UAV Systems Engineering · Calculators

Engineering calculators

Five small tools for the Module 1 and Module 7 labs. Each shows its formula, runs in your browser, and states its assumptions. Results are estimates for learning and early sizing, not substitutes for measurement.

Established engineering Momentum theory, rigid-body mixing and frame conventions are textbook material.

M1-L4 · Thrust-to-weight ratio

Take-off mass (kg)Number of motorsMaximum static thrust per motorThrust unit

Thrust-to-weight ratio

T/W = \frac{n\,T_{\text{motor}}}{m\,g}

\text{hover fraction} = \frac{1}{T/W}, \qquad a_{z,\max} = \frac{n\,T_{\text{motor}} - m\,g}{m}

Largest tilt that still holds altitude

\theta_{\max} = \arccos\!\left(\frac{1}{T/W}\right)

Use the static thrust from the motor-propeller test data at the voltage you will fly. Thrust falls as the battery sags, so evaluate the margin at the end of a mission, not at full charge. Thrust per motor in gram-force is converted with $g=9.80665\,\mathrm{m}/\mathrm{s}^{2}$ .

M1-L5 · Hover endurance and battery energy

Airframe + battery mass (kg)Payload mass (kg)RotorsPropeller diameter (inches)Battery capacity (mAh)Cells in series (S)Usable fraction of energy (%)Rotor figure of merit (assumed)Motor + ESC efficiency (assumed)Avionics + payload power (W)

Hover thrust and total rotor disc area

T = m\,g, \qquad A = n\,\pi\left(\frac{D}{2}\right)^{2}

Ideal hover power

P_{\text{ideal}} = \frac{T^{3/2}}{\sqrt{2\rho A}}

Actuator-disc (momentum) theory with ρ = 1.225 kg/m³.

Electrical power

P_{\text{elec}} = \frac{P_{\text{ideal}}}{\mathrm{FM}\cdot\eta} + P_{\text{avionics}}

Battery energy and endurance

E_{\text{pack}} = C\,[\text{Ah}]\cdot N_{\text{cells}}\cdot 3.7\ \text{V}, \qquad t_{\text{hover}} = \frac{\text{usable fraction}\cdot E_{\text{pack}}}{P_{\text{elec}}}

Larger rotor area lowers ideal power for the same thrust, which is why large slow propellers are efficient at hover. The figure of merit (typically a fraction well below 1 for small rotors) and the drivetrain efficiency are assumptions: replace them with values from bench measurements. 3.7 $V$ per cell is a nominal LiPo value; real packs sag under load.

M1-L3 · Motor mixer

Total thrust T (N)Roll torque τx (N·m)Pitch torque τy (N·m)Yaw torque τz (N·m)Motor-to-motor arm length (m)Yaw torque / thrust ratio k (m)Max thrust per motor (N)

Wrench from the motor thrusts

\begin{aligned} T &= f_1 + f_2 + f_3 + f_4\\ \tau_x &= d\,(-f_1 + f_2 + f_3 - f_4)\\ \tau_y &= d\,(-f_1 + f_2 - f_3 + f_4)\\ \tau_z &= -k\,(f_1 + f_2 - f_3 - f_4) \end{aligned}

Body frame: x forward, y left, z up. Motors at (±d, ±d) with d = L/√2. Motors 1 and 2 spin counter-clockwise, motors 3 and 4 clockwise.

The mixer (inverse of the wrench equations)

\begin{aligned} f_1 &= \frac{T}{4} - \frac{\tau_x}{4d} - \frac{\tau_y}{4d} - \frac{\tau_z}{4k}\\ f_2 &= \frac{T}{4} + \frac{\tau_x}{4d} + \frac{\tau_y}{4d} - \frac{\tau_z}{4k}\\ f_3 &= \frac{T}{4} + \frac{\tau_x}{4d} - \frac{\tau_y}{4d} + \frac{\tau_z}{4k}\\ f_4 &= \frac{T}{4} - \frac{\tau_x}{4d} + \frac{\tau_y}{4d} + \frac{\tau_z}{4k} \end{aligned}

Motor numbering follows the PX4 quadrotor-X convention (1 front-right, 2 rear-left, 3 front-left, 4 rear-right). Real autopilots use the same linear idea with normalised outputs, then handle saturation by priority; check your autopilot’s airframe geometry file for the exact numbering and spin directions. The yaw torque ratio $k$ and thrust coefficients here are illustrative.

M1-L2 · Coordinate frame converter

Vector component 1Vector component 2Vector component 3Roll (deg)Pitch, nose up positive (deg)Yaw, from north clockwise (deg)

World frames

\mathrm{NED} = (\text{North, East, Down}), \qquad \mathrm{ENU} = (\text{East, North, Up})

\mathbf p_{\mathrm{ENU}} = (p_E,\ p_N,\ -p_D) = (y_{\mathrm{NED}},\ x_{\mathrm{NED}},\ -z_{\mathrm{NED}})

Body frames

\mathrm{FRD} = (\text{Forward, Right, Down}) \ [\text{PX4}], \qquad \mathrm{FLU} = (\text{Forward, Left, Up}) \ [\text{ROS REP-103}]

\mathbf v_{\mathrm{FLU}} = (x,\ -y,\ -z)_{\mathrm{FRD}}

Yaw

\psi_{\mathrm{ENU}} = \frac{\pi}{2} - \psi_{\mathrm{NED}}

From north-referenced clockwise to east-referenced counter-clockwise.

PX4 uses NED and FRD; ROS uses ENU and FLU (REP-103). Mixing them is the most common cause of a UAV that flies away from where it was told to go. The last output line rotates a body FRD vector into NED with ZYX Euler angles.

M7-L5 · Compute and power budget

Battery energy (Wh)Usable fraction (%)Ambient temperature (°C)Enclosure thermal resistance (°C per W, assumed)

Average power and endurance

P_{\text{avg}} = \sum_i P_i\cdot\text{duty}_i, \qquad t = \frac{\text{usable fraction}\cdot E_{\text{battery}}}{P_{\text{avg}}}

Enclosure temperature

T_{\text{enclosure}} \approx T_{\text{ambient}} + P_{\text{avg}}\,R_{\text{thermal}}

Enter the power of each consumer from its datasheet at the operating mode you will use, or measure it. The starting rows are placeholders to be replaced, not specifications of any product. The enclosure temperature is a one-resistor first-order estimate; real thermal design needs measurement under airflow and sun load (lesson M7-T13).