Kinematics

Kinematics describes motion without asking what causes it. Given a body that moves, we want to express its position, velocity and acceleration as functions of time, using the coordinate system best suited to the problem at hand. This purely geometric study is a prerequisite for kinetics, where forces enter the picture through Newton’s second law, but it also stands on its own in applications such as robotics, mechanism design and motion planning.

We develop particle kinematics first, treating bodies as points that translate without rotating. Position vectors, their time derivatives, and the resulting velocity and acceleration vectors are derived in Cartesian, polar and natural (tangent-normal) coordinate systems. Rather than memorizing a catalogue of kinematic formulas for special cases, we formulate each problem as an initial value problem and solve the governing differential equations directly, using SymPy for exact solutions and SciPy for numerical integration. This computational approach generalizes immediately to problems that admit no closed-form solution, which in practice means most problems worth solving.

We then apply the same vector and reference-frame machinery to a concrete robotics example: computing the end-effector position of a multi-link robotic arm through successive coordinate transformations. This forward kinematics problem demonstrates how rotation matrices and systematic frame-by-frame modelling extend naturally from the particle case to articulated systems. Rigid body kinematics, where angular velocity and the constraints of rigidity enter the description, completes the part.