A Long-Short-Term Mixed-Integer Formulation for Highway Lane Change Planning

TL;DR

The paper tackles optimal lane-change planning in structured multi-agent traffic by introducing a long-short-horizon motion planner (LSTMP) that decouples long-horizon transitions from short-horizon dynamics. It formulates both horizons as a single mixed-integer quadratic program (MIQP), with the long-horizon stage operating in continuous space (ST-space) via approximate reachability, Chebyshev-centering, and disjunctive gap selection, while the short-horizon stage provides a discrete-time trajectory capturing a potential lane change. The main contributions are the novel LSTMP formulation that keeps the number of binary variables near $O(N_{ ext{veh}}+N)$, a safe, consistent integration of STF and LTF, and comprehensive evaluations showing improved real-time performance and competitive or superior closed-loop behavior compared to state-of-the-art baselines. The framework is validated in deterministic and interactive SUMO/CommonRoad scenarios, highlighting practical impact for real-time highway planning with reliable safety properties and scalable computation.

Abstract

This work considers the problem of optimal lane changing in a structured multi-agent road environment. A novel motion planning algorithm that can capture long-horizon dependencies as well as short-horizon dynamics is presented. Pivotal to our approach is a geometric approximation of the long-horizon combinatorial transition problem which we formulate in the continuous time-space domain. Moreover, a discrete-time formulation of a short-horizon optimal motion planning problem is formulated and combined with the long-horizon planner. Both individual problems, as well as their combination, are formulated as MIQP and solved in real-time by using state-of-the-art solvers. We show how the presented algorithm outperforms two other state-of-the-art motion planning algorithms in closed-loop performance and computation time in lane changing problems. Evaluations are performed using the traffic simulator SUMO, a custom low-level tracking model predictive controller, and high-fidelity vehicle models and scenarios, provided by the CommonRoad environment.

Figures (12)

• Figure 1: Overview of the proposed MIQP formulation for motion planning, referred to as LSTMP. The MIQP consists of long-term and short-term planning formulations where the decision variables of both are coupled through consistency constraints. The short-term decision variables include a continuous point-mass model trajectory to approximate a single lane change. The long-term decision variables account for selecting gaps between SV on each lane.
• Figure 2: The first figure shows the enumeration of lanes and gaps and the rightmost three figures show the sets related to free spaces. SV are uniquely enumerated. Gaps are the free spaces on a lane w.r.t. SV and are enumerated according to the leading vehicles. An additional index is used for each frontmost gap. The sets $\mathcal{N}_l,\mathcal{S}^+_i$ and $\mathcal{S}^-_i$ define half-spaces in the SLT-space and are plotted in green for the position dimensions. The sets are tightened to include all configurations of the SV and ego vehicle to allow collision-free planning with a point-mass model. All leading vehicles on the same lane are considered to construct the set $\mathcal{S}^+_{i}$, since any slower vehicle requires all following vehicles to brake. For the following vehicle set $\mathcal{S}^-_{i}$ only the closest vehicle to gap $i$ is considered, since preceding ones are assumed to not influence leading vehicles.
• Figure 3: Construction of longitudinal ostacle-free space $\mathcal{S}_i^+$ for an SV with index $i$ and two leading vehicles. The left plot shows the nominal prediction sets $\mathcal{O}_i$. The right plot shows the blocking lower-bound set, enforced on the following vehicles. Red trajectories are plotted corresponding to samples of actually driven trajectories.
• Figure 4: Sketch of obstacle-free sets $\mathcal{F}^\mathrm{lc}$ (green) for lane changing related to three SV on the two lanes $l$ and $l+1$. The left plot shows the curvilinear space with coordinates $s$ and $n$. The right plot shows the ST-space. Three possible gaps with indices $g_2$, $g_3$, and $g_4$ on the consecutive lane $l+1$ are available for a transition from gap index $g_1$ and lane $l$.
• Figure 5: Visualization of SV sets $\mathcal{O}_{\{1,2,3\}}$ in the SLT-space and consecutive convex free regions between gap index $1$ and gap index $2$. The time sub-spaces $\mathcal{T}^\mathrm{lc-}=\{(t,s,n)|t\leq \tau_1-\overline{t}_\mathrm{lc}/2\}$, $\mathcal{T}^\mathrm{lc}=\{(t,s,n)|\tau_1-\overline{t}_\mathrm{lc}/2 \leq t\leq \tau_1+\overline{t}_\mathrm{lc}/2\}$ and $\mathcal{T}^\mathrm{lc+}=\{(t,s,n)|t\leq\tau_1+\overline{t}_\mathrm{lc}/2\}$ define the consecutive time-related spaces on the planning horizon. The set $\mathcal{F}^{+}_{1}\cap\mathcal{T}^\mathrm{lc-}$ is the obstacle-free space on the first lane before the lane change, $\mathcal{F}^\mathrm{lc}_{1,2}\cap\mathcal{T}^\mathrm{lc}$ is the free space during the lane change and $\mathcal{F}^{+}_{2}\cap\mathcal{T}^\mathrm{lc+}$ is the obstacle-free space on the next lane after the lane change. Rear vehicles in the same lane are ignored, i.e., a vehicle is always allowed to brake. The binary variables $\lambda_k$ determine, which set constraints are active for each $x_k$.

Theorems & Definitions (1)

• Definition 6.1