Category
Robotics & Controls
Robotics and controls formulas and calculators for control systems and manipulator kinematics used in automated and robotic system design. Includes feedback control, transfer function, and joint transformation equations.
34Formulas
2Subcategories
Updated Jul 2026
General
Robot Wheel Angular Velocity from Encoder Counts
omega = (delta_counts / CPR) * (2*pi / delta_t)
General
Odometry Heading Update
delta_theta = omega * delta_t
General
Odometry Position Update (Y Component)
delta_y = v * delta_t * sin(theta)
General
Odometry Position Update (X Component)
delta_x = v * delta_t * cos(theta)
General
DC Motor Torque-Speed Linear Approximation
tau = tau_stall * (1 - omega/omega_no_load)
General
PWM Duty Cycle to Average Voltage
V_avg = D * V_supply
General
Time-of-Flight (LiDAR) Distance Measurement
d = (c * t) / 2
General
Ultrasonic Sensor Distance Measurement
d = (v_sound * t) / 2
General
Accelerometer Tilt Angle
theta = arcsin(a_x / g)
General
Hobby Servo Pulse-Width to Angle Mapping
theta = (pulse_width - 1000) / (2000 - 1000) * 180
General
Stepper Motor Speed (RPM)
RPM = (steps_per_sec * 60) / steps_per_rev
General
Encoder Angular Resolution
delta_theta = 360 / CPR
General
Rise Time (2nd-Order System, Approx.)
t_r = 1.8 / omega_n
General
Peak Time (2nd-Order System)
t_p = pi / (omega_n * sqrt(1 - zeta^2))
General
Percent Overshoot (2nd-Order System)
%OS = exp(-zeta*pi / sqrt(1 - zeta^2)) * 100
General
Settling Time (2nd-Order System)
t_s = 4 / (zeta * omega_n)
General
Closed-Loop Gain (Negative Feedback)
G_cl = G / (1 + G*H)
General
Derivative Control Output
u_d = Kd * (de/dt)
General
Integral Control Output
u_i = Ki * integral(e dt)
General
Proportional Control Output
u = Kp * e
General
Torque Multiplication through Gear Reduction
tau_out = tau_in * N
General
Single-Link Robot Arm Gravity Torque
tau = m * g * L * cos(theta)
General
Torque for Angular Acceleration
T = I * alpha
General
Robot Path Curvature
kappa = 1 / R
General
Wheel Linear Speed from Angular Speed
v = omega * r
General
Robot Turning Radius
R = v / omega
General
Differential Drive Robot Angular Velocity
omega = (v_right - v_left) / L
General
Differential Drive Robot Linear Velocity
v = (v_right + v_left) / 2
General
Inverse Kinematics Elbow Angle (2-Link Planar Manipulator)
theta2 = acos((x^2 + y^2 - L1^2 - L2^2) / (2*L1*L2))
General
Forward Kinematics (2-Link Planar Manipulator)
x = L1*cos(theta1) + L2*cos(theta1 + theta2)
Kinematics
Rotational Torque
τ = Iα
Kinematics
Manipulator Jacobian Relation
v = J(q)·q̇
Control Systems
Closed-Loop Transfer Function
T(s) = G(s) / (1 + G(s)H(s))
Control Systems
PID Controller Output
u(t) = K_p·e(t) + K_i∫e(t)dt + K_d·de/dt