# MPC-Based Motion-Cueing Algorithm for a 6-DOF Driving Simulator with Actuator Constraints

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## Abstract

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## 1. Introduction

- Providing realistic motion cues to the driver or passenger sitting inside the simulator.
- Keeping the motion platform within its physical boundaries.

- Since the parameters of the filters are fixed, they must be designed for the worst-case maneuver. As a result, the algorithm does not use the available workspace for gentle maneuvers resulting in minimum motion.
- Tuning the filters is a complex task because the filter coefficients should be modified based on subjective participant feedback without taking into account meaningful physical quantities.
- Since there is no provision for incorporating the physical limits of the motion platform within the algorithm, the filters have to be tuned for each maneuver/participant to ensure that the motion platform remains within its physical limit.

## 2. Motion Platform

- Inertial Frame (IF)—is fixed to the ground and does not move with the motion platform. The origin coincides with the centroid of the fixed base of the platform (Point O in Figure 4). The positive x-axis points forward, in the direction of drive. The positive y-axis points to the right, while the positive z-axis points vertically downwards.
- Platform Frame (PF)—is fixed to the motion platform and moves with it. The origin coincides with the centroid of the moving plate (Point ${P}_{0}$ in Figure 4). Similar to the IF, the positive x-axis points forward, the positive y-axis points to the right, while the positive z-axis points in the downwards direction. Since the PF is body-fixed, its axes are only aligned with that of the IF when the platform has a zero roll, pitch and yaw angle.
- Driver Frame (DF)—is fixed to the driver’s head and moves with it. The origin coincides with the eyepoint of the driver (Point ${D}_{0}$ in Figure 4). The positive x-axis points forward, the positive y-axis points to the right, while the positive z-axis points in the downwards direction. Since DF is fixed to the driver’s eyepoint, its axes are only aligned with the IF when the platform has a zero roll, pitch and yaw angle.

## 3. System Model

- Vestibular system model to provide realistic motion cues.
- Motion platform model to manage the available motion workspace.

#### 3.1. Vestibular System Model

#### 3.1.1. Semi-Circular Canals

#### 3.1.2. Otolith Organ

#### 3.1.3. Complete Vestibular System Model

#### 3.2. Motion Platform Model

**Limiting motion workspace**—Forward kinematics is used to calculate the motion space (as per the current actuator position) in terms of the translational and angular displacement of the point ${P}_{0}$. Furthermore, the constraints are applied based on the current motion state and the same should be updated at each time step. As a result, the resulting motion space is a 6-dimensional complex body.**Limiting actuator workspace**—Inverse kinematics is used to determine the motion space directly in terms of the actuator positions. Subsequently, fixed constraints are added based on the permissible actuator length.

#### Actuator Kinematics

#### 3.3. Combined System Model

## 4. MPC Formulation

#### 4.1. Objective Function

#### 4.2. Constraints

- Constraint on the tilt rate ($\omega $).
- Constraints on the actuator positions (${l}_{i}$).

#### 4.3. Reference Generation

#### 4.4. Adaptive Weight-Based Tuning

#### 4.5. Optimization Problem

- Preparation step: The objective function is evaluated in the form of unknown state feedback ${x}_{0}$. The original QP problem is formulated and condensed into a smaller and denser QP.
- Feedback step: The state feedback ${x}_{0}$ is substituted and the QP is solved to obtain the control input.

## 5. Performance Indicators

#### 5.1. Indicators for Reference Tracking Performance

**Root Mean Square Error (RMSE)**calculates for each time step is added and the result is normalized so that the indicator can compare short and long signals fairly. RMSE is presented in Equation (47) and has the range of $[0,+\infty ]$.

**Correlation Coefficient (CC)**is the shape correlation between the reference and the actual signal given in Equation (48) with the range of $[0,+1]$. If the two signals are similar in shape, then the CC should be close to one, while it should be close to zero when there is low correlation [37]. This indicator can be particularly useful to signify if there are many missing or false cues.

**Estimated Delay (ED)**calculates the magnitude of delay between the reference and the actual signal. Since both the signals are not exactly equal, actual delay cannot be calculated. Therefore, it is estimated as the offset applied to the reference signal which maximizes the correlation coefficient. The range of ED indicator is $[0,+\infty ]$ and an ideal signal with no delay with respect to the reference should have ED equal to zero.

#### 5.2. Indicators for Workspace Use

**Interquartile range (IQR)**of the actuator length can be used to analyze how an MCA uses the available actuator workspace [38]. It is a measure of variability and is defined as the difference between the 75th and 25th percentile of the given sample. A high interquartile range denotes high usage of the actuator workspace.

## 6. Results and Discussion

#### 6.1. Simulation Setup

#### 6.2. Simulation Results

#### 6.2.1. Reference Tracking Performance: Translational Acceleration

#### 6.2.2. Reference Tracking Performance: Rotational Velocity

#### 6.2.3. Workspace Use: Actuator displacement

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Degree of Freedom | Threshold Value |
---|---|

Roll ${\omega}_{\varphi}$ | 3.0 deg/s |

Pitch ${\omega}_{\theta}$ | 3.6 deg/s |

Yaw ${\omega}_{\psi}$ | 2.6 deg/s |

Parameter | ${W}_{{\omega}_{rot}}$ | ${W}_{{\beta}_{rot}}$ | ${W}_{{\omega}_{tilt}}$ | ${W}_{{\beta}_{tilt}}$ | ${W}_{{y}_{v},\widehat{a}}$ | ${W}_{{x}_{v},\widehat{\omega}}$ | ${W}_{{l}_{i}}$ |

Weight | $0.1\times {10}^{-2}$ | $0.1\times {10}^{-2}$ | $0.1\times {10}^{-2}$ | $0.1\times {10}^{-2}$ | $2\times {10}^{-2}$ | $10\times {10}^{-2}$ | $0.1\times {10}^{-2}$ |

Motion | Excursion [m] | Velocity [m/s] | Acceleration [m/s^{2}] |
---|---|---|---|

Surge x | −0.51…0.63 | ±0.81 | ±7.1 |

Sway y | −0.51…0.51 | ±0.81 | ±7.1 |

Heave z | −0.42…0.42 | ±0.61 | ±10.0 |

Roll $\varphi $ | ±24.3 | ±35.0 | ±260.0 |

Pitch $\theta $ | −25.4…28.4 | ±38.0 | ±260.0 |

Yaw $\psi $ | ±25.0 | ±41.0 | ±510.0 |

Actuator | 1.297…1.937 | - | - |

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**MDPI and ACS Style**

Khusro, Y.R.; Zheng, Y.; Grottoli, M.; Shyrokau, B. MPC-Based Motion-Cueing Algorithm for a 6-DOF Driving Simulator with Actuator Constraints. *Vehicles* **2020**, *2*, 625-647.
https://doi.org/10.3390/vehicles2040036

**AMA Style**

Khusro YR, Zheng Y, Grottoli M, Shyrokau B. MPC-Based Motion-Cueing Algorithm for a 6-DOF Driving Simulator with Actuator Constraints. *Vehicles*. 2020; 2(4):625-647.
https://doi.org/10.3390/vehicles2040036

**Chicago/Turabian Style**

Khusro, Yash Raj, Yanggu Zheng, Marco Grottoli, and Barys Shyrokau. 2020. "MPC-Based Motion-Cueing Algorithm for a 6-DOF Driving Simulator with Actuator Constraints" *Vehicles* 2, no. 4: 625-647.
https://doi.org/10.3390/vehicles2040036