#
Design, Modeling, and Control of a Differential Drive Rimless Wheel That Can Move Straight and Turn^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Hardware

#### 2.1. Mechanical Design

#### 2.1.1. Rimless Wheels

#### 2.1.2. Torso

#### 2.2. Electronics

## 3. Controller

#### 3.1. Torso Pitch Controller

#### 3.2. Straight-Line Motion

#### 3.3. Turning Motion

## 4. Modeling and Simulation

#### 4.1. Steering Model

#### 4.2. Sagittal or Fore-Aft Plane Model

#### 4.2.1. Equations of Motion for the Stance Phase

#### 4.2.2. Support Transfer Conditions

#### 4.2.3. Equations of Motion for the Support Transfer Phase

- (1)
- Support transfer is through left and right side spokes simultaneously

- (2)
- Support transfer is through left side spoke only

- (3)
- Support transfer is through right side spoke only

#### 4.3. Motor Torque and Power Model

#### 4.4. Computer Simulation

## 5. Results

#### 5.1. Straight-Line Motion

#### 5.2. Turning Motion

#### 5.3. Parameter Studies

## 6. Discussion

^{2}, and ℓ is leg length). Outrunner, the 0.6 m tall, 6 legged rimless wheel had a Froude number of 5.14 and HexRunner, the 1.83 m tall, 6 legged rimless wheel had a Froude number of 4.82. Thus, RR2 is twice as slow as these other rimless wheels. To increase the speed of the rimless wheel we need to reduce the collisional losses. We can do this either by decreasing its mass or increasing the number of spokes. Alternately one can add more energy to the system during the single stance phase by increasing the pitch angle or increasing the distance of the center of mass from the center of the wheel.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- A video of the hardware prototype available at this YouTube link: https://youtu.be/SNxeP29ayhI (accessed on 15 July 2021).
- Supplementary materials including simulations, animation, mechanical design, and robot code are on github: https://github.com/pab47/RoadRunner2 (accessed on 15 July 2021).
- The MATLAB and Blender code with test examples are on github: https://github.com/EzAme/DR2 (accessed on 15 July 2021).

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**Figure 2.**Mechanical Design: (

**a**) A single rimless wheel with compliant spokes, and (

**b**) torso that includes the transmission using belt drives, the motors, the sensors, and the batteries.

**Figure 3.**Schematic of the electronics layout: The Raspberry Pi does all the high-level data management including saving the data and receiving inputs from the Odrive motor controller, the inertial measurement unit, and the joystick. The Teensy does the body pitch control using the body pitch data from the Raspberry Pi. The Odrive regulates the motor current based on the input from the Teensy.

**Figure 4.**Schematic of the feedback controller: The feedback controller controls the torso pitch using a Proportional-Integral-Derivative (PID) controller at 1000 Hz.

**Figure 5.**Steering model adapted from differential drive systems: A top view of the robot. (

**a**) The world or fixed frame ${X}_{0}-{Y}_{0}$ is shown to coincide with robot local frame. (

**b**) The local frame ${X}_{1}-{Y}_{1}$ is attached to the robot and moves with it. The steer or heading angles is $\beta $ and the point C is midway between two wheels have the coordinates ${x}_{c}^{1},{y}_{c}^{1}$ in the local frame and ${x}_{c}^{0},{y}_{c}^{0}$ in the world or fixed frame. For the steering model, a differential equation for the position of C, ${\dot{x}}_{c}^{0},{\dot{y}}_{c}^{0}$ and heading $\dot{\beta}$ is found.

**Figure 6.**Planar model: The robot is progressing from left to right. (

**a**) The robot parameters used in the simulation, (

**b**) the robot at an instant before support transfer, and (

**c**) the robot at an instant after support transfer. All state variables before and after support transfers are appended a superscript

^{−}and

^{+}respectively. The point P is the current spoke touching the ground and point Q is the spoke that will touch the ground on the next step. Note that the robot has 10 legs on each side but only 8 spokes are shown in the illustration.

**Figure 7.**Experimental data for straight line motion for 1 complete revolution or 10 steps of the rimless wheel: (

**a**) Torso pitch, (

**b**) absolute speed of the robot, (

**c**) motor torque, and (

**d**) power, all as a function of time. The mean is shown with black dashed line and the blue bands show one standard deviation. The simulation data is shown by red solid line. The measurements had considerable noise leading to a relatively large standard deviation.

**Figure 8.**Experimental results for robot turning (

**a**) Trajectory of the robot during turning; (

**b**) Shortest turning radius of $0.5$ m.

**Figure 9.**Simulation data for turning: (

**a**) Trajectory for circular turn, (

**b**) torso angle, (

**c**) torso velocity, (

**d**) absolute angle of the wheel, (

**e**) absolute angular speed of the wheel, (

**f**) torque on the torso, and (

**g**) total motor power.

**Figure 10.**Parameter studies: Effect on speed, torque, power, and COT for motor as a function of (

**a**) torso pitch angle, (

**b**) number of spokes, (

**c**) torso center of mass, and (

**d**) mass of the wheel. The current design of the rimless wheel is shown by a dot.

**Table 1.**Comparison of experiment (mean values) on concrete with simulation for torso pitch set-point of ${50}^{\circ}$.

Quantity | Experiment | Simulation | |
---|---|---|---|

Power | Pi & Sensors | 5 W | |

Teensy | 0.2 W | ||

Motors | 5.18 W | 5.17 | |

Total Power | 10.38 W | 10.37 W | |

Mean Torque | 1.41 Nm | 1.5 Nm | |

Mass | 6.9 kg | ||

Mean Velocity | 0.936 m/s | 0.897 m/s | |

ACOT (see Equation (25)) | 0.08 | 0.085 | |

TCOT (see Equation (26)) | 0.164 | 0.171 |

Surface | TCOT | Avg. Velocity (m/s) |
---|---|---|

Polished Concrete | 0.16 | 0.936 |

Polished Wood | 0.15 | 1.197 |

Indoor Running Track | 0.14 | 1.268 |

Outdoor Running Track | 0.13 | 1.280 |

Asphalt | 0.13 | 1.379 |

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**MDPI and ACS Style**

Sanchez, S.; Bhounsule, P.A. Design, Modeling, and Control of a Differential Drive Rimless Wheel That Can Move Straight and Turn. *Automation* **2021**, *2*, 98-115.
https://doi.org/10.3390/automation2030006

**AMA Style**

Sanchez S, Bhounsule PA. Design, Modeling, and Control of a Differential Drive Rimless Wheel That Can Move Straight and Turn. *Automation*. 2021; 2(3):98-115.
https://doi.org/10.3390/automation2030006

**Chicago/Turabian Style**

Sanchez, Sebastian, and Pranav A. Bhounsule. 2021. "Design, Modeling, and Control of a Differential Drive Rimless Wheel That Can Move Straight and Turn" *Automation* 2, no. 3: 98-115.
https://doi.org/10.3390/automation2030006