Nonlinear Control of a Robot-Trailer System Using a Hybrid Backstepping-linearizing Approach by Aditya Singh A thesis submitted to the Graduate Faculty of Auburn University in partial ful llment of the requirements for the Degree of Master of Science Auburn, Alabama August 04, 2012 Keywords: Nonlinear control, backstepping, mobile robots Copyright 2012 by Aditya Singh Approved by John Y. Hung, Chair, Professor of Electrical and Computer Engineering Thaddeus A. Roppel, Associate Professor of Electrical and Computer Engineering Andrew J. Sinclair, Associate Professor of Aerospace Engineering Abstract In this work, the author develops a nonlinear controller to stabilize an au- tonomous wheeled robot and trailer system. A dynamic model based on robot-trailer kinematics that has previously proven su cient for state feedback control is chosen for the ease of design. An iterative approach similar to backstepping is utilized to obtain the control input. In a manner reminiscent of feedback linearization, nonlinearities are cancelled at each step to obtain an equivalent linear system. This method is sig- ni cantly di erent from integrator backstepping method as no signal di erentiation is required. However, it is also di erent from the feedback linearization method as it does not require any coordinate transformation. This hybrid method is essentially a selective amalgamation of the two methods. In contrast to known state-of-the-art approaches, the proposed method stabilizes the system in both the forward and re- verse motion directions, without modeling modi cations. Simulation results suggest that the Hybrid Backstepping Controller(HBC)is su cient for regulating the trailer to the desired path from any initial condition. Experimental results con rm that the Hybrid Backstepping Controller(HBC) can control the robot-trailer system and can regulate the trailer over a typical geophysical surveying path. ii Acknowledgments The author would like to thank the members of his committee Dr. John Y. Hung, Dr. Thaddeus A. Roppel and Dr. Andrew J. Sinclair for their guidance. The author would like to specially thank Dr. John Y. Hung for the invaluable advice and encour- agement given during this research. This work would not have been possible without the funding and support provided by the U.S Army Corp of Engineers. Particular thanks goes to David Hodo, whose extensive previous work provided the basis for this research. Finally the author would like to thank Michael L. Payne for his invaluable support while performing the experiments. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Geophysical Surveying . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Nonlinear Control Strategies . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 An Innovative Approach . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Parameter De nitions . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Controller Gains . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Simulation of the Controller . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1 The Robot-trailer System . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Di erent Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 22 iv 4.4 Path Following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Comparison with Linear Controller . . . . . . . . . . . . . . . . . . . 25 4.5.1 Di erent Initial Conditions . . . . . . . . . . . . . . . . . . . . 25 4.5.2 Path Following . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.1 E ectiveness of the Hybrid Backstepping Controller(HBC) . . . . . . 30 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.1 Full dynamic model . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.2 Choosing Controller Gains . . . . . . . . . . . . . . . . . . . . 32 5.2.3 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.4 Design of a Nonlinear Estimator . . . . . . . . . . . . . . . . 33 5.2.5 E ectiveness on other problems . . . . . . . . . . . . . . . . . 33 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 v List of Figures 1.1 Geophysical surveying system towed by a human operator . . . . . . . . 2 2.1 Kinematic model of the robot-trailer system . . . . . . . . . . . . . . . . 5 3.1 Block Diagram of the Robot-Trailer System . . . . . . . . . . . . . . . . 8 3.2 System position with xed gains (simulated) . . . . . . . . . . . . . . . . 16 3.3 Lateral error and heading error (simulated) . . . . . . . . . . . . . . . . 17 3.4 System Position with Gain Scheduling (simulated) . . . . . . . . . . . . 18 3.5 Backing of trailer (Simulated) . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Autonomous geophysical surveying system described in this thesis . . . . 21 4.2 Robot-Trailer Position with xed gains (experimental) . . . . . . . . . . 22 4.3 Robot-Trailer Position with Gain Scheduling (experimental) . . . . . . . 23 4.4 Lateral Error And Heading Error (experimental) . . . . . . . . . . . . . 23 4.5 Backing of the trailer (experimental) . . . . . . . . . . . . . . . . . . . . 24 4.6 Plot of Robot and Trailer path . . . . . . . . . . . . . . . . . . . . . . . 26 4.7 Plot of robot-trailer with Linear controller . . . . . . . . . . . . . . . . . 28 4.8 Plot of robot-trailer with HBC . . . . . . . . . . . . . . . . . . . . . . . 28 4.9 Comparison of Lateral error . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.10 Comparison of Heading error . . . . . . . . . . . . . . . . . . . . . . . . 29 vi List of Tables 2.1 Robot-trailer system parameters . . . . . . . . . . . . . . . . . . . . . . 5 4.1 Comparison of the two controllers for di erent initial conditions . . . . . 26 4.2 Comparison of the two controllers while path following . . . . . . . . . . 27 5.1 Proposed gain scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 33 vii Chapter 1 Introduction 1.1 Geophysical Surveying Geophysical surveying is a eld that has grown tremendously in the past decade.With survey areas estimated in the millions of acres in the continental USA alone [1] [2] unexploded ordnance (UXO) detection is a suitable eld for robots. These surveys are carried out using a variety of sensitive sensor systems [3]. Currently these surveys are performed by highly-trained personnel towing sensors either manually (see Fig. 1.1) or by using a vehicle. These methods are both ine cient and costly for performing geophysical surveys. The nature of UXO surveying is such that it can expose the operator to signi cant dangers. Robotic systems [4], remotely-driven vehicles [5] and airborne surveying systems [6] are alternatives to the current method. Airborne surveying though fast has its own limitations. Remotely driven vehicles eliminate the danger to the operator, but it still requires an highly quali ed operator to be present to perform the surveys. Robotic systems are potentially the best solution to the above problem. Geophysical surveying sometimes require a complete coverage of a desired region. There are several approaches used to achieve complete coverage. Coverage paths [7] and co-operative robotics [8] are a few methods to attain complete cover- age. In [4] a linear controller is developed to perform geophysical surveying. As the dynamics of the linearized robot model are dependent on the point of linearization, the linear controller is insu cient for control over all operating conditions. 1 Figure 1.1: Geophysical surveying system towed by a human operator 1.2 Nonlinear Control Strategies Several nonlinear controller strategies have been examined for mobile robotic system in the recent years. Several di erent non linear control laws have been pro- posed. These include -synthesis robust controller [9], adaptive backstepping control design [10], fuzzy logic control [11], role switching [12], di erent controller for kine- matic and dynamic control inputs [13] and feedback linearization [14]. In [15] and [16] a global tracking law is developed for mobile robotic systems, but has constrained desired linear and angular velocities. In [17] and [10] these constraints are eliminated. However the backstepping method requires a lot of computational power thereby mak- ing it unsuitable for real time application.In [17] an unique method is proposed to get around this problem. A modular structure is proposed so that other dynamic control laws (like PID, feedback linearization) can be applied thereby reducing the compu- tation e ort. But none of these control laws are su cient to control a robot-trailer system. The addition of a trailer behind the robot presents several unique challenges 2 to the design of the control law, with one of the greatest challenge being that all forms of actuation are limited to the robot; it is a type of non-collocated control system. 1.3 An Innovative Approach This thesis introduces an innovative nonlinear approach to design a control law for the robot-trailer system. The basic idea is to drive each state variable to a desired value or trajectory while cancelling the nonlinearity. To some degree, the control approach has semblance to feedback linearization. The design is rst performed on the output state variable, and then repeated for remaining variables. Therefore, there is some likeness to the iterative nature of integrator backstepping. However, the described method requires no global coordinate transformation (escaping the chief di culty of feedback linearization), nor any di erentiation of signals (avoiding the drawback of integrator backstepping). This is similar to the control law proposed in [17]. However there are some major di erences between the two methods. The controller proposed in [17] linearizes the kinematics and dynamics of the system and the controller also requires some co-ordinate transformation. The proposed control law neither linearizes the system nor requires any kind of co-ordinate transformation. The proposed control law stabilizes system dynamics in both the forward and the backward directions, without design model modi cations. The above said control law is simulated and implemented on a robot-trailer system and has proven su cient to control the system for any initial condition. This work has also been submitted for peer review to the 2012 IEEE International Conference on Industrial Electronics (IECON-2012, Montreal, Canada). 3 Chapter 2 System Model Previous work has developed an autonomous robot-trailer system to tow geo- physical sensor arrays using a linear controller [4, 18]. The author uses this system as a research platform to explore a innovative nonlinear control law to control the robot-trailer system. In this thesis, the robot is a four wheeled di erential vehicle capable of making sharp turns. In fact, the vehicle can turn "in place" without any forward motion. 2.1 System Model A mathematical model of the robot trailer system has to be derived before a con- troller can be designed. Previous work [4] has shown that a kinematic model(Fig 2.1) is su cient to control the robot-trailer system. Dynamic e ects such as moment of inertia, wheel slip and rolling friction are neglected for the slow speed that the system is run. 2.1.1 Parameter De nitions A list of parameters and variables and their values used in this thesis is presented in Table 2.1 and Fig 2.1. (Note: The terms easting and northing are geographic Cartesian coordinates for a point.) 4 Figure 2.1: Kinematic model of the robot-trailer system Table 2.1: Robot-trailer system parameters Variable Description Value Lt Length of trailer tongue 3.3 m Lr Length of robot hitch 0 m Vt Velocity of trailer Vr Velocity of robot 1 m/s r Heading of the robot er Easting of the robot nr Northing of the robot t Heading of the trailer et Easting of the trailer nt Northing of the trailer terr Heading error of trailer Lerr Lateral error from path of trailer Hitch angle !r Yaw rate of robot 5 2.1.2 Kinematic Model The system can be represented by the following state equations. _er = Vr sin( r) (2.1) _nr = Vr cos( r) (2.2) _ r = !r (2.3) _ = Vr Lt sin( ) Lr!r Lt cos( ) !r (2.4) The vehicle model has two input - the angular velocity !r and linear velocity Vr. However as the goal is to control the trailer position, the following equations describe relationships between robot and trailer. et = er Lr sin(!r) Lt sin(!r + ) (2.5) nt = nr Lr cos(!r) Lt cos(!r + ) (2.6) t = r (2.7) Vt = Vr cos( ) Lr!r sin( ) (2.8) The above equations give the relationship between robot variables and trailer vari- ables. The set of equations is used to derive the dynamic equation of the trailer [19]. _et = Vt sin( t) (2.9) _nt = Vt cos( t) (2.10) _ t = Vr Lt sin( ) Lr!r Lt cos( ) (2.11) _ = Vr Lt sin( ) Lr!r Lt cos( ) !r (2.12) 6 Chapter 3 Controller Design 3.1 Introduction Path following control of robot-trailer system has been extensively studied in the past. Several types of nonlinear control have been developed that can control the robot on a desired path. But the application of nonlinear controller to the robot-trailer system has not been explored as much. In this thesis a new nonlinear control approach is tested to control the robot-trailer system to the path. This new approach is similar to the back stepping control, as the desired control input is backstepped through each integrator. However this approach di ers signi cantly from the backstepping approach as no signal di erentiation takes place. This approach also looks similar to feedback linearization method as it assumes a pseudo feedback at each step, but it di ers from feedback linearization as there is no coordinate transformation. This is a novel approach that coalesces the desired features of the two approaches. 3.2 Controller Design In this section, the control law for the robot-trailer system is derived. For path following the error model is required. The dynamic equation derived in Section 2.1.2 is used to derive the error model. For ease of controller implementation, the desired path at any moment in time is transformed by a nonlinear transformation to a path whose Universal Transverse Mercator (UTM) coordinates run from south to north. This ensures that the trailer lateral error is directly de ned by the trailer easting et. 7 Figure 3.1: Block Diagram of the Robot-Trailer System Hence, the error variables can be de ned as follows. Lerr = edesired et (3.1) terr = desired tk 2 (3.2) where k2 is a gain. The goal of the project is to control the trailer to the path, so only the lateral error and the heading error of the trailer are chosen as the controlled variables. This approach also provides more freedom to the robot as the robot does not need to follow the path. Velocity and yaw (turn) rate are the physically controllable variables of the robot. Though it is desired that the trailer velocity is 1 m/s, it was seen that while making tight turns, the robot speed tends to in nity. This is due to the fact that trailer speed is given by eqn (2.8), so for equal to 90 the robot speed goes to in nity. Hence the speed of the robot is xed to 1 m/s. The trailer speed is 1 m/s on straight segments and slightly less than 1 m/s on mild turns. However when making tight turns having a xed robot speed stops the system from becoming unstable. So only the yaw rate !r is a control input. The basic idea is to drive each integrator in Fig. 3.1 to a desired value while cancelling out the nonlinearities. This approach seems similar to integrator backstepping, but there is a major di erence: Instead of 8 taking the derivative at each integrator, a pseudo feedback is assumed across it. The proposed method eliminates the need for numerical di erentiation (which is highly sensitive to small uctuations in data) and coordinate transformation (which may not be possible in every case). The system model has three state variables, so the design requires three iterative steps. Desired Input Calculation of First Integrator In this step the value of the rst integrator has to driven to the desired value (Lateral error). This done by assuming a pseudo feedback across integrator 1. This gives rise to the following equations. _et = Lerr (3.3) Substituting (2.9) in (3.3) we get: Vt sin( t1) = Lerr (3.4) where t1 is the desired heading to eliminate the heading error. Solving for t1 gives: t1 = sin 1 L err Vt (3.5) For controlling the trailer on the path the heading error also has to be controlled. Let k1 and k2 be the gains on the heading error and the lateral error. The desired value of t is given by tdesired = k1 t1 +k2 desired (3.6) k1 and k2 can be chosen without any constraint as the sin 1 term limits the controller output. The value of sin 1 remains 1:517 for any value less than 1 and in the 9 same way the value of sin 1 remains 1:517 for any value greater than 1. This implies that the controller keeps the desired control output to +or 90 until the tdesired value is between 1:517 and 1:517. Desired Input Calculation of Second Integrator In this step the value of the Second integrator has to driven to the desired value (obtained in previous step) . This done by assuming a pseudo feedback across integrator 2. This gives rise to the following equations. tfeedback = k1 t1 +k2 desired t (3.7) or tfeedback = k1 t1 +k2 desired tk 2 : (3.8) Substituting (3.2) in (3.8) we get: tfeedback = k1 t1 +k2 terr (3.9) _ t = tfeedback (3.10) Substituting (2.11) in (3.10) we get: VrL t sin( ) Lr!rL t cos( ) = tfeedback (3.11) Recall the trigonometric identity asin(x) +bcos(x) = K sin(x+ ) (3.12) 10 where K =pa2 +b2 (3.13) = sin 1( bpa2 +b2): (3.14) Applying (3.12) we can rewrite (3.11) as K sin( 1 + ) = tfeedback (3.15) where K = 1L t ( p Vr2 +Lr2!r2 (3.16) = sin 1( VrV r2 +Lr2!r2 )): (3.17) In (3.15), variable 1 is the desired value of the hitch angle . Solving for 1 we get 1 = sin 1( tfeedbackLtp Vr2 +Lr2!r2 ) sin 1( Vrp Vr2 +Lr2!r2 ) (3.18) which is the desired value of integrator 3. Desired Input Calculation of Third Integrator To drive the integrator 3 to its desired value, the hitch angle error err is required. The err is given by the following equation: err = sin 1 tfeedbackLtp Vr2 +Lr2!r2 ! sin 1 Vrp Vr2 +Lr2!r2 ! (3.19) 11 Assuming that there exists a pseudo feedback across integrator 3, we get: _ = err: (3.20) Substituting (2.4) in (3.20) we get: VrL t sin( ) Lr!rL t cos( ) !r = err (3.21) Solving for !r we get: !r = err VrL t sin( ) Lr!rL t cos( ) (3.22) Substituting (3.19) in (3.22) we get !r = sin 1 tfeedbackLtp Vr2 +Lr2!r2 ! sin 1 Vrp Vr2 +Lr2!r2 ! VrL t sin( ) Lr!rL t cos( ) (3.23) Substituting (3.9) in (3.23) we get: !r = sin 1( k1 t1 k2 terrp Vr2 +Lr2!r2 Lt) sin 1( Vrp Vr2 +Lr2!r2 ) VrL t sin( ) Lr!rL t cos( ) (3.24) 12 Finally substituting (3.5) in (3.24) yields the control input in terms of lateral error and heading error: !r = sin 1 k 1(sin 1( LerrVt )) k2 terrp Vr2 +Lr2!r2 ! sin 1 Vrp Vr2 +Lr2!r2 Lt ! VrL t sin( ) Lr!rL t cos( ) (3.25) Since the arcsine function sin 1 is valid over the interval ( 2; 2), the sign of lateral error has to be ipped when the robot faces in the opposite direction. 3.2.1 Controller Gains To calculate the controller gains the Butterworth gains and Linear Quadratic Regulator (LQR) gains were examined [20]. After examining both the methods it was observed that better control was obtained using LQR method. Hence LQR method was chosen to obtain the value of k1 and k2. For LQR calculations the system is assumed to be a three state linear system. This assumption is made as all the nonlinearities of the system are cancelled. It was seen that with di erent weights placed on the two states, di erent output characteristics can be obtained. For example if the weight on lateral error is greater than that on the heading error, the trailer regulates faster to the path but oscillates on the path. In contrast, if the weight on the heading error is greater than lateral error, the system does better line following but takes a long time for the system to regulate onto the path if the initial errors are large. Hence two di erent gains were calculated. Gain 1 was calculated with greater weight placed on lateral error while Gain 2 was calculated with greater weight placed on heading error. 13 Using the lqr() command in MATLABR , LQR gains k1 and k2 were calculated for di erent weight matrices. The Q matrix for Gain 1 Qx = 2 66 66 64 100 0 0 0 10 0 0 0 1 3 77 77 75 Ru = 1 (3.26) which resulted in closed loop poles at s1 = 2:0563 (3.27) s2 = 1:2746 +|1:7996 (3.28) s3 = 1:2746 |1:7996: (3.29) The following gains are obtained k1=10 k2=10 The Q matrix for Gain 2 Qx = 2 66 66 64 10 0 0 0 100 0 0 0 1 3 77 77 75 Ru = 1 (3.30) which resulted in closed loop poles at s1 = 0:3164 (3.31) s2 = 2:2853 +|2:1847 (3.32) s3 = 2:2853 |2:1847: (3.33) The following gains are obtained k1=3.1623 k2=11.4415 14 3.2.2 Gain Scheduling As discussed in the previous section, it was observed that the controller gains that provided the best path following did not produce the fastest regulation to the path when the initial error were large. Gain scheduling was performed to retain the best characteristics of Gain 1 and Gain 2 [21]. Another design issue was to obtain the gain scheduling point. As the controller and the system were non linear the linear methods (like Nyquist criteria) to determine the stability could not be used to obtain the scheduling point. Hence by experimental and simulation result it was seen that the scheduling point was to be set at 4 meters (lateral Error) for the Gain 2 to be able to smoothly bring the trailer on the path without any oscillation 3.3 Simulation of the Controller To test the validity of the control approach, the system is simulated with the proposed Hybrid Backstepping Controller(HBC). The system was simulated for vari- ous initial conditions. The system was simulated with xed gains and also using gain scheduling. The system was simulated for approximately 50 seconds. Fixed gain performance As can be seen in Fig. 3.2, the robot and trailer have an initial orientation that is 270 from the path and the lateral error is 6 m. It can be seen that the robot and trailer take a long north distance to regulate onto the path but is able to follow the path better when it reaches the path. In Fig. 3.3, the lateral error and heading error of the system is plotted. It can be seen that the system does not have any steady state error and the system shows good tracking. 15 Figure 3.2: System position with xed gains (simulated) 16 Figure 3.3: Lateral error and heading error (simulated) Gain scheduled performance Now the simulation is run with gain scheduling. In Fig. 3.4, the initial position is same as in Fig. 3.2. But it can be clearly seen that the robot-trailer system makes a sharper turn and regulates quicker to the path. The path following is as good as before but the regulation to the path is faster than with a xed gain. Trailer backing performance One of the greatest features of this control law is that it can control the trailer in both directions without any design modi cations. In contrast, other known methods require some additional design modi cations. For example [22] introduces a virtual robot that pulls the robot in the reverse direction. In Fig. 3.5, the backing per- formance of the controller is demonstrated. Initially the robot and trailer have an orientation of 135 to the path. The initial lateral error is 6 m. It can be seen that 17 Figure 3.4: System Position with Gain Scheduling (simulated) the controller is su cient to control the system while backing the trailer onto the path. 3.4 Conclusion These simulations show that the Hybrid Backstepping Controller(HBC) is su - cient in controlling the system for any initial condition. Further more they demon- strate that gain scheduling provides more dynamic control than xed gains. The simulation also suggest that the controller is dexterous at backing the trailer onto the path. 18 Figure 3.5: Backing of trailer (Simulated) 19 Chapter 4 Experimental Results 4.1 The Robot-trailer System The Autonomous Robot-Trailer system used for this research is built on a Seg- way Robotics Mobility Platform (RMP) 400. This is a four-wheeled, di erential-drive robot capable of carrying ample payload [23]. The RMP 400 communicates using a Universal Serial Bus (USB) interface. The inputs to RMP 400 are turn and veloc- ity command that are sent in counts. The RMP 400 has inbuilt speed and yaw controller that controls the speed of individual motors. A NovAtel SPANTM Global Navigation Satellite System/Inertial Navigation System (GNSS/INS) is present on the robot [24]. Using real-time kinematic (RTK) position corrections from a base station,the GNSS/INS provides accurate knowledge of the vehicle position and ori- entation [25]. Another Novatel Global Positioning System(GPS) antenna is mounted on the trailer. This provides the position information of the trailer. The position and velocity obtained from the GPS are very accurate [26]. A rotary encoder positioned at the trailer hitch of the vehicle provides the angle of the trailer with respect to the robot. The orientation of the trailer is calculated from the orientation of the robot provided by the GNSS/INS and the hitch angle provided by the rotary encoder. This provides more accurate orientation than the orientation provided by the trailer GPS alone. To minimise the electromagnetic interference to geophysical sensors , the trailer is constructed using berglass. For the purpose of this research the orientation from the trailer GPS, hitch angle information from the rotary encoder and the orientation information calculated from the hitch angle and the orientation information from the GNSS/INS is used. 20 Figure 4.1: Autonomous geophysical surveying system described in this thesis Figure 4.1 shows the robot-trailer system. As it can be seen in the gure the robot-trailer system has a hitch at the center of the robot; hence the robot hitch length (Lr) is zero.The linear controller [4] previously developed on this robot trailer system provides the scope for a direct comparison between the two controllers. 4.2 Experimental Setup Experiments were performed at Auburn University Solar House eld. Several di erent paths and initial conditions were chosen to demonstrate the capabilities of the Hybrid Backstepping Controller(HBC) to regulate the robot-trailer system to the desired survey lines. 21 Figure 4.2: Robot-Trailer Position with xed gains (experimental) 4.3 Di erent Initial Conditions The e ectiveness of the control law is tested for di erent initial conditions. This is done to show the versatility of this control law in controlling the robot-trailer system to the path. Fixed gain performance is also compared to the Gain scheduling performance. Fixed gain performance In Fig. 4.2, the robot and trailer have an initial orientation of 180 with respect to the desired heading and the trailer has a lateral error of 9 m. It can be seen that robot-trailer system shows good tracking of the path with minimal steady state error. It can be seen that controller automatically performs both clockwise turns (see Fig. 3.2) as well as counterclockwise turns (see Fig. 4.2), depending on the shortest route to the path. 22 Figure 4.3: Robot-Trailer Position with Gain Scheduling (experimental) Figure 4.4: Lateral Error And Heading Error (experimental) 23 Figure 4.5: Backing of the trailer (experimental) Gain scheduled performance In Fig. 4.3, the robot and trailer have an initial orientation of 180 with respect to the desired path { the same condition as for the xed gain experiment. However, the initial lateral error is increased from 9 m to 10 m. It can be seen that with gain scheduling the robot-trailer system makes a sharper turn and is able to get on the desired path faster than in the xed-gain case (compare to Fig. 4.2). Fig. 4.4 shows the plot of lateral error and heading error for the gain-scheduled case. It can be seen that the Hybrid Backstepping Controller(HBC) seems to control the system well. The robot and trailer are initially located at (0;0) and (0;3:3), respectively. Their initial headings are 270 from the path. The robot-trailer system moves to the right and downward in the plot. 24 Trailer backing performance In Fig. 4.5, the trailer backing performance of the controller is shown. Though the controller is able to back the trailer onto the path there are some oscillations in the position. The oscillation of the trailer is about 0.50 m. It is also seen that the amplitude of the oscillations reduce slowly. 4.4 Path Following The path following capability of the controller is examined in this case. The trailer is place on the path with a very small lateral error and a small heading error. It can be seen in the Figure 4.6 that the system is able to follow the desired path. It is capable of making the turns. In Figure 4.6 it can be seen that the controller is able to control the robot trailer system on the path. It can be further seen that the controller is able to make the turns. Hence the controller appears to be able to perform path following. 4.5 Comparison with Linear Controller To put the performance of the Hybrid Backstepping Controller(HBC) into per- spective, the performance of the Hybrid Backstepping Controller(HBC) is compared to the full state feedback controller. 4.5.1 Di erent Initial Conditions The two controllers are examined for di erent initial conditions. It is apparent that the Hybrid Backstepping Controller(HBC) can control the system over a wide range of initial conditions where the linear controller is inadequate. 25 Figure 4.6: Plot of Robot and Trailer path Table 4.1: Comparison of the two controllers for di erent initial conditions Initial Condition Full State Feedback HBC With very small heading and Can perform path following Can perform path following lateral error With large lateral error and Causes the system to become Can regulate the system back very small heading error unstable if lateral error onto the path for any lateral is greater than two meters error. With large heading error and Causes the system to become Can regulate the system back very small lateral error unstable if heading error onto the path for any heading is greater than 30 degrees. error. With large lateral and heading Causes the system to become Can regulate the system back error unstable thereby causing the onto the path without trailer to jackknife jackkni ng the trailer. With initial orientation of Causes the trailer to Can smoothly turn the robot- the robot in opposite jackknife trailer around without direction to the path jackkni ng the trailer Backing the trailer onto the Cannot control the trailer Can back the trailer onto path in reverse direction the path. 26 Table 4.2: Comparison of the two controllers while path following Characteristic Full State Feedback HBC Settling time Shorter larger lateral error Similar Similar Heading error Similar Similar Lateral error coming Slightly Smaller Slightly Greater out of a turn Heading error coming Similar Similar out of a turn 4.5.2 Path Following In this section the path following capability of the two controllers is examined. The nonlinear and linear controller is used to control the system on the same path to get a direct comparison of path following performance. The initial lateral and heading error are kept very small. Fig 4.7 shows the path following performance of the linear controller and Fig 4.8 shows the path following performance of the non linear controller. It can be seen that though both controllers are able to control the trailer on the path, but the Hybrid Backstepping Controller(HBC) takes longer to stabilize after coming out of the turn. In Fig 4.9 the comparison of the lateral error for the two controller is shown. It can be seen that both the controllers have similar type of lateral error characteristics. However it can be seen that the lateral error after coming out of the turn for the Hybrid Backstepping Controller(HBC) is not as good as that of the linear controller. Fig 4.10 shows the heading error of the two controller. The heading error characteristics are similar for the two controllers. However the performance of the linear controller while path following is better than that of the Hybrid Backstepping Controller(HBC). 27 Figure 4.7: Plot of robot-trailer with Linear controller Figure 4.8: Plot of robot-trailer with HBC 28 Figure 4.9: Comparison of Lateral error Figure 4.10: Comparison of Heading error 29 Chapter 5 Conclusions The conclusions drawn from the simulations are veri ed by the experimental results. The innovative nonlinear controller presented in this thesis is shown to be su cient to control the robot-trailer system. The selective amalgamation of the integrator backstepping method and feedback linearization not only avoids the design di culties of both the approaches but also provides excellent control. The Hybrid Backstepping Controller(HBC) is seen adequate in controlling the robot-trailer system on the path. It is also seen that the Hybrid Backstepping Controller(HBC) is capable of controlling the robot-trailer system from any initial condition. In contrast to the state-of-art-methods, this control law is able to control the system in both the forward and reverse direction without any modi cation.The author believes that this hybrid control law can be applied to other systems with similar success. 5.1 E ectiveness of the Hybrid Backstepping Controller(HBC) While path following The Hybrid Backstepping Controller(HBC) is able to control the robot-trailer system on the path and is able to make sharper turns than the state feedback con- troller. However it is seen that the while coming out of turns, Hybrid Backstepping Controller(HBC) takes longer to regulate the system onto the path. This is an unde- sirable characteristic. The author believes that this can be corrected by ne tuning the gains. 30 For di erent initial condition The Hybrid Backstepping Controller(HBC) is capable to handle any initial con- dition. It is capable of handling any initial heading or lateral error, while the linear controller is only capable of controlling if the heading and lateral error are small. This is a very desirable characteristic of the controller. This gives the system freedom that is not possible with the linear controller. While backing trailer The simulation suggested that the Hybrid Backstepping Controller(HBC) will smoothly back the trailer onto the path. However the experimental data showed that though Hybrid Backstepping Controller(HBC) is able to back the trailer onto the path there are some oscillations and it would take a long time for the oscillations to die down. Di erent gains were tried to make the system converge quickly, however the optimal gain could not be found. The author believes that this is due to the un- modeled dynamic e ects present in the experimental setup. Despite the de ciencies, the Hybrid Backstepping Controller(HBC) is able to control the system over a wide variety of conditions. 5.2 Future Work The Hybrid Backstepping Controller(HBC) has been shown to control the robot- trailer system accurately on the path and also control from any initial condition. However several improvements can be made to the controller to improve path following ability and backing capabilities of the trailer. 5.2.1 Full dynamic model As discussed in Chapter 2, the dynamic model used is only based on kinematics of the system. While a kinematic model is seen to be su cient for linear controller to 31 control the system on the path, the exclusion of dynamic e ects can have a pronounced e ect on the controller performance while backing the trailer. This is due to the fact that while backing the trailer, the system is inherently unstable and the unmodeled dynamics further reduces the e ectiveness of the controller. Future work will include the modeling of dynamics like wheel slip, moment of inertia and rolling friction. Future work will also incorporate the delays in the system. Modeling these dynamics should improve the ability of the controller to control the robot-trailer system while backing the trailer. 5.2.2 Choosing Controller Gains Butterworth gains and LQR method were examined for this thesis. The LQR tuning should be improved for better path following control. Di erent weight values were examined during the design process, but better gains are required to reduce the oscillations after the turns. Optimal gains for backing the trailer on to the path need to be investigated. Some other methods for choosing controller gains should also be examined. 5.2.3 Gain Scheduling A more complex gain scheduling should also be examined, as it is apparent the gain sets currently used do not provide as good path following as the linear controller. It is the authors belief that a separate gain should be used while path following. This would improve the path following performance of the controller. However the gains that provide the best path following may not be able to regulate the system onto the path fast enough when the errors are large. Hence by having more sets of gains all the desired features can be retained. Proposed three set gains can be seen in Table 5.1 32 Table 5.1: Proposed gain scheduling Gain Schedule Lateral error values Gain 1 For large lateral errors > 4 meters Gain 2 For medium lateral errors 0.5 meters to 4 meters Gain 3 (Similar to Linear For small lateral errors < 0.5 meters gain values) 5.2.4 Design of a Nonlinear Estimator The nonlinear controller used in this thesis require all the state variables. How- ever this also introduces multiple sources of measurement noise. For example the hitch angle values obtained from the rotary encoder has more noise than the GPS values: this introduces error in the controller. This problem can be reduced by estimating some of the states that cannot be measured as accurately as others. 5.2.5 E ectiveness on other problems The e ectiveness of the control law should be veri ed on other types of vehicles like Ackermann drive robot trailer system. The author believes that the control law should be su cient to control the Ackermann drive robot. 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