Reference Documentation

This section contains theoretical reference documentation for the skelarm robot kinematics and dynamics library. The content is derived from the project's foundational lessons.

The arm is modeled around a fixed base link of length \(l_0\) (the zeroth link), followed by the actuated links. So a "two-link arm" holds three links in total; in code, links[0] is the base and links[1:] are the movable joints.

Contents

1. Kinematics
   • Forward kinematics of a two-joint arm.
   • Analytic inverse kinematics (elbow-up / elbow-down).
2. Differential Kinematics
   • From a two-joint arm to the general \(n\)-joint recursion.
   • Recursive position, velocity, and acceleration.
   • Jacobian matrix and centripetal/Coriolis basis (with endpoint velocity/acceleration helpers).
3. Inverse Dynamics
   • Newton-Euler equations and mass properties.
   • Backward force and torque balance recursion.
   • Static-equilibrium validation.
4. Forward Dynamics
   • Lagrangian formulation.
   • System Inertia Matrix (\(H\)) and Bias Force Vector (\(b\)).
   • Equation of Motion: \(H\ddot{q} + b = \tau + J_E^{T} f_E\).
5. Numerical Methods
   • Linear Equation Solver (Gaussian Elimination).
   • Least-squares, pseudoinverse, and damped least-squares solves.
   • ODE Solver (Euler, Runge-Kutta).
6. Numerical Inverse Kinematics
   • Newton-Raphson, pseudoinverse, SR inverse, and Levenberg-Marquardt IK.
   • Sugihara-style residual-based damping for robust numerical IK.
7. Trajectory Tracking Control
   • Task-space trajectory planning and joint-space tracking.
   • PD, inverse-dynamics feedforward, and computed torque control.
8. Reaching Control
   • Reaching as a trajectory-tracking task.
   • Virtual spring-damper reaching and Seto/Sugihara online reference shaping for human-like motion.
9. Trajectory Filtering & Interpolation
   • Linear, natural cubic spline, and barycentric Lagrange interpolation.
   • First-order and Butterworth low-pass smoothing, applied zero-phase.