Sub Riemannian geodesics have constant speed

Question

I saw many times that Riemannian geodesics have constant speed. To see this, one considers the Levi Civita connection $\nabla$, associated with the Riemannian metric $g$ on $M$ and notes that $$\frac{d}{dt}g(\dot{\gamma},\dot{\gamma})=2g(\nabla_{\dot{\gamma}}\dot{\gamma},\dot{\gamma})=0.$$ I was wondering if this is the case also in the sub Riemannian setting. Given a sub Riemannian cometric $g^*$ and Hamiltonian $H=\frac{1}{2}\|\cdot\|_{g^*}$, is it true that normal (or maybe also singular/abnormal) sR geodesics on $TM^*$ have constant speed? It is a common procedure to look at geodesics in the sphere bundle $STM^*$ but I don't know how it is always possible to reparametrize the curve to be in $STM^*$.