Car following: Comparing distance-oriented vs. inertia-oriented driving techniques

DRIVING BEHAVIOR IN CAR-FOLLOWING: COMPARING DISTANCE-ORIENTED VS. INERTIA-ORIENTED DRIVING TECHNIQUES

Context and Objectives

Car-following (CF) behavior is a fundamental component of vehicular traffic that has been modeled for over 60 years from both engineering and psychophysiological perspectives. Classical CF models assume that drivers maintain a constant safety distance as an axiomatic parameter defining a unique (or natural) Normative Driving Behavior (NDB). This article challenges this fundamental premise, arguing that drivers can adapt their driving in both reactive and proactive ways. The authors propose two orthogonal CF techniques: (1) Driving to keep Distance (DD), which emphasizes maintaining constant distance to the lead vehicle, and (2) Driving to keep Inertia (DI), which emphasizes maintaining constant speed regardless of the lead vehicle's oscillations. The central objective is to determine which technique is more efficient in terms of fuel consumption, safety, and traffic stability.

The authors build on the Nagoya experiment (Sugiyama et al., 2008), which demonstrated how phantom traffic jams form even in the absence of physical bottlenecks. Inspired by longitudinal wave mechanics in nature and concepts from cybernetic control theory, they propose that drivers can consciously modulate their actions to dampen—rather than amplify—the oscillations transmitted by the lead vehicle.

Theoretical Framework

The article reviews three conceptual frameworks for CF behavior through the history of psychology:

1. Stimulus-Response Framework (1950s-1960s)

The most influential model is the Gazis-Herman-Rothery (GHR) model, which describes the acceleration of the following vehicle as a linear function of the velocity difference relative to the lead vehicle: an(t) = λΔVn(t-τn). This framework assumes drivers are essentially reactive, sensitive to specific stimulus variables from the lead vehicle. Subsequent developments incorporated concepts such as desired safety distance (Kometani & Sasaki, 1959; Gipps, 1981) and optimal velocity models (Bando et al., 1995), which refined but maintained the fundamental stimulus-response logic.

2. TOTE Unit (Test-Operate-Test-Exit) (1960s-present)

Miller et al. (1960) introduced a cognitive paradigm postulating that humans maintain mental representations of ideal versus actual behavior, and incongruence between these states motivates action. Applied to CF, this suggests drivers can be more than reactive followers; they can establish hierarchies of actions and maintain proactive longitudinal guidance strategies. This framework also recognizes that even at constant speed, there exists a natural, systematic oscillation of velocity (~1 m/s around the mean) consubstantial to the regulation process, as demonstrated by Wille (2005, 2011).

3. Mental Models (1983-present)

Johnson-Laird (1983) proposes that humans generate complex internal models of their environments. Physical mental models are structural analogs of specific referents, while conceptual models are abstract. Applied to CF, this means drivers can develop and maintain different conceptualizations of how to follow a vehicle (e.g., maintaining distance vs. maintaining inertia) and can switch between them based on instructions or context.

Wave Analysis and Compensation

The article develops a fundamental mathematical metaphor based on wave mechanics. Following Fourier decomposition, any complex oscillation can be represented as a sum of simple waves. In the CF context, the movement pattern of car n in the platoon is described as: ωn = ωn-1 + iωn, where ωn is the wave of movement of car n, ωn-1 is the wave of the preceding vehicle, and iωn is the "imaginary" wave (mental) enacted by the driver through their driving strategy. Under DD, the driver attempts to replicate the distance pattern of the lead vehicle, thereby amplifying oscillations (transmitting perturbations). Under DI, the driver maintains constant speed, compensating for and damping the lead vehicle's oscillation.

Methodological Design

Participants

A total of N = 113 participants (M age = 21.1 years, SD = 2.04) were recruited across three studies:

• Study 1: N = 44 (24 women, 20 men; M age = 23.3 years; 84.1% university education; M driving years = 4.07)
• Study 2: N = 44 (37 women, 4 men; M age = 20.7 years; 68.2% university education; M driving years = 2.81)
• Study 3: N = 25 (12 women, 13 men; M age = 21.3 years; 100% university education; M driving years = 2.68)

All participants were licensed drivers, mostly university students, recruited via posters at nearby shops, driving schools, and restaurants.

Experimental Design

All three studies employed a repeated-measures design controlling for order. The independent variable was driving technique (DD/DI) as a within-subject factor. The order of technique presentation (DD/DI or DI/DD) was the between-subject factor. The laboratory provided a booth for task execution with an adjacent monitoring room with two-way glass.

Materials and Simulator

A 3D driving simulator called React Follower (Impactware, 2014), based on UNITY software, was developed and ran on a standard PC. The simulator allowed external control of parameters (speed, frequency of acceleration-deceleration cycles) via XML. Participants controlled their vehicle by pressing up/down arrows on a keyboard (acceleration/deceleration). The simulator ran on an HP TouchSmart iq522es with 23-inch screen, NVIDIA GeForce 9300m GS video card, 4 GB RAM, Intel Core 2 Duo T6400 processor at 2.00 GHz, under Windows 7.

Three scenarios were presented:

• Scenario A: Driving alone on the road (control)
• Scenario B: Driving behind another vehicle at constant speed of 3 m/s (10.8 km/h) (control)
• Scenario C: Driving behind a vehicle with sinusoidal stop-and-go cycles with mean speed of 3 m/s (experimental scenario, only condition reported)

Procedure

Participants received verbal instructions about DD or DI (assigned in random order). The DD instruction was: "In the simulated driving scenario you will enter, you will see a vehicle ahead of you and it will not move at a constant speed. Sometimes it will go faster or slower. We ask you to travel behind that vehicle as closely as possible without risking a crash." The DI instruction was: "In the simulated driving scenario, you will see a vehicle ahead of you and it will not move at a constant speed. Sometimes it will go faster or slower. We ask you to travel smoothly behind the vehicle and maintain a constant speed, without letting the lead vehicle move too far away."

The task lasted 4 minutes of straight-road simulation driving. Initial adaptation to CF (12-18 seconds) was regular thereafter. After completing each technique, self-assessment scales (SAM) were administered regarding affective and personality responses.

Dependent Variables and Instruments

The simulator automatically recorded:

• Accelerations (total number)
• Decelerations (total number)
• Crashes (total number)
• Average speed (m/s)
• Speed dispersion (m/s) - standard deviation
• Distance to leader (meters)
• Distance dispersion (meters) - standard deviation
• Virtual fuel consumption (liters) - gross estimate based on speed variations per frame

Analysis

Repeated-measures ANOVAs were conducted with two levels of driving orientation (DD/DI), two presentation orders (DD/DI vs. DI/DD), and three studies as between-subject factors. Both main effects and second-order interactions were analyzed.

Results

Overall Summary of Results

Data were subjected to repeated-measures ANOVA. Comparing DD/DI means for all factors (accelerations, decelerations, crashes, etc.) yielded statistically significant differences: Study 1 (p < .001), Study 2 (p < .001), Study 3 (p < .005).

Table of means summary by variable:

Variable	Study 1 (DD/DI)	Study 2 (DD/DI)	Study 3 (DD/DI)
Accelerations (n)	147.9 / 90.1	158.3 / 106.6	230.9 / 55.0
Decelerations (n)	108.5 / 55.8	100.7 / 62.6	134.2 / 31.2
Crashes (n)	3.7 / 0.3	2.9 / 0.2	1.56 / 0.36
Fuel consumption (l)	19.4 / 15.0	18.6 / 15.1	19.7 / 13.9
Distance to leader (m)	6.6 / 11.9	7.7 / 17.6	9.25 / 19.4
Distance dispersion (m)	3.95 / 4.74	4.37 / 5.40	4.57 / 6.08
Speed (m/s)	3.08 / 3.05	3.07 / 3.03	3.07 / 3.03
Speed dispersion (m/s)	2.57 / 1.44	2.54 / 1.44	2.24 / 0.99

Punctual Actions: Accelerations, Decelerations, Crashes

Accelerations and Decelerations: Overall, more accelerations (M = 131.42) than decelerations (M = 82.22) occurred, F(1107) = 71.52, p < .0001, ηp² = .401, consistent with acceleration/deceleration asymmetry observed in real driving. Comparing DD vs. DI, significantly more accelerations and decelerations occurred under DD (M = 146.81) than under DI (M = 66.83), F(1107) = 87.39, p < .0001, ηp² = .450. A factor interaction nuances this: more accelerations (M = 179.0) than decelerations (M = 114.6) under DD compared to DI (accelerations M = 83.8; decelerations M = 49.8), but decelerations differ more proportionally.

The main effect of Study was not significant (p > .54), but the number of accelerations/decelerations differed significantly for DD and DI considering each study, F(2107) = 10.3, p < .0001, ηp² = .160. DD/DI differences were less extreme in S-1 (DD: M = 128.2; DI: M = 72.9) and S-2 (DD: M = 129.5; DI: M = 84.6) than in S-3 (DD: M = 182.8; DI: M = 42.9), yielding a second-order interaction.

Crashes: Significantly more crashes occurred under DD (M = 2.72) than under DI (M = 0.31), F(1107) = 56.7, p < .0001, ηp² = .346. Study presented a marginal effect on crashes, F(2107) = 2.79, p < .07, ηp² = .049, modulated by interaction with driving technique. Differences were larger in S-1 (DD: M = 3.66; DI: M = 0.32) and S-2 (DD: M = 2.91; DI: M = 0.23) than in S-3 (DD: M = 1.59; DI: M = 0.38).

Referential Measures: Distance to Lead Vehicle

Average distance: Average distance to the lead vehicle differed significantly by technique, being greater under DI (M = 16.26) than under DD (M = 7.84), F(1107) = 138.43, p < .0001, ηp² = .564. Study also presented significant differences, F(2107) = 10.06, p < .0001, ηp² = .158: S-2 (M = 12.68) and S-3 (M = 14.26) did not differ from each other (p > .19), but both differed from S-1 (M = 9.22; p < .001). These factors yielded a significant interaction, F(2107) = 5.35, p < .01, ηp² = .091: differences between S-1 vs. S-2/S-3 were more acute for DI.

Order also was significant as a between-subject factor, F(1107) = 4.81, p < .05, ηp² = .043, with distance being globally greater under DI/DD order (M = 13.10) vs. DD/DI order (M = 11.01). This interaction with driving technique was significant, F(1107) = 7.96, p < .01, ηp² = .069: when subjects followed DI-DD order, distance was greater under DI (M = 18.31) than when order was DD-DI (distance under DI, M = 14.21), while distance under DD was always similar (DI-DD, MDD = 7.87; DD-DI, MDD = 7.81).

Distance dispersion: Distance dispersion to the leader was significantly greater under DI (M = 5.40) than under DD (M = 4.30), F(1107) = 28.63, p < .0001, ηp² = .211. Study also showed differences, F(2107) = 3.74, p < .05, ηp² = .065: S-2 (M = 4.89) and S-3 (M = 5.31) did not differ (p > .25), nor did S-1 and S-2 (M = 4.35; p > .08), but S-1 and S-3 did (p < .01).

Referential Measures: Speed

Average speed: Average speed differed depending on technique, being greater under DD (M = 3.08) than under DI (M = 3.04), F(1107) = 46.66, p < .0001, ηp² = .304. Study showed differences, F(2107) = 5.72, p < .005, ηp² = .097: S-2 (M = 3.05) and S-3 (M = 3.05) did not differ (p > .63), but both differed from S-1 (M = 3.07; p < .01). Order presented significant differences, F(1107) = 12.60, p < .001, ηp² = .110: drivers starting with DD (M = 3.07) drove faster overall than those starting with DI (M = 3.05). An interaction nuanced this: when subjects began with DI, speed was first low under DI (M = 3.02) and then higher under DD (M = 3.07); however, when they began with DD, speed was equally high under both techniques (DD: M = 3.08; DI: M = 3.06).

Speed dispersion: Speed dispersion measures showed a very strong main effect, F(1107) = 305.43, p < .0001, ηp² = .741. Dispersion was clearly higher under DD (M = 2.45) than under DI (M = 1.29). Study presented differences, F(2107) = 8.51, p < .001, ηp² = .137: S-1 (M = 2.01) and S-2 (M = 1.99) did not differ (p > .82), but both differed from S-3 (M = 1.63; p < .001). A second-order interaction was observed, F(2107) = 3.18, p < .05, ηp² = .056.

Overall Measures: Fuel Consumption

Virtual fuel consumption differed significantly, being greater under DD (M = 19.23) than under DI (M = 14.65), F(1107) = 429.4, p < .0001, ηp² = .801. This represents approximately 24% less fuel consumption under DI. This effect was modulated by Study, F(2107) = 8.39, p < .0001, ηp² = .136: fuel consumption under DD vs. DI was more extreme in S-3 than in S-1/S-2 (difference in S-3 = 5.86 l; S-2 = 3.53 l; S-1 = 4.34 l). A second-order interaction also occurred, F(2107) = 4.50, p < .01, ηp² = .078.

Study 3: Platoon Stability Measures

In Study 3, eight virtual "robot" cars practicing the traditional DD approach were programmed to follow the driver subject (who was unaware of this). The simulator recorded distances from the leader to the 8th car and from the subject to the 8th car. Average distance from leader to 8th car was similar under DD and DI (DD: M = 117.3 m, SD = 1.93; DI: M = 118.95 m, SD = 8.75). However, distance from subject to leader was larger under DI. Most importantly, measuring distances from subject's car to 8th car under DD vs. DI (DD: M = 108.03 m, SD = 1.93; DI: M = 99.55 m, SD = 3.69) yielded significant differences: F(1,23) = 30.32, p < .001. DI furnished platoon stability and optimized road space.

Discussion and Conclusions

Primary Findings

Statistical analyses confirm the main results regarding characterization of performance and operative indicators. All three studies showed significant differences in these factors, always in the same direction depending on the technique employed.

First finding: All drivers can drive under DD/DI mode when following an oscillating vehicle and maintain the technique permanently as requested (not revert solely to DD or another "natural" driving mode after a while).

Second finding: Drivers assume these techniques easily after a 10-second instruction (a few sentences or a short video).

Third finding: Differences in behavioral and operative terms (accelerations, decelerations, crashes, speed, distance to leader, fuel consumption, etc.) are statistically significant. DD and DI techniques are essentially orthogonal modes.

Fourth finding: DI consistently leads to approximately 20% less fuel consumption than DD.

Theoretical Implications

With proper instruction, DI drivers can be determinants and act proactively as a bottom-up element against the oscillatory nature of traffic flow. Following Wiener (1950), each driver's role can be essential in bringing order to the natural entropy of dynamic systems such as traffic flow.

The empirical findings can be formalized through the wave equation presented: ωn = ωn-1 + iωn, where ωn is the wave corresponding to movement of car "n" in the platoon, ωn-1 is the wave of the preceding vehicle, and iωn is the "imaginary" wave enacted by mental endeavor (flow ordering strategy) corresponding to car "n" in the platoon. Human and automated drivers can move according to the same CF strategies as other animals in Nature. For example, pine processionary larvae (T. pityocampa) can turn in a circle one after another for 12 consecutive hours before disaggregating.

Role of Training and Education

Results showed differences in Studies as a factor. Although subjects in S-1 to S-3 received the same main instructions, S-2 included a short video about how phantom traffic jams emerge and how to prevent them by applying DD or DI, with the main recommendation embedded (written) at the video's end. Descriptions in S-1 and S-3 were more direct, even more directive, than S-2's instruction. Before the experiment proper, subjects in S-1 and S-2 were invited to check distance to the lead car (some purposely crashed to verify the limit), while subjects in S-3 were left to their own perceptions. Although DD/DI differences were sound in all studies, differences among studies (number of crashes, distance to leader, speed variations, fuel consumption) seemed related to instruction procedures. More research on the role of instruction procedures and demographics is warranted.

Effects of Presentation Order

Although the technique type applied first may have affected assimilation or contrast of magnitudes (speed, distance), the global set analyzed (including psychophysiological and self-report measures) points to a statistical effect that tends to compensate under different conditions (sometimes assimilation, sometimes contrast). Order, as such, is not a theoretical variable, just a methodological control. If perceived as relevant, this effect should be theoretically analyzed and deliberately introduced.

Study Limitations

Participant Population: Most subjects were young university drivers. This weakness may be a strength: acquiring both techniques was easy despite their inexperience. Does experience (habit) improve or worsen performance, especially under DI? Future studies should check.

Driver Heterogeneity: Endogenous factors (gender, age, personality, individual differences, transient states, specific travel goals and timing) may introduce considerable changes in CF. Although experimental manipulation of DD and DI conditions led to sound differences, subjects were heterogeneous too. Figure 8 shows a proficient driver's performance under DD and DI: this skilled driver performed as requested, but other subjects were less good, with DI output that was a "mild" version of DD (too many ups and downs in speed, close distance to leader).

Simulator and Materials: The simulator used PC keys, not a realistic driving environment with gas/brake pedals. The lead vehicle's speed (10.8 km/h) was set up thinking of a critical, jammed situation to gain clear observation of effects. Recently, Carrasco (2017) tested DI on a circular track (30 m radius) with six real cars, similar to the Nagoya experiment. The first was an automated car driving with acceleration cycles (until 25 km/h) and deceleration (to full stop); the second was the subject car (no instruction in Trial 1, and a short DI instruction in Trial 2); other cars were followers with no instruction. In only three complete loops, driving with no instruction equated to DD and reproduced backward soliton wave instability; driving with DI instruction kept free flow behind the subject's car. However, a wider range of speeds should be tested.

Lead Vehicle Oscillation: The lead vehicle's oscillation was constant (harmonic wave), so DD/DI instruction was easy to apply. But most studies under the Action Point paradigm analyze how followers adapt to a relatively stable leader. When the lead's speed is non-constant, the follower attempts to achieve desired spacing but the process is continuously perturbed by the bias produced by the leader.

Implications for Traffic Policy

Currently, drivers are supposed to practice CF in the "natural" (desired) way. Yet, rather than assuming "naturally endowed" CF behavior, drivers are taught DD: roadway "capacity" has been designed considering couplings of speed-safety distance and number of expected drivers (per lane and kilometer); then driving schools teach DD, and road signs reinforce it as does surrounding traffic—not always safely. What would result if car drivers learned DI instead, to aim for uniform speed like teamsters do? Despite our studies' narrow speed range, we know drivers can learn and apply DI. The challenges now are determining driver ability to apply DI under differing speed-distance CF contexts and calculating the gains if DI becomes commonplace.

Fundamental Traffic Flow Diagram

Figure 9 presents two extremes of the fundamental traffic flow diagram, inspired by Smeed's classic accounts (1968). The black curve shows the typical relationship between velocity and flow under DD. Point A is maximum flow at the speed limit (e.g., 120 km/h). Forced traffic begins at B. Maximum flow is attained at M. B and M coincide. Ideally, maximum flow at the corresponding speed (90 km/h) should be kept, but, given the oscillatory nature of traffic flows (reaction time, summing waves), this state cannot last long; a jam occurs and speed and flow decrease.

The green curve represents DI. A' is maximum flow at the speed limit (e.g., 120 km/h). Forced traffic begins (~20–25 km/h) at B'. Maximum flow is attained at M' (~70 km/h). B' and M' are not coincident so M' is not as precarious as M and can last much longer. The bad news: M' is lower than M, so capacity seems undermined; but M will not last long. The good news: DI should promote a stable flow lasting longer. Angle Φ represents level of flow stability: as the efficiency factor increases the maximum flow decreases, but gets more stable.

Integration of Human and Automated Driving

Conceptual models need more physical and mathematical complexity to depict the road network (multiple lanes, curves, hills, various speeds, overtaking, merging). However, as such analysis progresses, and with the right driver training and education (in a growing ICT context), the role of individual drivers in modern traffic deserves review. Small, simple changes may effect global transformations if we all adopt them. Washing hands, for instance, revolutionized sanitation. Teaching drivers DI may similarly transform traffic flows. This pertains to automated cars too, which may be programmed to leave extra space with the vehicle ahead. If humans grasp the basic principles of flow stability, they will also understand how automatons may drive. Longitudinal mechanical waves are instruments of Nature that serve different types of movement, and robots are not free of such allegiance. Perhaps under this common stance integrating human and automated driving will not be difficult.

Significance and contribution

Este trabajo contribuye significativamente a la comprensión del seguimiento vehicular y la dinámica del flujo de tráfico. Proporciona evidencia empirica robusta (N = 113, tres estudios, diseño de medidas repetidas) demostrando que los conductores pueden rápidamente aprender y aplicar técnicas alternativas de conducción que producen resultados mensurablemente diferentes en eficiencia de combustible (~20% de reducción), estabilidad de velocidad, y seguridad. Los hallazgos tienen implicaciones importantes para políticas de educación vial, diseño de redes de tráfico, y sostenibilidad ambiental al sugerir que cambios relativamente simples en los comportamientos individuales de los conductores podrían producir transformaciones globales en eficiencia de tráfico.

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Significance and contribution

This work significantly contributes to understanding car-following and traffic flow dynamics. It provides robust empirical evidence (N = 113, three studies, repeated-measures design) demonstrating that drivers can rapidly learn and apply alternative driving techniques that produce measurably different results in fuel efficiency (~20% reduction), speed stability, and safety. The findings have important implications for driver education policy, traffic network design, and environmental sustainability by suggesting that relatively simple changes in individual driver behaviors could produce global transformations in traffic efficiency.

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