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WRAP: Fixtureless Wrench-aware Multi-Robot Assembly Planning

Valentin N. Hartmann, Huang Su, Yijiang Huang, Stelian Coros

arXiv:2609.29407v1cs.RO

TL;DR

Assembly planning often relies on fixtures or restricted strategies, while force-aware coordination across multiple robots is challenging. WRAP searches for multi-robot action plans that support assembly forces, and its ablations show that omitting wrench constraints or considering gravity alone lowers search times and shortens plans.

  • Problem

    Robotic assembly planning can rely on fixtures or restricted sequential and top-down strategies, while often ignoring forces needed for contact-rich tasks.

    Source: §I · p. 1

  • Method

    WRAP searches over robot actions and parameterizations, using assembly-step feasibility checks to plan force-supporting grasps and handovers.

    Source: §I · p. 1, C. Task planning · p. 3

  • Results

    Without wrench constraints, or with gravity alone, planning times are significantly lower and plans are shorter because fewer constraints need to be satisfied.

    Source: 2) Scaling · p. 5

  • Takeaways & Limitations

    WRAP continues searching after finding a valid task plan to improve it, but retains the initially chosen assembly sequence.

    Source: Plan improvement · p. 4

    The classical search may require testing all actions and parameterizations to certify infeasibility, which does not scale to large assemblies and many robots.

    Source: §VI · p. 7

Abstract

from arXiv · show

Assembly using robots often requires specially designed fixtures, or relies on top-down only assembly strategies. Using multiple robots, we can avoid using fixtures and make robotic assembly more flexible. Planning assembly sequences for multiple robots is challenging due to the high number of possible task assignments and orders. In addition, we need to reason over forces that occur during the assembly process, e.g., to decide if multiple robots are required for support, or if external support such as a table should be used. We present Wrap, a multi-robot assembly planner for multi-part assemblies, given the inter-part ordering-dependencies, the part meshes, and their initial state. We formulate a linear program to reason about valid grasps for supporting the forces that occur during assembly. The search leverages the assembly sequence, and greedily finds a feasible solution per assembly step by computing a heuristic via a cheap backwards search, and using the heuristic in the more expensive forward search. We then solve the multi-robot, multi-goal motion planning problem, and for execution, we split the plan into contact-rich assembly skills, and free space motion. We benchmark the planner on a variety of multi-part assemblies, and apply the planner to groups of robots differing in size and kinematics. We validate the work both in a physics simulation, and in real. Videos and code are available at https://www.vhartmann.com/wrap.

I. INTRODUCTION

WRAP targets assembly planning that often depends on fixtures, restricted assembly strategies, and manual planning, while accounting for forces that can make contact-rich tasks difficult. It jointly plans robot actions around assembly dependencies, extending prior work by incorporating force-aware multi-robot assignment and sequencing.

I. INTRODUCTION

  • WRAP computes the grasps and handovers needed to support assembly forces, enabling fixtureless assembly with multiple robots.The authors demonstrate plans in simulation and on a real robot system.Source: §I · p. 1

A. Assembly Sequence Planning

  • Part-centric assembly planning reasons about part motions, while WRAP jointly decides robot assignments, grasps, handovers, set-downs, and wrench balancing within the action sequence.Prior approaches often use robots only as a feasibility filter or select contacts after inferring a sequence [11, 12, 13].Source: A. Assembly Sequence Planning · p. 1

B. Multi-Robot Assembly Systems and Planning

  • Earlier multi-robot planners leave gaps WRAP addresses: some fix holder and inserter roles, omit force modeling, or abstract mating operations away.Fabrica integrates sequence, grasp, and motion planning but disallows regrasping and reorientation and uses a grasp-stability heuristic for mating forces.Source: B. Multi-Robot Assembly Systems and… · p. 2

III. PROBLEM FORMULATION

The problem is to find a parameterized multi-robot action sequence for building an assembly, then configurations and a collision-free trajectory that satisfy its task constraints. The assembly is represented by parts, possible final relative poses, and a dependency graph of assembly steps.

III. PROBLEM FORMULATION

  • An assembly dependency graph represents subassemblies as vertices and links preceding subassemblies to the assembly steps that use them.A step takes its parent subassemblies as input and produces a new subassembly; multiple possible final relative poses allow for symmetric parts.Source: §III · p. 2
  • Each assembly action must avoid collisions and provide grasp wrench capabilities that satisfy the wrench constraint arising from the required assembly force.Mated parts can sustain only a convex set of wrenches.Source: §III · p. 2
  • The planner must find which robot manipulates which object, when and how, how assemblies are supported, and a collision-free trajectory that includes assembly controllers.The action sequence is parameterized, and configurations must satisfy the constraints at task boundaries.Source: §III · p. 2

IV. MULTI-ROBOT ASSEMBLY PLANNING

WRAP searches over feasible robot actions and their parameterizations, then plans and postprocesses motion for the resulting action plan. Its scope in this formulation is prehensile manipulation.

IV. MULTI-ROBOT ASSEMBLY PLANNING

  • WRAP formulates feasible action-sequence planning as a search over robot actions and their parameterizations, considering prehensile manipulation.Source: §IV · p. 2
  • Before searching, WRAP samples valid grasps and stable placement poses for subassemblies in the dependency graph.Source: §IV · p. 2
  • For a found action plan, WRAP computes a motion plan, postprocesses it for dynamic constraints, and replaces contact-rich segments with local controllers for interaction forces.Source: §IV · p. 2

A. Force constraints

WRAP checks whether an assembly step can achieve static equilibrium given grasp and interface wrench capabilities. It also tests robustness to bounded variations in the external wrench, with table support available when a part is on the table.

A. Force constraints

  • A linear program checks whether grasp and part-interface wrench capabilities can achieve static equilibrium for an assembly step.The force constraints are illustrated alongside an assembly action in Fig. 3.Source: A. Force constraints · pp. 2–3, Figure 3 · p. 3
  • The equilibrium check accounts for each considered part, the robots acting on it, connected parts, and the external wrench, including gravity.Source: A. Force constraints · p. 3
  • Robust satisfaction requires the linear program to hold at every vertex of the bounded external-wrench set, including transverse force components and an application-point offset.The model can use table support when a part rests on the table; a brace can withstand possible lateral deviations that might otherwise topple a component resisting a nominal press.Source: Robust satisfaction · p. 3, A. Force constraints · p. 3

B. Preprocessing

WRAP preprocesses stable resting orientations and grasps, then searches for robot actions milestone by milestone using a backward heuristic to guide forward planning.

B. Preprocessing

  • Stable placements use orientations that keep the subassembly’s center of mass within its supporting points, while grasps are sampled and checked for sufficient finger-pad overlap or vacuum support area.Vacuum grasps are sampled normal to the surface.Source: Stable placement orientations · p. 3, Grasp sampling · p. 3

C. Task planning

  • The forward search state records completed subassemblies, their placement poses, and each robot’s grasp on a part.Source: C. Task planning · p. 3
  • Milestone search plans robot actions for one assembly step at a time, backtracks when needed, and widens the grasp and placement sets if the current resolution yields no solution.Source: C. Task planning · p. 4
  • A backward Dijkstra search applies inverse actions from valid goals, using projected grasp classes and cheap collision and wrench checks to estimate feasibility.A grasp class is feasible when any represented grasp meets the corresponding geometric and wrench constraints.Source: Backward Search · p. 4
  • Forward planning uses the heuristic in a greedy lazy shortest-path search and tests action configurations with inverse kinematics.IK constraints depend on the action, and the solver enforces minimum clearance between robots.Source: Forward Search · p. 4, Inverse kinematics (IK) · p. 4
  • After finding a valid plan, large-neighborhood search widens its fill window and combines greedy and exact A* searches, but retains the initially chosen assembly sequence.Joint keyframe optimization then selects configurations to reduce IK-branch changes and joint motion while increasing clearance; it does not change the task sequence.Source: Plan improvement · p. 4, Joint Keyframe Optimization · p. 4

D. Motion planning

The motion-planning problem is formulated as multi-modal, multi-goal, multi-robot path planning; motion-planning timeouts feed back into task planning.

D. Motion planning

  • WRAP formulates motion planning as a multi-modal, multi-goal, multi-robot path-planning problem.Source: D. Motion planning · p. 4
  • Because a task plan may lack a feasible motion plan, WRAP includes feedback from motion planning to task planning.Source: D. Motion planning · p. 4
  • The motion planner cannot certify infeasibility, so on timeout it suspends the search, stores the task plan, and returns to it with more planning effort later.Source: D. Motion planning · p. 4

E. Retiming and Execution

Execution postprocessing retimes each robot’s path independently while coordinating manipulation events and limiting deviations from the original timing.

E. Retiming and Execution

  • Retiming keeps each robot on its geometric path but lets it advance independently to satisfy kinodynamic constraints for hardware execution.Source: E. Retiming and Execution · p. 5
  • Manipulation events and potential collisions constrain synchronization, while reserved skill durations let unrelated robots continue moving up to a maximum deviation.Source: E. Retiming and Execution · p. 5
  • If a manipulation skill overruns its reserved duration, participating robots wait at the event while unrelated robots continue until synchronization is restored.Unlike APEX-MR, WRAP directly optimizes and certifies continuous path phases.Source: E. Retiming and Execution · p. 5

V. EXPERIMENTS

The experiments use new assemblies and modified benchmark objects, with repeated runs and grasp-set settings adjusted for complex meshes.

A. Benchmark Suite and Setup

  • The new stool has five parts and snap-fit legs, and the eight-part chair requires assembling its backrest separately before joining the modules.Source: A. Benchmark Suite and Setup · p. 5
  • The seven-part Cross tests scaling with repeated reorientations, while the eight-part Cube is used only for scaling analysis.Source: A. Benchmark Suite and Setup · p. 5
  • The benchmark suite also includes seven objects from, with initial and on-table assembly fixtures removed, making assembly harder and requiring multiple handovers before final assembly.Source: A. Benchmark Suite and Setup · p. 5
  • Simulation uses a 10 N insertion force and typically repeats each experiment 10 times; Fabrica models use 64 grasps instead of 20 and disable grasp widening.Source: A. Benchmark Suite and Setup · p. 5

1) Computation times:

Across computation-time, scaling, and ablation comparisons, WRAP is usually faster than the optimistic planner, but its greedy search can yield worse solutions. Runtime grows roughly linearly with part count, while adding robots increases search time; relaxing wrench checks lowers planning time and plan length.

1) Computation times:

  • Against an optimistic planner similar to, WRAP is usually faster but sometimes finds worse solutions.Table II compares search times and solution costs; Table III breaks down WRAP’s computation time.Source: 1) Computation times · p. 5, Table II · p. 6
  • Grasp-database computation, forward search, and motion planning account for the most significant time differences; wrench-LP and heuristic computation are negligible in most cases.The paper attributes higher times on Fabrica assemblies to the backend’s required convex decomposition, which makes collision checking more expensive.Source: 1) Computation times · p. 5, Table III · p. 6

2) Scaling:

  • Compute time scales roughly linearly with part count, but adding robots significantly increases it because more pick assignments and handovers enter the search.Figure 5 compares scaling for two objects with two, three, and four robots.Source: 2) Scaling · p. 5, Figure 5 · p. 6
  • Without wrench checks, or with gravity alone, search times are significantly lower and plans are shorter because fewer constraints must be satisfied.Figure 5 also varies mating forces by scaling factor λ.Source: 2) Scaling · p. 5, Figure 5 · p. 6

3) Ablations:

  • WRAP finds initial solutions fastest on average, while counting and optimistic searches are often competitive; flat lazy search runs out of memory on every problem.The ablations include flat search, a counting heuristic, exact local fills, and a finer projection.Source: 3) Ablations · pp. 5–6
  • Optimal forward search in local fills produces better solutions than WRAP’s greedy search, but takes longer.Table II reports search times and plan costs, with means and standard deviations over 10 runs.Source: 3) Ablations · p. 6, Table II · p. 6

4) Convergence and Optimality:

WRAP usually finds solutions earlier than other planners, and continued search often brings its cost to parity or better, though some assemblies remain difficult to optimize. The section also reports simulation failures without wrench-aware planning and demonstrates extensions to other robot setups and assembly motions.

4) Convergence and Optimality:

  • WRAP usually finds solutions earlier than other planners, but starts with worse cost and often reaches the same or better cost with continued optimization.Fig. 6 plots cost convergence for Cross, Stool, and Chair.Source: 4) Convergence and Optimality · p. 6, Figure 6 · p. 6
  • For Chair, optimizing within a chosen assembly sequence may miss the optimum, while fixed milestone end-states can require longer search horizons to escape.Chair is identified as a counterexample and consists of many parts and submodules.Source: 4) Convergence and Optimality · p. 6

5) Generalizations:

  • Without changes to WRAP, the authors demonstrate a Cross assembly with four seven-DoF OpenArm robots and a stool assembled using screw motions.The stool has five parts, with its four legs attached by screw motions rather than linear insertions.Source: 5) Generalizations · p. 6

B. Simulation

  • The computed paths are tested in MuJoCo under assembly wrenches and robot dynamics, with wrench-limit utilization reported by gripper type and execution phase.To address real-to-sim difficulties, parts are rigidly attached to grippers and mating wrenches are applied externally.Source: B. Simulation · p. 6
  • Both approaches that omit assembly-wrench checks fail to complete an assembly, with common failures including unsupported assembly, sliding parts, and unsuitable vacuum-gripper orientation.WRAP also fails once on Cube because of object misalignment.Source: B. Simulation · p. 6

C. Real world execution

The real-robot demonstration assembles stool legs with two UR5e robots, using calibrated gripper and insertion wrench limits. The discussion identifies search scalability and execution assumptions as boundaries, while the conclusion summarizes WRAP’s fixtureless, wrench-aware planning scope.

C. Real world execution

  • Table IV reports simulation success and mean peak support utilization over 10 runs, split by execution phase and gripper type.Bold utilization values exceed the limit; a dash marks an absent gripper type.Source: Table IV · p. 7
  • Two UR5e robots on a stationary Husky execute a seven-action plan on a halved stool, including a handover and two leg insertions.A joint-space velocity controller tracks the path, and compliant control performs insertion; accurate calibration made vision-based recovery unnecessary.Source: C. Real world execution · pp. 6–7
  • Real-robot planning uses a nominal 25 N force after test insertions measured maxima of 17.32 N axial and 5.38 N transverse force.Gripper wrench limits use the more conservative of manufacturer and measured slippage limits; vacuum assembly permits only z-force because of deflection.Source: C. Real world execution · p. 7, Table V · p. 7

VI. DISCUSSION & LIMITATIONS

  • The classical search does not scale to large assemblies and many robots because certifying infeasibility requires testing all possible actions and parameterizations.The authors suggest focusing on subsets of the state space rather than considering all robots from the outset.Source: §VI · p. 7
  • The current approach is limited to prehensile actions with deterministic effects, and larger execution failures require replanning, which is left for future work.The authors also note that part rigidity may fail under load, so internal stress should be included alongside external forces.Source: §VI · p. 7

VII. CONCLUSION

  • WRAP computes task assignments and sequences from an assembly dependency graph, accounts for assembly forces, avoids fixtures, and produces asynchronous motion plans executable on real robots.Source: §VII · p. 7
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