Potential Field Path Planning

Why potential fields matter

Almost every mobile robot faces the same real-time question thousands of times a second: given where I am, where the goal is, and what my sensors see right now, which way should I move? A search-based planner answers that by reasoning about the whole map. A potential field answers it with a single vector-arithmetic step — no graph, no tree, no map required. That combination of near-zero compute cost and direct sensor coupling is why the method has survived four decades in production code.

• Real-time obstacle avoidance. Khatib built it precisely so a manipulator could dodge a moving object between servo cycles, at rates a global planner cannot match.
• Sensor-direct. A LiDAR or sonar return is itself a repulsive source; the field can be computed straight from raw range data without building a map first.
• Local layer of a hybrid stack. On real platforms it (or its close cousin the Dynamic Window Approach) is the fast local planner underneath a global route from A*, Dijkstra, or an RRT.
• Swarms and formations. Inter-robot repulsion plus goal attraction gives flocking and collision-free formation keeping almost for free.
• Smooth commands. The gradient is a continuous vector, so the output is a naturally smooth velocity rather than a jagged waypoint list.
• Teaching clarity. "Roll downhill toward the goal, but the obstacles are bumps" is the most intuitive planner there is — a gateway to configuration-space thinking.

How it works, step by step

Model the robot as a point in configuration space at position q (in the plane, q = (x, y)). Build a scalar field U(q) as the sum of an attractive part and a repulsive part, then move opposite to its gradient.

1. Attractive potential. A bowl centered on the goal q_goal. A parabolic (quadratic) well U_att = ½·ζ·‖q − q_goal‖² gives a spring-like pull; far from the goal engineers switch to a conic well so the force does not grow without bound.
2. Repulsive potential. Around each obstacle, a barrier that is zero beyond an influence radius ρ₀ and rises sharply as the distance ρ to the obstacle shrinks — the field ignores obstacles you are nowhere near.
3. Superposition. U(q) = U_att(q) + Σ U_rep,i(q). Fields simply add, which is why the method scales trivially with obstacle count.
4. Take the gradient. The artificial force is F(q) = −∇U(q) = F_att + Σ F_rep,i. It points "downhill," toward the goal but bent away from nearby obstacles.
5. Command the robot. Treat F as a desired velocity (v = k·F) or, in Khatib's operational-space form, as a force/acceleration on the manipulator's end-effector. Clamp to the robot's speed limit.
6. Repeat every control tick. Re-read the sensors, recompute F, move. No state is carried between steps — the planner is memoryless and purely reactive.

The governing equations

The attractive force from a quadratic well, capped by a switch distance d* to a conic profile beyond it:

F_att(q) = −ζ (q − q_goal) for ‖q − q_goal‖ ≤ d*, and F_att(q) = −ζ d* (q − q_goal) / ‖q − q_goal‖ beyond it.

The repulsive force from a single obstacle, active only inside the influence radius:

F_rep(q) = η ( 1/ρ − 1/ρ₀ ) (1/ρ²) ∇ρ when ρ ≤ ρ₀, and F_rep(q) = 0 when ρ > ρ₀.

The commanded motion is the superposed, negated gradient:

F(q) = F_att(q) + Σ_i F_rep,i(q) = −∇U(q),   v_cmd = k · F(q).

Symbols, meaning, and typical units

Symbol	Meaning	Typical value / units
q, qgoal	Robot and goal position in configuration space	metres (m), planar (x, y)
U(q)	Total scalar potential	arbitrary "energy" units (J-like)
F = −∇U	Artificial force / motion vector	N (or unitless direction · gain)
ζ (zeta)	Attractive gain (spring constant)	~0.5–5 N/m
η (eta)	Repulsive gain	~0.1–2 N·m³ (scenario-tuned)
ρ	Shortest distance from robot to obstacle surface	metres (m)
ρ₀	Obstacle influence radius (repulsion cutoff)	~0.3–1.5 m
d*	Switch distance, quadratic → conic well	~1–3 m
k	Force-to-velocity scaling gain	tuned per platform

Worked example: the force at a point

Take a planar robot at q = (0, 0) m, a goal at q_goal = (10, 0) m, and one circular obstacle whose nearest surface point is 0.8 m away at bearing directly toward the goal. Use ζ = 1.0 N/m, η = 0.5 N·m³, ρ₀ = 1.5 m, and stay inside the switch distance so the well is quadratic.

• Attractive force: F_att = −ζ (q − q_goal) = −1.0 · (0−10, 0−0) = (+10, 0) N — a strong pull straight toward the goal.
• Repulsive force: ρ = 0.8 m < ρ₀, so it fires. Magnitude = η (1/ρ − 1/ρ₀)(1/ρ²) = 0.5 · (1/0.8 − 1/1.5) · (1/0.64) ≈ 0.5 · (1.25 − 0.667) · 1.5625 ≈ 0.456 N, directed away from the obstacle, i.e. (−0.456, 0) N along the line of approach.
• Net force: F = (10 − 0.456, 0) ≈ (9.54, 0) N. The obstacle sits dead ahead, exactly between robot and goal, so both forces are collinear — the robot slows but keeps pushing straight in.

This is exactly the geometry that breeds trouble: shrink the obstacle gap and crank η, and the repulsion can grow to cancel the attraction. When the two become equal and opposite the net force is zero — a local minimum — and the robot freezes short of the goal even though a clear path exists a metre to either side.

Common misconceptions and failure modes

• "Gradient descent always reaches the goal." False for the basic field. Concave (U-shaped) obstacles, a symmetric obstacle on the goal line, and narrow throats all create local minima where ∑∇U = 0 short of the target.
• "It plans a path." It does not. The basic method has no memory and no lookahead; it emits one vector per tick. What looks like a path is the integral of many reactive steps, and it is not guaranteed to exist.
• "Bigger repulsive gain is safer." Large η makes the robot skittish, causes it to refuse narrow-but-valid passages, and worsens oscillation as it bounces across the influence boundary ρ₀.
• "Obstacles far away still matter." No — repulsion is exactly zero beyond ρ₀. This locality is a feature (cheap, sensor-direct) but means the robot is blind to what it must eventually route around.
• "It is complete." It is not. Unlike a resolution-complete grid search or a probabilistically complete RRT, the basic field can fail to find a path that demonstrably exists.
• "Oscillation is a tuning bug." Partly structural: the repulsive gradient is steep and switches abruptly at ρ₀, so discrete time steps in a narrow corridor produce limit cycles. Damping, potential smoothing, or a vortex/rotational field are the real cures.

Escaping the trap: navigation functions and beyond

Because the trap is fundamental to naive gradient descent, escape strategies fall into two camps — patch the descent, or replace the field with one that has no bad minima.

Potential-field variants vs. a sampling planner

Method	Local minima?	Needs a map?	Complete?	Typical cost
Basic APF (Khatib)	Yes — traps possible	No (sensor-direct)	No	µs, ~1 kHz
APF + random walk / wall-follow	Escapes, not guaranteed	No	No	µs–ms
Navigation function (Rimon–Koditschek)	None (single minimum)	Yes (sphere world + diffeomorphism)	Yes*	ms (offline build)
Harmonic potential (Laplace)	None (max principle)	Yes (grid BVP)	Yes*	ms–s (grid solve)
RRT / RRT* (sampling)	N/A — global search	Yes	Probabilistically complete	ms–s

*Guaranteed under the stated geometric assumptions (known, well-modelled workspace).

• Navigation function. Rimon and Koditschek (1992) constructed a potential with a single minimum at the goal and no spurious ones on a "sphere world," using a tunable exponent κ; arbitrary domains are mapped to a sphere world by a diffeomorphism. Gradient descent then reaches the goal from almost every start — completeness restored, at the price of a known map.
• Harmonic fields. Solve Laplace's equation ∇²U = 0 over the free space with the goal at low potential and obstacle boundaries at high potential. The maximum principle forbids interior minima, so descent always succeeds; the cost is a numerical boundary-value solve on a grid.
• Vortex / rotational fields. Add a swirl component so the robot circulates around an obstacle instead of stalling in front of it — cheap, keeps reactivity, tames oscillation.
• Hybrid stacks. The industry answer: a global planner (A*, Dijkstra, RRT) supplies a route that dodges concave traps, and a potential field or DWA does the fast, sensor-driven local avoidance along it.

Potential fields vs. RRT: local reflex vs. global search

The two live at opposite ends of the planning spectrum. A potential field is a reflex: from the current sensor snapshot it computes one force in microseconds, needs no model, and runs at control-loop rate — but it is local and can be trapped. An RRT is a search: it grows a tree over the whole configuration space, is probabilistically complete, and finds a route around a concave obstacle a field would stall in — but it needs a model, costs milliseconds to seconds, and outputs a path that still requires a tracking controller. Real robots use both: the sampling planner picks the route, the field keeps you off the walls between waypoints.