A two-wheeled balancing robot is a real-world example of an inverted pendulum, where the heavy body is above the wheel axle instead of hanging below it. This arrangement is naturally unstable because even a small tilt causes gravity to create a torque that makes the robot fall farther. The robot stays upright by constantly sensing its tilt and driving its wheels to move the base back under its center of mass.
This same idea appears in self-balancing scooters, robotic platforms, and control systems labs.
Key Facts
- For small angles, gravitational torque on the body is approximately τ = m g L θ.
- An inverted pendulum is unstable because a small tilt produces a torque in the same direction as the fall.
- Tilt angle θ and angular velocity ω are often measured using an IMU with accelerometers and gyroscopes.
- A common feedback law is u = Kp θ + Kd ω, where u is the motor command.
- Wheel acceleration shifts the contact point so the robot can move its base under the center of mass.
- Stable balancing requires fast sensing, fast computation, and motor response faster than the falling motion.
Vocabulary
- Inverted pendulum
- A pendulum with its mass above the pivot point, making the upright position unstable without active control.
- Center of mass
- The average position of an object's mass, where gravity can be treated as acting on the whole object.
- IMU
- An inertial measurement unit that uses sensors such as accelerometers and gyroscopes to estimate motion and orientation.
- Feedback control
- A control method that uses measured error, such as tilt angle, to decide how to change the system's input.
- Motor torque
- The turning effect produced by a motor that causes the wheels to rotate and accelerate the robot.
Common Mistakes to Avoid
- Treating the upright robot as naturally stable is wrong because the center of mass is above the axle, so gravity amplifies small tilts instead of correcting them.
- Using only tilt angle and ignoring angular velocity is wrong because the controller also needs to know how fast the robot is falling to avoid late or excessive corrections.
- Driving the wheels opposite the needed direction is wrong because the base must move under the center of mass, not simply away from the tilt.
- Assuming stronger motors always fix balance is wrong because slow sensors, delayed computation, or poor feedback tuning can still make the robot oscillate or fall.
Practice Questions
- 1 A balancing robot has mass 2.0 kg, center of mass 0.25 m above the axle, and tilt angle 0.10 rad. Using τ = m g L θ with g = 9.8 m/s², find the approximate gravitational torque.
- 2 A controller uses u = Kp θ + Kd ω with Kp = 18, Kd = 3, θ = 0.050 rad, and ω = 0.40 rad/s. Calculate the motor command u.
- 3 A robot is leaning forward and beginning to fall forward. Explain which direction the wheels should accelerate and why this helps restore balance.