching the web for a working example with numbers plugged in and boiling down to the final coordinates is scarce with Google searches.

我不是数学专业；我只是一个探索 AI 问题的爱好者.

I'm not a math major; I am just an enthusiast exploring AI problems.

如何计算问题的解释和分步示例是我需要的，因为数学不是我的强项.以下是一些示例数据:

An explanation and step by step example of how to calculate the problem is what I need as mathematics are not my strong point. Below is some sample data:

• 点 1:x=39，y=28，距离=8
• 点 2:x=13，y=39，距离=11
• 第 3 点:x=16，y=40，距离=8

任何使用我的示例数据的示例将不胜感激.一旦我能够围绕数学进行编程，这将是非常简单的.

Any example using my sample data would be greatly appreciated. The programming to this will be very straight forward once I can wrap my head around the mathematics.

推荐答案

作为维基百科三边形文章 描述，您通过连续计算来计算 (x,y) 坐标:e_x, i, e_y, d, j, x, y.您必须熟悉向量表示法，例如，e_x = (P2 - P1)/‖P2 - P1‖ 表示:

As the Wikipedia trilateriation article describes, you compute (x,y) coordinates by successively calculating: e_x, i, e_y, d, j, x, y. You have to be familiar with vector notation, so, for example, e_x = (P2 - P1) / ‖P2 - P1‖ means:

• e_x,x = (P2_x - P1_x)/sqrt((P2_x -P1_x)² + (P2_y - P1_y)²)
• e_x,y = (P2_y - P1_y)/sqrt((P2_x -P1_x)² + (P2_y - P1_y)²)

• e_x,x = (P2_x - P1_x) / sqrt((P2_x - P1_x)² + (P2_y - P1_y)²)
• e_x,y = (P2_y - P1_y) / sqrt((P2_x - P1_x)² + (P2_y - P1_y)²)

您的数据是:

• P1 = (39, 28);r₁ = 8
• P2 = (13, 39);r₂ = 11
• P3 = (16, 40);r₃ = 8

• P1 = (39, 28); r₁ = 8
• P2 = (13, 39); r₂ = 11
• P3 = (16, 40); r₃ = 8

计算步骤为:

1. e_x = (P2 - P1)/‖P2 - P1‖
2. i = e_x(P3 - P1)
3. e_y = (P3 - P1 - i · e_x)/‖P3 - P1 - i · e_x‖
4. d = ‖P2 - P1‖
5. j = e_y(P3 - P1)
6. x = (r₁² - r₂² + d²)/2d
7. y = (r₁² - r₃² + i² + j²)/2j - ix/j

1. e_x = (P2 - P1) / ‖P2 - P1‖
2. i = e_x(P3 - P1)
3. e_y = (P3 - P1 - i · e_x) / ‖P3 - P1 - i · e_x‖
4. d = ‖P2 - P1‖
5. j = e_y(P3 - P1)
6. x = (r₁² - r₂² + d²) / 2d
7. y = (r₁² - r₃² + i² + j²) / 2j - ix / j

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