Slipcompensated odometry for tracked vehicle on loose and weak slope
 Genki Yamauchi^{1}Email authorView ORCID ID profile,
 Keiji Nagatani^{2},
 Takeshi Hashimoto^{1} and
 Kenichi Fujino^{1}
Received: 10 August 2017
Accepted: 21 October 2017
Published: 2 November 2017
Abstract
Odometry is widely used to localize wheeled and tracked vehicles because of its simplicity and continuity. Odometric calculations integrate the wheel or track’s rotation speed. The accuracy of position thus calculated, is affected by slippage between the ground and the wheel or track. When traveling on a loose slope, the localization accuracy of the odometry decreases remarkably due to slippage. To improve its accuracy in such environments, terramechanics focus on estimating the interaction between a vehicle and the ground. However, because these formulas are complicated and governed by many terrainspecific parameters, they are difficult to use in unknown environments. In this study, we propose slip estimation methods targeted toward use in unknown environments. We consider four types of slippage, based on the slippage direction and maneuver type. Longitudinal and lateral slippage occurring during straight maneuvering are derived by approximating the terramechanics slip model. In contrast, for turning maneuvers, longitudinal slippage is derived from an empirical equation for the relationship between slip ratio and input velocity, and lateral slippage is obtained from a regression function. We also proposed slipcompensated odometry, which applies the slip model to the kinematics of a skidsteering vehicle. To evaluate the proposed slip model and slipcompensated odometry, we conduct several experiments with a skidsteering tracked vehicle on an indoor sandy slope. Experimental results confirmed that position estimation accuracy was improved by introducing slipcompensated odometry compared to conventional odometry.
Keywords
Introduction
Robotic volcano explorations aim at reducing the damage caused by volcanic eruptions, and have recently attracted considerable attention from research community. When a volcano erupts once, disaster may be caused by pyroclastic and debris flows [1]. Volcanic observation, therefore, assumes great importance from the viewpoint of reducing the risk of damage by issuing a warning to nearby inhabitants in the likely event of an eruption. However, the area within a few kilometers of a volcano crater is restricted after an eruption on account of measures taken to prevent secondary disasters, limiting the available information regarding such areas. Although fixed observation systems have been established on some active volcanoes, they are prone to severe damage by the shock caused by an eruption. This has given rise to the need of robotic explorations wherein robots traverse into the concerned area and visually survey the current state of active volcanoes.
Implementation of such robotic explorations requires high terrainability and good localization accuracy on part of the ground vehicles. Tracked vehicles are well suited to operations performed on such difficult terrains as volcanic fields. Accuracy of localization is also an important attribute of mobile robots in terms of not only mapping the target area but also controlling themselves. Odometry is a localization method in which a ground vehicle integrates the velocity of its wheel or track, and which guarantees positional continuity, in contrast with other position estimation methods, such as global navigation satellite systems (GNSS) and visual odometry. However, odometry may generate estimation errors owing to slip when the vehicle travels on an uneven terrain. Tracked vehicles typically employ skidsteering, which enables the right and left tracks to rotate at different speeds to change the vehicle’s orientation. Skidsteering tracked vehicles are, therefore, especially prone to slippage when turning.
To estimate this slippage, several studies have adopted terramechanics [2–5] and internal measurement unit (IMU)based [6–8] approaches. The terramechanicsbased approach considers the interaction between the wheel or the track and the terrain, with parametric values that are dependent on the terrain and the robot [9, 10]. The interaction represents the force acting on the robot and terrain, and the deformation of the terrain surface. Many soilspecific parameters, such as the pressuresinkage moduli [10], shear deformation modulus, and soil characteristics [11], are necessary when employing the terramechanicsbased approach. Generally, parameter determination requires largescale experimentation; therefore, this method is mainly used in wellknown environments.
Internal measurement unitbased approaches, on the other hand, efficiently utilize inertial information, such as angular velocity and acceleration. Theoretically, velocity and motion can be estimated by integrating only the inertial information. However, large errors in the estimated position occur due to noise, IMU bias, and limited sampling rates. Thus, many researchers employ IMU in combination with sensorsbased technologies, such as GNSS, encoders, and cameras [12–14]. For example, Endo et al. proposed an empirical formula for estimating the longitudinal slippage derived from the robot’s angular velocity, obtained by IMU, and rotational velocity, measured by the encoder [7]. This method can be applied to a skidsteering tracked vehicle during its rotation on rigid ground. However, this approach only targets longitudinal slippage during rotation on rigid ground and does not consider other slippage, such as the lateral slippage that occurs while moving straightly.

Longitudinal slippage during straight maneuvers: Longitudinal slippage is estimated based on the terramechanics force interaction model between the track and the terrain. We propose a simplified slippage model that uses approximation to reduce terraindependent parameters.

Lateral slippage during straight maneuvers: Lateral slippage is also estimated based on the terramechanics force interaction model between the track and the terrain. We propose an approximation slippage model with one terrain and robot dependent parameter.

Longitudinal slippage during turning maneuvers: Skidsteering slippage is a complex phenomenon, and it requires many parameters for estimating the terramechanicsbased slippage. As mentioned above, the empirical slip estimation formula [7] was confirmed for rigid flat ground. We verify whether the formula can be applied to loose and weak slopes.

Lateral slippage during turning maneuvers: To estimate lateral slippage while a robot is turning, regression analysis has been used along with training data obtained from the environment. The training data include inertial information and robot position.
Slipcompensated odometry including longitudinal and lateral slippage
Slip model for a skidsteering tracked vehicle
The kinematics defined in Eqs. 5–7 exploit the slip ratio and slip angle to calculate the longitudinal and lateral translational velocities. In this paper, we introduce four slip model types for estimating the slip ratio and slip angle, as mentioned in “Introduction” section.
For translational maneuvers, an approximated force model that considers resistance from terrain deformation has been introduced. Based on this force model, it is assumed that the resistances acting on each track are equivalent. Therefore, only the slip ratio and slip angle at the robot’s center of gravity need to be considered.
For turning maneuvers, the slip ratio is estimated by a previous method [7] that has been verified on a rigid plane; therefore, we confirmed its applicability on loose and weak slopes. On the other hand, a regression function is employed to estimate the slip angle. The regression function is trained by running data offline to identify the partial coefficients.
Slippage during straight maneuvering
Force model of the robot
Slip model for straight motion
Slip ratio for straight motion
Slip angle during straight motion
Slippage during turning maneuvers
The previous subsections describe the simplified slip ratio and slip angle models for steady straight motion based on force interactions on a weak terrain. In this section, we introduce a slippage estimation method for a robot rotating on a slope. Turning motions are complex phenomena that can cause longitudinal and lateral slippage. Furthermore, the slippage may differ at various points on the track. For instance, the lateral slippage at the front edge of the track is larger than that at its middle. The force model of turning motions is complex and the accurate prediction of its behavior requires several parameters. Therefore, it is difficult to use the force model for a real unknown field. In this research, we introduce a slippage estimation method based on a nonforce model during turning maneuvers.
Slip ratio during turning maneuvers
Slip angle during turning maneuvers
The lateral forces acting on the track during turning motion include not only the weight, as described for straight maneuvers, but also rotation resistance owing to the turning force, because a skidsteering track vehicle should slip to turn. The lateral forces consist of the bulldozing resistance acting on the sides of the tracks, shear resistance that rotates the robot and acts on the bottom of the tracks, and the weight of the robot acting parallel to the slope.
Verification experiments
Slip models for each direction and maneuver
Slip ratio  Slip angle  

Straight maneuver  \(\alpha = \frac{R_c}{C} + \frac{W}{C} \theta _{pitch}\)  \(\beta = \frac{W}{C'} \theta _{roll}\) 
Turning maneuver  \(\frac{\alpha _l}{\alpha _r} = sgn\left( v_r \cdot v_l \right) \left( \left \frac{v_r}{v_l} \right \right) ^n\)  \(\beta = \sum \nolimits _{i=0}^{7} a_i X_i\) 
To confirm these slip estimation methods, three experiments were conducted on a loose sandy field, as shown in Fig. 5. The first experiment was conducted to confirm the slip model for straight maneuvering. The robot traveled along a straight line on an inclined slope to the robot’s tilt roll and pitch angles independently. The second experiment was conducted to verify the conventional slip ratio estimation method [7] and the proposed slip angle estimation method during turning motion of the robot. The third experiment was conducted to evaluate slipcompensated odometry using the proposed slip model during turning motion of the robot.
Testbed specifications
Mass  25 kg 
Dimensions  503 mm × 686 mm × 522 mm 
Tread  393 mm 
Track width  150 mm 
Track length  600 mm 
Center of gravity height from the bottom of the track  150 mm 
Slip parameter n [7]  0.847 
Experiment I: Slip model verification on straight maneuvers
Experiment II: Slip model verification for turning maneuvers
This experiment was performed to confirm the method for estimating the slip ratio and slip angle during turning maneuver: whether conventional slip ratio estimation (Eq. 27) can be applied on the weak slope, and regression function for slip angle estimation (Eq. 29). In this experiment, the robot was set parallel to the long side of the field, and rotated through an angle of about \(90^{\circ }\) as shown in Fig. 12. The tilt angle of the field was varied: \(0^{\circ }, 5^{\circ }, 10^{\circ }\), and \(15^{\circ }\). The velocity of the robot was set to 5 cm/s, and the angular velocity was set at \(10^{\circ },/s 20^{\circ }\)/s, and \(30^{\circ }\)/s.
Result of regression analysis
Explanatory variable  Regression coefficient  p value 

Intercept (\(a_0\))  − 5.72  \(2.36 \times 10^{4}\) 
Input velocity (\(a_1\))  31.2  \(1.18\times 10^{4}\) 
Input angular velocity (\(a_2\))  344  \(1.42\times 10^{26}\) 
Gyro angular velocity (\(a_3\))  − 37.9  \(1.55\times 10^{5}\) 
Roll angle (\(a_4\))  − 127  \(6.13\times 10^{40}\) 
Pitch angle (\(a_5\))  82.7  \(1.07\times 10^{32}\) 
Yaw angular displacement (\(a_6\))  − 7.04  \(5.36\times 10^{16}\) 
Slope angle (\(a_7\))  0.321  0.0480 
Result of regression analysis
Number of training data  1283 
Coefficient of determination \(R^2\)  0.753 
Experiment III: Slipcompensated odometry for turning maneuvers
The coefficients of the slip estimation model were identified in the previous experiment. In this experiment, the slipcompensated odometry during turning maneuvers was evaluated using this slip estimation method verified in the previous experiments. The experimental setup was used as in previous experiments. The slope angle was set to \(8^{\circ }\) and \(15^{\circ }\). The velocity of the robot was set to 5 cm/s, and the angular velocity was set to \(20^{\circ }\)/s and \(30^{\circ }\)/s.
The estimated slip angle result at a slope angle of \(15^{\circ }\) is shown in Fig. 14. The vertical and horizontal axes represent the slip angle and running time, respectively. The blue dots indicate the ground truth of the slip angle obtained from the motion capture camera, and the red line indicates the estimated slip angle.
Discussion
The slip ratio during straight maneuvering had linear relationship with the slope angle near the origin. The line was identified by the slope, from \(0^{\circ }\) to \(20^{\circ }\) as shown in Fig. 9, based on the proposed slip model (Eq. 22) by the leastsquares method. The slip ratio at \(25^{\circ }\) deviatesd from the model compared to the other results because the proposed slip model was derived from secondorder approximation of shear stress near the origin; hence, the result far from the origin deviates away from the expected value. Shearstress approximation is also used in the slip angle model of straight maneuvers; thus, the error in the slip angle experiment was large at dots far from the origin. On the other hand, slip ratio during traversing remains almost the same at various slope angles. These results indicate that the slip ratio and slip angle can be estimated independently from the posture of the robot.
Slip parameter n in Eq. 13 was linear on the logarithmic graph despite the changing slope angle, and was identified as 0.873 in Experiment II. The value of n on the slope was similar to that on the rigid floor. These results indicate that the parameter does not change in this experiment using the same robot, despite environmental changes.
The slope angle is not an identical parameter because it can be calculated using Eq. 29. However, it is considered that the rotation behavior changes due to the slope angle and that it is important to include slope angle as an explanatory variable. In fact, the coefficient of determination \(R_2\) increased about 10% when including the slope angle as an explanatory variable.
Conclusion and future work
In this paper, we proposed a slip model for skidsteering tracked vehicles on loose and weak slopes. Estimation methods in the model were divided by slippage directions and motion types. During straight motion, the slippage model was based on the interaction between the track and terrain, and was linear to the roll and pitch angles of the robot. During turning motions, we verified that a previous empirical formula for slip ratio estimation on rigid ground can be useful for application to weak slopes. The slip angle was estimated using multiple linear regression function; the regression model explained 75.3% of the observed data. To include this slip model in the kinematics of a skidsteering vehicle, we proposed slipcompensated odometry for tracked vehicles, and confirmed that odometry accuracy was improved compared to that of conventional odometry.
In this research, slip model parameters were identified offline using position and velocity information acquired by a motion capture camera. To apply the slip model for odometry to an unknown environment, online parameters estimation should be considered in future work. In addition, the use of subtracks could prove effective in improving the terrainability of the vehicle over weak terrains. We intend to investigate the applicability of the proposed method to tracked vehicle with subtracks.
Declarations
Authors’ contributions
GY and KN conceived the ideas underlying this study. GY conducted experiments and data analysis and drafted this article as the corresponding author. KN, TH, and KF analyzed and revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This work was supported in part by a GrantinAid for JSPS Research Fellow (26\(\cdot\)3767).
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