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 Oct 2026
Kinematics
Robot Wheel Angular Velocity from Encoder Counts
omega = (delta_counts / CPR) * (2*pi / delta_t)
Kinematics
Odometry Heading Update
delta_theta = omega * delta_t
Kinematics
Odometry Position Update (Y Component)
delta_y = v * delta_t * sin(theta)
Kinematics
Odometry Position Update (X Component)
delta_x = v * delta_t * cos(theta)
Kinematics
DC Motor Torque-Speed Linear Approximation
tau = tau_stall * (1 - omega/omega_no_load)
Kinematics
PWM Duty Cycle to Average Voltage
V_avg = D * V_supply
Kinematics
Time-of-Flight (LiDAR) Distance Measurement
d = (c * t) / 2
Kinematics
Ultrasonic Sensor Distance Measurement
d = (v_sound * t) / 2
Kinematics
Accelerometer Tilt Angle
theta = arcsin(a_x / g)
Kinematics
Hobby Servo Pulse-Width to Angle Mapping
theta = (pulse_width - 1000) / (2000 - 1000) * 180
Kinematics
Stepper Motor Speed (RPM)
RPM = (steps_per_sec * 60) / steps_per_rev
Kinematics
Encoder Angular Resolution
delta_theta = 360 / CPR
Control Systems
Rise Time (2nd-Order System, Approx.)
t_r = 1.8 / omega_n
Control Systems
Peak Time (2nd-Order System)
t_p = pi / (omega_n * sqrt(1 - zeta^2))
Control Systems
Percent Overshoot (2nd-Order System)
%OS = exp(-zeta*pi / sqrt(1 - zeta^2)) * 100
Control Systems
Settling Time (2nd-Order System)
t_s = 4 / (zeta * omega_n)
Control Systems
Closed-Loop Gain (Negative Feedback)
G_cl = G / (1 + G*H)
Control Systems
Derivative Control Output
u_d = Kd * (de/dt)
Control Systems
Integral Control Output
u_i = Ki * integral(e dt)
Control Systems
Proportional Control Output
u = Kp * e
Kinematics
Torque Multiplication through Gear Reduction
tau_out = tau_in * N
Kinematics
Single-Link Robot Arm Gravity Torque
tau = m * g * L * cos(theta)
Kinematics
Torque for Angular Acceleration
T = I * alpha
Kinematics
Robot Path Curvature
kappa = 1 / R
Kinematics
Wheel Linear Speed from Angular Speed
v = omega * r
Kinematics
Robot Turning Radius
R = v / omega
Kinematics
Differential Drive Robot Angular Velocity
omega = (v_right - v_left) / L
Kinematics
Differential Drive Robot Linear Velocity
v = (v_right + v_left) / 2
Kinematics
Inverse Kinematics Elbow Angle (2-Link Planar Manipulator)
theta2 = acos((x^2 + y^2 - L1^2 - L2^2) / (2*L1*L2))
Kinematics
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