# Design of a Spherical Rover Driven by Pendulum and Control Moment Gyroscope for Planetary Exploration

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Spherical Robots: State of the Art

#### 1.1.1. Barycenter Offset

#### 1.1.2. Shape Transforming

#### 1.1.3. Angular Momentum

#### 1.1.4. Other Spherical Robots

#### 1.1.5. Contribution of This Work

- An original design of the power transmission for a hybrid BO-CMG spherical robot.
- A method for the optimization of the size of the gyroscopes according to the specifications and the available volume in the shell.
- A control architecture with fuzzy gain scheduler for tracking straight trajectories.

## 2. Mission Specifications

## 3. Functional Analysis of the Driving Mechanism

#### 3.1. Analysis of the BO Principle

#### 3.2. Analysis of the CMG Principle

#### 3.3. Combining BO and CMG

^{th}of the radius, as noted in [1]. In Figure 5 is considered $\theta =0$, as this is the initial condition. However, the gyroscopic torque’s useful part changes with tilt angle, as from Equation (2). As the gyroscopes tilt, the effective distance decreases until it matches the initial value $a$. Yet, the torque for climbing a step decreases as the robot ascends. Also, ${a}^{*}$ is not solely determined by gyroscopic torque but also by the system’s total mass.

## 4. Design

#### 4.1. Overview of the Previous Design Based on 2-DOF Pendulum

#### 4.2. New Design Based on Pendulum-CMG Mechanism

#### 4.2.1. Design of CMG

**Flywheel Dimensioning**

**Drag forces on a high-speed spinning flywheel**

_{a}air density, ω angular velocity and ${C}_{M}$ drag torque coefficient (${C}_{M}=3.87R{e}^{-\frac{1}{2}}$ if $Re<5\xb7{10}^{4}$, ${C}_{M}=0.146R{e}^{-\frac{1}{5}}$ if $Re\ge 5\xb7{10}^{4}$). Table 3 reports the data used for the drag torque computation.

**Spinning motors**

**Gyroscope assembly**

#### 4.2.2. Pendulum Actuators

#### 4.2.3. Electrical Integration

**Battery dimensioning**

**CMG impact on autonomy**

#### 4.3. Design Result and Considerations

## 5. Modelling and Control

#### 5.1. Two-Dimensional Analytical Model

#### 5.1.1. Two-Dimensional Kinematics

_{i}and s

_{i}are the cosine and the sine of the generalized variable q

_{i}, respectively.

#### 5.1.2. Two-Dimensional Dynamics

_{p}is the total inertia of the pendulum.

_{M}is the motor torque, and ${\mathsf{\tau}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{n}\mathrm{c}}$ are the torques due to external non-conservative forces, I

_{n}is equal to:

_{1}and β

_{2}.

#### 5.2. Straight-Trajectory Control Strategy

_{b}. It can be demonstrated that a proper tuning of this value can make the inner control loop much faster than the external one, allowing to ignore the motor internal dynamics. Specifically, a higher ω

_{b}leads to a wider bandwidth of the current loop. However, a trade-off must be made, as increasing this value also leads to an increase in the requested armature voltage.

#### 5.2.1. The Viscous Friction Problem

_{1}and β

_{2}. At the same time, the absence of a damping term causes the model to be unstable. The problem has been solved by including the damping effect in the control input, where the pendulum angular speed is multiplied by a constant term β and subtracted to the speed control output. This solution allowed to stabilize the system regardless of the presence of the viscous friction in the plant model. Referring to Figure 12, the input to the plant is equal to:

#### 5.2.2. The Speed Controller

- The input is the error between the desired and actual angular velocity; the outputs are the three PID gains. Both the input and the output variables are identified through “Linguistic Variables”: “error” was used to refer to the input signal, and “P-gain”, “I-gain” and “D-gain” to the output ones.
- The linguistic values associated with the error are “Ze”, “S”, “M”, “L” and “XL”.
- The ones associated with the outputs are:
- P-gain: “XS”, “S”, “M”, “L” and “XL”;
- I-gain: “S” and “M”;
- D-gain: “S” and “M”.
- The rules of the fuzzy gain scheduler are depicted in Figure 16. The “centroid defuzzification method” was adopted.

_{1}, namely, k

_{p}= 4, and k

_{i}= 10. As already mentioned, to control the activation of the P${\mathrm{I}}^{\mathrm{*}}$ controller, a set/reset block has been incorporated before it. This block enables the controller to operate only when the error is close to zero. Specifically, if the speed reference is greater than 4 rad/s, the controller starts working when the error is less than the 8% of the speed reference, while it is deactivated if it overcomes the 20%. For speed references below 4 rad/s, the percentage error is calculated relative to this velocity. The use of different threshold values (8% and 20%) for activating and deactivating the controller avoids chattering issues.

## 6. Results

#### 6.1. Simulation Environment

#### 6.2. Analysis of the Performance of the Control Strategy

#### 6.2.1. Test with the Analytical Model

#### 6.2.2. Test with the Multibody Model

#### 6.3. Analysis of the Performance of the CMG for Step Overcoming

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Conceptual design of a BO-CMG spherical robot. (

**b**) Front view of CMG activated, with schematics of gyroscopic torque components.

**Figure 4.**Free body diagram of the robot while climbing a step (static equilibrium condition). (

**a**) system; (

**b**) pendulum; (

**c**) spherical shell.

**Figure 5.**MSH-over-radius curve against Maximum Gyroscopic Torque. Total mass and barycenter position of the robot from [1].

**Figure 6.**(

**a**) Driving mechanism and pendulum. (

**b**) Assembly of the SR with the layer of harmonic steel sheets [1].

**Figure 7.**(

**a**) Front view of the gyroscope system; (

**b**) cross-section; (

**c**) 3D of the assembly. (1) First half of the case, (2) shaft, (3) second half of the case, (4) jaw coupling, (5) spacer, (6) small angular contact ball bearing, (7) big angular contact ball bearing, (8) motor plate, (9) C-plate for tilting motor connection, (10) flange

_{1}for main structure connection, (11) flange

_{1}for tilting motor-gyroscope connection, (12) tilting motor, (13) angular contact ball bearings for tilting axis positioning, (14) flange

_{2}for tilting motor-gyroscope connection, (15) flange

_{2}for main structure connection, (16) motor-to-C-plate connection plate, (17) closing Plate and (18) Spinning Motor.

**Figure 10.**Two-dimensional representation of the robot climbing a slope. q1 and q2 are the generalized variables.

**Figure 11.**Block schemes of the control strategy: (

**a**) control applied to the analytic model; (

**b**) control applied to the real robot.

**Figure 15.**PID gains membership functions: (

**a**) k

_{p}membership function; (

**b**) k

_{i}membership function; (

**c**) k

_{d}membership function.

**Figure 17.**Linear speed of the analytical robot model vs input speed profile. (

**a**) Step; (

**b**) trapezoidal profile.

Target | Earth, Moon, Mars |

Terrain | Clay, Sand, Grass, Mineral, Water |

Min. Step height | 100 mm |

Min. Slope | 15° |

Min. Velocity | 2.5 m/s |

Min. Acceleration | 0.5 m/s^{2} |

Max Diameter | 0.5 m |

Max Mass | 25 kg |

Autonomy | 1 h |

Computed | Final Design | |
---|---|---|

${m}_{fl}$ | $3.2\mathrm{k}\mathrm{g}$ | $3.35\mathrm{k}\mathrm{g}$ |

${I}_{fl}$ | $9.75\xb7{10}^{-3}\mathrm{k}\mathrm{g}\xb7{\mathrm{m}}^{2}$ | $9.8\xb7{10}^{-3}\mathrm{k}\mathrm{g}\xb7{\mathrm{m}}^{2}$ |

${R}_{ext}$ | 66 mm | 66 mm |

${R}_{int}$ | 42 mm | 40 mm |

${h}_{fl}$ | 51 mm | 51 mm |

${\rho}_{a}$ | $1.204\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ | Air density at 20 °C |

$\omega $ | $8420\mathrm{r}\mathrm{p}\mathrm{m}$ | Angular speed |

$l$ | $0.051\mathrm{m}$ | Thickness of the rotor |

${R}_{ext}$ | $0.066\mathrm{m}$ | External radius |

$\mu $ | $1.81\xb7{10}^{-5}\mathrm{k}\mathrm{g}/\mathrm{m}\mathrm{s}$ | Dynamic viscosity of air at 20 °C |

${R}_{int}$ | $0.005\mathrm{m}$ | Radius of the flywheel shaft |

${\tau}_{nom,s}$ | $13.96\mathrm{N}\mathrm{m}$ | Nominal torque at the shaft |

${i}_{r}$ | $81:1$ | Gear ratio |

${P}_{r}$ | $75\mathrm{W}$ | Output power at the gearbox |

${\tau}_{nom,r}$ | $7.5\mathrm{N}\mathrm{m}$ | Nominal torque at the gearbox |

${\tau}_{max,r}$ | $22\mathrm{Nm}$ | Maximum torque at the gearbox |

${\eta}_{D}$ | 0.72 | Efficiency of the gearbox |

${P}_{m}$ | $104\mathrm{W}$ | Output power of the motor |

${\tau}_{nom,m}$ | $0.129\mathrm{N}\mathrm{m}$ | Nominal torque of the motor |

${\tau}_{max,m}$ | $0.276\mathrm{N}\mathrm{m}$ | Maximum torque of the motor |

$R$ | $250$mm | Radius of the sphere |

${M}_{tot}$ | $22\mathrm{k}\mathrm{g}$ | Total mass of the system |

${M}_{sf}$ | $6\mathrm{k}\mathrm{g}$ | Sphere mass |

${m}_{p}$ | $16\mathrm{k}\mathrm{g}$ | Pendulum mass |

$a$ | $87.5\mathrm{m}\mathrm{m}$ | Robot COM—sphere center distance |

$l$ | $120\mathrm{m}\mathrm{m}$ | Pendulum COM—sphere center distance |

${\tau}_{90}$ | $18.88\mathrm{N}\mathrm{m}$ | Torque to raise the pendulum at 90° angle |

$\omega $ | $8000\mathrm{r}\mathrm{p}\mathrm{m}$ | Flywheel spinning velocity |

$\mathsf{\Omega}$ | $15\mathrm{r}\mathrm{p}\mathrm{m}$ | Flywheel tilting velocity |

${\mathrm{I}}_{fl}$ | $9.76\xb7{10}^{-3}\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}$ | Flywheel inertia |

${\tau}_{G}$ | $25.8\mathrm{N}\mathrm{m}$ | Maximum gyroscopic torque |

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## Share and Cite

**MDPI and ACS Style**

Melchiorre, M.; Colamartino, T.; Ferrauto, M.; Troise, M.; Salamina, L.; Mauro, S.
Design of a Spherical Rover Driven by Pendulum and Control Moment Gyroscope for Planetary Exploration. *Robotics* **2024**, *13*, 87.
https://doi.org/10.3390/robotics13060087

**AMA Style**

Melchiorre M, Colamartino T, Ferrauto M, Troise M, Salamina L, Mauro S.
Design of a Spherical Rover Driven by Pendulum and Control Moment Gyroscope for Planetary Exploration. *Robotics*. 2024; 13(6):87.
https://doi.org/10.3390/robotics13060087

**Chicago/Turabian Style**

Melchiorre, Matteo, Tommaso Colamartino, Martina Ferrauto, Mario Troise, Laura Salamina, and Stefano Mauro.
2024. "Design of a Spherical Rover Driven by Pendulum and Control Moment Gyroscope for Planetary Exploration" *Robotics* 13, no. 6: 87.
https://doi.org/10.3390/robotics13060087