Robotics & Automation Intermediate Level 18 min

Differential Drive Mobile Robots: Forward Kinematics & Odometry

Unicycle mobile robot kinematic model, wheel radius and track baseline, forward kinematics from encoder ticks to Cartesian pose (x, y, theta), and dead reckoning drift.

Prerequisites: Trigonometry, 2D coordinate frames, and numerical integration.

1. Overview & Concept Definition

A mathematical kinematic framework describing non-holonomic mobile robots propelled by two independently driven wheels sharing a common axis of rotation.

2. Engineering & Industrial Significance

Standard locomotion topology for indoor mobile service robots, robotic vacuum cleaners, warehouse automated guided vehicles (AGVs), and research platforms (TurtleBot).

3. Physical Working Principle & Transducer Dynamics

Robot linear velocity $v$ is the arithmetic average of left and right wheel velocities. Angular velocity $\omega$ is proportional to the difference between wheel velocities divided by track width $L$. By integrating encoder measurements over time, the robot continuously tracks its Cartesian position $(x, y)$ and heading $\theta$.

4. Governing Equations & Mathematical Models

Linear & Angular Velocity: $v = \frac{v_R + v_L}{2}, \quad \omega = \frac{v_R - v_L}{L}$

Right & Left Wheel Speeds: $v_R = v + \frac{\omega L}{2}, \quad v_L = v - \frac{\omega L}{2}$

Discrete Runge-Kutta Odometry Update: $\Delta \theta = \omega \Delta t$

$\Delta x = v \Delta t \cos\left(\theta + \frac{\Delta \theta}{2}\right), \quad \Delta y = v \Delta t \sin\left(\theta + \frac{\Delta \theta}{2}\right)$

5. Practical Implementation

// C++: Differential Drive Odometry Accumulator
#include <math.h>

struct Pose2D { float x; float y; float theta; };

Pose2D updateOdometry(Pose2D current, float dLeftMeters, float dRightMeters, float trackWidthMeters) {
    float dCenter = (dRightMeters + dLeftMeters) / 2.0f;
    float dTheta = (dRightMeters - dLeftMeters) / trackWidthMeters;

    Pose2D next;
    next.theta = current.theta + dTheta;
    // Normalize theta to [-pi, pi]
    next.theta = atan2f(sinf(next.theta), cosf(next.theta));

    float avgTheta = current.theta + (dTheta / 2.0f);
    next.x = current.x + dCenter * cosf(avgTheta);
    next.y = current.y + dCenter * sinf(avgTheta);
    return next;
}

6. Common Engineering Pitfalls & Troubleshooting

Frequent Engineering Pitfalls

Ignoring wheel slip: Odometry relies on the assumption of rolling without slipping. Rapid accelerations or slick floors cause severe positional drift.
Incorrect wheelbase measurement: An error of 2mm in measured track width $L$ causes systemic angular drift that rotates calculated trajectory away from actual trajectory.

Diagnostic & Troubleshooting Checklist

Calibrate track width: Command robot to rotate precisely 360° on the spot 10 times. Compare encoder estimated angle with true physical rotation to calculate effective wheelbase factor.
Fuse odometry with a yaw gyroscope using a Kalman Filter to eliminate rotational drift.

7. Applications & Physical Constraints

TYPICAL APPLICATIONS

Warehouse automated guided vehicles (AGVs)
SLAM (Simultaneous Localization and Mapping) mapping robots
Agricultural ground scouting robots

PHYSICAL LIMITATIONS

Non-holonomic constraint: Cannot move sideways directly ($\dot{y}_{robot} = 0$ in robot body frame); requires turning maneuvers.

Pinout & Interface Specs

PIN / SIGNAL	FUNCTION / ELECTRICAL SPEC
LEFT_PWM	Left motor H-bridge speed signal
LEFT_DIR	Left motor direction signal
RIGHT_PWM	Right motor H-bridge speed signal
RIGHT_DIR	Right motor direction signal
ENC_L_A / ENC_L_B	Left wheel quadrature encoder lines
ENC_R_A / ENC_R_B	Right wheel quadrature encoder lines

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