Differential-Algebraic Equations (DAEs) & Baumgarte Stabilization for Constrained 5-Link Robotics

1. Closed Kinematic Loops & DAEs

Unlike serial robot arms whose equations of motion are Ordinary Differential Equations (ODEs), parallel and closed-loop manipulators introduce algebraic constraint equations $\Phi(q) = 0$: $$\begin{bmatrix} M(q) & \Phi_q^T \\ \Phi_q & 0 \end{bmatrix} \begin{bmatrix} \ddot{q} \\ \lambda \end{bmatrix} = \begin{bmatrix} \tau - C(q, \dot{q})\dot{q} - G(q) \\ -\dot{\Phi}_q \dot{q} \end{bmatrix}$$

2. Baumgarte Numerical Stabilization

Direct numerical integration of second-order constraint accelerations leads to constraint violation drift over time ($\|\Phi(q)\| > 0$). We apply Baumgarte stabilization: $$\ddot{\Phi} + 2\alpha \dot{\Phi} + \beta^2 \Phi = 0$$ Choosing critical damping $\alpha = \beta = 20\,\text{s}^{-1}$ guarantees exponential convergence of position and velocity constraint violations back to zero.