Keywords

1 Introduction

Due to the strong demand for decarbonization toward sustainable society, x-electric vehicles (xEVs) have been introduced to the global market. On the other hand, it is known that torsional resonance of the drive shaft caused by motor torque change in the general drivetrain structure of automobiles in which the drive source is connected to the left and right wheels via differential gear and drive shafts, as shown in Fig. 1. At the same time, because of the inevitable gear intervention, there is a dead zone called backlash, as shown in Fig. 2.

Fig. 1.
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Control target (drivetrain)

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Fig. 2.
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An example of backlash of a gear.

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Fig. 3.
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Block diagram of wheel-speed observer.

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When implementing vibration suppression control in automobiles, it is often necessary to apply control methods that are within the scope of classical control and offer good visibility and performance. In the past, several vibration suppression control methods have been reported such as devising classical PI control [1], resonance ratio control using a disturbance observer [2], etc. These methods are based on the assumption that only the motor rotation speed is observable. Recently, however, a high vibration suppression effect by sensing the speed on the tire side has also been reported [3]. The simplest and most prospective method is to calculate feedback motor torque based on the difference between motor speed and tire speed multiplied by a proportional gain [4]. In this paper, therefore, an observer is introduced to estimate the tire rotation speed.

In addition, a backlash compensation method is developed to reduce noise and vibration in gear retightening in the backlash region. In backlash compensation, a method to set the torque to zero for a certain period of time at zero crossing has been reported [5]. This method has the problem that it takes time to pass through the backlash, which may lead to discomfort for the driver. This paper proposes a new torque compensation method that can solve this problem.

2 Vibration Suppression Control

2.1 Method

First, this section describes wheel-speed observer. The observer is configured as shown in Fig. 3 by equating a state equation (\(\dot{{\boldsymbol{x}}}={\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{B}}u, y = {\boldsymbol{C}}{\boldsymbol{x}}\)) from the motor torque to the rotation speed of the motor and tires. The load side changes depending on whether the tire is gripping or slipping, so the tire slip judgment \(s\) is included and the \({\boldsymbol{A}}\) and \({\boldsymbol{K}}\) matrices are switched (\({{\boldsymbol{A}}}_{\mathbf{g}\mathbf{r}\mathbf{i}\mathbf{p}}\leftrightarrow {{\boldsymbol{A}}}_{\mathbf{s}\mathbf{l}\mathbf{i}\mathbf{p}}, {{\boldsymbol{K}}}_{\mathbf{g}\mathbf{r}\mathbf{i}\mathbf{p}}\leftrightarrow {{\boldsymbol{K}}}_{\mathbf{s}\mathbf{l}\mathbf{i}\mathbf{p}}\)). \({\boldsymbol{C}}\) and \({\boldsymbol{C}}{^{\prime}}\) are the matrices that obtains observed output \({\omega }_{m}\) and \({\omega }_{w}\) respectively from the state vector \({\boldsymbol{x}}\).

Figure 4 shows the block diagram of the vibration suppression control method. The method obtains estimated tire speed \(\widehat{{\omega }_{w}}\) from the observer and feed-backs motor torque by the difference from the motor speed \({\omega }_{m}\). The observer parameters are switched by the tire slip judgement \(s\). The difference between the motor speed and the estimated tire speed is then calculated considering the reduction-gear ratio.

Fig. 4.
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Block diagram of speed difference feedback with observer.

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Fig. 5.
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Experimental result of vibration suppression on high-μ road.

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2.2 Experimental Validation

The effectiveness of the vibration suppression control is verified through driving tests on an experimental vehicle. In the driving test, acceleration was applied by an external step-shape acceleration command from the creep driving on high-\(\mu \) surface. Figure 5(a) shows the motor torque, and acceleration started at the 0.5 s. Compared to the case without control (blue dot line), the proposed method (orange solid line) generates a torque modification to reduce vibrations in the motor rotation speed. The motor rotation speed at that time is shown in Fig. 5(b). While vibration occurs without control, the vibration is reduced by the proposed method. However, there is one fluctuation in the motor rotation speed right after 0.5 s, which is due to backlash.

Next, constant acceleration of 50 Nm was applied on low-\(\mu \) (slippery) surface. Figure 6(a) shows the motor torque. Figure 6(b) shows the motor rotation speed, the tire rotation speed and observer output, which are converted to motor rotation shaft. In this test, the drive wheels started to slip at just before 1.8 s. When the tire slip started, there is an unexpected spike in the output of the observer (yellow solid line). To prevent the affect of this spike, the vibration suppression control was stopped for 0.2 s after the start of tire slip and restarted just before 2.0 s. Compared with the case without control (1.8–2.0 s), the vibration of motor speed is reduced after 2.2 s as shown in Fig. 6(b).

These results indicate a damping effect in both tire gripping and slipping conditions, and remaining issues with backlash-induced vibration and observer switching. The method for reducing vibration caused by backlash is presented in the next section.

Fig. 6.
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Experimental result of vibration suppression when tire slips.

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Fig. 7.
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Simulation result of backlash compensation method.

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3 Backlash Compensation

3.1 Method

This section describes a backlash compensation method. Figure 7 illustrates the concept of the control method. In this figure, the torque command value from a driver’s gas-pedal operation (dashed line) changes from a negative value to a positive value (zero crossing) at time 1. The proposed method stores the value of the motor speed at time 1 as the reference motor speed. Then time 2 is defined as the time when the difference between the motor speed and the reference speed exceeds a predetermined value. At time 2, the compensation torque is calculated based on the backlash width that is known in advance, so that the motor speed at the exit of the backlash is the same as at time 1. After time 2, the torque command value is replaced by the compensation torque.

3.2 Simulation Analysis

First, the effectiveness of the proposed method is verified through simulation. The upper figure of Fig. 8 shows an example of motor speed fluctuation when passing through the backlash, and the lower figure shows the motor torque. The motor speeds were calculated by a vehicle plant model with the input of motor torque. The blue dot lines show the conventional method without torque compensation, while the orange solid lines show the results with the backlash compensation method. In this test, the vibration suppression control shown in the previous section was not combined to confirm the effectiveness of the backlash compensation method alone. The vibration (collision) at the end of the backlash occurs with an amplitude of 8.5 rad/s with the conventional method. On the other hand, the proposed method reduces the amplitude to 3.0 rad/s (about 65% of reduction compared with the conventional method), confirming the effectiveness of the proposed method.

Fig. 8.
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Simulation result of backlash compensation method.

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3.3 Experiment

Second, the effectiveness of the proposed method is also verified through driving tests on an experimental vehicle, with the same test procedure as simulation. In the driving test, acceleration was applied by an external ramp-shape acceleration command from the creep driving. The upper figures of Fig. 9 show the motor speed (dashed line as command value and solid line as actual value), and the lower figures show the motor torque. Figure 9(a) illustrates the result of conventional method without backlash compensation while Fig. 9(b) is the result of proposed method. In Fig. 9(a), there is no deviation between the dashed and solid lines except for a slight delay, while in Fig. 9(b), torque change due to the proposed method can be seen at 0.75 – 0.79 s. The vibration (collision) at the end of the backlash occurs with an amplitude of about 16.3 rad/s with the conventional method (Fig. 9(a)). On the other hand, the proposed method (Fig. 9(b)) reduces the amplitude to about 5.7 rad/s (about 65% of reduction compared with the conventional method), confirming the effectiveness of the proposed method.

Fig. 9.
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Experimental result of backlash compensation method.

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4 Conclusion

This paper proposed a vibration suppression control that suppresses the torsional resonance of the drive shaft caused by motor torque change in the general drivetrain structure of automobiles. The proposed method calculates the feedback torque from the difference between the motor speed and the tire speed multiplied by a proportional gain, with estimating the tire rotation speed by an observer. The effectiveness of the proposed method was verified through driving tests, which confirmed a damping effect in both tire gripping and slipping conditions. This paper also proposed a backlash compensation method for reducing noise and vibration in gear retightening in the backlash region. The method observes the fluctuation of the motor speed after the torque zero-crossing and determine the motor torque so that the motor rotation speed returns to the speed at the entrance of the backlash. The effectiveness of the proposed method was verified through simulation and driving tests, which confirmed 65% reduction of vibration at the end of backlash. The future works will be to improve the accuracy of the observer behavior at the start of tire slip, and to combine the vibration suppression and the backlash compensation.