Linearizing Robotic Manipulator's Dynamics Using Koopman Operator and Applying Generalized Predictive Control

In this work, an Adaptive Linear Quadratic Regulator (LQR) is proposed for the tracking control of a 3 DOF articulated robotic manipulator. The main objective of this work is to drive the robotic manipulator to its equilibrium point from any initial position of the end effector of a robotic manipulator in an adaptive way even if the system dynamics consist of modelling uncertainties. The conventional LQR is able to provide a good measure of robustness against disturbances and achieve good tracking performance. However, it will give degraded tracking performance in case of the presence of model uncertainties. Therefore LQR tracking performance is enhanced by using a Model Reference Adaptive Controller (MRAC) in the literature for enabling a 2 DOF system to achieve good tracking against modelling uncertainties. But it has not been applied for a 3 DOF system till date. Therefore in this work, it is proposed to be applied for such a system for the first time to realize an adaptive control. The forward and inverse kinematics level equations are formulated using an indirect form of the Denavit Hartenberg (DH) convention. The dynamics of robotic manipulator are derived using the Euler-Lagrange formulation. The dynamics of manipulator with the LQR control are considered to be the reference dynamics for the MRAC control and the modelling uncertainty will be added to the system dynamics to test its performance. The MRAC augmented LQR provides faster convergence with improved stability against modelling uncertainties.

The inverse dynamics of a robotic manipulator is instrumental in precise robot control and manipulation. However, acquiring such a model is challenging, not only due to unmodelled non-linearities such as joint friction, but also from a machine learning perspective (e.g., input space dimension, amount of data needed). The accuracy of such models, regardless of the learning techniques, relies on proper excitation and exploration of the robot’s configuration space, in order to collect a rich dataset. This study aims to provide rich data in learning the inverse dynamics of a serial robotic manipulator using supervised machine learning techniques. We propose a method, called Max-Information Configuration Exploration (MICE), to incrementally explore and generate information-rich data via computing parameters of a trajectory set. We also introduce a new set of excitation trajectories that explores robot’s configuration through imposed stable limit cycles in robot joints’ phase space while satisfying feasibility constraints and physical bounds. We benchmark MICE against state-of-the-art in terms of data quality and learning accuracy. The proposed methodology for data collection, model learning, and evaluation, is validated with a KUKA IIWA14 robotic arm where the results prove significant improvement over traditional approaches.

Radial basis function network (RBFN) is used in this paper for predefined trajectory control of both one-link and two-link robotic manipulators. The updating equations for the RBFN parameters were derived using the gradient descent principle. The other advantage of using this principle is that it shows the clustering effect in distributing the radial centres. To increase the complexity, the dynamics of robotic manipulator is assumed to be unknown, and hence, simultaneous control and identification steps were performed using the RBFNs. The performance of the RBFN is compared with the multilayer feed-forward neural network (MLFFNN) in terms of mean square error, tolerance to disturbance and parameter variations in the system. The efficacy of RBFN as a controller and identification tool is verified by performing the simulation study, and the results obtained reveal the superior performance of RBFN over MLFFNN in both identification and control aspects for one-link and two-link robotic manipulators.

Enhanced tracking control for n-DOF robotic manipulators: A fixed-time terminal sliding mode approach with time delay estimation

The operating speed and precision of robot manipulators are limited by the dynamic response limitations of their actuators. The kinematics and dynamics of robot manipulators with revolute joints are highly nonlinear. As a result, an even greater demand is placed on the dynamic response of the prime actuators since the actuating torques required for accurate tracking of a desired trajectory must contain a significant number of higher harmonics of the joint trajectory harmonics. In this paper, a systematic method is presented for optimal integration of active materials based actuators into the structure of robot manipulators for the purpose of minimizing the higher harmonic components of the required actuating torques. Through computer simulation, it is shown that with low harmonic joint trajectories and by minimizing the higher harmonic components of the actuating torques with properly sized and positioned smart actuators, robot manipulators can operate at higher speeds, with greater tracking precision and with minimal vibration and control problems.

Further results on adaptive iterative learning control of robot manipulators

Online optimal control of quadruped robots would enable them to adapt to varying inputs and changing conditions in real time. A common way of achieving this is linear model predictive control (LMPC), where a quadratic programming (QP) problem is formulated over a finite horizon with a quadratic cost and linear constraints obtained by linearizing the equations of motion and solved on the fly. However, the model linearization may lead to model inaccuracies. In this paper, we use the Koopman operator to create a linear model of the quadrupedal system in high dimensional space which preserves the nonlinearity of the equations of motion. Then using LMPC, we demonstrate high fidelity tracking and disturbance rejection on a quadrupedal robot. This is the first work that uses the Koopman operator theory for LMPC of quadrupedal locomotion.

Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

A new nonlinear optimal control method is proposed for solving the problem of control and stabilization of a two-link robotic manipulator where the first link is rigid and the second link is flexible. The dynamic model of this robot describes both the rotational motion of the two links and the vibration modes which are excited at the flexible link. The control and stabilization problem of this 2-DOF rigid-flexible robot is nontrivial due to complex nonlinear dynamics and underactuation. To apply the proposed nonlinear optimal control method, the dynamic model of the two-link rigid-flexible robotic manipulator undergoes first approximate linearization around a temporary operating point that is updated at each iteration of the control algorithm. The linearization takes place through first-order Taylor series expansion and through the computation of the Jacobian matrices of the flexible manipulator’s state-space description. For the approximately linearized model of the two-link rigid-flexible robot, an H-infinity feedback controller is designed. Actually, the H-infinity controller stands for the solution of the optimal control problem for the rigid-flexible robot under uncertainty and external perturbations. For the computation of the feedback gains of the H-infinity controller, an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control algorithm are proven through Lyapunov analysis. First, it is shown that the control scheme achieves H-infinity tracking performance which signifies robustness for the control loop of the two-link rigid-flexible robot under uncertainties and external perturbations. Next, it is also shown that the control loop of the 2-DOF rigid-flexible robot is globally asymptotically stable. The proposed control method achieves fast and accurate tracking of setpoints under moderate variations of the control inputs. The concept of the proposed nonlinear optimal control method has been to approximate locally and at each sampling interval the nonlinear dynamics of the rigid-flexible robotic manipulator by an equivalent linear state-space description and to solve the H-infinity control problem for the linearized model. Finally, the control inputs are applied to the initial nonlinear dynamics and stability proof is given for the initial nonlinear state-space model.

This paper investigates iterative learning control based on passivity for two-degree-of-freedom (2DOF) robot manipulators with antagonistic bi-articular muscles. Firstly, a brief summary of dynamics of 2DOF robot manipulators with antagonistic bi-articular muscles is given. Next, an error dynamics of the bi-articular manipulator for iterative learning control that has an output strictly passivity property is constructed. Then, we propose an iterative learning control law for the biarticular manipulator. The proposed torque input does not need the parameters for the accurate models. Convergence analysis of the closed-loop system is carried out based on passivity. Finally, simulation results are presented in order to confirm the effectiveness of the proposed control law.

Although providing exceptional asymptotic performance, trial-and-error based reinforcement learning suffers from being data inefficient, i.e., requiring an excessive amount of trial data for training. In this paper, we propose a data-efficient model-based reinforcement learning algorithm based on the Koopman operator theory. By representing the environment dynamics as a linear dynamical system in a high-dimensional space, the Koopman operator theory allows incorporating effective optimal control methods to produce high-quality trials thus accelerating learning. However, when applying the theory for reinforcement learning, with the sparse and unevenly distributed trial data, it is difficult to learn globally linear representations thus leading to serious model bias. To overcome this problem, we devise a local Koopman operator approach that is tailored for the setup of reinforcement learning. By coupling with deep neural networks and the linear quadratic regulator control, we propose the first Koopman operator model-based reinforcement learning algorithm called deep Koopman reinforcement learning (DKRL). The simulation experiment results show the proposed method can outperform existing approaches and can learn robotics control tasks with a small amount of data.

This book is the second edition of a textbook published in 1996 by McGraw Hill and originates from a graduate level course given by the authors at the University of Naples. The topics include kinematics, statics and dynamics of robot manipulators together with trajectory planning and active control. There are only minor additions the first edition, which are mainly the use of quaternion to describe the orientation of the end effector and a short description of a closed chain architecture for a manipulator (parallelogram arm). The book is largely devoted to serial manipulators, with special developments about active control including adaptative control, robust controls and stability analysis. Another strength of this book is the great number of problems proposed at the end of each chapter, together with a list of references related to it. The fundamental features covered by the text are illustrated on simple examples of serial manipulators (two-link planar arm, parallelogram arm) including analytical results and numerical tests.The book has nine chapters followed by three appendices. The first appendix is devoted to linear algebra, the second recalls some fundamental aspects of rigid body mechanics and the third gives some basic principles of feedback control of linear systems.Chapter one is an introduction to the study of robot manipulators, giving an interesting classification of their architectures, the corresponding workspace and describes the tasks for which theyare used. After some standard examples of industrial manipulators, bibliographical reference texts are proposed, including textbooks on modelling and control of robots, general books on robotics, specialized texts, scientific robotic reviews and some international conferences on robotics.Chapters two, three and four are devoted to mechanical modelling of robot manipulators. The fundamental basics of kinematics are given in chapter two, including the representation of finite rotations by Euler angles or unit quaternions, homogeneous transformations, Denavit-Hartenberg parameters and workspace. The direct and inverse kinematical problems are solved in analytical form for some typical manipulator structures. The differential kinematics of robots are presented in chapter three, with an introduction to the geometric and analytic Jacobian matrices, kinematic singularities and redundancy. The inverse kinematic problem is presented, with special attention to the case of redundant robots where the solution is obtained by a linear optimization problem leading to the introduction of the pseudo-inverse Jacobian matrix and to the solution of several objectives such as avoidance of collision with an obstacle or moving away from singularities. Several inverse kinematics algorithms are given wih an interesting application to a three-link planar arm. Finally, a property of kineto-statics duality is deduced from the principle of virtual work applied to an equilibrium configuration of the robot. Chapter four is a standard presentation of the derivation of the dynamical model by Lagrange formulation and then by the Newton-Euler method. In the Lagrange formulation method, the linearity with respect to inertial parameters is shown and a detailed formulation of the dynamical model is obtained for a two-link Cartesian arm, a two-link planar arm and a parallelogram arm. The problem of dynamic parameter identification is also briefly presented from a numerical point of view. The recursive algorithm constructed from the Newton-Euler formulation is presented and illustrated by considering a two-link planar arm. Finally, the operational space dynamic model is introduced.In chapter five, paths and trajectory planning in joints and in operational spaces are presented; several classical methods of interpolation are described.Chapter six is an extensive study of active control of manipulators. Several methods are presented, involving classical independent joint control, non-linear centralized control, robust control and adaptative control. Both joint-space control and operational-space control are studied together with stability analysis by using Liapounoff functions. An interesting application to the two-link planar arm already used shows the comparison between various control schemes.Chapter seven deals with interaction control of serial manipulators with the working environment. Several strategies involving compliance control, impedance control, force control and hybrid control are presented.Chapter eight describes the actuators and the sensors used in robotics. Several types of servomotors (electric and hydraulic) are presented, together with the model giving their input/output relationship. Several kinds of sensors are also described including encoders, tachometers, force and vision sensors.The last chapter gives a short presentation of the functional architecture of a robot's control system, including characteristics of the programing environment and the hardware architecture.In conclusion, the book provides a good insight about simulation and control of robot manipulators, with a detailed study of the various control strategies and several interesting and pedagogical applications. This book is an excellent review of the standard knowledge needed not only for graduate students but also for researchers interested in robot manipulators.M Pascal

Computational kinematics and dynamics of robot manipulators are dealt with. A recursive method based on the vector form of Rodrigues' equation is presented for the computation of the associated coordinate transformations. The method allows for forward, backward, and two-way recursions and is applied to the computations of the Jacobian matrices and dynamic equations for mechanical manipulators. The computational complexities of the resulting equations are also evaluated and compared to some of the existing methods in each case. It is shown that the algorithms presented have certain computational advantages over most of the existing methods.

Implementing Model Predictive Control (MPC) for large-scale nonlinear systems is often computationally challenging due to the intensive online optimization required. To address this, various reduced-order linearization techniques have been developed. The Koopman operator linearizes a nonlinear system by mapping it into an infinite-dimensional space of observables, enabling the application of linear control strategies. While Artificial Neural Networks (ANNs) can approximate the Koopman operator in a data-driven manner, training these networks becomes computationally intensive for high-dimensional systems as the lifting into a higher-dimensional observable space significantly increases data size and complexity. In this work, we propose a technique, combining Proper Orthogonal Decomposition (POD) with an efficient ANN structure to reduce the training time of ANN for large order systems. By first applying POD, we obtain a low order projection of the system. Subsequently, we train the ANN with an efficient structure to approximate the Koopman operator, significantly decreasing training time without sacrificing accuracy.

This paper presents a dynamic model for the robotic manipulators with flexible links by means of the Finite Element Method and Lagrange’s formulation. By the concept of the Equivalent Rigid Link System (ERLS), the generalized coordinates are selected to represent the total motion as a large motion and a small motion. Two sets of coupled nonlinear equations are obtained where the equations representing small motions are linear with respect to the small motion variables. An example is presented to illustrate the importance of the flexibility effects.