FIR filters for online trajectory planning with time- and frequency-domain specifications

In this paper, the use of FIR (Finite Impulse Response) filters for planning minimum-time trajectories for robots or automatic machines under constraints of velocity, acceleration, etc. is presented and discussed. In particular, the relationship between multi-segment polynomial trajectories, i.e. trajectories composed of several polynomial segments, each one possibly characterized by constraints on one or more specific derivatives (i.e. velocity, acceleration, jerk, etc.), and FIR filters disposed in a cascade configuration is demonstrated and exploited in order to design a digital filter for online trajectory planning. The connection between analytic functions and dynamic filters allows a generalization of these trajectories, usually obtained by second- or third-order polynomial functions (e.g. trapezoidal velocity and double S velocity trajectories), to a generic order with only a modest increase of the complexity. As a matter of fact, the computation of trajectories with higher degree of continuity simply requires additional FIR filters in the chain. Moreover, the modular structure of the planner provides a direct frequency characterization of the motion law. In this way, it is possible to define the trajectories by considering constraints expressed in the frequency-domain besides the classical time-domain specifications, such as bounds on velocity, acceleration, and so on. Two examples illustrate the main features of the proposed trajectory planner, in particular with respect to the problems of multi-point trajectories generation and residual vibrations suppression

Online trajectory planner with constraints on velocity, acceleration and torque

A filter for trajectories smoothing is presented. The filter provides an output reproducing the input trajectory if this is compliant with given constraints on the velocity, the acceleration and the torque required for tracking it. Otherwise, the filter approximates the input profile guaranteeing that the output trajectory satisfies all the kinematic and dynamic limits. The tracking of the input signal is optimal in the sense that at each time instant the limit value of one among velocity, acceleration and torque is reached. The filter, based on a variable structure controller, is designed in the continuous-time domain but can be implemented by discretization as a sampled system. It can therefore used in mechatronic and robotic applications driven by digital controllers in order to filter trajectories planned without considering the above mentioned constraints or to generate realtime smooth trajectories by simply providing basic inputs such as step or ramp functions

A Repetitive Control Scheme Based on B-Spline Trajectories Modification

In many applications of interest in industrial robotics, tasks are cyclic and must be repeated over and over. In this context, it seems natural to exploit the intrinsic properties of repetitive control schemes, where the cyclic nature of "disturbances" and/or unmodeled dynamic effects can be exploited to reduce the tracking errors. In this paper, we propose a new repetitive control scheme, where the main idea consists in the modification of the reference trajectory in order to compensate for the periodic undesired effects. By exploiting the dynamic filters for the B-spline generation, it is possible to integrate the trajectory planning within a repetitive control scheme able to modify in real-time the reference signal with the aims of nullify interpolation errors. By means of an extensive experimental activity on a servo mechanism pros and cons of the proposed approach are analyzed

A repetitive control scheme for industrial robots based on b-spline trajectories

In this paper, a novel repetitive control scheme is presented and discussed. The general framework is the control of repetitive tasks of robotic systems or, more in general, of automatic machines. The key idea of the proposed scheme consists in modifying the reference trajectory provided to the plant in order to compensate for external loads or unmodelled dynamics that cyclically affect it. By exploiting the dynamic filters for the B-spline trajectory planning, it has been possible to integrate the trajectory generation within a repetitive control scheme able to modify in real-time the reference signal with the aims of nullify interpolation errors. Experimental results obtained controlling two joints of a standard industrial manipulator are reported, showing the effectiveness of the proposed method

Optimal Trajectories for Vibration Reduction Based on Exponential Filters

In this paper, a new type of trajectory, based on an exponential jerk, is presented along with filters for their online generation. The goal is to generalize constant jerk trajectories, widely used in industrial applications, in order to reduce vibrations of motion systems. As a matter of fact, constant jerk trajectories do not assure a complete vibration suppression when the damping of the resonant modes is not negligible. The values of the parameters (decay rate and duration) of the jerk impulses that allow residual vibration cancellation are derived in an analytical way as a function of the dynamic characteristics of the plant. Comparisons with the well-known input shaping techniques and with system-inversion-based filters show the advantages of the proposed method in terms of robustness with respect to modeling errors, smoothness of the resulting trajectory, and time duration of the motion under velocity and acceleration constraints

Qualitative graphical representation of Nyquist plots

In this paper, the procedure for manually drawing the Nyquist plot of a generic transfer function is revised and, in particular, two novel parameters (\u394\u3c4\u394\u3c4, \u394p\u394p), which allow to simplify this process, are presented. Thanks to these parameters, the analysis of the frequency response at low and high frequencies is considerably enhanced, with a very little effort. These parameters allow to predict initial and final directions of the polar curve in the vicinity of initial and final points and consequently the sectors of the complex plane where the plot starts/ends. In many cases it is possible to obtain a qualitative Nyquist plot, able to correctly predict the stability properties of the closed-loop system, by simply joining the initial and final tracts found with the proposed procedure. Moreover, the analysis based on these parameters can aid to correctly interpret the plots obtained with computer programs which often, in particular when poles at the origin are present, hide the behavior of the frequency response in the area close to the origin of the complex plane

Multidimensional Trajectories Generation with Vibration Suppression Capabilities: the Role of Exponential B-splines

In this paper, exponential B-spline trajectories are presented and discussed. They are generated by means of a chain of filters characterized by a truncated exponential impulse response. If properly tuned, the filters applied to a vibrating plant are able to cancel the oscillations and in this sense the resulting splines are optimized with respect to the problem of vibrations suppression. Different types of exponential B-spline are illustrated, with one or more exponential filters in the chain, and the procedure for the interpolation of a given set of desired via-points, with a proper choice of the control points, is shown. As a matter of fact, exponential B-splines, generated by means of dynamic filters, combine the vibration suppression capability of input shapers and smoothing filters with the possibility of exactly interpolating some via-points. The advantages of these curves are experimental proved by considering the motion of a spherical pendulum connected to the flange of an industrial robot

Feedforward control of Variable Stiffness Joints robots for vibrations suppression

This paper presents a new feedforward controller based on a continuous-time finite impulse response filter, designed to minimize the vibrations that usually affect robot manipulators with elastic joints. In particular, Variable Stiffness Joints (VSJ) robots are considered, since they are usually characterized by a very low level of damping which makes the problem of the oscillations quite important. The proposed approach allows to simplify the overall control structure of VSJ robots, which is based on a decentralized control of each servomotor, imposing the desired position and the desired stiffness at each joint, and on a novel feedforward control, filtering the reference signals. After analyzing some of the filter properties and the method for the parameters choice, experimental results on a VSJ robot demonstrate the importance of the proposed filtering action for minimizing vibrations and oscillations

Repetitive Control Meets Continuous Zero Phase Error Tracking Controller for Precise Tracking of B-spline Trajectories

In this paper, a novel repetitive control scheme is presented and discussed, based on the so-called B-spline filters. These dynamic filters are able to generate a B-spline trajectory if they are fed with the sequence of control points defining the curve. Therefore, they are ideal tools for generating online reference signals with the prescribed level of smoothness for driving dynamic systems, possibly together with a feedforward compensator. In particular, a Continuous Zero Phase Error Tracking Controller (ZPETC) can be used for tracking control of non-minimum phase systems but because of its open-loop nature it cannot guarantee the robustness with respect to modeling errors and exogenous disturbances. For this reason, ZPETC and trajectory generator have been embedded in a repetitive control scheme that allows to nullify interpolation errors even in non-ideal conditions, provided that the desired reference trajectory and the disturbances are periodic. Asymptotic stability of the overall control scheme is proved mathematically and experimental validation based on a non-minimum phase system is performed. Different models of the same physical system have been identified and used in the implementation of this model-based control scheme, allowing a real evaluation of the relationship between control system performance and model accuracy

Improving the Accuracy of Industrial Robots via Iterative Reference Trajectory Modification

In this paper, a novel repetitive control (RC) scheme is presented and discussed. The general framework is the control of repetitive tasks of robotic systems or, more in general, of automatic machines. The key idea of the proposed scheme consists in modifying the reference trajectory provided to the plant in order to compensate for external loads or unmodeled dynamics that cyclically affect it. By exploiting the fact that uniform B-spline trajectories can be generated by means of dynamic filters, the trajectory planning phase has been integrated within an RC scheme that is able to modify in real time the reference signal in order to nullify the tracking errors occurring at the desired via-points. Because of this mechanism, the control scheme is very suitable for the application to industrial plants with off-the-shelf, unmodifiable controllers. Experimental results obtained with a standard industrial manipulator both in joint space and in workspace show the effectiveness of the proposed method