In this work, we study angle-based localization and rigidity maintenance control for multi-robot networks under sensing constraints. We establish the first equivalence between angle rigidity and bearing rigidity considering \textit{directed} sensing graphs and \textit{body-frame} bearing measurements in both $2$ and $3$-\textit{dimensional space}. In particular, we demonstrate that a framework in $\mathrm{SE}(d)$ is infinitesimally bearing rigid if and only if it is infinitesimally angle rigid and each robot obtains at least $d-1$ bearing measurements ($d \in \{2, 3\}$). Building on these findings, this paper proposes a distributed angle-based localization scheme and establishes local exponential stability under switching sensing graphs, requiring only infinitesimal angle rigidity across the visited topologies. Then, since angle rigidity strongly depends on the robots' spatial configuration, we investigate rigidity maintenance control. The \textit{angle rigidity eigenvalue} is presented as a metric for the degree of rigidity. A decentralized gradient-based controller capable of executing mission-specific commands while maintaining a sufficient level of angle rigidity is proposed. Simulations were conducted to evaluate the scheme's effectiveness and practicality.

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