A ballbot is a dynamically-stable mobile robot designed to balance on a single spherical wheel (i.e., a ball). Through its single contact point with the ground, a ballbot is omnidirectional and thus exceptionally agile, maneuverable and organic in motion compared to other ground vehicles. Its dynamic stability enables improved navigability in narrow, crowded and dynamic environments.
The ballbot works on the same principle as that of an inverted pendulum. As such, the ballbot can be broken down into three major components: The pendulum, the control system, and the drive unit. The pendulum component, and therefore the control system, can further be broken down from a single three dimensional coordinate system to pair of two dimensional coordinate systems.
Because the pendulum is broken down into two dimensional models, the control systems computes the required X and Y axis components to control the system. Using a linear quadratic regulator and discrete state space controller, the control system can use the angle from vertical to produce optimized control for the desired states. These states can be summed to represent a resultant vector in the XY plane.
This resultant vector can be converted to motor speeds though kinematic equations for the interaction between the motors, wheels, and ball.
CAPSTONE project by Adam Woo, Graham Goodwin, and Chloe Desjardins
In essence, a ballbot is an omnidirectional inverted pendulum. This means that the ballbot can be broken down into two inverted pendulums that travel along the X and Y axis. The angle of the robot along these axis at any given time can be used to calculate the necessary velocity required to keep the pendulums upright.
The X and Y components can be summed to a desired resulting vector, which can be used to maintain the robots verticality in 3D space.
The state equations for an inverted pendulum can be represented as:
Where the variables are:
These state space equations are used to produce a state space controller, where the output of the controller, , is the desired ball velocity along the X or Y axis.
These velocities can be plugged into a set of kinematic equations that calculate the required speed of each motor. The motor speeds much sum all torques around the Z axis to 0 for our system. These equations can be represented as such:
The scalar accounts for the angle of the wheels relative the the center of the ball. This shifts the resultant velocity to the top surface of the ball.
To solve for the motor speeds, the inverse of the 3x3 matrix was found. When arranged to solve for the desired motor speeds, the equations look as follows:
These equations can be used to calculate the desired speed of each motor in m/s, which can then be converted to steps/s based on the length of each step. The steps/s is the frequency at which pulses will be sent to the motor drivers.
Matlab was used for all of the control system modeling.
Here is a list of the selected hardware:
| Component | Part | Description |
|---|---|---|
| Microcontroller | Teensy 3.6 | Faster processor speed, easy plug and play setup, and an extensive community make the Teensy 3.6 an ideal choice for the microcontroller over less powerful options like an Arduino Mega, or a less supported board. |
| IMU | MPU6050 | The 6 degrees of freedom supported by the MPU6050 allow for accurate angle measurements for a very cheap price. |
| Ball | 10" Stainless Steel Mirror Ball w/ Plastidip | A 10" steel ball provides a round ridged surface strong enough to support the robots chassis. The ball is coated in Plastidip, a rubberlike spray that will increase the friction on the ball. |
| Wheels | 3x 3.25" VexPro Omniwheel | Omniwheels allow for free movement perpendicular to the wheel. This allows for the development of holonomic movement systems. |
| Motors | 3x Nema 23 Stepper Motor | Stepper motors provide the high level of precision and torque at low speeds that the system will need to remain stable. |
| Motor Drivers | 3x TB6600 Stepper Motor Driver | These drivers allow the stepper motors to be controlled with only two data pins; one for direction and one for speed. |
| Battery | Gens ace 5000mAh 11.1V LiPo | The battery must be at least 9V. This battery was a highly rated battery with a large capacity, sufficient voltage, and relatively low weight. |
| Universal Battery Elimination Circuit | BW® RC Servo BEC UBEC 3A 5V | An UBEC is a voltage regulating circuit used to take power from the battery to the microprocessor. |
The driver was manually set to the following:
- Microstepping: 1/8
- Current: 2.8A
The step pins output a square wave of varying frequencies. Pins 2, 3, and 5 where chosen because they all run on different timers. Therefore, the frequencies can be adjusted independently.
All programming was done using the Atom text editor, Arduino IDE, and Teensyloader programs.
Libraries
- discreteSSC.h by Adam Woo
- quaternionFilter.h by Kris Winer
- Math.h
- Wire.h
- TimerOne.h
- FrequencyTimer2.h
- TimerThree.h
The ballbot provided limited stability for the 2D control systems. While this could be for several reasons, a likely reason for the instability is that there are forces not accounted for by a two dimensional model of the system.
The longest recorded stability of the robot was approximately 10 seconds.