Acrobatic Control of a 2D Quadrotor

Acrobatic control of a 2D quadrotor refers to the ability of the quadrotor to perform complex, dynamic maneuvers such as flips and turns. In this project, we designed and implemented two controllers for a 2D quadrotor to enable it to perform acrobatic maneuvers: a Linear-Quadratic Regulator (LQR) controller and an iterative LQR (iLQR) controller with added line search. The LQR controller was designed to maintain a predefined position and follow a predefined trajectory, while the iLQR controller was developed to perform acrobatic maneuvers and avoid local minima.

Following Trajectory using LQR	Doing Full Flip using iLQR

Tracking Controller using LQR

For the Linearized Quadratic controller, we must compute the value of A and B at each stage of the trajectory as it is dependent on the state of the system.

To make quadrotor follow a trajectory first system dynamics have to be linearized around the desired states and desired controls are different time steps.

Linearized system dynamics can be written as - $$\bar{x_{n+1}} = A_n\bar{x_{n}} + B_n\bar{u_{n}}$$ where,

$A_n = \frac{\partial f}{\partial x}$ linearized around $x_{n}^\star, u_{n}^\star$,

$B_n = \frac{\partial f}{\partial u}$ linearized around $x_{n}^\star, u_{n}^\star$,

$x_{n}^\star, u_{n}^\star$ are the desired state and desired control to follow the trajectory.

Cost function for controller can be written as -

$$\sum_{n=0}^{N} (x_n - x^\star_n)^TQ_n(x_n-x^\star_n) + (u_n-u^\star_n)^TR_n(u_n-u^\star_n)$$

To make quadrotor follow a circle desired state at different time steps

$$x_n^\star = \begin{bmatrix}\cos{\frac{2\pi1000}{n}} \ 0\ \sin{\frac{2\pi1000}{n}} \ 0\ 0\ 0\end{bmatrix}$$

desired control was taken to as -

$$u_n^\star = \begin{bmatrix}\frac{mg}{2}\\ \frac{mg}{2}\end{bmatrix}$$

Plots of States for following circular trajectory

without perturbations	with perturbations

Plots of controls for following circular trajectory

without perturbations	with perturbations

Acrobatic Controller using iLQR

The cost function computes the total cost of trajectory $x$ with control trajectory $u$. It consists of two parts, i.e. cost to go and terminal cost. This cost function will be used in the next steps to compute $Q, R, q, r$ using Hessian and Jacobian.

The equation of time-varying cost function is given by :

The function can be found in the ipynb file with the following name:

 compute_cost(z, u, horizon_length):

The quadratic approximation is used to find the value of Q, q, R, and r. Q and R are the hessian while q and r are the jacobians of cost function J. Where, $$Q = \frac{\partial^2 J}{\partial x^2} $$ $$R = \frac{\partial^2 J}{\partial u^2} $$ $$q = \frac{\partial J}{\partial x} = Q * (x^* - x_{desired})$$ $$r = \frac{\partial J}{\partial u} = R * (u^* - u_{desired})$$

The function can be found in the ipynb file with the following name:

 get_quadratic_approximation_cost(z, u, horizon_length):

The function for iLQR algorithm can be found in the ipynb file with the following name:

 solve_iLQR(A, B, Q, R, q, r, N):

Plots for doing full flip without perturbations

States	Controls