[LQR] iLQR/DDP algorithm for Non-linear trajectory optimization
Differential Dynamic Programming (DDP) is an indirect method which optimizes over the unconstrained control-space. It uses a 2nd-order Taylor series approximation of the cost-to-go in Dynamic Programming (DP) to compute Newton steps on the control trajectory.
- iLQR: Only keeps the first-order terms (Gauss-Newton approximation), which is similar to Riccati iterations, but accounts for the regularization and line-search required to handle the nonlinearity.
- DDP: Second-order terms included (Newton approximation).
The iLQR/DDP controller solves the following finite-horizon optimization (Non-linear trajectory optimization) problem:
\[ \begin{aligned} \min_{x_{1:N},u_{1:N-1}} \quad & \sum_{i=1}^{N-1} \bigg[ \frac{1}{2} (x_i - x_{ref})^TQ(x_i - x_{ref}) + \frac{1}{2} u_i^TRu_i \bigg] + \frac{1}{2}(x_N- x_{ref})^TQ_f(x_N- x_{ref})\\ \text{s.t.} \quad & x_1 = x_{\text{IC}} \\ & x_{i+1} = f(x_i, u_i) \quad \text{for } i = 1,2,\ldots,N-1 \end{aligned} \]
Installation
To use this project, install it locally via:
git clone https://github.com/phatcvo/iLQR-DDP.gitThe dependencies can be installed by running:
pip install -r requirements.txtTo execute the code, run:
python3 main.py