Control of the Gyrover: a single-wheel gyroscopically stabilized robot

Abstract - The Gyrover is a single-wheel gyroscopically stabilized mobile robot developed at Carnegie Mellon University. An internal pendulum serves as a counter weight for a drive motor that causes fore/aft motion, while a large gyroscope on a tilt mechanism provides for lateral balance and steering actuation. In this paper, we develop a detailed dynamic model for the Gyrover and use this model in an extended Kalman filter to estimate the complete state. A linearized version of the model is used to develop a state feedback controller. The design methodology is based on a semi-definite programming procedure which optimizes the stability region subject to a set of linear matrix inequalities that capture stability and pole placement constraints. Finally, the controller design combined with the extended Kalman filter are verified on the robot prototype. Keywords: Gyroscope; symbolic modeling; robot control; linear matrix inequalities.

INTRODUCTION

The concept of a single-wheel gyroscopically stabilized robot was originally proposed by Brown and Xu. The idea is to take advantage of the inherent dynamic stability of a single wheel, but augment it with a mechanical gyroscope to affect steering and low-speed balance. The self-stabilizing dynamics of a single wheel can be illustrated as follows. Consider a single wheel rolling down a hill. When the wheel leans laterally, gyroscopic precession causes it to turn in the direction it is leaning and the 'centrifugal' forces resulting from the curved motion path tend to right the wheel.'Brown and Xu point out that it is paradoxical that those factors that produce static stability may actually contradict dynamic stability. A four-wheeled car has excellent static stability but is prone to roll-over when it hits a bump or takes a curve at high velocity.

Past research on the Gyrover focussed entirely on the mechanical design. After some initial tests to verify the concept, a simplified dynamic model was developed to weigh the different design characteristics: static stability versus high-speed dynamic responsiveness, slope climbing ability, etc. Based on this model, several generations of Gyrovers have been built with gradually increasing sophistication, reliability and performance.

So far, the Gyrover has been controlled using a remote control transmitter that allows the user to control the voltage of the drive motor and the angle of the tilt mechanism. Due to the coupling between the fore/aft and lateral motions and the lack of attitude sensing on the Gyrover, the user has to develop a feeling for the dynamics of the robot, estimate its current attitude by visual inspection and provide the appropriate input commands. Because of the self-stabilizing dynamics of the Gyrover, it is relatively easy for a novice user to keep it from falling over, especially when moving at moderate and high speeds. However, it is much more challenging to track a desired trajectory and nearly impossible to control the robot when it is out of sight.

To use the Gyrover for inspection tasks in which fine control in remote locations is required, we need to develop a controller that relieves the user from stability concerns and provides an intuitive control interface. Au and Xu recently developed a decoupled linear state feedback controller based on a simplified model of the Gyrover. Simulation results demonstrate this controller's ability to balance the Gyrover laterally. This paper presents the development of a more general controller based on a comprehensive dynamic model.

GYROVER SYSTEM

Overall description

The Gyrover is a single-wheel robot that is stabilized and steered with an internal, mechanical gyroscope. The Gyrover can stand and turn in place, move deliberately at low speed, climb moderate grades, and move stably at high speeds even on rough terrain. It has a relatively large rolling diameter which facilitates motion over rough terrain, and a single track and narrow profile for obstacle avoidance. It can be completely enclosed for protection from the environment.

The Gyrover consists of four rigid bodies connected to each other through a 3-d.o.f. kinematic chain: the wheel, the pendulum, the tilt mechanism and the gyroscope.

Tire and wheel. The wheel is the only element that is in direct contact with the environment. It consists of a rim and two polycarbonate domes that connect the rim to the axle. The Gyrover uses a lightweight, 16 inch rim, tire and innertube of the type used in racing wheelchairs.

Pendulum. The main body of the Gyrover hangs as a pendulum from the axle of the wheel. The pendulum includes a DC motor and transmission that drive the wheel shaft. With gravity acting as reaction torque, this drive mechanism generates forward acceleration and braking for the Gyrover. The forward drive system uses a two-stage, toothed belt transmission system with an approximate gear ratio of 13:1.

Gyroscope. The stabilizing gyroscope is the heart of the Gyrover mechanism. The angular momentum of the rotating mass provides stability and a reference against which the Gyrover wheel can be tilted by the tilt motor or “servo”. The gyroscope is housed in a fiberglass and aluminum housing, rotating on precision ball bearings and mounted in rubber vibration isolators. An integrated brushless DC motor spins the gyroscope to operating speed, controlled by a speed-control unit mounted outside the housing. It maintains a constant angular velocity of "approximately 15 000 r.p.m. Because the motor is too small to generate any sudden change in angular velocity, we do not use this degree of freedom for control purposes. In the remainder of the paper, we will therefore assume that the angular velocity of the gyroscope is constant. The gyro requires about 1 min to accelerate to operating speed (longer on recent versions with higher speed spin motors) and about 20 min to spin down to a stop after the power is removed.

Gyroscope tilt servo. The tilt servo controls the relative angle of the gyroscope spin axis with respect to the wheel axis and pendulum. This rotation axis is perpendicular to the main axle and is located below the axle on the sagittal plane. The servo is a very high torque unit that provides the torque to cause the wheel to lean relative to the gyroscope. This torque, acting to balance the wheel against gravity, is what leads to the yaw precession that produces the steering effect. For example, when the forward velocity is zero, one can rotate the Gyrover to the left by leaning it slightly to the left. The gyroscopic effect stops the Gyrover from falling over and simultaneously induces a positive rotation around the vertical axis steering the robot to the left.

Computer and custom I/O board. A custom-built circuit board contains the control computer and flashdisk, the interface circuitry for the radio system and servos, components and logic to control power for the actuators, and an interface for the on-board sensors. The on-board computer, a Cardio(TM) 486 PC 100 MHz, can be operated as a conventional PC by connecting a standard keyboard, monitor and mouse. It operates using the QNX(TM) real-time operating system. It also includes a radio system for remote control (JR Model XP783A), that can operate independently of the computer control system.

Sensors and instrumentation. A number of on-board sensors have been installed on the Gyrover to measure its state.

Dynamics

The development of the state estimator and controller of the Gyrover builds on the dynamic equations. The dynamics of the Gyrover is described by a set of highly coupled nonlinear differential equations. The derivation of the dynamic equations for the Gyrover presented here is based on the Newton-Euler approach. Previous derivations of the dynamic equations were based on a Lagrangian approach with simplifying geometric assumptions for simulation purposes. In our derivation, we make the following assumptions:

• All the components are rigid bodies.

• The wheel rolls without slipping.

• The friction model for the contact between the wheel and the floor, and between the drive motor and transmission includes Coulomb and viscous friction.

• The angular velocity of the gyroscope is constant.

• The wheel and gyroscope are axially symmetric.

• The floor is flat and horizontal.

• The wheel remains in contact with the ground.

Unlike the Newton – Euler dynamics for fixed-base manipulators, the Gyrover dynamics cannot be calculated numerically in an iterative fashion. For fixed-base manipulators, the acceleration of the base is known and fixed, so that the accelerations of the distal links can be computed sequentially. Once all the accelerations are known, the reaction forces can be computed in an inward iteration from the end-effector towards the base. However, since the accelerations of the wheel of the Gyrover are not fixed but depend on the accelerations of the internal degrees of freedom, one cannot evaluate the Newton-Euler equations numerically. Instead, the complete dynamics need to be derived symbolically after which the contact constraints can be imposed.

Both kinematic and force constraints need to be considered at the contact point. Rolling without slipping imposes constraints on the wheel accelerations.