Gyroscopic Stabilization of Unstable Vehicles: Configurations, Dynamics, and Control

We consider active gyroscopic stabilization of unstable bodies such as two-wheeled monorails, twowheeled cars, or unmanned bicycles. It has been speculated that gyroscopically stabilized monorail cars would have economic advantages with respect to birail cars, enabling the cars to take sharper curves and traverse steeper terrain, with lower installation and maintenance costs. A two-wheeled, gyro-stabilized car was actually constructed in 1913. The dynamic stabilization of a monorail car or two-wheeled automobile requires that a torque acting on the car from the outside be neutralized by a torque produced within the car by a gyroscope. The gyroscope here is used as an actuator, not a sensor, by using precession forces generated by the gyroscope. When torque is applied to an axis normal to the spin axis, causing the gyroscope to precess, a moment is produced about a third axis, orthogonal to both the torque and spin axes. As the vehicle tilts from vertical, a precession-inducing torque is applied to the gyroscope cage such that the resulting gyroscopic reaction moment will tend to right the vehicle. The key idea is that motion of the gyroscope relative to the body is actively controlled in order to generate a stabilizing moment. This problem was considered in 1905 by Louis Brennan [1]. Many extensions were later developed, including the work by Shilovskii [2], and several prototypes were built. The differences in the various schemes lie in the number of gyroscopes employed, the direction of the spin axes relative to the rail, and in the method used to produce precession of the spin axes. We start by deriving the equations of motion for a case where the system is formed of a vehicle, a load placed on the vehicle, the gyroscope wheel, and a gyroscope cage. We allow for track curvature and vehicle speed. We then derive the equations for a similar system with two gyroscopes, spinning in opposite directions and such that the precession angles are opposite. We linearize the dynamics about a set of equilibrium points and develop a linearized model. We study the stability of the linearized systems and show simulation results. Finally, we discuss a scaled gyrovehicle model and testing.