Calibration of Differential Wheeled Mobile Robot with Trajectory Measuring Device

1. Introduction

Wheeled mobile robots (WMRs) are currently widely used worldwide. WMRs are primarily used for autonomous movement on flat, structured surfaces in warehouses, industrial plants, logistics, industrial automation, and services. Of all WMRs types, the Differential Wheeled Mobile Robot (DWMR) is the most widely used. DWMRs move using independent drives on two wheels located on the same axis on either side of the robot’s chassis. The wheel drives are controlled independently, allowing the robot to change direction based on wheel speed without steering. This feature of DWMRs provides sufficient maneuverability and controllability, making them well-suited for precise movements indoors. DWMRs are designed with one or more caster wheels for stable movement [1].

One of the important challenges for DWMRs is improving localization accuracy. Localization plays a fundamental role in DWMR navigation, allowing them to determine their position and orientation in the environment [2]. To achieve precise localization, various methods of obtaining data from various sources and sensors are typically used. The most common method is odometry. Odometry is the simplest and most widely used method for estimating the relative movement of a DWMR and is based on measuring the angular velocity of its wheels. The main requirement for the correct application of odometry is no of wheel slippage. This assumes that wheel rotations can be converted into the linear movement of the DWMR relative to the surface. However, in real-world DWMR operating conditions, odometry has errors that accumulate over time. The main causes of this error are wheel slippage, uneven surface terrain, and design flaws in the DWMR manufacturing process.

The odometry method exhibits both systematic and non-systematic errors [3]. Systematic odometry errors are caused by manufacturing inaccuracies in the DWMR design, such as deviations from the nominal wheel dimensions, misalignment of the axles and chassis dimensions, or limitations in sensor capabilities, resolution, and sampling frequency. Non-systematic odometry errors occur during DWMR motion due to wheel slippage on low-friction surfaces. In this case, the wheels lose traction, causing uneven wheel rotation and resulting in odometry inaccuracies. These two types of odometry errors complicate precise robot localization, requiring odometry calibration.

Odometry calibration is necessary to determine and adjust the kinematic parameters of the DWMR motion model to improve localization accuracy. Accurate calibration of the odometry system minimizes systematic errors, resulting in more reliable estimates of the DWMR position and orientation. The paper [3] demonstrated that systematic errors, which have a greater impact on odometry and accumulate continuously, are typically the primary causes of positioning errors on smooth indoor surfaces, where DWMRs primarily operate.

Numerous papers are devoted to odometry calibration methods. Paper [4] provides an overview of existing odometry methods and their comparative analysis. Paper [5] presents the types of DWMR calibrations. Most of the works on odometry calibration are devoted to minimizing systematic errors, since they remain constant during movement. Systematic errors affecting the DWMR orientation have a cumulative effect, since, once they occur, they cause errors in the trajectory of the mobile robot [6,7]. Odometry calibration is carried out in two modes: offline and online. Offline odometry calibration uses a DWMR kinematic model and certain parameters for calibration [8,9]. Offline calibration allows for initial calibration to correct DWMR positioning errors [10,11,12,13,14,15]. The University of Michigan Benchmark (UMBmark) odometry accuracy method for calibrating DWMRs is presented in [3]. The UMBmark method involves experimenting with a bidirectional square DWMR trajectory, with clockwise and counterclockwise motion, to prevent odometry error compensation. The method calculates trajectory errors caused by wheelbase and wheel diameter errors. For DWMRs, wheelbase error causes a rotational error, which is corrected by adding a correction factor to the wheelbase. Wheel diameter error causes the robot’s trajectory to deviate from a straight path, which is corrected by adding a correction factor to the ratio of the two wheel diameters. In [16,17,18], the UMBmark odometrical accuracy method was used to calibrate the DWMR to correct for systematic positioning errors. In [19], an experimental method for calibrating the DWMR, as opposed to the UMBMark method, was proposed. This method is based on the kinematic model of the DWMR and determines errors in motion along a straight-line trajectory, unlike the UMBMark method, which calculates motion along a square trajectory.

Optical motion capture (OMC) is currently widely used for calibrating DWMRs in laboratory settings, representing one of the most accurate methods available [20]. OMC calibration yields high-precision ground-truth coordinates with sub-millimeter accuracy (up to ±0.03 mm). It facilitates both the calibration of kinematic parameters (correcting errors in wheel radius and wheelbase length) and extrinsic calibration (aligning the robot’s coordinate system with that of the OMC). However, OMC has certain drawbacks: high cost and the requirement for a specially equipped laboratory zone free of obstacles that could obstruct the cameras. Deploying OMC in an actual automotive factory presents significant challenges, such as line-of-sight obstructions (occlusions) caused by equipment, columns, and other robots, as well as glare from metallic surfaces and the floor.

This paper proposes an improvement to the linear path DWMR calibration method [19] by utilizing a new trajectory measurement device designed for application in real-world automotive factory environments.

2. Materials and Methods

2.1. DWMR Kinematics

According to Figure 1, the kinematic equations of DWMR look like [19]

$$\dot{x}\left(t\right)\cos \left(\theta \left(t\right)\right)+\dot{y}\left(\mathrm{t}\right)\sin \left(\theta \left(t\right)\right)=R\left({\dot{\theta }}_L\right(t)+{\dot{\theta }}_R(t\left)\right)/2$$

(1)

$$\dot{\theta }\left(t\right)=R\left({\dot{\theta }}_L\right(t)-{\dot{\theta }}_R(t\left)\right)/b$$

(2)

The third equation is related to the fact that the DWMR cannot move in the lateral direction, defined as

$$\dot{x}\left(\mathrm{t}\right)\sin \left(\theta \right(t\left)\right)-\dot{y}\cos \left(\theta \left(t\right)\right)=0$$

(3)

In Figure 1, the following notations are introduced: b is the nominal distance between the two driving wheels, R is the radius of the wheels, X and Y are the coordinates of the global coordinate system, x and y are the coordinates of the system attached to the body at point B, which has coordinates x_B, y_B relative to the global coordinate system. The angle θ is the orientation of the robot, here it is the angle between the global coordinate system and the attached coordinate system, ${\dot{\theta }}_R$ ,${\dot{\theta }}_L$ are the angular velocities of the right and left wheels, respectively, ρ_B is the polar radius of point B, φ_B is the polar angle of point B.

2.2. Calibration Method of DWMR with a Linear Path Test

The widely known paper [3] presents the UMBmark method for measuring the odometric accuracy of DWMR. The UMBmark method provides reliable and valid results for DWMR. The method is based on the fact that positioning errors are mainly caused by the unequal diameters of the two wheels and the uncertainty in the wheelbase length. A simpler calibration method using a linear path test and a small amount of computation for DWMR is presented in [19]. The proposed method uses the equation (4) for solving the inverse problem of DWMR and is aimed at eliminating “lateral” and “longitudinal” errors.

$$\left[\begin{array}{c}{\dot{\theta }}_L\left(t\right)\\ {\dot{\theta }}_R\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{2}{D_L}}\cos \left(\theta \right(t\left)\right)& {\displaystyle \frac{2}{D_L}}\sin \left(\theta \right(t\left)\right)& -{\displaystyle \frac{b}{D_L}}\\ {\displaystyle \frac{2}{D_R}}\cos \left(\theta \right(t\left)\right)& {\displaystyle \frac{2}{D_R}}\sin \left(\theta \right(t\left)\right)& {\displaystyle \frac{b}{D_R}}\end{array}\right]\left[\begin{array}{c}\dot{x\left(t\right)}\\ \dot{y}\left(t\right)\\ \dot{\theta }\left(t\right)\end{array}\right]$$

(4)

here D_R and D_L are the nominal diameters of the right and left wheels.

Figure 2 shows the target trajectory and its associated variables. In Figure 2, the robot is directed to move along a desired path (A), in Cartesian coordinates {XY}[19]. Due to systematic errors, regardless of their source, the robot follows the actual path (B) shown in Figure 2. The actual position is measured relative to the desired end point O. The angle λ is the orientation error of the mobile platform.

To reduce positioning error, the DWMR must be maintained on a desired trajectory (A) by making adjustments during movement. A lateral correction factor F_lat is introduced, which reflects the ratio of the angular velocities of the wheels and is determined by the following equation [19]

$$F_{\mathit{lat}}={\displaystyle \frac{D_L}{D_R}}+{\displaystyle \frac{4\mathit{l\lambda }}{D_R{\theta }_L}},$$

(5)

$$\lambda ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\tan }^{-1}\left({\displaystyle \frac{\delta y_i}{\delta x_i}}\right)$$

(6)

where n is the number of tests, and δx_i and δy_i are the longitudinal and lateral errors measured in each test, as shown in Figure 2.

Also, here a second coefficient is introduced, called the longitudinal correction coefficient F_lon, which is defined as follows [19]:

$$F_{\mathit{lon}}={\displaystyle \frac{L}{\sqrt{{(L-\delta x)}^2+{\left(\delta y\right)}^2}}}$$

(7)

where L is the length of the path,

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

(8)

The correction factors F_lat , F_lat are introduced into equation (4) and the following equation is obtained [19]

$$\left[\begin{array}{c}{\dot{\theta }}_L\\ {\dot{\theta }}_R\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{2F_{\mathit{lat}}{F}_{\mathit{lon}}}{D_L}}\cos \left(\theta \right)& {\displaystyle \frac{2F_{\mathit{lat}}{F}_{\mathit{lon}}}{D_L}}\sin \left(\theta \right)& -{\displaystyle \frac{{\mathit{lF}}_{\mathit{lat}}{F}_{\mathit{lon}}}{D_L}}\\ {\displaystyle \frac{2F_{\mathit{lon}}}{D_R}}\cos \left(\theta \right)& {\displaystyle \frac{2F_{\mathit{lon}}}{D_R}}\sin \left(\theta \right)& {\displaystyle \frac{l{F}_{\mathit{lon}}}{D_R}}\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]$$

(9)

This DWMR calibration method is based on kinematic equations and does not make any assumptions about the sources of robot motion error [19].

2.3. The Proposed Method

The advantage of the DWMR calibration method discussed above for correcting systematic errors along a linear path test is its simplicity, as it only calibrates a single linear motion, rather than the square path required by the UMBmark method. A disadvantage of this method is the need for constant manual trajectory measurement during numerous test experiments, which is time-consuming and introduces certain errors.

In this paper, we propose a calibration method with continuous measurement of the DWMR trajectory parameters. Let us write equation (4) in the form

$$\left[\begin{array}{c}{\dot{\theta }}_L\left(t\right)\\ {\dot{\theta }}_R\left(t\right)\end{array}\right]={\displaystyle \frac{1}{R}}\left[\begin{array}{ccc}\cos \left(\theta \right(t\left)\right)& \sin \left(\theta \right(t\left)\right)& -b\\ \cos \left(\theta \right(t\left)\right)& \sin \left(\theta \right(t\left)\right)& b\end{array}\right]\left[\begin{array}{c}\dot{x\left(t\right)}\\ \dot{y}\left(t\right)\\ \dot{\theta }\left(t\right)\end{array}\right]$$

(10)

Let us define the linear velocities of the wheels as $v_L\left(t\right)={\dot{\theta }}_L\left(t\right)R$, $v_R\left(t\right)={\dot{\theta }}_R\left(t\right)R$. Let us consider the movement of DWMR in a straight path, only forward with a constant speed v, then we have

$$v_L\left(t\right)=v_R\left(t\right)=v=\mathrm{const},\dot{\theta }\left(t\right)=0$$

(11)

Equations for rectilinear motion

$$\left\{\begin{array}{c}x\left(t\right)=x\left(0\right)+\mathit{vtcos}\left(\theta \left(t\right)\right)\\ y\left(t\right)=y\left(0\right)+\mathit{vtsin}\left(\theta \left(t\right)\right)\end{array}\right.$$

(12)

Let us differentiate equations (12)

$$\left\{\begin{array}{c}\dot{x}\left(t\right)=\mathit{vcos}\left(\theta \left(t\right)\right)-\mathit{vtsin}\left(\theta \left(t\right)\right)\dot{\theta }\left(t\right)\\ \dot{y}\left(t\right)=\mathit{vsin}\left(\theta \left(t\right)\right)+\mathit{vtcos}\left(\theta \left(t\right)\right)\dot{\theta }\left(t\right)\end{array}\right.$$

(13)

Figure 3 shows the trajectories of DWMR: the desired trajectory and the actual trajectory. As can be seen from the figure, the actual trajectory has longitudinal δx_i and transverse δy_i errors, and as a result of these errors, the robot orientation angle θ_i has an error of δθ_i = φ_Bi(t_d). Figure 3 shows the notations δx_i = L − x_Bi(t_d), δy_i = y_Bi(t_d) - longitudinal and lateral errors measured in each i- th test, as shown in Figure 3, L is the length of the desired trajectory of DWMR, t_d is the final time of the robot’s movement.

To measure the parameters of the DWMR trajectory, a Trajectory Measuring Device (TMD) was developed, consisting of two sensors: an absolute angular encoder BRT50-C2M 17 bit D400-RT1 [21] (see Figure 4a) and a rope displacement sensor BRT38-5m-C2M4096- RT1 [21] (see Figure 4b). The absolute angular encoder is designed to measure the rotation angle of the TMD and the maximum resolution of a single turn is 17 bits, and the accuracy can reach 0.07 degrees.

The rope displacement sensor is designed to measure the distance to the attachment point of the rope end on DWMR. The sensor is compact, easy to install, and easy to use. For linear positioning, the sensor features displacement feedback. Linear accuracy is ±0.1%, and repeat accuracy is ±0.01%. The maximum measurement length is 5000 mm. Accuracy verification of the BRT38-5m-C2M4096-RT1 sensor are reported in Appendix A,

TMD device is shown in Figure 5. The end of the rope is attached to the center between the DWMR wheels, as shown in Figure 6.

Using the TMD device, the trajectory of point B of the DWMR is determined in polar coordinates ρ_B = ρ_B(φ_B). Here ${{\rho }_B=\rho }_B\left(t\right)$ is the polar radius determined using the BRT38-5m-C2M4096-RT1 rope displacement sensor, φ_B=φ_B(t) is the polar angle determined using the BRT50-C2M 17-bit D400-RT1 IP54 absolute angular encoder.

In the Cartesian coordinate system, the position of point B (see Figure 1) is determined using the equations

$$\left\{\begin{array}{c}{x_B\left(t\right)=\rho }_B\left(t\right)\cos {\phi }_B\left(t\right)\\ {y_B\left(t\right)=\rho }_B\left(t\right)\sin {\phi }_B\left(t\right)\end{array}\right.$$

(14)

where 0 ≤ ρ_B(t) ≤ ∞, 0 ≤ φ_B(t) ≤ 2π , 0 ≤ t ≤ t_d.

The average longitudinal δx and lateral δy errors of the DWMR trajectory are calculated using the equations

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

(15)

where n is the number of robot trials.

The average final position error δr is calculated using the equation

$$\delta r=\sqrt{({\delta x)}^2+({\delta y)}^2}$$

(16)

To evaluate the DWMR calibration, we use the average error of its position at the end point of the trajectory δr. Since δr is the average error of the DWMR position at the end point of the trajectory relative to its desired position, the percentage improvement in the average final position error δr_mp is determined according to equation (17), comparing its values δr_BF before and after δr_AF calibration:

$${\delta r}_{\mathit{mp}}={\displaystyle \frac{{(\delta r}_{\mathit{BF}}-{\delta r}_{\mathit{AF}})}{{\delta r}_{\mathit{BF}}}}\times 100$$

(17)

2.4. DWMR Calibration Algorithm

The angle φ_Bi=φ_Bi(t) determined during the i-th test of DWMR movement in a straight line, according to Figure 3, is equal to the robot orientation angle θ_i(t)

$${\mathsf{\theta }}_i\left(t\right)={\mathsf{\phi }}_{\mathrm{Bi}}\left(t\right)\dot{{\theta }_i}\left(t\right)={\dot{\phi }}_{\mathit{Bi}}\left(t\right)$$

(18)

When the DWMR moves in a straight line, the orientation angle θ_i(t) is ideally 0. Due to systematic errors, regardless of the source, the DWMR follows the actual trajectory shown in Figure 3.

Angle δθ_i = φ_Bi(t_d) is the DWMR orientation error. The average error of the DWMR orientation angle δθ is calculated using the equation

$$\delta \theta ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\delta {\theta }_i$$

(19)

The experimental graph of the orientation angle θ = θ(t) corresponding to the average error δθ, shown in Figure 7, is a straight line that is described by the equation

θ(t) = kt

(20)

$$k={\displaystyle \frac{\delta \theta }{t_d}}$$

(21)

The derivative of θ(t) is equal to

$$\dot{\theta }\left(t\right)=k$$

(22)

We substitute (20) and (22) into equation (13) and determine $\dot{x}\left(t\right)$ and $\dot{y}\left(t\right)$, which we substitute into equation (10). Taking into account (20) and (2), we obtain the values of the angular velocities of the wheels of the mobile robot ${\dot{\theta }}_L\left(t\right)$ and ${\dot{\theta }}_R\left(t\right)$.

We determine the deviation of the angular velocities of the wheels as

$$∆={\displaystyle \frac{1}{2}}\mathit{max}\left|{\dot{\theta }}_L\left(t\right)-{\dot{\theta }}_R\left(t\right)\right|$$

(23)

We carry out calibration according to the following algorithm:

$$\mathrm{If}{\dot{\theta }}_L\left(t\right)>{\dot{\theta }}_R\left(t\right)\mathrm{then}{\dot{\theta }}_L\left(t\right)={\dot{\theta }}_L\left(t\right)-∆\mathrm{and}{\dot{\theta }}_R\left(t\right)={\dot{\theta }}_R\left(t\right)+∆\mathrm{else}{\dot{\theta }}_L\left(t\right)={\dot{\theta }}_L\left(t\right)-∆\mathrm{and}{\dot{\theta }}_R\left(t\right)={\dot{\theta }}_R\left(t\right)-∆.$$

(24)

3. Results

3.1. Experimental Testing of DWM

The DWMR MRTC prototype shown in Figure 8 was developed and manufactured at the Institute of Mechanics and Machine Science (Almaty). Specifications of the DWMR MRTC prototype: dimensions (L × W × H; cm) 115 × 80 × 39, weight 180 kg, wheel radius 8 cm, and wheelbase 73.5 cm. The DWMR MRTC features two drive wheels and four freely rotating wheels. The mobile robot is capable of autonomous navigation and route planning.

Experimental studies of the movement of the DWMR MRTC prototype were conducted at the production facility of the SaryArkaAvtoProm LLP automobile plant (Kostanay, Kazakhstan) (see Figure 9). Five experimental tests (i=5) of the DWMR MRTC prototype motion along a straight-line trajectory of length L=4000 mm was conducted at a speed of v=0.4 m/s. Table 1 shows the longitudinal δx_i, lateral δy_i and angular δθ_i errors of the DWMR MRTC prototype motion trajectory.

Average errors: longitudinal δx = 90 mm, lateral δy = 284 mm, and angular error δθ = 4.496 degrees of the DWMR MRTC prototype trajectory. The average final position error of the DWMR MRTC prototype is δr = 298 mm. Figure 10 shows the graphs of the angular δθ_i errors of the DWMR MRTC prototype trajectory for the i-th test. Figure 11 shows the graphs of the DWMR MRTC prototype trajectory for the i-th test.

The coefficient k is determined by formula (21), then k = 0.4496 deg/s and the deviation of the angular velocities of the wheels according to formula (23) Δ = 0.06 rad/s. The DWMR MRTC prototype calibration was performed using algorithm (24) and the following data were obtained: the DWMR MRTC prototype trajectory errors after calibration: longitudinal δx_C = 24 mm, lateral δy_C= 110 mm, and angular error δθ_C= 1.65 degree. The final position error of the DWMR MRTC prototype after calibration is δr_C= 112 mm. The graph of the angular error is shown in Figure 11, denoted by the letter C.

The percentage improvement in the average final position error δr_mp is determined using equation (17).

$${\delta r}_{\mathit{mp}}={\displaystyle \frac{(\delta r-{\delta r}_C)}{\delta r}}\times 100={\displaystyle \frac{(298-112)}{298}}\times 100=65.4\%$$

Let us carry out a calculation using the obtained experimental data (see Table 1) using the DWMR calibration method to correct systematic errors along a linear path test [19]. As a result of calculations using equations (5-7), we obtain the values F_lat = 1.02 ,F_lon = 1.024 , λ = 0.09 rad, which we substitute into equation (4) and determine the angular velocities of the wheels ${\dot{\theta }}_L\left(t\right)$ and ${\dot{\theta }}_R\left(t\right)$.

We performed a calibration by setting the given angular velocities on the MRTC DWMR and obtained the following data: trajectory errors after calibration: δx_F= 12 mm, lateral δy_F= 140 mm, and angular error δθ_F = 2.07 degree. The final position error of the MRTC DWMR after calibration is δr_F= 140 mm. The angular error graph is shown in Figure 10, denoted by the letter F.

The percentage improvement in the average final position error δr_mp is determined using equation (17).

$${\delta r}_{\mathit{mp}}={\displaystyle \frac{(\delta r-{\delta r}_F)}{\delta r}}\times 100={\displaystyle \frac{(298-140)}{298}}\times 100=53\%$$

4. Discussion

In this paper, we proposed a method for calibrating a DWMR using continuous measurement of its trajectory parameters. A trajectory measurement device (TMD) was developed to measure the DWMR trajectory parameters. Using a TMD during DWMR calibration significantly reduces time costs and eliminates the need for manual measurements to determine DWMR trajectory errors during multiple test experiments. Compared to OMC, the TMD is well-suited for DWMR calibration in an automotive plant environment—specifically where test areas do not require special preparation and where the use of OMC chambers is not feasible. Most importantly, the TMD is inexpensive and easy to use. The experiment was conducted using the DWMR MRTC4 prototype at the SaryArkaAvtoProm LLP automobile plant (Kostanay, Kazakhstan). Calibration using the developed method resulted in 65.4% reduction in odometry errors. The MRTC DWMR prototype was also calibrated using a well-known linear path systematic error correction method [19], which resulted in 53% reduction in odometry errors. Therefore, both calibration methods demonstrated relatively good results in reducing odometry errors. During the calibration of the MRTC DWMR prototype, a low odometry error recovery rate (65.4%) led to recommendations for improving the robot’s control system. Overall, this paper makes a significant contribution to the development of sensor systems and calibration methods for mobile robots with differential drives. The calibration method proposed in this paper has practical significance and can be widely applied due to its easy implementation for all types of mobile robots. Video materials and experimental data depicting experimental tests are presented in the [22].