Title 1: Construction and Representation of Unified Feature Manifold Space Subtitle 1 Illustration: Definition of Feature Manifold Based on Product Space/Fiber Bundle Image Type: Conceptual Diagram (Combination of 3D and 2D). Visual Core: Left: A three-dimensional Lie group manifold (e.g., a twisted surface) representing "pose space," with a trajectory curve drawn on it. Right: A regular two-dimensional/three-dimensional Euclidean space grid representing "process parameter space" (force/speed), with each grid point labeled with parameter values. Middle: A fused structure, visualized as a "fiber bundle": with the left manifold as the "base space," a small Euclidean space (like a fiber) "grows" from each point on it, representing a complete set of process parameter choices attached to each pose point. Alternatively, a "product space" diagram can be used to show how the two spaces are combined into a higher-dimensional space like a Cartesian product. Key Labels: Base Space (Pose Manifold), Fiber (Process Parameter Space), Feature Manifold Point (a point on the fiber). Subtitle 2 Illustration: Unified Encoding Mechanism for Process and Trajectory Information Image Type: Information Encoding Diagram. Visual Core: An abstract feature manifold point (which can be represented as a luminous data sphere or a high-dimensional cube) is located at the center. Two clear decoding paths emanate from this point: Path 1 points to a pose model diagram of a robot/tool (including position and orientation). Path 2 points to a process parameter dashboard diagram (displaying specific force values, speed values, and curves). At the same time, arrows flowing to the point indicate that the above two types of information are jointly encapsulated in this single manifold point. Key Labels: Unified Encoding, Pose Information (x, y, z, Rx, Ry, Rz), Process Information (F, n). Subtitle 3 Illustration: Introduction of Geometric Tools such as Tangent Space and Covariant Derivative Image Type: Differential Geometry Diagram. Visual Core: Shows a curve (representing a trajectory) on the feature manifold (a smooth surface). Select a point P on the curve and draw the tangent plane at that point (i.e., the tangent space T_pM), with multiple tangent vectors on the plane. Demonstrate the concept of covariant derivative: a vector field along the direction of the curve, where one vector is "parallel transported" to an adjacent point, showing the change of its direction under the intrinsic geometry of the manifold. The difference between the ordinary derivative in Euclidean space (direction unchanged) and the covariant derivative on the manifold (direction affected by the curvature of the surface) can be shown by comparison. Key Labels: Point P, Tangent Space T_pM, Tangent Vector, Covariant Derivative ∇_v u, Manifold M. Title 2: Geometric Representation and Quantitative Analysis of Force-Speed Conformal Trajectories Subtitle 1 Illustration: Curvature and Torsion Geometric Representation of Trajectory Smoothness Image Type: Curve Geometry Analysis Diagram. Visual Core: Draw a distinct force-speed conformal trajectory curve on the feature manifold surface. Select multiple points on the curve at intervals, and draw a Frenet frame at each point: tangent vector T (pointing to the direction of motion), normal vector N (pointing to the "
