A Constraint Embedding Approach for Dynamics Modeling of Parallel Kinematic Manipulators with Hybrid Limbs

The paper explicitly derives the local constraint embedding and the block maps H and Ḣ (eqs. (17)–(20)) and states the submersion-based parameterization for each fundamental cycle, matching the model’s part (a) . The inner Newton loop (Algorithm 2) and the nested/compound IK schemes (Algorithms 1 and 3) in the paper align with the model’s part (b) descriptions . The reduction of the limb EOM via virtual power (yielding M̄̄ = H̄ᵀ M̄ H̄ and C̄̄ = H̄ᵀ(M̄ Ḣ̄ + C̄ H̄)) agrees with part (c) . However, in assembling task-space dynamics, the paper gives C_t = Σ_l F̄ᵀ( C̄̄ F̄ + M̄̄ Ḟ ) + P_pᵀ G_p M_p P_p (eq. (41)) , while the model states C_t = P_pᵀ G_p M_p P_p + Σ_l(F̄ᵀ C̄̄ F̄ − F̄ᵀ M̄̄ F̄ L̇_t), which both changes the sign and drops the required L_t^{-1} factor implied by Ḟ = −L_t^{-1} L̇_t F̄. This is dimensionally inconsistent and contradicts the paper’s formula. Hence the paper’s derivation is correct, and the model’s part (d) is incorrect in the Coriolis/centrifugal term.