Abstract
Wheeled robots on rough terrain are needed to effectively change wheel control strategies since optimal slip and maximum traction levels differ depending on soil types such as sandy soil, grassy soil or firm soil. In a view point of wheel control, this paper focuses on a prediction method of optimal control parameters such as optimal slip ratio and traction coefficient acting on wheels to maximize traction or minimize energy consumption. In this paper, optimal control parameter (OCP) models based on surface reaction index (SRI) are experimentally derived using characteristic data from wheel-soil interaction through indoor experiments by a testbed for analysis of wheel-soil interaction on three types of soil; grass, gravel and sand. For estimating surface reaction index (SRI), actual traction coefficient, including information of motion resistance, is observed by a state estimator which is constructed from longitudinal wheeled robot dynamics. The actual traction coefficient and slip ratio on wheels are employed to estimate surface reaction index (SRI) by a numerical method on the basis of derived optimal models. The proposed algorithm is verified through outdoor driving experiments of a wheeled robot on various types of soil.
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Keywords
- Optimal control parameter
- Maximum traction coefficient
- Optimal slip ratio
- Tractive efficiency
- Surface reaction index
- Vehicle dynamics
- State observer
- Soil identification
- Wheeled robot
- Rough terrain
1 Introduction
Outdoor wheeled robots have tried to overcome obstructions of moving on rough surfaces of terrains, in order to fulfill important tasks regarding the purpose of exploration, reconnaissance, rescue, etc. To reach such goals, wheeled robots should have abilities to deal with two kinds of characteristic changes on rough terrains; a change of soil types (slippery or non-slippery) and surface shapes (flat or steep). Both the terrain characteristic changes are crucial factors in the decision regarding optimal wheel slip or traction as a control parameter of a wheel controller since tractive force of a wheel is differently exerted on a surface according to such changes [1–6]. In case of changing surface shapes, it is relatively easy for wheeled robots to realize the level of the change by motion sensors like inertial measurement units (IMU). On the contrary to this, it is not such an easy undertaking to judge a type of soil where a robot is operated in spite of using various sensors mounted on a robot. To solve this issue, many researches related to soil identification have been introduced in the field of robotics.
The studies on soil identification based on proprioceptive sensor data, not including dynamic state information of a moving robot, have been proposed. As proprioceptive sensors, the vibration information of an accelerometer or IMU and the current information of wheel motors were used to make the data signals, which are transformed into soil feature data in frequency domain using a Fast Fourier Transform (FFT). The soil feature data were classified into one of pre-learned soil models by a support vector machine (SVM) [6, 7] or a probabilistic neural network (PNN) [8, 9]. The performance of identifying a soil type was verified through driving simulations or real driving experiments on rough terrains. However, these algorithms have physical limitations on real applications of wheeled robots. First of all, the vibration and current information is strongly influenced by a robot speed and also a surface shape. Therefore, although two robots move on the same type of soil, it might indicate the result of identifying one into another soil type depending on a robot speed and a surface shape.
With wheel-soil interaction models for planetary rovers on loose soils, the algorithms for soil identification and for optimal wheel control were proposed. In [6], the purpose of soil identification is to estimate the maximum traction through optimization of a traction force model, based on observed roverwheel torque and sinkage. And in [7], the purpose of soil identification is to estimate key soil parameters, cohesion c and internal friction angle \(\phi \) which can be used to compute maximum shear stress related to maximum traction of wheels. To identify distinct type of soil, in these researches, proprioceptive sensor data are needed to be measured or estimated, such as the vertical load, torque, wheel angular speed, wheel linear speed and sinkage. The algorithms were demonstrated using experimental data from a four-wheeled robot in an outdoor Mars-analogue environment. However, these methods cannot be utilized for some wheeled robots like military vehicles which are sometimes operated on hard surfaces such as grass or firm soil, where the sinkage does not occur because the force equations become zero. On loose soils, it is also not easy to be employed since it is difficult to precisely estimate sinkage by vision or distance sensors. To solve these problems, this paper proposes an algorithm to estimate optimal control parameters; maximum traction coefficient and optimal slip ratio on rough surfaces with various soil types from a hard surface through a loose surface, based on estimating surface reaction index (SRI) without estimating wheel sinkage.
2 Parameter Modeling for Optimal Control
2.1 SRI-Based Parameter Model
Brixius equation is well-known as one of empirical methods, which express tractive characteristics of bias-ply pneumatic tires on a variety of soil types in outdoor environments [11, 12]. To meet the purpose of this paper, conventional Brixius equation is changed into a function of wheel slip ratio S and surface reaction index (SRI) K which can be measured or estimated by on-board sensors in real-time, as shown in (3)–(6). In (1), slip ratio is a key state variable and it is expressed as a function of the linear velocity \(V_{x}\) [m/s] and the circumference velocity \(\omega R_{w}\) (m/s).
where \(R_{w}\) (m) is the wheel radius and \(\omega \) (rad/s) is the wheel angular velocity. Surface reaction index (SRI) K is also a crucial variable for soil identification. For movement of wheeled robots, driving surface should have an enough reaction that surface can endure contact force of wheels without considerable soil failure. Therefore, Surface reaction index (SRI) K means a degree that soil surface reacts to wheel movement (e.g. wheel torque). If surface reaction index (SRI) K is a high value, than a wheel can have high torque to go forward on the surface. Surface reaction index (SRI) K is actually estimated on a real-time system of a robot by an estimation algorithm for soil identification in this paper.
Figure 1 shows forces acting on a driving wheel during a wheel-terrain interaction by wheel torque T (Nm) and normal load W (N). In (2), drawbar pull \(F_{DP}\) (N) is expressed by difference of gross traction \(F_{GT}\) (N) and motion resistance \(F_{MR}\) (N).
Based on conventional Brixius equation, gross traction \(F_{GT}\) and motion resistance \(F_{MR}\) are as follows:
By (2), drawbar force \(F_{DP}\) is defined as:
where \(C_1\), \(C_2 \), \(C_3 \), \(C_4 \), \(C_5\), and \(C_6\) are Brixius constants and the values are determined by a nonlinear regression technique. Equation (5) is divided by normal load W as follows: (upper sign: \(S > 0\), lower sign: \(S< 0\))
Equation (6) represents traction-slip curves according to slip ratio S and surface reaction index (SRI) K.
2.2 OCP Model Derivation from Actual Soil-Wheel Interaction
For model derivation of optimal control parameters, indoor experiments to acquire force data (\(F_{DP}\), \(F_{GR}\), and \(F_{MR})\) in Fig. 1 were conducted on three types of soil: sand, gravel and grass where surface reaction index (SRI) K is different, as shown in Fig. 2. In the system of the testbed, the maximum angular velocity is 4.5 rad/s and the maximum linear velocity is 32 cm/s. Experimental slip conditions were controlled at 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6. From measured data of the testbed, Eqs. (3)–(6) can be completed based on surface reaction index (SRI) K of each soil type. Brixius constants in the equations are calculated by a nonlinear regression technique using a statistics program, SPSS as follows: \(C_{1}=1.3\), \(C_{2}=0.01\), \(C_{3}=7.058\), \(C_{4}=0.04\), \(C_{5}=-5\), \(C_{6}=4\). Surface reaction index (SRI) K are also given: 50 (sand), 80 (gravel) and 200 (grass), respectively.
Using the given Brixius constants (\(C_1\), \(C_2\), \(C_3\), \(C_4\), \(C_5\), and \(C_6\)) and surface reaction index (SRI) K, graphs of relation between wheel traction and slip were drawn about the four types of soil from (6), as shown in Fig. 3. Actually, a curve in between grass and gravel was not acquired from the indoor experiments. When watching the gap between the curves, it is possible to expect that there exists another soil type which is harder than gravel or softer than grass. The expected soil type (EST) seems to have surface reaction index (SRI) of \(K=120\). On all the curves, wheel traction is changed by increasing wheel slip. And wheel traction indicates that it has the maximum value at peak points on the curves having a particular slip ratio. In this paper, the point is named optimal slip ratio for maximum traction, \(S_{T}\). And \(S_{T}\) points can be calculated by partially differentiating the traction-slip equation (6) with respect to slip ratio S. Therefore the optimal slip model for maximum traction and also the maximum traction coefficient model are defined as functions of surface reaction index (SRI) K by (7) and (8), respectively.
In another case, Eqs. (3) and (5) can be employed for analysis of wheel tractive efficiency of (9). Equation (9) represents a degree of generated drawbar pull \(F_{DP}\) when gross traction \(F_{GT}\) acts on wheels. From Eqs. (3) and (5), the curves of tractive efficiency are described as shown in Fig. 4. All tractive efficiency on soil types increases rapidly until reaching peak points near 0.1 of the slip ratio and decreases dramatically after that. In this paper, the slip ratio is called optimal slip ratio for TE, \(S_{E}\) and it means that wheeled robots can minimize energy consumption if the robots keep wheel slip at \(S_{E}\) while moving on rough terrains.
To derive an optimal slip model for maximum TE, it is possible to partially differentiate the TE equation (9) with respect to slip ratio S. However, there is complexity for partial differentiation of (9) where the nonlinear equation (3) and (5) are included. For simplification, the \(S_{E}\) model is derived as a linear equation of surface reaction index (SRI) K from real peak points on each curve on the basis that the points of maximum TE moving on the curves at near 0.1 of slip ratios. Derived \(S_{E}\) model is as follows:
Consequently, Eqs. (7), (8) and (10) can be illustrated as OCP models depending on surface reaction index (SRI) K in figure. In a real application of wheeled robots, firstly, a wheeled robot estimates surface reaction index (SRI) K on driving surface in real-time. And then, the robot can predict optimal parameters to control wheels of the robot for efficient driving in a certain mission on rough terrains. As an example, Fig. 5 describes optimal values; maximum traction coefficient \(\mu _{T}\), optimal slip ratio for traction \(S_{T}\) and for TE \(S_{E}\) calculated from the optimal control parameter (OCP) models based on surface reaction index (SRI) \(K=120\). Derived OCP models include a wide range of soil types from a hard surface like asphalt through a loose surface like sand. Once surface reaction index (SRI) K is estimated in the range from zero to infinity, optimal control parameters are determined and used to optimally adjust wheel rotations according to the control purpose.
3 Proprioceptive SRI Estimation for Prediction of Optimal Control Parameters
In this section, a method for estimation of surface reaction index (SRI) K was suggested. As shown in Fig. 6, surface reaction index (SRI) K can be simply determined through observing actual traction coefficient \(\mu \) and slip ratio S on the traction-slip curve as shown in Fig. 3. The estimator of the actual traction coefficient is developed based on wheeled robot dynamic models. Actual slip ratios of wheels can be calculated by (1). Acquired real information of the traction coefficient and the slip ratio are employed to estimate surface reaction index (SRI) K on the traction-slip curve in Fig. 3 by a numerical method.
3.1 Estimation of Real Traction Coefficient
The real traction coefficient estimator developed in this paper, which does not cause a huge computational burden or require derivations of sensor signals, is based on a Kalman filter using wheeled robot dynamics. Specially, the real traction coefficient estimator was constructed in consideration with the effects of motion resistance by soil flow and by surface shape. Soil flow is generated by wheel movement and the amount of soil flow differs according to soil type in spite of the same wheel torque.
If wheel movement increases soil flow, then wheel could be obstructed to go forward. And surface shape on a rough terrain is geologically changed. By changing surface shape, wheel could be disturbed by gravity force in the reverse direction to the forward direction of wheel. Two effects of soil flow and surface shape cause the motion resistance of a wheeled robot. Therefore, the effects should be observed in order to only estimate pure soil characteristics without misestimating physical phenomenon of wheel-soil interaction. The real traction coefficient estimator includes the terms related to compensating these effects in motion equations.
The motion equation of the robot on the \(\mathrm{X}_{\mathrm{R}}\)-\(\mathrm{Y}_{\mathrm{R}}\)-\(\mathrm{Z}_{\mathrm{R}}\) robot coordinates described in Fig. 7 is
where \(\psi \) is the yaw rate; \(I_{z}\) represents the moment of inertia of the robot; a and b are the distances from the center of mass of the robot to the rear axle and the front axle, respectively. And \(M_{z,R}\) is the resistance moment about \(\mathrm{Z}_{\mathrm{R}}\)-axis and it is defined as:
where \(\mu _{y,R}\) is the lateral motion resistance coefficient on \(\mathrm{Y}_{\mathrm{R}}\)-axis and \(F_{z}\) is the normal forces on wheels. The subscript i indicates that 1 is the left-rear wheel, 2 is the left-front wheel, 3 is the right-front wheel and 4 is the right-rear wheel.
The motion equation for the wheel is as follows:
where T is the wheel torque; \(I_{\omega }\) is the moment of inertial of a wheel; \(F_{x}\) is the longitudinal traction on \(\mathrm{X}_{\mathrm{R}}\)-axis; \(F_{x,R}\) and \(mg_{xi}\) are the motion resistance on \(\mathrm{X}_{\mathrm{R}}\)-axis by soil flow and by surface shape, respectively, which are the terms for compensating the effects of soil flow and surface shape. The terms of (13) can be obtained as follows:
where \(\mu \) is the longitudinal traction coefficient on wheels and \(\mu _{x,R}\) is the motion resistance coefficient on \(\mathrm{X}_{\mathrm{R}}\)-axis. In (11)–(14), the normal force \(F_{z}\) is calculated by 3-dimensional normal force dynamics defined as:
where m is the robot mass; h is the height from the surface to the center of mass of the robot; c and d are the distances from the center of mass of the robot to the left wheels and the right wheels; \(\dot{V}_x \), \(\dot{V}_y\) and \(\dot{V}_z\) are the acceleration; \(g_{x}\), \(g_{y}\) and \(g_{z}\) are the gravity force on the \(\mathrm{X}_{\mathrm{R}}\)-\(\mathrm{Y}_{\mathrm{R}}\)-\(\mathrm{Z}_{\mathrm{R}}\) robot coordinates, respectively. The gravity force is defined by (20)
where \(\mathbf{R}_{\mathbf{x}}\) and \(\mathbf{R}_{\mathbf{y}}\) are the rotation matrices about \(\mathrm{X}_{\mathrm{G}}\) and \(\mathrm{Y}_{\mathrm{G}}\)-axis, \(\mathbf{G}_{{ G}}\) is the gravity force vector on the global coordinate system. From (16)–(19), the equations are transformed into a form of a matrix as follows:
where
The normal forces are calculated by (25) defined as:
From (11)–(15), the states for the Kalman filter are defined as follows:
where
The measurements are
where
Equations (11)–(15) and (26)–(29) are integrated to build the following state-space system with process noise \(\mathbf{w}(t)\) and measurement noise \(\mathbf{v}(t)\) as follows:
where \(\mathbf{A}(t)\), \(\mathbf{B}(t)\) and \(\mathbf{H}(t)\) are defined in (32)–(34), and their \(\mathbf{I}_{i\times k}\) and \(\mathbf{O}_{i\times k}\) denote an \(i\times k\) identity matrix and a zero matrix, respectively. Equation (30) is discretized using zero-order hold for being applicable to the discrete-time Kalman filter as follows:
The algorithm of the discrete-time Kalman filter is
where \(\mathbf{W}_{k}\) and \(\mathbf{V}_{k}\) represent the covariance matrices of \(\mathbf{w}(t)\) and \(\mathbf{v}(t)\). The estimator includes the motion equations for the wheeled robot, but the traction coefficients \(\mu _{i}\) are considered to be unknown parameters to be estimated. And also, the longitudinal motion resistance coefficients \(\mu _{x,Ri}\) are included in the estimator in order to observe the change of surface shapes and of soil types.
3.2 Numerical Method-Based Estimation of Surface Reaction Index
From derived actual traction coefficient and slip ratio, surface reaction index (SRI) K is simply estimated by a numerical method. The numerical update rule of surface reaction index (SRI) K is defined as:
where \(K_{n+1}\) is the updated value of surface reaction index; \(K_{n}\) is the previous value of surface reaction index; \(\lambda \) is the learning rate selected in the range between 1 and 0; \(\eta _{E}\) is the learning weight defined as:
where \(\mu _{ref}\) is the reference value derived by the estimator of real traction coefficient; \(\mu _{e}\) is the arbitrary value from (40) based on the traction-slip curve in (6); \(S_{a}\) is the actual slip ratio of a robot and E is the error model by (38). The reference value \(\mu _{ref}\) is integrated to consider actual tractive coefficient \(\mu \) with actual motion resistance \(\mu _{x,R}\) related to the change of a surface shape and a soil type in (39). The arbitrary value \(\mu _{e}\) is calculated from the derived traction-slip model by entering previous surface reaction index \(K_{n}\) and actual slip ratio \(S_{a}\) as shown in (40). As initial surface reaction index \(K_{0}\) is selected as 250, the algorithm is iteratively worked until the error E becomes under 0.1.
4 Experimental Verification of Soil Identification and OCP Estimation
For verifying the proposed algorithm, a wheeled robot was employed on five types of terrains; a sandy slope (15\(^\circ \)), a rough sandy soil, a gravel surface, a firm soil and a grassy surface as shown in Fig. 8. The robot size is 50 cm long, 40 cm wide and 30 cm high. The weight of the robot is 160 N and it can move at max speed 2 m/s. To implement the proposed algorithm, it is most important to estimate slip ratio between the linear velocity of the robot and the circumference velocity of the wheels. In this paper, additional wheel with an encoder was used to measure the forward velocity of the body. And the circumference velocity of the wheels was acquired from the motor encoder of wheels. Also, the 3-axis accelerations, the 3-axis angles (roll, pitch and yaw) and angular rates on the \(\mathrm{X}_{\mathrm{R}}\)-\(\mathrm{Y}_{\mathrm{R}}\)-\(\mathrm{Z}_{\mathrm{R}}\) robot coordinates are measured by IMU.
At first, the performance of the suggested algorithm was confirmed through the driving experiment at robot speed 0.5 m/s on the sandy slope in Fig. 8a containing the information of a surface shape. Figure 8 shows estimated normal forces of each wheel. The subscripts of \(F_{z}\) mean that rf is the right-front wheel, rr is the right-rear wheel, lr is the left-rear wheel and lf is the left-front wheel, respectively. In Fig. 9, after 2 s, the robot is faced with an uphill slope, and thereby the normal forces on the front wheels decrease and the normal forces on the rear wheels increase. And from about 7–10 s, the robot moves on a downhill sandy slope. By the effect of the slope, the wheel slip data display different tendencies on wheels each other. In Fig. 10, from about 2–7 s, the front wheel slips occur more than the rear wheel slips since the front wheels lose the normal forces by the change of surface shape. From Figs. 8 and 9, it can be confirmed how the changes of surface shapes influence the robot dynamic states.
Figures 11 and 12 show the estimated traction coefficients with or without compensating the motion resistance regarding the surface shapes on the sandy slope. Figure 11 represents values of a combined model between the traction coefficient \(\mu \) and the motion resistance coefficient \(\mu _{x,R}\). Figure 12 indicates only the traction coefficient \(\mu \). From the results of the estimated actual traction coefficient and actual slip ratio, surface reaction index (SRI) K on the sandy slope was estimated by the numerical method as shown in Figs. 13 and 14. The convergence time was average 0.01 s every samples. Figure 13 displays the flow of surface reaction index (SRI) K in the vicinity of the desired area of surface reaction index (SRI) K of sand in contrast with Fig. 14. In Fig. 14, the estimated K is gradually decreasing during the whole time. From these results in Figs. 13 and 14, it can be verified that the suggested algorithm improves the performance of soil identification in spite of the change of surface shape. Figure 15 describes the results of estimating optimal control parameters from the estimated K on the sandy slope. Actually, the pre-experimental data were placed on about \(K=50, \mu _{T}=0.4\), \(S_{T}=0.26\) and \(S_{E}=0.12\). In Fig. 15, it is considered that the outdoor experimental sandy surface had more moisture, in that time, than the indoor experimental sand surface though the estimated optimal control parameters indicates slightly higher values than the pre-experimental data.
As other driving experiments at robot speed 1 m/s on the four types of soil in Fig. 8b–e, Fig. 16 describes the results of estimating K depending on soil types. From 0 to 1 second, there are error values by the initial measurement errors of wheel slip since the slip ratio is very sensitive when the robot moves at low speed. After 1 s, surface reaction index of soils was estimated to almost suit the pre-experimental data according to soil types. From Figs. 17 through 19, they show the results of estimating optimal control parameters on each soil type (Fig. 18).
Additionally, there were outdoor experiments in order to verify robustness of the proposed algorithm about the change of robot speed. Speed of the robot was controlled to maintain a designated longitudinal velocity at 0.5, 1 and 1.5 m/s on sandy surface, gravel surface, grassy surface and asphalt surface, where surface reaction is different. Figures 20, 21, 22, 23 and 24 show estimation results of surface reaction index (SRI) K which represents a certain soil type. In Figs. 20, 21, 22, 23 and 24, the estimation result was divided into three regions according to surface reaction index (SRI) K; High (\(K: 250{-}170\)), Middle (\(K: 170{-}70\)), Low (\(K:70{-}30\)). That is the reason why it is relatively low cost to properly identify a soil type in a real application rather than to precisely classify a soil type. High region means that this indicates hard surface (like asphalt). Low region represents soft surface (like sand). And Middle region expresses intermediate characteristic of surface between asphalt and sand. On sandy surface, regardless the change of robot speed, estimated surface reaction index (SRI) K at each robot speed converged on the line of 50 in Low region, which represents sandy surface, after the longitudinal distance 0.5 m which is similar to the robot size (50\(\,\times \,\)40\(\,\times \,\)30) because of initial estimation error. Likewise, on gravel surface, the proposed algorithm estimated K to be nearly placed on the Middle region as intermediate characteristic of surface between High and Low region. Meanwhile, as hard surface, the wheeled robot resulted in estimated K which is mostly included in High region on gassy surface, firm soil and asphalt surface.
5 Conclusions
This paper proposed an algorithm for identifying a soil type and acquiring optimal control parameters, such as maximum traction coefficient and optimal slip ratio to maximize traction or minimize energy consumption, based on estimation of surface reaction index. In this paper, the optimal models for wheel traction and slip were derived through indoor experiments using the testbed for analysis of wheel-terrain interactions on three types of soil; grass, gravel and sand. For estimating surface reaction index, actual traction coefficient, including information of motion resistance by soil flow and surface shape, was observed by the DKF-based state estimator related to wheeled robot dynamics. The actual traction coefficient and slip ratio on wheels were employed to estimate surface reaction index by the numerical method on the basis on derived optimal models. The proposed algorithm was verified through real driving experiments of the wheeled robot on various types of soil. From the evaluation of the estimation results, it could confirm that the suggested algorithm shows enough performance to identify soil types and to predict optimal control parameters on rough terrains.
In future works, the proposed algorithm needs some effort to show a wide range of estimation performance in all directions for driving. Regarding robot steering, force data on the x-y direction depending on slip angle of tire should be measured or analyzed through similar experiments to Fig. 2. And the dynamic equations used in the proposed algorithm should be changed into 3-dimensional dynamic equations to be employed for the real application in 3-dimensional environments.
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Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2015R1D1A3A03020805).
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Kim, J., Lee, J. (2016). Soil Identification and Control-Parameter Estimation for Optimal Driving of Wheeled Robots on Rough Terrain. In: Filipe, J., Gusikhin, O., Madani, K., Sasiadek, J. (eds) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol 370. Springer, Cham. https://doi.org/10.1007/978-3-319-26453-0_8
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