In recent years, the rapid advancement of electric vehicle (EV) technology and autonomous driving systems has heightened the demand for precise and reliable trajectory planning, particularly during lane change maneuvers. As a key component of intelligent transportation systems, trajectory planning must balance safety, comfort, and computational efficiency. Traditional polynomial-based methods, while offering continuity and differentiability, often struggle with multi-constraint optimization in dynamic environments. This paper addresses these limitations by introducing an improved polynomial function-based algorithm that incorporates dynamic weight factors and constraint optimization models. We focus on applications in China EV markets, where the integration of autonomous features is accelerating. The proposed method aims to enhance trajectory smoothness, reduce computational overhead, and improve overall performance for electric vehicles operating in complex scenarios.
The foundation of our approach lies in redefining polynomial functions to better suit lane change trajectory planning. Standard polynomial models, such as zero-order (constant), first-order (linear), or even fifth-order polynomials, exhibit inherent trade-offs between flexibility and stability. Lower-order polynomials lack the nonlinearity required for smooth transitions, while higher-order ones may introduce oscillations. To overcome this, we propose an enhanced polynomial base function defined as:
$$ x(t) = \sum_{i=0}^{n} a_i \cdot w(t) \cdot t^i $$
where \( x(t) \) represents the lateral position of the electric vehicle at time \( t \), \( a_i \) are coefficients to be determined, \( n \) is the polynomial order, and \( w(t) \) is a dynamic weight function that adjusts the influence of different polynomial terms over time. This weight function prioritizes lower-order terms near the start and end points to ensure stability and higher-order terms during mid-transition for agility. Such adaptability is crucial for China EV applications, where urban driving conditions often involve frequent lane changes.
To further optimize the algorithm, we integrate dynamic weight factors that respond to real-time vehicle states. The weight factor \( \omega(t) \) is computed as:
$$ \omega(t) = \alpha \cdot \exp(-\beta |d(t)|) + \gamma \cdot \frac{v(t)}{v_{\text{max}}} $$
Here, \( \alpha \) and \( \beta \) are tuning coefficients, \( d(t) \) denotes the deviation from the desired trajectory, \( v(t) \) is the current speed of the electric vehicle, \( v_{\text{max}} \) is the maximum allowed speed, and \( \gamma \) is a speed influence factor. This formulation dynamically adjusts the trajectory based on situational factors, such as proximity to obstacles or speed variations, which is essential for the safety of autonomous electric vehicles in China’s diverse traffic environments. For instance, at higher speeds, the weight factor emphasizes path tracking accuracy to prevent overshooting.
Constraint modeling is another critical aspect of our algorithm. We categorize constraints into safety and kinematic types. The safety constraint, which assesses collision risk, is expressed as:
$$ C_s(t) = k_1 \cdot \frac{d_{\text{min}} – d(t)}{\tau} + k_2 \cdot \left( \frac{a_y(t)}{a_{y,\text{max}}} \right)^2 \leq 0 $$
where \( d_{\text{min}} \) is the minimum safe distance (e.g., 8 m at 60 km/h), \( \tau \) is a prediction time constant (1.2 s), \( a_y(t) \) is the lateral acceleration, \( a_{y,\text{max}} \) is the maximum allowable lateral acceleration (2.5 m/s²), and \( k_1 \), \( k_2 \) are weight coefficients (0.6 and 0.4, respectively). This equation quantitatively links collision risk with vehicle motion, enabling real-time safety evaluations for electric vehicles. Kinematic constraints, such as steering angle limits (±35°) and steering rate limits (±15°/s), are derived from the electromechanical characteristics of EV steering systems, ensuring feasible trajectories under physical boundaries.
Convergence analysis of the algorithm confirms its reliability. The optimization problem is structured to be convex, guaranteeing a unique global solution. We employ a trust-region iterative method with adaptive penalty terms to handle constraints smoothly. The convergence rate follows:
$$ \| x_k – x^* \| \leq c \cdot k^{-p} $$
where \( x_k \) is the solution at iteration \( k \), \( x^* \) is the optimal solution, \( c \) is a positive constant, and \( p > 1 \) denotes the convergence order. This ensures that the algorithm efficiently reaches a feasible solution, which is vital for real-time applications in autonomous electric vehicles.
Trajectory smoothing is optimized through a multi-objective function that minimizes curvature and curvature change rate. The smoothness objective is defined as:
$$ J_{\text{smooth}} = \int \left( \lambda_1 \cdot \left( \frac{d^2 y}{dt^2} \right)^2 + \lambda_2 \cdot \left( \frac{d^3 y}{dt^3} \right)^2 \right) dt $$
where \( y \) is the lateral position, and \( \lambda_1 \), \( \lambda_2 \) are adaptive weight coefficients. Initially set to 0.6 and 0.4, these weights are adjusted dynamically: during acceleration phases (0-30% of the trajectory), \( \lambda_1 \) increases if lateral acceleration exceeds 1.8 m/s² or curvature surpasses 0.03 m⁻¹, emphasizing curvature control. In deceleration phases (30-100%), \( \lambda_2 \) gradually rises to 0.7, focusing on curvature change rate reduction. This分段优化 strategy enhances ride comfort for electric vehicle occupants, particularly in China EV scenarios where smooth maneuvers are prioritized. The variational approach transforms this into an Euler-Lagrange equation:
$$ \lambda_1 \cdot \frac{d^4 y}{dt^4} – \lambda_2 \cdot \frac{d^6 y}{dt^6} = 0 $$
Solving this sixth-order ordinary differential equation requires six boundary conditions, including position, velocity, and acceleration at start and end points, ensuring seamless trajectory transitions.
To improve computational efficiency, we adopt a hierarchical optimization structure that decomposes the problem into dynamic weight and smoothness subproblems. The computational complexity is analyzed as:
$$ O(n \cdot k \cdot m) $$
where \( n \) is the number of trajectory points, \( k \) is the iteration count, and \( m \) is the number of constraints. By employing recursive weight updates and simplified calculations, we reduce the planning time significantly. For example, the weight update uses a momentum-based递推 formula:
$$ w(t+1) = w(t) + \eta \cdot \Delta J[w(t)] + \mu \cdot (w(t) – w(t-1)) $$
with a learning rate \( \eta \) initially set to 0.2 and momentum factor \( \mu = 0.3 \). This approach minimizes oscillations and accelerates convergence, making it suitable for the rapid decision-making required in China EV autonomous systems.
Simulation validation was conducted using a MATLAB/Simulink platform that integrates vehicle dynamics and lane change scenarios. Parameters were based on a typical electric vehicle model, reflecting common specifications in the China EV market. The table below summarizes key simulation parameters:
| Parameter | Value | Unit |
|---|---|---|
| Vehicle Mass | 1850 | kg |
| Wheelbase | 2.78 | m |
| Front/Rear Track Width | 1.58/1.59 | m |
| Maximum Steering Angle | ±35 | ° |
| Steering Rate Limit | ±15 | °/s |
| Maximum Lateral Acceleration | 2.5 | m/s² |
| Lane Change Lateral Displacement | 3.75 | m |
| Motor Rated Power | 160 | kW |
| Motor Peak Torque | 310 | Nm |
| Battery Capacity | 70 | kWh |
| Simulation Time Step | 0.02 | s |
Performance comparisons were made against traditional fifth-order polynomial and B-spline methods. The improved algorithm demonstrated superior results across multiple metrics, as shown in the following table:
| Evaluation Metric | Improved Algorithm | Fifth-Order Polynomial | B-spline Curve |
|---|---|---|---|
| Maximum Lateral Acceleration (m/s²) | 2.1 | 2.8 | 2.5 |
| Maximum Curvature Change Rate (m⁻¹/s) | 0.042 | 0.068 | 0.055 |
| Trajectory Generation Time (ms) | 18.5 | 26.8 | 31.2 |
| Average Tracking Error (m) | 0.12 | 0.21 | 0.16 |
| Energy Consumption (kWh per maneuver) | 0.086 | 0.102 | 0.095 |
| Convergence Iteration Count | 8 | 12 | 15 |
The improved algorithm achieved a 23.5% enhancement in trajectory smoothness, a 31.2% increase in computational efficiency, and an 18.7% reduction in maximum lateral acceleration compared to the fifth-order polynomial method. These improvements are critical for electric vehicles in China, where energy efficiency and passenger comfort are key selling points. Additionally, the integration of motor efficiency terms in the objective function reduced energy consumption by 15%, aligning with the sustainability goals of China EV manufacturers.
In conclusion, our research presents a robust trajectory planning algorithm tailored for autonomous electric vehicles. By leveraging improved polynomial functions with dynamic weight factors and constraint optimization, we achieve a balance between smoothness, safety, and computational performance. The algorithm’s adaptability to various driving conditions, including those prevalent in China EV ecosystems, makes it a viable solution for future autonomous systems. Future work will explore real-world implementation and further optimization for multi-vehicle scenarios.