Integrated Lateral‑Longitudinal Control for Autonomous Vehicles Using Linear Time‑Varying MPC

In classic autonomous‑driving architectures the lateral and longitudinal controls are solved by two independent algorithms, a so‑called “separated” scheme. Although feasible, this approach does not reflect human driving behavior nor the tight coupling between the two directions.

The article proposes an integrated lateral‑longitudinal control solution that models the vehicle dynamics with coupled terms, builds joint constraints, and evaluates performance with a combined metric.

Problems of the separated scheme

Reduces problem size but ignores cross‑direction coupling.

Vehicle steering geometry limits coupling, making longitudinal control appear independent.

Separate performance metrics can lead to imbalance between lateral and longitudinal tracking.

Design ideas of the integrated scheme

Use a coupled vehicle model (e.g., a two‑wheel kinematic model) to capture dynamic interactions.

Formulate joint constraints (e.g., combined position bounds, maximum centripetal acceleration).

Define a unified performance index that penalizes errors in speed, longitudinal position, lateral position, and yaw angle.

Because the coupled model introduces nonlinearities (cross‑products, trigonometric terms), the authors adopt a Linear Time‑Varying Model Predictive Control (LTV‑MPC) framework. The nonlinear model is linearized around the current state and previous control input using a first‑order Taylor expansion, then discretized for real‑time solving.

Key steps:

1. Modeling : State vector x includes vehicle position, yaw, and longitudinal speed; control vector u contains steering angle and acceleration command.

2. Linearization : Perform first‑order expansion at the operating point x_k, u_k to obtain Δx_{k+1}=A_k Δx_k + B_k Δu_k. The linearized model is then discretized with a zero‑order hold.

3. Joint constraints : Example – maximum centripetal acceleration a_c = v^2 / R ≤ a_{c,max} is expressed as a linear inequality after linearization.

4. Objective function : Minimize a quadratic cost J = Σ (Q_x Δx_k^2 + R_u Δu_k^2) where the weight matrices reflect the importance of both lateral and longitudinal errors.

5. Optimization : The resulting problem is a linear‑constrained quadratic program solved at each control horizon to obtain the optimal control sequence. Only the first control input is applied, then the process repeats.

The proposed LTV‑MPC achieves a balance between lateral and longitudinal tracking: in high‑speed cornering the controller may deliberately reduce speed to improve lateral accuracy, reflecting a more human‑like driving strategy.

Conclusion

The integrated approach overcomes the limitations of the separated scheme by accounting for vehicle coupling, constructing joint feasible sets, and providing coordinated performance optimization. Experimental results (not shown) indicate improved model fidelity, faster computation, and better overall tracking compared with traditional linear‑time‑invariant MPC.