A simulation and animation tool for education in multivariable control is presented. The purpose of the tool is to support studies of various aspects of multivariable dynamical systems and design of multivariable feedback control systems. Different ways to use this kind of tool in control education are also presented and discussed.
The task of generating time optimal trajectories for a six degrees of freedom industrial robot is discussed and an existing convex optimization formulation of the problem is extended to include new types of constraints. The new constraints are speed dependent and can be motivated from physical modeling of the motors and the drive system. It is shown how the speed dependent constraints should be added in order to keep the convexity of the overall problem. A method to, conservatively, approximate the linear speed dependent constraints by a convex constraint is also proposed. A numerical example proves versatility of the extension proposed in this paper.
This paper discusses the aspects of controllability in the iteration domain for systems that are controlled using iterative learning control (ILC). The focus is on controllability for a proposed state space model in the iteration domain and it relates to an assumption often used to prove convergence of ILC algorithms. It is shown that instead of investigating controllability it is more suitable to use the concept of target path controllability (TPC), where it is investigated if a system can follow a trajectory instead of the ability to control the system to an arbitrary point in the state space. Finally, a simulation study is performed to show how the ILC algorithm can be designed using the LQ-method, if the state space model in the iteration domain is output controllable. The LQ-method is compared to the standard norm-optimal ILC algorithm, where it is shown that the control error can be reduced significantly using the LQ-method compared to the norm-optimal approach.
Control of a flexible joint of an industrial manipulator using H_{∞} design methods is presented. The considered design methods are i) mixed-H_{∞} design, and ii) H_{∞} loop shaping design. Two different controller configurations are examined: one uses only the actuator position, while the other uses the actuator position and the acceleration of end-effector. The four resulting controllers are compared to a standard PID controller where only the actuator position is measured. The choices of the weighting functions are discussed in details. For the loop shaping design method, the acceleration measurement is required to improve the performance compared to the PID controller. For the mixed-H_{∞} method it is enough to have only the actuator position to get an improved performance. Model order reduction of the controllers is briefly discussed, which is important for implementation of the controllers in the robot control system.
Control of a flexible joint of an industrial manipulator using H_{∞} loop shaping design is presented. Two controllers are proposed; 1) H_{∞} loop shaping using the actuator position, and 2) H_{∞} loop shaping using the actuator position and the acceleration of end-effector. The two controllers are compared to a standard PID controller where only the actuator position is measured. Using the acceleration of the end-effector improves the nominal performance. The performance of the proposed controllers is not significantly decreased in the case of model error consisting of an increased time delay or a gain error.
State estimation of a flexible industrial manipulator is presented using experimental data. The problem is formulated in a Bayesian framework where the extended Kalman filter and particle filter are used. The filters use the joint positions on the motor side of the gearboxes as well as the acceleration at the end-effector as measurements and estimates the corresponding joint angles on the arm side of the gearboxes. The techniques are verified on a state of the art industrial robot, and it is shown that the use of the acceleration at the end-effector improves the estimates significantly.
A sensor fusion method for state estimation of a flexible industrial robot is developed. By measuring the acceleration at the end-effector, the accuracy of the arm angular position, as well as the estimated position of the end-effector are improved. The problem is formulated in a Bayesian estimation framework and two solutions are proposed; the extended Kalman filter and the particle filter. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The technique is also verified in experiments on an ABB robot, where the dynamic performance of the position for the end-effector is significantly improved.
An estimation-based iterative learning control (ILC) algorithm is applied to a realistic industrial manipulator model. By measuring the acceleration of the end-effector, the arm angular position accuracy is improved when the measurements are fused with motor angle observations. The estimation problem is formulated in a Bayesian estimation framework where three solutions are proposed: one using the extended Kalman filter (EKF), one using the unscented Kalman filter (UKF), and one using the particle filter (PF). The estimates are used in an ILC method to improve the accuracy for following a given reference trajectory. Since the ILC algorithm is repetitive no computational restrictions on the methods apply explicitly. In an extensive Monte Carlo simulation study it is shown that the PF method outperforms the other methods and that the ILC control law is substantially improved using the PF estimate.
The iterative learning control (ILC) method improvesperformance of systems that repeat the same task several times. In this paper the standard norm-optimal ILC control law for linear systems is extended to an estimation-based ILC algorithm where the controlled variables are not directly available as measurements. The proposed ILC algorithm is proven to be stable and gives monotonic convergence of the error. The estimation-based part of the algorithm uses Bayesian estimation techniques such as the Kalman filter. The objective function in the optimisation problem is modified to incorporate not only the mean value of the estimated variable, but also information about the uncertainty of the estimate. It is further shown that for linear time-invariant systems the ILC design is independent of the estimation method. Finally, the concept is extended to non-linear state space models using linearisation techniques, where it is assumed that the full state vector is estimated and used in the ILC algorithm. It is also discussed how the Kullback-Leibler divergence can be used if linearisation cannot be performed. Finally, the proposed solution for non-linear systems is applied and verified in a simulation study with a simplified model of an industrial manipulator system.
The norm-optimal iterative learning control (ilc) algorithm for linear systems is extended to an estimation-based norm-optimal ilc algorithm where the controlled variables are not directly available as measurements. A separation lemma is presented, stating that if a stationary Kalman filter is used for linear time-invariant systems then the ilc design is independent of the dynamics in the Kalman filter. Furthermore, the objective function in the optimisation problem is modified to incorporate the full probability density function of the error. Utilising the Kullback–Leibler divergence leads to an automatic and intuitive way of tuning the ilc algorithm. Finally, the concept is extended to non-linear state space models using linearisation techniques, where it is assumed that the full state vector is estimated and used in the ilc algorithm. Stability and convergence properties for the proposed scheme are also derived.
A sensor fusion method for state estimation of a flexible industrial robot is presented. By measuring the acceleration at the end-effector, the accuracy of the arm angular position is improved significantly when these measurements are fused with motor angle observation. The problem is formulated in a Bayesian estimation framework and two solutions are proposed; one using the extended Kalman filter (EKF) and one using the particle filter (PF). The technique is verified on experiments on the ABB IRB4600 robot, where the accelerometer method is showing a significant better dynamic performance, even when model errors are present.
A method to find the orientation and position of a three degree-of-freedom (DOF) accelerometer mounted on a six DOF industrial robot is proposed and evaluated on experimental data. The method consists of two consecutive steps, where the first is to estimate the orientation of the sensor using data from static experiments. In the second step the sensor position relative to the robot base is identified using sensor readings when the sensor moves in a circular path and where the sensor orientation is kept constant in a path fixed coordinate system. Once the accelerometer position and orientation are identified it is possible to use the sensor in robot model parameter identification and in advanced control solutions.
A novel method to find the orientation and position of a triaxial accelerometer mounted on a six degrees-of-freedom industrial robot is proposed and evaluated on experimental data. The method consists of two consecutive steps, where the first is to estimate the orientation of the accelerometer from static experiments. In the second step the accelerometer position relative to the robot base is identified using accelerometer readings when the accelerometer moves in a circular path and where the accelerometer orientation is kept constant in a path fixed coordinate system. Once the accelerometer position and orientation are identified it is possible to use the accelerometer in robot model parameter identification and in advanced control solutions. Compared to previous methods, the accelerometer position estimation is completely new, whereas the orientation is found using an analytical solution to the optimisation problem. Previous methods use a parameterisation where the optimisation uses an iterative solver.
This paper summarizes previous work on tool position estimation on industrial manipulators, and emphasize the problems that must be taken care of in order to get a satisfied result. The acceleration of the robot tool, measured by an accelerometer, together with measurements of motor angles are used. The states are estimated with an extended Kalman filter. A method for tuning the covariance matrices for the noise, used in the observer, is suggested. The work has been focused on a robot with two degrees of freedom.
The performance of a non-linear filter hinges in the end on the accuracy of the assumed non-linear model of the process. In particular, the process noise covariance $Q$ is hard to get by physical modeling and dedicated system identification experiments. We propose a variant of the expectation maximization (EM) algorithm which iteratively estimates the unobserved state sequence and $Q$ based on the observations of the process. The extended Kalman smoother (EKS) is the instrument to find the unobserved state sequence. Our contribution fills a gap in literature, where previously only the linear Kalman smoother and particle smoother have been applied. The algorithm will be important for future industrial robots with more flexible structures, where the particle smoother cannot be applied due to the high state dimension. The proposed method is compared to two alternative methods on a simulated robot.
The performance of a non-linear filter hinges in the end on the accuracy of the assumed non-linear model of the process. In particular, the process noise covariance Q is hard to get by physical modeling and dedicated system identification experiments. We propose a variant of the expectation maximization (EM) algorithm which iteratively estimates the unobserved state sequence and Q based on the observations of the process. The extended Kalman smoother (EKS) is the instrument to find the unobserved state sequence. Our contribution fills a gap in literature, where previously only the linear Kalman smoother and particle smoother have been applied. The algorithm will be important for future industrial robots with more flexible structures, where the particle smoother cannot be applied due to the high state dimension. The proposed method is compared to two alternative methods on a simulated robot.
An H_{∞} synthesis method for control of a flexible joint, with non-linear spring characteristic, is proposed. The first step of the synthesis method is to extend the joint model with an uncertainty description of the stiffness parameter. In the second step, a non-linear optimisation problem, based on nominal performance and robust stability requirements, has to be solved. Using the Lyapunov shaping paradigm and a change of variables, the non-linear optimisation problem can be rewritten as a convex, yet conservative, LMI problem. The method is motivated by the assumption that the joint operates in a specific stiffness region of the non-linear spring most of the time, hence the performance requirements are only valid in that region. However, the controller must stabilise the system in all stiffness regions. The method is validated in simulations on a non-linear flexible joint model originating from an industrial robot.
This paper gives a short summary of an industrial development work on model-based motion control. This development has resultet in high robot motion performance simultaneously with an efficient use of the installed drive system of the robot.
This paper presents a data-driven approach to diagnostics of systems that operate in a repetitive manner. Considering that data batches collected from a repetitive operation will be similar unless in the presence of an abnormality, a condition change is inferred by comparing the monitored data against an available nominal batch. The method proposed considers the comparison of data in the distribution domain, which reveals information of the data amplitude. This is achieved with the use of kernel density estimates and the Kullback–Leibler distance. To decrease sensitivity to disturbances while increasing sensitivity to faults, the use of a weighting vector is suggested which is chosen based on a labeled dataset. The framework is simple to implement and can be used without process interruption, in a batch manner. The approach is demonstrated with successful experimental and simulation applications to wear diagnostics in an industrial robot gearbox and for diagnostics of gear faults in a rotating machine.
The CDIO (Conceive Design Implement Operate) Initiative is explained, and some of the results at the Applied Physics and Electrical Engineering program at Linköping University, Sweden, are presented. A project course in Automatic Control is used as an example. The projects within the course are carried out using the LIPS (Linköping interactive project steering) model. An example of a project, the golf playing industrial robot, and the results from this project are also covered.
Drive cycle following is important for concept comparisons when evaluating vehicle concepts, but it can be time consuming to develop good driver models that can achieve accurate following of a specific velocity profile. Here, a new approach is proposed where a simple driver model based on a PID controller is extended with an Iterative Learning Control (ILC) algorithm. Simulation results using a nonlinear vehicle and control system model show that it is possible to achieve very good cycle following in a few iterations with little tuning effort. It is also possible to utilize the repetitive behavior in the drive cycle to accelerate the convergence of the ILC algorithm even further.
This paper presents a gain scheduling control of a nonlinear system in which the reference trajectory is given in advance. Multiple frozen operating times are chosen on the reference trajectory and a linear time invariant model is obtained at each operating time. A linear parameter varying model is then constructed by interpolating the region between the neighbor frozen operating times. A gain scheduling state feedback law is designed by a linear matrix inequality formulation. The effectiveness is demonstrated in a numerical simulation of a traing control of a two-link robot arm.
A frequency domain analysis method of a second order iterative learning control (ILC) algorithm is considered. Using the notion of iterative systems bounds for stability are presented in the frequency domain for the second order term. The bounds are found using a geometrical approach based on the special structure of the transfer matrix in the iterative system. Two examples are included showing how the analysis method can be used in an application
A synthesis algorithm for the filters in a first order ILC is presented and applied on an industrial robot. The proposed ILC synthesis method is evaluated using two experiments on the robot. The first is a one-axis experiment where the system can be seen as a single servo. A modeling experiment is done to give input to the synthesis algorithm and then ILC is applied to the single axis showing a dramatic improvement in trajectory following. In the second experiment ILC is applied to a more complex multi axes motion where the robot draws a circle in a plane. The evaluation of the result is done using a pen mounted on the robot and it is evident that also on the arm-side an improved motion can be achieved. In both experiments the error converges to a stable level in about 5 iterations. Since a model is desired for the synthesis, an extra iteration has to be done for the modeling experiment. In this particular case a good path following can therefore be achieved after 6 iterations.
An introduction to Iterative Learning Control (ILC) is given. The basic principle behind ILC in both open loop and closed loop problems is explained. A general class of algorithms for updating of the ILC input signal is presented and the choice of the filters in the update algortihm is discussed with respect to convergence, robustness and disturbance sensitivity.
Control of a flexible mechanical system using Iterative Learning Control (ILC) is studied using a linear two-mass model. The available signals are position of the first mass and acceleration of the second mass. An ILC algorithm using an estimate of the position of the second mass is evaluated in simulations showing promising properties.
Design of Iterative Learning Control (ILC) algorithms using optimization is considered. By forming a quadratic criterion in the control error and the input signal using a nominal model of the system an ILC algorithm is derived. Special attention is paid to the frequency domain properties of the algorithm and to how it is able to handle non-minimum phase systems. A numerical example and an experiment carried out on an real industrial robot are presented.
Iterative learning control (ILC) based on minimization of a quadratic criterion in the control error and the input signal is considered. The focus is on the frequency domain properties of the algorithm, and how it is able to handle non-minimum phase systems. Experiments carried out on a commercial industrial robot are also presented.
The disturbance properties of high order iterative learning control (ILC) algorithms are considered. An error equation is formulated, and using statistical models of the load and measurement disturbances an equation for the covariance matrix of the control error vector is derived. The results are exemplified by analytic derivation of the covariance matrix for a second order ILC algorithm.
Iterative learning control applied to a simplified model of a robot arm is studied. The iterative learning control input signal is used in combination with conventional feed-back and feed-forward control, and the aim is to let the learning control signal handle the effects of unmodeled dynamics and friction. Convergence and robustness aspects of the choice of filters in the updating scheme of the iterative learning control signal are studied.
An optimization approach to Iterative Learning Control (ILC) is considered. The ILC algorithm is formed by minimizing a quadratic criterion in the control error and input signal. A frequency domain interpretation of the derived updating algorithm is given. Experiments carried out on an ABB IRB 1400 are presented.
Some aspects of the use of learning control for improved performance in robot control systems are studied. The learning control signal is used in combination with conventional feed-back and feed-forward control. The effects of disturbances, unmodeled dynamics and friction are studied theoretically and in simulations of a simplified model of a robot arm. Convergence and robustness aspects of the choice of filters in the updating scheme of the learning control signal are studied.
Some fundamental limitations of causal and Current Iteration Tracking Error (CITE) discrete time Iterative Learning Control (ILC) algorithms are stud- ied using time and frequency domain convergence criteria. Of particular interest are conditions for monotone convergence, and these are evaluated using a discrete- time version of Bode's integral theorem. A relation between the frequency domain convergence conditions and the time-domain monotone convergence criterion is also discussed.
Iterative learning control (ILC) is applied to a robot arm with joint flexibility. The ILC algorithm uses an estimate of the arm angle, where the estimate is computed using measurements of the motor angle and the arm angular acceleration. The design of the ILC algorithm is evaluated experimentally on a laboratory scale robot arm with good results.
Iterative learning control (ILC) is applied to a laboratory scale robot arm with joint flexibility. The ILC algorithm is based on an estimate of the arm angle, where the estimate is formed using measurements of the motor angle and the arm angular acceleration. The design of the ILC algorithm is based on a model obtained from system identification. The ILC algorithm is evaluated experimentally on the robot arm with good results.
Iterative learning control (ILC) is applied to a laboratory scale robot arm with joint exibility. The ILC algorithm is based on an estimate of the arm angle, where the estimate is formed using measurements of the motor angle and the arm angular acceleration. The design of the ILC algorithm is based on a model obtained from system identication. The ILC algorithm is evaluated experimentally on the robot arm with good results.