A robot navigates using a potential field where U_attractive(q) = 0.5 * ||q - q_goal||^2 and U_repulsive(q) = 1/(2*d_obs^2) where d_obs is distance to nearest obstacle. The robot is at position q = [1, 1], the goal is at q_goal = [2, 2], and the nearest obstacle is 0.5 meters away. What is the gradient direction the robot should move?
AToward the goal at 45 degrees; the attractive force dominates and the repulsive force is negligible
BAway from the obstacle first, then toward the goal; the large inverse-squared repulsive gradient dominates
CThe exact direction requires computing -∇U, balancing both attractive and repulsive gradients; it depends on the relative magnitudes of the gradients at this location
DThe robot cannot move because the attractive and repulsive forces are in conflict
This reflects the fundamental design trade-off in robotics: simple, fast local methods (potential fields) versus slower, complete global methods (RRT, PRM, grid-based A*). Modern systems typically layer them — global planner provides waypoints, local potential field controller reactively guides between waypoints while avoiding immediate obstacles. This hybrid approach gets computational speed of potential fields with the completeness of global planning.