Derivation of the Turning Radius Formula for a Differential-Drive Robot

1. Understanding the Geometry

Consider a differential-drive robot with:

• Axle Track ((d)): The distance between the two wheels.
• Robot’s Center: The midpoint between the wheels.
• Turning Center: The point around which the robot rotates.

When the robot turns:

• The left wheel follows a circular path with radius:
  [ R_{\text{left}} = R + \frac{d}{2} ]
• The right wheel follows a circular path with radius:
  [ R_{\text{right}} = R - \frac{d}{2} ] where (R) is the distance from the robot’s center to the turning center.

2. Relating Linear and Angular Velocities

For circular motion, the linear velocity (v) is related to the angular velocity (\omega) and the radius (r) by: [ v = \omega , r ]

Thus, the linear speeds for the two wheels are:

• Left Wheel: [ v_{\text{left}} = \omega \left(R + \frac{d}{2}\right) ]
• Right Wheel: [ v_{\text{right}} = \omega \left(R - \frac{d}{2}\right) ]

3. Expressing the Difference in Speeds

Subtract the equation for the right wheel from that of the left wheel: [ v_{\text{left}} - v_{\text{right}} = \omega \left[\left(R + \frac{d}{2}\right) - \left(R - \frac{d}{2}\right)\right] ] Simplify the expression: [ v_{\text{left}} - v_{\text{right}} = \omega , d ] Solve for (\omega): [ \omega = \frac{v_{\text{left}} - v_{\text{right}}}{d} ]

4. Expressing the Average Speed

Add the two equations for the left and right wheels: [ v_{\text{left}} + v_{\text{right}} = \omega \left[\left(R + \frac{d}{2}\right) + \left(R - \frac{d}{2}\right)\right] ] Simplify: [ v_{\text{left}} + v_{\text{right}} = \omega (2R) ] Solve for (R): [ R = \frac{v_{\text{left}} + v_{\text{right}}}{2\omega} ]

5. Substituting (\omega) into the Expression for (R)

Substitute the expression for (\omega) from step 3 into the equation for (R): [ R = \frac{v_{\text{left}} + v_{\text{right}}}{2 \left(\frac{v_{\text{left}} - v_{\text{right}}}{d}\right)} ] Simplify by multiplying the numerator and denominator: [ R = \frac{v_{\text{left}} + v_{\text{right}}}{\frac{2(v_{\text{left}} - v_{\text{right}})}{d}} = \frac{d}{2} \times \frac{v_{\text{left}} + v_{\text{right}}}{v_{\text{left}} - v_{\text{right}}} ]

6. Final Formula

The turning radius (R) is: [ \boxed{R = \frac{d}{2} \times \frac{v_{\text{left}} + v_{\text{right}}}{v_{\text{left}} - v_{\text{right}}}} ]

---

Important Notes:

• Straight Movement: When (v_{\text{left}} = v_{\text{right}}), the denominator is zero, indicating the robot moves in a straight line (i.e., (R \to \infty)).
• Turning Direction: The sign of (v_{\text{left}} - v_{\text{right}}) determines the direction of the turn.
• Axle Track Impact: A larger axle track (d) will increase the turning radius for a given set of motor speeds.