Trajectory Optimization and Stabilization
MIT 6.821: Underactuated Robotics
Spring 2024, Lecture 12
Follow live at https://slides.com/d/WGEGF9Y/live
(or later at https://slides.com/russtedrake/spring24-lec12)
Image credit: Boston Dynamics
https://en.wikipedia.org/wiki/Stall_(fluid_dynamics)
Dimensional Analysis
- Bird or plane...
- with mass \(m\), wing area \(S\), operating in a fluid with density \(\rho\)
- which requires a distance \(x\) to slow from \(V_0\) to \(V_f\)
- Distance-averaged drag coefficient:
\langle C_D \rangle = \frac{2m}{\rho S x} \ln \left( \frac{V_0}{V_f}\right)
| Vehicle | Average C_D |
|---|---|
| Boeing 747 | 0.16 |
| X-31 | 0.3 |
| Cornell Perching Plane | 0.25 |
| Common pigeon | 10 |
- A few (very rough) reference points:
Experiment Design
- Glider (no propellor)
- Flat plate wings
- Dihedral (passive roll stability)
- Offboard sensing and control
System Identification
- Nonlinear rigid-body vehicle model
- Linear actuator model (+ saturations, delay)
- Real flight data (no wind tunnel)
Lift Coefficient
Drag Coefficient
Dynamic Model
- Planar dynamics
- Aerodynamics fit from data
- State: \( {\bf x} = [x, y, \theta, \phi, \dot{x}, \dot{y}, \dot\theta] \)
- Control: \( {\bf u} = \dot\phi \)
- Enters motion capture @ 6m/s
- Perch in < 3.5m away
- Entire trajectory < 1s
Check out the perching.ipynb
- Direct collocation
- Open-loop simulation
- Finite-horizon stabilization
| Vehicle | Average C_D |
|---|---|
| Boeing 747 | 0.16 |
| X-31 | 0.3 |
| Cornell Perching Plane | 0.25 |
| Common pigeon | 10 |
| Our glider | 1.1 |
| Cobra maneuver (Mig) | 0.9 |
