Bounded zero-dynamics conjecture for spline path following with redundancy resolution

Let $x$ denote the system state, $u$ the control input, and let $x_c$ denote the configuration variables subject to joint limits. Let $r(x_c)$ be the redundancy-resolution objective defined by

$$r(x_c)_i = -\frac{u_{i_{\max}}-u_{i_{\min}}}{x_{c_{i_{\max}}}-x_{c_{i_{\min}}}}(x_{c_i}-x_{c_{i_{\min}}})+u_{i_{\max}},$$

for $i\in\{1,\ldots,N\}$. Under the feedback-transform control law

$$u = \beta^\dagger(x)(v-\alpha(x)) + (I_{n\times n}-\beta^\dagger(x)\beta(x))r(x),$$

where $\beta^\dagger(x)=W^{-1}\beta(x)^\top(\beta(x)W^{-1}\beta(x)^\top)^{-1}$, and with redundant dynamics $\dot{\zeta}=f_\zeta(x,u)$, assume that the underlying Euler–Lagrange system has inherent damping. Bounded zero-dynamics conjecture. Using this control input and redundancy objective results in bounded redundant dynamics, while achieving path following and maintaining joint limits. This conjecture concerns the claimed behavior of the redundancy-resolution scheme: the source reports that it was tested and verified on two examples and on the experimental platform, but does not provide a general proof.