Contact-Explicit Motion Planning

• Contact-explicit motion planning is a framework that explicitly models physical contacts, including timing, location, and force, to synthesize robust robot trajectories.
• It employs hybrid approaches that combine discrete contact mode scheduling with continuous optimization, enhancing performance in manipulation, locomotion, and human-robot collaboration.
• Key challenges include managing high-dimensional contact spaces, overcoming nonconvex dynamics, and ensuring real-time adaptation in dynamic and uncertain environments.

Contact-explicit motion planning encompasses methodologies for synthesizing robot trajectories that leverage explicit models of physical contact—location, timing, and mode—between the robot and its environment or other agents. Unlike collision-free or contact-implicit paradigms, contact-explicit planning directly parameterizes, constrains, or optimizes over contact configurations and associated wrenches, accommodating the hybrid, discontinuous, and high-dimensional structure of interaction-rich robotic behaviors. This class of approaches has become foundational in manipulation, locomotion, human-robot collaboration, and multi-modal task planning.

1. Foundational Principles and Problem Formulation

Contact-explicit motion planning addresses the intrinsic hybridness of contact-rich robotics, requiring explicit reasoning over both discrete contact modes (which surfaces/links are in contact, their types) and continuous robot trajectories and wrenches. The planning variable set typically spans:

• Discrete contact modes, often formalized as tuples or symbolic states identifying active interface pairs and contact type (e.g., unilateral/patch) (Ciebielski et al., 16 Aug 2025).
• Time-varying contact locations (possibly continuous parameters along body/outlines) and forces (Leve et al., 2024).
• Robot configurations, object states, and controls across a planning horizon.

Optimization objectives combine geometric goals (e.g., end-effector pose), motion smoothness, energy consumption, and explicit contact costs (e.g., force minimization, maintaining desired normal force, contact-tolerance metrics) (Song et al., 9 Oct 2025, Ling et al., 13 Mar 2025).

Constraints are manifold and may include:

• Rigid dynamic equations (possibly whole-body, including floating base) (Ciebielski et al., 16 Aug 2025).
• Non-penetration/contact complementarity ($$\phi(q)\geq 0, \; \lambda_n \geq 0, \; \lambda_n\phi(q) = 0$$) (Mastalli et al., 2019, Landry et al., 2019).
• Friction-cone inequalities and actuator capacity (Aceituno-Cabezas et al., 2019, Fakhari et al., 2020).
• Joint and workspace limits; task- or environment-specific requirements.

Contact transitions are modeled as mode changes, explicit switches, or transitions governed by complementarity, mixed-integer, or graph-structured formalisms (Aceituno-Cabezas et al., 2019, Ciebielski et al., 16 Aug 2025).

2. Contact Mode and Surface Parameterizations

Contact modes are modeled as discrete variables capturing the allocation and type of each robot-environment or inter-body interface (Ciebielski et al., 16 Aug 2025). The set of all possible contact interfaces (e.g., links, body patches) and partners (objects, environment) is enumerated, and the planning state records, for each, whether a contact is active and its modality (e.g., sticking, sliding, unilateral, bilateral). Symbolic actions transition between contact modes (e.g., make/break specific contacts).

Continuous contact parameterizations extend expressiveness and tractability. Recent work uses explicit parametrizations of robot and object surfaces to permit contact anywhere along continuous manifolds, enabling "whole-body" contact planning with variables such as arc-lengths along fitted surface curves ($\phi_a, \phi_u \in [0, 1]$), circumventing combinatorial explosion from linkwise discretization (Leve et al., 2024).

This principled explicitness enables the direct encoding of contact creation, sticking, sliding, or rolling, and—combined with velocity/force variables—enables smooth relaxation and switching of contact modes as optimization variables, with complementarity constraints or penalized switching (Ciebielski et al., 16 Aug 2025, Leve et al., 2024).

3. Optimization-based Contact-Explicit Planning Frameworks

Contact-explicit planners typically employ either unified (all-in-one) nonlinear programming or hierarchical schemes:

• Hierarchical frameworks: Separate high-level discrete contact or mode planning (e.g., tree search over contact sequences or skeletons) from low-level continuous optimization over robot trajectories, injecting the symbolic plan into the continuous stage as fixed constraints (Chen et al., 2021, Ciebielski et al., 16 Aug 2025).
• Single-stage (all-in-one) optimization: Formulate a mixed-integer nonlinear program (MINLP), mathematical program with complementarity constraints (MPCC), or bilevel QP/NLP in which all contact mode, timing, force, and motion variables are co-optimized (Landry et al., 2019, Aceituno-Cabezas et al., 2019).

Dynamics and force constraints may include:

• Full rigid-body equations with floating base/object states, accumulating contact wrenches across all possible contacts (Ciebielski et al., 16 Aug 2025).
• Quasi-static approximations for object manipulation or pushing (Graesdal et al., 2024, Fakhari et al., 2020).
• Embedded QPs for frictional contact forces at every knot point, solved inside each optimizer iteration and differentiated via KKT conditions for bilevel approaches (Landry et al., 2019).
• Sequential convexification and linearization for tractable iterative convex subproblem sequences (SafeTrack, ICOP) (Zhao et al., 2020).

Sampling-based integration is also common: for hybrid discrete/continuous problem spaces, sampling planners are extended with constraint projection steps, special mode samplers for transitions (e.g., sliding contact sampler for constrained manifold union) (Khoury et al., 2020), and adaptive per-link or per-mode cost heuristics regulating exploration of contact-rich subsets of the configuration space (Nechyporenko et al., 2023).

4. Practical Algorithms, Scheduling, and Hierarchical Search

Contact-explicit planning requires tractable enumeration/pruning of high-dimensional discrete mode spaces. Approaches to contact mode scheduling and search include:

• Acyclic symbolic graph search and pruning: Directed graphs over symbolic contact mode states are built to a fixed depth, pruned by feasibility checks and manipulation skeletons (e.g., grasp→lift→place→release) to prevent redundant or impossible mode transitions. Monte Carlo Tree Search and best-first expansions find low-cost, dynamically feasible action sequences, with the continuous planning layer resolved only for promising mode sequences (Ciebielski et al., 16 Aug 2025).
• Hierarchical procedure: A coarse initial trajectory (e.g., centroidal dynamics, unscheduled contacts) is optimized to provide a warm start for full-dynamics, contact-rich refinement (Mastalli et al., 2019).
• Transition-constraint sampling: Uniformly sampling between modes (contact/no contact/goal contact) and projecting samples onto hybrid constraint-manifolds ensures probabilistically complete exploration of contact transitions without explicit enumeration (Khoury et al., 2020).
• Mixed-integer convex optimization: MIQCQP-based planners encode contacts, gait, and switching via binary variables, bounding the complexity via convex relaxations and clever constraint structuring (e.g., per-foot and per-surface contact) (Aceituno-Cabezas et al., 2019).

5. Adaptive Reactivity, Online Inference, and Contact in Human-Robot Collaboration (HRC)

Contact-explicit planning under dynamic, uncertain, or interactive conditions integrates physical feedback and online inference to adapt trajectories during execution. Examples include:

• Torque-based contact force inference: Online optimization-based estimation of contact location and intent from joint torque residuals is performed at control-cycle rates, followed by immediate motion re-planning (Song et al., 9 Oct 2025).
• Path-deformation frameworks: Adaptive planners encode the executed trajectory as a deformation of a nominal path, with windowed bump-function updates based on estimated contact forces, ensuring smooth real-time corrections (Song et al., 9 Oct 2025).
• Reinforcement learning with explicit contact rewards: RL-based planners optimize for both efficiency and safety by including contact force into the reward function, with energy-based control barrier functions (CBF) enforcing deterministic safety at deployment. This dual explicitness allows robots to maintain safe contact with humans while optimizing for task success (Mulkana et al., 3 Dec 2025).
• Cost maps from environment semantics: VLM-based planners automatically assign cost values to environment regions based on inferred semantic contact-tolerance, integrating these explicit metrics into the sampling/rejection logic of standard roadmap planners to bias motion toward "harmless" contacts (Ling et al., 13 Mar 2025).

6. Applications, Performance, and Experimental Validation

Contact-explicit planners have demonstrated efficacy in scenarios demanding nuanced mode transitions, discontinuous contact schedules, or whole-body environmental exploitation:

• Humanoid and manipulation: Synthesis of physically consistent, torque-limited loco-manipulation behaviors (e.g., bimanual pick-and-place, climbing with mixed hand/foot contacts), with practical solve times (∼1 min) and robust acyclic plan generation (Ciebielski et al., 16 Aug 2025).
• Manufacturing and surface tasks: Iterative convex planners maintain strict contact along predefined surface paths while enforcing collision-avoidance (sub-mm errors, planning times <12 s for 40+ waypoints in confined weld grinding) (Zhao et al., 2020).
• Dexterous finger-gaited manipulation: Hybrid planning via contact sequence trees and contact-implicit optimization achieves 7× speed-up over pure CITO, with an order-of-magnitude reduction in object drop errors (Chen et al., 2021).
• Whole-body planar manipulation: Continuous explicit surface representations yield 99% iteration reduction and 96% planning time improvement versus previous contact-rich manipulation approaches (Leve et al., 2024).
• Climbing and legged locomotion: Mixed-integer convex planners reliably synthesize robust, non-flat, multi-gait trajectories in <2 s for HyQ quadruped, outperforming non-convex or fixed-gait alternatives (Aceituno-Cabezas et al., 2019). Bi-level frameworks for rope-assisted climbing maintain tractability over large terrain graphs by decoupling patch selection and dynamic feasibility search (Malacarne et al., 29 Apr 2026).
• Aerial manipulation: Contact-aware trajectory planning with time-varying force variables enables precise motion and force tracking (${\sim}0.7$ N RMSE) for dynamic aerial writing tasks (Guo et al., 2024).
• Human-robot collaboration: Adaptive planners with online contact intent inference achieve <5 ms estimation and update latencies, maintain task success under perception occlusion, and produce safe, corrective path adjustments in sub-second windows (Song et al., 9 Oct 2025). RL-based frameworks guarantee $<10 \mathrm{N}$ contact in object handover tasks (Mulkana et al., 3 Dec 2025).

7. Challenges, Extensions, and Future Directions

Despite substantial advances, contact-explicit motion planning remains computationally challenging. Principal barriers and research frontiers include:

• Combinatorial explosion: The enumeration of all possible contact sequences/modes is exponentially complex; continuous parameterizations (Leve et al., 2024) and uniform transition sampling (Khoury et al., 2020) mitigate but do not eliminate this challenge for high-DOF or 3D cases.
• Local minima and nonconvexity: Even with convexification strategies, nonconvex dynamics and discrete switching remain hard; methods such as Shor–Parrilo SDP relaxations (Graesdal et al., 2024) and bilevel optimization (Landry et al., 2019, Malacarne et al., 29 Apr 2026) offer progress but may still encounter rounding or sensitivity failures in large instances.
• Dynamic and uncertain environments: Adapting plans in real-time as contact opportunities or environment geometry changes requires efficient re-optimization, robust state estimation, and possibly learned heuristics or policies.
• Integration of semantics and perception: The use of language and vision for explicit environment labeling (e.g., contact tolerance via VLM) points toward unified pipelines for semantic and physical planning (Ling et al., 13 Mar 2025).
• Whole-body, multi-contact, and 3D generalization: Moving beyond planar or isolated contacts to robust, efficient planning of whole-body, multi-site, dynamic contacts in full 3D remains an open challenge; preliminary results with continuous surface representations and complementarity-based switching are promising (Leve et al., 2024, Ciebielski et al., 16 Aug 2025).
• Verification and certification: Formal guarantees (e.g., through barrier functions or solution suboptimality bounds) are increasingly handled in deployment-focused RL and reachability control (Mulkana et al., 3 Dec 2025), but general analytic certification of contact-rich plans remains rare.

Contact-explicit planning frameworks constitute a crucial and active research area underpinning safe, reliable, and capable robotic interaction in cluttered, dynamic, or collaborative settings. By explicitly modeling the combinatorial structure and hybrid dynamics of contact, these approaches continue to expand the behavioral repertoire and robustness of autonomous systems in the real world.