Low-harmonic trajectory synthesis using trajectory pattern method (TPM) for high-speed and high-precision machinery

In high-speed and high-precision machinery, trajectories with high-frequency harmonic content are one of the main sources of reduction of operational precision. Trajectories with high-frequency harmonic content generally demand even higher-harmonic actuating forces/torques due to the nonlinear dynamics of such systems, which may excite natural modes of vibration of the system and/or be beyond the dynamic response limitation of the actuation devices. In this paper, a global interpolation algorithm that uses the trajectory pattern method (TPM) for synthesizing low-harmonic trajectories is presented. The trajectory synthesis with the TPM is performed with a prescribed fundamental frequency and continuous jounce boundary condition, which would minimize the number of high-harmonic components in the required actuation forces/torques and avoid excitation of the system modes of vibration. The minimal curvature variation energy method, Lagrange multiplier method, and contour error control are used to obtain smooth kinematic profiles and satisfy the trajectory accuracy requirements. As an example, trajectory patterns that consist of a fundamental frequency sinusoidal time function and its first three harmonics are used to synthesize the desired trajectories for a selected dynamic system. The synthesized trajectories are shown to cause minimal system vibration during its operation. A comparison with a commonly used trajectory synthesis method clearly shows the superiority of the developed TPM-based approach in reducing vibration and demand on the actuator dynamic response, thereby allowing the system to operate at higher speeds and precision.

A corner smoothing algorithm using Trajectory Pattern Method (TPM) for high-speed and high-quality machining

In micro-line segments machining, transition curves with high harmonic components are more prone to causing vibration issues in the feed drive system, which affects machining efficiency and quality severely. To construct low harmonic trajectories, this paper proposes a corner smoothing algorithm that uses the Trajectory Pattern Method (TPM). The transition curve construction and axial motion scheduling are performed with a specified fundamental frequency in one step, which reduces the smoothing process time and avoids excitation of natural modes of vibration of the system. The synthesized trajectories and axial kinematic profiles are all smooth and only contain the selected fundamental frequency and its first two odd harmonics, which minimizes the number of high harmonic components in the required actuation forces/torques and avoids excitation of the system modes of vibration. Linear programming is used to synthesize the trajectories. The proposed algorithm is shown to achieve near time-optimal trajectories. The provided experimental analysis and comparisons demonstrate that the proposed algorithm achieves smooth axial kinematic profiles with low harmonic contents, which would improve machining efficiency and quality.

A Systematic Method for Model Parameter Identification of Nonlinear Dynamics Systems Using Trajectory Pattern Method

In this paper, a new method is presented for model parameter identification of a large class of fully controlled nonlinear dynamics systems such as robot manipulators. The method uses trajectory patterns with feed-forward controls to identify model parameters of the system. The developed method ensures full system stability, does not require close initial estimated values for the parameters to be identified, and provides a systematic method of emphasizing on the estimation of the parameters associated with lower order terms of the system dynamics model and gradually upgrading the accuracy with which the model parameters, particularly those associated with the higher order terms of the system dynamics, are estimated. The developed method is based on Trajectory Pattern Method (TPM). In this method, for a pattern of motion the inverse dynamics model of the system is derived in algebraic form in terms of the trajectory pattern parameters. The structure of the feedback error with feedforward signal calculated with the estimated model parameters will then be fixed, and its measurement can be used to systematically upgrade the model parameter estimation. The mathematical proof of convergence of the developed method and results of its implementation on a robot manipulator with highly non-linear dynamics are provided.

Model Parameter Identification for Non-Fully Controlled Nonlinear Dynamics Systems Using Trajectory Pattern Method

In this paper, a new method is presented for model parameter identification of a large class of nonlinear dynamics systems that are not fully controlled. The method uses trajectory patterns with feed-forward controls to identify model parameters of the system. The developed method ensures full system stability, does not require close initial estimated values for the parameters to be identified, and provides a systematic method of emphasizing on the estimation of the parameters associated with lower order terms of the system dynamics model and gradually upgrading the accuracy with which the model parameters, particularly those associated with the higher order terms of the system dynamics, are estimated. The developed method is based on Trajectory Pattern Method (TPM). The mathematical proof of convergence of the developed method and results of its implementation on a typical system with highly non-linear dynamics are provided.

A Systematic Method for Model Parameter Identification of Nonlinear Dynamics Systems Using Trajectory Pattern Method

In this paper, a new method is presented for model parameter identification of a large class of fully controlled nonlinear dynamics systems such as robot manipulators. The method uses trajectory patterns with feed-forward controls to identify model parameters of the system. The developed method ensures full system stability, does not require close initial estimated values for the parameters to be identified, and provides a systematic method of emphasizing on the estimation of the parameters associated with lower order terms of the system dynamics model and gradually upgrading the accuracy with which the model parameters, particularly those associated with the higher order terms of the system dynamics, are estimated. The developed method is based on Trajectory Pattern Method (TPM). In this method, for a pattern of motion the inverse dynamics model of the system is derived in algebraic form in terms of the trajectory pattern parameters. The structure of the feedback error with feedforward signal calculated with the estimated model parameters will then be fixed, and its measurement can be used to systematically upgrade the model parameter estimation. The mathematical proof of convergence of the developed method and results of its implementation on a robot manipulator with highly non-linear dynamics are provided.

Model Parameter Identification of Nonlinear Dynamics Systems by Trajectory Pattern Method

In this paper, a new method is presented for model parameter identification of a large class of fully controlled nonlinear dynamics systems such as robot manipulators. The method uses trajectory patterns with feed-forward controls to identify model parameters of the system. The developed method ensures full system stability, does not require close initial estimated values for the parameters to be identified, and provides a systematic method of emphasizing on the estimation of the parameters associated with lower order terms of the system dynamics model and gradually upgrading the accuracy with which the model parameters, particularly those associated with the higher order terms of the system dynamics model are estimated. The developed method is based on Trajectory Pattern Method (TPM). In this method, for a pattern of motion the inverse dynamics model of the system is derived in algebraic form in terms of the trajectory pattern parameters. The system dynamics model parameters are then identified using a systematic algorithm which ensures system stability as well as accurate estimation of the model parameters associated with lower as well as higher order terms. Mathematical proof of convergence of the developed method and an example of its application are provided.

Optimal Simultaneous Kinematic, Dynamic and Control Design of High Performance Manipulators Based on Trajectory Pattern Method

An optimal simultaneous kinematic, dynamic and control design approach is proposed for high performance computer controlled machines such as robot manipulators. The approach is based on the Trajectory Pattern Method (TPM) and a fundamentally new design philosophy that such machines in general and ultra-high performance machines in particular must only be designed to perform a class or classes of motions effectively. In the proposed approach, given the structure of the manipulator, its kinematic, dynamic and control parameters are optimized simultaneously with the parameters that describe the selected trajectory pattern. In the example presented in this paper, a weighted sum of the norms of the higher harmonics appearing in the actuating torques and the integral of the position and velocity tracking errors are used to form the optimality criterion. The selected optimality criterion should yield a system that is optimally designed to accurately follow the specified trajectory at high speed. Other objective functions can be readily formulated to synthesize systems for optimal performance. The potentials of the developed method and its implementation for generally defined motion patterns are discussed.

Trajectory Pattern Method Based Trajectory Synthesis and Inverse Dynamics Formulation for Minimal Vibrational Excitation for Flexible Structures

A new approach to trajectory synthesis and formulation of the inverse dynamics model of flexible structures for point to point motions with minimal high frequency component of the actuating torques (forces) is presented. Trajectories are synthesized such that the flexible structure comes to rest undeformed at the completion of motion. The developed method is based on the Trajectory Pattern Method. In this approach, an appropriate trajectory pattern is selected and described in parametric form. The inverse dynamics model of the system is formulated in terms of the trajectory parameters. The trajectory patterns used are in terms of a number of basic sinusoidal time functions and their harmonics. The basic frequencies are selected such that the harmonics appearing in the actuating torques do not excite the natural modes of vibration of the system. For each motion, the trajectory parameters are determined for minimal amplitudes of the higher actuating torque harmonics, noting that from the vibration and control points of view, such trajectories are more desirable. The higher harmonics refers to the harmonics of the actuating torques with frequencies above the highest trajectory harmonic frequency. As an example, a flexible beam undergoing large displacements and rotations in a plane is considered. The effectiveness of the approach is illustrated by an example.