Urban service robots require route planners that are not only collision-free but also consistent with the energetic behavior of differential drive locomotion. Conventional grid planners such as A* are efficient and reliable for geometric navigation, but their usual cost structure prioritizes path length and does not explicitly account for heading changes, concentrated turns, or localization risk near obstacles. This study presents an energy-aware route-planning formulation for differential-drive mobile robots operating in dense polygonal urban environments. The path is encoded through twelve continuous internal waypoints and is evaluated using an interpretable energy proxy that combines translational distance, cumulative absolute rotation, squared rotation, and a clearance-dependent localization risk term. Collision avoidance, boundary compliance, and maximum turn feasibility are handled through a feasibility-first dominance rule, and the resulting constrained problem is solved using a Genetic Algorithm and Particle Swarm Optimization. A* with clearance inflated occupancy grids is included as a deterministic baseline. The final experiments used a dense urban scenario with sixteen polygonal obstacles, an A* grid resolution of 0.10 m, a robot radius of 0.20 m, a safety clearance of 0.10 m, 80 GA individuals, 80 PSO particles, 100 iterations, and 10 independent runs. All methods achieved a 100% feasibility rate. PSO obtained the lowest average energy proxy, 18.505, compared with 18.645 for A* and 18.645 for GA, and reduced average rotation from 4.136 rad to 4.079 rad. However, A* remained much faster, 5.860 s on average, compared with 174.574 s for GA and 175.869 s for PSO. The ranking analysis shows that GA produced the best aggregate score when route quality, time, clearance, and feasibility were weighted equally, while PSO produced the best route quality. External perturbation tests indicate that open-loop execution in narrow corridors is sensitive to bias and waypoint noise, which motivates closed-loop tracking, online replanning, and physical validation in future work.