This paper describes a method for finding the location of a rigid body such that N specified points of the body lie on N given planes in space. Of special interest is the case N = 6, which is the minimum number to fully constrain the body. This geometric problem arises in two seemingly disparate contexts: metrology, as a generalization of so-called "3-2-1" locating schemes; and robotics, as the forward kinematics problem for 6ES or 6SE parallel-link platform robots. For N = 6, the geometric problem can be formulated algebraically as 3 quadratic equations having, in general, eight possible solutions. We give a method for finding all eight solutions via an 8 x 8 eigenvalue problem. We also show that for N ≥ 7, the solution can be found uniquely as a linear least squares problem.
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机译：本文介绍了一种用于找到刚体的位置的方法，使得身体的n个特定点位于空间中的N给定平面上。特殊兴趣是n = 6的情况，这是完全限制身体的最小数量。这个几何问题出现在两个看似不同的上下文中：计量学，作为所谓的“3-2-1”定位方案的概括;和机器人，作为6次或6SE平行链路平台机器人的前瞻性运动问题。对于n = 6，几何问题可以在一般而言之的八种可能的解决方案中作为3个二次方程式制定成像。我们通过8 x 8特征值问题给出一种解决所有八种解决方案的方法。我们还表明，对于n≥7，可以独特地发现解决方案作为线性最小二乘问题。
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