Dual-Stage Cost Modeling Mechanism (DCMM)

• DCMM is a dual-stage framework that generates smooth, feasible trajectory candidates in Frenet space and refines them with DRL-guided Cartesian evaluation for semantic accuracy.
• It aligns geometric consistency and adaptive, user-centric evaluation through a weight transfer mechanism that reconciles heuristic planning with learning-based optimization.
• Empirical results show improved cost reduction, risk minimization, and faster training convergence compared to baseline methods within assistive navigation systems.

Searching arXiv for the cited DCMM and related framework papers to ground the article in the referenced literature. The Dual-Stage Cost Modeling Mechanism (DCMM) is a two-stage cost framework introduced within the Momentum-constrained Hybrid Heuristic Trajectory Optimization Framework (MHHTOF) for assistive navigation in visually impaired scenarios. In this formulation, trajectory planning is divided into a Frenet-frame stage for generating feasible and smooth candidates and a Cartesian-frame stage for reward-driven adaptive evaluation and selection. The mechanism is intended to reconcile a stated tension between “Frenet-only optimization,” which is efficient and geometrically convenient but often too rigid for semantic and user-centric preferences, and “Cartesian-only learning / DRL,” which can encode richer context and interpretability but may lose geometric consistency and behave unstably or opaquely (Zeng et al., 16 Apr 2026). A closely related but differently named two-stage architecture appears in a separate system-modeling context, where interface-based structural complexity is first bounded and then translated into cost through cost factors; that framework does not use the term DCMM, but it exhibits a comparable “structure → complexity bound → cost bound” pipeline (Chan, 30 Jun 2026).

1. Conceptual definition and problem setting

DCMM is presented as the bridge between heuristic trajectory generation and learning-based trajectory refinement inside MHHTOF. Its stated role is to make planning both “physically feasible and smooth” in structured path space and “semantically adaptive, interpretable, and user-centered” in decision space (Zeng et al., 16 Apr 2026). In the 2025 presentation of MHHTOF, DCMM is described as a “dual-stage cost modeling mechanism (DCMM) with weight transfer” that aligns semantic priorities across stages and supports human-centered optimization (Zeng et al., 19 Sep 2025).

The mechanism is motivated by a planning problem specific to visually impaired scenarios. The cited work states that existing assistive trajectory planners struggle with multi-objective optimization, generalization, and interpretability, and that neither purely heuristic Frenet optimization nor purely Cartesian learning is sufficient by itself (Zeng et al., 16 Apr 2026). The first is described as efficient and geometrically consistent but limited in global semantic awareness and social or contextual adaptability; the second is described as semantically expressive but weaker at preserving geometric consistency and stable motion generation (Zeng et al., 19 Sep 2025).

Within that problem setting, DCMM is not merely a scalar objective function. It is a coupled two-stage scheme in which the first stage secures feasible and smooth motion in structured environments, while the second stage incorporates adaptive semantic evaluation (Zeng et al., 16 Apr 2026). The 2025 version makes this coupling explicit by characterizing DCMM as a cross-stage cost alignment scheme through “weight transfer,” so that the heuristic generator and the DRL policy remain organized around the same human-centered objective family rather than optimizing unrelated criteria (Zeng et al., 19 Sep 2025).

2. First stage: Frenet-space cost modeling

The first stage of DCMM operates during trajectory sampling and generation in the Frenet coordinate system. The cost is given as

$J[d, s]=J_{d}[d]+k_{s} J_{s}[s], \quad k_{s}>0$

where $J_d[d]$ penalizes lateral irregularity, such as abrupt curvature changes or deviation from navigable boundaries, and $J_s[s]$ evaluates longitudinal consistency, emphasizing smooth velocity and feasible deceleration (Zeng et al., 16 Apr 2026). In the 2025 formulation, this same stage is described as a Frenet-frame multi-objective cost used during heuristic trajectory sampling cluster generation (Zeng et al., 19 Sep 2025).

The Frenet state is defined as

$$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$

with $s$ denoting longitudinal arc length, $d$ denoting lateral deviation, $\dot{s}$ and $\ddot{s}$ denoting longitudinal velocity and acceleration, $\dot{d}$ and $\ddot{d}$ denoting lateral velocity and acceleration, and $J_d[d]$0 and $J_d[d]$1 denoting lateral variation or curvature smoothness terms (Zeng et al., 16 Apr 2026). The spatial conversion is written as

$J_d[d]$2

and the Cartesian back-mapping is given by

$J_d[d]$3

The cited paper explicitly states that these transformations “guarantee geometric consistency and serve as the basis for dynamic feasibility and constraint enforcement in planning” (Zeng et al., 16 Apr 2026).

Trajectory generation in this stage uses polynomial interpolation. The 2026 account states that candidate segments are generated with “third-/fifth-order polynomial interpolation” and endpoint constraints, while the 2025 account specifies generation by fifth-order polynomial parameterization with third-order endpoint smoothing (Zeng et al., 16 Apr 2026, Zeng et al., 19 Sep 2025). The polynomial form is written as

$J_d[d]$4

with coefficients solved from boundary conditions (Zeng et al., 16 Apr 2026). The 2025 paper also gives the third-order endpoint smoothing construction to ensure continuity of position, velocity, and acceleration across sampled segments (Zeng et al., 19 Sep 2025).

A further continuity constraint is specified in the 2026 presentation:

$J_d[d]$5

where $J_d[d]$6, $J_d[d]$7 is the nominal spacing, and $J_d[d]$8 scales spacing based on perception or environmental complexity (Zeng et al., 16 Apr 2026). This stage therefore produces a trajectory cluster that is already smooth, dynamically plausible, and structured enough for subsequent evaluation.

3. Second stage: Cartesian-space adaptive evaluation

After candidate trajectories $J_d[d]$9 are projected into Cartesian space, DCMM re-evaluates them using a DRL-guided adaptive cost. The 2026 presentation gives

$J_s[s]$0

and the final selection rule

$J_s[s]$1

The 2025 version presents the same structure, describing $J_s[s]$2 as interpretable Cartesian-space metrics and $J_s[s]$3 as adaptive weights predicted or adjusted by the policy network according to environment context and user preference (Zeng et al., 19 Sep 2025).

The 2025 paper instantiates the stage-2 cost with seven interpretable terms:

$J_s[s]$4

with

$J_s[s]$5

corresponding to acceleration energy, jerk minimization, velocity optimality, path deviation, obstacle proximity penalty, risk field penalty, and social compliance (Zeng et al., 19 Sep 2025).

This second stage is the semantically adaptive component of DCMM. The 2026 description characterizes the evaluation metrics as including curvature continuity, obstacle clearance, and social compliance, with adaptive weights learned by the policy (Zeng et al., 16 Apr 2026). The 2025 description emphasizes that the stage explicitly evaluates safety margin, social behavior, and risk awareness rather than only geometric smoothness (Zeng et al., 19 Sep 2025). The decision mechanism is therefore not direct motor control, but candidate re-ranking under a learned, interpretable cost structure.

4. Weight transfer, adaptive weighting, and human-centered optimization

A central feature of DCMM is that the priorities used in the first and second stages are aligned rather than isolated. The 2025 paper names this explicitly as “weight transfer,” stating that heuristic sampling in the Frenet frame ensures feasibility and comfort via a multi-objective cost function, while reward-driven refinement in the Cartesian frame enhances trajectory quality, and that “A weight transfer mechanism aligns both stages” (Zeng et al., 19 Sep 2025). In practical terms, the first stage emphasizes smoothness, feasibility, comfort, and boundary adherence, while the second stage evaluates smoothness, obstacle clearance, social compliance, and risk, yielding continuity of semantic priorities across stages rather than a reset between heuristic planning and learning (Zeng et al., 19 Sep 2025).

The adaptive update rule for the DRL-controlled weights is given as

$J_s[s]$6

Both versions stress that the policy does not output low-level control directly. Instead, it adjusts the importance of each cost term, and the clipped update preserves interpretability and stability (Zeng et al., 16 Apr 2026, Zeng et al., 19 Sep 2025). The 2026 account specifically notes that this is crucial because the agent’s action is interpretable as a change in cost priorities rather than an opaque motor command (Zeng et al., 16 Apr 2026).

The observation used for weight adaptation is described as a cognitive state,

$J_s[s]$7

which the paper explains includes estimated motion states, obstacle proximity, perceived collision risk, feasibility indicators, and lane semantic cues (Zeng et al., 16 Apr 2026). Reward shaping is hierarchical:

$J_s[s]$8

with Task Guidance Layer, Behavior Guidance Layer, and Risk Avoidance Layer, and subrewards including task completion feedback, completion timing, reference alignment deviation, speed deviation penalty, goal distance reduction, cost-optimality bias, self-action risk, and environmental risk (Zeng et al., 16 Apr 2026).

On that basis, DCMM is described as human-centered. The cited material states that it lets the planner adaptively trade off safety, comfort, efficiency, and social compliance in real time (Zeng et al., 16 Apr 2026). For visually impaired users, the paper gives examples such as slowing down near dense obstacles, prioritizing smoother motion in complex geometry, and favoring safer, more conservative paths in uncertain environments (Zeng et al., 16 Apr 2026). The 2025 account further associates this with perceptual alignment, emphasizing structured motion cues, boundaries, and lane semantics (Zeng et al., 19 Sep 2025). A plausible implication is that human-centeredness in DCMM arises not from replacing analytical planning with learned control, but from constraining learned adaptation to occur through interpretable cost priorities.

5. Integration with HTSC, MTO, and residual-enhanced DRL

DCMM is embedded in MHHTOF together with the Heuristic Trajectory Sampling Cluster (HTSC), Momentum-Constrained Trajectory Optimization (MTO), and residual-enhanced DRL. HTSC generates a compact set of candidate trajectories in the Frenet frame using quintic polynomials and endpoint smoothing (Zeng et al., 16 Apr 2026). In the 2025 abstract, the framework is summarized as integrating “trajectory sampling generation, optimization and evaluation with residual-enhanced deep reinforcement learning (DRL),” with HTSC generated in the Frenet coordinate system using third-order interpolation with fifth-order polynomials and MTO constraints to ensure smoothness and feasibility (Zeng et al., 19 Sep 2025).

MTO refines these candidates using a physically motivated cost. The 2026 paper gives the Lagrangian

$J_s[s]$9

expanded as

$$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$0

and the optimization objective

$$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$1

The 2025 paper presents an analogous formulation in terms of motion energy, guidance forces, crowd interaction, smoothness regularization, and perceptual uncertainty (Zeng et al., 19 Sep 2025).

The residual-enhanced DRL module receives the DCMM-generated candidates and the cognitive observation vector, then learns adaptive weights $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$2 using PPO with residual blocks and LSTM temporal modeling (Zeng et al., 16 Apr 2026). The 2026 paper specifies the architecture as a policy network $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$3, two ResBlocks, LSTM $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$4, output $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$5, with a value or reward network of the same pattern and output $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$6 (Zeng et al., 16 Apr 2026). Both versions emphasize that DCMM is the bridge between deterministic heuristic optimization and adaptive policy learning: the first stage supplies the feasible candidate set, and the second stage supplies semantic ranking and dense learning signals (Zeng et al., 16 Apr 2026, Zeng et al., 19 Sep 2025).

6. Empirical results, interpretation, and relation to adjacent frameworks

The experimental evidence reported for MHHTOF supports the effectiveness of the combined mechanism that DCMM enables, although the papers also state that DCMM is not isolated in a standalone ablation (Zeng et al., 16 Apr 2026, Zeng et al., 19 Sep 2025). The 2026 paper reports that the proposed LSTM-ResB-PPO converges in about 520,000 steps, whereas baseline PPO needs about 1,000,000 steps to reach comparable stability (Zeng et al., 16 Apr 2026). The 2025 abstract summarizes this as stable policy performance in approximately half the training iterations required by the PPO baseline (Zeng et al., 19 Sep 2025).

Reported reward metrics in the 2026 account are Final performance (FP) 861.138 vs 822.301, Peak reward (PR) 877.502 vs 869.986, Converged performance (CP) 861.055 vs 855.022, and Minimum reward (MR) 851.776 vs 841.548 (Zeng et al., 16 Apr 2026). Stability is described as only 1 fluctuation for the proposed model, compared with 5 and 7 fluctuations for GRU and bi-GRU variants, and a stable episode length of 147 at about 520,000 steps (Zeng et al., 16 Apr 2026). The cost metrics reported are an average cost reduction from 0.166472 to 0.116065, corresponding to a 30.3% reduction, and a cost variance reduction from 0.102921 to 0.048067, corresponding to a 53.3% improvement (Zeng et al., 16 Apr 2026). Risk metrics include a reduction in ego risk from $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$7 to $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$8 and in obstacle risk from $$\left[s, \dot{s}, \ddot{s} ; d, \dot{d}, \ddot{d}, d^{\prime}, d^{\prime \prime}\right],$$9 to $s$0 (Zeng et al., 16 Apr 2026). The 2025 abstract summarizes the latter as lowering ego and obstacle risks by over 77% (Zeng et al., 19 Sep 2025).

Scenario-level evidence is also reported. The 2026 paper states that baseline trajectories may violate lane semantics or fail task completion, while the proposed method preserves lane adherence, yields smoother velocity and acceleration, lowers cumulative weighted cost, and stabilizes predicted action deviation (Zeng et al., 16 Apr 2026). The 2025 paper provides the example of DEU_Lengede-21_1_T-15, where the baseline failed to complete the goal and moved into oncoming traffic, while LSTM-ResB-PPO completed the task correctly with lower cumulative cost, 5.4286 vs 6.6384, and lower ego risk (Zeng et al., 19 Sep 2025).

These results support the claim that DCMM improves robustness and interpretability by splitting planning into two complementary layers: a Frenet-space cost model for geometric consistency, smoothness, and physical feasibility, and a Cartesian-space adaptive evaluation layer for semantically meaningful, user-centered re-ranking (Zeng et al., 16 Apr 2026). At the same time, the papers are careful about scope. The evidence is for the overall MHHTOF system rather than an isolated DCMM-only comparison (Zeng et al., 16 Apr 2026, Zeng et al., 19 Sep 2025).

A related but distinct perspective is offered by the system-architecture framework in “A Quantitative Framework for Estimating System Complexity and Cost via Component Interface Analysis” (Chan, 30 Jun 2026). That paper is not named “Dual-Stage Cost Modeling Mechanism,” but it presents a structurally similar two-layer pipeline in which complexity is estimated first from observable structure and interfaces through SCCM and cost is estimated second from that complexity plus cost factors through SCEM, supported by CCM and CCSM as the structural representation (Chan, 30 Jun 2026). A plausible comparative interpretation is that both uses of dual-stage modeling exploit a common methodological principle: a first stage imposes a structured, bounded representation of feasibility or complexity, and a second stage converts that structure into decision-relevant cost. The domains, however, are different. In MHHTOF, the stages operate across Frenet and Cartesian spaces for assistive trajectory planning; in the component-interface framework, the stages operate from architectural structure to bounded system cost estimation (Zeng et al., 16 Apr 2026, Chan, 30 Jun 2026).

In summary, DCMM denotes, in the MHHTOF literature, a two-stage and cross-space cost mechanism that aligns heuristic generation with DRL-guided selection through adaptive, interpretable weighting. Its defining characteristics are the use of a Frenet-frame multi-objective cost for feasible and smooth trajectory generation, a Cartesian-frame semantic evaluation cost for candidate selection, and a weight-transfer or weight-alignment process that preserves human-centered priorities across both stages (Zeng et al., 19 Sep 2025).