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Quadrotor Drone Thrust Allocation Simulator

A quadrotor turns four motor thrusts into total lift plus three body moments (roll, pitch, yaw) — four inputs, four degrees of freedom. This tool solves the X / + / H frame allocation matrix in real time and shows the thrust each motor needs and how much spare authority is left.

Parameters
Vehicle mass m
kg
Arm length L
m
Frame type
Motor layout. X is the modern default
Roll angle φ
°
Pitch angle θ
°
Yaw rate ψ̇
°/s
Vertical accel a_z
m/s²
Hover=0, climb +, descend −
Thrust coefficient k_T
N/(rad/s)²
Moment coefficient k_M
N·m/(rad/s)²
Results
—
Total thrust T (N)
—
Motor 1 thrust (N)
—
Motor 2 thrust (N)
—
Motor 3 thrust (N)
—
Motor 4 thrust (N)
—
Thrust margin (%)
—
Quadrotor plan view — thrust, yaw and attitude

Each circle is a motor; bar length is proportional to required thrust. The arrow at the centre shows yaw direction and the body tilt indicates roll and pitch.

Per-motor thrust allocation
Motor thrust vs roll angle φ
Theory & Key Formulas

$$T_{total} = \frac{m\,(g + a_z)}{\cos\phi\cos\theta},\qquad [T_1, T_2, T_3, T_4]^T = A^{-1}\,[T, M_\phi, M_\theta, M_\psi]^T$$

A is the 4x4 allocation matrix. For a symmetric quadrotor it inverts analytically, so per-motor thrust falls out as a closed-form expression of the desired total thrust and body moments.

$$M_\phi = I_{xx}\,\dot\omega_\phi,\quad M_\theta = I_{yy}\,\dot\omega_\theta,\quad M_\psi = I_{zz}\,\dot\omega_\psi,\qquad I_{xx}\!\approx\!I_{yy}\!\approx\!\tfrac{1}{2}mL^2,\ I_{zz}\!\approx\!mL^2$$

Required moments are inertia times angular acceleration. This tool uses a simple model with gain 5 on roll/pitch and 0.05 on the yaw-rate derivative to synthesise ω̇.

$$T_i = k_T\,\omega_i^2,\qquad \tau_i = k_M\,\omega_i^2$$

Propeller thrust T_i scales with the square of the angular speed ω_i, and the reaction torque τ_i scales the same way. Pairing CW and CCW propellers cancels τ_i and lets the controller produce yaw torque on demand.

Quadrotor Drone Thrust Allocation — 4-Motor Control Allocation

🙋
How does a drone move forward and yaw with just four fixed-pitch propellers? They do not tilt and there is no tail rotor like a helicopter.
🎓
Great question. A quadrotor does everything with differences in motor speed, not propeller direction. To move forward it slows the two front motors a little and speeds up the two rear ones; the body pitches nose-down and the total thrust vector tilts forward, so the horizontal component pulls it along. Yaw is trickier — two diagonal motors spin CW and the other two CCW, and when you raise the CW pair the residual reaction torque rotates the body CW. There is no separate yaw actuator at all.
🙋
So the rule that decides "how much to add or subtract from each motor" is the thrust allocation this tool simulates?
🎓
Exactly. It is called control allocation or "mixer" and sits in every flight controller — PX4, ArduPilot, Betaflight, Cleanflight, you name it. Inputs are the four desired channels: total thrust T plus the three body moments M_phi, M_theta, M_psi. Outputs are the four motor thrusts T1-T4. The mapping is a 4x4 allocation matrix A, and for a symmetric airframe its inverse is closed-form. So in theory it is just linear algebra — but the practical engineering starts right after.
🙋
What is so hard about a 4x4 linear system?
🎓
Two things. First, motors have limits. The linear inverse will happily return T_i below zero or above the maximum thrust, but a real motor cannot push negatively and cannot exceed its top speed. That is called saturation and it is a daily occurrence in aggressive flight or with heavy payloads. Mitigations are pseudo-inverse with clipping, QP-based allocation, and prioritized null-space allocation. Second, the matrix depends on geometry — the + and X frames differ by 45° in motor placement, so the coefficients change. If you flip the frame selector in this tool, you can watch T1-T4 redistribute for the same attitude command.
🙋
What is the "thrust margin" in the top-right results showing 66% at hover?
🎓
It is (motor_max − max(T_i)) / motor_max — basically how much more throttle you have on the most-loaded motor. A typical racing quadrotor has a thrust-to-weight ratio of 3-5, so it hovers near 25-33% throttle and keeps the rest as margin. This tool assumes each motor can deliver 3x hover thrust, which is why the default settings give exactly 66%. Drop below 20% and the red flag is up: a sudden manoeuvre, gust or battery sag will saturate a motor and the attitude collapses. Heavy machines like the DJI Inspire 2 fly with very tight margins, which is why their motion is so deliberate.
🙋
What changes when you go to a hexa (6 rotor) configuration? Just 1.5x the thrust?
🎓
The bigger gain is fault tolerance. A quad cannot fly with a dead motor; a hexa can keep flying with one motor out by reallocating among the remaining five (yaw authority drops). That is why DJI Matrice, Boston Dynamics LS3 and the agricultural AT-Drone pick hexa or octa — redundancy. The cost is weight, expense and serviceability, so consumer Mavics and Skydios stay quad. This tool simplifies hexa as a 4-motor equivalent, but you can still feel the qualitative difference in the allocation pattern.

Frequently Asked Questions

A quadrotor uses four motor thrusts T1-T4 to control four degrees of freedom: total thrust T and the three body moments (roll M_phi, pitch M_theta, yaw M_psi). The mapping is a 4x4 allocation matrix A such that [T; M_phi; M_theta; M_psi] = A * [T1; T2; T3; T4]. To find the per-motor thrust from a desired total thrust and moment vector, you multiply by the inverse matrix. This block is called the control allocation (or mixer) and sits between the attitude controller and the motor commands in every modern flight controller.
In a + frame the four motors sit on the body forward, back, left and right axes, so the front-minus-back motor difference directly produces a pitch moment and the right-minus-left difference produces a roll moment (coefficient 1/(2L)). In an X frame the body axes pass between the motors, so every motor contributes to both roll and pitch with an effective arm of L/sqrt(2). X is the modern default because the camera looking forward does not see any propeller in its field of view.
Every spinning propeller produces a reaction torque on the airframe opposite to its rotation (k_M * omega^2). With two propellers spinning CW and the other two CCW, the four reaction torques cancel at equal RPM. By increasing the CW pair and decreasing the CCW pair while keeping total thrust constant, a net CW yaw torque appears and the body slowly rotates. This is the principle every modern flight controller (PX4, ArduPilot, Betaflight) uses for the yaw channel.
Saturation is when an allocated thrust T_i exceeds the motor's maximum (typically 3x hover thrust) or falls below zero. The real motor cannot respond beyond its limit, so the requested moment is not produced and the attitude diverges. Transient saturation is acceptable, but aggressive manoeuvres or an overloaded vehicle with a low thrust-to-weight ratio make saturation a continuous condition and are a leading cause of crashes. Mitigations: (1) lighter vehicle, (2) higher-KV motors / larger props, (3) prioritized null-space allocation in software.

Real-World Applications

Consumer drones: DJI Mavic, Skydio 2 and Parrot Anafi are all X-frame quadrotors and run this exact allocation matrix inside the flight controller every control cycle (typically 1 kHz). On a payload-variable platform like the Mavic 3, the centre of gravity moves with battery state and gimbal orientation, so the controller must track a dynamically changing thrust margin.

Industrial and agricultural drones: DJI Matrice 350, Agras T50 and Yamaha FAZER R are mostly hexa or octa rotors with fault-tolerant control so they can land safely after a single motor failure. The large thrust margin you see in the "hexa" mode of this tool corresponds to the headroom such redundant designs deliberately keep. Crop sprayers also re-estimate mass online as the tank empties and recalibrate the allocation on the fly.

Racing / FPV drones: A 5-inch FPV quad has a thrust-to-weight ratio of 5-8, equivalent to 80%+ margin in this tool, which lets it sustain 100 deg/s roll and pitch without breaking. Betaflight's mixer is essentially the X-frame allocation matrix plus an "Air Mode" anti-saturation layer, a very different operating point from the conservative 3:1 assumption used here.

Urban eVTOL aircraft: Joby, Lilium and Volocopter are essentially giant multicopters with 6-36 distributed electric rotors solving the same allocation problem at a much larger scale. Safe re-allocation after a single rotor failure is a certification requirement, and modern eVTOL controllers use QP-based dynamic allocation that generalises the simple analytic mixer used in this tool.

Common Misconceptions and Pitfalls

The first trap is assuming "just invert the allocation matrix and you are done". As this tool shows, large attitude angles or rapid yaw commands push individual motor thrusts above the maximum or below zero. Linear algebra is silent about this; real implementations add a non-linear clamp loop — detect saturation, redistribute by priority, donate any leftover margin to other axes. The serene flight of a consumer DJI is partly a deliberate operating envelope chosen to keep that clamp loop unused.

The second pitfall is treating k_T and k_M as fixed constants. This tool does for simplicity, but in real flight k_T varies with (1) airspeed (the rear propellers fly in the wake of the front ones), (2) ground effect (lift grows close to the ground) and (3) battery sag. Betaflight's "Throttle Boost" and PX4's "battery-compensated throttle" are feed-forward terms that try to compensate for these model errors. Expect a 10-30% correction term on a real vehicle and instrument k_T in flight rather than trusting the nameplate.

The third is over-trusting "hexa or octa is automatically safe". Yes, the system tolerates a single motor failure, but as this tool's yaw model implies, yaw torque is produced by the difference of reaction torques. If two diagonally opposite propellers fail simultaneously, yaw authority collapses even though the vertical channel still works. The reason industrial machines pick octa over hexa is that, for any single failure, there always exists a remaining motor pattern that preserves yaw control. Redundancy is about failure modes and layout, not just the number of rotors.

How to Use

  1. Enter quadrotor mass in kg (typical range 0.5–2.5 kg for small multirotor systems)
  2. Input arm length in meters, measured from center hub to motor mount (typically 0.15–0.35 m)
  3. Set desired roll and pitch angles in degrees (±45° maximum for stable control allocation)
  4. Click Calculate to solve the 4x4 control allocation matrix and obtain individual motor thrusts
  5. Review Motor 1–4 thrust outputs in Newtons and verify thrust margin percentage remains above 5%

Worked Example

Quadrotor with mass 1.2 kg, arm length 0.25 m, roll angle 15°, pitch angle 10°. Total thrust required = 1.2 × 9.81 = 11.77 N compensated for tilt. Control allocation matrix inverts the 4×4 configuration to distribute thrust: Motor 1 = 3.05 N, Motor 2 = 2.98 N, Motor 3 = 3.12 N, Motor 4 = 2.87 N. If max motor rating is 4.2 N each, thrust margin = (4.2 − 3.12) / 4.2 × 100 = 25.7%.

Practical Notes

  1. Larger arm lengths increase moment arms; a 0.30 m arm requires proportionally less torque to achieve the same angular acceleration versus 0.20 m
  2. Angle inputs above ±30° risk exceeding maximum available thrust—check thrust margin indicator; below 5% indicates insufficient motor power
  3. Motor thrust imbalance suggests asymmetric mass distribution; rebalance battery or payload or calibrate ESC gain values
  4. X-frame configuration (45° motor spacing) differs from + configuration (90° spacing); verify frame geometry matches simulator selection for accurate allocation