Mobile robots: kinematics and navigation

Why it matters

Warehouse AGVs and AMRs, inspection crawlers, agricultural rovers and delivery robots all have to know how their wheel speeds turn into motion, where they are, and how to reach a goal without hitting anything. Mobile-robot kinematics and navigation is the fastest-growing area of industrial robotics and a common interview topic for mechatronics roles.

Key ideas

Pose. A planar mobile robot's pose is (x, y, θ): position of a reference point (usually midway between the drive wheels) and heading angle θ in the world frame.

Wheel types and drive configurations.

• Differential drive — two independently driven wheels on a common axle plus casters. Turns by running the wheels at different speeds; can spin on the spot. Most common for indoor robots.
• Skid steer — tracked or four-wheel version; turning involves deliberate slip, so odometry is poor.
• Ackermann (car-like) steering — steered front wheels; cannot turn on the spot; minimum turning radius set by the steering angle limit.
• Omnidirectional — Mecanum or omni wheels with rollers; can move sideways (holonomic).
• Legged robots step instead of rolling — better on rough ground, far more complex.

Holonomic vs non-holonomic. A differential-drive robot has 3 pose coordinates but can only command 2 velocities (v, ω); it cannot move sideways instantly (the rolling-without-slipping constraint ẋ·sin θ − ẏ·cos θ = 0). It is non-holonomic: it can still reach any pose, but needs manoeuvres (think parallel parking). Mecanum-wheel platforms are holonomic.

Instantaneous centre of rotation (ICR). All wheels of a rigid vehicle rotate about one point. For differential drive it lies on the axle line at distance R = v/ω from the centre; equal wheel speeds put it at infinity (straight line); equal and opposite speeds put it at the centre (spin in place).

Odometry (dead reckoning). Integrate wheel encoder counts to estimate pose. Errors from wheel slip, uneven floors, wheel diameter and track-width errors accumulate without bound, so odometry is fused with IMU, lidar, cameras or beacons.

Navigation stack.

1. Localisation — where am I? Kalman filter (EKF) fusing odometry with landmark observations; particle filters (Monte Carlo localisation) against a known map.
2. Mapping / SLAM — build the map while localising in it (lidar or visual SLAM); occupancy-grid maps are standard.
3. Global path planning — shortest collision-free path on the map: Dijkstra, A*, RRT/PRM for large spaces.
4. Local planning / obstacle avoidance — react to moving or unmapped obstacles: Dynamic Window Approach, potential fields, Vector Field Histogram.
5. Motion control — track the path with a controller (pure pursuit, PID on heading and cross-track error).

Industrial AGVs often use simpler infrastructure-based guidance: magnetic tape, wire, QR floor codes or laser reflectors.

Formulas

Differential drive (wheel radius r, track width L between wheel contact points):

v_R = r·ω_R, v_L = r·ω_L

v = (v_R + v_L) / 2, ω = (v_R − v_L) / L, R = v / ω = (L/2)·(v_R + v_L) / (v_R − v_L)

• ω_R, ω_L — wheel angular speeds (rad/s); v — forward speed (m/s); ω — yaw rate (rad/s); R — turning radius to the robot centre (m).

Inverse (for commanded v, ω): v_R = v + ω·L/2, v_L = v − ω·L/2

Pose update: ẋ = v·cos θ, ẏ = v·sin θ, θ̇ = ω

Exact arc over time t (constant v, ω, starting at θ₀ = 0): x = R·sin(ωt), y = R·(1 − cos(ωt)), θ = ωt

Encoder odometry: distance per count = 2π·r / (counts per wheel revolution)

Ackermann: R ≈ wheelbase / tan δ (δ — steering angle)

Worked examples

Example 1 (standard). A differential-drive robot has r = 0.05 m and L = 0.4 m. The right wheel turns at 12 rad/s and the left at 8 rad/s. Find v, ω and the turning radius.

1. v_R = 0.05 × 12 = 0.60 m/s; v_L = 0.05 × 8 = 0.40 m/s.
2. v = (0.60 + 0.40) / 2 = 0.50 m/s.
3. ω = (0.60 − 0.40) / 0.4 = 0.50 rad/s (counter-clockwise, turning left).
4. R = v / ω = 0.50 / 0.50 = 1.0 m.

Answer: v = 0.5 m/s, ω = 0.5 rad/s, R = 1.0 m.

Example 2 (GATE level). The robot of Example 1 starts at (0, 0) facing +x and keeps these wheel speeds for 2 s. (a) Find its final pose. (b) What wheel speeds would make it follow a 2 m radius circle at 0.5 m/s?

1. θ = ω·t = 0.5 × 2 = 1.0 rad (57.3°).
2. x = R·sin θ = 1.0 × sin 1.0 = 0.841 m.
3. y = R·(1 − cos θ) = 1.0 × (1 − 0.5403) = 0.460 m.
4. (b) ω = v / R = 0.5 / 2 = 0.25 rad/s. v_R = 0.5 + 0.25 × 0.2 = 0.55 m/s; v_L = 0.5 − 0.25 × 0.2 = 0.45 m/s.
5. Wheel rates: ω_R = 0.55 / 0.05 = 11 rad/s; ω_L = 0.45 / 0.05 = 9 rad/s.

Answer: (a) (0.841 m, 0.460 m, 1.0 rad); (b) ω_R = 11 rad/s, ω_L = 9 rad/s.

Common mistakes

• Dividing by L/2 instead of L in ω = (v_R − v_L)/L.
• Using wheel angular speed as robot speed — multiply by r.
• Integrating x and y with the heading at the start of a long step; use small steps or the exact arc formula.
• Treating odometry as drift-free — errors grow with distance travelled.
• Calling differential drive holonomic because it can reach any pose; holonomic means it can move in any direction instantly.
• Confusing global path planning (A*) with local obstacle avoidance (DWA).

For GATE ME

Expect numericals on differential-drive kinematics (v, ω, turning radius, wheel speeds for a desired path), odometry distance from encoder counts, and conceptual questions on holonomic constraints, localisation vs mapping, and path-planning algorithms. Practise the four differential-drive equations until they are automatic.

Quick check

1. v_R = v_L = 0.3 m/s. What is the robot's motion?
2. v_R = 0.2 m/s, v_L = −0.2 m/s, L = 0.4 m. Yaw rate?
3. Wheel radius 0.1 m, encoder 1000 counts/rev. Distance per count?
4. Can a differential-drive robot move directly sideways?
5. What does SLAM do?

Answers: 1. Straight line at 0.3 m/s; 2. 1 rad/s, spinning on the spot; 3. 0.628 mm; 4. No — it is non-holonomic; 5. Builds a map while simultaneously localising the robot in it.