Surface-Constrained Offline Warping with Contact-Aware Online Pose Projection for Safe Robotic Trajectory Execution

Abstract

Robotic manipulation tasks that require repeated tool motion along curved surfaces frequently arise in surface finishing, inspection, and guided interaction. In practice, nominal motion primitives are often designed independently of the deployment surface and later reused across varying geometries. Directly tiling such primitives onto nonplanar surfaces introduces geometric inconsistencies, leading to interpenetration, orientation discontinuities, and cumulative drift over repeated cycles. We present a two-stage framework that separates geometric embedding from execution-level regulation. An offline surface-constrained warping operator embeds a nominal periodic primitive onto curved surfaces through asymmetric diffeomorphic deformation of dual-track waypoints and axis-consistent orientation completion, producing a surface-adapted reference trajectory. An online contact-aware projection operator then enforces bounded deviation relative to this reference using FSR-driven disturbance adaptation and a conic orientation safety constraint. Experiments across multiple analytic surface families and real-robot validation on a sinusoidal surface demonstrate improved geometric continuity, reduced large orientation jumps, and robust contact maintenance compared with direct tiling. These results show that decoupling offline geometric remapping from lightweight online projection enables stable and repeatable surface-embedded trajectory execution under sensor-lite feedbacks.

Figures (8)

• Figure 2: Surface-constrained robotic trajectory execution with stable contact. A robotic end-effector follows a warped periodic trajectory over a curved surface using offline geometric warping and online contact-aware projection for stable and drift-free interaction, with applications in polishing, coating, inspection, and surgical assistance.
• Figure 3: Problem formulation and execution pipeline. A nominal periodic trajectory $\mathbf{T}^{\mathrm{nom}}$ is first adapted to a surface-embedded guide curve $\gamma(s)$ through the offline warping operator $\mathcal{W}_{\Phi}$, producing a surface-consistent warped trajectory $\mathbf{T}^{\mathrm{warp}}$ (orange). During execution, an online projection operator $\mathcal{E}$ constrains the executed pose $\mathbf{T}^{\mathrm{proj}}$ within a conic feasible set around the warped orientation while regulating position along gravity using contact-feedback measurements. This two-stage formulation separates offline geometric embedding from online safety-constrained execution.
• Figure 4: Offline surface-constrained warping pipeline. Step 1 extracts a single-period primitive and tiles its waypoints along a surface guide curve, separating contact ($\mathcal{I}_c$, green waypoints) and free-space ($\mathcal{I}_f$, orange waypoints) samples. Step 2 constructs a dual-track tool-axis parameterization using coupled base and tip waypoints. Step 3 applies a surface-constrained diffeomorphic deformation driven by local surface geometry. Step 4 constructs the final warped pose sequence $\mathbf T_k^{\mathrm{warp}}$ by combining the deformed waypoints (with contact-critical waypoints anchored) with axis-consistent orientation completion.
• Figure 5: Contact-aware online projection under FSR-driven disturbances. Force feedback perturbs the warped pose $\mathbf{T}_k^{\mathrm{warp}}$: low force drives downward motion, while high force drives upward motion. The disturbed pose is then projected into the conic feasible set, yielding $\mathbf{T}_k^{\mathrm{proj}}$ with bounded orientation deviation and stable contact.
• Figure 6: Collision-prone configurations under direct tiling. First row: tiled ($\mathbf T^{\mathrm{tile}}$). Second row: warped ($\mathbf T^{\mathrm{warp}}$). Direct tiling remains feasible on the convex surface but becomes collision-prone on the concave surface due to an unsafe local attitude. Offline warping produces a surface-consistent trajectory that avoids this failure mode.