Enhanced Sinusoidal Acceleration Model (SAM)

• Enhanced SAM is a lane-change trajectory model that represents maneuvers with four kinematic parameters (W, D, v₀, Δvₓ) to reconstruct smooth, continuous paths.
• It anchors predictions at the lane boundary crossing point, shifting from initiation to a state-aware continuation for enhanced physical plausibility.
• The model reduces output dimensionality by 80% and improves lateral RMSE across time horizons, showcasing efficiency in trajectory prediction.

The Enhanced Sinusoidal Acceleration Model (SAM) is a modified parametric lane-change trajectory model used for interpretable lane-change prediction in the hybrid architecture SAM-LLM. In that system, lane keeping is represented by discrete future coordinates, whereas lane-change maneuvers are represented by four kinematic parameters, $\{W,D,v_0,\Delta v_x\}$, from which a complete continuous trajectory can be reconstructed. The model is designed to retain the smoothness and physical plausibility associated with sinusoidal lane-change formulations while adapting the trajectory representation to a prediction setting anchored at the lane boundary crossing point $t_{insertion}$, rather than at maneuver initiation (Cao et al., 3 Sep 2025).

1. Concept and representational role

In SAM-LLM, the trajectory representation is explicitly hybrid:

$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$

where $I=0$ denotes keep lane, $I=1$ left lane change, and $I=2$ right lane change (Cao et al., 3 Sep 2025). The Enhanced SAM therefore applies only to lane-change cases; lane-keeping scenarios remain coordinate-based.

The stated motivation for this design is fourfold. First, each predicted quantity corresponds to an interpretable maneuver property. Second, the model yields a continuous and physically plausible trajectory rather than a sparse point list. Third, four parameters replace a coordinate sequence, making the output compact. Fourth, once the parameters are known, the trajectory can be reconstructed over the whole prediction horizon and sampled at arbitrary temporal resolution (Cao et al., 3 Sep 2025).

This representation is central to the paper’s distinction between predicting a trajectory directly and predicting a trajectory generator. For lane changes, the model does not emit raw $(x,y)$ samples; it emits the parameters of a continuous kinematic model. A plausible implication is that the representation shifts part of the prediction burden from token-level numeric precision to model-structured physical consistency.

2. Mathematical formulation

The paper first recalls the classical sinusoidal acceleration model for a complete lane change from maneuver initiation to completion:

$$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$

The Enhanced SAM modifies this formulation because prediction starts at the lane boundary crossing point rather than at $t_{start}$. The post-boundary lateral motion is modeled as

$$y(t) = v_0 t + \frac{W - 2v_0 D}{\pi} \sin\left(\frac{\pi t}{2D}\right).$$

The associated longitudinal velocity transition is

$t_{insertion}$0

For lane-changing cases, the parameters are fitted from ground-truth trajectories by least squares:

$t_{insertion}$1

The variables are defined in the paper as follows: $t_{insertion}$2 is lateral displacement, $t_{insertion}$3 maneuver duration, $t_{insertion}$4 the initial lateral velocity at boundary crossing, and $t_{insertion}$5 the longitudinal velocity change during the maneuver. The model is anchored at the insertion point $t_{insertion}$6, identified as the transition from observation to prediction and specifically as the lane boundary crossing point for SAM prediction (Cao et al., 3 Sep 2025).

The paper explicitly states that the modified SAM ensures

$t_{insertion}$7

However, the same source also notes a technical inconsistency between this statement and the printed lateral-position formula. Differentiating the provided equation yields

$t_{insertion}$8

and

$t_{insertion}$9

Under direct differentiation, $$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$0, but $$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$1 does not vanish unless the modeled interval is interpreted differently or the formula is regarded as approximate. This is a formal caveat in the printed specification rather than a disagreement about the authors’ stated design intent, which is endpoint smoothness (Cao et al., 3 Sep 2025).

3. Kinematic parameters and trajectory semantics

The Enhanced SAM uses exactly four predicted parameters for a lane change. Their meanings and typical scales are given directly in the source.

Parameter	Meaning	Typical values or effect
$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$2	Lateral displacement	typically $$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$3 m
$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$4	Maneuver duration	typically $$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$5 s
$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$6	Initial lateral velocity at boundary crossing	extracted at boundary crossing
$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$7	Longitudinal velocity change	acceleration, deceleration, or near-constant speed

$$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$8 controls the magnitude of cross-lane motion. In the paper it is described as typically $$\mathbf{T} = \begin{cases} \mathbf{T}_{coord} = \{(x_1, y_1), ..., (x_4, y_4)\} & \text{if } I = 0 \ \mathbf{T}_{param} = \{W, D, v_0, \Delta v_x\} & \text{if } I \in \{1, 2\} \end{cases}$$9 m, consistent with lane-width scale. Through the term $I=0$0, it shapes both the total lateral excursion and the curvature of the lateral path (Cao et al., 3 Sep 2025).

$I=0$1 controls the temporal scale of the maneuver and is described as typically $I=0$2 seconds. Larger $I=0$3 distributes the same lateral displacement over a longer interval, while smaller $I=0$4 yields a sharper maneuver. Because the longitudinal transition is also normalized by $I=0$5, this parameter affects both lateral unfolding and longitudinal velocity evolution (Cao et al., 3 Sep 2025).

$I=0$6 is the principal enhancement relative to the classical full-maneuver SAM. It is defined at the boundary crossing point, not at lane-change initiation. The paper emphasizes that this is crucial because the vehicle may already have nonzero lateral speed when prediction begins. The parameter therefore encodes how much of the maneuver is already underway at $I=0$7 (Cao et al., 3 Sep 2025).

$I=0$8 captures longitudinal acceleration or deceleration during the lane change through a linear transition model. Positive values correspond to speeding up, negative values to slowing down, and values near zero to approximately constant longitudinal speed. This keeps the model simple while separating lateral geometry from longitudinal speed adaptation (Cao et al., 3 Sep 2025).

The combination of these parameters yields a compact semantic description of a maneuver. This suggests that interpretability is not merely post hoc; it is structurally embedded in the output space.

4. Enhancement relative to classical SAM

The model is termed “enhanced” because it is not the direct classical SAM. The enhancement is task-specific and prediction-specific. The classical formula assumes access to the start of the entire lane change, whereas SAM-LLM predicts from the lane boundary crossing point $I=0$9 and therefore models only the continuation of the maneuver (Cao et al., 3 Sep 2025).

The paper identifies five concrete differences between the classical and enhanced forms. First, the temporal anchoring shifts from lane-change onset to lane boundary crossing. Second, the model covers only the second half of the lane change, as the authors explicitly state that they “require only the second half of the trajectory.” Third, $I=1$0 is introduced as an explicit condition, which is absent from the printed classical formula. Fourth, the modified model is designed for prediction from observed state rather than reconstruction from a known initiation event. Fifth, the lateral SAM is coupled to a simple longitudinal velocity-transition model via $I=1$1 (Cao et al., 3 Sep 2025).

The importance of this change is methodological. A lane-change model that assumes observation from maneuver initiation is poorly aligned with a forecasting setting in which the vehicle has already crossed the lane boundary at the start of prediction. The enhanced formulation is therefore a state-aware continuation model rather than a full-maneuver generator. A plausible implication is that the enhancement is less a new family of trajectory laws than a reparameterization of the lane-change segment most relevant to online prediction.

The source also describes a practical reconstruction pipeline. The model predicts intention $I=1$2; if $I=1$3, it outputs four future coordinate points, and if $I=1$4, it outputs $I=1$5. Lateral motion is then reconstructed from the enhanced SAM equation, and longitudinal velocity from $I=1$6. The paper does not print the integrated longitudinal position explicitly, but it notes that integrating this velocity gives the natural implied continuous longitudinal trajectory (Cao et al., 3 Sep 2025).

5. Integration with SAM-LLM

The Enhanced SAM is embedded in a hybrid LLM-plus-kinematic architecture. The full generated sequence is

$I=1$7

where $I=1$8 denotes input prompt tokens, $I=1$9 Chain-of-Thought reasoning, $I=2$0 intention, and $I=2$1 the trajectory representation (Cao et al., 3 Sep 2025). For lane changes, $I=2$2 is the parameter vector $I=2$3; for lane keeping, it is a set of coordinates.

Fine-tuning uses standard causal language modeling:

$I=2$4

The paper reports LoRA fine-tuning on Llama-2-7B with rank $I=2$5, targeting attention projection layers (Cao et al., 3 Sep 2025). This is described as parameter-efficient adaptation.

For supervision, the lane-change subset of the training data is converted from future trajectories into SAM parameters by a least-squares fitting procedure. The paper gives the preparation sequence explicitly: detect lane-changing scenarios, fit $I=2$6 to the observed future trajectory, then convert those fitted parameters into the training target text (Cao et al., 3 Sep 2025).

The output format includes a reasoning trace and a final structured prediction. The model therefore does not act only as a numeric regressor. It generates contextual reasoning, an intention label, and a trajectory representation whose semantics are physically interpretable. Within the scope of the paper, this is presented as the main mechanism by which explainability and output efficiency are improved together.

6. Empirical performance, interpretability, and limitations

The paper attributes several quantitative advantages to the SAM-based representation. It reports an overall intention prediction accuracy of $I=2$7, with per-class values of $I=2$8 for keep lane, $I=2$9 for left lane change, and $(x,y)$0 for right lane change (Cao et al., 3 Sep 2025). For lane-change trajectory quality, the reported aggregate values are:

Maneuver	Acc	Lat	Lon
Left Lane Change	97.43	0.286	1.539
Right Lane Change	98.61	0.264	1.223

The paper states that SAM-LLM usually improves lateral RMSE relative to coordinate baselines, while longitudinal RMSE is somewhat worse in some cases. The same source interprets this as consistent with the model’s design: the lateral component is physically structured by SAM, whereas longitudinal dynamics are represented only through a simple linear velocity change (Cao et al., 3 Sep 2025).

The strongest trajectory-quality evidence reported for the parametric formulation is pointwise lateral RMSE over a 1–4 s horizon. For left lane change, the paper gives $(x,y)$1, $(x,y)$2, $(x,y)$3, and $(x,y)$4 at 1 s through 4 s; for right lane change, it gives $(x,y)$5, $(x,y)$6, $(x,y)$7, and $(x,y)$8 (Cao et al., 3 Sep 2025). It further states that lateral-error improvements over LC-LLM range from $(x,y)$9 at 1 second to $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$0 at 4 seconds, and that SAM-LLM achieves the lowest lateral RMSE values across all time points and intention classes.

Resource efficiency is another central claim. Replacing 20 coordinates with four physical parameters yields an $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$1 reduction in output dimensionality and a $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$2 inference speedup, with runtime reduced from $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$3 ms for LC-LLM (20-pt) to $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$4 ms for SAM-LLM (Cao et al., 3 Sep 2025). These gains follow directly from the shift from sequence prediction to low-dimensional parameter prediction.

The paper also presents interpretability evidence. It claims that predicted SAM parameters form distinct, physically meaningful clusters for left and right lane changes, with visualizations such as $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$5 versus $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$6 and $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$7 versus $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$8 showing tight clustering and clear separation (Cao et al., 3 Sep 2025). This is presented as evidence that the outputs occupy physically sensible regions of parameter space.

The reported limitations are equally specific. First, the paper does not provide a separate ablation isolating individual SAM parameters or a controlled comparison against the unmodified classical SAM. Second, the longitudinal component is simpler than the lateral one and may underperform coordinate baselines on longitudinal RMSE. Third, the printed endpoint-acceleration claim is stronger than what the printed lateral-position equation strictly guarantees under direct differentiation (Cao et al., 3 Sep 2025). These caveats are intrinsic to any technical reuse of the model and delimit the strength of the current empirical evidence.

7. Acronym ambiguity and scope

The acronym “SAM” is overloaded across fields, and confusion about scope is common. In the lane-change prediction paper, SAM denotes a sinusoidal acceleration model adapted for post-boundary lane-change continuation (Cao et al., 3 Sep 2025). In fisheries stock assessment, by contrast, SAM refers to a state-space integrated stock assessment model whose 2025 extension adds penalized smoothing splines for age-dependent parameters (Vandeskog et al., 28 Feb 2025). In numerical analysis of highly oscillatory nonlinear Schrödinger equations, SAM denotes the Stroboscopic Averaging Method, a micro–macro multiscale integrator for periodic fast oscillations (Chartier et al., 2013).

These are unrelated models that share only an acronym. The stock assessment paper concerns smooth age effects in observation-related parameters such as $$y(t) = y_0 + \frac{-W}{2\pi}\sin\left(\frac{2\pi(t-t_{start})}{D}\right) + \frac{W(t-t_{start})}{D}.$$9, $t_{start}$0, and $t_{start}$1, not vehicle trajectories (Vandeskog et al., 28 Feb 2025). The NLS paper concerns stroboscopic averaging for oscillatory PDEs, not kinematic lane-change modeling (Chartier et al., 2013). The lane-change Enhanced SAM should therefore be understood narrowly as a task-adapted parametric maneuver model within SAM-LLM.

Within that scope, the Enhanced Sinusoidal Acceleration Model is best characterized as an interpretable, compact, continuous lane-change generator defined by $t_{start}$2, anchored at the lane boundary crossing point, and intended to replace direct coordinate-sequence prediction for lane changes. Its principal empirical strengths are compact output, smooth continuous lateral trajectories, strong long-horizon lateral accuracy, and lower inference cost; its principal technical cautions concern the simplicity of the longitudinal model and a mismatch between the printed endpoint-smoothness claim and the printed lateral-position equation (Cao et al., 3 Sep 2025).