Reduced coordinates

Steel’s state is the joint coordinates, State.joint_q and State.joint_qd, as for SolverFeatherstone. Tree joints hold exactly, and the diagonal of a contact row’s effective mass, $J_i\tilde H^{-1}J_i^\top$, already contains the full articulated inertia behind the contact: a fingertip contact feels the arm, the base and the implicit drives. What is left for the iterative solve is the coupling between contacts and the remaining joint-space rows (limits, drive rows, mimic and loop closures), a regime projected Gauss-Seidel handles well.

Figure 2Reduced against maximal coordinates.

The same chain with a heavy tip, released from horizontal. Left: the state is the joint angles, so the links keep their length by construction. Right: every link is a free body and each joint is a constraint row solved by Gauss-Seidel at the chosen iteration budget; the heavy tip stretches the chain, most at low iterations. A toy model of the idea, not Steel’s code. Tutorial: Joint types · Floating base · Mass ratio

Practical consequences:

• Write start states to joint coordinates. A pose written only to body_q or a velocity only to body_qd is overwritten on the first step. Set joint_q/joint_qd on the builder before finalize(), or compute them with newton.eval_ik. After writing joint coordinates into a state yourself, run newton.eval_fk before the first collision query so body poses match.
• Free bodies need a free joint. add_body adds one. Free-joint joint_qd is (v_com, omega) in the frame of the joint’s parent anchor, the world frame for an identity anchor.
• Bodies on no joint keep their body_q, enter contacts with zero velocity, and pass through to the output state.
• The cost moves to the front half. Mass-matrix assembly and factorization grow with a tree’s DOF count. Steel groups articulations by size, uses sparse factors that follow the tree when every articulated group shares one topology of at most 64 DOFs and the configuration allows it, and skips the factorization for articulations whose DOFs do not couple. parallel_tree and use_parallel_streams add parallelism across branches and across size groups.