A controller needs to know what to command, and for a robot arm that means understanding its geometry and its physics. Kinematics describes the motion of the robot, how joint angles relate to the position and orientation of the hand, without worrying about forces. Dynamics adds the forces and torques that actually cause that motion.
You'll start with forward kinematics: given the joint angles, where is the end-effector? Then you'll flip the question with inverse kinematics: given the pose you want, which joint angles get you there, and why there may be zero, one, or several answers. You'll also meet the Jacobian, the tool that links joint speeds to hand speed.
Finally you'll move on to robot dynamics. You'll see how the Lagrangian and Newton-Euler methods give the joint torques needed to produce a desired motion, including the mass, velocity, and gravity terms. These ideas carry straight into ROS 2 & Robot Software, motion planning, and simulation.
How to find where a robot's end-effector is from its joint values, using a simple planar arm to build the general product of exponentials formula for forward kinematics.
The Jacobian that maps joint velocities to the end-effector's velocity, and how each column is built from the joints between that joint and the base.
The inverse kinematics problem of finding joint positions for a desired end-effector pose, why it can have zero, one, or many solutions, and a worked analytic solution for a 2R arm.
How the Lagrangian method (kinetic minus potential energy) gives a robot's equations of motion, and what the mass matrix, velocity-product, and gravity terms mean.
The recursive Newton-Euler algorithm, with forward passes for link motion and backward passes for the joint forces and torques needed to produce a desired motion.