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We develop and commercialize autonomous machines, such as logistic robots and self-driving cars, around the globe. A critical challenge to our -- and any -- autonomous machine is accurate and efficient localization under resource constraints, which has fueled specialized localization accelerators recently. Prior acceleration efforts are point solutions in that they each specialize for a specific localization algorithm. In real-world commercial deployments, however, autonomous machines routinely operate under different environments and no single localization algorithm fits all the environments. Simply stacking together point solutions not only leads to cost and power budget overrun, but also results in an overly complicated software stack. This paper demonstrates our new software-hardware co-designed framework for autonomous machine localization, which adapts to different operating scenarios by fusing fundamental algorithmic primitives. Through characterizing the software framework, we identify ideal acceleration candidates that contribute significantly to the end-to-end latency and/or latency variation. We show how to co-design a hardware accelerator to systematically exploit the parallelisms, locality, and common building blocks inherent in the localization framework. We build, deploy, and evaluate an FPGA prototype on our next-generation self-driving cars. To demonstrate the flexibility of our framework, we also instantiate another FPGA prototype targeting drones, which represent mobile autonomous machines. We achieve about 2x speedup and 4x energy reduction compared to widely-deployed, optimized implementations on general-purpose platforms.