Unit 3: Multi Rotor Dynamics

I. Orientation — Principles and Conventions

A multirotor is a rigid-body aircraft whose independently controlled propellers generate the forces and torques required for translation and rotation. Its motion follows Newton–Euler mechanics: external forces determine linear acceleration, while external moments determine angular acceleration.

• Six degrees of freedom: The vehicle translates along three axes and rotates about three axes.
  • Position in the inertial frame is (\mathbf{p}=[x,y,z]^T), measured in metres.
  • Body velocity is (\mathbf{v}_B=[u,v,w]^T), measured in metres per second.
  • Angular velocity is (\boldsymbol{\omega}=[p,q,r]^T), measured in radians per second.
• Reference frames: The inertial frame is fixed to Earth, whereas the body frame moves with the aircraft. A common aerospace body convention uses (x_B) forward, (y_B) right, and (z_B) downward.
• Attitude variables: Roll (\phi), pitch (\theta), and yaw (\psi) describe orientation. The rotation matrix (\mathbf{R}_{IB}) converts body-frame vectors into inertial-frame vectors.
• Rigid-body assumption: The frame, motors, and payload maintain fixed relative positions. Structural flexing and propeller elasticity are neglected in the basic model.
• Actuation principle: Rotor speed changes produce thrust approximately proportional to the square of angular speed. Differences among rotor thrusts create control moments.
• External influences: Gravity, aerodynamic drag, wind, ground effect, and propeller–body interactions affect motion. Simplified analysis usually retains gravity and drag while treating other disturbances separately.
• Vehicle parameters: Mass (m) is measured in kilograms, inertia matrix (\mathbf{J}) in (\mathrm{kg\,m^2}), force in newtons, and torque in newton-metres.

II. Full Rigid-Body Dynamics — Six-Degree-of-Freedom Motion

A. Full Rigid-Body Dynamics

Full rigid-body dynamics couple the multirotor’s translational motion, rotational motion, and attitude kinematics.

• Translational equation: Newton’s second law in the inertial frame gives

m p̈ = m g + R_IB F_B + F_D + F_ext

Here (m) is mass, (\ddot{\mathbf{p}}) is inertial acceleration, (\mathbf{g}) is gravitational acceleration, (\mathbf{F}_B) is total body-frame propulsive force, (\mathbf{F}D) is drag, and (\mathbf{F}{ext}) represents other external forces.

• Rotational equation: Euler’s rigid-body equation is

J ω̇ + ω × (Jω) = τ_B + τ_ext

Here (\mathbf{J}) is the inertia matrix, (\dot{\boldsymbol{\omega}}) is angular acceleration, (\boldsymbol{\tau}B) is the control torque, and (\boldsymbol{\tau}{ext}) is the disturbance torque. The cross-product term represents gyroscopic coupling.

• Principal-axis model: For a symmetric vehicle, body axes approximately coincide with principal axes:

J = diag(I_x, I_y, I_z)

The quantities (I_x), (I_y), and (I_z) are moments of inertia about the roll, pitch, and yaw axes.

• Attitude kinematics: Euler-angle rates are related to body rates by

[φ̇ θ̇ ψ̇]ᵀ = E(φ,θ)[p q r]ᵀ

The matrix (\mathbf{E}(\phi,\theta)) performs the rate transformation. Euler angles become singular at (\theta=\pm90^\circ); quaternions avoid this singularity.

• Coupling: Tilting changes both attitude and the direction of thrust. Consequently, horizontal translation cannot occur without rotational motion first redirecting the thrust vector.

• Model limitations: The basic equations omit motor lag, blade flapping, flexible structures, and detailed airflow. These effects become important during aggressive flight or near aerodynamic boundaries.

III. Generating Force and Torque — Rotor-Based Control

A. Generating Force and Torque

A multirotor controls all six motions by varying rotor-generated thrust and reaction torque.

• Single-rotor thrust: A common static approximation is

T_i = k_f Ω_i²

Here (T_i) is rotor (i)’s thrust in newtons, (k_f) is its thrust coefficient, and (\Omega_i) is its angular speed in radians per second.

• Reaction torque: Rotor drag produces a torque opposing propeller rotation:

Q_i = k_m Ω_i²

Here (Q_i) is aerodynamic reaction torque and (k_m) is the moment coefficient. Counter-rotating propeller pairs normally cancel net yaw torque at equal speeds.

• Total thrust: For (n) aligned rotors,

T = Σ(i=1 to n) T_i

The resultant usually acts along the body’s vertical axis. In the downward-positive aerospace convention, upward propulsive force is (\mathbf{F}_B=[0,0,-T]^T).

• Roll and pitch moments: A rotor displaced from the centre of mass produces (\boldsymbol{\tau}_i=\mathbf{r}_i\times\mathbf{F}_i), where (\mathbf{r}_i) is its position vector. For arm length (l), a thrust difference (\Delta T) gives a moment of approximate magnitude (l\Delta T).

• Yaw moment: Increasing one rotational pair and decreasing the opposite pair changes the sum of reaction torques while approximately preserving total thrust.

• Control allocation: Rotor commands are obtained from a mixer:

[T τ_x τ_y τ_z]ᵀ = A[Ω_1² Ω_2² ... Ω_n²]ᵀ

Here (\tau_x,\tau_y,\tau_z) are roll, pitch, and yaw torques, and (\mathbf{A}) is the allocation matrix determined by rotor positions, rotation directions, and coefficients.

• Actuator constraints: Since (\Omega_i^2\geq0) and motor speed is bounded, requested forces and moments may be infeasible. Saturation can reduce attitude control authority.

IV. Hover — Static Vertical Equilibrium

A. Hover

Hover is the condition of nearly constant position and attitude, with total thrust balancing weight.

• Equilibrium condition: For level, stationary flight in still air,

T = mg

Here (T) is total upward thrust, (m) is mass, and (g\approx9.81\ \mathrm{m/s^2}) is gravitational acceleration.

• Per-rotor requirement: Identical rotors share the load equally:

T_i = mg/n

For a (2\,\mathrm{kg}) quadrotor, (mg=19.62\,\mathrm{N}), so each of four rotors supplies approximately (4.91\,\mathrm{N}).

• Torque balance: Equal thrust alone is insufficient; net roll, pitch, and yaw moments must also be zero. The centre of thrust should pass through the centre of mass.

• Vertical response: If (T>mg), the vehicle accelerates upward; if (T<mg), it accelerates downward. Under an upward-positive convention,

a_z = T/m - g

Here (a_z) is vertical acceleration.

• Hover power: Ideal induced power increases approximately as (T^{3/2}/\sqrt{2\rho A}), where (\rho) is air density and (A) is total effective rotor-disc area. Larger disc area reduces induced power for the same weight.

• Practical disturbances: Wind, payload imbalance, battery-voltage decline, sensor bias, and ground effect require continuous feedback corrections rather than fixed rotor speeds.

V. Forward Flight — Tilting the Thrust Vector

A. Forward Flight

Forward flight is produced by pitching or rolling the aircraft so that total thrust acquires a horizontal component.

• Thrust decomposition: For level-altitude flight at tilt angle (\alpha),

T_v = T cos α
T_h = T sin α

Here (T_v) is the vertical component, (T_h) is the horizontal component, and (\alpha) is the angle between total thrust and vertical.

• Altitude condition: Constant altitude requires

T cos α = mg

Therefore, total thrust must rise to (T=mg/\cos\alpha) when the vehicle tilts.

• Horizontal acceleration: Neglecting drag,

a_h = T sin α / m = g tan α

Here (a_h) is horizontal acceleration. With drag included, (a_h=(T\sin\alpha-D)/m), where (D) is opposing drag.

• Transition sequence: The controller first generates a pitch or roll moment, then arrests the angular motion at the desired tilt, and finally adjusts collective thrust to prevent loss of altitude.

• Steady flight: At constant horizontal velocity, acceleration is zero and (T\sin\alpha=D). Thus tilt is required even after the desired speed has been reached.

• Flight effects: Translational airflow changes rotor inflow and body drag. At higher speeds, rotor asymmetry, blade flapping, and attitude-dependent aerodynamic forces reduce the accuracy of the simple point-mass model.

VI. Maximum Tilt and Thrust — The Flight-Envelope Boundary

A. Maximum Tilt and Thrust

Maximum tilt is constrained by the thrust reserve needed to support weight while producing horizontal force.

• Thrust-limited angle: If (T_{\max}) is maximum total thrust, level altitude can be maintained only when

T_max cos α_max = mg
α_max = cos⁻¹(mg/T_max)

Here (\alpha{\max}) is the greatest sustainable tilt angle, provided (T{\max}\geq mg).

• Thrust-to-weight ratio: Define (\lambda=T_{\max}/(mg)). Then

α_max = cos⁻¹(1/λ)

A vehicle with (\lambda=2) has an ideal level-flight tilt limit of (60^\circ).

• Maximum horizontal force: At the altitude-preserving limit,

F_h,max = √(T_max² - (mg)²)

The corresponding drag-free acceleration is (a{h,\max}=F{h,\max}/m).

• Command priority: Near saturation, increasing attitude torque may require redistributing rotor thrust and can reduce available collective thrust. Flight controllers commonly prioritise attitude stability over exact position tracking.

• Practical margin: Continuous motor limits are lower than short-duration peak limits. Battery state, air density, propeller efficiency, payload, and motor heating therefore reduce usable (\alpha_{\max}).

• Geometric limit: At (90^\circ) tilt, thrust is entirely horizontal and provides no vertical support; level flight is impossible without another lifting force.

VII. Drag and Maximum Velocity — Steady-Speed Limitation

A. Drag and Maximum Velocity

Maximum forward velocity occurs when available horizontal thrust can no longer exceed aerodynamic drag.

• Quadratic drag model: At moderate or high airspeed,

D = ½ ρ C_D A_f V²

Here (D) is drag, (\rho) is air density, (C_D) is drag coefficient, (A_f) is frontal area, and (V) is airspeed.

• Direction: Drag opposes velocity relative to the surrounding air, not necessarily ground velocity. A headwind increases airspeed for a given ground speed, while a tailwind decreases it.

• Terminal force balance: In steady, level forward flight,

T cos α = mg
T sin α = D

Combining the equations gives (D=mg\tan\alpha).

• Maximum-velocity estimate: If the maximum sustainable horizontal force is (F_{h,\max}), the quadratic model gives

V_max = √(2F_h,max / (ρ C_D A_f))

This is an ideal estimate because (C_D), frontal area, and available thrust may change with attitude and speed.

• Low-speed behaviour: Linear drag, (D=cV), may approximate low-speed motion, where (c) is a damping coefficient in (\mathrm{N\,s/m}). Quadratic drag becomes more representative as inertial aerodynamic effects dominate.

• Power and propulsion limits: Maximum speed may be reached before the static-thrust limit because propeller thrust changes with inflow, motors have finite power, and voltage falls under load.

• Design implications: Streamlining lowers (C_DA_f); greater thrust-to-weight ratio raises available horizontal force. Both can increase maximum velocity, but added propulsion capacity also increases mass and energy consumption.