Position and Altitude of the Nao Camera Head from Two Points on the Soccer Field plus the Gravitational Direction

TL;DR

The paper tackles localizing a Nao robot on a RoboCup field using only two visible ground points and the gravity direction. It adopts Rational Trigonometry to model a tetrahedral viewpoint and derives explicit formulas for the height $H$ and ground coordinates, achieving the same accuracy as classical trig but with about $28{.}7\%$ faster computation on a NAO v6 platform, validated against OptiTrack ground truth. Metrological evaluation shows mean errors in the 3–6 cm range when gravity is inferred from field geometry, and the approach remains robust under limited field-of-view conditions. The work demonstrates that fast, landmark-based pose estimation can feed into SLAM pipelines, with practical impact for real-time robot soccer control and navigation.

Abstract

To be able to play soccer, a robot needs a good estimate of its current position on the field. Ideally, multiple features are visible that have known locations. By applying trigonometry we can estimate the viewpoint from where this observation was actually made. Given that the Nao robots of the Standard Platform League have quite a limited field of view, a given camera frame typically only allows for one or two points to be recognized. In this paper we propose a method for determining the (x, y) coordinates on the field and the height h of the camera from the geometry of a simplified tetrahedron. This configuration is formed by two observed points on the ground plane plus the gravitational direction. When the distance between the two points is known, and the directions to the points plus the gravitational direction are measured, all dimensions of the tetrahedron can be determined. By performing these calculations with rational trigonometry instead of classical trigonometry, the computations turn out to be 28.7% faster, with equal numerical accuracy. The position of the head of the Nao can also be externally measured with the OptiTrack system. The difference between externally measured and internally predicted position from sensor data gives us mean absolute errors in the 3-6 centimeters range, when we estimated the gravitational direction from the vanishing point of the outer edges of the goal posts.

Figures (5)

• Figure 1: The quadrances $Q_i$ and spreads $s_i$ of a triangle.
• Figure 2: The X-intersection detected on the Standard Platform League field.
• Figure 3: A line $L$ between two known points seen from a camera. The red points on the ground plane are the intersections of middle line and circle of which the quadrance $L$ is known. The red point on the top right is the camera point. Together with the foot of the altitude these points form a tetrahedron with two right angles. The altitude quadrance $H$ can be determined from quadrance $L$ and the spreads $p_1$, $p_2$, and $q_{12}$.
• Figure 4: Viewpoints from the 2004 and 2005 SLAM challenge locations
• Figure 5: Top view of the 2004 and 2005 SLAM challenge locations and a Nao robot with an OptiTrack rigid body marker