cart-link-swingup

The controller for the cart-link swing-up problem on a bounded track has been implemented and verified in `/app/work/controller.py`. ### Control Strategy The controller combines two complementary nonlinear and optimal control techniques: 1. **Energy-Shaping Swing-Up with Exact Partial Feedback Linearization:** - Computes total mechanical energy of the pole relative to the upright reference $E_0 = 0$: \[ E(\theta, \dot{\theta}) = \frac{1}{2} J \dot{\theta}^2 + m g l (\cos\theta - 1) \] - Calculates energy error $\Delta E = E - E_0$ and applies pumping acceleration proportional to $\Delta E \cdot \text{sign}(\dot{\theta} \cos\theta)$ with initial symmetry breaking. - Adds PD cart centering ($-K_{px} x - K_{dx} \dot{x}$) to keep cart motion bounded within the track ($|x| \le 1.4 \ll 2.4$). - Computes the horizontal actuator force $F$ matching desired acceleration $a_{\text{des}}$ via exact partial feedback linearization. 2. **Discrete-Time Linear Quadratic Regulator (DLQR) Catch & Stabilization:** - Switches to DLQR when the link enters the basin of attraction ($|\theta| < 0.45\text{ rad}$, $|\dot{\theta}| < 2.5\text{ rad/s}$). - DLQR gains $K_{\text{LQR}}$ stabilize $x \to 0, \dot{x} \to 0, \theta \to 0, \dot{\theta} \to 0$ with high convergence precision and zero steady-state error. ```101:144:/app/work/controller.py def controller(obs: dict) -> np.ndarray: """Compute the horizontal force action for the cart-link system. Args: …