MAPTOR: Multiphase Adaptive Trajectory Optimizer

Author: David Timothy

Contact: chendavidtimothy@gmail.com

A Python framework for trajectory and design optimization using optimal control. MAPTOR simultaneously optimizes system parameters and trajectories for vehicles, robots, spacecraft, and other dynamic systems.

When to Use MAPTOR

If you only need basic PATH planning (geometry-focused problems), use path planning algorithms:

• A*, Dijkstra, basic RRT, PRM
• Fastest for obstacle avoidance without complex dynamics

If you need TRAJECTORY optimization with simple constraints, faster methods exist:

• iLQR (iterative Linear Quadratic Regulator)
• ALTRO (Augmented Lagrangian Trajectory Optimizer)
• Faster convergence for dynamics-heavy, constraint-light problems

Use MAPTOR for complex DESIGN + TRAJECTORY problems:

• Multiple design parameters + trajectory optimization
• Complex nonlinear path constraints (obstacle avoidance, state bounds)
• Complex multibody dynamics with built-in SymPy Lagrangian mechanics integration
• Multiphase missions with automatic phase linking
• When you need the full flexibility of direct transcription

Quick Start: Pure Trajectory Optimization

import maptor as mtor

# Minimum time trajectory: reach target with bounded control
problem = mtor.Problem("Minimum Time to Target")
phase = problem.set_phase(1)

# Variables
t = phase.time(initial=0.0)  # Free final time
position = phase.state("position", initial=0.0, final=1.0)
velocity = phase.state("velocity", initial=0.0, final=0.0)
force = phase.control("force", boundary=(-2.0, 2.0))

# Dynamics and objective
phase.dynamics({position: velocity, velocity: force})
problem.minimize(t.final)

# Solve
phase.mesh([8], [-1.0, 1.0])
solution = mtor.solve_adaptive(problem)

if solution.status["success"]:
    print(f"Optimal time: {solution.status['objective']:.3f} seconds")
    solution.plot()

Example: Simultaneous Design and Trajectory Optimization

While the above shows basic trajectory optimization, MAPTOR also handles simultaneous design and trajectory optimization:

import maptor as mtor


# Engine sizing optimization with mass penalty
problem = mtor.Problem("Engine Sizing Optimization")
phase = problem.set_phase(1)

# Design parameter: maximum engine thrust capability
max_thrust = problem.parameter("max_thrust", boundary=(1000, 5000))

# Physical parameters
base_mass = 100.0  # kg (vehicle dry mass)
engine_mass_factor = 0.05  # kg per Newton (engine specific mass)
gravity = 9.81  # m/s²

# Mission variables
t = phase.time(initial=0.0)
altitude = phase.state("altitude", initial=0.0, final=1000.0)
velocity = phase.state("velocity", initial=0.0, final=0.0)
thrust = phase.control("thrust", boundary=(0, None))

# Engine cannot exceed design capability
phase.path_constraints(thrust <= max_thrust)

# Total vehicle mass increases with engine size
total_mass = base_mass + max_thrust * engine_mass_factor

# Vertical flight dynamics with gravity
phase.dynamics({altitude: velocity, velocity: thrust / total_mass - gravity})

# Objective: minimize mission time + engine mass penalty
engine_mass_cost = max_thrust * engine_mass_factor * 0.1  # Cost per kg of engine
problem.minimize(t.final + engine_mass_cost)

# Mesh configuration
phase.mesh([6], [-1.0, 1.0])


phase.guess(terminal_time=50.0)

# Solve with adaptive mesh refinement
solution = mtor.solve_adaptive(problem)

# Results
if solution.status["success"]:
    optimal_thrust = solution.parameters["values"][0]
    engine_mass = optimal_thrust * engine_mass_factor
    total_vehicle_mass = base_mass + engine_mass
    mission_time = solution.status["objective"] - engine_mass * 0.1

    print("Optimal Engine Design:")
    print(f"  Max thrust capability: {optimal_thrust:.0f} N")
    print(f"  Engine mass: {engine_mass:.1f} kg")
    print(f"  Total vehicle mass: {total_vehicle_mass:.1f} kg")
    print(f"  Mission time: {mission_time:.1f} seconds")
    print(f"  Thrust-to-weight ratio: {optimal_thrust / (total_vehicle_mass * gravity):.2f}")

    solution.plot()
else:
    print(f"Optimization failed: {solution.status['message']}")

#Output
#Optimal Engine Design:
#  Max thrust capability: 3535 N
#  Engine mass: 176.7 kg
#  Total vehicle mass: 276.7 kg
#  Mission time: 29.6 seconds
#  Thrust-to-weight ratio: 1.30

Example Applications:

• Aerospace: Optimize fuel capacity + ascent trajectory
• Robotics: Optimize actuator sizing + motion planning
• Autonomous Vehicles: Optimize battery capacity + route planning

Beyond Spatial Trajectories: MAPTOR also handles abstract optimal control problems where "trajectory" refers to the evolution of any system state over time (chemical processes, financial optimization, resource allocation).

Core Methodology

MAPTOR implements the Legendre-Gauss-Radau pseudospectral method with:

• Spectral accuracy: Exponential convergence for smooth solutions
• Adaptive mesh refinement: Automatic error control through phs-adaptive mesh refinement method
• Multiphase capability: Complex missions with automatic phase linking
• Symbolic computation: Built on CasADi for exact differentiation and optimization

Installation

pip install maptor

Requirements: Python 3.10+, NumPy, SciPy, CasADi, Matplotlib

Development Installation:

git clone https://github.com/maptor/maptor.git
cd maptor
pip install -e .

Documentation

Example Trajectories

The examples gallery demonstrates trajectory optimization across multiple domains:

Design + Trajectory Optimization

• 3DOF Manipulator Design: Simultaneous motor sizing and trajectory optimization with 5kg payload transport
• 2DOF Manipulator Design: Actuator investment vs. performance trade-offs with SymPy-generated dynamics

Advanced Trajectory Optimization

• Quadcopter Flight: Quadcopter dynamics with obstacle avoidance
• Overtaking Maneuver: Complex street scenario with dual moving obstacles
• Multiphase Vehicle Launch: Realistic rocket trajectory with stage separations and orbital insertion

Classical Benchmarks

• Hypersensitive Problem: Challenging optimal control benchmark with sensitive dynamics

Architecture

MAPTOR provides a layered architecture separating trajectory design from numerical implementation:

User API (Problem, solve_adaptive, solve_fixed_mesh)
         ↓
Trajectory Definition (States, controls, dynamics, constraints)
         ↓
Mathematical Framework (Radau pseudospectral method)
         ↓
Symbolic Computation (CasADi expressions and differentiation)
         ↓
Optimization (IPOPT nonlinear programming solver)

Key Design Principles:

• Intuitive API: Define trajectories naturally without numerical details
• Automatic differentiation: CasADi handles complex derivative computations
• Adaptive precision: Mesh refinement ensures solution accuracy
• Multiphase support: Complex missions with automatic phase transitions

Contributing

We currently do not accept code submissions, but we welcome issues and feedback reports from the trajectory optimization and optimal control community. Please use GitHub Issues to:

• Report bugs or unexpected behavior
• Request new features or enhancements
• Ask questions about usage or implementation
• Suggest improvements to documentation or examples
• Share feedback on your experience with MAPTOR

Your input helps improve MAPTOR for the entire community.

License

MAPTOR is licensed under the GNU Lesser General Public License v3.0. This allows use in both open source and proprietary applications while ensuring improvements to the core library remain open.

Citation

If you use MAPTOR in academic research, please cite:

@software{maptor2025,
  title={MAPTOR: Multiphase Adaptive Trajectory Optimizer},
  author={Timothy, David},
  year={2025},
  url={https://github.com/maptor/maptor},
  version={0.2.0}
}

References

MAPTOR builds upon established methods in computational optimal control:

Optimal Control Theory and Methods:

• Betts, J. T. (2020). Practical Methods for Optimal Control Using Nonlinear Programming, Third Edition. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611976199

Pseudospectral Methods:

• Agamawi, Y. M., & Rao, A. V. (2020). CGPOPS: A C++ Software for Solving Multiple-Phase Optimal Control Problems Using Adaptive Gaussian Quadrature Collocation and Sparse Nonlinear Programming. ACM Transactions on Mathematical Software, 46(3), Article 25. https://doi.org/10.1145/3390463

Adaptive Mesh Refinement:

• Haman III, G. V., & Rao, A. V. (2024). Adaptive Mesh Refinement and Error Estimation Method for Optimal Control Using Direct Collocation. arXiv preprint arXiv:2410.07488. https://arxiv.org/abs/2410.07488

Symbolic Computation Framework:

• Andersson, J. A. E., Gillis, J., Horn, G., Rawlings, J. B., & Diehl, M. (2019). CasADi -- A software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 11(1), 1-36. https://doi.org/10.1007/s12532-018-0139-4

Support

• Documentation: https://maptor.github.io/maptor
• Issues: GitHub Issues

Acknowledgments

MAPTOR implements methods from the computational optimal control literature, particularly pseudospectral collocation techniques and adaptive mesh refinement strategies. The framework leverages CasADi for symbolic computation and automatic differentiation.

Next Steps: Begin with the Quick Start Guide or explore the Examples Gallery to see MAPTOR applied to trajectory optimization problems in your domain.