Skew-symmetry of the manipulator dynamics and the passivity property

The rigid-body equation of motion of an $n$-DOF serial manipulator is written as

$$M(q)\,\ddot{q} + C(q,\dot{q})\,\dot{q} + g(q) = \tau,$$

where $M(q)$ is the symmetric positive-definite inertia matrix and $C(q,\dot{q})$ is the Coriolis/centrifugal matrix.

With a *particular* (Christoffel-symbol) choice of $C$, the matrix $\dot{M}(q) - 2C(q,\dot{q})$ is skew-symmetric.

(a) Derive or justify where this skew-symmetry property comes from (relate it to the kinetic energy and the structure of $C$). (b) Explain its physical meaning in terms of energy / passivity. (c) Explain concretely how it is exploited to prove global asymptotic stability of a manipulator controller (e.g. PD-plus-gravity-compensation or a passivity-based / Slotine–Li tracking controller). (d) State clearly whether the property holds for *every* valid factorization of $C$, and what that implies for someone who computes $C$ numerically.