The impact of load on the wheel rolling radius and slip in a small mobile platform
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Abstract
Automated guided vehicles are used in a variety of applications. Their major purpose is to replace humans in onerous, monotonous and sometimes dangerous operations. Such vehicles are controlled and navigated by applicationspecific software. In the case of vehicles used in multiple environments and operating conditions, such as the vehicles which are the subject of this study, a reasonable approach is required when selecting the navigation system. The vehicle may travel around an enclosed hall and around an open yard. The pavement surface may be smooth or uneven. Vehicle wheels should be flexible and facilitate the isolation and absorption of vibrations in order to reduce the effect of surface unevenness to the load. Another important factor affecting the operating conditions are changes to vehicle load resulting from the distribution of the load and the weight carried. Considering all of the factors previously mentioned, the vehicle’s navigation and control system is required to meet two opposing criteria. One of them is low price and simplicity, the other is ensuring the required accuracy when following the preset route. In the course of this study, a methodology was developed and tested which aims to obtain a satisfactory compromise between those two conflicting criteria. During the study a vehicle made in Technical University of Rzeszow was used. The results of the experimental research have been analysed. The results of the analysis provided a foundation for the development of a methodology leading to a reduction in navigation errors. Movement simulations for the proposed vehicle system demonstrated the potential for a significant reduction in the number of positioning errors.
Keywords
Automated guided vehicles (AGV) Odometry Positioning errors Wheel slip1 Introduction
The first AGVs were built in the 1950s and were used for inhouse transport by automotive sector plants and by wholesalers. Their present area of application is much wider and includes such applications as hospital logistics and healthcare. The correct operation of AGVs is dependent on the use of a suitable navigation system. This is particularly important in AGVs operating without fixed routes. Some routes are taskspecific and they change constantly. Among the basic navigation systems used in such applications is odometry. Odometry assumes the stability of certain values, such as the rolling radius, wheelbase, and a nonoccurrence of slippage. Under reallife conditions maintaining the aforementioned assumptions is often not possible. Therefore, odometry is prone to errors that are corrected with additional navigation systems or dedicated calculation algorithms.
The AGV postulated by the authors is an AGV designed for the transport of loads from warehouse buildings to trucks waiting outside. During each individual journey, the AGV may carry loads with a different overall weight. The distribution of each load may vary as well. All of the changes relating to the weight and distribution of the carried loads shall affect the working conditions of the wheels on the pavement surface. The specific operating conditions of the AGV, consisting of frequent changes in the working environment and its configuration, inhibit frequent corrections of positioning by fixed reference points and the use of additional systems, such as GPS or laser navigation. Under such variable operating conditions there is a need for the existence of route sections where odometry is the sole guidance system. After an analysis of the considerations mentioned above, the following solution was proposed. The first step shall be to use data from encoders connected to nondriven rear wheels in navigation. This solution eliminates errors due to wheel slippage which may occur while transferring braking or driving torque. The second step involved an investigation to determine the characteristics of the rolling radiuses under load. This relationship shall allow any necessary adjustments to be made to the calculation algorithm.
The paper is organised as follows. Section 2 lists the literature concerning navigation errors and the determination of the end position. The main causes of these errors and the methods used to alleviate them are described there too. Section 3 formulates the research problem, describes the research object and presents specific work conditions. Section 4 in the first part describes the behavior of the elastic wheel during movement. It presents such parameters related to the movement of the wheel as the rolling radius and a slip. Then it presents the movement equations and the odometry error analysis. In the final part of the section, a new positioning algorithm was proposed containing a block that corrects the effective radiuses of the wheels.
Section 5 presents a description of the experimental research concerning the AGV. A measuring system is presented and requirements concerning research methodology are formulated. Section 6 includes an analysis of the research results. Section 7 features simulation research of AGV movements, with an analysis of the results. Finally, Sect. 8 presents the conclusions which many be drawn from the results of the research.
2 Related work
Various navigation systems are used to guide and position AGVs. Such systems enable vehicle movement from the start point to the end point. Early solutions made use of vehicles which moved along a strictly defined route, set out by an induction, magnetic or optical loop. Due to the inherently low flexibility of such solutions (MartínezBarberá and HerreroPérez 2010) and an ever expanding area of applications, the majority of contemporary AGVs use a guidepath saved in the computer memory of the vehicle (Martinelli 2001). Such systems, based on calculation navigation and odometry, are often referred to as free ranging systems. In odometric systems, positioning is prone to errors. An extensive review with an analysis of such errors can be found in the following studies (Borenstein and Feng 1996; Chong and Kleeman 1997; Martinelli 2002a). There are various errors that affect positioning accuracy. They can be divided into systematic errors and nonsystematic errors. Systematic errors are the cumulative errors that are produced while determining the current position of the vehicle in motion (Borenstein and Feng 1996, Kelly 2004b), thus exacerbating the outcome. Depending on the pavement surface type, the share of systematic and nonsystematic errors in the positioning accuracy error may vary. Additional calculation errors in navigation may also be produced by the equations of odometry (Epton and Hoover 2012). Such equations describe any trajectory as a series of shorter segments. The accuracy of such an approximation is a function of sampling frequency and vehicle speed. Errors can be corrected using a number of methods. The early work in this area referred to systematic errors and described methods in which position adjustment and system calibration occur after the preset section of the route is covered and the end position error is determined. Among the early methods there is one (Borenstein and Feng 1996) which presents a dedicated calibration technique called UMBmark. This technique was the subject of ongoing modifications. One example is (Lee et al. 2011), where an additional new selection method of the test track for calibration was proposed.
The calibration of nonsystematic errors was the research area of authors (Chong and Kleeman 1997) and (Jung et al. 2014). Those studies used the model UMBmark, with further modifications implemented by the authors. The determination of the odometry errors was based on the determination of the AGV end position; this is also the area of interest in studies (Kelly 2004a) and (Martinelli 2002b). The second study uses observables which are measurable quantities related to a given robot motion. The study (Ravikumar et al. 2014) additionally undertakes the determination of AGV wheel load impact on the end position.
Developments in measuring techniques enabled online measurements and therefore the inmotion adjustment of position and calibration of the calculation algorithm. Various measuring techniques may be adopted, as reviewed briefly in the study (Smieszek et al. 2015). Presently, the most common are visual, laser and GPS measurements.
The first studies which refer to online adjustment described the use of ultrasonic sensors (Shoval and Borenstein 1999) and gyroscopes (Kelly 2004b). A further expansion of the range of measuring techniques enabled the adoption of the visual system calibration for error correction. The study (Antonelli et al. 2005) used a video camera erected above the research area. In the study (Chen and Jia 2014) there is an onboard camera, used to determine the AGV position against ceilingmounted lamps. A visual measuring technique was also used in the analysis of error buildup and a reduction in measurement noise. Descriptions of research studies on AGVs with cameras and inertia sensors are presented in (Knuth and Barooah 2013).
In the literature on online error correction, studies presenting laser rangefinders and scanners are well represented. One example of using a laser rangefinder is presented in the study (Epton and Hoover 2012). Measurement data enabled error correction calculations to be performed for the vehicle position originally determined by odometry. Also, 2D laser scanners are used to correct odometry errors, as described in studies (Nguyen et al. 2007; Zhao and Chen 2011). Those papers present new algorithms for the positioning (Nguyen et al. 2007) and the detection of lines and curves (Zhao and Chen 2011) in collections of measurement points. The majority of solutions employing additional measurement techniques use complex filtration methods. The study (Smieszek and Dobrzanska 2015) described the application of Kalman filters in the analysis of data from laser rangefinders. Filtered data enabled the precise determination of the vehicle position against the reference line and calibration of the navigation system. In the case of vehicles fitted out with two independently driven wheels, with incorrectly estimated values of the rolling radius, the actual movement pattern does not follow a straight line, but rather a curved line with a constant radius. The study (Dobrzanska et al. 2016) assessed the efficiency of various filter types in the determination of the reference line—a curve section. Another example of filter application is the study (Martinelli et al. 2007), assessing the efficiency of two filters the Augmented Kalman Filter and the Observable Filter. The same study also examines the SLAM problem. This problem was described in more detail in papers (Lee and Chung 2010) and (RodriguezLosada et al. 2006). The work discussed above assumed that both the wheel and the surface are rigid, that the contact surface is a point and that there is no slippage. Deviations from those assumptions were considered in the final adjustment of position and in the calibration of input data.
In recent years, papers have been published that account for actual movement conditions. Two groups may be isolated based on those studies. One of them deals with the method of slippage detection. The paper (Meng and Bischoff 2004) presents a slippage detection method consisting of a comparison of measurement data from the driven and nondriven wheel. The paper (Ojeda and Borenstein 2006) describes a slippage detection method employing values of route deviation angles determined by gyroscope. Studies (Palacin et al. 2006; Song et al. 2009) present optical methods used in the detection of wheel slip. They compare the circumferential speed of the wheel with the speed of vehicle determined using the optical method. Odometry errors due to slippage may also be determined by comparing data acquired from a group of robots working within a specific workplace. Examples of this are shown in papers (Glas et al. 2015) and (Li et al. 2016), discussing the slippage problem in detail. Slippage may be detected by models describing the kinematics and dynamics of movement (Antonelli and Chiaverini 2007; Gutirrez et al. 2002; Doh et al. 2006; Fernandez et al. 2014) and state observers (Song et al. 2007). The study (Iagnemma and Ward 2008) also accounted for the possibility of the AGV moving under various surface conditions and verifying the model with GPS technology. Papers (Konduri et al. 2014; Lindgren et al. 2002) determine the maximum driving torque transferable by a given wheel. Introducing this value to the guidance algorithm prevents sliding, thus reducing positioning errors.
The second group of studies related to the detection and determination of slippage are concerned with more complex tasks. They employ various models of friction and wheelsoft surface/wheelhard surface interactions. In the case of hard surfaces this problem is undertaken in the paper (Xiong et al. 2012) and in papers (Tian and Sarkar 2014; Zesu et al. 2012) employing the “magic formula” developed for cars. In the case of soft deformable surfaces, it uses various models describing surface deformation and wheelsurface interactions. Two areas of interest may be isolated within this group; the first one is allterrain vehicles, represented in papers (Lindgren et al. 2002; Wu et al. 2011; Ghotbi et al. 2016; Trojnacki and Dabek 2017). The other one is papers discussing the kinematics and dynamics of Moon and Mars rovers. Examples of such papers are (Xiao and Zhang 2016; Shirai and Ishigami 2015).
3 Object of research and its working conditions
3.1 Object of research

Mass of the vehicle ready to work with small batteries is 100 kg,

Maximum operating speed of 1 m/s,

Two DC motors were used to supply both the drive and steering,

The vehicle is able to carry loads of up to 200 kg,

Distance beetwen contact points of rear wheels 0.7 m,

Wheelbase 1.0 m,

Vehicle width 0.8 m,

Vehicle length 1.2 m.
The front wheel is driven and steered. It is not possible to ensure ideal conditions of wheelsurface contact in the working environment analysed. This refers in particular to the yard, where wheels may slip on coarse or dirty fragments during driving or braking. It was decided that measurement data from the nondriven rear wheels shall be used for positioning purposes. Those wheels were only transferring vertical load, and therefore their sliding due to the loss of traction when accelerating or decelerating could be excluded. In the case analysed, boxes containing goods were manually loaded onto the AGV in the warehouse. Their distribution on the surface of the AGV is often random, as shown in Fig. 1b. Such random distribution leads to a differentiation of the loads on individual wheels. There is a significant change in the load within a single work cycle, comprising AGV transfer from warehouse to truck and back. One half of the route is covered by the AGV with a load and the other half without a load. The cargo may not be exposed to excessive vibrations produced by the coarse pavement surface. The wheels provided suitable vibration damping properties but also demonstrated deformations. Initial experimental studies demonstrated a significant impact of the load on changes in wheel radius. Therefore, in the study presented, it was decided that the problem ought to be investigated more thoroughly, without going into too much detail concerning the physical phenomena occurring within the rubber band of the wheels. During the study, rolling radius characteristics under a load were determined. Computer simulations of motion, employing the above characteristics, demonstrated the potential for influencing the route and the final value of the positioning error.
3.2 The impact of load distribution on wheel load
The force balance along the Z axis gives
The above calculations, the results of which are shown in Fig. 3, were made for a static vehicle. In motion, vertical responses are shifted off the wheel contact centre. In variable and curvilinear motion, the vehicle is affected additionally by inertia forces, causing changes in individual wheel loads. Due to slow vehicle speeds and values of longitudinal accelerations and small shifts in reaction force off the wheel centre, additional changes in wheel load are small, and were disregarded in calculations. When analysing diagrams were obtained, it may be pointed out that depending on load position or weight on the vehicle surface, individual wheel loads may change to a large extent. Changes are most pronounced in Fig. 3a, where the effect of shifting fixed weight loads \(m_{L}\) along the axis y, i.e. from one side of the vehicle to another, is shown. In the scenario discussed, wheel load changes from 212 to 866 N i.e. over fourfold. The highest wheel load is observed in the scenario shown in Fig. 3b. Changes in wheel loads are high. Dedicated experimental research and computer simulations are required to explain their impact on vehicle motion and rolling radiuses.
4 Description of AGV motion
4.1 Radius and slip in a deformable wheel
In the range of small slip values s from \(\,0.025\) to 0.025 the curves for different vertical loads coincide. Such slip values occur under normal traffic conditions on hard surfaces without intense braking and acceleration. The graph shown in Fig. 4 is presented and described in many works (Genta 2012; Hirschberg et al. 2002).
4.2 Movement equations
 Various wheel radiuses due to production variations and deformations caused by variable wheel load. This error is referred to as radius error \(E_{\mathrm{r}}\):where \(r_{L}\) and \(r_{R}\) are current wheel radiuses.$$\begin{aligned} E_{\mathrm{r}}=\frac{r_{\mathrm{R}}}{r_{\mathrm{L}}} \end{aligned}$$(15)
 Wheelbase uncertainty—wheelbase is defined as the distance between contact points, in which there is no slip in the curvilinear motion of the two wheels of the moving vehicle and the surface. This error is referred to as wheelbase error \(E_{\mathrm{b}}\):where \(b_{a}\) is the current distance beetwen contact points of driven wheels, \(b_{n}\) is the normal distance beetwen contact points of driven wheels.$$\begin{aligned} E_{\mathrm{b}}=\frac{b_{\mathrm{a}}}{b_{\mathrm{n}}} \end{aligned}$$(16)
In the analysed case, the basic values used in the determination of the position of the AGV in odometry are wheel speeds \(v_{\mathrm{L}}\) and \(v_{\mathrm{R}}\). Such speed values are determined based on measurements of angular speed using encoders and rolling radiuses of wheels adopted for calculations. The values of rolling radiuses are not known accurately and they depend on wheel load and wheel slippage at the wheel surface contact area. Changes in the values of rolling radiuses may be the source of both systematic and nonsystematic errors.
Error \({\varDelta } O _{k}\) does not have a fixed value, it increases with distance covered. The error propagation principle applies here (Kelly 2004b). In the case of the current and the end position there are three positioning errors: the longitudinal distance error, going along the assumed trajectory \(e_x\), the error perpendicular to the assumed trajectory \(e_y\) and the orientation error \(e_{\theta }\).
In order to eliminate errors and improve the accuracy of position determination, an additional block was introduced. The additional block contains a formula that allows determining the effective radiuses of the wheels as a function of the load. This dependence was developed on the basis of experimental research described in the further part of the work. The current values of effective radiuses are entered into the calculation algorithm using the odometry equations.
5 Research and experiments
5.1 Scope and purpose of research
The theoretical estimation of the impact of load on a slip and rolling radius is a complex and difficult task. There are works (Genta 2012; Tian and Sarkar 2014; Zesu et al. 2012) where a “magic formula” is used to determine the wheel slip. This method makes it possible to determine such values as the rolling radius and a wheel slip on the basis of available literature data. However, the accuracy of determining these quantities based on literature data is insufficient from the point of view of the proposed calculation procedure shown in Fig. 7. Increasing the accuracy of the estimation requires additional experimental testing. In the considered range of slip changes \(s = \pm 2\)% Fig. 4, it can be seen that the relationship between the slip and load is linear. Taking this fact into account, it is possible to determine directly the dependence of the effective radius on the load without having to refer to a magic formula that requires the estimation of its additional characteristic parameters (Genta 2012). In connection with the above, the main objective of the experimental research described in the further part of this paper is the direct determination of the relationship between the load and the effective radius and the slippage of nondriven wheels. To achieve this goal, it was necessary to conduct a series of tests, collect results and their mathematical development. The dependencies obtained from the research can be directly applied in the calculating effective radiuses block Fig. 7 and concern only one type of wheel being the subject of the research.
The measurements were carried out in the corridor of rigid surface visible in the photo of Fig. 1b. During the tests the length of the distance \(S_{O}\) covered by the vehicle was measured. The value of the distance was measured by a laser range finder and ranged from 37 to 42 m.
Two stages of research can be isolated. Stage 1 was the development and verification of the measurement methodology. This stage consisted of conducting preliminary research aimed at the verification of the measurement system and determination of the requirements concerning vehicle motion. Stage 2 consisted of actual experimental research, during which the relationship between the effective radius and the vertical load was determined. The results of Stage 2 research are presented in Sect. 6.
5.2 Measurement methods
While determining the measurement methodology two variants of vehicle guidance were considered. One of them was manually hauling the AGV along the indicated line. In the other, the AGV was guided automatically along the preset trajectory, routed along the corridor axis. Irrespective of the vehicle guidance method (manual or automatic), it was difficult to obtain accurate rectilinear motion. The paper (Smieszek and Dobrzanska 2015) showed deviations from the assumed route in the automatic guidance scenario. In the automatic guidance scenario, with incorrectly assumed effective radiuses, the vehicle strayed from the route, as shown in Fig. 8a. Imperfections in the steering system caused, in turn, that the vehicle oscillated around the preset line defining the theoretical direction of travel.
In order to determine the effect of deviation from the target track on a measurement size, a series of computer simulations was run. The results are shown in Fig. 9. In cases from Fig. 9a, b it may be observed that the effect of the deviation on the relative error is quite large. To ensure accuracy the deviation had to be minimized. During manual hauling of the AGV along the set out line, the oscillations observed were much smaller than in the automatic guidance scenario. Such a guidance system was used in the main research.
6 Analysis of results
Using the results of experimental research, first the effect of load in changing the wheel radius of the static vehicle was determined. Later, tests were carried out with the vehicle in motion.
7 Simulation research
7.1 Description of the research
As part of the simulation research, it was decided to determine the benefits resulting from an application of the adopted method of determining the position on the basis of the scheme (Fig. 7) and to determine the impact of the dispersion of measurements made in experimental tests on the deviation from the running track set. During the simulation process many variants regarding the mass of transported cargo and its position on the platform were considered. Changing the size of the load mass and the position of its center of gravity influences the load on the wheels and thus their effective radiuses. Calculations carried out in Sect. 3.2, which took into account the impact of the mass of transported cargo and its position, showed that it was possible to change the load in the range from 200 to 1100 N. Using the regression equation from Fig. 11 and selected values of wheel loads from Fig. 3, the effective wheel radius for the considered cases was determined. The results of calculations are presented in Table 1.
Values of the effective radius determined for preset loads
Load F (N)  200  300  900  1100 

\(r_{eL}\) (m)  0.10080  0.10095  0.10185  0.10215 
\(r_{eR}\) (m)  0.10086  0.10100  0.10179  0.10205 
The results of calculations of effective radiuses for various loads presented in Table 1 were used in motion simulations where two cases of vehicle load were considered. The first of them assumed that the center of gravity of the transported load lies on the longitudinal symmetry axis of the vehicle \(x_v\) (Fig. 2). In this arrangement, the load on the rear wheels is symmetrical—the same. In the second case, the center of gravity of the transported cargo moved perpendicular to the axis of symmetry \(x_v\) along the y axis. In the final part of the chapter, simulations were carried out during which the influence of uncertainty was investigated—dispersion of experimental results on position error and trajectory of motion. This dispersion was determined based on a \(95\%\) confidence interval.
7.2 Symmetrical load of the rear wheels
In the first case of simulation calculations designed to determine the error of the final position as a result of changing the wheel load, it was assumed that the position of the center of gravity of the transported cargo moves along the longitudinal axis of the vehicle \(x_v\). The load on the right and left wheels is the same. Considering two extreme cases, i.e. an unloaded and fully loaded vehicle, the load of each wheel changes from 300 to 1100 N. Assuming that the effective radius was determined for a nonloaded truck, the maximum ratio of the unladen radius to the fully loaded wheel will be:
The final error is not big. With a symmetrical wheel load, changes in effective radiuses only affect the distance traveled along a straight line and on the curve. With smaller real effective radiuses, the vehicle will not overcome the assumed straight line and thus will move to the curve earlier. In the course of the curve motion, the angle of rotation equal to \(\pi /2\) will also not be achieved and the end section of the straight line will not be perpendicular to the initial direction of the motion. For error \(E_r=0.98\), the vehicle will cover 4.9 m instead of the 5 m straight section.
7.3 Nonsymmetrical load of the rear wheels
As shown by the results of the simulations presented in Figs. 13 and 14, the change in the position of the center of gravity of the transported load along the y axis perpendicular to the symmetry axis of the vehicle \(x_v\) generates large errors in the trajectory realized and when reaching the final position. This is a very unfavorable way of loading the load platform. At each loading process, it would be necessary to arrange the load so that its center of gravity lies on the axis of symmetry of the vehicle \(x_v\). In real conditions it is impossible. An improvement of accuracy in the implementation of the assumed route requires the use of the correction shown in the diagram (Fig. 7).
7.4 Errors after correction adjustment
An introduction of corrected values of rolling wheel radiuses to the calculation algorithm strongly limits the errors of the final position. The maximum \(e_x\) error from Fig. 14 reaching the value of 1.1m was reduced to 0.06 m after the correction was entered (Fig. 16). This error is very small in relation to the error in Fig. 14 and omitting it in the previously performed simulations has no major impact on the considerations conducted.
8 Conclusion
In a vehicle working under fixed, constant load, the correction of systematic errors only required the application of one of the methods described in papers (Borenstein and Feng 1996; Chong and Kleeman 1997; Jung et al. 2014). Such methods enable the determination of wheel effective radiuses upon the final correction of the position. In vehicles working under a variable load, designated for operation under conditions detailed in the article, the previously mentioned methods proved ineffective. Due to workplace configuration, the application of additional measuring techniques is inhibited. Studies and simulations demonstrated a significant impact of load on the effective radiuses of the wheels. After determining the relationship between the effective radius and the wheel load, and determining the vertical load force for the wheel, the determination of the actual value of the effective radius of the wheel shall be possible. Through the implementation of this value in the algorithm reduces the end position error and extends the length of the route which can be covered using the odometry guidance system. This is clearly visible when comparing Figs. 14 and 16. Under actual working conditions a degree of slippage is unavoidable, due to the variability of pavement surfaces. Implementing the corrections in the effective radiuses of wheels will not eliminate the need for additional measuring techniques in the final correction of AGV position. Such a correction may be, however, carried out at longer intervals and at more convenient points of the contemplated workplace operating area of the AGV.
Notes
References
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