Observer Design for Estimating Road Elevations at All Tire Contact Patches Using Only an Inertial Sensor

Abstract

This paper designs an observer for estimating road elevations at all tire contact patches using only an inertial sensor with high accuracy, comparable to that of laser scanning. The observer is constructed within the framework of the unknown input Kalman filter to estimate road elevations, which act as disturbances to vehicle dynamics. The model for the observer is based on vertical-pitch-roll dynamics, encompassing road elevations at all tire contact patches. The introduction of virtual measurements for all tires ensures the observability of the model in the inertial sensor-based observer without requiring both additional sensors and model simplification. Additionally, a bias model is added to compensate for sensor installation errors for practical realization. Experimental validation demonstrates that the proposed observer can estimate road elevation with high accuracy, regardless of vehicle speed and dynamics, even when utilizing only an inertial sensor, making it suitable for rapid and robust road maintenance.

1 Introduction

Road maintenance is of paramount importance for ensuring ride comfort, stable vehicle control, and even safety against road cracks or potholes [1, 2]. A critical initial step in road maintenance is the measurement of road bumpiness [3, 4]. Accurate measurement of the road profile typically employs direct-contact devices called profilometers, albeit at a significant cost in terms of both time and money.

Inertial sensors offer a more affordable alternative because road elevation variations affect inertial sensor signals. However, a challenge arises as road elevation changes are not directly observable using only inertial sensor signals. This limitation stems from the fact that the inertial sensor is attached to the sprung mass, with the excitation caused by road elevation changes being filtered by the unsprung masses. To address this, previous methods have either incorporated additional sensors at an increased cost [5] or utilized a vehicle model without unsprung masses, compromising accuracy [6].

Previous work by J. Gim et al. showcased the potential for road elevation estimation using only inertial sensor signals by introducing virtual measurement and synthesizing the Unknown input Kalman filter [7]. However, this research primarily focused on estimating longitudinal road elevations for vehicle localization, emphasizing consistency and repeatability, without extending its applicability to road maintenance, which requires high precision.

This paper proposes an observer design for the accurate estimation of road elevations at all tire contact patches, with a specific emphasis on its applicability to road maintenance. The vehicle model in the observer is extended to an eight-degree-of-freedom (8-DOF) model to encompass road elevation changes at all four tire contact patches. To ensure observability, virtual measurement is also extended for all tires, and a bias model is introduced to compensate for sensor installation errors. The effectiveness of the designed observer is validated by comparing it with directly measured road elevations using profilometers, establishing its potential utility in practical road maintenance applications.

2 Observer Design

Figure 1 shows the scheme of the proposed road elevation observer.

Fig. 1.

Scheme of the proposed observer for estimating road elevation changes using only inertial sensor signals.

Vehicle dynamics is excited by alterations in road elevation and the driver’s acceleration or brake input, and the inertial sensor mounted on the vehicle measures this vehicle dynamics in terms of accelerations and angular velocities. The discretized model for both vehicle dynamics and the inertial sensor are derived as:

$$ \begin{aligned} & x_{k + 1} = f\left( {x_{k} ,u_{k} ,d_{k - 1} } \right), \\ & y_{k} = h\left( {x_{k} ,d_{k - 1} } \right), \\ \end{aligned} $$

(1)

where vehicle state x $\in$ ℝⁿ encompasses the vertical displacements of sprung and unsprung masses and their rates, as well as the pitch and roll angles of the sprung mass and their rates. The known input u $\in$ ℝ¹ represents the longitudinal acceleration. The disturbance d $\in$ ℝ^q signifies the road elevations at all tire contact patches and their rates, which are the parameters to be estimated. The measurement y $\in$ ℝ^p consists of only the inertial sensor signals.

The designed observer employs an 8-DOF vehicle model, incorporating vertical, pitch, and roll dynamics representing the responses excited by road elevation changes at all tire contact patches, as illustrated in Fig. 2 [8]. The measurement model y includes four inertial sensor signals—vertical acceleration, lateral acceleration, pitch rate, and roll rate signals—to design a road profile observer based only on inertial sensors.

Fig. 2.

8-DOF vehicle model.

A observer framework is based on the unknown input Kalman filter, which estimate disturbances with rapid responses, provided the system satisfies the following two conditions [9]:

$$ rank\left( {\left[ {\begin{array}{*{20}c} {H_{k} } \\ {H_{k} F_{k} } \\ \vdots \\ {H_{k} F_{k}^{n - 1} } \\ \end{array} } \right]} \right) = n\,\,\,and\,\,\,rank\left( {H_{k} G_{k} } \right) = rank\left( {G_{k} } \right) = q, $$

(2)

where $F_{k} = \frac{\partial }{{\partial x_{k} }}f\left( {x_{k} ,u_{k} ,d_{k} } \right),\,\,G_{k} = \frac{\partial }{{\partial d_{k} }}f\left( {x_{k} ,u_{k} ,d_{k} } \right),\,\,H_{k} = \frac{\partial }{{\partial x_{k} }}h\left( {x_{k} } \right)$.

However, the employed 8-DOF vehicle model does not satisfy the observability conditions, meaning that the road profile cannot be estimated using only inertial sensor signals. This is because the inertial sensors are not responsive to changes in the road profile that are longer than the wheelbase, such as hills. The concept of virtual measurement involves continuously monitoring low-pass-filtered road profiles as zero, under the assumption that road profiles longer than wheelbase should be flat. Therefore, four virtual measurements, corresponding to each tire contact patch, are augmented into the measurement model involving the only inertial sensor signals:

$$ \begin{aligned} & y_{aug,k} = \left[ {y_{k} \,\,\,\overline{R}_{fl,k} \,\,\,\overline{R}_{fr,k} \,\,\,\overline{R}_{rl,k} \,\,\,\overline{R}_{rr,k} } \right]^{T} = \left[ {y_{k} \,\,\,0\,\,\,0\,\,\,0\,\,\,0} \right]^{T} , \\ & {\text{where}}\,\,\overline{R}_{ij,k} = \alpha R_{ij,k - 1} + (1 - \alpha )\overline{R}_{ij,k - 1} , \\ \end{aligned} $$

(3)

$\overline{R}_{ij}$ and R_ij are the virtual measurements and the vertical road heights at all tire contact patches, where i denotes the front or rear tires, and j are left or right side. α determines how much the estimated road height influences the virtual measurement. Consequently, the model guarantees observability by augmenting the original states with four virtual measurements for each tire contact patch.

Finally, a bias model is incorporated to compensate inertial sensor installation errors. Inertial sensor signals involve the sensor biases with constant or slowly changing dynamics. The final model for a road profile observer is designed with:

$$ \begin{aligned} & x_{final,k} = \left[ {z_{s,k} \,\,\,\dot{z}_{s,k} \,\,\,\theta_{k} \,\,\,\dot{\theta }_{k} \,\,\,\phi_{k} \,\,\,\dot{\phi }_{k} \,\,\,z_{fl,k} \,\,\,\dot{z}_{fl,k} \,\,\,z_{fr,k} \,\,\,\dot{z}_{fr,k} \,\,\,z_{rl,k} \,\,\,\dot{z}_{rl,k} \,\,\,z_{rr,k} \,\,\,\dot{z}_{rr,k} } \right. \\ & \left. {R_{fl,k} \,\,\,R_{fr,k} \,\,\,R_{rl,k} \,\,\,R_{rr,k} \,\,\,\overline{R}_{fl,k} \,\,\,\overline{R}_{fr,k} \,\,\,\overline{R}_{rl,k} \,\,\,\overline{R}_{rr,k} \,\,\,b_{{\ddot{z}_{s} ,k}} \,\,\,b_{{\dot{\theta },k}} \,\,\,b_{{\dot{\phi },k}} \,\,\,b_{{\ddot{y}_{s} ,k}} } \right]^{T} , \\ & u_{final,k} = \ddot{x}_{s,k} , \\ & d_{final,k} = \left[ {\dot{R}_{fl,k} \,\,\,\dot{R}_{fr,k} \,\,\,\dot{R}_{rl,k} \,\,\,\dot{R}_{rr,k} } \right], \\ & y_{final,k} = \left[ {\ddot{z}_{s,k} \,\,\,\dot{\theta }_{k} \,\,\,\dot{\phi }_{k} \,\,\,\ddot{y}_{s,k} \,\,\,\overline{R}_{fl,k} \,\,\,\overline{R}_{fr,k} \,\,\,\overline{R}_{rl,k} \,\,\,\overline{R}_{rr,k} } \right]. \\ \end{aligned} $$

(4)

The proposed observer synthesizing the designed observable model into the unknown input Kalman filter estimates the vehicle states and the one-time-step delayed road profiles at all tire contact patches.

3 Validation

Figure 3 shows the validation results through comparative analysis of road elevations measured by laser sensors on profilometers for four distinct cases. The estimated results demonstrate that the designed observer can accurately estimate road elevations for all tire contact patches using only inertial sensor signals, with high accuracy similar to that achieved by laser scanning. This accuracy remains consistent across different vehicle types, vehicle speeds, and driver’s acceleration intentions.

Fig. 3.

Estimated results of the designed observer for four cases.

4 Conclusion

This paper proposes an observer for estimating road elevation changes at all tire contact patches using only inertial sensors, achieving accuracy comparable to that obtained with laser sensors. Therefore, the designed observer can be an alternative to direct contact measurement approaches requiring special devices with a short time horizon.