Manipulator Kinematics

Joints ↔ poses. Forward kinematics with DH parameters, inverse kinematics, Jacobians, singularities and statics.

• Forward Kinematics & Denavit–Hartenberg Parameters — Where is the gripper for given joint angles? Learn the forward kinematics problem, link and joint parameters, the Denavit–Hartenberg (DH) convention (θ, d, a, α) and the modified DH variant, assigning frames step by step, building the DH table, multiplying link transforms, planar arm closed-form kinematics, typical 6-axis arm structures (spherical wrist), the product-of-exponentials alternative, and kinematic calibration — with NumPy code computing the pose of a 3-link arm with DH matrices and checking it against the closed form.
• Inverse Kinematics: Analytical & Numerical Solutions — Which joint angles put the tool where we want? Learn the inverse kinematics (IK) problem, existence (workspace) and multiplicity of solutions (elbow up/down, wrist flip, shoulder left/right), the analytical 2R solution with the law of cosines and atan2, kinematic decoupling for 6-axis arms with spherical wrists, numerical IK (Newton–Raphson with the Jacobian, gradient descent, damped least squares/Levenberg–Marquardt), joint limits, redundancy and null-space optimisation — with NumPy code comparing the analytical and numerical solutions of a 2R arm.
• Partial Differentiation — Partial derivatives measure change in one variable while holding others fixed. Learn first and higher-order partials, mixed derivatives and Clairaut's theorem, the multivariable chain rule, implicit differentiation, homogeneous functions and Euler's theorem, and applications to thermodynamics and fields.
• Velocity Kinematics, Jacobians, Singularities & Statics — How joint speeds map to tool speeds — and forces back to torques. Learn the manipulator Jacobian (geometric and analytical), deriving it for a 2R arm and from DH frames, end-effector velocities (v = Jq̇), inverse velocity control (resolved-rate), singularities (boundary, interior, wrist) and their consequences, manipulability and condition number, static force mapping τ = JᵀF, stiffness and force ellipsoids, and payload/torque checks — with NumPy code computing determinants near singularity, tip velocities and joint torques for a tip load.