The proliferation of multi-motor distributed electric drive technology marks a revolutionary shift in automotive chassis design, effectively decoupling wheel control constraints and enabling unprecedented freedom in vehicle dynamics management. Currently, the dynamic analysis and control of such distributed drive electric vehicle systems predominantly rely on deterministic models. These models form the bedrock for sophisticated torque-vectoring algorithms, which are widely implemented to enhance stability and performance. However, a significant challenge arises when these deterministic frameworks encounter real-world driving environments. In practice, the vehicle system is perpetually subject to a multitude of uncertainties—including random perturbations in system parameters (like tire pressure and wear), unmodeled dynamics, and measurement inaccuracies in state variables and sensor data. These uncertainties are particularly pronounced in the complex, interactive process between the wheel and the road surface. Consequently, control systems derived from idealized, deterministic models often struggle to maintain optimal performance across diverse and unpredictable road conditions. This gap between theoretical models and practical application underscores a critical need to develop a new foundational theory for the dynamics and control of distributed drive electric cars, one that explicitly accounts for and adapts to systemic uncertainties.
Our research addresses this fundamental gap by focusing on the uncertainties inherent in the wheel-road interaction—the primary conduit for force generation in any ground vehicle. We move beyond deterministic modeling to construct a probabilistic framework for the wheel-ground contact forces. This framework incorporates key stochastic factors such as fluctuations in wheel normal load, non-uniform distribution of road adhesion coefficients, and variations in tire effective radius due to wear and deformation. By establishing this uncertainty-aware model, we aim to provide a more realistic representation of the forces acting on a distributed drive electric vehicle.
Building upon this model, we conducted extensive simulation experiments to analyze vehicle behavior under both hard-surface (e.g., asphalt) and soft-surface (e.g., soil, sand) conditions. A deep analysis of these results revealed a crucial and previously underexplored relationship: the interplay between the constraints imposed by the road surface and those imposed by the wheel-end control strategy. We conceptualize this relationship as the “Tension-Relaxation Theory.” In essence, when the road surface provides a firm, predictable, and “tight” constraint (high friction, low deformation), the control system should adopt a “relaxed” constraint on individual wheel speeds, prioritizing direct torque control. Conversely, when the road surface offers a “loose” or uncertain constraint (low friction, high deformation), the control system should “tighten” its constraint on wheel speeds, implementing a coordinated speed control mode to prevent excessive slip and maintain efficiency. This theory posits that a mismatch—applying a tight control constraint on a tight road constraint, or a loose control constraint on a loose road constraint—leads to internal energy dissipation, known as parasitic power, and can induce destabilizing yaw moments.
To substantiate this theory and derive a practical control mechanism, we performed a detailed statistical disturbance analysis. We quantified the impact of various uncertainty sources on three key performance indices for the electric vehicle car: Parasitic Power (the wasted energy due to kinematically incompatible wheel motions), Traction Efficiency (the ratio of effective propulsion power to total wheel power), and Additional Yaw Moment (unwanted rotational force caused by uneven longitudinal tire forces). This analysis allowed us to formalize the intrinsic link between the probabilistic state of wheel-road contact and the optimal choice of control mode (torque vs. speed). The outcome is the proposal of a novel, Mode-Switching Drive Control Mechanism. This mechanism dynamically selects the most appropriate control mode based on real-time estimates of road constraint conditions, effectively applying the Tension-Relaxation Theory in practice.
The core of our methodological approach lies in developing uncertainty-integrated dynamic models for the distributed drive electric vehicle. The longitudinal dynamics of the vehicle body and each individual wheel are governed by the following equations. The vehicle longitudinal motion is described by:
$$ M \dot{v}_x = \sum_{i=1}^{4} F_{x,i} – F_i – F_w $$
$$ F_i = Mg \sin\theta, \quad F_w = \frac{1}{2} \rho C_d A v_x^2 $$
where \( M \) is the total vehicle mass, \( v_x \) is the longitudinal velocity, \( F_{x,i} \) is the longitudinal tire force at wheel \( i \), \( F_i \) is the grade resistance, \( F_w \) is the aerodynamic drag, \( \rho \) is air density, \( C_d \) is the drag coefficient, \( A \) is the frontal area, and \( \theta \) is the road slope. The rotational dynamics for each wheel are:
$$ J_w \dot{\omega}_i = T_{m,i} – F_{x,i} R $$
where \( J_w \) is the wheel’s moment of inertia, \( \omega_i \) is the wheel angular speed, \( T_{m,i} \) is the motor torque applied at the wheel, and \( R \) is the effective tire radius.
The critical innovation is in modeling the longitudinal tire force \( F_x \). Instead of a deterministic value, we model it as a probabilistic function \( F_x = f(s) \cdot \mathbf{u} \cdot \mathbf{N} \), where \( s \) is the wheel slip ratio, \( \mathbf{u} \) represents stochastic road adhesion parameters, and \( \mathbf{N} \) represents stochastic wheel-terrain contact parameters. For a hard surface, we use a modified Burckhardt model under uncertainty:
$$ F_{x,hard} = \widetilde{\mu}_{max} \cdot \left[c_1 (1 – e^{-c_2 s}) – c_3 s \right] \cdot \widetilde{F}_z $$
Here, \( \widetilde{\mu}_{max} \sim \mathcal{N}(\mu_{\mu}, \sigma_{\mu}^2) \) represents the uncertain peak adhesion coefficient, and \( \widetilde{F}_z \sim \mathcal{N}(\mu_{F_z}, \sigma_{F_z}^2) \) represents the uncertain normal load. The tire’s effective radius is also considered uncertain: \( \widetilde{R}_e \sim \mathcal{N}(\mu_{R}, \sigma_{R}^2) \), influenced by load \( \Delta F_z \), pressure \( \Delta P \), and wear \( \Delta R_W \):
$$ \Delta R = \Delta R_L + \Delta R_P + \Delta R_W = \left( \frac{R_{Lnom}}{1 + k_{load}\Delta F_z} – R_{Lnom} \right) + \left( \frac{R_{Pnom}}{1 + k_{p}\Delta P} – R_{Pnom} \right) + 0.01 R_e $$
For a soft, deformable surface, we employ a terramechanics-based model under uncertainty. The traction force is given by:
$$ F_{x,soft} = A \widetilde{\tau}_{max} \left[ 1 – \frac{k}{sl} \left( 1 – e^{-sl/k} \right) \right] $$
where \( A \) is the contact area, \( l \) is the contact length, \( k \) is the shear deformation modulus, and \( \widetilde{\tau}_{max} \) is the uncertain maximum shear stress of the terrain. This maximum stress is related to stochastic soil parameters: cohesion \( \widetilde{C} \sim \mathcal{N}(\mu_C, \sigma_C^2) \) and internal friction angle \( \widetilde{\phi} \sim \mathcal{N}(\mu_{\phi}, \sigma_{\phi}^2) \), as well as the uncertain normal load \( \widetilde{F}_z \):
$$ \widetilde{\tau}_{max} = \widetilde{C} + \frac{\widetilde{F}_z}{A} \tan(\widetilde{\phi}) $$
The local variability of soil parameters can be modeled using a probability clustering method, leading to a non-Gaussian distribution for \( \widetilde{\tau}_{max} \) that accurately reflects real-world terrain heterogeneity.
To evaluate performance and illustrate the need for mode-switching in this electric vehicle car, we define our key metrics mathematically. Parasitic Power \( P_p \) estimates energy loss due to slip and kinematic incompatibility, influenced by radius and slip uncertainty:
$$ P_p \propto \widetilde{F}_z \cdot \widetilde{s} \cdot \widetilde{R}_e \cdot \left( \frac{K_g}{\mu_0 K_0} \right) $$
Traction Efficiency \( \eta_t \) is the ratio of useful power to total wheel power:
$$ \eta_t = \frac{ \sum_{i=1}^{4} (F_{x,i} \cdot v_x) }{ \sum_{i=1}^{4} (T_{m,i} \cdot \omega_i) } $$
Additional Yaw Moment \( \Delta M_z \) arises from unbalanced longitudinal forces:
$$ \Delta M_z = \frac{B_f}{2} (F_{x,fr} – F_{x,fl}) + \frac{B_r}{2} (F_{x,rr} – F_{x,rl}) $$
where \( B_f \) and \( B_r \) are the front and rear track widths.
Our simulation analysis, comparing pure Speed Control (SC) and pure Torque Control (TC) modes, clearly demonstrates the limitations of a single-mode strategy and validates the Tension-Relaxation concept. The results are summarized in the table below:
| Condition | Control Mode | Avg. Parasitic Power (W) | Avg. Traction Efficiency | Avg. |Add. Yaw Moment| (Nm) |
|---|---|---|---|---|
| Hard Surface (High-μ) | Speed Control (SC) | ~100.7 | High | Large (Peak >2500) |
| Torque Control (TC) | ~1.5 | High | ~0 | |
| Soft Surface (Low-μ) | Speed Control (SC) | Moderate | >99% | Moderate |
| Torque Control (TC) | Low | Low (~66% under accel) | ~0 |
The table reveals a critical pattern: On hard surfaces with “tight” road constraints, Torque Control excels—it minimizes parasitic losses and prevents unwanted yaw moments by allowing wheels to find their natural speeds. Speed Control, which tries to enforce a “tight” speed constraint on an already tight system, causes large internal power circulation and significant destabilizing yaw moments. Conversely, on soft surfaces with “loose” road constraints, Speed Control is superior for maintaining high traction efficiency. It prevents wheels from digging in and achieving excessive, inefficient slip. Torque Control, applying a “relaxed” constraint here, leads to poor efficiency as wheels easily spin. This dichotomy is the empirical foundation of our Tension-Relaxation Theory.
To operationalize this theory, we derived a Mode-Switching Drive Control Mechanism based on statistical disturbance analysis. The core is a Mode-Switching Factor \( K \) calculated from the variance of performance metrics under different assumed dominant uncertainties:
$$ K = C_x \cdot d^{-1} \cdot f_E(T, W) \cdot \text{Penalty}(\widetilde{s}, \omega, R’) $$
$$ f_E(T, W) = \lambda_1 \frac{\text{var}(T_{E,soft}, W_{hard})}{\text{var}(T_{E,hard}, W_{soft})} + \lambda_2 \frac{\text{var}(T_{E,hard}, W_{soft})}{\text{var}(T_{E,soft}, W_{hard})} $$
where \( C_x \) is tire longitudinal stiffness, \( d \) is a tuning coefficient, \( T_E \) and \( W \) are traction efficiency and parasitic power, var() denotes variance, \( \lambda \) are weighting gains, and the Penalty() function suppresses undesirable states like high slip. The switching logic is elegantly simple:
- If \( K \geq K_T \): Uncertainties in effective radius \( \Delta R_e \) and speed tracking error \( \Delta n_e \) are dominant. This indicates a “tight” road constraint scenario. → Switch to Torque Control (TC) Mode.
- If \( K < K_T \): Uncertainties in normal load \( \Delta F_z \) and adhesion coefficient \( \Delta \mu \) are dominant. This indicates a “loose” road constraint scenario. → Switch to Speed Control (SC) Mode.
The threshold \( K_T \) is calibrated based on specific vehicle and road characteristics.
We validated the proposed mode-switching mechanism in a challenging simulation scenario featuring a road with a sudden adhesion coefficient jump (e.g., from asphalt to ice-patch and back). The results conclusively demonstrate its superiority over fixed-mode strategies for the distributed drive electric car.
| Performance Metric | Speed Control Only | Torque Control Only | Mode-Switching Control (Proposed) |
|---|---|---|---|
| Avg. Parasitic Power (W) | ~2.97 | ~5.14 | ~2.22 |
| Avg. |Add. Yaw Moment| (Nm) | High (Peak ~2642) | ~0 | Very Low (~9.6) |
| Avg. Traction Efficiency | >99.2% | Variable (Low on soft) | Stable >99% |
The proposed system successfully navigated the changing road conditions. On high-friction segments, it operated in efficient Torque Control mode, eliminating parasitic power and yaw moment. Upon detecting the low-friction patch (via a change in the calculated \( K \) factor), it seamlessly switched to Speed Control mode, preventing traction efficiency from collapsing. This adaptive behavior resulted in the lowest overall parasitic power, near-zero added yaw disturbance, and consistently high traction efficiency throughout the maneuver. Compared to the best single-mode controller for a given metric, the mode-switching mechanism reduced parasitic power by over 13.9% and slashed the additional yaw moment by more than 54.1%, while matching the best-case traction efficiency.
In conclusion, our research addresses a fundamental challenge in the control of distributed drive electric vehicles by formally incorporating system uncertainties into the longitudinal dynamics analysis. We introduced a probabilistic wheel-road interaction model that accounts for realistic stochastic variations in contact parameters. Through rigorous simulation and analysis, we discovered and formulated the “Tension-Relaxation Theory,” which describes the essential relationship between road-imposed constraints and controller-imposed constraints. This theory guided the development of a data-driven, mode-switching control mechanism that dynamically selects between torque and speed control based on real-time statistical analysis of system performance. Validation under complex, variable-grip driving conditions confirms that this adaptive approach significantly outperforms fixed control modes, effectively minimizing parasitic energy losses and stabilizing vehicle yaw motion while maintaining optimal traction efficiency. This work enriches the theoretical foundation for distributed drive electric car control and paves the way for more robust, efficient, and intelligent vehicle dynamics management systems capable of handling the inherent uncertainties of real-world driving. Future work will extend this framework to consider a broader range of uncertainties, such as dynamic load transfer and combined longitudinal-lateral-vertical coupled dynamics, and will focus on developing advanced actuator allocation algorithms for holistic vehicle performance optimization.