Path Planning and Trajectory Optimization: MATLAB Report

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Path Planning and Trajectory Optimization: MATLAB Report

 

1. Abstract

Summarize the goals of the project (e.g., designing collision-free paths, optimizing trajectories for efficiency/smoothness), methods used (e.g., A*, RRT, optimal control), and key results (e.g., computation time, path quality).

2. Introduction

  • Objective: Develop algorithms to plan paths and optimize trajectories for robots, drones, or autonomous vehicles.
  • Challenges: Obstacle avoidance, dynamic constraints, real-time computation.
  • Applications: Autonomous navigation, industrial robotics, UAVs.
  • Tools: MATLAB’s Robotics System Toolbox, Optimization Toolbox, and custom algorithms.

3. Theoretical Background

3.1 Path Planning Algorithms

  1. Grid-Based Methods:
    • A Algorithm*: Heuristic search for shortest path on grids.
    • Dijkstra’s Algorithm: Guarantees shortest path without heuristics.
  2. Sampling-Based Methods:
    • Rapidly Exploring Random Trees (RRT): Probabilistic exploration for high-dimensional spaces.
    • Probabilistic Roadmaps (PRM): Precomputes collision-free paths.
  3. Artificial Potential Fields: Attraction to goal and repulsion from obstacles.

3.2 Trajectory Optimization

  1. Optimal Control: Minimize cost functions (time, energy, jerk).
  2. Convex Optimization: Quadratic programming (QP) for smooth trajectories.
  3. Spline Interpolation: Cubic/Bézier splines for continuous curvature.
  4. Model Predictive Control (MPC): Receding horizon optimization.

4. Methodology

4.1 Path Planning in MATLAB

Case 1: A Algorithm on a Grid Map*

matlab
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% Create a binary occupancy grid (1 = obstacle, 0 = free)  
map = binaryOccupancyMap(10, 10, 1);  
setOccupancy(map, [3 3; 4 4; 5 5], 1); % Add obstacles  

% Define start and goal  
start = [1, 1];  
goal = [10, 10];  

% Create A* path planner  
planner = plannerAStarGrid(map);  

% Plan path  
path = plan(planner, start, goal);  

% Visualize  
show(planner);

Case 2: RRT for UAV Path Planning

matlab
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% Define state space (2D position and velocity)  
ss = stateSpaceDubins([0 100; 0 100], 'TurningRadius', 1);  

% Create RRT planner  
planner = plannerRRT(ss, @distanceDubins, @steerDubins);  

% Plan path from [0,0] to [80,80]  
path = plan(planner, [0 0 0], [80 80 0]);  

% Smooth trajectory using splines  
smoothedPath = smooth(path, 'Spline');

4.2 Trajectory Optimization

Case 1: Minimum-Jerk Trajectory

matlab
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% Define waypoints and time vector  
waypoints = [0 0; 2 3; 5 4; 10 0];  
t = linspace(0, 10, size(waypoints,1));  

% Fit quintic spline  
traj = cubicpolytraj(waypoints', t, linspace(0,10,100));  

% Plot trajectory  
plot(traj(1,:), traj(2,:), 'LineWidth', 2);

Case 2: Convex Optimization with fmincon

matlab
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% Minimize energy: ∫(u(t)^2) dt  
costFunc = @(u) sum(u.^2);  

% Constraints (e.g., dynamics x_dot = Ax + Bu)  
A = [0 1; 0 0];  
B = [0; 1];  

% Solve using fmincon  
options = optimoptions('fmincon', 'Algorithm', 'sqp');  
u_opt = fmincon(costFunc, u0, [], [], [], [], lb, ub, @(u) dynamicsConstraint(u, A, B), options);

5. Results and Analysis

5.1 Path Planning

  • A vs. RRT*: Compare path length, computation time, and success rate in cluttered environments.
  • Figure: Grid map with A* path vs. RRT exploration tree.

5.2 Trajectory Optimization

  • Smoothness: Compare jerk-minimized vs. linear trajectories.
  • Energy Efficiency: Optimal control vs. heuristic paths.
  • Figure: Time vs. position/velocity/acceleration plots.

5.3 Dynamic Environments

  • MPC Performance: Ability to replan trajectories with moving obstacles.

6. Discussion

  • Algorithm Trade-offs:
    • A*: Optimal but computationally heavy for large grids.
    • RRT: Fast for high-D spaces but suboptimal.
  • Optimization Challenges: Non-convexity, real-time constraints.
  • MATLAB Tools: Strengths (pre-built functions) vs. limitations (customization).

7. Conclusion

  • Summarize the effectiveness of MATLAB for prototyping path planning and optimization.
  • Highlight applications in robotics and autonomous systems.
  • Suggest future work (e.g., machine learning integration, ROS compatibility).

8. Appendix

MATLAB Code

Include full scripts for:

  • A* grid planning.
  • RRT with Dubins curves.
  • Trajectory optimization using fmincon or trajectoryOptimalControl.

References

  1. LaValle, S. M. Planning Algorithms. Cambridge University Press, 2006.
  2. MATLAB Robotics System Toolbox Documentation.
  3. Boyd, S. Convex Optimization. Cambridge University Press, 200

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