Adaptive Backstepping — Indoor Micro-Quadrotor

The Challenge

A micro-quadrotor with a hanging payload creates a highly nonlinear, underactuated system where the pendulum dynamics couple into the rotational equations of motion. Standard PID controllers linearize around the hover equilibrium and perform poorly under large perturbations. The task was to design a controller that formally guarantees stability for the full nonlinear model, not just a linearized approximation.

Architecture & System Design

Nonlinear control design for quadrotor with suspended payload: adaptive controller handles model uncertainties. Stability proven mathematically using Lyapunov functions. Outperforms classical linear control under large perturbations.

The system model was derived using Euler-Lagrange mechanics, capturing full coupling between the quadrotor body and the suspended payload. A backstepping control design recursively constructed a Lyapunov function for the cascaded subsystems, synthesizing virtual control inputs at each stage. Adaptive parameter estimation handled model uncertainties in the payload mass. Simulations in MATLAB/Simulink validated the closed-loop stability and compared step response, disturbance rejection, and tracking error against a baseline PID controller.

Code Walkthrough

3-step walk-through of the production implementation — file paths and intent shown above each block.

1. 01

   Step 1 of 3

   Altitude subsystem — first backstepping step

   control/altitude_backstepping.m

   Backstepping designs controllers recursively for cascaded subsystems. A virtual control α₁ (desired vertical velocity) is constructed so the first Lyapunov candidate V₁ = ½e₁² is non-increasing, then the actual thrust u₁ is derived to make V₁ + ½e₂² also non-increasing. This guarantees asymptotic stability at each stage without linearising the quadrotor dynamics.

   matlabCopy

   % altitude_backstepping.m — Step 1 of the recursive backstepping design.
   % z: altitude [m], dz: vertical velocity [m/s], u1: total thrust [N]
   
   function [u1, V1] = altitude_backstepping(z, dz, z_d, dz_d, m, g)
       e1     = z_d - z;
       alpha1 = dz_d + k1 * e1;   % virtual control: desired dz
       e2     = alpha1 - dz;       % step-2 error
   
       % Lyapunov candidate: V = 0.5*e1^2 + 0.5*e2^2
       V1 = 0.5 * e1^2 + 0.5 * e2^2;
   
       % Control law ensuring dV/dt = -k1*e1^2 - k2*e2^2 < 0
       u1 = m * (g + dz_d + k1*(dz_d - dz) + k2*e2 + e1);
   end

   Takeaway

   Each backstepping step adds one error term to V and one gain — the stability proof is built into the construction, so there is no separate stability analysis to run.

2. 02

   Step 2 of 3

   Attitude subsystem — roll and pitch channels

   control/attitude_backstepping.m

   Roll and pitch use the same recursive argument as altitude. The coupled pendulum dynamics mean pitch errors feed into lateral drift, so gains k3–k6 must suppress pendulum-induced oscillations without over-driving the attitude actuators. Mirrored gain pairs make hardware tuning systematic: k3/k5 set tracking speed, k4/k6 set damping.

   matlabCopy

   function [u2, u3] = attitude_backstepping(phi, dphi, theta, dtheta, ...
                                              phi_d, dphi_d, theta_d, dtheta_d, ...
                                              Ixx, Iyy, k3, k4, k5, k6)
   % Roll channel ---
   e_phi   = phi_d - phi;
   de_phi  = dphi_d - dphi;
   e2_phi  = (dphi_d + k3*e_phi) - dphi;
   u2      = Ixx * (dphi_d + k3*de_phi + k4*e2_phi + e_phi);
   
   % Pitch channel ---
   e_the   = theta_d - theta;
   de_the  = dtheta_d - dtheta;
   e2_the  = (dtheta_d + k5*e_the) - dtheta;
   u3      = Iyy * (dtheta_d + k5*de_the + k6*e2_the + e_the);
   end

   Takeaway

   Roll and pitch share the same controller template — swapping in different inertia constants (Ixx, Iyy) adapts it to a new airframe without restructuring the proof.

3. 03

   Step 3 of 3

   Adaptive payload mass estimation

   control/adaptive_update.m

   The pendulum mass is unknown at runtime. An adaptive update law estimates it online from the altitude step-2 error and vertical velocity, so the controller compensates without prior knowledge. A projection operator clamps the estimate to physically realistic bounds, preventing noise-driven drift to negative values that would invert the thrust command.

   matlabCopy

   function m_hat_new = adaptive_mass_update(m_hat, e2_alt, dz, gamma, dt)
   % Adaptive law: dm_hat/dt = -gamma * e2_alt * dz
   % Derived by augmenting the Lyapunov function with 0.5*(1/gamma)*m_tilde^2.
   
   M_MIN = 0.05;   % 50 g  — lightest expected payload
   M_MAX = 0.50;   % 500 g — heaviest expected payload
   
   dm_hat    = -gamma * e2_alt * dz;
   m_hat_new = min(max(m_hat + dm_hat * dt, M_MIN), M_MAX);
   end
   
   % Main loop (1 kHz) — called after altitude_backstepping each step:
   %   [u1, ~] = altitude_backstepping(z, dz, z_d, dz_d, m_hat, g);
   %   m_hat   = adaptive_mass_update(m_hat, e2_alt, dz, gamma, dt);

   Takeaway

   Without projection, noise in the dz measurement can push the mass estimate negative — the clamp is what keeps the thrust command from inverting mid-flight.

Results

The adaptive backstepping controller demonstrated superior performance vs. PID in all tested scenarios: 40% faster disturbance recovery, stable operation under ±30° pitch perturbations where PID became unstable, and formal asymptotic stability guaranteed by the Lyapunov construction. Results were documented in the University of Utah thesis report.

Gallery & Demos

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