Preliminary Investigation of Mechanical Impedance Characteristics During Lane Change Maneuver

Abstract

Driving support systems have been actively developed. These systems use itself vehicle and surrounding environmental information. There is few research about that system using driver’s physiological information. However, it is discomfort for drivers because electrodes are needed to attach to drivers. Therefore, we focused on mechanical impedance derived from driver’s arm, which are inertia, viscosity, and stiffness. The impedance can be estimated from steering wheel angle, angular velocity, angular acceleration, and torque around a steering wheel shaft where drivers always grip. In this paper, driving simulator experiments were conducted to investigate dynamic characteristics of the impedance in a situation of single lane change without a steering support system, and the time-varying impedance were estimated by using Kalman filter. As results, moment of inertia did not change so much. On the other hand, viscosity and stiffness decreased while steering the wheel.

1 Introduction

Driver assistance system (DAS) partially cover driving tasks of cognition, judgement and operation instead of human [1]. Though the DAS becomes widely spreading to us, a problem is a conflict between human and system intention. Therefore, a support by the system should not be bothersome, and its timing is appropriate for human [2]. Existing DAS uses surrounding and vehicle information [3]. The surrounding information is dynamic information of other vehicles and pedestrians and road alignments acquired by some sensors. The vehicle information is, for example, position, speed, and direction of its own vehicle. Not only surrounding and vehicle information but driver information that is physiological information of driver, is necessary to dissolve the conflict between the system and human. If human’s driving intention can be estimated from the driver information, it is applicable to haptic-shared-control [4] which is one of a DAS having a mechanism of reaction force to the driver via a steering wheel or pedals. Existing studies estimated driver’s steering intentions using their electroencephalogram [5] and eye direction [6]. Drivers are required body constraint and mental stress to obtain the physiological information, it is necessary for the system that can collect driver information with low constraint of humans and estimate driver’s intention.

Characteristics of human’s movement can be expressed by mechanical impedance which is inertia, viscosity and stiffness [7]. In order to begin to move an arm fast, humans rise their arm viscoelasticity by contracting their muscles [8]. In a steering situation, drivers make their arm muscles firm or soft according to vehicle speed and curvature of road [8]. Tanaka et al. estimated human’s arm mechanical impedance mechanically equivalent around a steering shaft at a static condition and proposed steering control system using the impedance [9]. In this method it is feasible with existing electronics power steering system, since only steering wheel angle and torque are measured to estimate the impedance, and not constrain driver just griping a wheel. If the impedance in a dynamic situation can be obtained, it is possible to support the drivers in various driving situations.

The objective of this study is to estimate driver’s mechanical impedance dynamically unless any constraint like physiological measurement, and develop a steering control system. In this paper, driving simulator (DS) experiments were conducted to investigate dynamic characteristics of mechanical impedance during single lane change situation without DAS.

2 Dynamics and Impedance Identification

2.1 Mechanical Model of Human-Steering System

The equation of motion of a steering system is expressed as

$$ M_{s} \ddot{\theta } + B_{s} \dot{\theta } = \tau_{h} + \tau $$

(1)

where M_s and B_s are moment of inertia and rotational viscosity due to structural properties around a steering shaft, respectively. The θ is steering wheel angle. The τ_h and τ are operational torque by human and external torque, respectively.

It is hypothesize that a driver controls steering by impedance control [10] as used in robotics, equation of motion of a human-steering system is expressed as

$$ M\ddot{\theta } + B\dot{\theta } + K(\theta - \theta_{v} ) = \tau $$

(2)

where M, B, and K are target moment of inertia, viscosity, and stiffness for impedance control, respectively. The θ_v is a target steering wheel angle by human control, so-called virtual trajectory. It is ill-posed problem to solve Eq. (2) for the impedance M, B, and K, since θ_v is unknown. To resolve this problem, perturbation method was adopted.

If small displacement Δθ is occurred by small perturbation torque Δτ within a short duration, Eq. (2) becomes

$$ M(\ddot{\theta } + \Delta \ddot{\theta }) + B(\dot{\theta } + \Delta \dot{\theta }) + K(\theta + \Delta \theta - \theta_{v} ) = \tau + \Delta \tau $$

(3)

By subtracting Eq. (2) from Eq. (3), following equation not including θ_v is obtained.

$$ M\Delta \ddot{\theta } + B\Delta \dot{\theta } + K\Delta \theta = \Delta \tau $$

(4)

The $\Delta \ddot{\theta },\,\Delta \dot{\theta },\,\Delta \theta$, and $\Delta \tau$ can be obtained by applying a high-frequency-pass filter to $\Delta \ddot{\theta },\,\Delta \dot{\theta },\,\Delta \theta$ and $\Delta \tau$, respectively.

2.2 Dynamic Identification of Impedance

In order to identify mechanical impedance, state space model in discrete time system was constructed.

$$ \varvec{x}[i + 1] = \varvec{x}[i] + \varvec{v}[i] $$

(5)

$$ y[i] =\varvec{\theta}^{T} [i]\varvec{x}[i] + w[i] $$

(6)

where i denotes i-th sample, and system output y is Δτ. The vectors x and θ are defined as following equations, respectively. The v and w are system noise vector and observation noise, respectively.

$$ \varvec{x}[i] = [M[i]\,\,\,B[i]\,\,\,K[i]]^{T} $$

(7)

$$ \varvec{\theta}[i] = [\Delta \ddot{\theta }[i]\,\,\,\Delta \dot{\theta }[i]\,\,\,\Delta \theta [i]]^{T} $$

(8)

In order to identify time-varying impedance x, Kalman Filter was constructed. Estimated x is updated by following equations.

$$ \varvec{M}[i] = \varvec{R}[i - 1] + \varvec{L} $$

(9)

$$ \varvec{k}[i] = \frac{{\varvec{M}[i]\varvec{\theta}[i]}}{{\sigma_{v}^{2} +\varvec{\theta}^{T} [i]\varvec{M}[i]\varvec{\theta}[i]}} $$

(10)

$$ \varvec{x}[i] = x[i - 1] + \varvec{k}[i](y[i] -\varvec{\theta}^{T} [i]\varvec{x}[i - 1]) $$

(11)

$$ \varvec{R}[i] = \left( {\varvec{I} - \varvec{k}[i]\varvec{\theta}^{T} [i]} \right)\varvec{M}[i] $$

(12)

3 Lane Change Experiment

Driving simulator (DS) experiments were conducted to investigate impedance characteristics during single lane change without a steering assistance system.

3.1 Apparatus

An outline of the DS is shown in Fig. 1. A steering wheel shaft was connected to a direct drive motor (SGMCS-14C3C41, Yasukawa Electric) to create perturbation and reaction torque. A motor-driver (SGDV-2R8A01B, Yasukawa Electric) was connected to a Windows PC via a DA board (PEX-361216, Interface) and a counter board (PCI-6205C, Interface). The torque and wheel angle and torque were measured at 1 [kHz] by a self-made C++ language program.

Fig. 1.

Outline of driving simulator used in experiments.

A display (LCD-M4K431XDB, IO Data) in front of a driving seat showed an experimental scene created by Unity. Vehicle behavior according to participants’ steering operation was calculated by using CarSim (Mechanical Simulation). Figure 2 shows experimental appearance.

Fig. 2.

Experiment appearance.

3.2 Experimental Conditions

Course of Lane Change.

Figure 3 shows an experimental single-lane-change course [11] used in this experiments. Red traffic cones as red squares in Fig. 3 were placed to set course dimensions as shown in Fig. 3. One set of single-lane-change course was repeated by 200 m interval.

Fig. 3.

Dimensions of single-lane-change course. (Color figure online)

Reaction and Perturbation Torques.

Reaction torque τ_c to participants’ steering operation was generated by impedance control as the steering system shows a following characteristic.

$$ M_{s} \ddot{\theta } + B_{d} \dot{\theta } + K_{d} \theta = \tau_{h} + \tau_{p} $$

(13)

where B_d and K_d are target impedance for control, and τ_p is a perturbation torque. The τ in Eq. (1) is τ_c + τ_p. From Eqs. (1) and (13), the reaction torque τ_c is to be

$$ \tau_{c} = \left( {B_{s} - B_{d} } \right)\dot{\theta } - K_{d} \theta $$

(14)

In this experiment, B_d and K_d were set to 0.5 [Nm s/rad] and 2.0 [Nm/rad], respectively. The structural impedance M_s and B_s were 0.0456 [kgm²] and 0.2486 [Nm s/rad] identified by other preliminary experiments, respectively.

The perturbation torque τ_p was +1.0 [Nm] or −1.0 [Nm] created as maximum length sequence by a primitive polynomial f (x) = x⁷ + x + 1. The perturbation torque was changed for each 30 [ms] according to created sequence.

Procedures.

Participants sat on the seat and adjusted its position and backrest angle, then fasten four point seat belt. They were asked to grip the steering wheel at 10 and 2 o’clock, and not to re-grip the wheel during a lane change trial. They drove the DS in several sets of single-lane-change for practices. After the practices, they drove 5 test trials. Vehicle speed was set to constant 80 [km/h], therefore participants were not to need to step gas and brake pedals.

Signal Processing.

At first, recorded data were interpolated for each 1 [ms] because of fluctuation of a sampling time. Second, low-pass filter with a cutoff frequency at 33 [Hz] were applied to them in remove aliases due to the perturbation torque. Third, in order to remove the effects of virtual trajectory θ_v, the data were filtered by high-pass filter with a cutoff frequency at 1 [Hz]. These processed data were used for Kalman filter.

Parameters of Kalman Filter.

Table 1 shows the parameters of Kalman filter to identify the impedance, which are variance of observation noise $\sigma_{v}^{ 2}$, covariance matrix L of system noise v, and an initial value of a posteriori covariance matrix R[0].

Table 1. Parameters of Kalman filter

3.3 Results and Discussions

Figures 4, 5, 6, 7 and 8 show measured data of 5 trials for each subject. The graphs are torque, steering wheel angle, steering wheel angular velocity, and steering wheel angular acceleration in order from the top for each figure. Time of 0 [s] at horizontal axes express start time of steering for the lane change.

Fig. 4.

Measured data for participant A.

Fig. 5.

Measured data for participant B.

Fig. 6.

Measured data for participant C.

Fig. 7.

Measured data for participant D.

Fig. 8.

Measured data for participant E.

Figures 9, 10, 11, 12 and 13 show estimated impedance with the steering wheel angle of 5 trials for each participant. The graphs show the steering wheel angle, estimated moment of inertia, viscosity, and stiffness in order from the top. The inertia not change so much, the range is from 0.04 to 0.08 [kgm²] for all participants. In previous research, it has been reported that the inertia are in the range of 0.05 to 0.07 [kgm²] when static holding of a steering wheel with both hands [12]. Therefore, the estimated inertia in Figs. 9, 10, 11, 12 and 13 seems valid.

Fig. 9.

Estimated impedance for subject A.

Fig. 10.

Estimated impedance for subject B.

Fig. 11.

Estimated impedance for subject C.

Fig. 12.

Estimated impedance for subject D.

Fig. 13.

Estimated impedance for subject E.

The viscosity in Figs. 9, 10, 11, 12 and 13 tend to decrease around the start time of steering. Moreover, the viscosity while steering are decreasing or keep lower. It is thought that participants softened their arm muscles since the impedance is to be a resistance to movement. After the half of steering, the viscosity tend to increase gradually. The participants had been raising their muscle viscosity in order to maintain the wheel angle at 0 [deg] to run straight. The stiffness in Figs. 4, 5, 6, 7 and 8 tend to change as similar to the viscosity.

4 Conclusions

In this paper, in order to investigate the dynamic characteristics of impedance during lane-change maneuver, DS experiments were conducted without a steering support system. As the results, the moment of inertia did not change so much. On the other hand, both viscosity and stiffness tended to decrease or be low while steering the wheel. After a half time of steering, they gradually increased.

As future works, DS experiments with a steering support system will be conducted to compare dynamic characteristics of impedance with and without the system.