A Novel Estimating Algorithm of Critical Driving Parameters for Dual-Motor Electric Drive Tracked Vehicles Based on a Nonlinear Observer and an Adaptive Kalman Filter

Abstract

High-speed dual-motor electric drive tracked vehicles (DDTVs) have emerged as a research hotspot in the field of tracked vehicles in recent years due to their advantages in fuel economy and the scalability of electrical equipment. The emergency braking of a DDTV at high speed can lead to slipping or even yawing (which is caused by a large deviation of forces at each track directly), posing significant challenges to the vehicle’s stability and safety. Therefore, the accurate real-time acquisition of critical driving parameters, such as the longitudinal force and vehicle speed, is crucial for the stability control of a DDTV. This paper developed a novel estimating algorithm of critical driving parameters for DDTVs equipped with conventional sensors such as rotary transformers at PMSMs and onboard accelerometers on the basis of their dynamics models. The algorithm includes a sensor signal preprocessing module, a longitudinal force estimation method based on a nonlinear observer, and a speed estimation method based on an adaptive Kalman filter. Through hardware-in-loop experiments based on a Speedgoat real-time target machine, the proposed algorithm is proven to estimate the longitudinal force of the track and vehicle speed accurately, whether the vehicle has stability control functions or not, providing a foundation for the further development of vehicle stability control algorithms.

1. Introduction

Tracked vehicles are widely used in agriculture, engineering, exploration, and the military with the advantages of low ground pressure and good passability. High-speed dual-motor electric drive tracked vehicles have a structure that allows the engine to operate stably at the optimal working condition for most of the time with the capability of regenerative braking, thus significantly decreasing the overall fuel consumption, extending the driving radius, and improving the scalability of on-vehicle electronic equipment; hence, attention has been drawn to the research of tracked vehicles in recent years [1,2,3]. Due to the large weight and relatively high speed of over 70 km/h of tracked vehicles, track locking and vehicle yawing during high-speed braking may lead to loss of control or even rollover [4,5], which is extremely dangerous. The earliest study on the stability control of tracked vehicles involves a simple traction control method to relieve the slippage during vehicle startup by controlling the engine torque [6], but since then, relevant research has remained scarce. When conducting research on the braking stability control strategies of DDTVs, obtaining critical driving parameters reliably in real time is of great importance. The deviation of the longitudinal force on each track is the driving force of the vehicle yawing, and the track’s wrapping speed might deviate significantly from the actual vehicle speed when braking or in an aggressive driving condition. Therefore, it is necessary to establish a method for estimating these two parameters based on conventionally equipped onboard sensors.

As for longitudinal force estimation, traditional ways usually depend on tire models. In recent years, with the development of smart tires, researchers have begun to try to install three-axis acceleration sensors [7], force sensors [8], wireless piezoelectric sensors [9], and image recognition devices [10] on the tires and estimate the longitudinal force of each tire through machine learning, reference-model-based real-time calculating, and other means. However, the wheel–ground relationship faced by ground vehicles is more complex, and it is more suitable to estimate the force based on the vehicle motion equations and vehicle attitudes [11]. Although current vehicles are generally equipped with GPS equipment, the relatively low refresh rate of GPS data makes it unsuitable for dynamic control systems. Additionally, when the vehicle is traveling through buildings, mountains, or complex electromagnetic environments, it is difficult to ensure communication with navigation satellites. When developing classical braking stability control strategies like ABS or EBD, vehicle manufacturers determined the reference speed mainly using the maximum wheel speed method, slope method, or comprehensive method [12]. These methods are simple to implement, but the determined reference speed often deviates significantly from the actual speed and has poor adaptability to different ground conditions. Kinematic-based estimation methods usually estimate the speed by directly integrating the signals from onboard sensors such as accelerometers [13], which have almost no requirements for the vehicle model but may lead to large errors due to the accumulation of noise. Observers such as nonlinear full-order vehicle speed observers, reduced-order vehicle speed observers, and cascade vehicle speed estimation observers based on Look-up tables or mechanistic models are commonly used in the estimation of driving state parameters for road vehicles [14,15,16,17]. However, all these observers are developed from model-based estimation methods and have a strong dependence on vehicle parameters, such as the wheel–road interaction force, which is greatly influenced by the accuracy of the tire model. Commonly used models, such as the Lugre model [18] and the Uni-Tire Model [19,20], which integrate various tire and road contact states into an analytical form determined by several parameters, could quickly and accurately calculate the friction force between the tire and the road and the vehicle’s slip ratio, paving the way for the application of the above estimation methods. The Kalman filter is widely used in inertial navigation and is a basic method for estimating vehicle driving state parameters. This method is an estimation method with a prediction-correction process, which predicts the current state through the state prediction equation of the previous moment and corrects the prediction result using external measurement values through the measurement update equation [21]. The Kalman filter-based estimation of vehicle speed has been explored, and good results can be achieved by adjusting the filter gain through methods such as rule-based adaptive algorithms, fuzzy logic control, and AI algorithms [22,23,24,25,26].

However, as a significant subset of off-road vehicles, tracked vehicles differ significantly from wheeled ones in structures and dynamics. The longitudinal force exerted by the ground on the tracked vehicle depends on the force conditions of each track plate and its connecting parts; thus, the longitudinal force is hard to obtain by arranging sensors on road wheels or between track plates in engineering practices. At present, research on track force estimation is more common in the estimation of tension force [27], and there is relatively less research on longitudinal force conducted by the ground. In the meantime, due to the lack of a universal analytical model that can describe various track–ground contact conditions, relevant methods are difficult to apply to tracked vehicles. Hence, exploring longitudinal force using dynamics-based observer and speed estimation methods based on the Kalman filter for DDTVs is worthwhile.

Therefore, this paper presents a novel algorithm for estimating the longitudinal force on each track and the longitudinal speed of the DDTV. The organization of this paper is as follows. In Section 2, the basic features of a DDTV are introduced. In Section 3, the signal preprocessing method of onboard sensors is presented. Section 4 and Section 5 introduce the longitudinal force estimation method based on a nonlinear observer and the longitudinal vehicle speed estimation method based on the adaptive Kalman filter, respectively. Section 6 verifies the effectiveness of the proposed algorithm through hardware-in-loop experiments based on Speedgoat. Finally, the conclusion is given in Section 7. The nomenclature and abbreviations are listed in Abbreviation section.

2. Basic Features of a DDTV

2.1. Structure Overview of a DDTV

The hybrid powertrain of a DDTV and its control systems, in this research, is a series HEV system, and its structure is shown in Figure 1. It could be divided into a front power chain, composed of a diesel generator set and an energy storage unit, and a rear power chain, composed of two PMSMs coupled with a coupling mechanism, mechanical brakes, and electrohydraulic retarders. The front and rear power chains are power coupled by a DC bus, and there is no mechanical connection between them. The diesel generator set and PMSMs are controlled by their controllers, respectively, and the energy storage unit is controlled by a DC/DC convertor. The mechanical brakes, electrohydraulic retarders, and the controllers mentioned above work cooperatively under the control of the transmission control unit.

2.2. Dynamic Models

A tracked vehicle is a self-paving vehicle, its steering is caused by different speeds of two tracks. The longitudinal motion and the steering motion of tracked vehicles are normally modeled separately during their dynamic analysis. Since the negative effects of slipping and sliding usually appear at braking conditions in high-speed straight-line movement, a suitable longitudinal dynamic model is an important foundation for controlling a DDTV effectively. The walking structure of a tracked vehicle and a wheeled vehicle varies greatly; a DDTV can be mainly divided into three parts as shown in Figure 2, which are the vehicle body, the tracks, and the wheels. Thus, the rolling of the road wheels, idlers, and support rollers, the rotational motion of the tracks, the rolling of the drive wheels, and the longitudinal motion of the vehicle body should be taken into consideration when analyzing the vehicle dynamics.

By setting the forward direction of the vehicle and the corresponding rotation direction of the driving wheel as positive, the dynamics of the longitudinal motion of half of the vehicle and the rolling of the drive wheel can be denoted as follows:

$$\left\{\begin{array}{c}{F}_{\mathrm{x}}^i+F_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}+M{\dot{v}}_{\mathrm{x}}=0\\ J^i{\dot{\omega }}^i+F_t^{i}r+T_{\mathrm{b}}^i=0\end{array}\right.$$

(1)

where $F_{\mathrm{x}}^i$ is the track–ground resultant force acting on a single track, $F_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}$ is half of the total air drag, $M$ is half of the DDTV’s total mass, $v_{\mathrm{x}}$ is the longitudinal speed of the vehicle, $J^i$ is the rotational inertia of the drive wheel, with the transmission system attached to it, of one side, and ${\mathsf{\omega }}^i$ is its angular speed. In this equation, $F_t^i$ is the force applied on the drive wheel by the track, $r$ is the radius of the drive wheel, and $T_b^i$ is the braking torque applied on the drive wheel by the transmission system through a half-shaft. The i in Equation (1) refers to left or right, which means one side of the DDTV laterally.

The tracks are the key equipment that converts the braking torque on the drive wheel into the braking force of the vehicle. They link the rotating motion of the drive wheel with the linear motion of the vehicle and drive the rolling of the road wheels, support wheels, and idler. Since the weight of the track and the moment of inertia of these wheels are not so small as to be negligible, the detailed dynamic analysis of the track and these wheels for calculating $F_t^i$ in actual time is relatively complicated, so the motion of the track and these wheels can be considered equivalent to the movement of the drive wheel by calculating the equivalent rotational inertia (ERI) on the axis of the drive wheel based on the energy conservation theory. The drive wheel, road wheels, support wheels, and idler share the same linear speed, and the track can be separated into four parts, which are the front part, rear part, upper part, and bottom part as shown in Figure 2. The slip or slide of a DDTV is significant when accelerating or braking in a rather big acceleration, and the slip ratio of a single track ${\lambda }^i$ can be denoted as follows when braking:

$${\lambda }^i={\displaystyle \frac{v_{\mathrm{x}}-{\omega }^{i}r}{v_{\mathrm{x}}}}$$

(2)

According to [28] (Appendix B), the ERI of the wheels and the track considered with the slip ratio ${\lambda }^i$ can be written as follows:

$${J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{r}\mathrm{w}}}^i=n_{\mathrm{r}\mathrm{w}}{J}_{\mathrm{r}\mathrm{w}}^i{\displaystyle \frac{r^2}{r_{\mathrm{r}\mathrm{w}}^2}}$$

(3)

$${J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{s}\mathrm{r}}}^i=n_{\mathrm{s}\mathrm{r}}{J}_{\mathrm{s}\mathrm{r}}^i{\displaystyle \frac{r^2}{r_{\mathrm{s}\mathrm{r}}^2}}$$

(4)

$${J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{I}}}^i=J_{\mathrm{I}}^i{\displaystyle \frac{r^2}{r_{\mathrm{I}}^2}}$$

(5)

$$\left\{\begin{array}{c}\begin{array}{c}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{t}\_\mathrm{f}}}^i=m_{\mathrm{t}\_\mathrm{f}}{r}^2\left[1+{\left({\displaystyle \frac{1}{1-{\lambda }^i}}\right)}^2-{\displaystyle \frac{2\cos {\phi }_{\mathrm{A}}}{\left(1-{\lambda }^i\right)}}\right]\\ {J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{t}\_\mathrm{r}}}^i=m_{\mathrm{t}\_\mathrm{r}}{r}^2\left[1+{\left({\displaystyle \frac{1}{1-{\lambda }^i}}\right)}^2-{\displaystyle \frac{2\cos {\phi }_{\mathrm{D}}}{\left(1-{\lambda }^i\right)}}\right]\\ {J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{t}\_\mathrm{u}}}^i=m_{\mathrm{t}\_\mathrm{u}}{r}^2{\left({\displaystyle \frac{2-{\lambda }^i}{1-{\lambda }^i}}\right)}^2\end{array}\\ {J_{\mathrm{e}\mathrm{r}\mathrm{i}}^{\mathrm{t}\_\mathrm{b}}}^i=m_{\mathrm{t}\_\mathrm{b}}{r}^2{\left({\displaystyle \frac{{\lambda }^i}{1-{\lambda }^i}}\right)}^2\end{array}\right.$$

(6)

where $J_{\mathrm{r}\mathrm{w}}^i$, $J_{\mathrm{s}\mathrm{r}}^i$, and $J_{\mathrm{I}}^i$ are the rotational inertia of the road wheels, support rollers and idlers separately, $n_{\mathrm{r}\mathrm{w}}$, $r_{\mathrm{r}\mathrm{w}}$, $n_{\mathrm{s}\mathrm{r}}$, $r_{\mathrm{s}\mathrm{r}}$, and $r_{\mathrm{I}}$ are the quantity and the radius of them. In Equation (6), $m_{\mathrm{t}\_\mathrm{f}}$, $m_{\mathrm{t}\_\mathrm{r}}$, $m_{\mathrm{t}\_\mathrm{u}}$, and $m_{\mathrm{t}\_\mathrm{b}}$ are the masses of the front, rear, upper, and bottom parts of the track, respectively, ${\phi }_{\mathrm{A}}$ is the approach angel, and ${\phi }_{\mathrm{D}}$ is the departure angel.

The braking torque $T_b^i$ is the combination of all the torques produced in the rear power chain of the DDTV’s transmission system, which is symmetrical about the coupling mechanism, so the torque and motion of the left side as an example is shown in Figure 3. The coupling mechanism with a transmission ratio of $i_{\mathrm{C}}$ has a characteristic parameter $k_{\mathrm{C}}$, and the transmission ratio of the wheel-side reducer is $i_{\mathrm{R}}$; thus, the braking torque on the drive wheels $T_{\mathrm{b}}^L$ and $T_{\mathrm{b}}^R$ can be written as follows:

$$\left\{\begin{array}{c}{T}_{\mathrm{b}}^L=i_{\mathrm{R}}[T_{\mathrm{e}\mathrm{r}}^L+T_{\mathrm{m}\mathrm{b}}^L+i_{\mathrm{C}}({\displaystyle \frac{2+k_{\mathrm{C}}}{2}}{T}_{\mathrm{m}}^L-{\displaystyle \frac{k_{\mathrm{C}}}{2}}{T}_{\mathrm{m}}^R)]\\ T_{\mathrm{b}}^R=i_{\mathrm{R}}[T_{\mathrm{e}\mathrm{r}}^R+T_{\mathrm{m}\mathrm{b}}^R+i_{\mathrm{C}}({\displaystyle \frac{2+k_{\mathrm{C}}}{2}}{T}_{\mathrm{m}}^R-{\displaystyle \frac{k_{\mathrm{C}}}{2}}{T}_{\mathrm{m}}^L)]\end{array}\right.$$

(7)

where $T_{\mathrm{e}\mathrm{r}}^L$, $T_{\mathrm{e}\mathrm{r}}^R$, $T_{\mathrm{m}\mathrm{b}}^L$, $T_{\mathrm{m}\mathrm{b}}^R$, $T_{\mathrm{m}}^L$, and $T_{\mathrm{m}}^L$ are the braking torques of the electrohydraulic retarder, the mechanical brake, and the PMSM on each side separately.

Based on the analysis above, the longitudinal dynamics of the DDTV can be amended with the consideration of the ERI and the slip ratio as follows:

$$\left\{\begin{array}{c}{F}_{\mathrm{x}}^i+F_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}+M{\dot{v}}_{\mathrm{x}}=0\\ J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i\dot{\omega }-F_X^{i}r+T_{\mathrm{b}}^i=0\end{array}\right.$$

(8)

where the ERI of half the DDTV $J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i$ can be achieved by summing up $J^i$ and all the ERIs in Equations (3)–(6).

2.3. Onboard Sensor Layouts

Like most mass-produced vehicles, the DDTV is equipped with an accelerometer to obtain its driving status. Additionally, each PMSM of the DDTV that is connected to the drive wheels by the coupling mechanism and wheel-side reducers (which are non-slip mechanical structures) is equipped with a high-precision rotary transformer; thus, the DDTV can obtain the vehicle acceleration, drive wheel angular speeds, and angular accelerations relatively accurately in real time. The movement of the drive wheel is the result of the dual PMSMs driven through the coupling mechanism and the wheel-side reducers, so the angular speeds of the drive wheel can be calculated as follows in real time based on the angular speeds of the PMSM through Equation (9):

$$\left[\begin{array}{c}{\omega }^L\\ {\omega }^R\end{array}\right]={\displaystyle \frac{1}{i_{\mathrm{R}}{i}_{\mathrm{C}}}}\left[\begin{array}{cc}{\displaystyle \frac{2+k_{\mathrm{C}}}{2(1+k_{\mathrm{C}})}}& {\displaystyle \frac{k_{\mathrm{C}}}{2(1+k_{\mathrm{C}})}}\\ {\displaystyle \frac{k_{\mathrm{C}}}{2(1+k_{\mathrm{C}})}}& {\displaystyle \frac{2+k_{\mathrm{C}}}{2(1+k_{\mathrm{C}})}}\end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{m}}^L\\ {\omega }_{\mathrm{m}}^R\end{array}\right]$$

(9)

where ${\omega }^L$, ${\omega }^R$, ${\omega }_{\mathrm{m}}^L$, and ${\omega }_{\mathrm{m}}^R$ are the angular speeds of the left drive wheel, the right drive wheel, the left PMSM, and the right PMSM.

3. Signal Preprocessing

The output signals of the sensors contain actual signals and noises, while the high-frequency vibrations and sudden torque fluctuations of the vehicle may cause a drastic increase in measurement noise while measuring acceleration and angular speeds. If the raw signals of the sensors are directly used for vehicle control, the control efficiency will be distinctly reduced or even will not be able to reach a steady state. Therefore, before the longitudinal force and speed estimation, the acceleration signal measured by the accelerometer and the angular speed signals measured by the rotary transformers should be preprocessed to reduce the influence of noise.

3.1. Basic Principles of a Kalman Filter

A Kalman filter is a predictive-corrective algorithm with numerical solutions, consisting of a prediction process composed of time update equations and a correction process composed of measurement update equations. For a discrete system with mutually independent zero-mean process noise ${\omega }_{\mathrm{s}}$ and measurement noise $n_{\mathrm{m}}$ that follow Gaussian distributions, written as Equation (10), the time update equations and measurement update equations can be written as Equations (11) and (12).

$$\left\{\begin{array}{c}x(\mathrm{k}+1)=Ax\left(\mathrm{k}\right)+Bu\left(\mathrm{k}\right)+{\omega }_{\mathrm{s}}\left(\mathrm{k}\right)\\ y(\mathrm{k}+1)=Cx(\mathrm{k}+1)+n_{\mathrm{m}}(\mathrm{k}+1)\end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{\widehat{x}}^{-}(\mathrm{k}+1)=A\widehat{x}\left(\mathrm{k}\right)+Bu\left(\mathrm{k}\right)\\ P^{-}(\mathrm{k}+1)=AP\left(\mathrm{k}\right)A^T+Q\left(\mathrm{k}\right)\end{array}\right.$$

(11)

$$\left\{\begin{array}{c}K(\mathrm{k}+1)=P^{-}(\mathrm{k}+1)C^T{\left(C{P}^{-}\right(\mathrm{k}+1)C^T+R(\mathrm{k}+1\left)\right)}^{-1}\\ \widehat{x}(\mathrm{k}+1)={\widehat{x}}^{-}(\mathrm{k}+1)+K(\mathrm{k}+1)\left(y\right(\mathrm{k}+1)-C{\widehat{x}}^{-}(\mathrm{k}+1\left)\right)\\ P(\mathrm{k}+1)=(I-K(\mathrm{k}+1\left)C\right)P^{-}(\mathrm{k}+1)\end{array}\right.$$

(12)

The meanings of each symbol in the equations is shown in

Table 1. Meanings of the symbols in the Kalman filter.

Symbol	Meaning	Symbol	Meaning
$x$	system state	$A$	state transition matrix
$y$	measured values of system output	$B$	input coupling matrix
$u$	system input	$C$	measurement sensitivity matrix
$P$	covariance matrix of state estimation uncertainty	$R$	covariance matrix of measurement noise
$Q$	covariance matrix of process noise	$K$	Kalman gain

, where $I$ means the identity matrix.

3.2. Kalman Filtering of the Onboard Sensors

The state equation and output equation of the vehicle acceleration can be denoted as follows:

$$\left\{\begin{array}{c}{\dot{a}}_{\mathrm{x}\mathrm{t}}={\omega }_{\mathrm{a}\mathrm{x}}\\ a_{\mathrm{x}\mathrm{m}}=a_{\mathrm{x}}+n_{\mathrm{a}\mathrm{x}}\end{array}\right.$$

(13)

where $a_{\mathrm{x}\mathrm{t}}$ is the actual acceleration, $a_{\mathrm{x}\mathrm{m}}$ is the measured value of the accelerometer, ${\omega }_{\mathrm{a}\mathrm{x}}$ is the process noise of the variation change, which contains the statistical characteristics of the vehicle jerk, and $n_{\mathrm{a}\mathrm{x}}$ is the measurement noise of the accelerometer, and their covariance can be determined by tests. Assuming that these noises are mutually independent and zero mean and follow Gaussian distributions, the mathematical model of the accelerometer can be discretized as follows:

$$\left\{\begin{array}{c}{a}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)=a_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+{\omega }_{\mathrm{a}\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\\ a_{\mathrm{x}\mathrm{m}}(\mathrm{k}+1)=a_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)+n_{\mathrm{a}\mathrm{x}}\left(\mathrm{k}\right)\end{array}\right.$$

(14)

Thus, the standard Kalman filter of the vehicle accelerometer can be established with $A=1$, $B=0$, $C=1$, $D=0$, $Q=cov({\omega }_{\mathrm{a}\mathrm{x}},{\omega }_{\mathrm{a}\mathrm{x}})$, and $R=cov(n_{\mathrm{a}\mathrm{x}},n_{\mathrm{a}\mathrm{x}})$.

The angular speed $\omega$ of the drive wheel can be collected and calculated by the rotary transformer through Equation (9), and the angular acceleration $\alpha$ can be derived in the same way. Thus, the state equation and output equation of the drive wheel’s angular speed and acceleration can be denoted as follows:

$$\left\{\begin{array}{c}\left[\begin{array}{c}{\dot{\omega }}_{\mathrm{t}}\\ {\dot{\alpha }}_{\mathrm{t}}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{t}}\\ {\alpha }_{\mathrm{t}}\end{array}\right]+\left[\begin{array}{c}{\omega }_{\mathsf{\omega }}\\ {\omega }_{\mathsf{\alpha }}\end{array}\right]\\ {\omega }_{\mathrm{m}}=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{t}}\\ {\alpha }_{\mathrm{t}}\end{array}\right]+n_{\omega }\end{array}\right.$$

(15)

where ${\omega }_{\mathrm{t}}$ and ${\alpha }_{\mathrm{t}}$ are the actual angular speed and acceleration at the drive wheel and ${\omega }_{\mathrm{m}}$ is the drive wheel’s angular speed calculated by using the measured value of the rotary transformers. As for ${\omega }_{\mathsf{\omega }}$ and ${\omega }_{\mathsf{\alpha }}$, they are the process noises of the angular speed and acceleration, separately, where the influences of the angular jerk are included. The measurement noises of rotary transformers equivalent to the drive wheel can be expressed as $n_{\omega }$, and the covariance can be determined by tests. Assuming that these noises are mutually independent, zero mean, and follow Gaussian distributions, the mathematical model can be discretized as follows by marking the sampling interval as ${\tau }_{\mathrm{s}}$:

$$\left\{\begin{array}{c}\left[\begin{array}{c}{\omega }_{\mathrm{t}}\left(\mathrm{k}+1\right)\\ {\alpha }_{\mathrm{t}}\left(\mathrm{k}+1\right)\end{array}\right]=\left[\begin{array}{cc}1& {\tau }_{\mathrm{s}}\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{t}}\left(\mathrm{k}\right)\\ {\alpha }_{\mathrm{t}}\left(\mathrm{k}\right)\end{array}\right]+\left[\begin{array}{c}{\omega }_{\mathsf{\omega }}\left(\mathrm{k}\right)\\ {\omega }_{\mathsf{\alpha }}\left(\mathrm{k}\right)\end{array}\right]\\ {\omega }_{\mathrm{m}}\left(\mathrm{k}+1\right)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{\omega }_{\mathrm{t}}\left(\mathrm{k}+1\right)\\ {\alpha }_{\mathrm{t}}\left(\mathrm{k}+1\right)\end{array}\right]+n_{\omega }\left(\mathrm{k}+1\right)\end{array}\right.$$

(16)

Thus, the standard Kalman filter of the rotary transformer can be established with $A=\left[\begin{array}{cc}1& {\tau }_{\mathrm{s}}\\ 0& 1\end{array}\right]$, $B=0$, $C=\left[\begin{array}{cc}1& 0\end{array}\right]$, $D=0$, $Q=\left[\begin{array}{cc}cov({\omega }_{\omega },{\omega }_{\omega })& 0\\ 0& cov({\omega }_{\mathsf{\alpha }},{\omega }_{\mathsf{\alpha }})\end{array}\right]$, and $R=cov(n_{\omega },n_{\omega })$.

4. Estimation Method of Longitudinal Force Based on Nonlinear Observer

4.1. Design of the Observer

The difference in the longitudinal force on each side of the tracked vehicle is the driving force of the vehicle steering, so the accurate estimation of longitudinal force in the braking process is of great significance to the stability control of a DDTV. After establishing the corresponding experimental models of all the equipment in the rear power chain of the DDTV’s transmission system, the braking torque acting on the drive wheel can be calculated through Equation (7). Ignoring the viscous resistance and loss of the torque output by the PMSM, mechanical brake, and electro-hydraulic retarder through the coupling mechanism, the dynamics of the drive wheel can be converted to the following form based on preprocessed vehicle acceleration, angular speed, and angular acceleration:

$$F_{\mathrm{x}}^i={\displaystyle \frac{1}{r}}\left(T_{\mathrm{b}\mathrm{m}}^i-J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i{{\widehat{\alpha }}_{\mathrm{t}}}^{\mathrm{i}}\right)+{\Omega }_{\mathrm{f}}^i$$

(17)

where $T_{\mathrm{b}\mathrm{m}}^i$ is the calculated torque on the drive wheel based on vehicle control signals and experimental models of the rear power chain, ${{\widehat{\alpha }}_{\mathrm{t}}}^i$ is the angular acceleration of the drive wheel output by the standard Kalman filter, ${\Omega }_{\mathrm{f}}^i$ is the uncertainty caused by the ERI and the measurement error of relevant variables like torques, angular acceleration, and so on. The main aim of designing an observer for estimating the longitudinal force is to minimize the influence of these uncertainties on longitudinal force observations.

According to the reference of the state observer proposed in [29], a PID observer for the longitudinal force on the track can be denoted as follows:

$${\widehat{F}}_{\mathrm{x}}^i={\displaystyle \frac{T_{\mathrm{b}\mathrm{m}}^i-J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i{\widehat{\alpha }}_{\mathrm{t}}^i}{r}}-{\eta }_1{e}_{\mathsf{\omega }\mathrm{i}}+{\eta }_2\int e_{\mathrm{F}\mathrm{i}}\mathrm{d}t$$

(18)

where ${\eta }_1$ and ${\eta }_2$ are designed parameters, $e_{\mathsf{\omega }\mathrm{i}}$ and $e_{\mathrm{F}\mathrm{i}}$ are the deviation between the estimated value and the actual value of the drive wheel’s angular speed and longitudinal force on the track, which can be denoted separately as follows:

$$e_{\mathsf{\omega }\mathrm{i}}={\omega }^i-{\widehat{\omega }}^i$$

(19)

$$e_{\mathrm{F}\mathrm{i}}=F_{\mathrm{x}}^i-{\widehat{F}}_{\mathrm{x}}^i$$

(20)

By substituting Equations (18) and (19) into (20), the error of the longitudinal force observer can be derived as follows:

$$e_{\mathrm{F}\mathrm{i}}=-{\eta }_2\int e_{\mathrm{F}\mathrm{i}}\mathrm{d}t+{\eta }_1{e}_{\mathsf{\omega }\mathrm{i}}+{\Omega }_{\mathrm{f}}^i$$

(21)

Then the dynamic equation of the observer error can be obtained by taking the derivative of Equation (21) with respect to time in the following form:

$${\dot{e}}_{\mathrm{F}\mathrm{i}}=-{\eta }_2{e}_{\mathrm{F}\mathrm{i}}+{\eta }_1{\dot{e}}_{\mathsf{\omega }\mathrm{i}}+{\dot{\Omega }}_{\mathrm{f}}^i$$

(22)

${\widehat{\omega }}^i$ in Equation (19) is the output value of an auxiliary drive wheel angular speed observer designed for longitudinal force estimation, which is denoted as follows:

$${\dot{\widehat{\omega }}}^i={\displaystyle \frac{1}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i}}(T_{\mathrm{b}\mathrm{m}}^i-r{\widehat{F}}_{\mathrm{x}}^i+{\eta }_3\int e_{\mathsf{\omega }\mathrm{i}}\mathrm{d}t+r{\eta }_2\int e_{\mathrm{F}\mathrm{i}}\mathrm{d}t)$$

(23)

where ${\eta }_3$ is a designed parameter as well. This observer can be written in the following form by taking the definition of ${\widehat{F}}_{\mathrm{x}}^i$ in Equation (18) into Equation (23):

$${\dot{\widehat{\omega }}}^i={{\widehat{\alpha }}_{\mathrm{t}}}^{\mathrm{i}}+{\displaystyle \frac{1}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i}}(r{\eta }_1{e}_{\mathsf{\omega }\mathrm{i}}+{\eta }_3\int e_{\mathsf{\omega }\mathrm{i}}\mathrm{d}t)$$

(24)

The measurement error of the angular speed and angular acceleration at the drive wheel is comparatively small because of the high precision of the rotary transformer and the preprocessing of the Kalman filter; thus, the uncertainty ${\Omega }_{\mathrm{f}}^i$ is mainly composed of the deviation between the actual torque output by the electromechanical hydraulic braking system to the drive wheel and the one from the mathematical model based on test data used for calculation. Accurate and reasonable experimental modeling can make this deviation zero mean, so the effect of ${\Omega }_{\mathrm{f}}^i$ can be eliminated by integrating it. Therefore, $e_{\mathsf{\omega }\mathrm{i}}$ can be replaced by ${{\widehat{\omega }}_{\mathrm{t}}^i-\widehat{\omega }}^i$, and $\int e_{\mathrm{F}\mathrm{i}}\mathrm{d}t$ can be used instead by integrating the difference of ${\displaystyle \frac{T_{\mathrm{b}\mathrm{m}}^i-J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i{\widehat{\alpha }}_{\mathrm{t}}^i}{r}}$ at the current step and the estimated ${\widehat{F}}_{\mathrm{x}}^i$ at the previous step in practice.

4.2. Error Analysis of the Observer

An observer is meaningful only when its observation error is bounded. Equation (17) can be rewritten into the expression of ${\widehat{\alpha }}_{\mathrm{t}}^i$ as follows:

$${\widehat{\alpha }}_{\mathrm{t}}^i={\displaystyle \frac{1}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i}}\left[T_{\mathrm{b}\mathrm{m}}^i-r(F_{\mathrm{x}}^i-{\Omega }_{\mathrm{f}}^i)\right]$$

(25)

Equation (26) can be deduced by subtracting Equation (23) from Equation (25), while Equation (27) can be deduced by substituting ${\dot{e}}_{\mathrm{F}\mathrm{i}}$, defined in Equation (22), into the derivation of Equation (26) as follows:

$$J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i{\dot{e}}_{\mathsf{\omega }\mathrm{i}}=-r{e}_{\mathrm{F}\mathrm{i}}+r{\Omega }_{\mathrm{f}}^i-{\eta }_3\int e_{\mathsf{\omega }\mathrm{i}}\mathrm{d}t-r{\eta }_2\int e_{\mathrm{F}\mathrm{i}}\mathrm{d}t$$

(26)

$$J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i{\ddot{e}}_{\mathsf{\omega }\mathrm{i}}+r{\eta }_1{\dot{e}}_{\mathsf{\omega }\mathrm{i}}+{\eta }_3{e}_{\mathsf{\omega }\mathrm{i}}=0$$

(27)

These equations can be rewritten in the expression of the state space in the following form:

$$\left[\begin{array}{c}{\dot{e}}_{\mathsf{\omega }\mathrm{i}}\\ {\ddot{e}}_{\mathsf{\omega }\mathrm{i}}\end{array}\right]=\underset{\kappa }{\underbrace{\left[\begin{array}{cc}0& 1\\ {\displaystyle \frac{-{\eta }_3}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i}}& {\displaystyle \frac{-r{\eta }_1}{J_{\mathrm{e}\mathrm{r}\mathrm{i}}^i}}\end{array}\right]}}\left[\begin{array}{c}{e}_{\mathsf{\omega }\mathrm{i}}\\ {\dot{e}}_{\mathsf{\omega }\mathrm{i}}\end{array}\right]$$

(28)

where $\kappa$ is a Hurwitz matrix when ${\eta }_1$ and ${\eta }_3$ are both greater than zero and Equation (28) is exponentially stable; thus, the observation error of the angular speed and angular acceleration at the drive wheel both approach zero. The dynamic equation of the longitudinal force observation error defined in Equation (22) can be expressed asymptotically as follows under the above conditions:

$${\dot{e}}_{\mathrm{F}\mathrm{i}}={-\eta }_2{e}_{\mathrm{F}\mathrm{i}}+{\dot{\Omega }}_{\mathrm{f}}^i$$

(29)

The algebraic solution of Equation (29) is as follows:

$$e_{\mathrm{F}\mathrm{i}}\left(t\right)=e^{{-\eta }_{2}t}{e}_{\mathrm{F}\mathrm{i}}\left(0\right)+{\displaystyle \frac{{\dot{\Omega }}_{\mathrm{f}}^i}{{\eta }_2}}$$

(30)

where $e^{{-\eta }_{2}t}{e}_{\mathrm{F}\mathrm{i}}\left(0\right)$ exponentially converges to zero for any ${\eta }_2>0$, and the error of this longitudinal force observer is bounded with the boundary of ${\displaystyle \frac{{\dot{\Omega }}_{\mathrm{f}}^i}{{\eta }_2}}$. Thus, an appropriately large value of ${\eta }_2$ is of benefit for reducing the observation error when selected during a parameter adjustment.

5. Longitudinal Speed Estimation Based on an Adaptive Kalman Filter

5.1. Kalman Filtering of the Longitudinal Vehicle Speed

After preprocessing with the Kalman filter designed in Section 3, the angular speed and angular acceleration at the drive wheel of both sides can be denoted as ${{\widehat{\omega }}_{\mathrm{t}}}^{\mathrm{L}}$, ${{\widehat{\alpha }}_{\mathrm{t}}}^{\mathrm{L}}$, ${{\widehat{\omega }}_{\mathrm{t}}}^{\mathrm{R}}$, and ${{\widehat{\alpha }}_{\mathrm{t}}}^{\mathrm{R}}$. Taking the average value of these filtered outputs on both sides, ${\widehat{\omega }}_{\mathrm{a}}$ and ${\widehat{\alpha }}_{\mathrm{a}}$, to estimate the longitudinal speed of the DDTV, the linear speed and acceleration at the drive wheel can be denoted as follows:

$$\left[\begin{array}{c}{\widehat{v}}_{\mathsf{\omega }}\\ {\widehat{a}}_{\mathsf{\omega }}\end{array}\right]=r\left[\begin{array}{c}{\widehat{\omega }}_{\mathrm{a}}\\ {\widehat{\alpha }}_{\mathrm{a}}\end{array}\right]$$

(31)

Severe slipping or sliding may cause significant negative impacts on the stability of the DDTV when running at a rather high speed, but when the speed drops to near zero, whether the drive wheel is locked or not does not affect the stability anymore. Therefore, the main aim of longitudinal speed estimation is to precisely estimate the actual speed at high speed, especially when the vehicle starts braking heavily. In order to unify the relationship between the actual speed and the linear speed of the drive wheel during slipping and sliding, the definition of the slip ratio can be extended to the slide state (not the same as the slide ratio) in the following form:

$$\lambda =\left\{\begin{array}{c}{\displaystyle \frac{v_{\mathrm{x}}-v_{\mathsf{\omega }}}{v_{\mathrm{x}}}}, v_{\mathrm{x}}>0\\ 0, v_{\mathrm{x}}=0\end{array}\right.$$

(32)

According to the equation above, $\lambda <0$ means the vehicle is sliding and $\lambda >0$ means it is slipping. Thus, the relationship of the linear speed of the drive wheel and the longitudinal speed of the DDTV can be expressed as follows:

$$v_{\mathsf{\omega }}=v_{\mathrm{x}}-\lambda v_{\mathrm{x}}$$

(33)

Regarding the observed value ${\widehat{v}}_{\mathsf{\omega }}$ by the rotary transformer as the measured value of the vehicle longitudinal speed and the observed value ${\widehat{a}}_{\mathrm{x}\mathrm{t}}$ by the accelerometer as the state of the system, the state equation and output equation of the speed estimation system can be written as follows:

$$\left\{\begin{array}{c}{\dot{v}}_{\mathrm{x}\mathrm{t}}={{\widehat{a}}_{\mathrm{x}\mathrm{t}}+\omega }_{\mathrm{v}\mathrm{x}\mathrm{t}}\\ {\widehat{v}}_{\mathsf{\omega }}={(1-\lambda )v}_{\mathrm{x}\mathrm{t}}+n_{\mathrm{v}\mathrm{x}\mathrm{m}}\end{array}\right.$$

(34)

where $v_{\mathrm{x}\mathrm{t}}$ is the actual speed, ${\omega }_{\mathrm{v}\mathrm{x}\mathrm{t}}$ is the process noise of the speed variation, which is mainly influenced by the measurement noise of longitudinal acceleration, $n_{\mathrm{a}\mathrm{x}\mathrm{m}}$ is the measurement noise of the vehicle speed, which is mainly influenced by the measurement noise of the drive wheel’s angular speed. Denoting the sampling interval as ${\tau }_{\mathrm{s}}$, the model of vehicle speed estimation can be discretized as follows:

$$\left\{\begin{array}{c}{v}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)=v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+{\tau }_{\mathrm{s}}{\widehat{a}}_{\mathrm{x}\mathrm{t}}+{\omega }_{\mathrm{v}\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\\ {\widehat{v}}_{\mathsf{\omega }}\left(\mathrm{k}+1\right)=(1-\lambda )v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)+n_{\mathrm{v}\mathrm{x}\mathrm{m}}\left(\mathrm{k}+1\right)\end{array}\right.$$

(35)

Hence, the measurement sensitivity matrix of this vehicle speed estimation model $C=1-\lambda$ has a time-varying structure. In practice, the expression of $\lambda$ can be denoted as follows, according to Equation (32):

$$\widehat{\lambda }(\mathrm{k})=\left\{\begin{array}{c}{\displaystyle \frac{{\widehat{v}}_{\mathrm{x}}\left(\mathrm{k}\right)-{\widehat{v}}_{\mathsf{\omega }}\left(\mathrm{k}\right)}{{\widehat{v}}_{\mathrm{x}}\left(\mathrm{k}\right)}}, {\widehat{v}}_{\mathrm{x}}(k)>0\\ 0, {\widehat{v}}_{\mathrm{x}}\left(k\right)=0\end{array}\right.$$

(36)

The value of $\widehat{\lambda }(\mathrm{k})$ is determined by the previous speed estimation, which means the structural change of the model in Equation (35) is determined by the output of the filter. This coupling relationship will aggravate the influence of the initial error of the filter, resulting in the filter output not being able to converge stably to the actual speed, so it is necessary to seek other methods. At the same time, it should be noted that the real slip ratio $\lambda$ is not able to be obtained accurately, so, in practice, the previous speed $v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$ in Equation (35) can only be substituted for the estimated value ${\widehat{v}}_{\mathrm{x}}\left(\mathrm{k}\right)$ with a slip ratio estimation error. In order to meet the requirements of Kalman filtering, the process noise ${\omega }_{\mathrm{v}\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$ should be replaced by ${\omega }_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$, which equals ${\omega }_{\mathrm{v}\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+(v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)-{\widehat{v}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right))$, and the measurement noise $n_{\mathrm{a}\mathrm{x}\mathrm{m}}(\mathrm{k}+1)$ should be replaced by $n_{\mathrm{x}\mathrm{m}}(\mathrm{k}+1)$, which equals $-\lambda v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)+n_{\mathrm{v}\mathrm{x}\mathrm{m}}\left(\mathrm{k}+1\right)$; thus, the discretized model of longitudinal speed estimation can be adapted as follows:

$$\left\{\begin{array}{c}{v}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)=v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+{\tau }_{\mathrm{s}}{\widehat{a}}_{\mathrm{x}\mathrm{t}}+{\omega }_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\\ {\widehat{v}}_{\mathsf{\omega }}\left(\mathrm{k}+1\right)=v_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)+n_{\mathrm{x}\mathrm{m}}\left(\mathrm{k}+1\right)\end{array}\right.$$

(37)

Assuming that ${\omega }_{\mathrm{x}\mathrm{t}}$ and $n_{\mathrm{x}\mathrm{m}}$ are both zero mean and satisfy the Gaussian distribution, the Kalman-filtered longitudinal speed can be denoted as follows when the covariance of the noises above are, separately, $Q$ and $R$:

$$\begin{array}{l}{\widehat{v}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}+1\right)=\underset{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{m}1}{\underbrace{{\widehat{v}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+{\tau }_{\mathrm{s}}{\widehat{a}}_{\mathrm{x}\mathrm{t}}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\underset{\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{m}2}{\underbrace{{\displaystyle \frac{P\left(\mathrm{k}\right)+Q\left(\mathrm{k}\right)}{P\left(\mathrm{k}\right)+Q\left(\mathrm{k}\right)+R\left(\mathrm{k}+1\right)}}\left[{\widehat{v}}_{\mathsf{\omega }}\left(\mathrm{k}+1\right)-\left({\widehat{v}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)+{\tau }_{\mathrm{s}}{\widehat{a}}_{\mathrm{x}\mathrm{t}}\right)\right]}}\end{array}$$

(38)

The initial value of this equation is ${\widehat{v}}_{\mathrm{x}\mathrm{t}}\left(0\right)={\widehat{v}}_{\mathsf{\omega }}\left(0\right)$, item1 represents the part of the estimation using the measured vehicle acceleration from the accelerometer, and item2 represents the part of the estimation using the measured angular speed from the rotary transformer. Thus, the error covariance update equation of the longitudinal speed estimation can be denoted as follows:

$$P\left(\mathrm{k}+1\right)={\displaystyle \frac{R\left(\mathrm{k}+1\right)(P\left(\mathrm{k}\right)+Q\left(\mathrm{k}\right))}{P\left(\mathrm{k}\right)+Q\left(\mathrm{k}\right)+R\left(\mathrm{k}+1\right)}}$$

(39)

Due to the values of ${\omega }_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$ and $n_{\mathrm{x}\mathrm{m}}\left(\mathrm{k}\right)$ being influenced by the error of the slip ratio estimation, and that they are not the actual process noise and measurement noise, their covariance values $Q\left(\mathrm{k}\right)$ and $R\left(\mathrm{k}+1\right)$ are both time-varying quantities.

5.2. Adaptive Adjustment Algorithm of the Filter Parameter

When the DDTV is running stably on the ground with good adhesion conditions and almost no slip between the tracks and ground, $Q$ $R$ can be used as a constant for Kalman filtering. However, the influence of the slip ratio cannot be ignored in real driving conditions, especially when braking heavily at high speed. Therefore, it is necessary to adjust the values of $Q$ and $R$ that are adaptively used in the filter to ensure that the filter can follow the changes and maintain good working efficiency. According to the definition of a Kalman filter, the values of $Q$ and $R$ represent the confidence in the vehicle speed updating by the state equation and the confidence in the vehicle speed updating by the output equation, respectively. Their basic variation rules are as follows:

(1)

Slipping of the DDTV has a relatively great influence when the absolute value of the slip ratio estimation $\widehat{\lambda }\left(\mathrm{k}\right)$ is large at the last step. Updating by the output equation is not quite reliable in this situation, while the confidence in updating by the state equation is high, so the value of R is relatively large.

(2)

When the difference in the estimated linear acceleration at the drive wheel ${\widehat{a}}_{\mathsf{\omega }}\left(\mathrm{k}\right)$ and the estimated vehicle longitudinal acceleration ${\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$ is rather large at the last step, the vehicle slipping is aggravated, and the value of R and the confidence in the state equation are similar to the previous situation.

(3)

When the estimated vehicle longitudinal acceleration ${\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)$ is small at the last step, the errors of the slip ratio estimation and sensor noise are greatly influenced when updated by the state equation, so the value of $Q$ needs to be relatively large.

Due to the rules above, an adaptive law of adjusting $Q$ and $R$ can be designed as follows:

$$R(\mathrm{k}+1)=R_v(\mathrm{k}+1)+R_a(\mathrm{k}+1)$$

(40)

$$\left\{\begin{array}{c}{R}_v(k+1)=0.05+50\left|\widehat{\lambda }(\mathrm{k})\right|\\ R_a\left(\mathrm{k}+1\right)=30{\displaystyle \frac{\left|{\widehat{a}}_{\mathsf{\omega }}\left(\mathrm{k}\right)-{\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\right|}{g}}\end{array}\right.$$

(41)

$$Q\left(\mathrm{k}\right)=\left\{\begin{array}{cc}0.01+{\displaystyle \frac{0.01}{\left|{\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\right|}}& ,\left|{\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\right|>0.01\\ 1.01& ,\left|{\widehat{a}}_{\mathrm{x}\mathrm{t}}\left(\mathrm{k}\right)\right|\le 0.01\end{array}\right.$$

(42)

where $R_v(\mathrm{k}+1)$ and $R_a(\mathrm{k}+1)$ represent the influence of velocity and acceleration separately. Under normal conditions, the absolute value of the slip ratio $\left|\widehat{\lambda }(\mathrm{k})\right|$ is very close to zero, and this value rarely exceeds 0.2, even if a sudden slip occurs, unless the vehicle meets an extreme slip, which would result in the vehicle losing control. Thus, the adjusting rule of $R_v(\mathrm{k}+1)$ needs a small constant item and a rather large coefficient of $\left|\widehat{\lambda }(\mathrm{k})\right|$, and the adjusting rule of $R_a\left(\mathrm{k}+1\right)$ needs a rather large coefficient, in the same way, to make $R(\mathrm{k}+1)$ large enough.

The state estimation error covariance $P\left(\mathrm{k}+1\right)$ can be updated automatically without outside intervention along the iteration of the prediction process under normal circumstances. However, due to the existence of sensor noises and estimation errors, a negative estimate of the vehicle’s longitudinal speed may occur when the speed is low, which is impossible when the DDTV is moving forward. Thus, the filter should be initialized in the following form to make the output conform to the reality:

$$if {\widehat{v}}_{\mathrm{x}\mathrm{t}}(\mathrm{k}+1)<0, then {\widehat{v}}_{\mathrm{x}\mathrm{t}}(\mathrm{k}+1)=0 \& P\left(\mathrm{k}+1\right)=P\left(0\right)$$

(43)

6. Hardware-in-Loop Experiments

6.1. Experiment Setups

Hardware-in-loop experiments connect the real hardware with the virtual models running on a real-time host through communication interfaces, providing a closed-loop experiment environment for the hardware or its algorithms to be experimented with that is easy to modify, avoiding the high cost required by experimenting in a real environment and reducing the risk caused by the immature early control strategy, making it possible to develop control strategies and the controlled object at the same time. The accuracy of estimating the longitudinal force and speed of a DDTV is critical when driving on low-adhesion ground with an intense way of driving, which is a great danger with respect to the field experiment, especially when the DDTV itself is in an immature prototype development stage. Therefore, a hardware-in-loop experiment is the most suitable way of testing and improving the relevant algorithms, taking the experimental effect and safety into account along with the affordable costs.

The experiment system is mainly composed of a TCU prototype subsystem, which receives the driving input through analog signals, and a real-time simulating subsystem, which is managed by an experiment management system through Ethernet, as shown in Figure 4. The TCU prototype subsystem is based on a real TCU equipped with I/O modules, vehicle control algorithms, and the proposed estimation algorithm. The real-time simulating subsystem is based on a Speedgoat performance real-time target machine equipped with an IO602 four-channel CAN interface I/O module. This subsystem contains powertrain models, vehicle dynamic models, and sensor models built in MATLAB/Simulink R2022a according to experiments and is connected to the TCU prototype with a CAN bus, which is exactly the same as in a real DDTV.

The slipping trend is relatively serious when the DDTV is braking on low-adhesion ground, which could cause a large difference between the actual vehicle speed and the linear speed at the drive wheel; thus, the experimental conditions were designed so that the DDTV performs progressive braking with a slip control strategy, full braking with a slip control strategy, and full braking without a slip control strategy, respectively, until the DDTV has stopped when running on muddy ground at the speed of 80 km/h.

In the experiment, the actual longitudinal speed and longitudinal force are set to be calculated results of the real-time dynamic models, and the sensor outputs are set to be the combination of the actual ones mentioned above and typical sensor noise by experiment. The main parameters of the DDTV for the experiment are listed in

Table 2. The main parameters of the DDTV for the experiment.

Parameter	Value	Units
Vehicle mass, $2M$	52,000	$\mathrm{k}\mathrm{g}$
Drive wheel radius, $r$	0.309	$\mathrm{m}$
Frontal area, $A$	5.36	${\mathrm{m}}^2$
Rotational inertia of drive wheel and power train, $J$	158.8	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$
Rotational inertia of road wheel, $J_{\mathrm{r}\mathrm{w}}$	23.7	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$
Rotational inertia of support roller, $J_{\mathrm{s}\mathrm{r}}$	13	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$
Rotational inertia of idler, $J_{\mathrm{I}}$	31.2	$\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$
Quantity of road wheels, half vehicle, $n_{\mathrm{r}\mathrm{w}}$	6	-
Quantity of support rollers, half vehicle, $n_{\mathrm{s}\mathrm{r}}$	3	-
Approach angle, ${\phi }_{\mathrm{A}}$	27.3	$°$
Departure angle, ${\phi }_{\mathrm{D}}$	35.6	$°$
Aerodynamic drag, ${\mathrm{C}}_{\mathrm{D}}$	0.78	-
Ratio of the wheel-side reducer,	4.59	-
Ratio of the coupling mechanism	2.2	-
Rated power of the PMSM	485	kW
Rated speed of the PMSM	3000	rpm
Max speed of the PMSM,	9000	rpm
Allowable overload factor of the PMSM	1.3	-

.

6.2. Experiment Results and Analysis

Muddy ground is a typical low-adhesion track–ground condition with a largest adhesion coefficient of around 0.2; thus, the slipping of the DDTV is supposed to be severe during heavy braking on this ground. The actual speed calculated in dynamic models, the linear speed calculated from the preprocessed sensor model output, and the estimated speed from the adaptive Kalman filter are selected to test the effectiveness of the longitudinal speed estimation. Meanwhile, the actual longitudinal force from dynamic models is the calculated longitudinal force, which is the result of ignoring ${\Omega }_{\mathrm{f}}^i$ when substituting the output of the sensor model into Equation (17) (which is the calculated value from using the sensor output and vehicle dynamic models that represent a typical way of obtaining the longitudinal force in real time onboard), and the estimated longitudinal force from the nonlinear observer is selected to test the effectiveness of the longitudinal force observation. The calculated difference is the deviation between the calculated results and the actual ones, the absolute difference between the linear speed and the actual speed, and the estimated difference is the deviation between the estimated results and the actual ones and the estimated error of the longitudinal force, which is the deviation of the estimated and actual values as a percentage of the actual values; these values are also taken into account and visualized to show the effectiveness of the proposed estimation algorithm. The experiment results of braking with a slip control strategy are shown in Figure 5 and Figure 6.

Figure 5 and Figure 6 above represent the experiment results of progressive braking and full braking with a slip control algorithm, respectively. From Figure 5a and Figure 6a, it can be observed that the ideal uniform motion would result in the accelerometer measuring a purely mean-zero measurement noise, thereby making the estimated vehicle speed essentially consistent with the drive wheel’s linear speed. However, the presence of the slip ratio during the actual vehicle operation would lead to a discrepancy between the drive wheel’s linear speed and the actual vehicle speed, resulting in a deviation between the estimated and actual vehicle speeds under ideal uniform motion conditions. Consequently, at the onset of braking, the estimated vehicle speed is almost coincident with the drive wheel’s linear speed. However, as braking commences and the braking intensity continually increases, the confidence in the measured vehicle acceleration rises while the confidence in the measured angular speed declines, and the estimated vehicle speed gradually converges with the actual vehicle speed. The estimated difference converges rapidly to zero since the braking commences and maintains a very low value throughout the entire braking process. The maximum value of the difference between the linear and actual speeds, 18.24 km/h and 22.16 km/h, appears in 5.488 s and 2.486 s in Figure 5c and Figure 6c, separately, and the estimated difference and estimated error values are 1.18 km/h, 2.08%, 2.57 km/h, and 3.86% at the same time, reduced by 93.53% and 87.82% compared with the former values, respectively. The calculated longitudinal forces in Figure 5b and Figure 6b are obtained by directly substituting the sensor signals with the noise into Equation (17). It can be discerned from Figure 5b,d and Figure 6b,d that the estimated longitudinal forces can effectively track the variations in the actual longitudinal forces, reducing the impact of the measurement noise on the estimation of longitudinal forces by approximately 65%.

The longitudinal force and vehicle speed estimating method proposed in this paper primarily serves the anti-lock braking control of tracked vehicles. However, in order to inspect the performance of the estimation method under extreme conditions such as wheel lockup, experiments were also conducted for emergency braking scenarios without anti-lock braking control strategies. The experiment results are shown in Figure 7.

From Figure 7a,c, it is evident that, although the drive wheel quickly locked up after the initiation of braking, which led to a significant deviation between the measured vehicle speed used in the filter and the actual vehicle speed, the estimated vehicle speed can still follow the actual vehicle speed with a relatively low difference due to the low confidence in the measured angular speed in the adaptive Kalman filter for the vehicle speed during emergency braking. The difference in the linear and actual speeds reached the maximum value of 68.68 km/h at 1.805 s, while the estimated difference and estimated error were still maintained at 3.71 km/h and 5.42%; thus, the difference was reduced by 94.59% compared to the former values. The performance of the longitudinal force observer is illustrated in Figure 7b,d. Prior to the drive wheel lockup, the performance of the longitudinal force observer was consistent with normal operating conditions. Once the drive wheel is locked up, there is no relative motion between the static and dynamic friction plates of the mechanical brakes; then, the braking torque is composed of the integral of static friction force between the braking plates, resulting in the absence of the linear relationship between the actual braking torque and the hydraulic cylinder thrust of the mechanical brake. At this point, the estimated error of the braking torque is $T_{\mathrm{b}\mathrm{m}}^i-F_{\mathrm{x}}^{i}r$, which makes the uncertainty caused by the estimated error in the dynamic relationship described by Equation (17) extremely large, leading to a significant deviation between the calculated longitudinal force and the actual value. However, the observer introduces compensation for the deviation between the observed angular speed and the measured angular speed, constraining the boundary of the observation error by the differential of the uncertainty ${\dot{\Omega }}_{\mathrm{f}}^i$. When the vehicle stays undisturbed during braking, the increase in the torque estimation error caused by wheel lockup is a constant; thus, the boundary of the observer’s observation error is not affected, theoretically, as shown in Figure 7e, where the observer can still operate stably most of the time. However, it is noteworthy that the torque estimation error is discontinuous at the moment of the drive wheel lockup, which could significantly impact the stability of the observer. Additionally, during actual driving, the vehicle is constantly subjected to disturbances from unbalanced loads on both sides, and these disturbances can severely compromise vehicle stability in high-speed braking and lockup, thus negatively affecting the performance of the observer.

To sum up, the longitudinal speed estimation method based on an adaptive Kalman filter is not only applicable to DDTVs equipped with anti-lock braking systems but also operates with considerable accuracy even in the event of drive wheel lockup. In the meantime, although drive wheel lockup adversely affects the performance of the longitudinal force observer, it remains functional for the majority of the time.

7. Conclusions

Accurate longitudinal speed and force acquisition is a critical foundation of controlling the electric drive tracked vehicle stably during harsh driving conditions. In order to improve the drive control effect and enhance the maneuverability, a novel estimating algorithm that only relies on onboard sensors was presented. This algorithm comprises a preprocessing module for sensor signals based on a standard Kalman filter, a nonlinear longitudinal force observer, and an adaptive Kalman filter for longitudinal vehicle speed. The adaptive Kalman filtering used for longitudinal vehicle speed estimation has been experimentally verified to be applicable not only to tracked vehicles equipped with anti-lock braking control but also to maintain relatively high accuracy in the event of drive wheel lockup. The observed values provided by the nonlinear longitudinal force observer can still remain accurate to the actual ones even when its performance is negatively impacted by drive wheel lockup. The effectiveness of the proposed algorithm has been tested through HIL experiments, which would provide assurance for the development of braking stability control strategies for high-speed dual-motor electric drive tracked vehicles. On this basis, the optimization algorithms of parameters related to adaptive algorithms, as well as the corresponding stability control algorithms for DDTVS and the associated experimental techniques, will be the focus of future research directions.