Problem Bank
This section presents engineering-style worked problems in spacecraft dynamics, orbital mechanics, and guidance–navigation–control systems.
Each problem is structured to connect: problem context → model → solution → code → engineering insight.
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Interactive Problem Set
These problems show how solver choice, timestep, energy drift, and structure preservation affect spacecraft simulation.
ODE45 in GNC: Dynamics, Stability, and Control
This flagship interactive problem explores a fundamental issue in guidance– navigation–control simulation: a solver such as ODE45 will faithfully integrate any equations it is given, even if the underlying model is incorrect.
The problem builds progressively from incorrect rotational dynamics to PD-controlled motion, stability mapping, high-gain numerical sensitivity, and a chaser–target control extension.
- Why missing ω breaks second-order dynamics
- How wrong and correct models diverge under the same solver
- How PD gains shape stability and oscillation regions
- When instability is numerical vs physical
- Why solver choice does not replace correct modeling
ODE45 Beyond Dynamics: Events, Orbits, Drag, and Solver Limits
This advanced interactive problem extends ODE45 beyond basic dynamics, showing how real spacecraft simulations combine control, event-based switching, orbital propagation, drag modeling, and solver behaviour.
The problem demonstrates that simulation is not only about integrating equations, but also about handling hybrid systems, environmental effects, and numerical limits.
- Compare Euler vs ODE45-style integration for chaser–target control
- Implement event-based mode switching (approach → rendezvous → dock)
- Understand energy drift in orbit propagation
- Explore drag modeling using ballistic coefficient
- See how stiffness appears in fast–slow dynamics
Fixed-Step Solver: Dynamics, Stability, and Control
This interactive problem reveals a critical engineering truth: even when the mathematical model is correct, a fixed timestep can still distort the physics.
The problem explores discrete-time dynamics, Euler integration, sampled-data control, timestep-driven instability, phase lag, and aliasing.
- Why missing ω breaks discrete state updates
- How Euler integration deviates from continuous motion
- Why digital PD behaves differently from continuous control
- How timestep affects stability and oscillation
- What numerical instability looks like
- How aliasing creates false dynamics
Fixed-Step Solver: Events, Orbits, and Drag
This interactive problem extends fixed-step simulation beyond control, showing how timestep choice affects mission logic, event detection, orbit propagation, and drag modelling.
The problem demonstrates that fixed-step errors are not only numerical — they directly affect mission behaviour and system interpretation.
- How timestep affects chaser–target convergence
- Why events are detected late or missed
- How orbit propagation gains false energy
- How drag errors grow with timestep
- Why stiff systems fail under fixed-step integration
Runge–Kutta Methods in GNC & Space Mechanics: Accuracy, Stability, and Solver Choice
This interactive problem compares Euler, RK4, and RK7 methods to show how solver order affects control response, orbit propagation, energy drift, and timestep sensitivity.
The problem demonstrates that higher-order integration improves numerical accuracy, but does not fix incorrect physics, poor timestep selection, or weak control design.
- Compare Euler, RK4, and RK7 on the same dynamic systems
- See how timestep size changes accuracy and stability
- Explore orbital energy drift under different integration methods
- Check how solver order affects control response
- Understand when adaptive RK methods help or become inappropriate
This problem connects:
integration order → timestep sensitivity → control response → orbital accuracy → engineering judgmentRunge–Kutta for Space Mechanics: Orbit Propagation and Energy Conservation
This interactive problem focuses on orbit propagation using Euler, RK4, and RK7 methods, showing how numerical integration affects trajectory prediction and long-term energy conservation.
The problem demonstrates why an ideal two-body orbit should be checked against physical invariants such as specific mechanical energy, especially when timestep size is changed.
- Compare Euler, RK4, and RK7 orbit propagation
- Visualize orbit distortion caused by numerical error
- Check long-term specific mechanical energy conservation
- Measure trajectory prediction error over multiple orbits
- Explore step-size sensitivity for orbital accuracy
This problem connects:
two-body motion → numerical integration → energy conservation → trajectory prediction → timestep sensitivitySymplectic Integrators in GNC & Space Mechanics: Preserving Physics, Not Just Accuracy
This interactive problem introduces symplectic integration as a structure-preserving approach for conservative systems such as oscillators and ideal two-body orbits.
The problem compares Euler, RK4, RK7-style, and Symplectic Leapfrog methods to show that numerical accuracy and physical structure preservation are not always the same thing.
- Why Euler can inject false energy into conservative systems
- How RK methods reduce numerical error
- Why symplectic methods preserve long-term energy behaviour
- How oscillator and orbit examples reveal different solver behaviour
- Why symplectic methods are not ideal for damped control problems
Symplectic Orbit Propagation: Energy Conservation and Long-Term Stability
This interactive problem focuses on long-term two-body orbit propagation, comparing Euler, RK4, RK7-style, Symplectic Leapfrog, and Velocity Verlet methods.
The problem shows why preserving orbital structure can be more important than simply increasing solver order when simulating conservative space systems.
- Why Euler can create false outward or inward orbital spirals
- How RK methods improve accuracy but may still drift over long durations
- Why symplectic methods better preserve conservative orbital structure
- How energy drift and radius drift reveal numerical error
- How timestep choice affects long-term orbit prediction
These problems focus on model validation, simulation mistakes, controller timing, and debugging logic.
Fixing a Spacecraft GNC Model: From Broken Dynamics to Stable Control
This interactive lab turns a simplified spacecraft GNC model into a debugging exercise where incorrect assumptions are tested and corrected.
The workflow mirrors real engineering practice: validating state propagation, checking system interfaces, tuning control, and testing system response.
- Compare incorrect and correct rotational dynamics
- Check whether SatelliteBus exposes full state
- Understand why attitude control needs both θ and ω
- Explore how PD gains affect stability and damping
- Test chaser–target response behaviour
Why Your Controller Works in Discrete Time but Fails in Continuous Simulation
This problem investigates a subtle but critical issue in spacecraft control simulation: a controller that appears stable in a fixed-step simulation may behave differently when placed inside an adaptive solver such as MATLAB ODE45.
The controller gains and spacecraft dynamics remain unchanged. The only change is how time is handled — from fixed-step updates to adaptive timestep integration.
- Why does a stable PD controller begin to oscillate under adaptive stepping?
- What assumptions are hidden in fixed-step simulations?
- How does irregular timestep affect damping and control timing?
- What is the correct way to model a digital controller inside a continuous system?
This case study connects:
controller design → numerical integration → simulation mismatch → instability → engineering fixThese problems build the control storyline from PID to optimal, nonlinear, constrained, and adaptive control.
When PD Is Not Enough: Bias, Integral Action, and Windup
This interactive problem extends the PD controller case study by introducing a constant disturbance torque, revealing steady-state pointing error.
- Why PD cannot eliminate steady-state error
- How integral action removes bias
- Why actuator saturation creates windup
- How anti-windup restores behaviour
LQR Control: Optimal Feedback vs Manual Gain Tuning
This interactive problem compares manually tuned PD control with Linear Quadratic Regulator (LQR) design. Instead of selecting gains through trial-and-error, LQR computes feedback gains by minimizing a cost that balances pointing accuracy, angular-rate motion, and control effort.
- Why manual gain tuning becomes difficult for competing objectives
- How PD can be written as state feedback
- What the LQR cost function represents physically
- How Q and R weights change system behaviour
- Why “optimal” depends on what you choose to penalize
Actuator Saturation & Anti-Windup: When Controllers Hit Physical Limits
This interactive problem reveals a fundamental reality of spacecraft control: even well-designed controllers can fail when actuators reach their physical limits. It explores how torque saturation distorts control response, why PID controllers suffer from integral windup, and how anti-windup strategies restore stability.
- Why commanded torque differs from applied torque under saturation
- How high gains can demand unrealistic actuator authority
- Why integral action leads to windup and overshoot
- How anti-windup prevents stored control error
- How saturation affects recovery time and stability
Quaternion-Based Nonlinear Control: Full 3D Spacecraft Attitude Stabilization
This interactive problem extends spacecraft control from single-axis and linearized systems to full 3D nonlinear attitude dynamics using quaternions. It shows how spacecraft can be stabilized from arbitrary orientations without singularities.
Unlike Euler-angle control, quaternion-based control works globally and allows smooth large-angle rotations, making it suitable for real spacecraft systems.
- Why Euler angles fail near singular configurations (gimbal lock)
- How quaternions represent rotation without singularity
- How quaternion error defines the shortest rotation to target
- How nonlinear PD control stabilizes full 3D motion
- How angular velocity damping shapes convergence
Model Predictive Control: Prediction, Constraints, and Control Under Limits
This interactive problem introduces Model Predictive Control as a control strategy that predicts future system behaviour and chooses control actions while respecting actuator and state constraints.
Unlike classical feedback control, MPC explicitly accounts for limits before the system violates them. This makes it powerful for spacecraft, robotics, and constrained aerospace systems.
- Why feedback control alone may react too late
- How prediction horizon affects future planning
- How control effort penalties reduce aggressive commands
- How actuator and state constraints shape the solution
- Why MPC applies only the first control action before replanning
Adaptive Control: Model Uncertainty, Changing Dynamics, and Real-Time Gain Adjustment
This interactive problem introduces adaptive control for systems whose dynamics are uncertain or changing. It shows why fixed-gain controllers can degrade when the assumed spacecraft inertia differs from the true inertia.
The problem compares baseline feedback control with adaptive correction, showing how control behaviour can update in response to model mismatch, disturbance torque, and changing dynamics.
- Why fixed gains can fail when the model is wrong
- How true inertia and assumed inertia create parameter mismatch
- How adaptive correction responds to tracking error
- Why adaptation rate must be tuned carefully
- How uncertainty affects control torque and settling behaviour
From ideal simulation to real-world behaviour under disturbance, energy limits, and system constraints.
Why Your Controller Fails in Wind Disturbance
This interactive problem shows why a controller that appears stable in an ideal aircraft simulation can become oscillatory or fragile when wind gusts are introduced.
The case study uses a simplified 2D longitudinal aircraft model with velocity, altitude, and pitch states. The same controller is tested in two conditions: no wind and wind disturbance.
- Compare ideal no-wind response with disturbed flight response
- Introduce step gust and sinusoidal gust inputs
- Observe altitude, pitch, velocity, and wind input plots
- Use sliders for controller gain, gust strength, gust frequency, damping, and simulation time
- See why controllers designed without disturbances are fragile
When the Controller Asks for More Than the Aircraft Can Give
This interactive problem builds on the wind-disturbance case by adding physical actuator limits. The aircraft is exposed to gusts, while the controller tries to reject altitude and pitch error using elevator or thrust correction.
The key issue is that the controller may calculate a large command, but the real actuator can only apply a limited value. The result is control clipping, degraded recovery, and possible performance collapse.
- Compare ideal control with physically limited control
- Show controller demand versus actuator capacity
- Visualize control clipping under wind disturbance
- Measure saturation percentage and recovery status
- Connect aircraft limits to the existing anti-windup problem
Same Controller, Different Timing, Different Behaviour
This interactive problem shows how real-time effects can change controller behaviour even when the aircraft model and control gains remain unchanged.
The case compares continuous control, fast discrete control, slow discrete control, and discrete control with sensor delay. It shows how sample time, latency, and sample-and-hold behaviour affect altitude, pitch, and command response.
- Compare continuous, discrete-fast, discrete-slow, and delayed control
- Visualize sample-and-hold command behaviour
- Show how sensor delay creates late correction and overshoot
- Measure altitude error, overshoot, recovery status, and timing risk
- Connect aircraft timing effects to the existing discrete-vs-continuous page
A Stable Controller Can Still Drain the Mission
This interactive problem introduces endurance thinking for high-altitude long-endurance aircraft. The controller must maintain altitude and pitch under wind disturbance, but every control action consumes energy.
The case compares an aggressive controller and a conservative controller. The aggressive controller tracks altitude better but drains battery faster, while the conservative controller saves energy but may allow larger deviations.
- Add battery level as a dynamic state
- Model power usage as baseline power plus control effort
- Compare aggressive and conservative control strategies
- Visualize altitude, pitch, control effort, battery level, and power usage
- Connect stability, energy consumption, and mission endurance
Endurance Is a Systems Problem — Not Just Control or Dynamics
This interactive problem ties together wind disturbance, controller effort, actuator limits, solar charging, battery drain, and long-duration mission survival for a simplified high-altitude platform aircraft.
The aircraft must survive repeated day–night cycles while maintaining station keeping under wind disturbance. A controller that performs well over seconds may still fail over 48–96 hours because energy balance becomes the limiting factor.
- Model lateral deviation under wind disturbance
- Add PD control with actuator saturation
- Track battery recharge and drain over day–night cycles
- Compare conservative, balanced, and aggressive controllers
- Diagnose battery collapse, station-keeping loss, saturation, and slow energy decline
Why a Controller Tuned for One Aircraft Can Fail on the Real Aircraft
This interactive problem compares a tuned nominal PID controller with a conservative robust controller for a simplified high-altitude aircraft under mass uncertainty, drag variation, wind disturbance, actuator limits, and sensor noise.
The nominal PID may perform beautifully on the expected model, but the robust controller is designed to remain safer when the real aircraft does not match the model assumptions.
- Compare nominal PID and robust/conservative control
- Add mass and drag/damping uncertainty
- Include wind disturbance, actuator saturation, and sensor noise
- Visualize deviation, control effort, windup, and energy use
- Use robustness maps and Monte Carlo outcomes to compare survival
These problems show how incorrect state representation — not incorrect physics — leads to wrong orbits, failed simulations, and misleading control behaviour.
When the Orbit Is Wrong but the Equations Are Right
This interactive debugging lab explores how small mistakes in orbital-element interpretation can produce a completely different orbit — even when the equations and propagation are correct.
- How degree/radian mistakes distort orbital geometry
- Why Ω, ω, and ν cannot be interchanged
- How circular and equatorial orbits create singularities
- How to diagnose errors using energy, plane, and position metrics
Why Your Orbit Is Wrong in MATLAB but Correct in STK
This problem investigates one of the most common and dangerous orbit-analysis mistakes: comparing states that are not expressed in the same coordinate frame.
A propagated state only makes physical sense when the coordinate frame, epoch, Earth rotation model, and transformation chain are handled consistently.
- Why a TLE state is naturally associated with TEME
- Why ECI, TEME, and ECEF are not interchangeable
- How Earth rotation creates apparent ground-track errors
- Why MATLAB and STK may disagree even when both are “correct”
- How to debug frame, epoch, and unit mismatches systematically
Same Orbit, Different Time, Completely Different State
This problem shows how a spacecraft state can become wrong even when the orbital elements, frame, and propagator are correct — simply because the epoch or propagation start time is inconsistent.
The key idea is that an orbit state is time-dependent. A satellite in LEO moves several kilometres every second, so even a small timing mistake can create a large position error.
- Why the TLE epoch matters
- How time offsets create along-track phase error
- Why the same orbit at two different times is not the same state
- How MATLAB/STK comparisons fail when start times differ
- How to debug UTC, epoch, timestep, and propagation-duration mistakes
These problems bridge orbital mechanics and control, focusing on relative motion, rendezvous, and formation flying dynamics. They show how small errors evolve in orbit and why control is essential.
Relative Motion in Orbit: Bounded Motion, Drift, and Why Docking Is Hard
This interactive problem introduces Clohessy–Wiltshire (CW) relative motion, showing how a chaser spacecraft behaves near a target in circular orbit. Even without control, the motion is not intuitive — small offsets can lead to oscillation, drift, or long-term separation.
The problem builds physical intuition using the LVLH frame, then explores how initial conditions shape the trajectory. It highlights why velocity errors dominate long-term behaviour and why orbital dynamics must be respected before control design.
- Why relative motion is not straight in orbit
- How LVLH frame couples radial and along-track motion
- Difference between bounded motion and drifting trajectories
- Why small velocity errors create large long-term drift
- How mean motion (orbit altitude) changes relative behaviour
- Where CW linear approximation works and where it breaks
Docking Is a Control Problem, Not Just a Guidance Problem
This interactive problem upgrades relative motion into a full control challenge. A chaser spacecraft must not only reach the target, but do so safely — without oscillation, overshoot, or excessive control effort.
Using Clohessy–Wiltshire dynamics as the plant, the problem explores how PD control, actuator limits, and initial conditions affect docking behaviour in orbit.
- Why “move toward the target” fails in orbital motion
- How PD control stabilizes relative dynamics
- Docking vs oscillation vs drift behaviour
- Effect of control tuning (Kp, Kd) on stability and safety
- Impact of actuator saturation on real-world performance
- Why safe docking requires both position and velocity control
Formation Flying Stability: When Small Errors Become Large Separation
This interactive problem shows why formation flying is not just about placing spacecraft near each other. Small relative position and velocity errors can grow into large drift unless the formation geometry is designed and controlled carefully.
Using CW relative dynamics, the problem explores passive motion, active station-keeping, correction burns, disturbances, safety thresholds, and Monte Carlo sensitivity to initial errors.
- How small velocity errors create long-term formation drift
- Difference between passive bounded motion and active station-keeping
- How correction frequency affects drift and Δv cost
- Why disturbances and drag mismatch degrade formation stability
- How safety thresholds classify bounded, warning, and unsafe regions
- How Monte Carlo trials reveal sensitivity to initial errors
These problems connect orbital assumptions, maneuver design, and mission-level interpretation.
Designing a Hohmann Transfer: Beyond the Equation
This worked example develops the classical two-impulse Hohmann transfer as a complete engineering walkthrough — connecting assumptions, mathematical modeling, step-by-step solution, and Python implementation.
The problem includes symbolic formulation and practical cases, including a LEO orbit raise and an Earth parking orbit to GEO transfer.
- What assumptions make the Hohmann transfer Δv-optimal?
- How the two burns change orbital energy
- Why GEO transfer is more expensive than small orbit raises
- How to implement the calculation cleanly in Python
These problems are designed not as isolated exercises, but as a connected engineering workflow — linking modelling assumptions, numerical methods, control design, and mission-level interpretation.