# Design Requirements for Personal Mobility Vehicle (PMV) with Inward Tilt Mechanism to Minimize Steering Disturbances Caused by Uneven Road Surface

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vehicle Characteristics, Design Parameters and Configuration of This Report

#### 2.1. Vehicle Characteristics and Design Parameters

- Focused Requirements on Steady Characteristic;

- -
- Straight running ability on slanted roads (lateral vehicle movement and/or steering pull)
- -
- Straight running ability on rutted roads (lateral vehicle movement and/or steering pull)

- Design Parameters to Derive to Prevent the PMV from Lateral Vehicle Movement and/or Steering Pull;

- -
- Tire camber characteristics (normalized camber stiffness (D
_{y}), pneumatic trail by camber angle (e_{γ})) - -
- Front wheel steering axis (caster angle (ξ), caster trail (T
_{ξ}), kingpin angle (ψ), kingpin offset (D_{ψ}))

#### 2.2. Configuration of This Report

- Introduction
- Vehicle Characteristics, Design Parameters and Configuration of this Report
- 2.1.
- Vehicle Characteristics and Design Parameters
- 2.2.
- Configuration of this Report

- Materials and Methods
- 3.1.
- Vehicle and Tire Specifications Used in this Report
- 3.1.1.
- Vehicle Specifications
- 3.1.2.
- Tire Specification
- 3.1.3.
- Consideration of Generally Given Tire Camber Characteristic

- 3.2.
- Steering Axis Design Parameters to Derive
- 3.2.1.
- Steering Axis Design Parameters to Derive (Four Unknowns)
- 3.2.2.
- Reduce the number of parameters to Derive (to Two Unknowns) by Previous Studies Related to Steering Axis Geometry
- 1.
- Requirements for Caster Trail Considering Tire Lateral Force on the Road Surface [16]
- 2.
- Kingpin Offset Requirements Considering the Longitudinal Braking Force in the Contact Surface on Straight Running

- Results
- 4.1.
- Derivation of Two Steering Axis Design Parameters from Two Requirements on Steady Characteristics
- 4.1.1.
- A Method to Minimize Steering Disturbance Caused by Uneven Road Surface for Personal Mobility Vehicle (PMV) with Inward-tilting Mechanism
- 1.
- Minimization of Steering Disturbance due to Reaction Force against the Vertical Load when Standing Upright
- 2.
- Maintaining Zero Steering Disturbance due to Reaction Force against the Vertical Load even if the Tire Contact Point Moves Laterally when the Vehicle is Tilted Inward

- 4.1.2.
- Specification Setting Procedure for Minimizing Steering Disturbance due to the Reaction Force against the Vertical Load

- 4.2.
- Vehicle Stability during Disturbance Caused by Uneven Road Surface in the Market, Using the Method to Minimize Steering Disturbance
- 4.2.1.
- Lateral Force Balance on Slanted Road
- 4.2.2.
- Straight Running Stability on Slanted Road
- 4.2.3.
- Straight Running Stability on Rutted Road

- Discussion (Insight into Dynamic Phenomenon Analysis for the Future)
- Conclusions

## 3. Materials and Methods

#### 3.1. Vehicle and Tire Specifications Used in This Report

#### 3.1.1. Vehicle Specifications

- TRA: target roll angle
- A: user amplification factor
- δ: tire-steered angle
- v: vehicle speed
- l: wheel base

#### 3.1.2. Tire Specifications

#### 3.1.3. Consideration of Generally Given Tire Camber Characteristics

_{y}) is obtained by dividing the camber thrust by the camber angle and the vertical load. Therefore, at D

_{y}(/rad) ≈ 1 or D

_{y}(/deg) ≈ π/180 = 0.0174, the camber thrust alone provides the centripetal force necessary to turn. Table 2 shows that such characteristics are generally given to motorcycle tires.

_{y}(/rad) < 1. This will be discussed later also in Section 4.2.

#### 3.2. Steering Axis Design Parameters to Derive

#### 3.2.1. Steering Axis Design Parameters to Derive (Four Unknowns)

_{ξ}) and kingpin offset (D

_{ψ}) at ground level as shown in Figure 4. Alternatively, the tire radius (R) can be used to set T

_{off}= T

_{ξ}− Rsinξ and D

_{off}= D

_{ψ}− Rsinψ at the height of the wheel axis as independent values.

#### 3.2.2. Reduce the Number of Parameters to Derive (to Two Unknowns) by Previous Studies Related to Steering Axis Geometry

- Requirements for Caster Trail Considering Tire Lateral Force on the Road Surface [16]

_{ξ}).

_{ξ}.

_{ξ}) offsets the pneumatic trail (e

_{γ}) caused by the camber angle; the additional torque proportional to the roll angle of the vehicle is added to the steering wheel in order to offset the hysteresis of the steering torque; and finally the additional torque proportional to the prospected lateral acceleration of the vehicle, proportional to the square of the vehicle speed and the steering wheel angle, is added to obtain an appropriate slope of the steering torque. The appropriate value of the added torque, which is proportional to the prospected lateral acceleration, should be determined based on the human sense by driving the vehicle. However, this value is not related to this report. It is only concerned to offset the pneumatic trail (e

_{γ}) due to the camber angle with the caster trail (T

_{ξ}).

_{ξ}to be equal to the absolute value of the pneumatic trail (e

_{γ}) due to the camber angle (γ) (T

_{ξ}= 26.7mm) also in this report and proceed with the following study.

- MT: steering wheel torque
- M
_{z}: tire aligning moment - B
_{1}: aligning torque adjustment factor - F
_{y}: lateral force - PLA: provisional lateral acceleration
- e
_{γ}: tire pneumatic trail on camber angle - B
_{2}: hysteresis adjustment factor - r
_{s}: steering ratio - φ: tilt angle

- 2.
- Kingpin Offset Requirements Considering the Longitudinal Braking Force in the Contact Surface on Straight Running

_{ψ}) is the requirement to keep straight running, as shown in the Equation (3), obtained by the simplified moment around the steering axis assumed to be nearly vertical due to the longitudinal force on the ground surface. This condition is shown in Figure 7. In automobiles, it is a constant value of about −21mm, and in motorcycles, it is always 0 mm. However, in the case of PMVs with active inward-tilting mechanism, an inward-tilting angle that is proportional to the square of the vehicle speed and the steering angle is inevitably generated from Equation (1). Since a proportional lateral force by tire camber angle is generated, the requirement to run straight on one-side braking is the kingpin offset value, which is dependent on vehicle speed, as shown in Figure 7.

## 4. Results

#### 4.1. Derivation of Two Steering Axis Design Parameters from Two Requirements on Steady Characteristics

#### 4.1.1. A Method to Minimize Steering Disturbance Caused by Uneven Road Surface for Personal Mobility Vehicles (PMVs) with Inward-Tilting Mechanism

_{ξ}), and kingpin offset at ground level (D

_{ψ}), caster trail (T

_{ξ}) and kingpin offset (D

_{ψ}) have already been assumed according to Section 3.2, therefore, in this section the caster angle (ξ) and kingpin angle (ψ) are derived from the two target characteristics of minimizing steering disturbance due to the reaction force against the vertical load when standing upright, and of maintaining zero steering disturbance due to the reaction force against the vertical load even if the tire contact point moves laterally when the vehicle is tilted inward.

- Minimization of Steering Disturbance due to Reaction Force against the Vertical Load when Standing Upright

_{1}(x

_{1}, y

_{1}, 0), the other point is P

_{2}(x

_{2}, y

_{2}, z

_{2}), and |P

_{1}− P

_{2}| = 1, the four parameters ξ, ψ, T

_{ξ}, and D

_{ψ}are represented by four coordinate values x

_{1}, y

_{1}, x

_{2}, and y

_{2}as follows.

_{z}to the x-y plane, the projection length L

_{x}to the y-z plane, and the projection length L

_{y}to the z-x plane of the line segment P

_{1}− P

_{2}are as follows.

_{1}− P

_{2}| = 1, L

_{x}is the efficiency of the moment around the steering axis created by the vertical force and by the longitudinal force, L

_{y}is by the lateral force, and L

_{z}is by the efficiency. L

_{x}, L

_{y}and L

_{z}are also represented by four coordinate values x

_{1}, y

_{1}, x

_{2}, and y

_{2}.

_{0}(x

_{0}, y

_{0}, 0) in Figure 8. This reaction force axis and the steering axis are generally in a skewed position. Since the reaction force axis is vertical, the distance between the two axes is equal to the distance d between the point P

_{0}and the steering axis P

_{1}− P

_{2}in planar view as shown in Figure 8.

_{1}− P

_{2}in planar view is set to ax + by + c = 0,

_{S}around the steering axis due to the reaction force against the vertical load F

_{z}is expressed by Equation (7) as the product of these.

_{0}are (0, 0, 0) and d is d

^{2}= c

^{2}/(a

^{2}+ b

^{2}). By substituting Equations (5) and (6) into Equation (7), the moment around the steering axis (M

_{S}) becomes Equation (8).

_{S}= 0 is shown in Equation (9) by modifying c = x

_{2}y

_{1}− x

_{1}y

^{2}= 0, and Figure 9 is obtained by substituting T

_{ξ}= 26.7 mm and D

_{ψ}= 5 mm.

_{h}. If the coordinates of P

_{h}are (0, 0, h), h is expressed by Equation (10).

- 2.
- Maintaining Zero Steering Disturbance Due to Reaction Force against the Vertical Load Even If the Tire Contact Point Moves Laterally When the Vehicle Is Tilted Inward

_{0}, P

_{1}, P

_{2}move to P

_{0}

^{*}, P

_{1}

^{*}, P

_{2}

^{*}.

_{φ}moves from P

_{0}to the inner side of the vehicle by rφ, and becomes P

_{φ}(0, y

_{φ}, 0) = (0, rφ, 0). Assuming that the steering axis P

_{1}

^{*}− P

_{2}

^{*}during tilt is a

^{*}x + b

^{*}y + c

^{*}= 0,

^{*}is expressed as

_{z}

^{*}of the line segment P

_{1}

^{*}− P

_{2}

^{*}onto the x-y plane is as follows.

_{s}= F

_{z}d

^{*}L

_{z}

^{*}= F

_{z}(brφ + c

^{*}) = 0 is brφ + c

^{*}= 0.

_{h}

^{*}are (0, rφ, h), where h corresponds to r shown as Figure 12. Equation (11) is obtained regardless of φ.

#### 4.1.2. Specification Setting Procedure for Minimizing Steering Disturbance Due to the Reaction Force against the Vertical Load

_{ξ}= 0.0267 m from Section 3.2.2 and D

_{ψ}= 0.005 m from Section 3.2.2.

- (1)
- Caster angle (ξ) = 25 deg is obtained from r = 0.057 m in order to maintain the d value during tilt.
- (2)
- In order to set d = 0 when standing upright, the kingpin angle (ψ) = 5 deg can be obtained from ξ = 25 deg.

#### 4.2. Vehicle Stability during Disturbance Caused by Uneven Road Surface in the Market, Using the Method to Minimize Steering Disturbance

#### 4.2.1. Lateral Force Balance on Slanted Road

_{z}) and roll inertia (I

_{x}). However, it transitions relatively quickly to the balanced state above, under the condition both β and φ are small.

- x: longitudinal direction
- y: lateral direction
- v: vehicle speed
- m: vehicle mass
- g: gravitational acceleration
- l: wheel base
- Tr: front tread
- G.C.: gravity center
- l
_{f}: front distance from G.C. - l
_{r}: rear distance from G.C. - φ: slant angle (≈tilt angle)
- GCH: gravity center height
- K
_{y}: cornering stiffness - C
_{y}: normalized cornering stiffness

- Q
_{y}: camber stiffness - D
_{y}: normalized camber stiffness

_{y}) required to realize the state shown in Figure 3 is obtained by Equation (18). The balance of the yaw moment is expressed by equation (19), but this equation is nothing more than the indication of the vehicle’s center of gravity (GC) position (Equation (15)). This relationship is equivalent to the balanced state of a motorcycle running straight on a slanted road.

_{y}) required to maintain straight running on a slanted road is obtained by Equation (22). Using the same tire crown radius (CR = 57mm) as in Reference (15), the necessary normalized camber stiffness (D

_{y}) for the slant angle (φ) is almost constant as shown in Figure 17. When φ = 10deg, D

_{y}= 1.522 × 10

^{−2}/deg, and this value is almost equal to the value of D

_{y}required for turning only with camber stiffness (Q

_{y}), which is obtained in Reference [14]. In other words, for a PMV that tilts inward when turning, by giving appropriate tire characteristics, it is possible to achieve both turning characteristics that do not produce a slip angle (β) during turns and straight running ability on slanted roads.

#### 4.2.2. Straight Running Stability on Slanted Road

_{ξ}

^{*}) is slightly smaller than the caster trail (T

_{ξ}) as a suspension specification because the tire contact point is slightly higher than the original position due to the lateral movement of the tire contact point. Expressed as an equation, it is smaller by CR(1 − cosφ)sinξ as shown in Equation (23), however, the difference is very small that it can be ignored compared to T

_{ξ}as shown in Figure 19.

_{z}

^{*}) against the road surface is equal to the moment around the steering axis when tilts are inward due to the reaction force of the vertical load (F

_{z}) in Section 4.1.1. This is already zero, since the steering axis is followed the specification that satisfies the minimization requirement in Section 4.1.1. The steering moment (M

_{z}

^{*}) around the steering axis due to the lateral force is given by Equation (24). Since CR(1 − cosφ)sinξ is negligibly small compared to T

_{ξ}, M

_{z}

^{*}is effectively zero. In other words, as long as the concept of the steering axis setting shown in Section 4.1.2 is followed, when running on a transverse slanted road, the steering moment is not generated at the same time as the lateral force balance and the yaw moment balance.

#### 4.2.3. Straight Running Stability on Rutted Road

## 5. Discussion (Insight into Dynamic Phenomenon Analysis for the Future)

- -
- The dynamic behavior (1 to 2 Hz) of the entire vehicle is considered to be an accumulation of static balances, so in principle it is not affected by disturbances. Therefore, verification in this frequency domain should be entrusted to the confirmation phase with real vehicles.
- -
- The possibility of vehicle response that cannot be explained by static balance due to transient characteristics of tire force and moment (affected by running speed, but about 5Hz at 36km/h, for example), which are not included in the assumptions of this report. Considering the response frequency of the vehicle, the transient vehicle response becomes a point of interest. Therefore, analysis with a dynamic model of the vehicle that incorporates a dynamic tire model is required. The authors are preparing this type of analysis as the next step of the research.
- -
- Unsprung vibrations (approximately 10 Hz or higher), which are not considered in this report, may be transmitted to the entire vehicle in a vibratory manner. In order to reproduce even unsprung vibration, a dynamic model that considers the mass, inertia, and rigidity of each part of the vehicle, including the unsprung mass, is required. Furthermore, at practical speeds (e.g., 72 km/h), the transient characteristics of tires overlap with the frequency range, so analysis using a fairly advanced multi-degree-of-freedom model is required. Analysis of the steering axis arrangement from this point of view is difficult in the short term, and at the same time, it seems to deviate from the research scope of straight running performance of the vehicle on slanted roads and rutted roads.

## 6. Conclusions

- -
- Based on the characteristics of general motorcycle tires, the centripetal force required for turning is obtained mainly by the camber angle of the tires to the ground (camber thrust). This means the lateral force of each tire is balanced on every angle of transverse slant as mentioned in Section 3.1.3. Therefore, steering torque is focused in order to avoid the disturbances on uneven road surface.
- -
- Four unknown steering axis design parameters (caster angle (ξ), caster trail (T
_{ξ}), kingpin angle (ψ), kingpin offset (D_{ψ})) to derive in order to minimize the steering torque disturbances, are reduced into two unknowns (caster angle (ξ), kingpin angle (ψ)) by previous studies of requirements for caster trail (T_{ξ}) considering tire lateral force and kingpin offset (D_{ψ}), requirements considering the longitudinal braking force as mentioned in Section 3.2.2. - -
- Two unknown steering axis design parameters (caster angle (ξ), kingpin angle (ψ)) are derived from two vehicle requirements on steady characteristics, as minimization of steering disturbance due to reaction force against the vertical load when standing upright and maintaining zero steering disturbance even if the tire contact point moves laterally when the vehicle is tilted inward, as mentioned in Section 4.1. This derivation is a new knowledge from a completely unique point of view.
- -
- The lateral force balance and minimized steering torque free from the disturbance of each wheel on transverse slant of road surface as mentioned in Section 4.2.1 and Section 4.2.2. Then the free from the disturbance of each wheel on transverse slant gives free from every kind of rut of road surface that is the combination of various transverse slant angles as mentioned in Section 4.2.3.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Dimensions of model vehicle [6].

**Figure 2.**Typical tire cross section (Motorcycle, PMV) [16].

**Figure 3.**Motorcycle tire model is used in CarMaker [13].

**Figure 12.**Steering axis to minimize M

_{s}by F

_{z}on tilted condition. The * shows when the tire is tilted inward.

**Table 1.**Specifications of model vehicle [6].

Item | Unit | Value | Item | Unit | Value |
---|---|---|---|---|---|

Total length | m | 2.645 | Total mass | kg | 369.8 |

Total width | m | 0.880 | Front mass distribution | kg | 222.1 |

Total height | m | 1.445 | Rear mass distribution | kg | 147.7 |

Wheel base | m | 2.020 | Roll moment of inertia | kgm^{2} | 58.8 |

Front distance from gravity center | m | 0.807 | (Roll moment of inertia of sprung mass) | kgm^{2} | 43.0 |

Rear distance from gravity center | m | 1.213 | Pitch moment of inertia | kgm^{2} | 197.3 |

Front tread | m | 0.850 | (Pitch moment of inertia of sprung mass) | kgm^{2} | 118.0 |

Gravity center height | m | 0.358 | Yaw moment of inertia | kgm^{2} | 187.3 |

Steering gear ratio | - | 16.0 | (Yaw moment of inertia of sprung mass) | kgm^{2} | 102.3 |

Item | Value | Item | Value |
---|---|---|---|

Tire radius: R | 0.242 m | Crown radius: CR | 0.057 m |

Cornering stiffness: K_{y} | 427.4 N/deg | Camber stiffness: Q_{y} | 17.77 N/deg |

Normalized K_{y}: C_{y} | 42.74 × 10^{−2}/deg | Normalized Q_{y}: D_{y} | 1.777 × 10^{−2}/deg |

Pneumatic trail on slip angle: e_{β} | 0.0136 m | Pneumatic trail on camber angle: e_{γ} | −0.0267 m |

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**MDPI and ACS Style**

Haraguchi, T.; Kaneko, T.
Design Requirements for Personal Mobility Vehicle (PMV) with Inward Tilt Mechanism to Minimize Steering Disturbances Caused by Uneven Road Surface. *Inventions* **2023**, *8*, 37.
https://doi.org/10.3390/inventions8010037

**AMA Style**

Haraguchi T, Kaneko T.
Design Requirements for Personal Mobility Vehicle (PMV) with Inward Tilt Mechanism to Minimize Steering Disturbances Caused by Uneven Road Surface. *Inventions*. 2023; 8(1):37.
https://doi.org/10.3390/inventions8010037

**Chicago/Turabian Style**

Haraguchi, Tetsunori, and Tetsuya Kaneko.
2023. "Design Requirements for Personal Mobility Vehicle (PMV) with Inward Tilt Mechanism to Minimize Steering Disturbances Caused by Uneven Road Surface" *Inventions* 8, no. 1: 37.
https://doi.org/10.3390/inventions8010037