The Road to Learning Explainable Inverse Kinematic Models: Graph Neural Networks as Inductive Bias for Symbolic Regression

TL;DR

The paper tackles the IK problem for robotic manipulators with varying link lengths by learning IK with a Graph Neural Network (GNN) that serves as an inductive bias for subsequent symbolic regression (SR). It introduces GraphMatic, a graph-based framework where manipulators are modeled as graphs and IK is learned via Message Passing Neural Networks, with direct-estimation (DE) and reference-guided (RG) formulations, and local vs full connectivity variants. Through extensive synthetic data generation using forward kinematics and collision checks across 3, 5, and 6 DOF, the study shows that DE succeeds for simple 3 DOF but struggles for higher DOF, while RG achieves high accuracy and near-unit $R^2$ for 3–5 DOF, with extrapolation challenges at 6 DOF. The authors argue that GraphMatic can provide a powerful inductive bias to drive SR-based discovery of analytical IK equations, addressing explainability and extrapolation concerns, and outline future work to enhance generalization and reliability for real-world deployment. The IK problem is defined as finding $oldsymbol{ heta}=g^{-1}(oldsymbol{X})$ from the end-effector pose $oldsymbol{X}=[oldsymbol{p},oldsymbol{o}]^ op$, where FK gives $oldsymbol{X}=g(oldsymbol{ heta})$, and the approach integrates GNNs with SR to enable explainable, data-driven IK across manipulators within the same DOF family but varying link lengths.

Abstract

This paper shows how a Graph Neural Network (GNN) can be used to learn an Inverse Kinematics (IK) based on an automatically generated dataset. The generated Inverse Kinematics is generalized to a family of manipulators with the same Degree of Freedom (DOF), but varying link length configurations. The results indicate a position error of less than 1.0 cm for 3 DOF and 4.5 cm for 5 DOF, and orientation error of 2$^\circ$ for 3 DOF and 8.2$^\circ$ for 6 DOF, which allows the deployment to certain real world-problems. However, out-of-domain errors and lack of extrapolation can be observed in the resulting GNN. An extensive analysis of these errors indicates potential for enhancement in the future. Consequently, the generated GNNs are tailored to be used in future work as an inductive bias to generate analytical equations through symbolic regression.

Figures (6)

• Figure 1: Graph projected on a manipulator with 5 DOF. Nodes $v_i$ and $v_j$ are connected by an edge $e_{i,j}$. Solid lines indicate neighbourly connections between nodes. Dashed lines indicate the additional edges for full connectivity, using node $v_2$ as an example to avoid visual clutter. More edges are added for all other nodes in the same way.
• Figure 2: Error distribution of the $5\%$ test samples with the highest error of RG-F-3-2-22.
• Figure 3: Orientation error distribution analysis for RG-F-3-2-22
• Figure 4: Error distribution of the $5\%$ test samples with the highest error of RG-N-5-4-35.
• Figure 5: Distribution of the $5\%$ largest end-effector position errors for 6 DOF RG on the test set.