In this report we explore the application of the Lagrange-Newton method to the SAM (smoothing-and-mapping) problem in mobile robotics. In Lagrange-Newton SAM, the angular component of each pose vector is expressed by orientation vectors and treated through Lagrange constraints. This is different from the typical Gauss-Newton approach where variations need to be mapped back and forth between Euclidean space and a manifold suitable for rotational components. We derive equations for five different types of measurements between robot poses: translation, distance, and rotation from odometry in the plane, as well as home-vector angle and compass angle from visual homing. We demonstrate the feasibility of the Lagrange-Newton approach for a simple example related to a cleaning robot scenario.

翻译：本报告探讨了拉格朗日-牛顿方法在移动机器人平滑与建图（SAM）问题中的应用。在拉格朗日-牛顿SAM中，每个位姿向量的角度分量通过方向向量表示，并借助拉格朗日约束进行处理。这与典型的高斯-牛顿方法不同——后者需要在欧几里得空间与适用于旋转分量的流形之间反复映射变量变化。我们推导了机器人位姿间五类不同测量量的方程：平面内里程计产生的平移、距离和旋转，以及视觉归航中的归航向量角与罗盘角。我们通过一个与清洁机器人场景相关的简单实例，验证了拉格朗日-牛顿方法的可行性。