## Abstract

This article investigates the theoretical Cramér-Rao bounds on estimation accuracy of longitudinal vehicle dynamics parameters. This analysis is motivated by the value of parameter estimation in various applications, including chassis model validation and active safety. Relevant literature addresses this demand through algorithms capable of estimating chassis parameters for diverse conditions. While the implementation of such algorithms has been studied, the question of fundamental limits on their accuracy remains largely unexplored. We address this question by presenting two contributions. First, this article presents theoretical findings which reveal the prevailing effects underpinning vehicle chassis parameter identifiability. We then validate these findings with data from on-road experiments. Our results demonstrate, among a variety of effects, the strong relevance of road grade variability in determining parameter identifiability from a drive cycle. These findings can motivate improved experimental designs in the future.

## 1 Introduction

This article explores the impact of drive cycle characteristics on the identifiability of longitudinal vehicle chassis parameters. From a conventional engineering perspective, vehicle chassis parameter estimation is an important exercise for identifying and controlling relevant vehicle models. For example, online mass estimation has significant importance to active vehicle safety systems [1]. Similarly, drag estimation can provide utility in quantifying the benefits associated with a heavy-duty vehicle’s participation in a vehicle platoon [2]. Furthermore, as connected and automated vehicle (CAV) systems are studied in greater detail, online parameter estimation algorithms will become increasingly important, as the function of CAV system optimization will likely depend on accurate knowledge of parameters including vehicle mass, drag, and rolling resistance coefficients. This consideration becomes even more important when considering vehicles with highly variable loads, including freight trucks [3,4]. Korayem et al. [5] present a recent review of parameter and state estimation for such heavy-duty vehicles, particularly in cases where the vehicle is towing a trailer. Their review partitions relevant literature into two groups: (1) model-based estimation methods and (2) nonmodel-based estimation methods. Viehweger et al. [6] provide an experimental comparison of a diverse set of such algorithms’ state estimation accuracy. For a review of relevant state estimation techniques for both passenger and heavy-duty vehicles, Guo et al. [7] present a precise summary of the current state of the art.

Historically, relevant literature presents numerous applications of parameter estimation algorithms for both passenger and heavy-duty vehicles. For example, research by Bae et al. [8] apply a recursive least-squares algorithm (RLS) to obtain online estimates of longitudinal parameters including vehicle mass from experimental data. Fathy et al. explore online mass estimation using a supervisory algorithm, which identifies predominantly longitudinal vehicle motion [9]. Vahidi et al. [10] estimate both vehicle mass and road grade through RLS utilizing multiple forgetting factors. Rhode and Gauterin [11] evaluate a generalized version of total RLS based on its experimental performance in estimating longitudinal parameters including rolling resistance coefficients and vehicle mass. Recent work by Paulsson and Asman [12] further explores the application of RLS for a similar context in a real-time experimental protocol. Lin et al. [13] recently explore vehicle mass estimation with a novel representation of uncertainty. Mahyuddin et al. [14] also present an adaptive observer that estimates vehicle mass and road grade online.

Vehicle chassis parameter estimation is not limited to the use of longitudinal vehicle dynamics. For example, Rajamani and Hedrick [15] show that suspension dynamics can be utilized to effectively estimate vehicle chassis parameters. Pence, Fathy, and Stein [16] demonstrate that vehicle mass can be estimated inexpensively from base excitation suspension dynamics. Reina et al. [17] utilize lateral vehicle dynamics with model-based estimation to estimate vehicle states and mass. Recent work by Hernandez and Hyun [18] utilizes sensor fusion to estimate the distribution of masses of heavy-duty vehicles traversing a known path. Korayem et al. [19] also estimate heavy-duty vehicle trailer mass using data-driven modeling approaches.

Exploring factors that affect parameter identifiability dates back to a landmark paper by Bellman and Astrom [20]. Examining limitations related to the identifiability of model parameters provides an opportunity to observe new insights related to the nature of such estimation problems. Such insights can address fundamental questions like “What conditions of an on-road experiment impose the biggest limitations on chassis parameter estimation accuracy?”, or “In what ways does the evolution of the drive cycle affect the identifiability of vehicle chassis parameters?” We can also attempt to quantify the impact of the effects that are present in the scenarios posed by such questions, including, for example, quantifying the impact of terrain variability on vehicle chassis parameter identifiability. The previous work in Ref. [21] makes some progress in addressing these questions, but overall they remain relatively unexplored. This is where our paper’s motivation lies, especially considering that new advancements on this front can motivate improvements to experimental design protocols and data selection algorithms. These improvements can be utilized within connected and automated vehicle systems to improve system-wide performance via more accurate parameter estimation schemes.

We should note that, in addition to providing extensive work regarding the estimation algorithms themselves, the literature does provide some recognition of the impact of the design of a vehicle experiment on the accuracy of the resulting chassis parameter estimates. The SAE testing standard for longitudinal vehicle chassis parameter estimation outlines several basic experimental conditions that facilitate estimation accuracy [22]. Within another context, the work by Muller et al. [23] demonstrates that the addition of speed bumps to a road test allows relevant suspension parameters to be estimated with greater precision. While such work recognizes effects that facilitate estimation accuracy, there is a need to provide theoretical justification for and quantification of such observations. Specifically, a theoretical Fisher information analysis of the maximum achievable parameter estimation accuracy for any unbiased estimator has not been presented within this context.

The Fisher information metric is an effective means for quantifying parameter identifiability [24]. The invertibility of the Fisher information matrix indicates whether the experiment in question can yield uniquely identifiable parameter estimates. Specifically, the Cramér-Rao theorem states that the best parameter estimation covariance achievable by an unbiased estimator is given by the inverse of the matrix formulation of Fisher information.

In this article, we present a theoretical analysis that reveals the underlying effects and limitations on the best achievable estimation accuracy for longitudinal vehicle dynamics. For an appropriate estimation algorithm, our analysis leverages Fisher information to mathematically show the form of the Cramér-Rao lower bounds on longitudinal parameter estimates in a simple linear estimation case and in a more nuanced nonlinear regression problem. We use these expressions to show fundamental insights that can motivate improved experiment design. We differentiate this paper’s theoretical contributions from our previous work in Ref. [21], which conducted an experimental, numerical Fisher analysis of on-road driving.

The remaining organization of this article is as follows: Section 2 details our formulation of the longitudinal vehicle dynamics. Section 3 describes the format of the Fisher information metric corresponding to the longitudinal dynamics. In Sec. 4, we derive an analytic Fisher matrix for a linear estimation case. We describe our methods for obtaining analytic expressions for Fisher information with nonlinear least-squares in Sec. 5, including validation of our derivations through simulation and on-road experiments. Finally, Sec. 6 summarizes and synthesizes insights from the article’s findings.

## 2 Longitudinal Vehicle Dynamics Model

*x*is the vehicle’s position,

*v*is the vehicle velocity, $v\u02d9$ is the vehicle acceleration,

*F*is the propulsion force, and

*β*is the road grade. Table 1 presents other relevant parameters of this model.

Value | Description | Units |
---|---|---|

m | Vehicle mass | kg |

C_{d} | Drag coefficient | — |

μ | Rolling resistance coefficient | — |

A_{ref} | Frontal area | m^{2} |

ρ | Air density | kg/m^{3} |

g | Gravitational constant | m/s^{2} |

Value | Description | Units |
---|---|---|

m | Vehicle mass | kg |

C_{d} | Drag coefficient | — |

μ | Rolling resistance coefficient | — |

A_{ref} | Frontal area | m^{2} |

ρ | Air density | kg/m^{3} |

g | Gravitational constant | m/s^{2} |

This model is relatively simple, but past work has utilized it with great efficacy to estimate relevant vehicle chassis parameters [8,12–14]. More complex dynamical models can consider a greater plurality of potential effects. However, for the purposes of keeping our theoretical Fisher information analysis relatively simple, we elect to utilize the basic but valuable formulation of longitudinal dynamics shown in Eqs. (1) and (2).

## 3 Formulation of Fisher Information

Fisher information is a metric that quantifies how much information a data set contains about a set of relevant model parameters [24]. In formulating our Fisher analysis, we first define relevant system output and longitudinal chassis parameters from the model described by Eqs. (1) and (2). We specifically select two subsets of vehicle mass *m*, drag coefficient *C*_{d}, and rolling resistance coefficient *μ* to be the chassis parameters, which we assume are being estimated in our linear and nonlinear Fisher information analysis.

Fisher information analysis requires selection of a measured output variable. In this paper, we use the propulsion force as the measured output for the linear derivation, and longitudinal vehicle velocity as our measured output for our nonlinear derivation. In both cases, the selection of the output variable is made to facilitate intuitive solution of the mathematical analysis and to match our experimental capabilities. Other variables including vehicle acceleration can also be applied as potential outputs in Fisher analysis. However, since our experimental setup utilizes both a final-drive torque sensor and a GPS velocity measurement system, we elect to use propulsion force and velocity as the basis for our analyses.

*θ*. In the case of linear least-squares, the Fisher information matrix is computed as follows:

*R*is the regressor matrix and

*σ*

^{2}is the sensor variance associated with output measurement noise for an assumed white, independent, and identically distributed noise process. This indicates that estimation accuracy depends not only on how we excite the system but also on how accurately sensors can measure their intended signals.

*i*indicates the time-step and

*j*indicates the parameter and is the sensitivity of the output with respect to the parameters. This sensitivity can be approximated via finite differences:

*v*(

*t*,

*θ*) and

*v*(

*t*,

*θ*+

*ɛ*) are simulated with a computer using the known road grade and propulsion force signals. In Sec. 4, we derive analytic continuous-time expressions for the output sensitivities given by (5). We therefore approximate the summations in the Fisher information matrix with integrals when using these analytic expressions:

*T*is the final time and

*f*is the sampling frequency.

_{lb}(

*θ*) is the best achievable parameter estimation error covariance for any unbiased estimator. The diagonal terms of this covariance matrix are the Cramér-Rao lower bounds of the estimation variances for each parameter estimate in

*θ*. These estimation error variances quantify the accuracy of the estimator. This article’s simulated and experimental identifiability analyses compare the error bounds obtained from Fisher analyses using both numeric and analytic methods in the nonlinear least-squares case.

*ρ*and

*A*

_{f}, both of which we assume are known.

In this article, we operate under the assumption that the parameters in question do not vary in time once the vehicle starts maneuvering. This is a fair assumption for several reasons. While a heavy-duty vehicle might experience significantly variable mass over time, the mass will not appreciably change in a single uninterrupted segment of driving. Rolling resistance, likewise, can be assumed constant if road and tire conditions do not change significantly during driving. The drag coefficient could experience a time-varying effect if, for instance, the vehicle in question participates in a platoon, as platooning scales down the effective drag coefficients of the following vehicles. The methods presented in this article can also be extended to cases where parameters are time-varying.

## 4 Linear Estimation Case

We perform Fisher analysis for both the full nonlinear vehicle parameter estimation problem and a simplified linear case.

*β*is sufficiently small such that

*μ*is known. This is a fair assumption: vehicle mass can vary significantly (e.g., heavy-duty vehicles), and aerodynamic drag can likewise change due to inter-vehicle interactions (i.e., platooning), but rolling resistance tends to stay fairly constant. By assuming

*μ*and

*β*are known, we can reformulate the longitudinal model to be conducive to application of ordinary least-squares as follows:

*x*

_{2}(

*t*) =

*v*

^{2},

*y*=

*F*,

*θ*

_{1}=

*m*, and $\theta 2=1/2\rho CdAf$. By using Eq. (7) to reformulate the Fisher matrix in continuous time, we obtain the following expression:

*v*(

*t*

_{0}) =

*v*(

*t*

_{f}) and $elevation|t0=elevation|tf$. Now, filling in the values for

*x*

_{1}(

*t*) and

*x*

_{2}(

*t*) as follows:

*u*=

*v*

^{2}and $dv=v\u02d9dt$, we transform this expression into:

This resulting expression allows us to infer several insights about chassis parameter identifiability.

First, our ability to estimate the drag coefficient is largely dependent on the kinetic energy of the experiment squared. This supports the existing empirical testing standard [22], which dictates a vehicle be brought to high velocity and then coasted down slowly.

Vehicle mass identifiability is subject to a host of more varied, interacting effects. A new insight from Eq. (24) is that rolling resistance squared also affects the identifiability of the mass parameter. Specifically, the larger the rolling resistance coefficient the easier it will be for one to estimate mass from measurements of the vehicle’s propulsive force.

Perhaps the most important factor that affects the mass parameter identifiability is the interplay between vehicle acceleration and gravitational acceleration. When expanding this integral out, it is clear the increased levels of road grade variablility (as represented by the integral of

*g*^{2}*β*^{2}) improve mass identifiability. However, this is only true insofar as the vehicle’s acceleration deviates from pure gravitational acceleration. In the worst case, vehicle acceleration $v\u02d9=\u2212g\beta (t)$ or is purely dependent on gravity, which can lead the mass estimation error to be considerably large. This makes intuitive sense. Consider that if one lets the vehicle accelerate and decelerate purely due to gravity up and down a series of hills, the vehicle will accelerate nearly the same way regardless of its mass. This makes it almost impossible to estimate vehicle mass unless the control input forces the vehicle to accelerate in ways which depart from pure gravitational acceleration. In this case, mean square road grade (the integral of road grade squared) will only improve estimation errors if the vehicle is controlled in such a way as to suppress the dependence of vehicle acceleration on road grade and gravity.

These insights provide theoretical support to existing empirical testing standards. In the remainder of this article, we will explore these effects with more comprehensive theoretical and experimental analyses. Specifically, we will first derive analytic expressions for the Fisher information matrix when conducting full nonlinear least-squares to estimate vehicle mass, drag coefficient, and rolling resistance coefficient. From these derivations, we corroborate the results of this exploratory linear analysis.

## 5 Nonlinear Least-Squares Analysis

Now, we consider a full nonlinear least-squares problem where we estimate the vehicle mass, drag, and rolling resistance coefficient. Simultaneous estimation of these three parameters is a more complicated process. By deriving analytic expressions for the Fisher information matrix, we can design simulation studies that replicate the conditions we found ideal in the previous section. In this section, we show that the effects that improve estimation accuracy for the full (i.e., 3 parameter) problem are similar to those we found in the previous section, where we only estimate vehicle mass and drag coefficient. We also show experimental validation of our analytic Fisher information matrix which reveals strong correlation between *C*_{d} and *μ* sensitivities.

As described in Sec. 3, our format of Fisher information utilizes sensitivity expressions of vehicle velocity with respect to perturbations in *m*, *C*_{d}, and *μ* parameters. Namely, we use Eqs. (4) and (5) exactly. In this section, analytic expressions for the sensitivity of vehicle velocity with respect to perturbation in *m*, *C*_{d}, and *μ* parameters are derived. Then, we assemble the full Fisher information matrix via Eq. (4).

### 5.1 Obtaining the Nominal Velocity Trajectory.

*β*

_{0}= 0 and

*F*

_{0}being the force required for our vehicle to maintain a constant predetermined speed

*v*

_{0}across a flat terrain. Next, we can subtract (26) from the expression we get by plugging in Eqs. (27) and (28) into Eq. (25) to obtain the following equation for $\delta v\u02d9n=v\u02d9n\u2212v\u02d90$:

*v*

_{0}

*δv*when neglecting higher-order terms, where

*v*

_{0}is the constant speed obtained by applying the constant propulsion force

*F*

_{0}to the vehicle subject to a flat terrain:

*δv*

_{n}(

*t*) yields the expression for perturbed vehicle velocity relative to steady-state, with respect to an arbitrary time-varying terrain profile and propulsion force:

*v*

_{0}to obtain the nominal linearized velocity for arbitrary time-varying terrain and propulsion force. This overall expression is essential to the analytic derivation of the sensitivity of velocity with respect to perturbations in each chassis parameter

*m*,

*C*

_{d}, and

*μ*.

### 5.2 Sensitivity Derivations.

Section 5.3 details analytic calculations for the sensitivity of velocity with respect to perturbations in the longitudinal chassis parameter *m*. The same procedure we use for *m* can be applied to the other chassis parameters, but for the purpose of brevity, we omit those derivations.

*δv*

_{n}(

*t*) is represented by Eqs. (31) and (32) in Sec. 4.1. Upon dividing through by

*ɛ*

_{m}, we obtain a simple time-varying first-order ordinary differential equation, which we can solve directly for the sensitivity

*δv*(

*t*)/

*ɛ*

_{m}:

*v*(

*t*) is sourced from Eq. (36).

We use the matlab symbolic algebra toolbox to analytically solve this ordinary differential equation (ODE) for *δv*(*t*)/*ɛ*_{m}. For the purpose of brevity, we omit the final, lengthy, analytic expression for *δv*(*t*)/*ɛ*_{m}.

### 5.3 Intuition From Derivations.

*α*(

*t*) includes additional effects, appears several times in the sensitivity expressions (and thus the resulting Fisher information matrix). This term originates from the perturbed velocity expression we solve for at the beginning of this section. Considering that $\delta F(t)m$ is a propulsion-induced acceleration term, this shows a commonality between the linear and nonlinear cases in terms of how road grade affects parameter identifiability. In fact, for the mass sensitivity equation, the interaction between propulsion force and gravitational acceleration is squared, similarly to the upper left hand entry of the linear Fisher information matrix (24). This indicates that similar insights prevail in the nonlinear estimation case, namely, that the interaction between acceleration and gravitational forces is important in determining parameter identifiability from an experiment. The remainder of this article presents simulation studies that validate our sensitivity expressions under various driving conditions.

### 5.4 Simulation and Experimental Validation of Sensitivity Expressions.

To validate our analytic sensitivity expressions from our nonlinear least-squares analysis, we utilize both a simulation study and an experimental analysis. First, in our simulation study, we compute a representative driving segment with sufficient excitation. Particularly, we want the vehicle to experience a larger range of velocities to validate that the linearization approximations we made in our nonlinear derivations do not significantly hamper the accuracy of our results. Then, we simulate longitudinal driving along that segment subject to the same longitudinal model parameterization given in Ref. [21] corresponding to a Volvo VNL300 heavy-duty vehicle, namely, *m* = 8875 kg, *C*_{d} = 0.49, and *μ* = 0.0056528. By using the data from this simulated driving, we compute the parametric sensitivity signals using expressions given in Sec. 5.2 of this article and those we obtain through Eq. (5). For our analytic expressions, we take the operating points as (1) the average measured velocity and (2) the average measured wheel force. Figure 1 demonstrates this comparison for 20 min of simulated driving. The sensitivities plotted in Fig. 1 are not normalized. As we can see, even with a relatively large range of velocities, the requisite approximations we make in our analytic derivations end up having little effect on the accuracy of our final results. Furthermore, the Cramér-Rao bounds we obtain from this simulation using numerical and analytic sensitivities are in strong agreement. Table 2 lists the estimation error bounds from both numerical and analytic sensitivities.

Estimation error | Numerical (sim) | Analytic (sim) | Numerical (exp) | Analytic (exp) |
---|---|---|---|---|

Mass m (kg) | 9.5311 | 9.5623 | 26.91 | 28.89 |

Drag coefficient C_{d} (–) | 0.0090 | 0.0087 | 0.0108 | 0.0091 |

Rolling eesistance μ (–) | 0.0004124 | 0.0004053 | 0.00086 | 0.00078 |

Estimation error | Numerical (sim) | Analytic (sim) | Numerical (exp) | Analytic (exp) |
---|---|---|---|---|

Mass m (kg) | 9.5311 | 9.5623 | 26.91 | 28.89 |

Drag coefficient C_{d} (–) | 0.0090 | 0.0087 | 0.0108 | 0.0091 |

Rolling eesistance μ (–) | 0.0004124 | 0.0004053 | 0.00086 | 0.00078 |

For our experimental study, we use data from an instrumented heavy-duty vehicle driving on-road to validate our analytic sensitivity expressions. The experimental vehicle is instrumented with a final-drive torque sensor, GPS, and a CAN interface, which logs data through a Simulink real-time machine. Our use of GPS to measure vehicle velocity decreases the requirement for consideration of wheel slip in our dynamical model. The precise details of this experimental setup can be referenced in Ref. [21]. This setup allows us to record values of wheel torque, road grade, velocity and acceleration, braking indicator, and other useful signals. To conduct our experimental comparison, we cut the data using the braking indicator to avoid unmodeled braking dynamics. The longest segment of data with no braking was approximately 5 min long. Then, we compute the sensitivities of the vehicle velocity with respect to each nominal parameter value both numerically (via Eq. (6)) and using our analytic expressions from the previous section. With both sets of sensitivities, we compute Fisher information using Eq. (4). Figure 2 demonstrates the results of this comparison. Overall, we observe strong agreement between numeric and analytic sensitivity signals for highway driving. Upon computing the Fisher information matrices for both the analytic and numeric cases, our resulting Cramér-Rao bounds also agree. This validates the accuracy of our derivations and shows that the necessary approximations have minimal effect on the accuracy of our analytic expressions relative to a more computationally expensive and general approach for computing the sensitivities. Table 2 shows the Cramér-Rao bounds obtained from the above data sample. Overall, our Cramér-Rao bounds from these analyses do not deviate from the numerical results by a factor more than 3.45% in the simulation study (mean parameter estimation error deviation of 1.84%) and 15.74% in the experimental study (mean parameter estimation error deviation of 5.90%). These results show that the approximations we made in our analytic sensitivity derivations still manage to preserve a strong degree of accuracy in our final results.

One interesting distinction among these sensitivity plots is that in steady state, the sensitivities associated with rolling resistance and aerodynamic drag are consistently negative, whereas the sensitivity associated with vehicle mass continues to oscillate in both magnitude and sign. This makes intuitive sense for the following reason: one can expect a slight increase in either aerodynamic drag or rolling resistance to gradually slow a given vehicle down over time, resulting in a negative sensitivity. In contrast, a slight increase in the mass of a vehicle amplifies the contributions of inertial and gravitational effects to overall propulsion force, relative to resistive effects (such as rolling resistance or aerodynamic drag). This may result in either a slight increase or decrease in vehicle velocity depending on the shape of both the vehicle input force and road grade as a function of time.

## 6 Conclusion

This article presents two contributions. These include: (i) theoretical insights into the chassis parameter identifiability and (ii) two studies that illustrate the relationships among terrain variability, velocity, propulsion force, and longitudinal chassis parameter identifiability.

The first application of these insights pertains directly to the design of on-road experiments for longitudinal chassis parameter identifiability. Choosing routes for such experiments that possess the greatest degree of terrain variability could not only improve the accuracy of parameter estimates obtained from the experiment but could also enable researchers to avoid unnecessarily costly instrumentation. Perhaps more significant than the application to experimental design is the relevance of this article’s results to CAV systems research. Much of this research relies on a priori knowledge of relevant vehicle parameters, including vehicle mass, drag, and rolling resistance coefficients. For actual implementation of CAV system optimization, these parameters may in fact need to be estimated in real time with online parameter estimation algorithms, and such algorithms will need to interface with system-wide optimization. The effectiveness of CAV optimization algorithms will likely depend on the accuracy of these parameter estimates, and as a result choosing and weighing data from segments of road characterized by high terrain variability can improve the function of such algorithms.

## Acknowledgment

This work was supported by a National Science Foundation Graduate Research Fellowship #DGE-1752814 and by NSF Award #CMMI-1538300, NSF Award #CMMI-1351146, and ARPA-E Award #DE-AR0000801.

## Conflict of Interest

There are no conflicts of interest.