Trajectory Tracking

We consider planning and control problems for underactuated manipulators, a special instance of mechanical systems having fewer input commands than degrees of freedom. This class includes robots with passive joints, elastic joints, or flexible links. Structural control properties are investigated, showing that manipulators with passive joints in the absence of gravity are the most difficult to control. With reference to

Trajectory tracking is an important aspect of autonomous vehicles. The idea behind trajectory tracking is the ability of the vehicle to follow a predefined path with zero steady state error. The difficulty arises due to the nonlinearity of vehicle dynamics. Therefore, this paper proposes a stable tracking control for an autonomous vehicle. An approach that consists of steering wheel control and lateral control is introduced. This control algorithm is used for a non-holonomic navigation problem, namely tracking a reference trajectory in a closed loop form. A proposed future prediction point control algorithm is used to calculate the vehicle’s lateral error in order to improve the performance of the trajectory tracking. A feedback sensor signal from the steering wheel angle and yaw rate sensor is used as feedback information for the controller. The controller consists of a relationship between the future point lateral error, the linear velocity, the heading error and the reference yaw rate. This paper also introduces a spike detection algorithm to track the spike error that occurs during GPS reading. The proposed idea is to take the advantage of the derivative of the steering rate. This paper aims to tackle the lateral error problem by applying the steering control law to the vehicle, and proposes a new path tracking control method by considering the future coordinate of the vehicle and the future estimated lateral error. The effectiveness of the proposed controller is demonstrated by a simulation and a GPS experiment with noisy data. The approach used in this paper is not limited to autonomous vehicles alone since the concept of autonomous vehicle tracking can be used in mobile robot platforms, as the kinematic model of these two platforms is similar.

In this study, the method of direct-motion parallel parking is introduced both in theory and in simulations for the vehicle with and without a trailer. The results show that not only a vehicle with a trailer park in parallel using direct motion, but also a vehicle with a trailer park in parallel in direct motion after the final value of the orientation angle is satisfied. Simulation results are given to verify the effectiveness of the proposed method.

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Control of an industrial robot includes nonlinearities, uncertainties and external perturbations that should be considered in the design of control laws. This paper presents a control strategy for robot manipulators, based on the coupling of the fuzzy logic control with the so-called sliding mode control, SMC, approach. The motivation for using SMC in robotics mainly relies on its appreciable features, such as design simplicity and robustness. Yet, the chattering effect, typical of the conventional SMC, can be destructive. In this paper, this problem is suitably circumvented by adopting an adaptive fuzzy sliding mode control, AFSMC, approach with a proportional-integral-derivative, PID sliding surface. For this proposed approach, we have used a fuzzy logic control to generate the hitting control signal. Moreover, the output gain of the fuzzy sliding mode control, FSMC, is tuned on-line by a supervisory fuzzy system, so the chattering is avoided. The stability of the system is guaranteed in the sense of the Lyapunov stability theorem. Numerical simulations using the dynamic model of a 3 DOF planar rigid robot manipulator with uncertainties show the effectiveness of the approach in trajectory tracking problems. The simulation results that are compared with the results of conventional SMC with PID sliding surface indicate that the control performance of the robot system is satisfactory and the proposed AFSMC can achieve favorable tracking performance, and it is robust with regard to uncertainties and disturbances.

The paper describes a new and novel approach to control a nonholonomic wheeled mobile robot (WMR) robustly with reference to its trajectory tracking capability in the wake of introduced disturbances. The workspace of a mobile robot is not always ideal but more often than not, filled with disturbances (known or unknown) such as inherent friction, irregular surface terrain, uncertainties, and parametric changes. An intelligent active force control (IAFC) scheme incorporating fuzzy logic has been proposed in the study to counter the disturbances and consequently improve the trajectory tracking characteristic of the system. In the study, IAFC scheme is employed together with a resolved acceleration control (RAC) that has been shown to provide a very robust and accurate performance of the WMR. Fuzzy logic is explicitly used for the estimation of the inertia matrix that is required in the inner feedback control loop of the IAFC scheme. The robustness and effectiveness of the proposed control scheme are investigated considering various forms of loading and operating conditions. The IAFC scheme has also been compared to two other control methods for the purpose of benchmarking.

The design of reliable navigation and control systems for Unmanned Aerial Vehicles (UAVs) based only on visual cues and inertial data has many unsolved challenging problems, ranging from hardware and software development to pure control-theoretical issues. This paper addresses these issues by developing and implementing an adaptive visionbased autopilot for navigation and control of small and mini rotorcraft UAVs. The proposed autopilot includes a Visual Odometer (VO) for navigation in GPS-denied environments and a nonlinear control system for flight control and target tracking. The VO estimates the rotorcraft ego-motion by identifying and tracking visual features in the environment, using a single camera mounted on-board the vehicle. The VO has been augmented by an adaptive mechanism that fuses optic flow and inertial measurements to determine the range and to recover the 3D position and velocity of the vehicle. The adaptive VO pose estimates are then exploited by a nonlinear hierarchical controller for achieving various navigational tasks such as take-off, landing, hovering, trajectory tracking, target tracking, etc. Furthermore, the asymptotic stability of the entire closed-loop system has been established using systems in cascade and adaptive control theories. Experimental flight test data over various ranges of the flight envelope illustrate that the proposed vision-based autopilot performs well and allows a mini rotorcraft UAV to achieve autonomously advanced flight behaviours by using vision.

A decoupling trajectory tracking method for gliding reentry
vehicles is presented to improve the reliability of the guidance
system. Function relations between state variables and control
variables are analyzed. To reduce the coupling between control
channels, the multiple-input multiple-output (MIMO) tracking
system is separated into a series of two single-input
single-output (SISO) subsystems. Tracking laws for both velocity
and altitude are designed based on the sliding mode control (SMC).
The decoupling approach is verified by the Monte Carlo
simulations, and compared with the linear quadratic regulator
(LQR) approach in some specific conditions. Simulation results
indicate that the decoupling approach owns a fast convergence
speed and a strong anti-interference ability in the trajectory
tracking.

Citation: Zixuan Liang, Zhang Ren, Xingyue Shao. Decoupling
trajectory tracking for gliding reentry vehicles. IEEE/CAA
Journal of Automatica Sinica, 2015,  2(1): 115-120

This paper presents the design of a sliding mode control (SMC) for trajectory tracking of an unmanned aerial vehicle (UAV), quadrotor. A simplified model of the quadrotor is used for the controller design. The robustness of the controller is verified through simulations, and also through data analysis from the experiments in the 3DR Arducopter platform. The SMC algorithms are implemented in a microcontroller that communicates with a human machine interface (HMI), which monitors the behavior and stability of the state variables. The results show effectiveness of the control technique for maintaining stability in the quadrotor under different operating scenarios.

We propose a vision-based position control method, with the purpose of providing some level of autonomy to a quad-rotor unmanned aerial vehicle. Our approach estimates the helicopter X-Y-Z position with respect to a landing pad on the ground. This technique allows us to measure the position variables that are difficult to compute when using conventional navigation systems, for example inertial sensors or Global Positioning Systems in urban environment or indoor. We also present a method to measure translational speed in a local frame. The control strategy implemented is based on a full state feedback controller. Experimental results validate the effectiveness of our method.

This paper presents the design of a vision-based controller for an underactuated, unmanned aerial vehicle (UAV) equipped with a pan-tilt camera unit (PTCU) to achieve the objective of following a leader vehicle autonomously. The relative position and orientation information is obtained from the monocular camera utilizing homography-based techniques. The proposed controller, built upon Lyapunov design methods, achieves uniform ultimate bounded (UUB) tracking. As an extension, it is also demonstrated that the approach used in the development of the control strategy for the leaderfollower problem can be applied, with a few modifications, to the problem of trajectory tracking, where the desired trajectory is described as a sequence of images taken, for example, by the on-board camera during a previous flight.

Recently developed optimization and nonlinear control strategies are applied to ride a multi-body motorcycle model along a specified path with an associated velocity profile. The resulting control scheme is based on three main pillars: a dynamic inversion procedure to compute the input-state trajectories corresponding to a desired maneuvering task, an inverse optimal control heuristic for designing closed loop dynamics, and a maneuver regulation controller to overcome limitations of standard trajectory tracking scheme. The controller (virtual rider) is implemented on a commercial simulation software. Simulation results are presented.

The Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control (STARMAC), a fleet of quadrotor helicopters, has been developed as a testbed for novel algorithms that enable autonomous operation of aerial vehicles. This paper develops an autonomous vehicle trajectory tracking algorithm through cluttered environments for the STARMAC platform. A system relying on a single optimization must trade off the complexity of the planned path with the rate of update of the control input. In this paper, a trajectory tracking controller for quadrotor helicopters is developed to decouple the two problems. By accepting as inputs a path of waypoints and desired velocities, the control input can be updated frequently to accurately track the desired path, while the path planning occurs as a separate process on a slower timescale. To enable the use of planning algorithms that do not consider dynamic feasibility or provide feedforward inputs, a computationally efficient algorithm using space-indexed waypoints is presented to modify the speed profile of input paths to guarantee feasibility of the planned trajectory and minimum time traversal of the planned. The algorithm is an efficient alternative to formulating a nonlinear optimization or mixed integer program. Both indoor and outdoor flight test results are presented for path tracking on the STARMAC vehicles.

In this paper, the tracking control of a underactuated quadrotor aerial vehicle is presented where position and yaw trajectory tracking is achieved using feedback control system. The control design is complicated by considering parametric uncertainty in the dynamic modeling of the quadrotor aerial-robot. Robust control schemes are then designed using a Lyapunovbased approach to compensate for the unknown parameters in each dynamic subsystem model. Lyapunov-type stability analysis suggests a global uniform ultimately bounded (GUUB) tracking result.

This paper presents recent work concerning a small tiltrotor aircraft with a reduced number of rotors. After several attempts, the final design consists of two propellers mounted laterally. The direction of the thrust can be redirected by tilting the propellers laterally and longitudinally. A theoretical analysis of this mechanism proves its effectiveness and the experimental results show that this aerodynamical configuration is very promising. A model of the full birotor dynamics is also presented in this paper and a controller based on the backstepping procedure is synthesized performing also an autonomous flight.

The robotic arm used in minimally invasive surgery enters patient’s body through a port which constrains its end-effector translation along two axes. We aim to achieve the minimally-invasive operations using a general articulated robotic arm (GARA). The algorithm is applicable to articulated robotic arm independent of its design; given only end-link is constrained. Geometric transformations based on the constraints acting on the end-link coupled with kinematic-relations obtained using conventional techniques, were used to drive a simulated 6-DOF GARA for minimally-invasive operations. The method was verified by tracing predefined planar and 3D trajectories using this simulated arm. The mean deviation of the traced trajectories was of the order of 10-03 cm and the mean absolute error in maintaining remote center-of-motion (RCM) at the port was ~0 (< 10^-15 cm). The proposed method enabled a GARA to perform minimally-invasive operations without specialized design and with sufficient accuracy.

A new approach for navigation of mobile robots in dynamic environments by using Linear Algebra Theory, Numerical Methods, and a modification of the Force Field Method is presented in this paper. The controller design is based on the dynamic model of a unicycle-like nonholonomic mobile robot. Previous studies very often ignore the dynamics of mobile robots and suffer from algorithmic singularities. Simulation and experimentation results confirm the feasibility and the effectiveness of the proposed controller and the advantages of the dynamic model use. By using this new strategy, the robot is able to adapt its behavior at the available knowing level and it can navigate in a safe way, minimizing the tracking error.

We address the problem of position trajectorytracking and path-following control design for underactuated autonomous vehicles in the presence of possibly large modeling parametric uncertainty. For a general class of vehicles moving in either two or three-dimensional space, we demonstrate how adaptive switching supervisory control can be combined with a nonlinear Lyapunov-based tracking control law to solve the problem of global boundedness and convergence of the position tracking error to a neighborhood of the origin that can be made arbitrarily small. The desired trajectory does not need to be of a particular type (e.g., trimming trajectories) and can be any sufficiently smooth bounded curve parameterized by time. We also show how these results can be applied to solve the path-following problem, in which the vehicle is required to converge to and follow a path, without a specific temporal specification. We illustrate our design procedures through two vehicle control applications: a hovercraft (moving on a planar surface) and an underwater vehicle (moving in three-dimensional space). Simulations results are presented and discussed.

Industrial applications of medium-voltage drives impose increasingly stringent performance requirements, particularly with regards to harmonic distortions of the phase currents of the controlled electrical machine. An established method to achieve very low current distortions during steady-state operation is to employ offline calculated optimized pulse patterns (OPP). Achieving high dynamic performance, however, proves to be very difficult in a system operated by OPPs.

In this paper, the kinematic model of nonholonomic differential wheeled mobile robot steering system is established. Based on the model, a nonlinear feedback path tracking controller is proposed, which causes the closed loop system state equation for the robot to have equilibrium condition at the origin. Lyapunov candidate function is used to prove that the closed loop system is asymptotically stable at origin. Simulation results verify the usefulness of the tracking control approach.

The main goal of this study was to compare the performances of genetic algorithm (GA) and ant colony optimization (ACO) algorithm for PID controller tuning on a pressure control process. GA and ACO were used for tuning of the PID controller when predefined trajectory reference signal was applied. Offline learning approach was employed in both GA and ACO algorithms. Realized pressure process dynamic has nonlinear behavior, thus system was modeled by nonlinear auto regressive and exogenous input (NARX) type artificial neural network (ANN) approach. PID controller was also tuned by ZieglerÀNichols (ZÀN) method to compare the results. A cost function was design to minimize the error along the defined cubic trajectory for the GA-PID and ACO-PID controller. Then PID controller parameters (K p , K i , K d ) were found by GA-PID, ACO-PID algorithms, which were adjusted with their optimal parameters. It was concluded that both ACO and GA algorithms could be used to tune the PID controllers in the pressure process with excellent performance. This material is suitable for an engineering course on neural networks, genetic algorithm, ant colony optimization and process control laboratory. ß

Adaptive control algorithms are of interest in flight control systems design not only for their capability to improve performance and reliability but also for handling aerodynamic parameter uncertainties, external disturbances and modeling inaccuracies. In this paper, a direct adaptive sliding mode control is developed for the quadrotor attitude stabilization and altitude trajectory tracking. First, developed controller is applied without considering disturbances and parameter uncertainties. After, a centered white gaussian noise with some parameter uncertainties are added to the considered output vector, mass and inertia matrix, respectively. The synthesis of the adaptation laws is based on the positivity and Lyapunov design principle. Numerical simulations are performed showing the robustness of the proposed control technique.

In the face of large-scale parametric uncertainties, the single-model (SM)-based sliding mode control (SMC) approach demands high gains for the observer, controller, and adaptation to achieve satisfactory tracking performance. The main practical problem of having high-gain-based design is that it amplifies the input and output disturbance as well as excites hidden unmodeled dynamics, causing poor tracking performance. In this paper, a multiple model/control-based SMC technique is proposed to reduce the level of parametric uncertainty to reduce observercontroller gains. To this end, we split uniformly the compact set of unknown parameters into a finite number of smaller compact subsets. Then, we design a candidate SMC corresponding to each of these smaller subsets. The derivative of the Lyapunov function candidate is used as a resetting criterion to identify a candidate model that approximates closely the plant at each instant of time.

This paper presents a solution of the trajectory tracking problem for robotic manipulators using a recurrent high order neural network (RHONN) structure to identify the robot arm dynamics, and based on this model a discrete-time control law is derived, which combines block control and the sliding mode techniques. The block control approach is used to design a nonlinear sliding surface such that the resulting sliding mode dynamics is described by a desired linear system. The neural network learning is performed on-line by Kalman filtering. The local controller for each joint uses only local angular position and velocity measurements. The applicability of the proposed control scheme is illustrated via simulations.

This work proposes an adaptive control scheme applied to single link-flexible manipulators, which combines a feedback controller of the joint angle with an adaptive input shaper updated by an algebraic non-asymptotic identification. The feedback controller is designed to guarantee trajectory tracking of the joint angle, simplifying thus the input shaper, which can be designed for the arm dynamics only. The input shaper is updated by an algebraic identification of the natural frequency corresponding to the first vibration mode of the arm. In addition, the influence of the assumptions adopted to derive the algebraic identification on the performance of the estimation is studied. Finally, the proposed adaptive control strategy is implemented in practice.

In this paper, the flight formation control and trajectory tracking control design of multiple mini rotorcraft systems are discussed. The dynamic model of a mini rotorcraft is presented using the Newton-Euler formalism. Our approach is based on a leader/follower structure of multiple robot systems. The centroid of the coordinated control subsystem is used for trajectory tracking purposes. A nonlinear controller based on separated saturations and a multi-agent consensus algorithm is developed. The analytic results are supported by simulation tests. Experimental results include yaw coordination and tracking only.

Decision making and autonomy Altitude control Mission planning Guidance and control of UAVs Trajectory tracking and path following robust control applications Eectromechanical actuators a b s t r a c t This paper presents an integrated guidance and control design scheme for an unmanned air vehicle (UAV), and its flight test results. The paper focuses on the longitudinal control and guidance aspects, with particular emphasis on the terrain-following problem. An introduction to the mission, and the terrain-following problem is given first. Waypoints for climb and descent are defined. Computation of the reference trajectory in the vertical plane is discussed, including a terrain-following (TF) algorithm for real-time calculation of climb/descent points and altitudes. The algorithm is particularly suited for online computation and is therefore useful for autonomous flight. The algorithm computes the height at which the vehicle should fly so that a specified clearance from the underlying terrain is always maintained, while ensuring that the vehicle's rate of climb and rate of descent constraints are not violated. The output of the terrain-following algorithm is used to construct a smooth reference trajectory for the vehicle to track. The design of a robust controller for altitude tracking and stability augmentation of the vehicle is then presented. The controller uses elevators for pitch control in the inner loop, while the reference pitch commands are generated by the outer altitude control loop. The controller tracks the reference trajectory computed by the terrain-following algorithm. The design of an electromechanical actuator for actuating the control surfaces of the vehicle during flight is also discussed. The entire guidance and control scheme is implemented on an actual experimental vehicle and flight test results are presented and discussed.

a b s t r a c t Rather severe parametric uncertainties and uncertain nonlinearities exist in the dynamic modeling of a parallel manipulator driven by pneumatic muscles. Those uncertainties not only come from the time- varying friction forces and the static force modeling errors of pneumatic muscles but also from the inherent complex nonlinearities and unknown disturbances of the parallel manipulator. In this paper, a discontinuous projection-based adaptive robust control strategy is adopted to compensate for both the parametric uncertainties and uncertain nonlinearities of a three-pneumatic-muscles-driven parallel manipulator to achieve precise posture trajectory tracking control. The resulting controller effectively handles the effects of various parameter variations and the hard-to-model nonlinearities such as the friction forces of the pneumatic muscles. Simulation and experimental results are obtained to illustrate the effectiveness of the proposed adaptive robust controller.

This paper presents a novel methodology for the trajectory tracking control of nonholonomic wheeled mobile robots using multiple identification models. The overall control system includes two stages. In the first stage, a kinematic controller developed by using kinematic model provides the required linear and angular velocities of the robot for tracking a reference trajectory. In the second stage, the required velocities are taken as the inputs to an adaptive dynamic controller which uses multiple adaptive models for the parameter identification. The proposed adaptive dynamic controller is developed using a combined direct and indirect adaptive control approach where both prediction and tracking errors are used for identification. Simulation results show the effectiveness of the proposed combined direct and indirect control scheme and multiple models approach.

This paper deals with the position control of Vertical Take-Off and Landing (VTOL) Unmanned Aerial Vehicles (UAVs) without linear velocity measurements. We propose a multistage constructive procedure, exploiting the cascade property of the translational and rotational dynamics. More precisely, we consider the force as a virtual control input for the translational dynamics, from which we extract the required (desired) system attitude and thrust achieving the tracking objective. Thereafter, the control torque is designed to drive the actual attitude to the desired one. A nonlinear observer, as well as some instrumental auxiliary variables are used to obviate the need for the linear velocity. Global asymptotic stability of the overall closed loop system is achieved. Simulation results are provided to show the effectiveness of the proposed control scheme.

This paper addresses the trajectory tracking control of a nonholonomic wheeled mobile manipulator with parameter uncertainties and disturbances. The proposed algorithm adopts a robust adaptive control strategy where parametric uncertainties are compensated by adaptive update techniques and the disturbances are suppressed. A kinematic controller is first designed to make the robot follow a desired end-effector and platform trajectories in task space coordinates simultaneously. Then, an adaptive control scheme is proposed, which ensures that the trajectories are accurately tracked even in the presence of external disturbances and uncertainties. The system stability and the convergence of tracking errors to zero are rigorously proven using Lyapunov theory. Simulations results are given to illustrate the effectiveness of the proposed robust adaptive control law in comparison with a sliding mode controller.

This paper presents an approach to the control of the KNTU CDRPM using an integrated control scheme. The goal in this approach is achieving accurate trajectory tracking while assuring positive tension in the cables. By the proposed controller, the inherent nonlinear behavior of the cable and the target tracking errors are simultaneously compensated. In this paper asymptotic stability analysis of the close loop system is studied in detail. Moreover, it is shown that the integrated control strategy reduces the tracking error by 80% compared to that of a single loop controller in the considered manipulator. The closed-loop performance of the control topology is analyzed by a simulation study that is performed on the manipulator. The simulation study verifies that the proposed controller is not only promising to be implemented on the KNTU CDRPM, but also being suitable for other cable driven manipulators.

An integral sliding mode controller in the trajectory tracking problem for a nonholonomic mobile robot with kinematic disturbances is proposed. To circumvent the chattering phenomenon, a neural compensator is designed by a modeling technique of Gaussian radial basis function neural networks and does not require offline training. Stability analysis is guaranteed by the Lyapunov method. Simulation results of the proposed approach are provided.

We address the problem of trajectory-tracking and path-following control design for underactuated autonomous vehicles in the presence of possibly large modeling parametric uncertainty. For a general class of vehicles moving in either two or three-dimensional space, we demonstrate how adaptive switching supervisory control can be combined with a nonlinear Lyapunov-based tracking control law to solve the problem of global boundedness and convergence of the position tracking error to a neighborhood of the origin that can be made arbitrarily small. We also show how these results can be extended to the path-following problem, in which the vehicle is required to converge to and follow a path, without a special temporal specification, and once in the path, to satisfy a desired dynamic behavior. We illustrate our design procedures through two vehicle control applications: a hovercraft (moving on a planar surface) and an underwater vehicle (moving in three-dimensional space). Simulations results are presented and discussed.

This work presents a trajectory control for non-redundant serial-link manipulators that is valid for trajectories with ordinary singularities of codimension one and non-ordinary singularities of any codimension. For this purpose, several singularity classifications are considered and a procedure is developed in order to solve the indeterminate motion of non-ordinary singularities. The proposed trajectory control is validated by simulation and by experiments with the six-revolute (6R) industrial robot KUKA KR 15/2.

This paper presents the modeling, identification and control of the 7 degrees of freedom (DoFs) Mitsubishi PA-10 robot arm. The backdrivability, high accuracy positioning capabilities and zero backlash afforded by its harmonic drive transmission, make the PA-10 ideal for precise manipulation tasks. However, the lack of any technical knowledge on the dynamic parameters of its links and the non linear characteristics of friction at its joints, make the development of an accurate dynamic model of the robot extremely challenging. The innovation of this research focuses on the development of the full dynamic model of the PA-10 robot arm, the development of a new non linear model for the friction at its joints, the estimation of the stiffness characteristics of its joints and finally the full identification of the dynamic parameters of the robot arm. The accuracy of the full dynamic model identified is proved by an end-effector trajectory tracking task using a model-based inverse dynamic controller.

This paper presents an adaptive backstepping control design for a
class of unmanned helicopters with parametric uncertainties. The
control objective is to let the helicopter track some pre-defined
position and yaw trajectories. In order to facilitate the control
design, we divide the helicopter's dynamic model into three
subsystems. The proposed controller combines the backstepping
method with online parameter update laws to achieve the control
objective. The global asymptotical stability (GAS) of the
closed-loop system is proved by a Lyapunov based stability
analysis. Numerical simulations demonstrate that the controller
can achieve good tracking performance in the presence of
parametric uncertainties.

Citation: Bin Xian, Jianchuan Guo, Yao Zhang. Adaptive backstepping
tracking control of a 6-DOF unmanned helicopter.  IEEE/CAA
Journal of Automatica Sinica, 2015, 2(1): 19-24

This paper presents an approach to adaptive trajectory tracking of mobile robots which combines a feedback linearization based on a nominal model and a RBF-NN adaptive dynamic compensation. For a robot with uncertain dynamic parameters, two controllers are implemented separately: a kinematics controller and an inverse dynamics controller. The uncertainty in the nominal dynamics model is compensated by a neural adaptive feedback controller. The resulting adaptive controller is efficient and robust in the sense that it succeeds to achieve a good tracking performance with a small computational effort. The analysis of the RBF-NN approximation error on the control errors is included. Finally, the performance of the control system is verified through experiments.