Computer-Aided Design of Rational Motions for 4R and 6R Spatial Mechanism Synthesis

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.

Configuration space~(C-space) of a mechanism is a real variety describing the set of feasible configurations that it can attain. To understand the behavior of a mechanism, it is crucial to identify and scrutinize especially the singular points of its C-space. They usually appear when the variety intersects itself, leading to different branches of motion. There exist many approaches to detect those intersections if they are transversal. However, the problem remains challenging if there are tangential, cuspidal, inter-dimensional or a combination of these intersections. This paper exploits an approach acquired from tropical geometry to analyze the neighborhood of any point on C-spaces of 1-degree-of-freedom~(dof) mechanisms. This is done by finding the approximate rational parametrization of the curve(s) passing through the given point using Puiseux series. The proposed approach is shown to succesfully detect the transversal branchings in two foldable four bar mechanisms and a cusp in the configuration curve of the double Watt mechanism.

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.