Resource-limited robots face significant challenges in executing computationally intensive tasks, such as locomotion and manipulation, particularly for real-time optimal control algorithms like Model Predictive Control (MPC). This paper provides a comprehensive design space exploration to identify optimal hardware computation architectures for these demanding model-based control algorithms. We profile and optimize representative architectural designs, including general-purpose scalar CPUs, vector processors, and specialized accelerators. By characterizing kernel-level benchmarks and end-to-end robotic scenarios, including a hardware-in-the-loop evaluation on a fabricated RISC-V multi-core vector SoC, we present a quantitative comparison of performance, area, and utilization across distinct architectural design points. Our findings demonstrate that targeted architectural modifications, coupled with deep software and system optimizations, enable up to 3.71x speedups for MPC, resulting in up to 27% system-level power reductions while completing robotic tasks. Finally, we propose a code generation flow designed to simplify the complex engineering effort required for mapping robotic workloads onto specialized architectures.