VTOL Flight Control Simulation: Vertical Takeoff Tutorial

The Newton-Euler approach derives equations of motion by first drawing a free body diagram showing all external forces (contact and distant forces like gravity), then drawing an effect diagram showing inertia forces (mass times absolute acceleration of CG) and time rate of change of absolute angular momentum. The equations state: (1) Sum of external forces equals mass times derivative of absolute velocity of CG; (2) Sum of external moments about CG equals time rate of change of absolute angular momentum about CG. This provides the foundation for deriving the quadcopter's translational and rotational dynamics.

Euler's equations of motion for a rigid body relate the time derivatives of angular velocity components to external torques and principal moments of inertia. The three equations are: dω₁/dt = (I₃ - I₂)ω₂ω₃/I₁, dω₂/dt = (I₁ - I₃)ω₁ω₃/I₂, dω₃/dt = (I₂ - I₁)ω₁ω₂/I₃. These equations describe how a rigid body rotates under applied torques.

Euler's equations describe the rotational dynamics of a rigid body in its body-fixed frame, derived from the angular momentum rate equation, and are given by I₁ω̇₁ = (I₂ - I₃)ω₂ω₃ + L₁, I₂ω̇₂ = (I₃ - I₁)ω₃ω₁ + L₂, I₃ω̇₃ = (I₁ - I₂)ω₁ω₂ + L₃, where I₁, I₂, I₃ are the principal moments of inertia and L₁, L₂, L₃ are the components of the external torque; these equations show that when no external torque acts (free rigid body), the angular velocity components follow coupled nonlinear ordinary differential equations that exhibit interesting qualitative behaviors such as precession and tumbling depending on whether the body is oblate (I₃ > I₁ = I₂) or prolate (I₃ < I₁ = I₂).

The Newton-Euler equations describe the motion of vehicles using two coordinate systems: the body-fixed frame (with axes i, j, k) and the global frame (with axes X, Y, Z). For six-degree-of-freedom motion, the force equation is F = m(dV/dt) + ω × V, where ω = [P, Q, R] represents angular velocity, and the torque equation is τ = dH/dt + ω × H, where H = I × ω is the angular momentum. In the four-degree-of-freedom model used for vehicle roll dynamics, pitching motion (Q=0) and vertical movement (W=0) are neglected, simplifying the equations to capture essential roll dynamics while reducing computational complexity.

Dynamics combines kinematics (describing motion with acceleration, velocity, position, time) and kinetics (analyzing motion with forces, moments, torque). Newton's Second Law (F=ma) from 1687 established the foundation. Euler later extended these laws to rigid bodies in the 1750s by applying F=ma to the center of mass, recognizing that rigid bodies consist of particles at fixed distances. For rigid bodies, the sum of moments equals Iα, where I is mass moment of inertia (resistance to rotational motion about an axis). The parallel axis theorem states I = IG + md² for fixed parallel axes. Subscripts must match consistently: acceleration is always at the center of mass, and I and α must be about the same point. Work is defined as force integrated over distance, leading to work-energy principle: ∫F ds = ½mv² - ½mv₁². For rotation, ∫M dθ = ½Iω² - ½Iω₁². Combined, total mechanical energy is ∫F ds = ½mv² + ½Iω². Every term must be applied to the center of mass. The impulse-momentum theorem states that the integral of force over time equals change in linear momentum: ∫F dt = mv₂ - mv₁. Angular impulse is ∫M dt = h₂ - h₁. Choose the right tool: Newton-Euler for constant forces, work-energy for position-dependent forces, impulse-momentum for time-dependent forces.

Closed loop tuning is an iterative procedure for processes that cannot be opened. The method involves: (1) ensuring stable operation with minimal integral and derivative action, (2) making step changes to the setpoint, (3) increasing proportional gain until sustained oscillations occur, (4) recording the ultimate gain (PB*) and ultimate period (τ₀). Empirical formulas then provide controller settings based on these values. Examples include Ziegler-Nichols (PI: PB = 2.2PB*, Ti = 0.875τ₀) and Shinskey's alternative formulas. Settings serve as initial estimates requiring further refinement.

A closed-loop control system (feedback control system) consists of a reference signal (set-point), a controller that processes information, actuators that perform actions, a process or plant, and sensors that measure the output. The measured output is compared with the reference to calculate an error, which is fed back to the controller for correction. A PID controller combines three components: proportional (P) provides immediate response to current error, integral (I) eliminates steady-state error by accumulating past errors, and derivative (D) predicts future error based on rate of change to improve stability.

The closed loop transfer function is T(s) = G(s) / (1 + G(s)H(s)), where the denominator is the characteristic equation. Poles determine stability: left half-plane poles indicate stable systems. PID controllers (Proportional, Integral, Derivative) are widely used because they are simple, robust, and effective. The proportional term responds to error magnitude, the integral term eliminates steady state error, and the derivative term anticipates future errors. Proportional controllers decrease rise time but cause steady state error and amplify noise. Integral controllers eliminate steady state error but can cause actuator windup. Derivative controllers reduce overshoot but amplify high frequency noise. Root locus plots show how poles move as gain varies.

This section introduces feedback control systems using block diagrams, explaining key concepts including setpoint inputs, disturbances (unexpected inputs with unknown timing), and the role of PID controllers (proportional-integral-derivative). Tuning refers to selecting controller parameters (Kp, Ki, Kd) to achieve desired system behavior. Five fundamental objectives guide control system design: maintaining stability (no poles with positive real parts), eliminating steady-state error (offset), achieving rapid and smooth setpoint responses, minimizing disturbance effects, and ensuring robustness against modeling errors and changing operating conditions.

PID controllers combine three feedback mechanisms: proportional control (constant gain Kp) responds immediately to current error; integral control (K/s) eliminates steady-state error by accumulating past errors; derivative control (Ks) predicts future error based on current rate of change. The closed-loop transfer function Q(s) = H(s) / [1 + G(s)H(s)] shows how feedback modifies system dynamics, where the denominator becomes the original denominator plus the product of the original numerator and feedback network. To stabilize an unstable system, the feedback network must be chosen so all poles of the closed-loop transfer function have negative real parts. This mathematical framework enables systematic controller design, though tuning PID parameters requires careful adjustment to achieve optimal performance. The relationship between open-loop and closed-loop behavior provides the theoretical basis for understanding how feedback transforms system characteristics.

Translational dynamics follow Newton's second law: m*d²r/dt² = mg + R*F_B, where forces transform through the rotation matrix. For quadcopters, total thrust equals gravitational force for hover. Rotational dynamics use Euler's equations: I*dω/dt + ω×(I*ω) = τ_B, with diagonal inertia matrices for symmetric systems. The four-motor configuration enables differential thrust control—two motors clockwise, two counterclockwise—for independent roll, pitch, and yaw control. Hover requires total thrust balancing weight and counter-torque balancing between opposing rotors.

Quadcopter drones are rotorcraft that use four rotors to generate both lift and thrust simultaneously, unlike airplanes which rely on wings for lift; they overcome the torque problem created by spinning rotors by arranging rotors in opposing directions so that torque forces cancel out, and they achieve controlled flight through differential thrust—varying the speed of individual rotors to move in any direction (pitch, roll, yaw) without requiring traditional control surfaces.

Quadcopters generate thrust through four propellers, each contributing to overcome gravity. The thrust required equals mass times gravity divided by four. Propeller speed converts to thrust through a quadratic relationship, where thrust is proportional to the square of propeller speed. Power required equals spin rate multiplied by torque from air resistance, which scales with spin rate squared, resulting in a spin rate cubed relationship. When thrust is not equal on all propellers, torque imbalance causes rotation. The cross product of current and desired direction vectors creates a spin axis for rotation, enabling precise control of drone orientation.

All aircraft experience four forces: lift (upward force from propellers), gravity (downward force from Earth), thrust (forward force from propellers), and drag (backward force from air resistance). For vertical flight, lift must exceed gravity. For horizontal flight, thrust must exceed drag. Multirotor control works by varying individual propeller speeds: forward movement increases rear propeller speed while decreasing front speed, causing the front to dip. This differential thrust creates controlled horizontal movement.

Quadricóptero flight relies on precise motor speed control across four motors arranged diagonally. Opposite diagonal motors rotate in opposite directions (clockwise for motors 1 and 3, counterclockwise for motors 2 and 4) to cancel counter-torque forces that would otherwise cause unwanted spinning. Takeoff requires equal acceleration of all four motors above idle speed to generate sufficient collective thrust exceeding gravity. Directional movement is achieved through differential throttling: increasing rear motor speeds while decreasing front motor speeds tilts the vehicle forward for forward motion, with similar principles applied for backward, left, and right movements. Horizontal rotation (yaw) is controlled by asymmetric motor speed adjustments between diagonal pairs, creating net torque around the vertical axis.