Robotics Physics — Torque, Sensors and Control

Torque and gearing: the muscle math

Every robot joint is a torque problem. Torque ($\tau = rF$) — force times lever arm — is what actually rotates things. A motor's raw torque is usually small, so gearboxes do the heavy lifting: a 30:1 reduction multiplies torque ~30× while dividing speed by 30.

Mini-example: a robot arm holds a $20\ \mathrm{N}$ gripper $0.3\ \mathrm{m}$ from the shoulder joint. Required torque: $\tau = 0.3 \times 20 = 6\ \mathrm{N\cdot m}$. A hobby servo delivering $0.5\ \mathrm{N\cdot m}$ cannot do it directly — but through a 15:1 gearbox it supplies $7.5\ \mathrm{N\cdot m}$ (minus friction losses). Gearing is torque arbitrage.

Power ties it together: $P = \tau\omega$. You cannot gear your way past the motor's power — gearing trades speed for torque at (nearly) constant power.

Sensing: echoes, light and counts

Robots perceive through transducers — physics converting world into signal:

• Ultrasonic rangers chirp and time the echo: $d = v\Delta t/2$ with $v \approx 343\ \mathrm{m/s}$ in air. Cheap, but soft fabrics absorb sound and wide beams blur direction.
• Infrared sensors triangulate reflected light — fast and cheap, confused by sunlight and dark surfaces.
• Encoders count wheel or shaft rotations for odometry: precise locally, but wheels slip, so error accumulates.
• Lidar times laser pulses for centimeter-accurate 3D maps — the self-driving car's eyes.
• IMUs (accelerometer + gyroscope, like your phone) track orientation and motion between fixes.

No single sensor suffices; real robots fuse them, weighting each by its current reliability — the same sensor-fusion idea as in smartphones.

Balance: mass over base

A walking robot is an inverted pendulum that refuses to fall. The rule is geometric: the center of mass must stay above the support polygon (the area between the feet on the ground).

• Standing still: keep the mass centered over both feet.
• Walking: continuously throw the mass toward the next foothold — controlled falling, caught each step.
• Running/jumping: the mass leaves the support polygon entirely; landing must recapture it.

Humanoid robots compute this thousands of times per second (the 'zero moment point' criterion), shifting hips and arms to steer the mass. Wheeled robots cheat by keeping the mass low and the base wide — statically stable, no computation required.

Feedback control: the loop that makes it work

Open-loop commands ('turn the motor 90°') fail in the real world — friction, loads, and battery sag all interfere. Feedback control closes the loop:

1. Measure where the joint actually is (encoder).
2. Compute error = target − actual.
3. Correct with motor power proportional to the error.
4. Repeat hundreds of times per second.

PID control refines this: the P term reacts to present error, the I term eliminates persistent small errors, the D term damps oscillation by reacting to error's rate of change. Tune the three gains and a wobbly arm becomes surgical. It is the same loop you run balancing a broom on your palm — eyes measure, brain computes error, hand corrects — formalized into three numbers.

Variables and units

Common mistakes

Real-world applications

• Industrial robot arms position welds to fractions of a millimeter using servo torque control.

• Warehouse robots navigate with lidar, ultrasonic and encoder sensors fused together.

• Surgical robots scale down surgeons' hand motions for ultra-precise operation.

• Planetary rovers balance power, torque and traction on alien terrain.