This is a live 3D render of the racket's actual orientation — not a canned animation. Its shape comes directly from the geometry sliders. The rotation each frame comes from numerically solving the real rigid-body equations of motion (Euler's equations) for the spin it started with. If Control is off, you're watching the raw spin of the racket. If Control is on, an attitude controller is actively steering it toward the chosen target axis.

The Hamiltonian represents the total energy of the system, in this case, the racket's rotational kinetic energy. The dashed lines show the energy the racket would have if spinning purely about each of its three axes. The racket's actual energy (solid line) stays trapped between the smallest and largest of these forever. With Control on, the dashed line becomes the chosen target energy instead, and the solid line is no longer flat — it climbs or falls as the controller does real work to get there.

This display shows the spin rate (radians/second) about each of the racket's principal axes, over time — long, intermediate, and short. Near the intermediate axis, all three swing through large, coupled oscillations instead of settling down, which is the visible signature of the instability. Near a stable axis (long or short), one component dominates while the other two stay as small, bounded wobbles.

A live 3D render of the pendulum's orientation, driven by Euler's equations for a rigid body, with gravity, friction, and the control torque. Since this rod and bob pivot about a fixed point, gravity is constantly pulling the pendulum toward hanging straight down. If Control is off, you're watching it swing freely. If Control is on, an attitude controller actively cancels gravity and steers it to the chosen target.

The Hamiltonian represents the total energy of the system — for this pendulum, that's kinetic energy plus gravitational potential energy. With no friction and no control, this value stays constant, since gravity alone conserves it. Friction removes energy from the system slowly, causing the line to drift downward over time. With Control on, the dashed line is the target energy, and the solid line (energy of the system) climbs or falls as the controller does real work to get there.

This plot displays the spin rate (radians/second) about each of the pendulum's three principal axes, all measured through the pivot. The axial axis runs along the rod itself — spin about it is like twirling the rod between your fingers, not swinging it anywhere. It's nearly free, since only the bob's own small radius resists it. Transverse 1 and Transverse 2 are perpendicular to the rod and to each other, spanning the plane the bob can actually move in. By the bob's spherical symmetry they're identical to each other, and together they carry almost all of the pendulum's real swinging motion.

IDA-PBC (Interconnection and Damping Assignment Passivity-Based Control) is the technique steering the pendulum when Control is on. It reshapes the pendulum's energy landscape so the target orientation becomes the new low point. It cancels gravity's pull at the current orientation, then adds a spring-like pull toward the target (plus damping that removes leftover spin) so the pendulum settles instead of oscillating forever. The torque line shows the magnitude of that push; the work total is the real energy spent to get there.