Section 5

API reference

Public functions and classes, by module. All array arguments are NumPy float64, and twists are shape (6,) in the paper's convention ξ = (ω, v).

Types · `articulatearena.types`

`Twist`	Immutable shape-(6,) twist wrapper, raw array via `.data`.
`EndpointPair`	Unordered pair `{u, v}` of endpoint twists.
`Joint`	Unit screw twist `xi_hat`, limits `a`, `b` (±inf for continuous), and `joint_type`.
`LinkInertia`	Mass, center of mass, and 3×3 rotational inertia of a link.
`InnerProduct`	Protocol: callable from a shape-(6,) twist to its squared norm.

Representation · `new_equation.representation`

`screw_twist(axis_dir, point, joint_type, pitch=0)`	Unit screw twist of a one-DOF joint from a world-frame axis and a point on it.
`make_endpoint_pair(xi_hat, a, b)`	The unordered endpoint pair {a·ξ, b·ξ} (code names the limits `a`, `b` = paper's l⁻, l⁺).
`joint_to_endpoint_pair(joint)`	Endpoint pair of a typed `Joint`.

Joint metric · `new_equation.E`

`joint_metric(u1, v1, u2, v2, inner_product)`	The quotient metric E, the ℤ₂-swap minimum over endpoint pairings.
`training_loss(u1, v1, u2, v2, inner_product, epsilon=1e-9)`	Smooth swap-invariant surrogate for E² (differentiable objective).

Norms · `new_equation.inner_product`

`make_split_norm(alpha)`	Split norm ‖Δω‖² + α²‖Δv‖².
`make_kinetic_energy_norm(inertia)`	Kinetic-energy norm from a `LinkInertia`.
`make_riemannian_norm(metric)`	Norm from a user-supplied SPD 6×6 matrix.

Continuous joints · `new_equation.continuous`

`radial_compactify(u, kappa, inner_product=None)`	Radial map ϕ into the open unit ball.
`continuous_joint_repr(xi_hat, inner_product=None)`	Boundary endpoint pair of an unbounded joint.
`compactified_joint_metric(u1, v1, u2, v2, inner_product, kappa)`	E∘ϕ, the compactified metric.
`interval_term(a1, a2)`	Extended-real distance between limit values.

Tree metric · `new_equation.tree`

`cost_to_fixed(pair, inner_product)`	Contraction/expansion cost E(J, J₀).
`assignment_tree_distance(pairs1, pairs2, inner_product, unmatched_cost=None)`	Assignment relaxation over joint multisets (Hungarian, padded matrix).
`contract_small_edges(edges, inner_product, tol=1e-12)`	Contract exactly-fixed (zero-motion) edges of a labeled tree.
`assignment_matching(pairs1, pairs2, inner_product)`	Relaxation value together with its optimal matching.
`matching_realizable(edges1, edges2, matching)`	Linear-time check that a matching extends to an isomorphism of the contracted trees.
`exact_tree_distance(edges1, edges2, inner_product)`	Exact motion-aware tree edit distance. Returns `(distance, certified)`.

Skeleton distance · `new_equation.skeleton`

`topology_penalty(num_unmatched, num_relation_mismatches, lambda_topology, relation_weight=1)`	Constant-λ penalty for surplus joints and relation mismatches.
`skeleton_distance(cost_matrix, lambda_topology, ...)`	Assigned E cost plus topology penalty. Returns `(row_ind, col_ind, total)`.

Material motion · `new_equation.material_motion`

`se3_exp(xi_hat, theta)`	Rigid transform exp(θ ξ̂).
`config_discrepancy(...)`, `finite_joint_discrepancy(...)`	RMS displacement discrepancy of material points swept along two joints (used to calibrate E against physical motion).

Background math · `general_equation`

`exp_map.exp_map(xi_hat, q=1)`	se(3) exponential map.
`assignment.assign(cost_matrix)`	Hungarian optimal assignment.
`tree_edit.tree_edit_distance(...)`	Classical ordered tree edit distance (Zhang-Shasha) on `TreeNode`s.
`configuration.configuration_difference(...)`	L² configuration difference (Park 1995).

I/O · `read` and `write`

`read.config_loader.load_config(path=None)`	Flatten a grouped YAML config into the `Config` dataclass. Defaults to `configs/default.yaml`.
`read.urdf_loader.load_joint(urdf_path, joint_name)`	Typed `Joint` from a URDF, with the axis resolved to the world frame by forward kinematics.
`read.inertia_loader.load_link_inertia`, `load_joint_child_inertia`	`LinkInertia` of a link / of a joint's child link, for the kinetic-energy norm.
`read.case_loader.load_case`, `load_case_joints`	Prediction/ground-truth evaluation case referencing two URDF joints.
`write.report.Report_Generator`	JSON / Markdown evaluation reports.

Baselines · `previous_metrics.component_scores`

Component scores of prior evaluation templates (type, axis, origin, and limit errors with thresholded joint_success), kept for reproducing §3-style baseline numbers.

API reference

Types · articulatearena.types

Representation · new_equation.representation

Joint metric · new_equation.E

Norms · new_equation.inner_product

Continuous joints · new_equation.continuous

Tree metric · new_equation.tree

Skeleton distance · new_equation.skeleton

Material motion · new_equation.material_motion

Background math · general_equation

I/O · read and write

Baselines · previous_metrics.component_scores