Consider the bounded signals $e_1(t)$ and $z_2(t)$, for which the following hold:
$$ ||e_1(t)||_\infty \leq ||e_1(t)||_2 \leq ||e_1(0)||_2$$ $$ ||z_2(t)||_\infty \leq ||z_2(t)||_2 \leq ||z_2(0)||_2$$
Now consider the following signal:
$$ u(t) = \frac{m}{c_\phi \cdot c_\theta} \cdot \big[g - \ddot{r} + k_1z_2 - k_1^2e_1 + e_1 + k_2z_2\big] $$
where $m > 0$ corresponds to mass, $g$ is the gravity acceleration, $||\ddot{r}|| \leq C$, with $C>0$ being a known constant and $k_1,k_2>0$ are design parameters (controller gains) that can be selected. Assume also that the angles $|\phi|, |\theta| < \frac{\pi}{2}$ in order for $u(t)$ to be bounded ($c_\phi$ and $c_\theta$ correspond to $\cos(\phi)$ and $\cos(\theta)$, respectively).
I would like to investigate whether a condition $||u(t)||_\infty \leq u_{max} $ with $u_{max} > 0$ being known can hold. To do so, I followed the below analysis:
$$ \begin{equation} c_\phi c_\theta u = m\cdot [g - \ddot{r} + k_1z_2 -k_1^2e_1+ e_1 + k_2z_2] \Rightarrow \\ \Rightarrow ||c_\phi c_\theta u||_\infty \leq m\cdot \big[g + C + ||z_2(0)||_2\cdot (k_1 + k_2) + ||e_1(0)||_2\cdot (k_1^2 + 1)\big] \ \ \ (1) \end{equation} $$
However, the following holds: $$ ||c_\phi c_\theta u||_\infty \leq ||u||_\infty \ \ \ \ \big(\text{if } ||c_\phi||_\infty = ||c_\theta||_\infty = 1\big) $$
So, if the following holds: $$ ||u||_\infty \leq m\cdot \big[g + C + ||z_2(0)||_2\cdot (k_1 + k_2) + ||e_1(0)||_2\cdot (k_1^2 + 1)\big] $$
Then, $(1)$ also holds. Assuming that $||e_1(0)||_2,||z_2(0)||_2$ can be computed and by properly selecting the values of $k_1,k_2$, the following inequality holds (for a given $u_{max}$): $$ m\cdot \big[g + C + ||z_2(0)||_2\cdot (k_1 + k_2) + ||e_1(0)||_2\cdot (k_1^2 + 1)\big] \leq u_{max} $$
And as a result: $$ ||c_\phi c_\theta u||_\infty \leq ||u||_\infty \leq u_{max} $$
I would like some feedback on whether the analysis, and thus the concluding result, is correct or not. If not, where is the missing point ?
A sum of bounded signals is bounded. So, consider
$$u(t)=\sum_{i=1}^n\alpha_i v_i(t)$$ where $\alpha_i\in\mathbb{R}$ and $||v_i(t)||_\infty\le \bar v_i$. From the triangle inequality, we get that
$$ \begin{array}{rcl} ||u(t)||_\infty&=&\left|\left|\sum_{i=1}^n\alpha_i v_i(t)\right|\right|_\infty\\ &\le&\sum_{i=1}^n|\alpha_i|\cdot||v_i(t)||_\infty\\ &\le&\sum_{i=1}^n|\alpha_i|\bar v_i. \end{array} $$
So, if we apply to your case, we have that $$ \begin{array}{rcl} ||u(t)||_\infty&=&\left|\left|\frac{m}{c_\phi c_\theta} \cdot \big[g - \ddot{r} + k_1z_2 - k_1^2e_1 + e_1 + k_2z_2\big]\right|\right|_\infty\\ % &\le&\left|\frac{m}{c_\phi c_\theta}\right|\left(g+||\ddot{r}(t)||_\infty+|k_1+k_2|\cdot||z_2(t)||_\infty+|1-k_1^2|\cdot||e_1(t)||_\infty\right)\\ &\le&\left|\frac{m}{c_\phi c_\theta}\right|\underbrace{\left(g+C+|k_1+k_2|\cdot||z_2(0)||_2+|1-k_1^2|\cdot||e_1(0)||_2\right)}_{\mbox{$v_\max$}} \end{array} $$
This bound is fine if the angles $\theta$ and $\phi$ are constant. If they are time-varying, then we need to assume that $|\theta|\le a<\pi/2$ and $|\phi|\le a<\pi/2$. In such a case, we have that
$$ \begin{array}{rcl} ||u(t)||_\infty&=&\underbrace{\frac{m}{\cos(a)^2}\left(g+C+|k_1+k_2|\cdot||z_2(0)||_2+|1-k_1^2|\cdot||e_1(0)||_2\right)}_{\mbox{$u_\max$}}. \end{array} $$
The issue is that if we allow $\theta$ or $\phi$ to be arbitrarily close to $pi/2$, then there is no upper-bound on the norm of the signal. If the angles remain bounded away from this value, we can find the upper-bound shown above.