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Home/Engineering/Quadcopter 2D (Pitch)

Quadcopter 2D (Pitch)

A quadcopter's pitch dynamics are isolated in this side-view simulation, focusing on the core control problem of attitude stabilization. The model represents a cross-section of a quadcopter, showing two rotors that generate forces F₁ and F₂. The craft's center of mass is fixed in space, allowing it only to rotate (pitch) around this point. The fundamental physics is governed by Newton's second law for rotation: the net torque (τ) equals the moment of inertia (I) multiplied by the angular acceleration (α), or τ = Iα. The torque is produced by the differential thrust of the two rotors: τ = (F₂ - F₁) * d, where d is the horizontal distance from each rotor to the center of mass. To achieve stable hovering, a PD (Proportional-Derivative) controller is implemented. This controller calculates the required differential thrust based on two error signals: the proportional term responds to the current pitch angle error (θ_desired - θ_actual), and the derivative term responds to the angular velocity (ω). The combined output adjusts F₁ and F₂ symmetrically to create a correcting torque. Key simplifications include a 2D side view, a fixed center of mass (no translational motion), and idealized, instantaneous motor dynamics. By interacting with this model, students learn how differential thrust creates torque, the role of moment of inertia in rotational dynamics, and the practical function of a PD controller in stabilizing an unstable system—a foundational concept in aerospace robotics and control systems engineering.

Who it's for: Undergraduate engineering students studying dynamics, control systems, or robotics, as well as advanced high school physics students exploring rotational motion and feedback loops.

Key terms

  • Pitch (rotation)
  • Torque
  • Moment of Inertia
  • PD Controller
  • Differential Thrust
  • Rotational Dynamics
  • Angular Acceleration
  • Stabilization

PD (toy)

2.8
1.4
1.15
0.35

Two rotors in the plane: τ ∝ (F_R − F_L) drives pitch about a fixed pivot. Disturbance adds a small gust torque so the PD loop has something visible to reject.

Measured values

Goal θ0°
Modelfixed COM

How it works

Bare-bones pitch loop about a fixed point: proportional–derivative thrust split cancels tilt. Side view only — no translation.

Frequently asked questions

Why does the quadcopter only rotate and not move sideways in this simulator?
This is a key simplification to isolate the pitch control problem. The center of mass is fixed in the simulation, removing translational degrees of freedom. In a real 3D quadcopter, pitch rotation is coupled with forward/backward translation, but this model focuses purely on understanding how differential thrust creates and controls rotational motion.
What do the P and D gains in the controller actually do?
The Proportional (P) gain determines how aggressively the controller reacts to the current angle error—a higher P-gain creates a stronger corrective torque for a given error. The Derivative (D) gain reacts to the rate of change of the angle (angular velocity), providing damping to prevent overshoot and oscillation. Together, they form a classic PD controller for stable, responsive attitude control.
How is this 2D model related to a real quadcopter?
A real quadcopter controls pitch by differentially speeding up the front and rear rotors, exactly as shown for the two rotors in this side-view slice. The full 3D craft uses two such independent control axes (pitch and roll) for attitude control. This simulator captures the essential physics of one of those axes, making the core principle of torque generation and stabilization clear.
Why is the quadcopter inherently unstable without the controller?
Any small disturbance, like a gust of wind, creates a torque that causes the craft to tilt. Because the thrust vectors are aligned with the body, a tilt redirects the net force, creating a component that further amplifies the rotation—a positive feedback loop. The PD controller provides the necessary negative feedback to counteract this inherent instability.