I study the comfort problem that appears when an electric vehicle changes from traction to regenerative braking. In an electric vehicle, the electric machine does not act as an isolated torque source. Its torque is transmitted through a gearbox, intermediate shaft, half-shaft, tire, and vehicle body. Each of these components stores energy, dissipates energy, or introduces a nonlinear constraint. The resulting electromechanical coupling produces low-frequency torsional vibration, high-frequency torque ripple, backlash impacts, and transient shocks. These effects are weak in steady traction but become visible during rapid torque reversal. I therefore formulate a compensation controller that increases the effective damping of the electric vehicle driveline without changing its mechanical structure.
The motivation is practical. A driver expects an electric vehicle to deliver smooth deceleration when the accelerator pedal is released. If the regenerative torque is applied as a step, the half-shaft torque oscillates, the longitudinal acceleration oscillates, and the occupant perceives a shunt or surge. The problem is not the average braking force. It is the transient energy exchange among the motor rotor, gear pairs, shaft compliance, tire compliance, and vehicle inertia. I quantify this exchange, identify the dominant resonance bands, and design a half-shaft torque feedback law that suppresses the oscillation while retaining regenerative energy recovery.
1. System Description and Modeling Assumptions
I model a front-drive electric vehicle with a fixed-ratio reducer and an open differential. The propulsion source is a permanent-magnet synchronous machine. The reducer contains two helical gear stages. The half-shafts connect the reducer output to the wheels. The tire is represented by a torsional stiffness and damping in parallel with the vehicle longitudinal inertia. I neglect steering dynamics and lateral load transfer because the study concerns straight-line regenerative braking. I also neglect clutch compliance because the electric vehicle uses a fixed mechanical connection between the motor and the reducer. The model is deterministic and is used to compare baseline and compensated behavior under the same torque request.
I define the main variables as follows. The motor electrical angular speed is denoted by the symbol for electrical angular velocity. The mechanical motor speed is denoted by the symbol for mechanical angular velocity. The half-shaft speed at the wheel side is denoted by the symbol for vehicle-side angular velocity. The motor torque is the sum of the commanded torque and the torque ripple. The load torque at the wheel side is produced by rolling resistance, aerodynamic drag, road grade, and hydraulic braking. The regenerative torque is negative with respect to traction and is limited by the battery, inverter, and motor capability.
2. Permanent-Magnet Synchronous Machine Model
I use a flux-linkage model because the compensation study requires the dominant torque ripple to be represented. The voltage equations in the rotor reference frame are written as a coupled electrical subsystem. The machine has eight poles and forty-eight slots in the reference design. The slot and pole combination produces a rich harmonic spectrum, and the low-order current harmonics are transmitted to the driveline as torque pulsation.
$$
\begin{aligned}
u_d &= R_s i_d + \frac{d\psi_d}{dt} – \omega_e \psi_q,\\
u_q &= R_s i_q + \frac{d\psi_q}{dt} + \omega_e \psi_d,\\
\psi_d &= L_d i_d + \psi_f,\\
\psi_q &= L_q i_q,\\
T_e &= \frac{3}{2}P_n\left[\psi_f i_q + (L_d-L_q)i_d i_q\right],\\
J_m\frac{d\omega_m}{dt} &= T_e – T_L – B\omega_m.
\end{aligned}
$$
The electromagnetic torque contains a mean component and a harmonic component. The harmonic component is not an external disturbance in a strict sense because it originates inside the electric machine. However, from the viewpoint of the mechanical driveline, it behaves as a broadband excitation whose frequency scales with motor speed. I retain the dominant twenty-fourth and forty-eighth orders because their amplitudes are significant in the electric vehicle operating range. The machine parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Maximum power | 80 kW |
| Maximum torque | 180 N·m |
| Base speed | 4000 r/min |
| DC-link voltage | 300 V |
| Stator resistance | 0.008 Ω |
| Direct-axis inductance | 0.047 mH |
| Quadrature-axis inductance | 0.142 mH |
| Permanent-magnet flux linkage | 0.083 Wb |
| Pole pairs | 4 |
3. Gear Mesh, Backlash, and Shaft Compliance
I represent the gear mesh as a piecewise stiffness-damping contact. The backlash is the total clearance between the driving and driven flanks. When the relative displacement is inside the clearance band, the mesh force is zero. When contact occurs, the force is proportional to the elastic penetration and to the relative velocity. This piecewise behavior causes impact when the torque reverses. The impact is especially important in an electric vehicle because the motor can reverse torque much faster than a combustion engine.
$$
F_g =
\begin{cases}
k_g(\theta-\alpha)+c_g\dot{\theta}, & \theta>\alpha,\\
0, & |\theta|\le \alpha,\\
k_g(\theta+\alpha)+c_g\dot{\theta}, & \theta<-\alpha,
\end{cases}
$$
where the relative gear displacement, half-clearance, mesh stiffness, and mesh damping are represented by the corresponding symbols. The driving shaft and the intermediate shaft are modeled as torsional spring-damper elements. The transmitted shaft torque is the sum of an elastic term and a viscous term. The half-shaft is modeled in the same way because its compliance is the main path for low-frequency oscillation.
$$
T_{ds}=k_{ds}\left(\theta_{fd}-\theta_{ds}\right)+c_{ds}\left(\dot{\theta}_{fd}-\dot{\theta}_{ds}\right).
$$
The gear train has two stages. The first stage reduces speed with a ratio of 4.0, and the second stage reduces speed with a ratio of approximately 2.333. The overall ratio is about 9.33. The gear parameters are given in Table 2. The driveline stiffness and damping parameters are given in Table 3. I use these values consistently in the eigenanalysis, reduced-order model, and simulation.
| Parameter | First-stage driving gear | First-stage driven gear | Second-stage driving gear | Second-stage driven gear |
|---|---|---|---|---|
| Number of teeth | 36 | 144 | 36 | 84 |
| Helix angle | 20 deg | 20 deg | 20 deg | 20 deg |
| Normal pressure angle | 20 deg | 20 deg | 20 deg | 20 deg |
| Base radius | 34 mm | 136 mm | 34 mm | 79.5 mm |
| Parameter | Value |
|---|---|
| Intermediate-shaft stiffness | 85,943 N·m/rad |
| Motor-shaft stiffness | 100,000 N·m/rad |
| Half-shaft stiffness | 6,208 N·m/rad |
| Tire torsional stiffness | 18,000 N·m/rad |
| Motor-shaft damping | 2 N·m·s/rad |
| Intermediate-shaft damping | 0.9 N·m·s/rad |
| Half-shaft damping | 9.5 N·m·s/rad |
| Tire damping | 286.56 N·m·s/rad |
| Motor equivalent inertia | 0.03 kg·m² |
| First-stage driving gear inertia | 0.00016 kg·m² |
| First-stage driven gear inertia | 0.00125 kg·m² |
| Second-stage driving gear inertia | 0.00031 kg·m² |
| Second-stage driven gear inertia | 0.007 kg·m² |
| Tire equivalent inertia | 0.63 kg·m² |
| Vehicle equivalent inertia | 144 kg·m² |
4. Longitudinal Dynamics of the Electric Vehicle
I use a longitudinal model to connect the driveline torque to the occupant-perceived acceleration. The road load includes rolling resistance, aerodynamic drag, and grade resistance. The ground braking force is the sum of the front hydraulic braking force and the regenerative braking force referred to the wheel. For a front-drive electric vehicle, the front wheel brake force is the sum of the hydraulic part and the motor part. The regenerative torque is multiplied by the reducer ratio and the driveline efficiency, and divided by the rolling radius.
$$
\begin{aligned}
m_c\frac{dv}{dt} &= -F_f-F_a-F_i-F_{xb},\\
F_f &= m_c g f_r\cos\alpha_r,\\
F_a &= \frac{1}{2}\rho C_d A_f\left(v+v_w\right)^2,\\
F_i &= m_c g\sin\alpha_r,\\
F_{bf} &= F_{zr}+\frac{T_{reg}i_0 i_g \eta_t}{R},\\
F_{xb} &= F_{bf}+F_{br}.
\end{aligned}
$$
The vehicle and environment parameters are listed in Table 4. I select a mass of 1327 kg and a wheel radius of 0.3 m. The overall driveline ratio is 9.3. These values produce a realistic electric vehicle acceleration range and a realistic half-shaft torque level during regenerative braking.
| Parameter | Value |
|---|---|
| Vehicle mass | 1327 kg |
| Wheelbase | 2.69 m |
| Wheel rolling radius | 0.3 m |
| Overall final-drive ratio | 9.3 |
| Air density | 1.225 kg/m³ |
| Aerodynamic drag coefficient | 0.32 |
| Frontal area | 2.1 m² |
| Rolling resistance coefficient | 0.012 |
5. Natural Frequencies and Critical Speeds
I assemble the mass matrix and stiffness matrix of the electric vehicle driveline. The natural frequencies are obtained from the generalized eigenvalue problem. The eigenvalues determine which excitation orders can produce resonance. The mode shapes indicate whether the oscillation is concentrated at the motor, the reducer, the half-shaft, or the tire. I solve the undamped eigenproblem first and then use the damping matrix to estimate the modal damping. The eigenvalue equation is written as follows.
$$
\det\left(\mathbf{M}^{-1}\mathbf{K}-\omega_n^2\mathbf{I}\right)=0.
$$
The first six natural frequencies are listed in Table 5. The first three modes are the important ones for regenerative braking comfort. The first mode is a low-frequency driveline mode dominated by the half-shaft and vehicle inertia. The second and third modes contain reducer and shaft interaction. The higher modes are far above the torque reversal bandwidth and are not strongly excited by the mode-switch transient.
| Mode | Natural angular frequency (rad/s) | Natural frequency (Hz) | Dominant response |
|---|---|---|---|
| 1 | 7.20 | 1.15 | Half-shaft and vehicle inertia |
| 2 | 105.53 | 16.79 | Reducer and shaft coupling |
| 3 | 105.53 | 30.88 | Driveline torsion |
| 4 | 10,485.07 | 1,668.75 | Gear mesh and local compliance |
| 5 | 27,331.42 | 4,349.93 | High-frequency shaft mode |
| 6 | 54,613.76 | 8,692.05 | High-frequency structural mode |
The critical speed of the electric machine is obtained by equating the harmonic order frequency to a natural frequency. A harmonic order of twenty-four or forty-eight is considered because these orders have large amplitudes in the machine torque spectrum. The critical speed is expressed in motor revolutions per minute. When the electric vehicle reaches the corresponding vehicle speed, the harmonic excitation coincides with a driveline mode.
$$
n_c=\frac{60 f_n}{r P_n}.
$$
Table 6 summarizes the critical speeds. I convert the motor speed to vehicle speed using the wheel radius and the overall ratio. Several critical speeds fall inside the normal urban and suburban operating range of an electric vehicle. This is the reason the mode-switch transient can be uncomfortable even when the average braking torque is moderate.
| Harmonic order | Natural frequency (Hz) | Motor critical speed (r/min) | Approximate vehicle speed (km/h) |
|---|---|---|---|
| 24 | 1.15 | 4171.95 | 47.21 |
| 24 | 16.79 | 10,874.83 | 120.00 |
| 48 | 1.15 | 2085.96 | 23.58 |
| 48 | 16.79 | 5437.41 | 61.51 |
| 48 | 30.88 | 10,865.12 | 120.00 |
The three-dimensional spectral response of the baseline electric vehicle shows that the first three modes dominate the half-shaft torque. The response is not a single-frequency sine wave because the gear mesh and the motor ripple are modulated by speed. However, the dominant peaks remain aligned with the low-order natural frequencies. This justifies a reduced-order model for controller design. I do not need to include every high-frequency mode in the control law; I need the low-frequency mode that carries the transient comfort energy.
6. Influence of Gear Backlash
I vary the gear backlash while keeping the half-shaft stiffness and damping fixed. The backlash is expressed as an equivalent angular clearance. Five cases are examined: zero clearance, 3 degrees, 5 degrees, 7 degrees, and 10 degrees. The half-shaft torque oscillation grows with clearance, but the growth is not linear. The first contact after torque reversal produces an impact, and the subsequent contact loss produces a free-decay interval. The combination appears as a delayed and amplified oscillation.
| Backlash case | Peak half-shaft torque oscillation | Response delay | Qualitative comfort result |
|---|---|---|---|
| Zero clearance | Lowest | Shortest | Best baseline but not physically achievable |
| 3 degrees | Slightly higher | Slightly longer | Acceptable but visible ripple |
| 5 degrees | Moderate increase | Moderate | Noticeable shunt at torque reversal |
| 7 degrees | High | Long | Uncomfortable impact and delayed response |
| 10 degrees | Highest | Longest | Strong impact and prolonged oscillation |
The result supports a design principle for electric vehicle drivelines. Backlash cannot be eliminated in a production gearbox, and a very small backlash may increase manufacturing cost and friction. The control system should therefore tolerate a bounded clearance. The compensation law must not rely on immediate contact. It should shape the torque trajectory so that the gear flanks approach each other with low relative velocity. This is why the half-shaft torque feedback is preferable to a pure motor-speed feedback. The half-shaft torque contains the effect of backlash and shaft compliance before the impact reaches the vehicle body.
7. Influence of Half-Shaft Damping
I examine the half-shaft damping because it is the most direct passive parameter for the low-frequency electric vehicle driveline mode. I vary the half-shaft damping from 10 to 200 N·m·s/rad while keeping the backlash and stiffness fixed. The torque ripple rate and the decay time both decrease strongly. The result confirms that the first mode is underdamped and that additional damping in the half-shaft path is beneficial.
| Half-shaft damping (N·m·s/rad) | Torque ripple rate (%) | Decay time (s) | Observation |
|---|---|---|---|
| 10 | 290.50 | 1.29 | Strong low-frequency oscillation |
| 50 | Approximately 220 | Approximately 0.90 | Visible improvement |
| 100 | Approximately 160 | Approximately 0.55 | Fast transient suppression |
| 150 | Approximately 115 | Approximately 0.38 | Near-comfortable response |
| 200 | 91.24 | 0.28 | Best passive result in this study |
The passive solution is effective but not directly applicable to an existing electric vehicle. Increasing the physical half-shaft damping would require a redesign of the shaft, the bearings, and the mounting system. It would also increase drag and may reduce driveline efficiency. I therefore use the passive result as a target. The controller should synthesize the same increase in equivalent damping by modulating the motor torque, without adding mechanical hardware.
8. Reduced-Order Driveline Model
For controller design I reduce the driveline to two inertias connected by an equivalent spring and damper. The motor side includes the rotor and the reducer gears referred to the motor shaft. The vehicle side includes the wheel, tire, and vehicle equivalent inertia. The gear ratio transforms the motor-side quantities to the wheel side. The reduced model preserves the first low-frequency mode because this mode dominates the comfort response. The equations are written in equivalent coordinates.
$$
\begin{aligned}
J_{me}\dot{\omega}_m &= T_m-T_e,\\
J_{ve}\dot{\omega}_v &= T_e-T_L,\\
T_e &= k_e\left(\frac{\theta_m}{i_g}-\theta_v\right)+c_e\left(\frac{\omega_m}{i_g}-\omega_v\right).
\end{aligned}
$$
The equivalent inertias are obtained by referring all gear inertias to the motor shaft and adding the vehicle inertia at the wheel side. The equivalent stiffness and damping are obtained from a series combination of the motor shaft, intermediate shaft, half-shaft, and tire. The numerical values are given in Table 9. The equivalent damping is small compared with the critical damping of the first mode, which confirms the underdamped nature of the electric vehicle driveline.
$$
\begin{aligned}
J_{me} &= J_m+J_1+\frac{J_2+J_3}{i_1^2}+\frac{J_4}{i_1^2i_2^2},\\
J_{ve} &= J_o+J_v,\\
k_e &= \left(\frac{1}{k_i}+\frac{1}{i_1^2k_m}+\frac{1}{i_1^2i_2^2k_a}+\frac{1}{i_1^2i_2^2k_o}\right)^{-1},\\
c_e &= \left(\frac{1}{c_i}+\frac{1}{i_1^2c_m}+\frac{1}{i_1^2i_2^2c_a}+\frac{1}{i_1^2i_2^2c_o}\right)^{-1}.
\end{aligned}
$$
| Parameter | Value |
|---|---|
| Equivalent motor-side inertia | 0.0348 kg·m² |
| Equivalent vehicle-side inertia | 144.63 kg·m² |
| Equivalent torsional stiffness | 3590 N·m/rad |
| Equivalent torsional damping | 5.2 N·m·s/rad |
| Natural frequency of the reduced mode | 59.9 rad/s, 9.54 Hz equivalent |
| Damping ratio | 0.043 |
The state-space form is written with the motor speed, vehicle-side speed, and relative angle as states. The input is the motor torque, and the disturbance is the load torque. The output is the motor speed or the half-shaft torque, depending on the controller. The state-space equation is used to compute the transfer function and the pole-zero map.
$$
\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}u+\mathbf{D}d,\qquad
y=\mathbf{C}\mathbf{x}.
$$
The transfer function from motor torque to motor speed contains a pair of complex-conjugate poles close to the imaginary axis. The numerator contains a second-order term because the half-shaft torque is not a pure rigid-body response. The numerical transfer function obtained from the reduced model is shown below. The denominator determines the oscillation frequency and damping. The small damping ratio is the root cause of the prolonged half-shaft torque oscillation.
$$
G_{ma\_T}(s)=\frac{\omega_m(s)}{T_m(s)}
=\frac{33.3s^2+1.5s+2090}{s^2+5.2s+3590}.
$$
The poles are located at approximately minus 2.6 plus or minus 59.86 imaginary units. The real part is small in magnitude, so the decay is slow. The damping ratio is approximately 0.043. This is much lower than the value of 0.3 to 0.7 that would be desirable for a comfortable electric vehicle mode switch. A passive increase in half-shaft damping moves the poles to the left, but a production electric vehicle cannot easily adopt a much larger half-shaft damper. I therefore use active damping.
9. Half-Shaft Torque Estimation
The controller uses the half-shaft torque as the feedback variable. A direct torque sensor would add cost and may be difficult to install in a production electric vehicle. I therefore estimate the half-shaft torque from the motor speed, the wheel speed, and the half-shaft model. The estimate contains an elastic term and a damping term. The elastic term is obtained by integrating the relative speed between the motor-side equivalent speed and the wheel-side speed. The damping term is proportional to the same relative speed. The estimator is written as follows.
$$
\hat{T}_{hs}(t)
=k_{hs}\int_0^t\left(\frac{\omega_m}{i_g}-\omega_v\right)d\tau
+c_{hs}\left(\frac{\omega_m}{i_g}-\omega_v\right).
$$
The estimator requires the half-shaft stiffness and damping. These are known nominal parameters in the electric vehicle model. In practice they vary with temperature and load, so a conservative filter is used. The estimated torque is low-pass filtered to remove the high-frequency gear mesh component. The cutoff frequency is selected above the first three driveline modes but below the switching frequency and the gear mesh frequency. This preserves the comfort-relevant signal and rejects sensor noise.
$$
H_{lp}(s)=\frac{\omega_c}{s+\omega_c}.
$$
I use the estimated half-shaft torque in a feedforward-feedback architecture. The feedforward term comes from the baseline regenerative torque request. The feedback term is a PID function of the torque error. The proportional term increases the stiffness of the closed-loop response. The derivative term increases the effective damping. The integral term is set to zero in this study because the steady-state regenerative torque is already known and because integral action can cause windup during mode switching. The controller output is limited to the motor torque capability and is ramped at a safe rate.
$$
T_{reg}^{cmd}
=T_{reg}^{ff}
-K_P\left(\hat{T}_{hs}-T_{hs}^{ref}\right)
-K_I\int_0^t\left(\hat{T}_{hs}-T_{hs}^{ref}\right)d\tau
-K_D\frac{d}{dt}\left(\hat{T}_{hs}-T_{hs}^{ref}\right).
$$
The reference half-shaft torque is generated from the driver demand and the regenerative braking strategy. During the transition from traction to regenerative braking, the reference is a smooth negative torque. The controller does not change the average braking force. It changes the transient shape of the motor torque so that the half-shaft torque follows the reference with low overshoot. The PID gains are listed in Table 10. The integral gain is zero, so the controller is effectively a proportional-derivative controller with a feedforward path.
| Gain | Value | Function |
|---|---|---|
| Proportional gain | 15.71 | Shapes the closed-loop stiffness and reduces steady error |
| Integral gain | 0 | Avoids windup during mode switching |
| Derivative gain | 0.79 | Adds equivalent damping and suppresses overshoot |
| Torque command limit | 180 N·m | Protects the electric machine and battery |
| Torque rate limit | 500 N·m/s | Prevents impulsive torque reversal |
| Estimator filter cutoff | 40 Hz | Rejects gear mesh noise |
10. Simulation Procedure
I simulate the electric vehicle in a straight-line maneuver. The first interval is a short steady cruise. The accelerator pedal is then fully depressed for five seconds to accelerate the electric vehicle. After that, the pedal is released and the motor torque is commanded to negative one hundred newton-metres for regenerative braking. I compare two cases. The baseline case has no compensation. The compensated case uses the half-shaft torque estimator and the PID damping controller. Both cases use the same road load, the same gear backlash, and the same motor torque limit.
The evaluation uses four indicators. The first is the half-shaft torque ripple rate. The second is the maximum half-shaft speed difference. The third is the root-mean-square jerk. The fourth is the longitudinal acceleration trace. I also examine the speed at the end of the regenerative braking interval to check whether the controller harms energy recovery. The torque ripple rate and jerk are defined as follows.
$$
\delta_T=\frac{T_{\max}-T_{\min}}{T_{\rm ref}}\times 100\%.
$$
$$
J_{\rm rms}=\sqrt{\frac{1}{T}\int_0^T\left(\frac{d a_z}{dt}\right)^2dt}.
$$
11. Results and Discussion
The baseline electric vehicle accelerates to a maximum speed of approximately 71.07 km/h. With compensation, the maximum speed is approximately 71.66 km/h. The small increase is caused by reduced oscillation loss during the acceleration interval. During regenerative braking, the baseline half-shaft speed difference reaches 268.31 rad/s. With compensation, the maximum difference falls to 44.61 rad/s. This is an 83.4 percent reduction. The half-shaft speed trace becomes smoother, and the low-frequency oscillation is almost removed.
| Indicator | Baseline | Compensated | Improvement |
|---|---|---|---|
| Maximum vehicle speed | 71.07 km/h | 71.66 km/h | Comparable |
| Maximum half-shaft speed difference | 268.31 rad/s | 44.61 rad/s | 83.4% reduction |
| Half-shaft torque ripple rate | 353.8% | 206.57% | 41.61% reduction |
| RMS jerk during mode switching | 51.59 m/s³ | 11.81 m/s³ | 77.1% reduction |
| Peak longitudinal acceleration | -9.58 m/s² | Smooth transient | Severe oscillation removed |
| Vehicle speed at end of braking interval | 10.17 km/h | 12.33 km/h | Deceleration slope comparable |
The longitudinal acceleration trace explains the comfort improvement. In the baseline electric vehicle, the acceleration drops abruptly when the motor torque reverses. The occupant perceives a sharp negative peak followed by oscillation. In the compensated electric vehicle, the acceleration changes progressively. The peak negative acceleration is no longer a sharp spike. The jerk is therefore much lower. The root-mean-square jerk falls from 51.59 to 11.81 m/s³, which is a 77.1 percent reduction. This is the most direct indicator of occupant comfort in the present study.
The half-shaft torque trace shows the same behavior. In the baseline case, the torque oscillates for about one second after the mode switch. The peak-to-peak ripple is large, and the gear backlash produces a delayed contact impact. In the compensated case, the torque rises or falls in a controlled manner. The ripple rate is reduced from 353.8 percent to 206.57 percent, which is a 41.61 percent reduction. The remaining ripple is mainly the high-frequency motor torque harmonic and the gear mesh component. These components are less harmful to comfort because they are above the dominant body response band.
The spectral response confirms the mechanism. The baseline electric vehicle has sharp peaks at 1.15 Hz, 16.79 Hz, and 30.88 Hz. These peaks correspond to the first three driveline modes. The compensated electric vehicle has much lower peaks at the same frequencies. The PID derivative term increases the equivalent damping, so the poles move away from the imaginary axis. The resonance peaks are flattened. The controller does not shift the natural frequencies significantly because it does not change the mechanical stiffness. It adds damping, which is the correct remedy for an underdamped electric vehicle driveline.
| Frequency band | Baseline response | Compensated response | Interpretation |
|---|---|---|---|
| 1.15 Hz | Sharp low-frequency peak | Peak reduced toward zero | Half-shaft mode damped |
| 16.79 Hz | Reducer-shaft resonance | Strongly attenuated | Mode-switch energy reduced |
| 30.88 Hz | Torsional resonance | Strongly attenuated | Impact energy dissipated |
| Above 100 Hz | Gear mesh and harmonic content | Largely unchanged | Controller bandwidth does not excite high-frequency modes |
I also check the energy recovery. The compensated electric vehicle reaches 12.33 km/h at the end of the braking interval, while the baseline reaches 10.17 km/h. The difference is small, and the deceleration slope during the main braking phase is essentially the same. This means the compensation does not significantly reduce the regenerative energy. The small difference is caused by the different oscillation loss and by the torque limiting during the first part of the mode switch. The controller is not a brake-by-wire replacement. It shapes the transient torque while preserving the mean braking torque.
12. Robustness and Implementation Considerations
The proposed compensation is suitable for an electric vehicle because it uses existing speed measurements and a software torque command. The motor speed is available from the resolver or encoder. The wheel speed is available from the anti-lock braking system. The half-shaft torque is estimated from these signals. The controller does not require a physical torque sensor. The main implementation risks are parameter variation, signal delay, and model mismatch. I therefore include a low-pass filter, a torque rate limit, and a conservative derivative gain.
| Risk | Effect on compensation | Mitigation |
|---|---|---|
| Half-shaft stiffness variation | Bias in estimated torque | Use conservative nominal stiffness and feedforward adaptation |
| Half-shaft damping variation | Under- or over-damping | Limit derivative gain and use robust filter |
| Wheel-speed delay | Phase lag in feedback | Compensate with motor-speed derivative and low-pass filter |
| Gear backlash variation | Residual impact at torque reversal | Rate-limit torque command and use smooth reference |
| Road friction change | Wheel slip and estimator error | Integrate with ABS and traction control states |
| Sensor noise | Derivative amplification | Use filtered derivative and avoid pure differentiation |
The controller bandwidth must remain below the first high-frequency gear mesh mode. If the derivative gain is too large, the controller can excite the high-frequency modes and produce acoustic noise. The gain values in Table 10 are chosen to keep the closed-loop bandwidth within the comfort band. The torque rate limit is also important. A very fast torque reversal can produce a large impact even if the feedback gain is high. The rate limit gives the gear flanks time to approach each other with low relative velocity.
13. Design Implications for Electric Vehicle Drivelines
The study leads to several design implications. First, the electric vehicle driveline should be treated as an electromechanical system, not as a rigid transmission. The motor torque ripple and the gear backlash interact. A controller that only filters the motor current may reduce the harmonic source but may not suppress the low-frequency half-shaft oscillation. Second, the half-shaft torque is a better feedback variable than the motor speed for comfort control. The motor speed contains the rigid-body acceleration and the wheel dynamics, while the half-shaft torque contains the elastic energy that causes the shunt. Third, the controller should add damping rather than stiffness. Increasing stiffness can shift the resonance but does not remove the oscillation. Increasing damping reduces the peak response at the resonance.
Fourth, the design of the electric vehicle regenerative braking strategy should include a torque transition trajectory. A step change in regenerative torque is simple but uncomfortable. A smooth reference with a controlled rate is easier for the driveline to follow. The PID controller then corrects the residual error. This combination is more robust than a pure feedforward ramp because the feedback compensates for parameter variation and backlash. Fifth, the controller should be integrated with the hydraulic brake blending. The regenerative torque is limited by the battery and motor, and the hydraulic brake must take over when the regenerative torque is reduced. The comfort controller should not fight the hydraulic brake compensator.
| Method | Advantage | Limitation | Suitability |
|---|---|---|---|
| Increase physical half-shaft damping | Direct and passive | Requires redesign and adds drag | Limited for existing platforms |
| Motor torque filtering | Simple software change | May slow response and reduce energy recovery | Useful as a supplement |
| Active damping by speed feedback | No extra sensor | Phase lag and noise sensitivity | Moderate |
| Half-shaft torque feedback PID | Targets the comfort variable directly | Requires estimation and calibration | High for production electric vehicle control |
| Model predictive control | Handles constraints explicitly | High computational burden | Future work |
14. Discussion of Nonlinear Coupling
The electric vehicle comfort problem is not caused by a single nonlinearity. It is caused by the interaction of three nonlinearities. The motor torque ripple is a periodic excitation whose frequency depends on speed. The gear backlash is a piecewise contact nonlinearity that produces impact and dead zone. The half-shaft and tire compliance create a low-frequency energy storage path. The combination produces a response that is amplitude-dependent and speed-dependent. A linear controller can still be effective if it is designed around the dominant low-frequency mode and if the torque command is rate-limited. This is the reason the PID approach works in the present study.
The half-shaft torque feedback also provides a physical interpretation. The derivative term of the PID controller is proportional to the rate of change of the half-shaft torque. This term opposes the rapid buildup of elastic energy. It acts like a virtual damper placed across the half-shaft. The proportional term acts like a virtual spring. The virtual damper is the key element for comfort because it dissipates the mode-switch energy. The virtual spring improves the tracking of the reference torque. The feedforward term preserves the steady braking force. The combination is a virtual driveline mount that can be calibrated in software.
$$
T_{reg}^{cmd}
=T_{reg}^{ff}
-K_P e_T
-K_D\dot{e}_T,
\qquad
e_T=\hat{T}_{hs}-T_{hs}^{ref}.
$$
The virtual damping coefficient is not constant in a physical sense. It depends on the operating point, the estimator filter, and the torque limit. However, around the mode-switch operating point, the closed-loop response is approximately linear. The effective damping ratio increases from about 0.043 to a value that produces a well-damped response. The exact value depends on the derivative gain and the estimator bandwidth. The calibration objective is to minimize jerk without causing torque reversal chatter or high-frequency noise.
15. Limitations and Future Work
The present study is simulation-based. It uses a deterministic road load and a fixed half-shaft stiffness. In a real electric vehicle, the half-shaft stiffness changes with temperature, the gear backlash changes with wear, and the road friction changes with weather. The controller must therefore be robust to parameter variation. The next step is hardware-in-the-loop testing with a dynamometer and a real motor inverter. The second step is vehicle testing on a proving ground. The third step is integration with the battery management system so that the regenerative torque limit is respected during high state-of-charge and low-temperature conditions.
Another limitation is that the controller uses a single feedback variable. A multi-variable controller could use motor speed, wheel speed, and half-shaft torque simultaneously. A model predictive controller could include the backlash as a constraint and optimize the torque trajectory over a short horizon. However, the computational cost and calibration effort would be higher. For a production electric vehicle, a well-tuned PID with feedforward and filtering is attractive because it is simple, transparent, and easy to certify. I therefore regard the present controller as a practical baseline for future advanced control.
16. Conclusions
I have presented a comfort compensation control study for an electric vehicle during regenerative braking. The main conclusions are as follows. First, the electric vehicle driveline is an underdamped electromechanical system. The dominant low-frequency mode is close to the imaginary axis, and a step torque reversal excites a prolonged half-shaft torque oscillation. Second, the motor torque harmonics, gear backlash, and shaft flexibility are coupled. The twenty-fourth and forty-eighth torque orders can coincide with the first three natural frequencies at realistic electric vehicle speeds. Third, increasing half-shaft damping is an effective passive remedy. A damping increase from 10 to 200 N·m·s/rad reduces the torque ripple rate from 290.50 percent to 91.24 percent and shortens the decay time from 1.29 seconds to 0.28 seconds.
Fourth, a half-shaft torque feedback PID controller can synthesize the beneficial effect of increased damping without changing the mechanical structure. The controller uses a feedforward-feedback architecture, a torque estimator, a low-pass filter, and a torque rate limit. In the simulated mode switch, the half-shaft torque ripple rate falls by 41.61 percent, the maximum half-shaft speed difference falls by 83.4 percent, and the root-mean-square jerk falls by 77.1 percent. The longitudinal acceleration changes from a severe negative spike to a smooth transition. Fifth, the resonance peaks at 1.15 Hz, 16.79 Hz, and 30.88 Hz are strongly attenuated, while the average regenerative braking force and the deceleration slope remain comparable. The proposed method therefore improves electric vehicle ride comfort during regenerative braking without a significant energy recovery penalty.
| Performance index | Baseline electric vehicle | Compensated electric vehicle | Relative change |
|---|---|---|---|
| Half-shaft torque ripple rate | 353.8% | 206.57% | 41.61% lower |
| Maximum half-shaft speed difference | 268.31 rad/s | 44.61 rad/s | 83.4% lower |
| RMS jerk | 51.59 m/s³ | 11.81 m/s³ | 77.1% lower |
| Peak negative acceleration | 9.58 m/s² magnitude | Smooth transient | Severe spike removed |
| Low-frequency resonance peaks | Sharp at 1.15, 16.79, and 30.88 Hz | Strongly attenuated | Comfort band improved |
| Regenerative deceleration slope | Reference | Comparable | Energy recovery preserved |
In summary, I conclude that comfort-oriented regenerative braking in an electric vehicle should be treated as a damping problem. The average braking torque determines the stopping distance, but the transient torque trajectory determines the occupant perception. By estimating the half-shaft torque and feeding it back through a proportional-derivative compensator, the electric vehicle can achieve a smoother mode transition, lower jerk, and reduced half-shaft oscillation. The method is compatible with existing electric vehicle sensors and can be implemented as a software upgrade. It therefore offers a practical path toward improved regenerative braking comfort in future electric vehicle platforms.