Drone navigation and control rely on combining physical dynamics, empirical identification, and rigorous feedback design. The following 100 benchmark problems illustrate how system modelling, system identification, parameter estimation, frequency response analysis, and control theory interact across multirotor, fixed-wing, VTOL, and slung-load platforms.

1. System Modelling (Problems 1–20)

  1. 6-DOF Quadrotor Dynamics Formulation: Derive full 6-DOF translational and rotational equations of motion using Euler-Newton and Euler-Lagrange formulations in body-fixed and inertial reference frames.
  2. High-Speed Aerodynamic Drag & Blade Flapping: Model the aerodynamic drag and rotor blade flapping moments that introduce velocity-dependent translational-rotational coupling during high-speed flight.
  3. Slung-Load Cable Dynamics: Formulate the coupled differential equations of a quadrotor transporting a suspended point-mass payload via a flexible or rigid cable.
  4. BLDC Motor Electromechanical Dynamics: Model the first- and second-order dynamic response of brushless DC motors, including Back-EMF, motor resistance, and propeller rotational inertia.
  5. Gyroscopic Torque & Rotor Precession: Model the gyroscopic moments exerted on the airframe due to the high angular momentum of spinning propellers during rapid attitude maneuvers.
  6. Ground Effect Aerodynamics: Construct an empirical and analytical ground-effect model representing the thrust-increase non-linearity when hovering close to solid surfaces.
  7. Fixed-Wing Longitudinal & Lateral Decoupled Models: Derive linearized state-space representations for small-perturbation longitudinal ($u, w, q, \theta$) and lateral-directional ($v, p, r, \phi, \psi$) flight modes.
  8. Tilt-Rotor VTOL Transition Phase Dynamics: Model the non-linear thrust-vectoring and wing-lift aerodynamics during the transition from hover to forward wing-borne flight.
  9. Stochastic Wind Turbulence Modelling: Implement Dryden and von Kármán wind turbulence models as color-filtered white noise driving the airframe disturbance dynamics.
  10. ESC/Motor Dead-Band and PWM Delay: Model time delays, pulse-width modulation (PWM) quantization, and Electronic Speed Controller (ESC) dead-bands in the actuator loop.
  11. Coaxial Rotor Interference: Model the aerodynamic downwash interaction and efficiency loss in coaxial twin-rotor multirotor configurations.
  12. Morphing-Wing Variable Mass/Inertia Tensor: Formulate time-varying inertia tensor $J(t)$ dynamics for drones undergoing mid-flight structural transformation or arm folding.
  13. IMU Sensor Stochastic Noise Models: Model accelerometer and gyroscope measurement processes, including high-frequency motor vibration noise, scale-factor errors, and random walk bias.
  14. Underactuated Mapping (Position-to-Attitude): Formulate the non-linear virtual control mapping from 3D position force vectors to desired roll and pitch Euler angles under total thrust constraints.
  15. Battery Voltage SAG Impact on Thrust: Model the state-of-charge (SoC) dependent voltage drop of LiPo batteries and its downstream non-linear attenuation of maximum available rotor thrust.
  16. Tethered UAV Dynamics: Derive constraint equations and catenary cable forces for tethered multirotors receiving power or data from a ground station.
  17. Ducted-Fan Aerodynamic Distortion: Model inlet lip forces, momentum drag, and cross-wind pitching moments specific to ducted-fan UAV geometries.
  18. Hybrid Aerial-Ground Vehicle Dynamics: Formulate the hybrid dynamic modes switching between wheel-ground contact friction and aerial propeller thrust generation.
  19. Liquid Payload Sloshing Dynamics: Model liquid cargo inside a drone-mounted tank using an equivalent mechanical spring-mass-damper or pendulum model.
  20. Elastic Airframe Structural Resonance: Construct a flexible-body dynamic model capturing structural bending modes of long-arm carbon-fiber frames.

2. System Identification (Problems 21–40)

  1. Frequency-Domain Chirp Excitation: Design automated frequency-sweep (chirp) input signals for roll/pitch axes to identify open-loop attitude transfer functions from flight test data.
  2. ARX Black-Box Modeling: Estimate Autoregressive Exogenous (ARX) models for motor command-to-body rate dynamics using discrete-time system identification toolboxes.
  3. Subspace Identification (N4SID): Apply state-space subspace identification methods to extract multivariable linearized $A, B, C, D$ state matrices from multi-input multi-output (MIMO) flight logs.
  4. ESC/Motor Step-Response Identification: Identify transient time constants and dampings of individual rotor-ESC units using high-speed optical tachometer step-response tests.
  5. PRBS Input Design for Closed-Loop Identification: Synthesize Pseudo-Random Binary Sequences (PRBS) optimized to excite target quadrotor dynamics without driving the system into unstable flight regimes.
  6. Closed-Loop System Identification: Extract open-loop unstable quadrotor dynamics while flying under active stabilization feedback using indirect and joint input-output identification techniques.
  7. Non-parametric Frame Vibration PSD Analysis: Identify frame structural vibration frequencies by calculating Power Spectral Density (PSD) from high-frequency raw accelerometer telemetry.
  8. Fixed-Wing Aerodynamic Derivative Identification: Estimate stability and control derivatives ($C_{L\alpha}, C_{m\alpha}, C_{n\beta}, C_{l\delta a}$) from flight data using Output Error Methods (OEM).
  9. Vision-Based Black-Box Identification: Identify quadrotor dynamics using external motion-capture or onboard visual-inertial camera trajectory measurements as reference outputs.
  10. Maximum Likelihood Estimation (MLE) of Flight Dynamics: Formulate time-domain MLE cost functions to estimate flight parameters from flight experiments in turbulence.
  11. Grey-Box Aerodynamic Drag Modeling: Combine physical kinetic equations with experimental wind-tunnel data to fit unknown translational drag coefficients.
  12. Hammerstein-Wiener Propeller Saturation Modeling: Identify static input non-linearities (PWM saturation) and output non-linearities (thrust limits) bounding a linear dynamic core.
  13. Cross-Coupling Frequency Response Matrix Extraction: Extract frequency-dependent cross-axis interaction transfer functions (e.g., yaw-induced pitch coupling) from MIMO sweep data.
  14. In-Flight Wind Disturbance Identification: Estimate unknown external wind force vector dynamics by filtering residual accelerations not accounted for by nominal rotor thrust models.
  15. Control Surface Servo Actuator Identification: Identify high-order dynamic transfer functions of fixed-wing elevon and rudder servo actuators under aerodynamic loading.
  16. Swept-Sine Identification of Structural Modes: Isolate airframe resonance modes by subjecting a mounted drone to swept-sine shaker table tests.
  17. Physics-Informed Neural Network (PINN) Residual Identification: Train neural network models on flight data to identify unknown residual aerodynamics while preserving core Newtonian physical constraints.
  18. VTOL Transition Regime System ID: Identify time-varying local linear models along the transition corridor of a tilt-wing VTOL aircraft.
  19. Optical Flow Noise Identification: Characterize non-stationary measurement noise statistics of optical flow sensors across varying ground textures and altitudes.
  20. Hysteresis Identification in Control Surface Linkages: Model backlash and mechanical hysteresis in fixed-wing pushrod linkages using dynamic flight test residuals.

3. Parameter Estimation (Problems 41–60)

  1. Online RLS for Payload Mass Estimation: Implement Online Recursive Least Squares (RLS) with directional forgetting to estimate payload mass changes during flight.
  2. IMU Accelerometer and Gyro Bias Estimation: Formulate online estimation algorithms to isolate and track temperature-dependent zero-rate offsets in MEMS IMUs.
  3. EKF Center-of-Mass (CoM) Drift Estimation: Design an Extended Kalman Filter (EKF) to continuously estimate changes in the 3D position of the drone’s center of mass caused by asymmetric loading.
  4. Inertia Matrix Estimation via Pendulum Tests: Determine principal moments of inertia ($I_{xx}, I_{yy}, I_{zz}$) experimentally using bifilar and trifilar torsional pendulum setups.
  5. Online Thrust ($K_T$) & Torque ($K_Q$) Coefficient Tracking: Estimate real-time rotor thrust and torque coefficients as battery voltage degrades and propeller blades age.
  6. Magnetometer Hard/Soft-Iron Calibration: Estimate 3D ellipsoid deformation parameters using non-linear least squares to calibrate magnetometer readings against ambient frame magnetism.
  7. Adaptive Mass/Inertia Estimation during Aerial Pick-and-Place: Formulate real-time parameter estimators that update the system model instantaneously upon gripper contact and load pickup.
  8. Unscented Kalman Filter (UKF) for Aerodynamic Coefficients: Apply UKF state augmentation to estimate non-linear lift and drag coefficients in high-angle-of-attack maneuvers.
  9. BLDC Motor Parameter Estimation ($R_m, K_v$): Estimate electrical motor resistance and back-EMF constants using voltage and current sensing telemetry.
  10. Airspeed and Wind Vector Estimation: Estimate true airspeed, angle of attack ($\alpha$), and sideslip angle ($\beta$) by fusing pitot-tube data, IMU measurements, and GPS velocity vectors.
  11. Ground Effect Coefficient MAP Estimation: Use Maximum A Posteriori (MAP) estimation to quantify ground-effect lift multiplier coefficients as a function of ultrasonic/lidar height measurements.
  12. Propeller Damage Detection via Parameter Residuals: Identify individual rotor blade damage or missing tips by detecting anomalous drops in single-axis thrust coefficients.
  13. Tether Geometry Parameter Estimation: Estimate catenary curve shape parameters and tether tension vectors for stationary tethered hovering drones.
  14. Particle Filter Non-Gaussian Multipath Estimation: Estimate GPS pseudorange biases and multipath reflection parameter profiles in urban canyon environments using particle filtering.
  15. Sloshing Cargo Mass-Spring Parameter Extraction: Estimate the equivalent damping coefficient and natural frequency of liquid sloshing from IMU oscillation decay rates.
  16. Off-Board Sensor Latency Estimation: Formulate cross-correlation algorithms to estimate time delays between off-board visual tracking measurements and onboard IMU timestamps.
  17. Baro-Altitude Lapse Rate Parameter Estimation: Continuously estimate local atmospheric pressure-to-altitude scale factors and baseline drift using fused lidar/barometer data.
  18. Blade Flapping Derivative Estimation: Extract lateral and longitudinal blade flapping constants ($a_1, b_1$) from high-speed flight telemetry.
  19. Metaheuristic Parameter Optimization (PSO/GA): Optimize global aerodynamic derivatives for fixed-wing flight models using Particle Swarm Optimization (PSO) or Genetic Algorithms.
  20. Drag Polar Estimation ($C_{D0}, k$): Estimate parasitic drag and induced drag factors from glideslope and power-required flight test measurements.

4. Frequency Response Analysis (Problems 61–80)

  1. Open-Loop Roll/Pitch Bode Plot Analysis: Construct and analyze open-loop Bode plots for quadrotor roll and pitch channels to assess crossover frequency and high-frequency attenuation.
  2. Gain and Phase Margin Evaluation: Calculate classical stability margins (Gain Margin, Phase Margin) for inner attitude rate feedback loops to ensure stability against frame flexure.
  3. Nyquist Stability Analysis under Time Delay: Apply the Nyquist criterion to evaluate closed-loop stability margins when significant sensor processing or digital communication delays exist.
  4. Sensitivity & Complementary Sensitivity ($S(s), T(s)$) Shaping: Analyze disturbance rejection vs. noise sensitivity trade-offs using $S(s)$ and $T(s)$ peak magnitudes ($M_s, M_t$).
  5. Rotor Motor RPM Spectral Identification: Isolate motor harmonic frequencies across variable throttle ranges to tune dynamic notch filters.
  6. Servo Actuator Bandwidth Analysis: Determine the operational bandwidth of fixed-wing control surface actuators from frequency response response sweeps.
  7. Altitude-Hold Lead-Lag Compensator Frequency Design: Synthesize frequency-domain lead-lag compensators to stabilize barometric altitude control loops.
  8. RPM-Tracking Dynamic Notch Filter Design: Analyze phase-loss impact caused by narrow-band dynamic notch filters tracking motor RPM harmonics.
  9. Sensor Aliasing Analysis: Analyze sampling-frequency Nyquist limits relative to high-frequency motor vibrations to prevent IMU signal aliasing.
  10. Outer-Loop Trajectory Bandwidth Separation: Evaluate frequency separation ratios between inner attitude loops (high bandwidth) and outer position loops (low bandwidth) to prevent control interaction.
  11. Colocated vs. Non-Colocated Sensor/Actuator Response: Analyze pole-zero placements and phase lag when IMU sensors are placed away from the drone’s physical center of mass.
  12. Ducted-Fan Aero-Mechanical Resonance Analysis: Analyze the frequency response of ducted-fan structures subjected to cross-wind aerodynamic buffeting.
  13. Empirical Transfer Function Estimate (ETFE): Compute ETFE matrices from noisy flight data to validate theoretical transfer function models.
  14. Multi-Model Robustness via Overlaid Bode Plots: Evaluate gain and phase variations across multiple payload mass configurations using overlaid family-of-plant Bode diagrams.
  15. Digital Filter Phase-Lag Impact on Stability: Quantify phase margin degradation caused by low-pass Butterworth or FIR filters in the rate gyro feedback path.
  16. CIFER-Based Frequency-Domain Identification: Apply Comprehensive Identification from Frequency Responses (CIFER) methodologies to extract high-fidelity multivariable UAV models.
  17. Slung-Load Anti-Resonance Dynamic Analysis: Isolate payload pendulum resonance frequencies on frequency response plots to identify notch-filtering frequencies for sway suppression.
  18. Describing Function Analysis for Actuator Saturation: Analyze limit-cycle oscillations caused by actuator rate and deflection limits using describing function techniques.
  19. Cross-Axis Coupling Frequency Mapping: Measure frequency-dependent magnitude ratios of roll-input to yaw-motion cross-talk.
  20. Disturbance Attenuation Frequency Analysis: Measure closed-loop power rejection capabilities against low-frequency wind shear vs. high-frequency turbulence.

5. Classical & Modern Control Theory (Problems 81–100)

  1. PID Rate and Attitude Loop Synthesis: Design cascaded PID controllers with anti-windup algorithms for quadrotor roll, pitch, and yaw stabilization.
  2. Full 6-DOF Linear Quadratic Regulator (LQR): Formulate LQR controllers to minimize state deviations and actuator effort for a linearized hover model.
  3. Model Predictive Control (MPC) for Constrained Trajectories: Design explicit or online MPC algorithms enforcing input voltage limits and state path constraints during trajectory tracking.
  4. Sliding Mode Control (SMC) with Chattering Attenuation: Implement SMC with continuous boundary-layer approximations to handle aerodynamic model uncertainties without high-frequency control chatter.
  5. Feedback Linearization / Dynamic Inversion: Synthesize Non-linear Dynamic Inversion (NDI) flight controllers to cancel physical non-linearities during high-agility aerobatic maneuvers.
  6. **$\mathcal{H}\infty$ Robust Control for Aerodynamic Disturbance:** Design robust $\mathcal{H}\infty$ controllers to guarantee stability under norm-bounded model uncertainties and gust disturbances.
  7. Adaptive Control (MRAC / $\mathcal{L}_1$) under Motor Loss: Implement Model Reference Adaptive Control (MRAC) or $\mathcal{L}_1$ adaptive control to recover flight stability following a partial propeller failure.
  8. Underactuated Position Control via Backstepping: Design recursive backstepping controllers for translational motion, guaranteeing global asymptotic stability of position tracking.
  9. Cascaded Loop Architecture Design: Formulate multi-rate control loops separating fast inner rate dynamics ($1000\,\text{Hz}$) from slower outer position loops ($50\,\text{Hz}$).
  10. Linear Parameter-Varying (LPV) Control for VTOLs: Synthesize LPV gain-scheduled controllers parameterized by forward airspeed during VTOL transition flight.
  11. Linear Quadratic Gaussian (LQG) Control: Combine LQR state feedback with Kalman Filter state estimation for optimal control under Gaussian measurement and process noise.
  12. Dynamic Surface Control (DSC) for Non-linear Systems: Implement DSC to overcome the “explosion of terms” problem associated with standard backstepping control design.
  13. Control Allocation for Over-Actuated Hexarotors: Solve constrained quadratic programming problems to distribute desired control wrench vectors across over-actuated motor configurations.
  14. Active Disturbance Rejection Control (ADRC): Construct Extended State Observers (ESO) to estimate and reject total system disturbances (unmodeled dynamics + wind forces) in real time.
  15. Passivity-Based Control (PBC) for Slung Loads: Apply energy-shaping and passivity-based control to stabilize quadrotors while actively damping suspended payload oscillations.
  16. Incremental Nonlinear Dynamic Inversion (INDI): Implement INDI controllers relying on high-rate sensor measurements rather than explicit model dynamics to achieve high disturbance rejection.
  17. Gain-Scheduled PID for Fixed-Wing Envelopes: Synthesize airspeed-scheduled gain maps to maintain uniform handling qualities across stall-to-maximum-speed envelopes.
  18. Differential Flatness-Based Trajectory Tracking: Exploit the differential flatness property of quadrotors ($x, y, z, \psi$) to generate feedforward control signals for aggressive trajectory tracking.
  19. Consensus-Based Swarm Formation Control: Apply Graph Laplacian algebraic connectivity rules and consensus algorithms to maintain geometric multi-drone flight formations.
  20. Fault-Tolerant Control (FTC) for Emergency Landing: Design emergency control allocation strategies that sacrifice yaw control to maintain stable altitude and roll/pitch recovery after a complete single-rotor loss.