Robotics & Control

Making machines move: PID control, kinematics, torque, gear reduction and encoders

Robotics & Control brings together the core formulas used to make machines move accurately and predictably, from PID control and kinematics to torque, gear reduction, encoders, and motor dynamics. It’s a practical reference for students, exam prep, and engineers working on real robots, automation, and motion systems, where these calculations help tune performance, estimate position, and turn electrical input into controlled physical motion.

PID Controller

u(t) = K_p e + K_i \!\int e\,dt + K_d \frac{de}{dt}

Control output from proportional, integral and derivative of the error.

Degrees of Freedom

NEWIntermediate

DoF = 6n - \sum c

Independent motions of a mechanism from bodies and constraints.

Gear Reduction Torque

NEWIntermediate

\tau_{out} = \tau_{in} \times GR \times \eta

Output torque after gear reduction, allowing for efficiency.

Encoder Resolution

\theta = \frac{360}{N}

Smallest angle a rotary encoder can resolve.

DC Motor Torque

NEWIntermediate

\tau = k_t I

Torque of a DC motor is proportional to current.

Motor Back-EMF

V_b = k_e \omega

Voltage a spinning motor generates against its supply.

Forward Kinematics (2-Link)

x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2)

End-effector position from joint angles of a two-link arm.

Angular Acceleration Torque

NEWIntermediate

\tau = I\alpha

Torque needed to angularly accelerate a rotating mass.

Stepper Motor Step Angle

\theta = \frac{360}{S}

Angle turned per step from steps per revolution.

Microstepping Resolution

NEWIntermediate

\theta_m = \frac{\theta}{m}

Finer step angle from microstepping a stepper motor.

Wheel Odometry Distance

NEWIntermediate

d = 2\pi r \frac{C}{N}

Distance a wheeled robot travels from encoder counts.

Settling Time (approx.)

t_s \approx \frac{4}{\zeta \omega_n}

Time for a control response to settle within 2% of target.

Control Loop Rate

f = \frac{1}{T}

Frequency of a control loop from its cycle time.

Payload Holding Torque

\tau = m g L

Torque a joint must hold to support a payload at reach L.

Robot Battery Life

t = \frac{Q}{I}

Runtime from battery charge and current draw.