Kinematics (FK / IK)

$x = f(q)$

Chain each joint's local transform (from DH parameters, or as a product of exponentials of screw axes) to get the end-effector pose from the joint angles. Always well-defined — one set of joint angles gives exactly one pose.

$\dot x = J(q)\dot q$

$J$ is the matrix of partial derivatives of the pose with respect to each joint angle — it linearizes the FK map at the current configuration and changes as the arm moves.

A singularity occurs where $J(q)$ loses rank: some end-effector directions become momentarily unreachable no matter how fast the joints move (e.g. an arm fully extended).

Manipulability ($\propto \sqrt{\det(JJ^\top)}$) shrinks toward zero near a singularity — a scalar sense of how far the arm is from losing a direction of motion.

$\tau = J^\top F$

The same Jacobian that maps joint velocity to end-effector velocity also maps end-effector force/wrench back to joint torques — no separate force model needed.

Closed-form: exact algebraic solution for specific arm geometries (e.g. spherical wrist) — fast, but only exists for particular structures.

Numerical (Newton-Raphson / damped least squares): iterate $q \leftarrow q + J^+ (x_{des}-f(q))$, using the pseudoinverse (or a damped version near singularities) — general-purpose but iterative and only locally convergent.

An IK problem can have no solution (target outside the workspace), a finite number of solutions (e.g. elbow-up/down), or infinitely many (redundant arm).

An arm with more than 6 joints has a null space: joint motions $\dot q_n$ satisfying $J\dot q_n = 0$ that move the joints without moving the end-effector at all.

Resolved-rate control exploits this: $\dot q = J^+ \dot x_{des} + (I - J^+J)\dot q_0$, using the leftover freedom $\dot q_0$ for a secondary goal (e.g. avoiding joint limits or obstacles) while still tracking the primary task.