Efficient Learning of Mechanism Designs and Compositionality
Authors: João Pedro Gandarela, Thiago Rios, Stefan Menzel, André Freitas
Organizations: Idiap Research Institute, Switzerland · École Polytechnique Fédérale de Lausanne (EPFL), Switzerland · Honda Research Institute Europe, Germany · Department of Computer Science, University of Manchester, UK · National Biomarker Centre, CRUK-MI, University of Manchester, UK
Learning mechanism behaviour requires a representation of how connected parts jointly constrain motion. We study whether exposing reusable constraint subassemblies improves prediction from limited data. Assur-KB represents planar mechanisms through irreducible modules, boundary interfaces, and solve dependencies while retaining source geometry. Our theoretical results characterize transfer through module interfaces and establish physical limits arising from missing interface information and accumulated sensitivity. To test the representation claim directly, the experimental design compares matched mechanism-curve pairs across training budgets within the same multilayer-perceptron family. This controlled comparison establishes the utility of the complete representation in the evaluated learning setting. Assur-derived features provide a geometric-mean 1.8× advantage in sample efficiency over flattened adjacency and position inputs across budgets of 50-10,000 mechanisms. Separate experiments establish invariance under tested joint relabelings, higher valid-generation rates under dyadic composition, and a coarser structural index over 10.02 million mechanisms. In summary, the theory and experiments identify how physical constraint structure supports limited-data prediction and which interface conditions govern compositional reuse.
Figures & tables
Figure 1: Same mechanism and motion, different computational structure made explicit. (a) A driver angle moves the links and a tracer point P draws y(θ) . (b) The solve graph exposes an irreducible module G1 , its interface, and the order the local solves run in. (c) That module is then a move an agent can make, stored with the two bar lengths and the interface it needs, landing on any pair of already-solved joints, admissible by construction, and extending the solve order by one step. (d) The same record is what a predictor reads, so with the multilayer perceptron held fixed, module features and geometry reach a given error on the curve y(θ) from close to half the training data (§ 5.1 ).
Figure 2: Source formats(2x), acceptance test (1x) and record (1x). (a) Each corpus proposes modules through the oracle it already ships, so joint adjacency yields the classical dyad and link incidence reaches n -ary bodies, prismatic pairs, and boundary arity above two. (b) Both proposals then meet the same test (Eq. 3 ), a module counting as pinned-rigid exactly when three pebbles remain free and as minimal exactly when every proper subset of its internal joints is still flexible. (c) The record keeps the solve graph beside four statuses, so a structural pass never reads as motion.
Figure 3: Peeling proposes one module at a time against the part already solved. (a) Two known joints and two lengths place the unknown joint at a circle–circle intersection with generically two real roots, so a mechanism with m dyads stores m branch bits. (b) Two steps on a six-bar, where the ternary link ABC and the bar from B to ground form one module because the ternary body enters once and fixes both B and C , and the remaining dyad closes at D .
Figure 4: Solve-graph features are worth about a factor of two in training data. Mean symmetric Chamfer distance over ten stratified fold units, with the arrow reading the gain on the data axis and stars the sizes significant under the corrected paired test.
Figure 5: Solve graphs expose useful structure before any learned model is introduced. (a) Overall validity for compositional and matched adjacency sampling. (b) 10.02 million mechanisms yield fewer solve-graph fingerprints than bare adjacency hashes.
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: A planar 4-bar linkage decomposed into its driver link and an Assur dyad. Fixed pivots A and D are grounded. The dyad’s two external joints are B (position known from the driver) and D (fixed); its one internal joint is C .
Figure 7: A dyad with interface arity two (and classical order two under the one-pair-per-boundary convention). Two links, three joints (one internal, two external). Given p1 and p2 , the position of q is found by intersecting two circles, typically yielding two assembly modes.
Stored interface arity
Links n
Joints j
Int. joints
Boundary IDs
2
2
3
1
2
3
4
6
3
3
4
6
9
5
4
Appendix
Table 1: Illustrative interface arities stored by this work. The value counts distinct boundary-joint identifiers and agrees with classical Assur order only under a one-to-one boundary-pair convention. It is not classical Assur class, which is recorded only with a named convention and independent validation. The equality 3n=2j is a generic count, not an independence, assembly, or full-cycle certificate.
Figure 8: A solve graph localizes the coupled computation in a multi-group mechanism. Hollow joints are inputs and filled joints are outputs, and ground meets two moving links at the compound pin O2 . Numbers give one valid topological solve order. This is the six-bar of Figure 3 b carried through to its condensation DAG.
Figure 9: The change of representation in the higher-class regime, on a Stephenson-style mechanism. (a) The adjacency view: seven joints and a set of bars; no group structure, and the upper triangle cannot be told apart from one ternary rigid body. (b) The Assur view: the driver determines the joint above j0 , the class-II dyad G1 writes the joint between the two ground pivots, and the class-III group G2 (amber rigid body, three coupled internal joints) writes the upper three joints at once. The condensation DAG (below right) makes the cascade and the location of the coupling explicit: the rest of the mechanism factors into one small local solve per group, and only the three internals of G2 are coupled to each other.
Branch
N
A overall
B overall
A geom.
B geom.
A conn
B conn
A path
B path
A dof
B dof
theoretical
4
1.000
0.405
1.000
0.405
1.000
0.990
1.000
0.405
1.000
0.411
theoretical
5
1.000
0.204
1.000
0.204
1.000
0.974
1.000
0.204
1.000
0.277
theoretical
6
1.000
0.108
1.000
0.108
1.000
0.958
1.000
0.108
1.000
0.215
theoretical
7
1.000
0.061
1.000
0.061
1.000
0.940
1.000
0.061
1.000
0.178
theoretical
8
1.000
0.035
1.000
0.035
1.000
0.924
1.000
0.035
1.000
0.155
empirical
8
0.284
0.001
0.284
0.017
1.000
0.782
1.000
0.050
1.000
0.124
Appendix
Table 2: Experiment 1 validity rates from 50,000 samples per procedure per size. Procedure A is the compositional dyad sampler; procedure B is the matched adjacency sampler.
Quantity
Value
Mechanisms hashed
10,020,000
Distinct adjacency hashes DA
1,173,078
Distinct Assur fingerprints DF
908,186
Coarsening factor DA/DF
1.29
Mean mechanisms per Assur fingerprint
11.03
Largest Assur-fingerprint bucket
330,809
Appendix
Table 3: Experiment 2 hash and fingerprint summary. Adjacency hashes use the bare joint graph; solve-graph fingerprints use group-order counts and the order-labelled condensation DAG.
Train n
Adj. Chamfer
Assur Chamfer
Adj. − Assur
NB p
50
0.1822 ± 0.0290
0.1632 ± 0.0046
0.0190
0.4812
100
0.1602 ± 0.0038
0.1526 ± 0.0029
0.0076
0.4122
200
0.1473 ± 0.0022
0.1409 ± 0.0024
0.0064
0.3609
500
0.1359 ± 0.0022
0.1294 ± 0.0018
0.0065
0.3310
1,000
0.1274 ± 0.0014
0.1221 ± 0.0015
0.0052
0.2423
2,000
0.1196 ± 0.0008
0.1158 ± 0.0016
0.0038
0.0866
Appendix
Table 4: Experiment 3 learning curve for the 128-point output-joint coupler curve. Scores are mean symmetric Chamfer distances ± standard deviations over 10 stratified fold units. The p column is the one-sided Nadeau–Bengio-corrected paired-test probability for “Assur is better.”
Comparison
Block
Gain
Std.
Default leave-one-out
order_count
0.00032
0.00074
Default leave-one-out
dag_shape
0.00219
0.00057
Default leave-one-out
lengths
0.00022
0.00049
Default leave-one-out
branches
-0.00209
0.00061
Default leave-one-out
coupler_tri
-0.00016
0.00074
Default leave-one-out
coupler_shape
0.00017
0.00063
Appendix
Table 5: Experiment 3 feature-ablation summary at train n=5,000 over 10 fold units. Default rows report leave-one-out importance; candidate rows report add-one-block gain. Positive values mean lower Chamfer relative to the corresponding baseline.
N
Trials
Fingerprint
Assur
Assur struct.
Adjacency
4
760
1.000
1.000
1.000
0.050
5
580
1.000
1.000
1.000
0.005
6
620
1.000
1.000
1.000
0.000
7
620
1.000
1.000
1.000
0.000
8
640
1.000
1.000
1.000
0.000
9
620
1.000
1.000
1.000
0.000
Appendix
Table 6: Experiment 4 relabelling invariance: the fraction of trials left unchanged by a random joint permutation. 300 generated mechanisms, 20 permutations each. The adjacency column is the Experiment 3 flattened adjacency-plus-position baseline, which is keyed by joint index and is expected to change.
Figure 10: What local structure alone leaves unresolved. (a) Exchanging lengths between the boundary ports of a four-bar dyad changes its output arc, shown after normalizing both chords to unit length (Proposition 9 ). (b) Dyad chains with identical local Jacobian singular values have endpoint configuration sensitivities 1 and 2m at their reference poses (Theorem 7 ). Panel (a) shows why a module needs the correspondence between constraints and boundary ports; panel (b) shows why the directions in which modules transmit motion matter. Both are analytic constructions. The included script reproduces the figure and checks the constructions against circle-intersection solves.
Mechanism
Construction
Resource
Records
Typed
Pose
Coupler
Sliders,
Solve
Group
incidence
x0
curve
n -ary
order
status
LINKS
10M
×
✓
✓ †
×
×
×
Purwar Lab
3M
✓
✓ ‡
✓
✓
×
×
Assur-KB (this work)
12.99M
✓
✓
✓
✓
✓
✓
Appendix
Table 7: The two source corpora store what a mechanism is, and neither stores how it is built. Rows are LINKS ( Heyrani Nobari et al., 2022 ) , the Purwar Lab four-, six-, and eight-bar corpus ( Nurizada et al., 2024 ) , and the layer derived from them, with source counts as those publications report. Appendix E qualifies the two marked cells.
Quantity
LINKS
Purwar Lab
Total
Shards
101
304
405
Records
10,020,000
2,966,754
12,986,754
Unparsable lines
0
0
0
JSONL size (GB)
131.1
26.7
157.8
Mean extraction time (ms/record)
1.64
4.17
—
Total extraction time (CPU h)
4.57
3.43
8.00
Appendix
Table 8: Build totals per source. Extraction time is the decomposition cost recorded on each record, summed over the source.
LINKS
Purwar Lab
Quantity
min
med
mean
max
min
med
mean
max
Joints
6
13
12.91
20
5
11
9.49
11
Links
7
19
19.47
35
2
8
6.82
8
Ground joints
2
3
2.68
7
2
3
2.79
4
Groups (total)
4
10
10.23
18
1
4
3.57
5
Assur groups
3
9
9.23
17
0
3
2.57
4
Appendix
Table 9: Per-mechanism distributions. Groups include the ground group; DAG depth is the longest path in the condensation DAG.
LINKS
Purwar Lab
Quantity
count
%
count
%
Groups (total)
102,534,069
—
10,599,777
—
Ground groups
10,020,000
—
2,966,754
—
Assur groups
92,514,069
100.0
7,633,023
100.0
order 2 (dyads)
92,514,069
100.0
5,761,578
75.5
order 3
0
0.0
1,703,554
22.3
Appendix
Table 10: Module inventory by source. Percentages are shares of that source’s Assur groups (top block) or of its mechanisms (bottom block).
LINKS
Purwar Lab
Quantity
min
med
mean
max
min
med
mean
max
Internal joints
1
1
1.00
1
1
1
1.84
7
Boundary joints
2
2
2.00
2
2
2
2.27
4
Bodies
2
2
2.00
2
1
2
1.99
6
Carried points
0
0
0.00
0
0
0
0.28
2
Appendix
Table 11: Per-module distributions. Carried points are tracer points attached to a module’s links and are counted over all groups, including ground.
Shape key
Groups
% of Assur groups
c2/i1/e2/L2-3
1,975,188
25.9
c2/i2/e2/L2-3
1,577,230
20.7
c3/i1/e3/L3
969,112
12.7
c2/i1/e2/L2-2
787,543
10.3
c2/i3/e2/L3-3
538,901
7.1
c2/i3/e2/L2-4
338,516
4.4
Appendix
Table 12: The ten most frequent Purwar Lab shape keys, covering 90.1% of its Assur groups.
Quantity
LINKS
Purwar Lab
Bodies
195,048,138
20,698,934
binary
195,048,138
10,321,008
ternary
0
8,938,010
quaternary
0
1,439,916
n -ary bodies ( n>2 )
0
10,377,926
Mechanisms with an n -ary link
0
2,966,754
Appendix
Table 13: Typed incidence in the build. Body and joint counts are instances summed over mechanisms, not distinct types.
We present Linkify, a framework for learning from interface-augmented assembly graphs to enable context-aware part retrieval in mechanical assemblies. While recent generative AI methods for CAD have focused largely on isolated parts or monolithic assemblies, the rich geometric information at the interfaces between parts, where function is realized, remains underexplored. We address this gap by recomputing high-fidelity interface geometry for the Fusion 360 Gallery Assembly dataset, correcting missing and erroneous contacts, and generating point-cloud representations of local contact regions. Using this data, we construct assembly graphs whose nodes encode part geometry and whose edges encode interface geometry via a pretrained point-cloud encoder. On top of this representation, we train a Graph Attention Network based on GATv2 to solve a masked part prediction task: given an assembly with one part held out, the model predicts the class of the missing component from a large vocabulary of geometrically clustered parts, thereby approximating a realistic part-retrieval scenario. Compared to non-graph baselines such as logistic regression and k-nearest neighbors operating on aggregated node features, Linkify achieves higher Top-K accuracy and F1 scores. Ablation studies on graph connectivity, edge attributes, and attention mechanisms demonstrate that accurate contact computation and dynamic attention over interfaces are critical for performance. Our corrected interface dataset and training pipeline, released publicly, provide a foundation for future interface-aware models for assembly retrieval, validation, and generative design.
Anushrut Jignasu, Daniele Grandi
Iowa State University, USA · Autodesk Research, USA
Deep learning has shown strong potential for scientific discovery, but its ability to model macroscopic rigid-body kinematic constraints remains underexplored. We study this problem on spatial over-constrained mechanisms and propose O-ConNet, an end-to-end framework that infers mechanism structural parameters from only three sparse reachable points while reconstructing the full motion trajectory, without explicitly solving constraint equations during inference. On a self-constructed Bennett 4R dataset of 42,860 valid samples, O-ConNet achieves Param-MAE 0.276 +/- 0.077 and Traj-MAE 0.145 +/- 0.018 (mean +/- std over 10 runs), outperforming the strongest sequence baseline (LSTM-Seq2Seq) by 65.1 percent and 88.2 percent, respectively. These results suggest that end-to-end learning can capture closed-loop geometric structure and provide a practical route for inverse design of spatial over-constrained mechanisms under extremely sparse observations.
Haoyu Sun, Meng Zhao, Tianhao Wang +1
School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China.
Planar path synthesis requires mechanisms whose coupler curves match a prescribed trajectory; the mapping from curve to linkage is inherently one-to-many across four-, six-, and eight-bar topologies. We address this design problem with simulation-grounded evaluation on a curated corpus of over one million mechanisms, reporting Chamfer distance and dynamic time warping after forward kinematics and geometric alignment. We formulate synthesis as conditional autoregressive sequence modeling: joint coordinates are uniformly quantized to tokens and generated by a decoder-only transformer with a variational-autoencoder (VAE) latent of the target curve and an explicit mechanism-type token. Training combines token cross-entropy with a Gaussian-smoothed bin auxiliary loss that respects ordinal structure among bins. At inference, a bounded latent-noise schedule decodes all mechanism types at each noise level; we retain the top five candidates by geometric error, yielding diverse accurate families without dataset lookup. On held-out tests, aggregate mean Chamfer distance is 0.0132 and mean dynamic time warping is 0.153; a latent k-nearest-neighbor baseline that conditions on training-set neighbor latents in VAE space achieves matched-topology mean Chamfer distance 0.0071 and mean dynamic time warping 0.117 using the same decoder.
Anar Nurizada, Anurag Purwar
Computer-Aided Design and Innovation Lab, Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794-2300, USA