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

Yizhou Chen; Felicia Sang; Shicheng Peng; Xiaoyue Wu; Yeheng Zong; Yuzhen Chen; Anran Li; Bohao Zhang

cs.ROFeb 20, 2025

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

Authors: Yizhou Chen, Felicia Sang, Shicheng Peng, Xiaoyue Wu, Yeheng Zong, Yuzhen Chen, Anran Li, Bohao Zhang,

Organizations: Department of Robotics, University of Michigan, Ann Arbor, MI 48109, United States

Abstract

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/.

Figures & tables

Figure 1: As this paper illustrates, DEFT can be used in concert with a motion planning algorithm to autonomously manipulate BDLOs. The figures above illustrate the ability of DEFT to autonomously perform wire insertion across different BDLO topologies and insertion configurations. Starting from a random initial configuration, the system guides the BDLO tip into the target hole . The figures present two representative cases: Top: insertion of the main branch tip into the target hole; Bottom: insertion of a child branch tip in a BDLO with a different topology, while only the parent branch is grasped. Notably, DEFT recursively predicts the state of the wire throughout the process without relying on ground-truth or perception data at any stage.

Figure 2: An illustration of DER coordinate frames.

Figure 3: An illustration of a branched deformable linear object (BDLO) model. Left: An example configuration of BDLO manipulation. Right: Relevant notation and structure at a junction.

Figure 4: An illustration of $\hat{\bm{\Omega}}^{i}_{\Delta,t}(\hat{\mathbf{X}}_{t},\Delta\hat{\mathbf{X}}_{t})$

Figure 5: Algorithm Overview of DEFT. In the initialization stage (Section 3.1 ), DEFT begins by separating the BDLO into a parent DLO and one or more children DLOs. Each DLO is discretized into vertices and represented as elastic rods. This setup allows DEFT to capture the geometric and physical properties required for dynamic simulation. To improve computational efficiency, DEFT then predicts the dynamics of each branch in parallel (Section 3.5 ). During this process, analytical gradients are provided to solve ( 10 ), as detailed in Section 3.5 , ensuring efficient and stable convergence. Next, to address numerical errors, DEFT employs a GNN designed to learn the BDLO’s residual dynamics (Section 3.3 ). By modeling discrepancies between simulated and observed behavior, the GNN refines predictions and enhances overall accuracy. After integration, DEFT enforces constraints (Section 3.4 ) to enforce physical realism. Inextensibility constraints are applied to each branch, while junction-level constraints ensure proper attachment at branch junctions. Additionally, edge orientation constraints enable the propagation of dynamics across these junctions. Throughout the entire pipeline, all components remain fully differentiable, allowing for efficient parameter learning from real-world data.

Figure 6: The adjacency matrix $\mathbf{A}$ for the junction in Figure 3 captures both self-loops (diagonal entries) and inter-node connections (off-diagonal entries). By embedding each node’s local dynamics alongside its coupling to neighboring nodes, $\mathbf{A}$ enables a graph representation of the BDLO’s behavior.

Figure 7: An illustration of applying $\hat{\bm{\Omega}}_{\Delta}^{i}$ to rotate the $i$ -th segment.

Figure 8: An illustration of the BDLOs used to evaluate and compare DEFT against baseline methods. To enhance dataset diversity, we vary stiffness (measured on a relative scale), length, mass, number of vertices, and junction topology. In real-world wire harnesses, rigid components such as clamps and fixation elements are commonly attached along the wires ( Trommnau et al., 2019 ) . To reflect this, we construct BDLOs that integrate deformable segments with rigid components modeled as high-stiffness regions. This setup highlights DEFT’s ability to simultaneously capture deformable behavior and interactions with rigid structures.

	BDLO 1	BDLO 2	BDLO 3	BDLO 4	BDLO 5	BDLO 6
Topology	1	1	1	1	2	3
Branch Stiffness	$[M,\;M,\;M]$	$[S,\;S,\;S]$	$[M,\;S,\;R]$	$[M,\;R,\;R]$	$[M,\;M,\;M]$	$[H,\;H,\;H]$
Marker Count	$[13,\;4,\;3]$	$[12,\;4,\;4]$	$[12,\;2,\;3]$	$[12,\;2,\;2]$	$[12,\;3,\;3]$	$[12,\;4,\;3,\;2]$
Dataset 1 (s)	505	735	645	495	505	435
Dataset 2 (s)	440	N/A	445	N/A	N/A	N/A

Table 1: Topology and branch-wise properties of each BDLO used in the experiments. Topology configurations are illustrated in Figure 8 . Branch stiffness is defined on a relative scale with four levels: soft (S), medium (M), stiff (H), and rigid (R). Dataset durations for each BDLO are reported in the final two rows.

Figure 9: Experimental setup for data collection and manipulation using a dual-arm robotic system.

	Modeling Accuracy (RMSE, $10^{-2}\,\mathrm{m}$ )								Time
	Dataset 1						Dataset 2		( $10^{-2}\,\mathrm{s}$ )
Method	1	2	3	4	5	6	1	3	1
GCN ( Kipf and Welling, 2017 )	6.64 [-2pt] $\pm\,0.20$	6.52 [-2pt] $\pm\,0.43$	5.41 [-2pt] $\pm\,0.15$	4.30 [-2pt] $\pm\,0.11$	4.50 [-2pt] $\pm\,0.30$	6.82 [-2pt] $\pm\,0.04$	7.29 [-2pt] $\pm\,0.14$	4.35 [-2pt] $\pm\,0.18$	0.163 [-2pt] $\pm\,0.01$
Tree-LSTM ( Tai et al., 2015 )	5.32 [-2pt] $\pm\,0.12$	4.45 [-2pt] $\pm\,0.20$	2.50 [-2pt] $\pm\,0.08$	2.82 [-2pt] $\pm\,0.13$	5.06 [-2pt] $\pm\,0.07$	4.96 [-2pt] $\pm\,0.10$	7.13 [-2pt] $\pm\,0.09$	4.54 [-2pt] $\pm\,0.14$	2.02 [-2pt] $\pm\,0.26$
FEM ( Nealen et al., 2006 ) : $\Delta_{t}$ : $10^{-6}$ s, node: 60	3.11	3.70	2.29	2.84	2.97	3.42	2.17	2.10	493
FEM ( Nealen et al., 2006 ) : $\Delta_{t}$ : $10^{-6}$ s, node: 160	2.07	3.35	1.82	1.54	2.13	2.10	1.40	1.94	1211

Table 2: Left: Modeling results over a

5

s horizon. For learned methods, each RMSE entry reports the mean on the first line and the standard deviation on the second line, computed across evaluation trajectories and three independently trained seeds. Right: Computational time per inference step (

10^{-2}\,\mathrm{s}

). For learned methods, the mean and standard deviation across the three seeds are reported using the same format.

Figure 10: Visualization of the predicted trajectories using DEFT, a DEFT ablation that leaves out the constraint described in Theorem 3 , and Tree-LSTM. The ground-truth initial position of the vertices are colored in blue, the ground-truth final position of the vertices are colored in pink, and the gradient between these two colors is used to denote the ground truth location over time. The predicted vertices are colored as green circles (DEFT), orange circles (DEFT ablation), and light red circles (Tree-LSTM), respectively. A gradient is used for these predictions to depict the evolution of time, starting from dark and going to light. Note that the ground truth is only provided at t=0 s. The prediction is performed recursively without requiring additional ground-truth data or perception inputs throughout the process. Additional visualizations of BDLO 5 can be found in the Appendix A , Figure 13 .

Figure 11: Computational speed comparison: parallel programming versus a naive (sequential) implementation.

	Time ( $10^{-2}\,\mathrm{s}$ )
Method	1	2	3	4
Analytical Gradient	0.34 [-2pt] $\pm\,0.02$	0.27 [-2pt] $\pm\,0.01$	0.41 [-2pt] $\pm\,0.01$	0.46 [-2pt] $\pm\,0.01$
Numerical Gradient	0.74 [-2pt] $\pm\,0.02$	0.65 [-2pt] $\pm\,0.01$	0.84 [-2pt] $\pm\,0.04$	0.91 [-2pt] $\pm\,0.02$

Table 3: Computational time for ( 10 ) with Dataset 1. Each entry reports the mean on the first line and the standard deviation on the second line across three timing runs.

	Modeling Accuracy (RMSE, $10^{-2}\,\mathrm{m}$ ) $\downarrow$
	Dataset 1				Dataset 2
Method	1	2	3	4	1	3
W/O Constraints. 1 , 2 and 3	$7.66\times 10^{8}$	$6.10\times 10^{8}$	$6.56\times 10^{6}$	$1.77\times 10^{8}$	$3.81\times 10^{7}$	$1.23\times 10^{8}$
W/O Constraints. 2 and 3	$1.36\times 10^{2}$	$1.81\times 10^{2}$	$1.51\times 10^{2}$	$7.64\times 10^{1}$	$8.72\times 10^{1}$	$3.38\times 10^{2}$
W/O Constraint. 3	3.95 [-2pt] $\pm\,0.23$	3.49 [-2pt] $\pm\,0.14$	2.03 [-2pt] $\pm\,0.19$	2.15 [-2pt] $\pm\,0.08$	4.01 [-2pt] $\pm\,0.16$	4.88 [-2pt] $\pm\,0.25$
W/O Learning + System ID	2.46	3.19	1.83	1.81	1.85	2.43

Table 4: Ablation study with Dataset 1 and 2. For configurations containing a learned residual model, each cell reports the mean RMSE on the first line and the standard deviation on the second line, computed across evaluation trajectories and three independently trained seeds.

	Modeling Accuracy (RMSE, $10^{-2}\,\mathrm{m}$ ) $\downarrow$
Method	Before Training	After Training	Improvement
Replace with Newtonian Force	4.27	3.94 [-2pt] $\pm\,0.26$	0.33 [-2pt] $\pm\,0.26$
Without Momentum Conservation	3.60	3.17 [-2pt] $\pm\,0.18$	0.43 [-2pt] $\pm\,0.18$
DEFT	2.46	1.44 [-2pt] $\pm\,0.07$	1.01 [-2pt] $\pm\,0.07$

Table 5: Comparison with constraint-enforcement alternatives on Dataset 1, BDLO 1. For the after-training and improvement results, each cell reports the mean on the first line and the standard deviation on the second line across three independently trained seeds.

Method	Average RMSE ( $10^{-2}$ m) $\downarrow$	RMSE Reduction ( $10^{-2}$ m) $\uparrow$	Average Comp. Time ( $10^{-2}$ s) $\downarrow$
W/O Residual Learning	1.89	0.00	0.68
Replace GNN with MLP	1.80	0.09	0.71
DEFT	1.31	0.58	0.74

Table 6: Ablation study comparing GNN- and MLP-based residual-learning approaches. Modeling accuracy is averaged across the six BDLO/dataset evaluation settings in Table 4 . Computational time is reported as an average across the evaluated BDLOs.

Figure 12: Visualization of planning across three manipulation tasks using DEFT, a DEFT ablation without the constraint in Theorem 3 , and Tree-LSTM. The first two tasks are performed on BDLO 1, while the third task uses BDLO 5. In the shape matching task, the BDLO is highlighted in yellow for visibility. The DEFT model successfully completes all tasks, whereas both the ablation variant and Tree-LSTM fail to achieve successful execution.

	Modeling Accuracy $\downarrow$
Method	(RMSE, $10^{-2}\,\mathrm{m}$ )
Single-step Training	2.02 [-2pt] $\pm\,0.38$
Multi-step Training	1.44 [-2pt] $\pm\,0.07$

Table 7: Ablation study comparing multi-step and single-step training on Dataset 1 (BDLO 1). Each entry reports the mean RMSE on the first line and the standard deviation on the second line across three independently trained seeds.

	Success Rate with 95% Wilson CI
Methods	Shape Matching	Insertion Task 1	Insertion Task 2
Tree-LSTM	13/40 (32.5%) [-2pt] $[20.1,48.0]\%$	0/40 (0.0%) [-2pt] $[0.0,8.8]\%$	0/20 (0.0%) [-2pt] $[0.0,16.1]\%$
DEFORM +	0/40 (0.0%) [-2pt] $[0.0,8.8]\%$	38/40 (95.0%) [-2pt] $[83.5,98.6]\%$	2/20 (10.0%) [-2pt] $[2.8,30.1]\%$
DEFT (Ablation)	0/40 (0.0%) [-2pt] $[0.0,8.8]\%$	38/40 (95.0%) [-2pt] $[83.5,98.6]\%$	2/20 (10.0%) [-2pt] $[2.8,30.1]\%$
DEFT	36/40 (90.0%) [-2pt] $[76.9,96.0]\%$	40/40 (100.0%) [-2pt] $[91.2,100.0]\%$	16/20 (80.0%) [-2pt] $[58.4,91.9]\%$

Table 8: Success rates of planning tasks for different methods. Each cell reports the number of successful trials and the corresponding success rate on the first line, followed by the 95% Wilson confidence interval on the second line. Insertion Task 1 involves inserting the tip of the parent branch through the central opening of a target torus using BDLO 1. Insertion Task 2 involves inserting the tip of the second child branch through the central opening of the torus using BDLO 5.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Configuration	Value
MLP architecture	$16\!-\!256\!-\!256\!-\!256\!-\!3$ (3 hidden layers)
Hidden-layer activation	LeakyReLU after each hidden layer (negative slope $0.2$ )
Output activation	None (linear output)
Loss function	Mean squared error
Optimizer	Adam ( $\epsilon_{\mathrm{Adam}}=10^{-8}$ )
GNN learning rate	$10^{-4}$

Appendix

Table 9: Architecture and training configuration of the residual GNN in DEFT.

Figure 14: Constraint residuals at every simulation step of a

5

s open-loop rollout on BDLO 1. The panels report the maximum inextensibility residual across all branches, the junction-attachment residuals, and the parent–child orientation residuals. The corresponding squared constraint values remain below a tight user defined threshold throughout the rollout.

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