1. Introduction

2. Model Analysis and Simulation

2.1. Kinematical Analogy of Skid-Steering with Differential Drive

Figure 1 shows the kinematics schematic of a skid-steering robot. We consider the following model assumptions:

(1)

the mass center of the robot is located at the geometric center of the body frame;

(2)

the two wheels of each side rotate at the same speed;

(3)

the robot is running on a firm ground surface, and four wheels are always in contact with the ground surface.

Figure 1. The kinematics schematic of skid-steering mobile robot.

Figure 1. The kinematics schematic of skid-steering mobile robot.

We define an inertial frame$\left(\text{X},\text{Y}\right)$ (global frame) and a local (robot body) frame$\left(\text{x},\text{y}\right)$, as shown inFigure 1. Suppose that the robot moves on a plane with a linear velocity expressed in the local frame as$v={\left(v_x,v_y,0\right)}^T$ and rotates with an angular velocity vector$\omega ={\left(0,0,{\omega }_z\right)}^T$. If$q={\left(X,Y,\theta \right)}^T$ is the state vector describing generalized coordinate of the robot (i.e., the COM position, X and Y, and the orientation$\theta$ of the local coordinate frame with respect to the inertial frame), then$\dot{q}={\left(\dot{X},\dot{Y},\dot{\theta }\right)}^T$ denotes the vector of generalized velocities. It is straightforward to calculate the relationship of the robot velocities in both frames as follows [6]:

$$\left[\begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{ccc}cos\theta & -sin\theta & 0\\ sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{v}_x\\ v_y\\ {\omega }_z\end{array}\right]$$

(1)

Let${\omega }_i$,i = 1,2,3,4 denote the wheel angular velocities for front-left, rear-left, front-right and rear-right wheels, respectively. From assumption (2), we have:

$$\begin{array}{c}{\omega }_L={\omega }_1={\omega }_2, {\omega }_R={\omega }_3={\omega }_4\end{array}$$

(2)

Then the direct kinematics on the plane can be stated as follows:

$$\left[\begin{array}{c}{v}_x\\ v_y\\ {\omega }_z\end{array}\right]=f\left[\begin{array}{c}{\omega }_{l}r\\ {\omega }_{r}r\end{array}\right]$$

(3)

where$v=\left(v_x,v_y\right)$ is the vehicle’s translational velocity with respect to its local frame, and${\omega }_z$ is its angular velocity,$r$ is the radius of the wheel.

When the mobile robot moves, we denote instantaneous centers of rotation (ICR) of the left-side tread, right-side tread, and the robot body as$IC{R}_l$,$IC{R}_r$ and$IC{R}_G$, respectively. It is known that$IC{R}_l$,$IC{R}_r$ and$IC{R}_G$ lie on a line parallel to thex-axis [7,16]. We define thex-y coordinates for$IC{R}_l$,$IC{R}_r$ and$IC{R}_G$ as$\left(x_l,y_l\right)$,$\left(x_r,y_r\right)$, and$\left(x_G,y_G\right)$, respectively.

Note that treads have the same angular velocity${\omega }_z$ as the robot body. We can get the geometrical relation:

$$y_G=\frac{v_x}{{\omega }_z}$$

(4)

$$y_l=\frac{v_x-{\omega }_{l}r}{{\omega }_z}$$

(5)

$$y_r=\frac{v_x-{\omega }_{r}r}{{\omega }_z}$$

(6)

$$x_G=x_l=x_r=-\frac{v_y}{{\omega }_z}$$

(7)

From Equations (4)–(7), the kinematics relation (3) can be represented as:

$$\left[\begin{array}{c}{v}_x\\ v_y\\ {\omega }_z\end{array}\right]=J_{\omega }\left[\begin{array}{c}{\omega }_{l}r\\ {\omega }_{r}r\end{array}\right]$$

(8)

Where the elements of matrix${\text{J}}_{\mathrm{\omega }}$ depend on the tread ICR coordinates:

$$J_{\omega }=\frac{1}{y_l-y_r}\left[\begin{array}{cc}-y_r& y_l\\ x_G& -x_G\\ -1& 1\end{array}\right]$$

(9)

If the mobile robot is symmetrical, we can get a symmetrical kinematics model (i.e., the ICRs lie symmetrically on thex-axis and$x_G$ = 0), so matrix$J_{\omega }$ can be written as the following form:

$$J_{\omega }=\frac{1}{2y_0}\left[\begin{array}{cc}{y}_0& y_0\\ 0& 0\\ -1& 1\end{array}\right]$$

(10)

where$y_0=y_l=-y_r$ is the instantaneous tread ICR value. Noted that$v_l={\omega }_{l}r, v_r={\omega }_{r}r$, for the symmetrical model, the following equations can be obtained:

$$\{\begin{array}{l}{v}_x=\frac{{\omega }_{l}r+{\omega }_{r}r}{2}=\frac{v_l+v_r}{2}\hfill \\ v_y=0\hfill \\ {\omega }_z=\frac{-{\omega }_{l}r+{\omega }_{r}r}{2y_0}=\frac{-v_l+v_r}{2y_0}\hfill \end{array}$$

(11)

Noted$v_y=0$, so that$v_G=v_x$. We can get the instantaneous radius of the path curvature:

$$R=\frac{v_G}{{\omega }_z}=\frac{v_G}{{\omega }_z}=\frac{v_l+v_r}{-v_l+v_r}{y}_0$$

(12)

A non-dimensional path curvature variable$\text{λ}$ is introduced as the ratio of sum and difference of left- and right-side’s wheel linear velocities [1], namely:

$$\lambda =\frac{v_l+v_r}{-v_l+v_r}$$

(13)

and we can rewrite Equation (12) as:

$$R=\frac{v_l+v_r}{-v_l+v_r}{y}_0=\lambda y_0$$

(14)

We use a similar index as in Mandow’s work [2,7], then an ICR coefficient$\text{χ}$ can be defined as:

$$\chi =\frac{y_l-y_r}{B}=\frac{2y_0}{B}, \chi \ge 1$$

(15)

where$B$ denotes the lateral wheel bases, as illustrated inFigure 1. The ICR coefficient$\chi$ is equal to 1 when no slippage occurs (ideal differential drive). Note that the locomotion system introduces a non-holonomic restriction in the motion plane because the non-square matrix$J_{\omega }$ has no inverse.

It is noted that the above expressions also present the kinematics for ideal wheeled differential drive vehicles, as illustrated inFigure 2. Therefore, for instantaneous motion, kinematic equivalences can be considered between skid-steering and ideal wheel vehicles. The difference between both traction schemes is that whereas the ICR values for single ideal wheels are constant and coincident with the ground contact points, tread ICR values are dynamics-dependent and always lie outside of the tread centerlines because of slippage, so we can know that less slippage results in that tread ICRs are closer to the vehicle.

Figure 2. Geometric equivalence between the wheeled skid-steering robot and the ideal differential drive robot.

Figure 2. Geometric equivalence between the wheeled skid-steering robot and the ideal differential drive robot.

The major consequence of the study above is that the effect of vehicle dynamics is introduced in the kinematics model. Although the model does not consider the direct forces, it provides an accurate model of the underlying dynamics using lump parameters:$IC{R}_l$ and$IC{R}_r$. Furthermore, from assumptions (1) and (3), we get a symmetrical kinematics model, and an ICR coefficient$\text{χ}$ from Equation (15) is defined to describe the model. The relationship between ICR coefficient and the vehicle motion path and velocity will be studied.

2.2. Dynamic Model for Kinematics Parameters Relationship

2.2.1. Skid-Steering Mobile Robot Dynamic Model

InSection 2.1, the effect of vehicle dynamics is introduced in the kinematics model. This section develops dynamic models of a skid-steering wheeled vehicle for the cases of 2D motion. Using the dynamic models, the relationship between the ICR coefficients and the path and velocity of the vehicle motion will be studied in a simulation.

In contrast to dynamic models described in terms of the velocity vector of the vehicle [4], the dynamic models here are described in terms of the angular velocity vector of the wheels. This is because the wheel velocities are actually commanded by the control system, so this model form is particularly beneficial for motion simulation.

Figure 3. Forces and moments acting on a wheeled skid-steering vehicle during a steady state turn.

Figure 3. Forces and moments acting on a wheeled skid-steering vehicle during a steady state turn.

As inFigure 3, following Wong’s model [16], the dynamic model is given by:

$$\{\begin{array}{l}{F}_{xfr}+F_{xrr}+F_{xfl}+F_{xrl}-R_x-m\frac{v_G^2}{R}sin\beta =0\hfill \\ F_{yfr}+F_{yrr}+F_{yfl}+F_{yrl}=m\frac{v_G^2}{R}cos\beta \hfill \\ M_d-M_r=0\hfill \end{array}$$

(16)

where$v_G$ is the vehicle velocity, and$\beta$ is the angle between the vehicle velocity andx-axis on the local frame.$F_{xfr}, F_{xrr}, F_{xfl}, F_{xrl}$ are the longitudinal (friction) forces and$F_{yfr}, F_{yrr}, F_{yfl}, F_{yrl}$ are the lateral forces.$R_{fr}, R_{rr}, R_{fl}, R_{rl}$ are external motion resistances on the four wheels.$M_d$ is the drive moment and$M_r$ is the resistance moment.

Based on the wheel-ground interaction theory [16,17], the shear stress${\tau }_{ss }$ and shear displacementj relationship can be described as:

$${\tau }_{ss }=p\mu \left(1-e^{-\frac{j}{K}}\right)$$

(17)

where$p$ is the normal pressure,$\mu$ is the coefficient of friction and$K$ is the shear deformation modulus.

Figure 4 depicts a skid-steering wheeled vehicle moving counterclockwise (CCW) at constant linear velocity$v$ and angular velocity$\dot{\varphi }$ in a circle centered at$O$ from position (1) to position (2). The four contact patches of the wheels with the ground are shadows inFigure 4.L andC are the patch-related distances. In the inertial X-Y frame, we define that$j_{fr}, j_{rr}, j_{fl}, j_{rl}$ and${\gamma }_{fr}, {\gamma }_{rr}, {\gamma }_{fl}, {\gamma }_{rl}$ are respectively the shear displacements and sliding velocity angle (opposite direction of sliding velocity). The readers can refer to Yu’s work [9] for a detailed analysis. The longitudinal sliding friction and lateral force of the four wheels can be expressed as follows:

$$\{\begin{array}{l}{F}_{xfr}={{\displaystyle \int }}_{C/2}^{L/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_r{\mu }_r\left(1-e^{-\frac{j_{fr}}{K_r}}\right)sin\left(\pi +{\gamma }_{fr}\right)d{x}_{r}d{y}_r\hfill \\ F_{yfr}={{\displaystyle \int }}_{C/2}^{L/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_r{\mu }_r\left(1-e^{-\frac{j_{fr}}{K_r}}\right)cos\left(\pi +{\gamma }_{fr}\right)d{x}_{r}d{y}_r\hfill \end{array},$$

(18)

$$\{\begin{array}{l}{F}_{xrr}={{\displaystyle \int }}_{-L/2}^{-C/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_r{\mu }_r\left(1-e^{-\frac{j_{rr}}{K_r}}\right)sin\left(\pi +{\gamma }_{rr}\right)d{x}_{r}d{y}_r\hfill \\ F_{yrr}={{\displaystyle \int }}_{-C/2}^{-L/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_r{\mu }_r\left(1-e^{-\frac{j_{rr}}{K_r}}\right)cos\left(\pi +{\gamma }_{rr}\right)d{x}_{r}d{y}_r\hfill \end{array},$$

(19)

$$\{\begin{array}{l}{F}_{xfl}={{\displaystyle \int }}_{C/2}^{L/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_l{\mu }_l\left(1-e^{-\frac{j_{fl}}{K_l}}\right)sin\left(\pi +{\gamma }_{fl}\right)d{x}_{l}d{y}_l\hfill \\ F_{yfl}={{\displaystyle \int }}_{C/2}^{L/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_l{\mu }_l\left(1-e^{-\frac{j_{fl}}{K_l}}\right)cos\left(\pi +{\gamma }_{fl}\right)d{x}_{l}d{y}_l\hfill \end{array},$$

(20)

$$\{\begin{array}{l}{F}_{xrl}={{\displaystyle \int }}_{-L/2}^{-C/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_l{\mu }_l\left(1-e^{-\frac{j_{rl}}{K_r}}\right)sin\left(\pi +{\gamma }_{rl}\right)d{x}_{l}d{y}_l\hfill \\ F_{yrl}={{\displaystyle \int }}_{-L/2}^{-C/2}{{\displaystyle \int }}_{-b/2}^{b/2}{p}_l{\mu }_l\left(1-e^{-\frac{j_{rl}}{K_r}}\right)cos\left(\pi +{\gamma }_{rl}\right)d{x}_{l}d{y}_l\hfill \end{array},$$

(21)

where$p_l$,${\mu }_l$ and$K_l$ are respectively the normal pressure, coefficient of friction, and shear deformation modulus of the left wheels, and$p_r$,${\mu }_r$ and$K_r$ are the ones of the right wheels, respectively. With the other parameters directly measured or given, such as mass of vehicle, m, patch-related distances,L andC, width of wheel,b,$p_l$ and$p_r$ can be determined by$mg/2\left(L-C\right)b$ when a uniform normal pressure distributions assumption used ($p_l=p_r$). We share some parameters:${\mu }_l, {\mu }_r$,$K_l$ and$K_r$ as in Yu’s research [9], because the same platform (Pioneer P3-AT robot) and similar lab surface are used for the simulation.

Figure 4. Motion of the skid-steering mobile robot wheel element on the lab surface (firm ground) from position (1) to (2).

Figure 4. Motion of the skid-steering mobile robot wheel element on the lab surface (firm ground) from position (1) to (2).

In Equation (16), the rolling resistance is denoted as${\mu }_{roll}$. We can obtain the resistance force, such that:

$$R_x=mg{\mu }_{roll}$$

(22)

2.2.2. Dynamic Simulation

This section describes the dynamic simulation that has been used to get the relationship between the treads’ ICRs and path. This model has been simulated for computing the treads’ ICR positions for the Pioneer P3-AT robot. We set all of the key parameters for the model as listed inTable 1.

Table 1. The parameters for Pioneer P3-AT robot and terrain dynamic model.

Table 1. The parameters for Pioneer P3-AT robot and terrain dynamic model.

Key Parameters	Symbol	Value
Mass of robot (kg)	m	31
Width of robot (m)	B	0.40
Length of robot (m)	L	0.31
Length of C (m)	C	0.24
Radius of tire (m)	R	0.11
Width of wheel (m)	b	0.05
Shear deformation modulus (m)	${\mathit{K}}_{\mathit{l}}\mathit{,}{\mathit{K}}_{\mathit{r}}$	0.00054
Coefficient of rolling resistance	${\mathit{\mu }}_{\mathit{r}\mathit{o}\mathit{l}\mathit{l}\mathit{,}\mathit{l}\mathit{a}\mathit{b}}$	0.0371
Coefficient of friction, of${\mathit{\mu }}_{\mathit{s}\mathit{a}}$	${\mathit{\mu }}_{\mathit{s}\mathit{a}}$	0.4437
Coefficient of friction, of${\mathit{\mu }}_{\mathit{o}\mathit{p}}$	${\mathit{\mu }}_{\mathit{o}\mathit{p}}$	0.3093

The values of the ICR coefficient$\text{χ}$ and nondimensional path curvature variable$\text{λ}$ are computed by solving the non-linear optimization problem with Equations (16)–(22):

$$\underset{\lambda ,\chi }{min}{\displaystyle \sum }_{i=1}^N\left[\Delta {F_{xi}}^2+\Delta {F_{yi}}^2\right]$$

(23)

where$i$ denotes theith of N simulated data. When the skid-steering wheeled vehicle is in a constant velocity circular motion, a set of different commanded turning radii is given by:

$$R=\lambda R_0\left(m\right), \lambda =0, 1, 2 ,\dots ,7$$

(24)

So we can use Equation (16) to get$\text{χ}$ with respect to a special$\lambda$. The simulated results of$\text{χ}$vs.$\lambda$ are shown inFigure 5.

Figure 5. The dynamic simulated results of$\chi$vs.$\lambda$ with Pioneer P3-AT robot using parameters inTable 1.

Figure 5. The dynamic simulated results of$\chi$vs.$\lambda$ with Pioneer P3-AT robot using parameters inTable 1.

InFigure 5, we can find that$\chi$ decreases as$\lambda$ increases, so there is a relationship between these two parameters. In this simulation, we must note that these results are obtained by assuming that the skid-steering mobile robot runs on the firm road and with special values of$K$,${\mu }_{sa}$, and${\mu }_{op}$. Because these terrain parameters are difficult to determine, an exact relationship needs further experimental identification.

3. Laser-Scanner-Based Experimental Kinematics Method

3.2. Laser-Scanner-Based Localization Method and Experiment Setup

The skid-steering mobile robot—a Pioneer P3-AT robot shown inFigure 6—is used for all testing in this research [20], the parameters for the Pioneer P3-AT robot are listed inTable 1 inSection 2.2.2. The P3-AT robot is driven by two motors on each side, and the two wheels of the same side are connected by one chain, so the two wheels of each side rotate at the same speed.

In all of the experiments the field is faced with a tile. Final drive shaft speeds of the motors on the robot are measured using two optical encoders. The optical encoders produce 2048 pulses per resolution, and the interface chip provides quadrature encoding, producing a change of 8192 counts for one revolution, or 0.0439° per count. A YL-100il wireless series port module, which can transmit data transparently, is selected as wireless transmission part.

Figure 6. Skid-steering mobile robot platform—a Pioneer P3-AT mobile robot.

Figure 6. Skid-steering mobile robot platform—a Pioneer P3-AT mobile robot.

A laser scanner-based localization method is used for the position and heading measurements of the robot. An overview of the proposed localization method and the robot control system are demonstrated inFigure 7. In order to localize the robot a thin plate (indicated as (1) inFigure 7) is mounted on top of the Pioneer P3-AT robot in the symmetric plane. A Sick LMS400-10000 laser scanner (3) is installed at the same height, the data of which are used to localize the robot. We use a laptop to control the Pioneer P3-AT robot (4) through a wireless serial port communication. The speed range of the robot is about 0–0.6 m/s.

Figure 7. An overview of the proposed localization method based on a Laser Scanner. (1) The thin plate; (2) Pioneer P3-AT robot; (3) Sick LMS400 Laser Scanner; (4) Laptop software panel.

Figure 7. An overview of the proposed localization method based on a Laser Scanner. (1) The thin plate; (2) Pioneer P3-AT robot; (3) Sick LMS400 Laser Scanner; (4) Laptop software panel.

The Sick LMS400-1000 laser scanner [21], has a large dynamic measurement range of 0.7 m to 3 m with 3 mm systematic error. The field of view of the laser scanner is 70° with$0.1°$angular resolution and 270 Hz–500 Hz scanning frequency. The measurement data from the laser scanner are sent to the laptop using an Ethernet connection. The data of the LMS400 regarding the plate are extracted by a clustering method. A line is fitted to the points. The line slope angle is equal to the heading angle, and coordinates of its center are equal to the coordinates of robot geometric center. A median filter, an edge filter and a mean filter are applied to the computed coordinates of robot to smooth the data as much as possible. Measurements of position and heading from the plate can be updated at 135 Hz after filtering. The calculations of position and heading from the plate are shown below.

Figure 8. Position and heading measurement from the plate based on LMS400-1000 Laser Scanner for a real trajectoryΩ of the robot’s center.

Figure 8. Position and heading measurement from the plate based on LMS400-1000 Laser Scanner for a real trajectoryΩ of the robot’s center.

InFigure 8, the laser scanner is at origin point O in the inertial frame$\left(\text{X},\text{Y}\right)$, the actual width of the thin plate is$L_{pa}=290.0 \text{mm}$, that is from point$P_{Start}$ to$P_{end}$ (the solid blue line) in the laser scanner image region. And$P_{mid}$ is the middle point of plate. The red dot dash line$\Omega$ is the planning trajectory of the robot’s center. The coordinates of$P_{mid}$ are the position of the robot’s center coordinates in inertial frame. The angle${\theta }_{pa}$ betweenX-axis and line$\overline{P_{Start}{P}_{end}}$, is the heading angle of the mobile robot. We can get point data from the laser scanner from polar coordinates to orthogonal coordinates:

$$P\bullet x=\rho cos\alpha , P\bullet y=\rho sin\alpha$$

(28)

where$\rho$ and$\alpha$ are the distance and corresponding angle of point$P$ measured by the laser scanner, respectively. The measured width of the thin plate$\widehat{L_{pa}}$ is:

$$\widehat{L_{pa}}=\sqrt{{(P_{end}\bullet x-P_{Start}\bullet x)}^2+{(P_{end}\bullet y-P_{Start}\bullet y)}^2}$$

(29)

All the data on the plate acquired by the laser scanner can be written in orthogonal coordinates:

$$P\bullet x_i={\rho }_i cos{\alpha }_i, P\bullet y_i={\rho }_i sin{\alpha }_i, i=1, 2,\dots ,n$$

(30)

The date can be fitted as a line, and a least-square method is applied to get the heading angle of the line (the same as the robot heading):

$$P\bullet y=kP\bullet x+a$$

(31)

$$P\bullet \overline{x}=\frac{{{\displaystyle \sum }}_{i=1}^{n}P\bullet x_i}{n}, P\bullet \overline{y}=\frac{{{\displaystyle \sum }}_{i=1}^{n}P\bullet y_i}{n}$$

(32)

$$\widehat{k}=\frac{{{\displaystyle \sum }}_{i=1}^{n}P\bullet x_i\bullet P\bullet y_i-nP\bullet \overline{x}\bullet P\bullet \overline{y}}{{{\displaystyle \sum }}_{i=1}^n{\left(P\bullet x_i\right)}^2-n{\left(P\bullet \overline{x}\right)}^2}$$

(33)

$$\widehat{a}=P\bullet \overline{y}-\widehat{k}P\bullet \overline{x}$$

(34)

The line slope angle is equal to the heading angle, considering that the field of view of the laser scanner is 70° ($0.389\pi$):

$$\widehat{{\theta }_{pa}}={tan}^{-1}\widehat{k}, \widehat{{\theta }_{pa}}\in \left[-0.194\pi , 0.194\pi \right]$$

(35)

so we can get the position and heading$q={\left(X,Y,\theta \right)}^T$ from the plate based on the laser scanner. The rotation angle$\varphi$ of the mobile robot during interval$t_{end}-t_0$ is:

$$\varphi =\widehat{{\theta }_{pa}}\left(t_{end}\right)-\widehat{{\theta }_{pa}}\left(t_0\right)$$

(36)

A set of experiments are executed to get the relationship between the ICR coefficient and the radius of path curvature and speed of the robot. On each path, the skid-steering wheeled vehicle is in constant velocity circular motion. The encoders equipped on the left and right motors give the left and right wheels’ moving distance, that is${{\displaystyle \int }}^{\text{}}{\omega }_{l}rdt$ and${{\displaystyle \int }}^{\text{}}{\omega }_{r}rdt$. The laser scanner measures the rotation angle change during entire cycle of motion. As inFigure 9, the robot moves clockwise from position (1) to (3), and the laser scanner get the distance data between the thin plate and the laser scanner center at a time interval of 3.7 ms. With Equations (15), (25) and (36), we can get the ICR coefficient$\text{χ}$.

Figure 9. The laser scanner measurement during an entire cycle$\Omega$ run from (1) start to (3) end with the mobile robot.

Figure 9. The laser scanner measurement during an entire cycle$\Omega$ run from (1) start to (3) end with the mobile robot.

3.4. Results

Data are collected in real time and processed off line with MATLAB.Figure 10a shows a comparison of the measured width of the plate from the laser scanner and the actual value. The measured ones, shown as a dot line red line, lies close enough to the actual value (a solid blue line). InFigure 10b, the mean difference between measured width and actual value is$\text{ shown}$. InTable 2, it shows that the mean value of measured width$\mu$ is 0.289 m and the standard deviation$\sigma$ is 0.003 m, and the maximum error is −0.01 m, corresponding to an extremely small ICR value estimatation error. The results demonstrate the effectiveness and the feasibility of the proposed laser scanner-based method.

Figure 10. (a) A comparison of the measured width of the plate from the laser scanner and the actual value; (b) Error between measured width and actual one.

Figure 10. (a) A comparison of the measured width of the plate from the laser scanner and the actual value; (b) Error between measured width and actual one.

Table 2. Mean value and errors of measured width of the plate from the laser scanner.

Table 2. Mean value and errors of measured width of the plate from the laser scanner.

Test Parameters	$\mathit{\mu }$ (m)	$\mathit{\sigma }$ (m)	Max. Error (m)
Width of the Plate (${\mathit{L}}_{\mathit{p}\mathit{a}}$)	0.2890	0.003	−0.01

Figure 11 shows the position and heading and velocity of robots calculated by the laser scanner-based localization algorithm during the experiment for speed of 0.25 m/s and curvature radius of 0.475 m.

Figure 11. Position and velocity of robots calculated by the proposed algorithm. (a) position and heading; (b) velocity.

Figure 11. Position and velocity of robots calculated by the proposed algorithm. (a) position and heading; (b) velocity.

Figure 12 depicts that the ICR coefficient$\text{χ}$ for various path curvatures and vehicle velocity. InTable 3, the mean values$\text{μ}$ and standard derivations$\text{σ}$ of the ICR coefficient$\text{χ}$ are shown. It shows that the maximum relative error is less than 1%, and the maximum$\text{σ}$ is no larger than 0.01, so the differences between different$\text{χ }$with respect to corresponding$\text{λ}$ can be distinguished, with this ICR coefficient measurement accuracy. Furthermore we can find thatFigure 12a represents the similar trends as inFigure 5. By increasing the path curvature (increasing$\text{λ}$), the ICR coefficient$\text{χ}$ decreases. This implies a specific behavior for$\text{χ}$ with respect to$\text{λ}$. Meanwhile, inFigure 12b, the ICR coefficient$\chi$ remains almost constant with increasing velocityv, at certain$\text{λ}$.

Figure 12. The calculated ICR coefficientχ for various: (a) path curvatures with$v=0.1, 0.2, \dots 0.5 m/s$; and (b) robot speeds with$v=0.1, 0.2, \dots 0.5 m/s$ with$\lambda =0, 1, 2, \dots , 7$.

Figure 12. The calculated ICR coefficientχ for various: (a) path curvatures with$v=0.1, 0.2, \dots 0.5 m/s$; and (b) robot speeds with$v=0.1, 0.2, \dots 0.5 m/s$ with$\lambda =0, 1, 2, \dots , 7$.

Table 3. The calculated ICR coefficient$\chi$ for various$\lambda$.

Table 3. The calculated ICR coefficient$\chi$ for various$\lambda$.

Parameters$\mathit{\lambda }$	$\mathit{\mu }$	$\mathit{\sigma }$	Max. Relative Error
0	1.4662	0.0033	0.34%
1	1.4480	0.0065	0.59%
2	1.4394	0.0055	0.62%
3	1.4341	0.0068	0.71%
4	1.4215	0.0080	−0.92%
5	1.4232	0.0050	0.52%
6	1.4165	0.0077	0.77%
7	1.4115	0.0043	0.30%

In order to reveal the relationship, an approximate function is then used to define$\chi$ as a function of the non-dimensional path curvature variable$\lambda$:

$$\chi \left(\lambda \right)=1+\frac{a}{1+b{\left|\lambda \right|}^{\frac{1}{2}}} , \lambda \in \left[0, 10\right]$$

(44)

wherea andb are determined by a curve fit of the experimental data,$a>0, b>0$. We run 10 sets of experiments with various$\lambda$ on the lab surface. Numerical values of parameters$a=0.4728,$$b=0.0538$ are obtained using a nonlinear least-square algorithm for the function given in Equation (44), and substituted as:

$$\chi \left(\lambda \right)=1+\frac{0.4728}{1+0.0538{\left|\lambda \right|}^{\frac{1}{2}}} ,\lambda \in \left[0, 10\right]$$

(45)

Figure 13 shows$\chi$ versus$\lambda$ using the approximating function and experimental data when$v=0.3 \text{m}/\text{s}$. The solutions obtained from the simulated and experimental methods are also summarized in this figure. From a qualitative standpoint, experimental results are coherent with those obtained in simulation. The simulation data are a little bigger than experimental results because of uncertain model parameters, e.g., the coefficient of rolling resistance, coefficient of friction, and shear modulus.

Figure 13. Experimental data curve, simulation data curve and data-fitted curve for$\chi$ with respect to$\lambda$, when$v=0.3 m/s$.

Figure 13. Experimental data curve, simulation data curve and data-fitted curve for$\chi$ with respect to$\lambda$, when$v=0.3 m/s$.

3.5. Dead-Reckoning Validation

To verify the developed kinematics model with the relationship between ICR coefficient$\chi$ and$\lambda$, an example path is estimated by two alternative kinematics models: the default P3-AT symmetric model with that$\chi$ is a constant value of 1.5, and the proposed approximating model with that$\chi$ changes with the nondimensional path curvature variable$\lambda$. The example path is different from the ones used in the former model experimental procedure, but it runs in the same environment.

During the dead-reckoning validation, with Equation (11), we have the default P3-AT kinematics model:

$$\{\begin{array}{l}{v}_x=\frac{v_l+v_r}{2}\hfill \\ v_y=0\hfill \\ {\omega }_z=\frac{-v_l+v_r}{2y_0}=\frac{-v_l+v_r}{2B\chi }\hfill \end{array} ,\chi \equiv 1.5$$

(46)

and with Equations (11) and (13), the proposed kinematics model is:

$$\{\begin{array}{l}{v}_x=\frac{v_l+v_r}{2}\hfill \\ v_y=0\hfill \\ {\omega }_z=\frac{-v_l+v_r}{2y_0}=\frac{-v_l+v_r}{2B\chi }\hfill \end{array},\chi \left(\lambda \right)=1+\frac{0.4728}{1+0.0538{\left|\lambda \right|}^{\frac{1}{2}}}$$

(47)

Consider that the center of the robot moves as the following velocity inputs during 6 s. The acceleration is below 3$\text{m}/{\text{s}}^2$, which is nearly the maximum acceleration for the P3-AT mobile robot and the velocity is less than 0.5$\text{m}/\text{s}$, which is nearly the effective maximum velocity of the robot.

$$\{\begin{array}{l}{v}_l=0, v_r=0, t=0;\\ v_l=0.2, v_r=0.15, 0<t\le 1;\\ v_l=0.2, v_r=0.1, 1<t\le 2;\\ v_l=0.3, v_r=0.1, 2<t\le 3;\\ v_l=0.4, v_r=0, 3<t\le 4;\\ v_l=0.3, v_r=0.2, 4<t\le 5;\\ v_l=0.3, v_r=0.25, 5<t\le 6;\\ v_l=0, v_r=0, t>6;\end{array}$$

(48)

Figure 14. Dead-reckoning performance using proposed model and P3-AT model for an example path with the mobile robot.

Figure 14. Dead-reckoning performance using proposed model and P3-AT model for an example path with the mobile robot.

Figure 14 shows the estimation of the path based on dead-reckoning (only drive shaft encoders are used) according to the default P3-AT model and the proposed model, and the LMS400 laser scanner data provide the actual position and heading. It can be seen that the proposal model achieves better dead reckoning estimation accuracy than that of the default P3-AT model.

Cartesian error and its norm are depicted inFigure 15. The proposed model has a smaller position error within 0.03 m, and a smaller angle error within 0.1 rad. Also, mean squared performance values obtained in the validation path are show inTable 4.

Figure 15. Dead-reckoning errors using the proposed model and the P3-AT model for an example path with the mobile robot. (a) Magnitude of position errors inX axis andY axis; (b) position magnitude errors and heading errors.

Figure 15. Dead-reckoning errors using the proposed model and the P3-AT model for an example path with the mobile robot. (a) Magnitude of position errors inX axis andY axis; (b) position magnitude errors and heading errors.

Table 4. Mean squared performance values of position and heading errors.

Table 4. Mean squared performance values of position and heading errors.

NO.	$\mathrm{∆}\mathit{x}$ (m)	$\mathrm{∆}\mathit{y}$ (m)	$\mathrm{∆}\Phi$ (rad)
P3-AT model	0.0169	0.0101	0.0438
Proposed model	0.0273	0.0119	0.0778

Meanwhile, estimated ICR coefficient$\text{χ}$ during the example path is presented inFigure 16. In fact, the ICR coefficient$\text{χ}$ varies as the path changes (it is not a constant value of 1.5).

Figure 16. ICR coefficient$\text{χ}$ using the proposed model and the P3-AT model for an example path with the mobile robot.

Figure 16. ICR coefficient$\text{χ}$ using the proposed model and the P3-AT model for an example path with the mobile robot.

4. Conclusions/Outlook

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