DEFT: Differentiable Branched Discrete Elastic Rods for Modeling Furcated DLOs in Real-Time

The gist: DEFT introduces Differentiable discrete branched Elastic rods for modeling Furcated DLOs in real-Time, a novel framework that combines a differentiable physics-based model with a learning framework to accurately model BDLO dynamics, achieve efficient computation for real-time inference, and enable planning to demonstrate dexterous BDLO manipulation<ref:2502.15037#pg2>

Modeling Branched Deformable Linear Objects (BDLOs)

The paper addresses the fundamental challenge of accurately modeling Branched Deformable Linear Objects (BDLOs) because junction points create complex force interactions and strain propagation patterns that cannot be adequately captured by simply connecting multiple single-DLO models<ref:2502.15037#pg3> The framework decomposes a BDLO into a hierarchy of DLOs, where the parent DLO is defined as the minimum-length path through the BDLO that connects the two end-effectors<ref:2502.15037#pg5> All other branches diverging from this primary path are modeled as children DLOs that maintain their physical connection to the parent<ref:2502.15037#pg5> Junction points are modeled using a graph representation where the adjacency matrix A captures both self-loops (diagonal entries) and internode connections<ref:2502.15037#pg5> The parent and children DLOs are each modeled as DLOs using DDER and inherit all previously introduced notation<ref:2502.15037#pg5>

DEFT Framework Overview

Algorithm 1 outlines the DEFT process, which involves several steps to predict the state of the wire recursively without relying on ground truth or perception data at any point in the process<ref:2502.15037#pg7> The steps include:

1. Associate grasped vertices with ut<ref:2502.15037#pg7>

2. Calculate Material Frames Γ(Xt, θt)<ref:2502.15037#pg7>

3. θ∗t(Xt, α) = arg minθt P(Γ(Xt, θt), α)<ref:2502.15037#pg7>

4. Vˆt+1 = Vˆt − ∆tM−1 ∂P (Γ(Xt,θ∗t)(Xt,α),α) ∂Xt▷ (1)<ref:2502.15037#pg7>

5. Xˆt+1 = Xˆt + ∆tVˆt+1 + GNN ▷ §4.3<ref:2502.15037#pg7>

6. Constraints enforcement on Xˆt+1 ▷ §4.4<ref:2502.15037#pg7>

7. Vˆt+1 = (Xˆt+1 − Xˆt)/∆t ▷ Velocity Update<ref:2502.15037#pg7>

DEFT simulates the behavior of a BDLO when given predictions of the vertices and their velocities under a specified input<ref:2502.15037#pg7>

Physics-Informed Learning Components

The framework combines differentiable physics models with a learning framework to improve sample efficiency and generalizability<ref:2502.15037#pg5> The DDER model builds upon DER theory to model three distinct deformation modes of DLOs: bending, twisting, and stretching while being able to auto-tune model parameters from real-world dataset<ref:2502.15037#pg7> The potential energy P(Γ(Xt, θt), α) arises from bending and twisting effects<ref:2502.15037#pg7> To compute the gradient of (3) with respect to α, DDER employs solvers from Theseus [27], which use PyTorch’s automatic differentiation<ref:2502.15037#pg7> A specialized Graph Neural Network (GNN) architecture is used for residual learning to correct numerical integration errors over long horizons<ref:2502.15037#pg7> The GNN processes BDLOs as graphs where vertices become nodes and edges capture both intra-branch connections and junctions through an adjacency matrix A<ref:2502.15037#pg7>

Constraint Enforcement for Realism

DEFT applies constraints to maintain branch inextensibility, junction attachment, and proper propagation of deformation effects<ref:2502.15037#pg7>

1. Inextensibility Enforcement: Theorem 3 computes the correction ∆xˆi explicitly by solving an optimization problem that penalizes deviations of each segment’s length from its undeformed length<ref:2502.15037#pg7>

2. Attachment at the Junction: Theorem 4 computes the correction ∆xˆp,i and ∆xˆc,1 explicitly to ensure the child branch remains attached to the parent branch<ref:2502.15037#pg7>

3. Edge Orientation Constraints: Theorem 5 models each edge as a rigid body and enforces an orientation constraint by requiring that the constraint function C p,iO(Xˆ, ∆Xˆ) is approximately satisfied to ensure each junction remains almost rigid<ref:2502.15037#pg7> These constraints are applied iteratively in Algorithm 2 to modify the prediction until all constraints are satisfied<ref:2502.15037#pg7>

Computational Efficiency and Validation

DEFT achieves efficient computation for real-time inference through several optimizations<ref:2502.15037#pg7>

1. Analytical Gradient Derivation: The analytical gradient of P(Γ(Xt, θt), α) with respect to θt is derived in Theorem 2, which is approximately twice as fast as using numerical gradients<ref:2502.15037#pg7>

2. Parallel Programming: DEFT constructs a computational graph that computes dynamics at junctions through constraints as post-processing, enabling efficient parallel execution of Algorithm 1 via PyTorch’s batch-wise operations<ref:2502.15037#pg7>

The framework demonstrates superior performance compared to state-of-the-art methods in accuracy, computational speed, and planning success rate<ref:2502.15037#pg7> Experiments show DEFT outperforms prior methods in accuracy, speed, and sample efficiency while enabling practical applications such as 3-D shape matching, multi-branch grasping, and thread insertion<ref:2502.15037#pg7> The system successfully completes more complex manipulation tasks than other baselines<ref:2502.15037#pg7>

Conclusion

DEFT delivers accurate modeling and realtime prediction of BDLO dynamic behavior during manipulation over extended time horizons<ref:2502.15037#pg7> It demonstrates that the combination of auto-tuning the material properties and residual learning provides substantial improvement, illustrating DEFT’s ability to leverage real-world data for parameter identification and residual learning<ref:2502.15037#pg7> The framework successfully integrates with planning and control frameworks for 3D shape matching and thread insertion in real-world scenarios<ref:2502.15037#pg7> DEFT achieves superior accuracy while maintaining sample efficiency and computational speed<ref:2502.15037#pg7> The paper represents, to the best of our knowledge, the first quantitative study of BDLO modeling and manipulation in real-world settings<ref:2502.15037#pg7> It successfully completes more complex manipulation tasks than other baselines<ref:2502.15037#pg7> The framework is implemented in PyTorch for seamless integration with deep learning workflows<ref:2502.15037#pg7> The authors would like to gratefully thank the support by Ford Motor Company<ref:2502.15037#pg7>

Limitations

While DEFT demonstrates strong performance in modeling and controlling BDLOs, several areas for improvement remain<ref:2502.15037#pg7> First, DEFT assumes the locations and geometries of the target insertion holes are known<ref:2502.15037#pg7> Second, DEFT occasionally fails at insertion tasks due to neglecting the gripper orientation during BDLO manipulation<ref:2502.15037#pg7> Lastly, DEFT does not explicitly handle interactions between deformable objects and their environment, such as contact forces and friction<ref:2502.15037#pg7>

References

[1] J. Sanchez, J.-A. Corrales, B.-C. Bouzgarrou, and Y. Mezouar. Robotic manipulation and sensing of deformable objects in domestic and industrial applications: a survey<ref:2502.15037#pg8>

[2] H. Wang, O. Salunkhe, W. Quadrini, D. Lamkull, F. Ore, B. Johansson, and J. Stahre. Overview of computer vision techniques in robotized wire harness assembly: Current state and future opportunities<ref:2502.15037#pg9>

[3] M. Saha and P. Isto.

1. Differentiable physics based modeling for real-time prediction of branched objects allows robots to accurately predict how flexible branched structures behave during manipulation, enabling reliable execution of assembly operations without relying on ground truth or perception data at any point in the process.

2. Integration of a Graph Neural Network for residual learning corrects numerical integration errors over long horizons, improving accuracy by learning residual dynamics specifically for BDLOs, which is crucial for open-loop predictions.

3. The framework enables dexterous manipulation tasks such as 3-D shape matching and thread insertion, allowing autonomous systems to complete more complex manipulation tasks than other baselines in real-world scenarios.

4. The system achieves efficient computation through parallel programming of parent and child DLOs and the use of analytical gradients for solving the inner loop, resulting in a computational speed that is 8.8 times faster than DEFORM++ and 1952 times faster than the finest FEM baseline.

5. Constraints enforcement modules ensure physical realism by maintaining branch inextensibility, junction attachment, and proper propagation of deformation effects, preventing simulation instability when modeling complex cable systems.

Autonomous wire-harness assembly requires precise, reliable manipulation of branched cables, whose deformation is difficult to predict. Models for single Deformable Linear Objects (DLOs) do not extend directly to Branched DLOs (BDLOs), whose junctions couple bending, twisting, and stretching across branches. We introduce Differentiable discrete branched Elastic rods for modeling Furcated DLOs in real-Time (DEFT), a framework combining differentiable physics and learning for accurate BDLO dynamics, real-time inference, and dexterous manipulation planning. DEFT introduces momentum-conserving junction constraints to capture cross-branch coupling, supports mid-structure grasping, uses analytical gradients and branch-parallel computation for efficient parameter learning and inference, and incorporates a residual Graph Neural Network (GNN) to reduce long-horizon prediction errors. Experiments on six physical BDLOs with different topologies and materials demonstrate formulation-level generalization: the same formulation and residual-GNN architecture apply without redesign, with physical and learned parameters fitted separately for each BDLO. DEFT outperforms state-of-the-art physics-based and learning-based methods, achieving RMSEs of 0.93-2.14 cm over five-second recursive open-loop predictions and inference above 100 Hz. In real-world planning, DEFT achieves success rates of 36/40 for shape matching, 40/40 for parent-branch insertion, and 16/20 for child-branch insertion. These results demonstrate accurate, efficient, planning-compatible BDLO prediction for autonomous wire-harness manipulation. Project website: https://roahmlab.github.io/DEFT/.