Over the last two decades, the robotics community witnessed the emergence of various motion representations that have been used extensively, particularly in behavorial cloning, to compactly encode and generalize skills. Among these, probabilistic approaches have earned a relevant place, owing to their encoding of variations, correlations and adaptability to new task conditions. Modulating such primitives, however, is often cumbersome due to the need for parameter re-optimization which frequently entails computationally costly operations. In this paper we derive a non-parametric movement primitive formulation that contains a null space projector. We show that such formulation allows for fast and efficient motion generation with computational complexity O(n2) without involving matrix inversions, whose complexity is O(n3). This is achieved by using the null space to track secondary targets, with a precision determined by the training dataset. Using a 2D example associated with time input we show that our non-parametric solution compares favourably with a state-of-the-art parametric approach. For demonstrated skills with high-dimensional inputs we show that it permits on-the-fly adaptation as well.

翻译：在过去二十年中，机器人学界见证了多种运动表示的涌现，这些表示在行为克隆中得到了广泛应用，尤其用于紧凑编码和泛化技能。其中，概率方法因其对变化、相关性以及适应新任务条件的编码能力而占据重要地位。然而，调制这些基元通常较为繁琐，因为需要重新优化参数，这往往涉及计算成本高昂的操作。本文推导了一种包含零空间投影器的非参数运动基元公式。我们证明，该公式能够以计算复杂度O(n²)实现快速高效的运动生成，而无需涉及复杂度为O(n³)的矩阵求逆运算。这是通过使用零空间来跟踪次要目标实现的，其精度由训练数据集决定。通过一个与时间输入相关的二维示例，我们表明非参数解优于现有最先进的参数方法。对于具有高维输入演示的技能，我们还展示了其支持即时在线适应。