A Geometric Task-Space Port-Hamiltonian Formulation for Redundant Manipulators

We present a novel geometric port-Hamiltonian formulation of redundant manipulators performing a differential kinematic task $η=J(q)\dot{q}$, where $q$ is a point on the configuration manifold, $η$ is a velocity-like task space variable, and $J(q)$ is a linear map representing the task, for example the classical analytic or geometric manipulator Jacobian matrix. The proposed model emerges from a change of coordinates from canonical Hamiltonian dynamics, and splits the standard Hamiltonian momentum variable into a task-space momentum variable and a null-space momentum variable. Properties of this model and relation to Lagrangian formulations present in the literature are highlighted. Finally, we apply the proposed model in an \textit{Interconnection and Damping Assignment Passivity-Based Control} (IDA-PBC) design to stabilize and shape the impedance of a 7-DOF Emika Panda robot in simulation.

💡 Research Summary

The paper introduces a novel geometric port‑Hamiltonian (pH) framework for fully actuated redundant manipulators that are required to perform a differential kinematic task η = J(q) · q̇, where q belongs to the configuration manifold Q, J(q) is a linear map (e.g., the analytic or geometric Jacobian), and η is a task‑space velocity‑like variable. Starting from the standard Lagrangian dynamics M(q) q̈ + h(q, q̇) + ∂q V(q) = τ, the authors first rewrite the system in the canonical pH form with state (q, p) and Hamiltonian H = K + V, where p = M(q) q̇. They then examine the geometric structure induced by the task map J(q). Because rank J = m ≤ n, the joint velocity space T_qQ can be orthogonally decomposed into the task‑space component v ∈ Ker⊥(J) and the null‑space component ν ∈ Ker(J). Using the dynamically consistent pseudo‑inverse J# M = M⁻¹Jᵀ(J M⁻¹Jᵀ)⁻¹, the task‑space velocity v is expressed as v = J# M η, leading to the definition of the task‑space inertia matrix Λ = (J M⁻¹Jᵀ)⁻¹. This matrix coincides with the well‑known operational‑space inertia (or mobility) tensor and captures the effective inertia seen at the end‑effector.

On the dual side, the generalized force τ is split into a task‑space component τ_F ∈ Im(Jᵀ) and a null‑space component τ₀ ∈ Ker(J M⁻¹) by exploiting the annihilator spaces Ann(Ker⊥(J)) = Ker(J M⁻¹) and Ann(Ker(J)) = Im(Jᵀ). This decomposition yields a clean power balance τᵀq̇ = τ_Fᵀv + τ₀ᵀν, separating the power exchanged with the task from that residing in the null‑space. The authors stress that the null‑space component is metric‑dependent, which is essential for preserving orthogonality of the kinetic energy terms.

To embed these structures into a pH model, the paper introduces an extended task velocity η_e = Ĵ(q) · q̇, where Ĵ(q) =