Nonlinear Control of Vehicles and Robots

Chapter 1 introduces some general definitions for autonomous vehicles and robots and it also gives a short historical review of the most important steps of robot and vehicle control development. The motivation of the research work summarized in this book is also presented here. The final part shortly outlines the structure of the book.

Chapter 2 focuses on the basic nonlinear control methods. First, the main classes of nonlinear systems are summarized, then some examples are presented how to find the dynamic model of simple systems and an overview of the stability methods is given. It is followed by an introduction to the concepts of some often used nonlinear control methods like input/output linearization, flatness control, backstepping and receding horizon control.

Chapter 3 focuses on the dynamic modeling of the robots and vehicles in unified approach. First, the kinematic and dynamic model of the rigid body is discussed which is the basis for the further investigations. For robots, the dynamic model is developed using Appell’s equation and Lagrange’s equation. For ground cars, a nonlinear dynamic model, two nonlinear input affine approximations and a linearized model is derived. For airplanes, first the usual navigation frames are shown. Then the kinematic and dynamic equations are presented considering aerodynamic and gyroscopic effects. Finally, the main flying modes, the trimming and linearization principle and the concept of the parametrization of aerodynamic and trust forces for identification purposes are outlined. For surface and underwater ships, first the rigid body equations are developed, then the hydrodynamic and restoring forces and moments are shown. A short introduction is given for wind, wave and current models. Finally, the results are summarized in the kinematic model and the dynamic models in both body frame and NED frame for ships.

Chapter 4 focuses on the most important control methods of industrial robots. From the control methods not using the dynamic model of the robot, first the decentralized three-loop cascade control is presented. This classical approach tries to apply linear control methods but its precision and allowable speed is limited. Then advanced control methods follow based on the robot dynamic model for high precision applications. The computed torque technique is practically nonlinear decoupling in joint space. In Cartesian space, first the transformation of sensory information and the realization of spatial generalized force by joint torques is discussed. It is followed by the nonlinear decoupling of the free motion in Cartesian space. The results of free motion are generalized in hybrid position and force control by using the operational space method. For the online identification of the robot parameters, the self-tuning adaptive control method is suggested. For the realization of nonsmooth path with sharp corners, a robust backstepping method is presented. The methods are illustrated in simulation results.

Chapter 5 focuses on the high level nonlinear control methods for cars (ground vehicles). The main tasks are path design, control algorithm development and control realization. These steps are demonstrated in the frame of the development of a collision avoidance system (CAS). From the large set of path design methods, the principle of elastic band is chosen. The car can be modeled by full (non-affine) or simplified (input affine) nonlinear models. Two nonlinear control methods are presented, the differential geometric approach (DGA), known as input-output linearization, and the nonlinear predictive control. For state estimation, Kalman filters and measurements of two antenna GPS and Inertial Measurement Unit (IMU) are used. The methods are illustrated through simulation results. Two software approaches are presented for the realization of the control algorithms. The first uses the MATLAB Compiler converting the whole control system to standalone program running under Windows or Linux. The second uses MATLAB/Simulink and Real-Time Workshop (RTW) to convert the control system for target processors.

Chapter 6 focuses on the predictive control of airplanes and the backstepping control of quadrotor helicopters. The airplane is an LPV system which is internally stabilized by disturbance observer and externally controlled by high level RHC. The control design is illustrated for the pitch rate control of an aircraft. The internal system is linearized at the beginning of every horizon and RHC is applied for the resulting LTI model. The RHC controller contains integrators. The management of hard and soft constraints and the use of blocking parameters are elaborated. Simulation examples show the disturbance suppressing property of disturbance observer and the efficiency of high level RHC. For the indoor quadrotor helicopter, two (precise and simplified) dynamic models are derived. The sensory system contains on-board IMU and on-ground vision subsystems. A detailed calibration technique is presented for IMU. The vision system is based on motion-stereo and a special virtual camera approach. For state estimation, two level EKF is used. The quadrotor helicopter is controlled using backstepping in hierarchical structure. The embedded control system contains separate motor and sensor processors communicating through CAN bus with the main processor MPC555 realizing control and state estimation. Simulation results are shown first with MATLAB/Simulink then during the hardware-in-the-loop test of the embedded control system where the helicopter is emulated on dSPACE subsystem.

Chapter 7 focuses on the dominant control methods of ships. First, the typical structure of the control system is shown. A short overview is given about the path design methods including Line-of-Sight guidance and wave filtering. The state estimation using GPS and IMU is discussed and observers are suggested to solve the problem. From the wide spread control methods the acceleration feedback with nonlinear PD, the nonlinear decoupling both in body and in NED frames, the adaptive feedback linearization and the MIMO backstepping in 6 DOF are presented including stability considerations. An overview is given about the most common actuators and the principles for control allocations. Simulation results are presented for the backstepping control of a multipurpose naval vessel including surge, sway, roll and yaw interactions in the nonlinear dynamic model.

Chapter 8 focuses on the formation control of unmanned ground and marine vehicles moving in horizontal plane. For stabilization of ground vehicles in formation, the potential field method is applied. The controller consists of three levels, the low level linearizing controller, the high level formation controller and the medium level stabilizing controller. The chosen potential function is structured such that each vehicle asymptotically approaches its prescribed position in the formation while a minimal distance to other vehicles is guaranteed. Simulation results illustrate the theory for UGVs. For stabilization of marine vehicles in formation, the passivity theory is applied. The control structure can be divided into synchronization level and the level of control subsystems stabilizing the different vehicles. The communication topology between the members of the formation is described by a graph. Only vehicles connected in the graph exchange their synchronization parameters. The stability of the formation is proven based on passivity theory and the Nested Matrosov Theorem. Simulation results illustrate the theory for UMVs.

The first part of the Chap. 9 presents the basic theoretical notions from the field of modeling and stability of nonsmooth systems. Afterward, it focuses on modeling of two nonsmooth nonlinearities that can influence the performances of mechanical control systems: friction and backlash. The most important static and dynamic friction models, that are applied in friction compensation algorithms, are discussed. After presenting the well known friction models from the literature, a novel, piecewise linearly parameterized model is introduced based on which the problem of friction compensation with unknown friction parameters can easily be solved. At the end of the chapter it is presented, how the backlash type nonlinearity influences the motion of mechanical systems.

Chapter 10 introduces a hybrid model for mechanical control systems with Stribeck friction and backlash, using which the stability of these systems can be analyzed. The model is applied for state feedback controller design. Two types of limit cycles, that can appear in these control systems due to wrong controller parameterization, are analyzed in the second part of the chapter, limit cycles around zero velocity and around Stribeck velocities.

Chapter 11 deals with the identification of nonsmooth mechanical nonlinearities and tracking control of mechatronic systems in the presence of friction. First, a friction and backlash measurement and identification method is presented for robotic manipulators that can be performed in closed loop using velocity controller. It is shown, how the procedure can be extended for hydraulic actuators. Second, a control algorithm is introduced for underactuated system that applies friction compensation. Afterward, an adaptive friction and payload compensation algorithm is presented for robotic systems. Both the friction identification and friction compensation algorithms are based on the piecewise linear friction model introduced in Sect. 9.​4.

The final chapter summarizes the introduced modeling and control techniques which were presented in this book. The need of development of such control algorithms which take into consideration the smooth and nonsmooth nonlinearities that appear in the model of robotic systems and vehicles was a motivating factor of this work. Such algorithms were presented that are theoretically founded and at the same time are implementable real-time. The second part of the chapter enumerates some possible future research directions.