In the tutorial

The loop relies on three conditions: robot motion must deform the object, the selected features must change as the object deforms, and the local quasistatic relation between robot motion and shape must remain well defined. The factorization

\vJ=\frac{\partial \vs}{\partial \vx}\,\frac{\partial \vx}{\partial \vq}

separates the first two. A controllability failure, or loss of control authority, occurs when \partial\vx/\partial\vq loses rank, so that robot motion along some direction no longer deforms the object; an observability failure, or loss of shape sensitivity, occurs when a motion of the object lies in the null space of \partial\vs/\partial\vx. Because the loop estimates only the product \vJ, both produce the same measured pairs, \dq_{k-1}\neq\mathbf{0} with \ds_k\approx\mathbf{0}, although their remedies differ.

Controllability. The corresponding singular value approaches zero: \sigma_{\min}\to0, w\to0, and the condition number grows, so the factor 1/\sigma_{\min} in the pseudoinverse produces large robot commands along a direction in which robot motion barely deforms the object. The damped least-squares inverse \vJh^{\top}(\vJh\vJh^{\top}+\mu^2\mathbf{I})^{-1} limits the gain of each inverted direction to at most 1/(2\mu). In the running example, as the rod is pulled taut, \sigma_{\min} of the exact Jacobian falls from 0.15 to 0.04 at 20% extension, w by a factor of 21, the condition number rises from 16 to 42, and \vpsi_{\min}, a unit vector, becomes increasingly a wrist rotation: the squares of its two wrist entries sum to 0.68 at the rest length and to 0.78 at 20% extension.

Observability. Changes in the object are not reflected in the measured features, because of the feature representation itself (an invariance) or of perception (e.g., occlusion). The consequences range from negligible, such as an unobserved rigid translation with curvature-based features, to mechanically critical: unobserved stretching can reduce control authority, and an unobserved axial twist of an untextured rod can accumulate torsional strain until the rod forms a loop (pigtailing), a mechanical instability.

Mechanical instability. A strictly stable equilibrium minimizes the potential energy \Pi(\vx;\vq) with a positive-definite Hessian \boldsymbol{\Pi}_{\vx\vx}. As the object is loaded, its stiffness along one deformation mode may decrease until the smallest Hessian eigenvalue vanishes,

\lambda_{\min}\!\big(\boldsymbol{\Pi}_{\vx\vx}\big)\;\to\;0,

Beyond this critical point, snap-through may produce a discontinuous transition, or under buckling the object follows another branch; the local relation \dot{\vs}=\vJ\vu fails at the transition. A very small \dq_{k-1} may then produce a large \ds_k and an abnormally large Broyden correction: the loop should reject such samples and re-estimate after the transition. In the running example, the smallest Hessian eigenvalue of the straight rod crosses zero at 5.8% compression.

Failure	Underlying cause	Observable signature	Mitigation
Controllability	\partial\vx/\partial\vq loses rank	weak singular directions; \dq\neq\mathbf{0}, \ds\approx\mathbf{0}	damped inverse; monitor \sigma_{\min} and w
Observability	motion in \Null(\partial\vs/\partial\vx)	weak singular directions; \dq\neq\mathbf{0}, \ds\approx\mathbf{0}	change feature map or viewpoint
Mechanical instability	quasistatic equilibrium loses regularity	small \norm{\dq} with large or poorly predicted \norm{\ds}	gate updates; re-estimate or avoid the transition

Try this

Controllability: drag the extension from 0 to 20% and observe \sigma_{\min}, w, the condition number, and the wrist share of \vpsi_{\min}.
Compare the pseudoinverse gains 1/\sigma_l with the damped gains \sigma_l/(\sigma_l^2+\mu^2), capped at 1/(2\mu).
Observability: run the three motions. The wrist motion of the taut rod and the rigid translation seen through curvature-based features give the same observable signature.
Mechanical instability: compress past 5.8%, where the straight rod's \lambda_{\min} crosses zero. Then rotate the wrists of the arch slowly: \lambda_{\min} of the current equilibrium approaches zero and the rod snaps through, with a spike of the response ratio.

The singular values of \vJh_k, the prediction error \vr_k, and the response ratio \norm{\ds_k}/\norm{\dq_{k-1}} can indicate these failures, but none of them separates ineffective actuation from an uninformative feature representation; that requires information from outside the Jacobian estimate.