Forward Kinematics (FK) computes end-effector pose $\mathbf{T}_{b,e} \in SE(3)$ given joint angles $\mathbf{\theta} \in \mathbb{R}^6$ using the **Product of Exponentials (PoE)** formulation over Lie Algebra $se(3)$:

where $\mathbf{S}_i$ are joint screw axes and $\mathbf{M} \in SE(3)$ is the home configuration pose. Inverse Kinematics (IK) computes $\mathbf{\theta}$ given target pose $\mathbf{T}_{\text{target}}$. Analytical solvers (IKFast) solve closed-form polynomial roots in $<2\mu\text{s}$ for arms with spherical wrists. For numerical IK, compute joint velocity updates $\mathbf{\dot{\theta}}$ via the Manipulator Jacobian $\mathbf{J}(\mathbf{\theta}) \in \mathbb{R}^{6 \times 6}$ ($\mathbf{\dot{x}} = \mathbf{J}(\mathbf{\theta})\mathbf{\dot{\theta}}$).

At a kinematic singularity, $\det(\mathbf{J}(\mathbf{\theta})) = 0$, causing pseudo-inverse $\mathbf{J}^+ = \mathbf{J}^T (\mathbf{J}\mathbf{J}^T)^{-1}$ entries to approach infinity ($\mathbf{\dot{\theta}} \to \infty$). To prevent wild joint motor spikes, implement **Damped Least Squares (DLS / Levenberg-Marquardt)** inverse kinematics:

where $\lambda$ is an adaptive damping factor adjusted based on the smallest singular value $\sigma_{\min}$ of $\mathbf{J}$:

Communication latency $T_{\text{delay}} > 20\text{ms}$ introduces positive phase shift into bilateral haptic feedback loops, causing energy growth and violent physical instability. To guarantee stability, implement **Wave Variable Transformation (Passivity Control)**:

where $b$ is the characteristic wave impedance. Wave variables encode force and velocity signals into passive energy flows, guaranteeing passivity ($\int_0^t P(\tau) d\tau \ge 0$) under arbitrary constant transmission delays.