Overview

Our project aimed to design an autonomous control system for a quadrotor drone capable of moving from any start pose to momentarily hold any goal pose. This involved addressing two major challenges: trajectory optimization and trajectory tracking.

Trajectory Optimization

Differential Flatness and iLQR
Leveraging the differential flatness property of quadrotors, we utilized convex trajectory optimization despite the nonlinear dynamics. We chose iLQR (Iterative Linear Quadratic Regulator) over non-convex solvers like SNOPT due to its ability to penalize errors in all six degrees of freedom while solving local convex optimizations.

Algorithm Tuning
Key adjustments were made to the base iLQR algorithm:

Trajectory Duration: A linear search over possible time intervals was performed to determine the optimal duration.
Cost Function: Reformulated to grow exponentially over time, emphasizing termination accuracy while ensuring progress throughout the trajectory.
Variable Time Step: Increased over time, enhancing long-horizon planning without additional computational burden.
Rotational Dynamics: Utilized Rodrigues' Formula for integrating discrete-time rotational dynamics and the vee map for computing rotational error angles, avoiding issues with rotational singularities.

Trajectory Tracking

Geometric SE(3) Controller
We initially considered finite horizon LQR but opted for the Geometric SE(3) controller proposed by Lee et al. due to its resilience to rotational singularities and nearly global exponential stability. This controller stabilizes the quadrotor to a desired position and body-frame x-direction without linearizing the dynamics.

Implementation and Challenges
The controller calculates the necessary force and moments to approach a trajectory setpoint, translating them into motor commands for the drone.

Switch to Finite Horizon LQR
After encountering persistent issues with the Geometric SE(3) controller, we implemented a finite horizon LQR controller. Though the results were mixed, the controller succeeded in approximately reaching the goal pose.

Conclusion and Future Work
Our project demonstrated the complexities of quadrotor control, particularly in handling rotations and ensuring stability. Future work will involve refining both the iLQR and finite horizon LQR approaches to improve performance.

Key Achievements

Developed a robust trajectory optimization framework using iLQR.
Implemented SE(3) state representation to handle rotational dynamics effectively.
Explored multiple control strategies, including Geometric SE(3) and finite horizon LQR.

Watch the Video

Drone Trajectory Optimization with DDP and SE(3)