Robotic systems with redundant degrees of freedom can achieve the same task outcome using multiple configurations, resulting in solution sets that form manifolds in the configuration space. Existing approaches typically exploit such redundancy locally through Jacobian-based techniques to compute individual solutions or trajectories. While effective for solution computation, these methods do not retain a representation of the geometry of the solution set itself. In this work, we adopt a representation-centric approach to estimate the geometric structure of the solution space. We consider solution manifolds induced by general task-defining maps and construct an implicit scalar field over the configuration space, whose zero-level set corresponds to the solution manifold. To this end, we generate samples in the neighborhood of the solution manifold using a Jacobian-guided exploration strategy, which efficiently captures its local and global structure. The resulting implicit representation is defined over the configuration space and naturally induces a continuous, distance field that encodes proximity to the solution manifold. Experiments on a planar three-link robot and a seven-degree-of-freedom Franka manipulator demonstrate the effectiveness of the proposed representation. Furthermore, the framework enables consistent modeling of solution spaces across families of tasks with continuous variation.

翻译：具有冗余自由度的机器人系统能够通过多种构型实现相同任务结果，从而在构型空间中形成解集流形。现有方法通常通过基于雅可比矩阵的技术局部利用此类冗余性，以计算单个解或轨迹。尽管这些方法在解计算方面有效，但未能保留解集本身的几何结构表示。在本工作中，我们采用以表示为中心的方法来估计解空间的几何结构。我们考虑由通用任务定义映射诱导的解流形，并在构型空间上构建隐式标量场，其零水平集对应于解流形。为此，我们通过雅可比引导的探索策略在解流形邻域生成样本，高效捕获其局部与全局结构。所生成的隐式表示定义于构型空间，自然导出一个连续的、编码与解流形邻近性的距离场。在平面三连杆机器人和七自由度Franka机械臂上的实验验证了所提出表示的有效性。此外，该框架能够对具有连续变化的任务族实现解空间的一致建模。