Why Dynamics Gets Hard Fast: Coupled Joints · 1/2
One pendulum is easy, an arm is not
A single pendulum swinging under gravity has dynamics that fit in one clean, well-known equation, torque depends on the pendulum's mass, its length, and the angle from vertical. It's a textbook example precisely because it's isolated, nothing else is moving that affects it.
A robot arm with several joints is a completely different kind of problem. Move the shoulder joint and you don't just move the shoulder, you swing the entire rest of the arm, elbow, wrist, and end effector, through space. That changes the effective inertia the shoulder motor has to fight, and it also changes the torque required at the elbow and wrist motors, since gravity and momentum are now acting on a mass distribution that has physically shifted.