Position Control Considering Slip Motion of Tracked Vehicle Using Driving Force Distribution and Lateral Disturbance Suppression

Complex slippage and nonholonomic constraints can disturb the precise movement of tracked vehicles. In this study, a position control system using driving force control and virtual-turning velocity control (VTVC) was established for a tracked vehicle. Slippage in the translational direction can be suppressed by controlling the driving force of the crawlers, estimated using the vehicle velocity. However, the driving forces interfere with each other in a turning motion. Therefore, a driving force distribution method was developed using the instantaneous turning center of a vehicle to decouple the forces. This distribution induced slippage in the direction of turning. In addition, a virtual reference was derived from the vehicle velocity and lateral disturbance, which described the effects of the nonholonomic constraints and skidding of the tracked vehicle. Next, a VTVC method was developed to suppress the lateral disturbance by controlling the turning velocity to follow the virtual reference. The experimental results confirm that the proposed approach ensures high-performance position tracking and adequate slippage of the tracked vehicle.


I. INTRODUCTION
A tracked vehicle can traverse diverse terrains because its crawlers ensure ground contact while traveling [1]- [5]. However, two significant disturbances to tracked vehicles hinder precise movement: 1) complex slippage of the crawler on the ground and 2) nonholonomic constraints of vehicle motion. Therefore, motion control for tracked vehicles must reflect slippage and nonholonomic constraints to ensure precise position control.
Compared to wheeled automobiles, tracked vehicles slip more frequently and are difficult to model. Various simulators describe the motion mechanisms of tracked vehicles, including slippage [6], [7]. A model and Kalman filter [8] can be employed to estimate the slippage of tracked vehicles and automobiles in real time. A multisensor fusion of location data and inertial measurements [9] and image processing [10] can also be applied. Several methods, such as backstepping [11], sliding mode control [12], model predictive control [13], observer-based robust control [14], and trajectory planning [15], have been proposed to handle slippage. These methods ensure system robustness against slippage but do not suppress slippage. Slippage is related to the frictional force acting between the crawler and the ground surface [16], [17], and it corresponds to the crawler driving force. Hence, slippage can be suppressed by controlling the driving force. Although the driving force control of automobiles has been investigated [18]- [21], few studies have considered tracked vehicles. The driving force has been estimated using observers based on the axle velocity [21], neglecting the slippage effect. In addition, the driving forces of the crawlers interfere with each other while controlling tracked vehicles using driving force feedback. Therefore, the driving force distribution that decouples the driving forces must be considered. The decoupled driving forces provide the necessary moment for turning. However, slippage can be induced in the direction of turning. Although this slippage facilitates the lateral movement of the tracked vehicle, it interferes with motion control, such as positioning. In addition, nonholonomic constraints are established in conventional mobile robots to nullify commands in the lateral direction. Recently, the effectiveness of position control, including a compensator for the effects of nonholonomic constraints, has been demonstrated for a wheeled mobile robot [22]. For tracked vehicles, an improvement in position control can be expected by suppressing the disturbance caused by the nonholonomic constraints while allowing adequate slippage.
We have previously developed a method for estimating vehicle velocity, including slippage, using a disturbance observer (DOB) and machine learning [23]. Moreover, we have established a driving control system based on the estimated driving force [24]. When turning is performed while controlling the driving force, the interference between the driving forces of the crawlers prevents motion. Therefore, a driving force distribution that induces slippage in the turning direction by decoupling the driving forces is developed in this study. We designed the distribution based on the instantaneous center of the vehicle. In addition, we analyzed the lateral disturbance of the tracked vehicle, adopting the skidding and nonholonomic constraints as the equivalent lateral disturbance. The equivalent lateral disturbance is estimated using a workspace observer (WOB) [25], [26], coordinate transformation, and selection matrix. Moreover, a virtual-turning velocity reference was derived from the equivalent lateral disturbance and vehicle velocity. Hence, a virtual-turning velocity control (VTVC) was established to suppress the equivalent lateral disturbance by ensuring that the actual turning velocity followed the virtual-turning velocity reference. A position control system was constructed by integrating the position control, driving force distribution, and VTVC. The resulting control system ensures high-performance position tracking by suppressing the lateral disturbance while allowing the tracked vehicle to slip appropriately. The proposed control method was validated experimentally.
The contributions of this study are as follows. 1) The proposed method controls the driving force estimated from the vehicle velocity, including slippage, to suppress slippage in the translational direction. 2) The proposed driving force distribution method decouples the interference of the crawler driving forces, induces adequate slippage in the turning direction, and improves the turning performance. 3) The proposed position control improves the lateral positioning performance while allowing for slippage in the turning direction induced by the distribution.
The remainder of this paper is organized as follows. Section II presents details of the tracked vehicle model. Section III describes the estimation and distribution of the driving forces. In Section IV, the position control with driving force distribution and VTVC is introduced. Section V presents the experimental results for the turning performance based on the proposed driving force distribution and the effects of the VTVC. Finally, we present the conclusions in Section VI.    Fig. 1 shows the tracked vehicle model, and Table 1 lists the model parameters. In Fig. 1 (a), the origin of the vehicle coordinates is the center of the vehicle. The axes are aligned in the translational and lateral directions of the vehicle. Position vector x representing the position (x, y) and turning angle (φ) of the vehicle in the world coordinates is defined by x = x y φ T . The turning angle is defined

II. TRACKED VEHICLE MODEL
The kinematics of the tracked vehicle is expressed aṡ The kinematic relationship between the vehicle velocity and crawler velocity is where λ i is the slip ratio, which represents the slip between the crawler and the road surface or between the sprocket and the crawler.
The equations of motion of the vehicle are given by where m and J v are the mass and moment of inertia of the vehicle, respectively. When the vertical pressure at the crawler contact surface is constant, the driving force can be quasi-experimentally expressed as follows [16], [17]: where µ and f (λ i , k, L) are the dynamic friction coefficient and a function with the slip ratio λ i , shear displacement coefficient k, and crawler length L, respectively, as arguments. From Fig. 1 (b), the equation of motion of the crawler is where J m is the motor-shaft conversion moment of inertia. Thus, the tracked vehicle is moved by the driving force that simultaneously behaves as a disturbance and acts on the motor.

III. ESTIMATION AND DISTRIBUTION OF DRIVING FORCE A. TRANSLATIONAL VELOCITY ESTIMATION NEURAL NETWORK
The crawler velocity, including slippage, was considered to estimate the driving force. We used a translational velocity estimation neural network (TVNN) to estimate the vehicle translational velocity [24]. The TVNN architecture is shown in Fig. 2. In Fig. 2, the hat symbol above a variable denotes the estimated value, andτ dis r andτ dis l are the disturbances to the right and left crawler motors, respectively, as estimated by a DOB [27], [28]. The inputs of the TVNN, except the estimated disturbance, namely, the motor velocities and translational acceleration, can be measured using the motor encoders and an accelerometer, respectively. Generally, motor disturbances include the reaction force applied to the motor by the ground surface when a driving force is generated. The reaction force includes the driving force related to the slip ratio, as expressed by (8). Therefore, the estimated disturbance reflects the nonlinear and complex slip information and is used as an input for the TVNN. The TVNN is a two-layer feedforward neural network. The hyperbolic tangent sigmoid and linearized transfer functions were used as the activation functions of the hidden and output layers, respectively. The TVNN can be trained using supervised machine learning or other methods. The trained TVNN estimates the translational velocityv x . By substituting the translational velocityv x and turning velocityφ (measured using a gyroscope) into the inverse kinematics, the estimated crawler velocityv i is obtained asv Here, the crawler velocity in the translational direction was estimated, considering the slippage in the direction of turning.

B. DRIVING FORCE OBSERVER
A driving force observer (DFOB) [23] was designed to estimate the driving force based on the estimated crawler velocity. A block diagram of the DFOB is presented in Fig. 3. The driving force estimated by the DFOB is expressed aŝ where J mn and g DF are the nominal motor-shaft conversion moment of inertia J m and DFOB cutoff frequency, respectively. The DFOB provides the driving force as a lowpass filtered value of the driving force calculated using (11). The driving force required for motion can be generated via feedback control of the estimated driving force, and slippage in the translational direction of the crawler can be suppressed.

C. DRIVING FORCE DISTRIBUTION
When controlling the vehicle turning and driving force simultaneously, generating the moment necessary for turning is difficult because the right and left driving forces interfere with each other. Thus, we distribute the driving force based on the inverse kinematics solution using a weight matrix. For a typical two-wheeled mobile robot, Jacobian matrix J , which establishes the relationship between the motor − − FIGURE 3. Block diagram of the DFOB. VOLUME 4, 2016 velocities (θ r ,θ l ) and vehicle velocities (v x ,φ), transforms into the following square matrix: By time-differentiating both sides of (13), the relationship between the motor accelerations (θ r ,θ l ) and vehicle accelerations (v x ,φ) can be derived. When the total driving force is calculated using the vehicle accelerations, the inverse Jacobian matrix J −1 can be used to evenly distribute the driving forces between the crawlers. However, the distribution causes interference from crawler driving forces. A weight matrix was introduced to design the distribution, as follows: Next, the driving force can be represented by the weight matrix and inverse Jacobian matrix, as follows: From (15), the weight matrix elements, w r and w l , are equivalent to the virtual inertias of the right and left crawlers, respectively. Therefore, by designing the weight matrix, the driving force reference of the crawler can be determined, and the force is distributed from the acceleration reference generated by the position controller. The weight matrix design involves the instantaneous turning center of the vehicle, as shown in Fig. 4. A line parallel to the Y v -axis, drawn on the tracked vehicle from the center of rotation, is defined as a nonslip line [29]. The instantaneous turning center is defined as a point on the nonslip line, and the velocity along the X v -axis at this point is the average of the crawler velocities. The position of the instantaneous turning  center of the vehicle coordinates The instantaneous turning center is handled as a virtual control point without slippage in the turning direction. In other words, in an actual situation, slippage in the turning direction is induced by controlling at this point. The equation of motion for the rotational direction at the instantaneous turning center can be expressed as From (5), (18), and (19), the distributed driving force necessary for motion is expressed as From (20), if the instantaneous turning center does not deviate from the vehicle center along the lateral direction ( v y ic = 0), the driving force should be equally distributed among the crawlers. Otherwise ( v y ic = 0), the deviation should be represented by the weight of the crawler driving forces. Thus, we designed the elements of the weight matrix as follows: The second term on the right-hand side of (20) indicates the disturbance in the world coordinates, compensated by the WOB, as described in Section IV.

A. CONTROL SCHEME
In this study, the position control system of a tracked vehicle was designed to achieve the intended motion and ensure robustness against disturbances. The proposed position control system is shown in Fig. 5  S, and T are the transformation matrices from Σ w to Σ v and from Σ v to Σ w , the selection matrix, and the command conversion matrix, respectively. The control system comprises a position controller C p (s), driving force controller C f (s), and virtual-turning velocity controller C v (s). The control system controls the position response (x res ) in tracking command (x cmd ). A DOB and WOB were adopted to construct the position control system with robustness against disturbances to the tracked vehicle. The WOB was also used to estimate the lateral disturbance, equivalent to the effects of skidding and nonholonomic constraints. We define this disturbance as the equivalent lateral disturbance. Based on the relationship between the equivalent lateral disturbance and the translational velocity, a virtual-turning velocity reference was determined to suppress the disturbance.

B. POSITION CONTROL
The position controller in the world coordinates was designed as follows: whereẍ ref p ,ẍ ref , andÿ ref represent the acceleration reference vector provided by the position controller and its X w and Y w direction components, respectively. The symbol, * , is an unused variable. The X w -and Y w -direction components are controlled by the position controller. The turning angle in the acceleration reference vector is controlled by the VTVC described in Section IV-C. The position controller C p (s) was designed using proportional-derivative control, as follows: where K pp and K pd are the proportional and derivative gains for the position, respectively.

C. VTVC BASED ON EQUIVALENT LATERAL DISTURBANCE
We compensated for the disturbances to the crawler motor, owing to the driving force and modeling errors using the DOB. Furthermore, we used the WOB to estimate disturbances to the vehicle in world coordinates (workspace). Fig. 6 shows a block diagram of the WOB. In Fig. 6, g W indicates the WOB cutoff frequency. The WOB estimates the disturbance in the acceleration dimension (ẍ dis = wẍdis wÿdis 0 T ) to the vehicle from the acceleration reference (ẍ ref ) and the velocity response (ẋ res ) in world coordinates. Among the estimated disturbances, disturbance compensation in the turning direction was not performed, given the interference with the VTVC. By using the transformation matrix w R v and selection matrix S, the disturbance estimated by the WOB is decomposed into translational disturbance vẍdis tr and lateral disturbance vẍdis la , as follows: The translational disturbance vẍdis tr can be suppressed by feeding them back to the acceleration reference in the world coordinates. The lateral disturbance vẍdis la is equivalent to the effects of lateral slippage and nonholonomic constraints. In this study, the lateral disturbance was defined as the equivalent lateral disturbance. The direct feedback of the equivalent lateral disturbance vẍdis la does not suppress it because of nonholonomic constraints. Therefore, we propose VTVC to suppress equivalent lateral disturbances.
In the VTVC, the equivalent lateral disturbance is indirectly suppressed by ensuring that the turning velocity follows the virtual-turning velocity reference, defined using VOLUME 4, 2016 the equivalent lateral disturbance and vehicle velocity. The vehicle should turn to generate a velocity opposite the direction of the disturbance to indirectly compensate for the equivalent lateral disturbance. For a sufficiently small slip angle φ , the following equation can be derived by transforming the equation of motion expressed by (6) using the turning velocity and slipping angular velocity: Therefore, the velocity reference to compensate for the equivalent lateral disturbance is defined as the virtual-turning velocity referenceφ ref vt as follows: In practice, oscillatory variation should be considered because of the division between velocity and acceleration. Therefore, the experimentally determined virtual-turning velocity gain K vt and thresholdφ lim vt are expressed bẏ The velocity controller is then designed as follows: whereẍ ref v andφ ref are the acceleration reference vector generated by the velocity controller and its turning direction component, respectively. The velocity controller C v (s) was designed using proportional control, as follows: where K vp is a scalar parameter representing the proportional gain in the velocity. The equivalent lateral disturbance is indirectly compensated for by ensuring that the turning velocity of the tracked vehicle follows the virtual-turning velocity reference.

D. DRIVING FORCE CONTROL
The acceleration reference vector in the world coordinates is obtained using the outputs of the position and velocity controllers and the translational direction disturbance, as follows:ẍ The acceleration reference in (34) is transformed into the vehicle coordinatesv ref using the command conversion matrix T . From this acceleration reference, the driving force reference F ref dr for each crawler is obtained using the distribution method described in Section III. The force controller was designed to feed back the driving force estimated by the DFOB and track the force reference, as follows: The force controller C f (s) was designed using proportionalintegral control, as follows: where K f p and K f i are the proportional and integral gains in the driving force, respectively.

A. EXPERIMENTAL CONDITIONS
We present the results of three experiments conducted to verify the effect of the driving force distribution on turning, the effect of the VTVC on the equivalent lateral disturbance suppression and position control, and the positioning performance for different trajectories. Fig. 7 shows the experimental setup, and Table 2 lists the specifications for the experimental setup. The experimentally tracked vehicle was driven by two direct-current motors with encoders. The tracked vehicle was equipped with an inertial measurement unit (IMU) to measure the acceleration and angular velocity. The Kalman filter was applied to the measured acceleration and angular velocity to estimate the turning angle. An augmented reality (AR) marker was installed on the surface of the vehicle. The vehicle position was measured by detecting the AR marker from the images captured using a camera. Based on previous trials, the number of neurons in the hidden layer of the TVNN was determined to be 15. The TVNN was trained using the Bayesian regularization error backpropagation method based on the velocity measured by detecting the AR marker as the supervisory data.

B. TURNING EVALUATION VIA DRIVING FORCE DISTRIBUTION
First, we evaluated the turning performance based on the driving force distribution. The proposed driving force distribution was compared with the method used in [30], described by a distribution matrix D consisting of a square Jacobian J  and an equivalent inertia matrix M n , as follows: The position command vector (x cmd = x cmd y cmd φ cmd T ) is expressed as a function of time t, where x cmd = [0 m 0 m πt/10 rad] T . The control parameters are listed in Table 3. For comparison, the position response, including the turning angle response, was fed back directly instead of the VTVC. Figs. 8 and 9 show the tracking performance in the turning direction using the distribution matrix D and the proposed driving force distribution, respectively. In Figs. 8 and 9, panels (a) and (b) represent the turning angle and driving force of the crawler, respectively. In Fig. 8 (a), poor turning performance is shown using the distribution matrix. As shown in Fig. 8 (b), the driving forces of the crawlers (F dr,r , F dr,l ) follow the references (F ref dr,r , F ref dr,l ). However, this result suggests that the driving forces of the crawlers changed simultaneously, indicating that the driving force interferes with each other and consequently hinder the generation of the moment necessary for turning. Conversely, the turning angle suitably follows the given command ( Fig. 9 (a)). In Fig. 9 (b), the left and right driving forces are generated separately. This result indicates that the driving force is distributed to prevent interference. Therefore, turning can be improved by decoupling the driving forces of the crawlers using the proposed driving force distribution.

C. POSITION CONTROL EVALUATION BY VTVC
Next, we evaluated the position-tracking performance and disturbance suppression of the proposed system. Cases with and without this control were compared and evaluated to       Table 4. To determine the slippage of the tracked vehicle, we compared the response coordinates (x res , y res ) calculated from the crawler motor encoders and IMU with the absolute coordinates (x meas , y meas ) obtained from the detected AR marker position.    lateral positioning. Conversely, the positioning performance was likely improved by suppressing the equivalent lateral disturbance using the VTVC. Furthermore, the performance of self-positioning estimation could be improved by appropriately suppressing slippage because the difference between the absolute and response coordinates decreased with the introduction of the VTVC. These results confirm that the proposed control system improves the positioning performance of the tracked vehicle while appropriately allowing for slippage in the turning direction.

D. TRACKING AND SELF-POSITIONING PERFORMANCES EVALUATION FOR DIFFERENT TRAJECTORIES
Finally, we evaluated the tracking and self-positioning performances for various trajectories. The position commands listed in Table 5 are expressed as trajectories for 20 s. The same control parameters were used in this evaluation, as listed in Table 4. The initial position vector x ini was set the same as the initial value of the position command vector. Here, the norms between the absolute coordinates (x meas , y meas ) and the position command (x cmd , y cmd ) at each sampling were treated as positioning errors. The norms between the position response (x res , y res ) and the absolute  coordinates (x meas , y meas ) at each sampling were treated as self-positioning estimation errors. Tables 6 and 7 list the data for the positioning errors and self-positioning estimation errors for each trajectory, respectively. Table 6 confirms that the positioning performance of the proposed system is comparable for different trajectories. Furthermore, Table 7 indicates that the self-position estimation errors decrease, independent of the trajectory. The slight difference between the absolute and response coordinates suggests that the slippage was suppressed sufficiently.

VI. CONCLUSION
We proposed a driving force distribution to control a tracked vehicle and VTVC to suppress the equivalent lateral disturbance. First, the crawler driving force control suppressed slippage in the translational direction of the tracked vehicle. Next, the proposed driving force distribution enabled the decoupling of the crawlers using the instantaneous turning center of the vehicle to induce adequate slippage in the turning direction. Furthermore, the VTVC suppressed the equivalent lateral disturbance. The experimental results confirmed that the position tracking performance was improved by suppressing the lateral disturbance while allowing the tracked vehicle to slip adequately.
The proposed position control method can be applied to mobile robots in general, where the occurrence of slippage cannot be disregarded. The position control proposed in this paper does not consider the case where there are obstacles in the path. In the future, we aim to study countermeasures such as a position control method that includes obstacle avoidance.