1 Introduction
Mobile robots in offroad terrain are increasingly used in wide range of nonindustrial applications such as agriculture, search and rescue, military, forestry, mining and security Kayacan et al. (2015); Li,Ming et al. (2009). However, guidance, navigation, and control of mobile robots require advanced control methods to alleviate the effects of unmodeled surface and soil conditions (e.g., snow, sand, grass), terrain topography (e.g., sideslopes, inclines), and complex robot dynamics. In the outdoor environment, it is sometimes arduous and/or almost impossible to obtain a priori model for such effects (i) modeling of robotterrain interactions is challenging, (ii) the soil condition is often not known ahead of time, and (iii) the identification of system parameters is a cumbersome process and must be recarried out for different terrains Kayacan et al. (2013, 2014b); Cui et al. (2017); Kayacan et al. (2018); Pan et al. (2018).
The initial studies on autonomous mobile robots used traditional controllers, e.g., proportionalintegralderivative, optimal, and model predictive controllers (MPCs) NormeyRico et al. (2001); Kanjanawanishkul and Zell (2009); Huynh et al. (2017); Amer et al. (2017). Proportionalintegralderivative controllers are convenient only for singleinputsingleoutput systems; however, mobile robots are multiinputmultioutput systems. As alternative methods, linear quadratic regulators and linear MPCs, which can be designed readily for multiinputmultioutput systems, were proposed for autonomous navigation of mobile robots in literatur Liao et al. (2017); Xiao et al. (2017); Seder et al. (2017). Since these methods require a linear system model to be designed, tracking errorbased control algorithms were developed Klancar and Skrjanc (2007); Skrjanc and Klancar (2017) in which the system model is linearized around the target path, and the total control input is obtained by the sum of feedback and feedforward control inputs Blazic (2011). The feedback control law is designed based on a nominal model, which is a priori model and cannot capture all the effects of uncertainties that are summarized in the previous paragraph. Moreover, the feedforward control law is derived taking the target path and nominal model into account so that it also cannot contain the effects of uncertainties. Since tracking errorbased models might not represent realtime systems behavior accurately, the traditional tracking errorbased control methods cannot ensure precise path tracking performance in the outdoor environment. Therefore, it can be concluded that a prerequisite for accurate tracking performance of modelbased controllers is the achievement of a precise mathematical model of the system to be controlled Kayacan et al. (2015). This shows that traditional approaches are not always inherently robust. Moreover, an amplitudesaturated output feedback control approach was proposed in Sun et al. (2017). In this approach, the output of the control input is limited by upper and lower bounds; however, the input rate cannot be limited. Furthermore, this approach requires to know the direct relation between the input and output, which is known in tracking errorbased control methods. In this paper, since traditional tracking errorbased controllers use an MPC as a feedback controller, we also use an MPC controller for a fair comparison with the previous works in literature.
To cope up with restrictions on traditional controllers, adaptive control approaches and MPC scheme with friction compensation were respectively proposed in Martins et al. (2008); S. et al. (2014), and successful results were reported for indoor applications. However, there is no evidence that these methods work well for outdoor applications where the uncertainty is so high, and soil conditions are changing. Robust trajectory tracking error modelbased predictive controller was designed for unmanned ground vehicles to overcome the limitations of the tracking errorbased controllers Kayacan et al. (2016). Although it exhibited robust control performance, it could not ensure precise path tracking performance, e.g., the tracking error was more than
cm. Moreover, learningbased nonlinear MPC algorithms were proposed in which online parameters estimators were used to update the system parameters in the system model for an articulated unmanned ground vehicle
Kayacan et al. (2015, 2014a, 2018). Although precise path tracking performance was reported in these studies, the required computation times is large, especially for embedded applications. Therefore, the purpose of this paper is to develop a high precision control algorithm for mobile robots, which must be computationally efficient and can learn the mobile robot dynamics by utilizing longitudinal and lateral error dynamics. Thus, the feedforward control action will be in charge of the overall control of the mobile robot in steadystate behavior.The main contribution of this study beyond state of the art is that a novel Tracking Error Learning Control (TELC) scheme is developed and implemented in realtime for the first time in literature. The first contribution of this paper is that the cost functions consisting of tracking error dynamics of the mobile robot are used to update the coefficients in the feedforward control law. Hence, TELC can learn mobile robot dynamics, and the feedback control action is removed from the overall control signal when the robot is ontrack. Therefore, the modelplant mismatch problem for outdoor applications cannot deteriorate the path tracking performance in the TELC scheme. The second contribution is that the stability of the TELC algorithm is proven that the TELC algorithm is asymptotically stable. The stability analysis shows the longitudinal and lateral error dynamics converge to zero if the learning coefficients are large enough. The third contribution is that it is proven that the TELC algorithm does not have any local minima so that it can reach the global minima. Moreover, it is shown that the feedforward control actions are bounded for a finite value for the coefficients in steadystate. Along with the theoretical results, this paper also presents path trackingtest results of the presented TELC algorithm on a mobile robot. TELC algorithm results in precise mobile robot path tracking performance when compared to the traditional tracking errorbased control method.
This paper is organized as follows: The tracking errorbased model is derived in Section 2. The traditional tracking errorbased control method is given in Section 3. The TELC algorithm is formulated in Section 4 while the update rules for the linear and angular velocity references are respectively derived in Sections 4.1 and 4.2, and the stability analysis is given in Section 4.3. Experimental results on a mobile robot are presented in Section 5. Finally, a brief conclusion of the study is given in Section 6.
2 Tracking ErrorBased System Model
In this paper, the mobile robot is illustrated in Fig. 1. The velocities of two driven wheels result in linear velocity and angular velocity with the distance between wheels , which are two control inputs of the mobile robot, . The traditional unicycle model is written for a mobile robot as follows:
(1) 
where and are the position, is the heading angle, is the linear velocity, is the angular velocity of the mobile robot.
The target path with respect to the inertial reference frame fixed to the motion ground is defined by a reference state vector
and a reference control vector . The error state expressed in the frames on the mobile robot is written as:(2) 
where is the transformation matrix formulated as below:
The tracking errorbased model is derived by taking the timederivative of the error state in (2) and the unicycle model in (1) into account as follows:
(3) 
where is the longitudinal error, is the lateral error and is the heading angle error.
The tracking errorbased model (2) is linearized around the target path ( ) as follows:
(4) 
Finally, it can be written in the statespace form as follows:
(5) 
where the state and control vectors are written as
e  (6)  
(7) 
where and .
Remark 1
The tracking errorbased system model is fully controllable when either the linear velocity reference or the angular velocity reference is nonzero, which is a sufficient condition.
3 Traditional Tracking ErrorBased Control
The traditional trackingerrorbased control algorithm for a mobile robot is formulated in this section. The total control input applied to the mobile robot is calculated as the summation of the feedback control action and the feedforward control action as follows:
(8) 
The feedback controller generates the differences between the actual and reference control inputs while the feedforward control actions are the reference control inputs. The traditional feedback and feedforward control actions are formulated in following subsections.
3.1 Feedback Control Action: Model Predictive Control
A mobile robot can be described by a linear continuoustime model:
(9) 
where e is the state vector and is the control input vector. The matrices A and B are found from the tracking errorbased system model in (2).
The input constraints for the mobile robot are defined for all as follows:
(10)  
(11) 
The cost function is written as follows:
(12) 
where and represent the prediction and control horizons, is the input change, and
is the matrix of the input vectors from sampling instant to sampling instant . Since the sampling time of the experiments is equal to millisecond, the prediction and control horizons are respectively equal to second and second. The positivedefinite weighting matrices and are defined as follows:
(13) 
The following plant objective function is solved at each sampling time for the MPC:
(14)  
subject to  
The convex optimization problem in (14) is solved for the current error state information in a receding horizon fashion. The steps for the implementation of the MPC algorithm are summarized as below:

Measure or estimate the current error states .

Obtain by solving the optimization problem in (14).

Calculate the feedback control action
The MPC problem is thereafter solved for the next sampling instant over shifted prediction and control horizons. The control input generated by the MPC is the feedback control action :
(15) 
In tracking error control scheme, the designed MPC minimizes the tracking error between the target path and the measured position of the mobile robot, and finds the differences between the actual and reference control inputs. Therefore, the feedback control inputs generated by the MPC are not the actual control inputs, which are sent to the mobile robot. We will formulate traditional feedforward control actions in the next subsection 3.2.
3.2 Feedforward Control Action
In openloop control, the feedforward control laws can only drive a mobile robot on a target path if there exist no uncertainties and initial state errors. Feedforward control inputs and are derived for a given target path (, ) by using the unicycle model (1).
The linear velocity reference, i.e., , for a mobile robot is derived for a target path defined in a time interval as follows:
(16) 
where the sign is the desired driving direction of the mobile robot ( for forward, for reverse).
The heading angle reference is derived from (1) as follows:
(17) 
where is the desired driving direction of the mobile robot ( for forward, for reverse) and the function is a fourquadrant inverse tangent function. The angular velocity reference, , for a mobile robot is derived by taking the timederivative of (17) for a given target path defined in a time interval as follows:
(18) 
Remark 2
The necessary condition in the path generation is that the target path must be twicedifferentiable, and the linear velocity reference must be nonzero, i.e., .
4 Tracking Error Learning Control
The tracking errorbased model is linearized around a target path by assuming that the longitudinal, lateral and heading angle errors are around zero. Therefore, the mismatch between the tracking errorbased model and real system might result in unsatisfactory control performance when the mobile robot is not ontrack. Moreover, there always exist uncertainties and unmodeled dynamics, which cannot be modeled, in outdoor applications.
In the TELC, the longitudinal and lateral error dynamics are used to train the learning algorithm, which is required to learn the uncertainties and unmodelled dynamics and to keep the system on track. The learning algorithm generates the references for the control inputs. In other words, it learns the dynamics of the real system.
Like (8), the control inputs consisting of the feedback and feedforward control actions are written as follows:
(19)  
(20) 
where and are the traditional feedback control actions formulated in Section 3.1, while and are new feedforward control actions that are updated by the tracking error learning algorithm. The new feedforward control actions are formulated as follows:
(21)  
(22) 
where and are the traditional feedforward control actions formulated in (16) and (18), respectively. and are the coefficients to update the linear velocity reference and and are the coefficients to update the angular velocity reference. TELC structure for a mobile robot is shown in Fig. 2. In the next subsection, we will formulate the update rules for these coefficients.
4.1 Linear Velocity
The requirement for the Lyapunov stability, i.e., , is satisfied for the selection of the coefficients for the linear velocity in (21). As can be seen from (2), the linear velocity appears in the channel of the longitudinal error . Therefore, we use the following squared firstorder longitudinal error dynamics as the cost function:
(23) 
where is a positive constant, i.e., . It is implied that if the cost function converges to zero, i.e., , then the robust control performance condition is satisfied so that the longitudinal error converges to zero.
Gradient descent, which is a firstorder iterative optimization algorithm for finding the minimum of a function, is used to minimize the cost function . In this approach, steps are taken proportional to the negative of the gradient of the closedloop error function, i.e., , to find the minimum of the cost function. The cost function is minimized to decide the coefficients in (21) as follows:
(24) 
where is the learning coefficient and positive, i.e.,
. It is rewritten by using the chain rule
(25) 
Equation (23) is inserted into the equation above, it is then obtained as
(26) 
Considering (2), is obtained and inserted into the equation above, it is obtained as
(27) 
If total control input for the linear velocity applied to the mobile robot (19) and the feedforward control action for the linear velocity (21) are inserted into (27), the adaptation of the coefficient for the linear velocity is written as follows:
(28) 
The bias term for the linear velocity is computed by using the same procedure and found as:
(29) 
4.2 Angular Velocity
First we take the timederivative of the lateral error so that the angular velocity can appear in the same channel with the lateral error . It is obtained considering (2) as follows:
(30) 
The requirement for the Lyapunov stability, i.e., , is satisfied for the selection of the coefficients for the angular velocity in (22). Therefore, we use the following squared secondorder lateral error dynamics as the same cost function as follows:
(31) 
where is a positive constant, i.e., . It is implied that if the cost function converges to zero, i.e., , then the robust control performance condition is satisfied so that the lateral error converges to zero.
As explained in Section 4.1, the gradient descent method is used to find the minimum of the cost function and to decide the coefficients in (22) as follows:
(32) 
where is the learning coefficient and positive, i.e., . It is rewritten by using the chain rule
(33) 
Equation (31) is inserted into the equation above, it is then obtained as
(34) 
Considering (2) and (30), is obtained and inserted into the equation above, it is obtained as
(35) 
If total control input for the angular velocity applied to the mobile robot (20) is inserted into (35) considering the feedforward control action for the angular velocity (22), the adaptation of the coefficient for the angular velocity is written as follows:
(36) 
The bias term for the angular velocity is computed by using the same procedure and found as:
(37) 
4.3 Stability Analysis
The summation of the cost functions used for the linear and angular velocities is used to formulate the Lyapunov function as follows:
(38) 
The Lyapunov function is positive semidefinite, i.e., . To check the stability of the trackingerror learning algorithm, the timederivative of the Lyapunov function above is taken as follows:
(39) 
It is rewritten by using the chain rule as follows:
(40)  
where, represents the derivative of the Lyapunov function with respect to the variables other than the coefficients in the formulations of the linear and angular velocities.
The timederivatives of the coefficienting terms in (24), (29), (32) and (37) are inserted into the aforementioned equation, then the timederivative of the Lyapunov function is obtained as follows:
(41)  
If the learning coefficients for the linear and angular velocity references are large enough, the timederivative of the Lyapunov function is negative, i.e., . This implies asymptotically stability of the learning algorithm.
4.4 Global Minima
The most important concern in the tracking error learning algorithm is that the system might reach some local minima and stay in these local minima. In this section, we will show that there are no local minima for the formulation of the trackingerror learning algorithm. If the second derivatives of the cost functions with respect to variables have the same sign, then the cost functions do not have a change in the curvature sign through the variables. This implies that the cost functions do not have local minima through these variables.
4.4.1 Linear Velocity
If we take the second derivative of the cost function for the linear velocity with respect to , it is obtained as follows:
(42) 
The chain rule is applied to the equation above
(43) 
First is obtained considering (2) and inserted into the equation above. Also, total control input for the linear velocity applied to the mobile robot (19) and the feedforward control action for the linear velocity (21) are inserted into the equation above. Then, (43) is obtained as follows:
(44) 
The second derivative of the cost function with respect to the bias coefficient is computed by using the same procedure as follows:
(45) 
Equations (4.4.1) and (45) show that the sign of the curvature of the cost function for the linear velocity (23) is always positive; therefore, there are no local minima, which indicate that the system reaches to the global minimum. After reaching the global minimum, since is a constant and is bounded, the coefficient update algorithms (4.1) and (29) show that coefficients converge to a finite value. A finite value for the coefficients in steadystate results in a bounded feedforward control action (21).
4.4.2 Angular Velocity
If we take the second derivative of the cost function for the angular velocity with respect to , it is obtained as follows:
(46) 
The chain rule is applied to the equation above
(47) 
First is obtained considering (2) and (30) and inserted into the equation above. Then, total control input for the angular velocity applied to the mobile robot (20) and the feedforward control action for the angular velocity (22) are inserted into the equation above. Then, (47) is obtained as follows:
(48)  
The second derivative of the cost function with respect to the bias coefficient is computed by using the same procedure as follows:
(49) 
Equations (48) and (49) show that the sign of the curvature of the cost function for the angular velocity (31) is always positive; therefore, there are no local minima, which indicate that the system reaches to the global minimum. After reaching the global minimum, since is a constant, and are bounded, the coefficient update algorithms (4.2) and (37) show that coefficients converge to a finite value. A finite value for the coefficients in steadystate results in a bounded feedforward control action (22).
5 Experimental Validation
5.1 Mobile Robot
The mobile robot, termed TerraSentia, is constructed entirely out of 3D printed construction as shown in Figs. 3 and 4, which leads to an extremely lightweight robot (14.5 lbs) and has been proven to be structurally resilient to field conditions during an entire season of heavy operation in Corn, Sorghum, and Soybean farms in Illinois Kayacan et al. (2018). We posit that this robot is an example of the potential of additive manufacturing (3D printing) in creating a new class of agricultural equipment that works in teams to replace, minimize, or augment traditional heavy farm equipment. In addition, lightweight equipment has several benefits, it is easier to manage, has better endurance, safer to operate in general, and leads to lower ownership cost. On the other hand, the ultralight robot described here mitigates many of these challenges, yet, it leads to a uniquely challenging control problem in uneven and soil terrain in crop fields. The difficulties in control arise from complex and unknown wheelterrain interaction and wheel slip.
The placement of hardware is illustrated in Fig. 5. One Global Navigation Satellite System (GNSS) antenna has been mounted straight up the center of TerraSentia, and the dualfrequency GPScapable realtime kinematic differential GNSS module (Piksi Multi, Swift Navigation, USA) has been used to acquire centimeterlevel accurate positional information at a rate of 5 Hz. Another antenna and module have been used as a portable base station and has transmitted differential corrections. There are four brushed 12V DC motors with a 131.25:1 metal gearbox (Pololu Corporation, USA), which are capable of driving an attached wheel at 80 revolutions per minute. A twochannel Halleffect encoder (Pololu Corporation, USA) for each DC motor is attached to measure velocities of the wheels. The Sabertooth motor driver (Dimension engineering, USA) is a twochannel motor driver that uses digital control signals to drive two motors per channel (left and right channel) and has a nominal supply current of 12 A per channel. An onboard computer (1.2GHz, 64bit, quadcore Raspberry Pi 3 Model B CPU) acquires measurements from all available sensors and sends control signals to the Sabertooth motor driver in the form of two Pulsewidth modulation signals.
All available measurements from all its onboard sensors (GNSS and encoders) are fed to a state estimator, i.e., extended Kalman filter, to estimate heading angle of the mobile robot. In every time instant, estimates are fed to calculate the errors with respect to the inertial reference frame fixed to the motion ground and the estimated heading angle is fed to the transformation matrix to calculate the tracking errors in the frame on the mobile robot. Then, tracking errors are fed to the TELC algorithm, which generates the linear and angular velocity references to track the target path. Then, the linear and angular velocity references are controlled by a proportionalintegralderivative type motion controller  in other words, the robot’s lowlevel controller  by using feedback from encoders attached to the motors to determine the required control signals in the form of the Pulsewidth modulation signal. Thus, the tracking of given reference command signals ensures that the robot’s desired velocities are maintained. The motion controller outputs the modified command signals to the Sabertooth Motor Driver which correlates the given control signals to the necessary output voltages needed by the DC motors.
5.2 State Estimation
An Extended Kalman Filter (EKF) is used for the state estimation in realtime because one GNSS antenna is not enough to obtain the heading angle of the mobile robot. We require heading angle information to calculate the heading angle error and also the rotation matrix. The inputs of the EKF are position information coming from the GNSS, and linear and angular velocities coming from the encoders and gyro. The outputs of the EKF are the position of the mobile robot on x and ycoordinate system and the heading angle of the mobile robot.
The discretetime unicycle model used for the implementation of the EKF is written as follows:
(50) 
where is the sampling interval. The general form of the estimated system model is written:
(51) 
where is the system model (5.2), is the measurement function and is the measurements. The difference between the system model and realtime system is the process noise and observation noise in the measurement model. These noises are assumed to be independent and zero mean multivariate Gaussian noises with covariance matrices and , respectively Goodarzi and Lee (2017):
(52) 
where the weighting matrices are defined ad follows:
(53)  
(54) 
5.3 Experimental Results
The experiment was carried out in offroad terrain and the GNSS is used as the ground truth measurements. The learning rate for the linear velocity is set to and for the coefficients and , respectively, while the learning rate for the angular velocity is set to and for the coefficients and , respectively. The positive constants and are set to . The sampling time of the experiment is equal to milliseconds.
An 8shaped path is used as the reference path for the mobile robot to evaluate the path tracking performance of the TELC algorithm. The 8shaped path consists of two straight lines and two smooth curves as illustrated in Fig. 6. The linear velocity reference is set to throughout the path generation. The angular velocity reference is zero for straight lines while it is set to for curved lines. Thus, the desired reference trajectory , and is generated by considering the unicycle model in (1) in the paper. The target and actual paths are shown in Fig. 6. The mobile robot controlled by the developed TELC algorithm can track the target path precisely.
The Euclidean errors for the traditional tracking error control and TELC algorithms are shown in Fig. 7. The mean values of the Euclidean errors for the traditional tracking error control formulated in Section 3 and TELC algorithms are respectively cm and cm. The TELC results in more precise mobile robot path tracking performance when compared to the traditional tracking error control method. This demonstrates the learning capability of the TELC algorithm in the outdoor environment.
The total linear velocity applied to the mobile robot is shown in Fig. 8. It is observed from the reference and measurements for the linear velocity that the lowlevel controller provides accurate tracking of the linear velocity reference despite the high noisy measurements. Moreover, the feedback control action generated by the MPC and the feedforward control action generated by the TELC algorithm are shown in Fig. 9. The feedback control action for the linear velocity is around zero as expected and the feedforward control action for the linear velocity takes the overall control action of the linear velocity of the mobile robot. Furthermore, the coefficients in the feedforward control action for the linear velocity are shown in Fig. 10. The nominal values for the coefficients and must be respectively and in the absence of uncertainties. These coefficients are different than the nominal values throughout experiments and varying due to the varying soil conditions in the outdoor environment.
The total angular velocity applied to the mobile robot is shown in Fig. 11. It is observed from the reference and measurements for the angular velocity that the lowlevel controller provides accurate tracking of the angular velocity reference despite the high noisy measurements. Moreover, the feedback control action generated by the MPC and the feedforward control action generated by the TELC algorithm are shown in Fig. 12. The feedback control action for the angular velocity is around zero as expected and the feedforward control action for the angular velocity takes the overall control action of the angular velocity of the mobile robot. Furthermore, the coefficients in the feedforward control action for the angular velocity are shown in Fig. 13. The nominal values for the coefficients and must be respectively and in the absence of uncertainties. These coefficients are different than the nominal values throughout experiments and varying due to the varying soil conditions in the outdoor environment. The angular velocity reference is equal to zero while the mobile robot is tracking straight lines. Therefore, the coefficient is constant, which can be seen from the update rule in (4.2). As a result, the only coefficient is updated to decrease the unmodeled effects in realtime.
6 Conclusions
In this paper, a novel TELC algorithm has been developed for precise path tracking and experimentally validated on a mobile robot in the outdoor environment. In case of the plantmodel mismatch, the TELC algorithm learns mobile robot dynamics by using the tracking error dynamics and updates the feedforward control actions. Therefore, the feedforward controller gradually eliminates the feedback controller from the control of the system once the mobile robot has been ontrack. The experimental results on the mobile robot show that the TELC algorithm ensures precise path tracking performance as compared to the traditional tracking errorbased control algorithm. The mean value of the Euclidean errors for TELC is 9 cm approximately in offroad terrain.
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