M. Vakil

A Constrained Lagrange Formulation of Multilink Planar Flexible Manipulator

In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.

Trajectory Tracking for the End-effector of a Class of Flexible Link Manipulators

A new controller for the end-effector trajectory tracking (EETT) of a class of flexible link manipulators which consists of a chain of rigid links with a flexible end-link (CRFE) is introduced. To design this new controller, a dynamic model of the CRFE is expressed in the singularly perturbed form; that is, decomposed into slow and fast subsystems. The states of the slow subsystem are the joints’ rotations and their time derivative, while the states of the fast subsystem are the flexible variables, which model the lateral deflection of the end-link, and their time derivative. For the slow subsystem, the new controller requires only ‘‘one’’ corrective torque in addition to the computed torque command of the rigid link counterpart of the CRFE for the reduction of the EETT error. This corrective torque is derived based on the concept of the integral manifold of the singularly perturbed differential equations. The need for only one corrective torque and its derivation are among the contributions of the new controller. To stabilize the fast subsystem, an observer-based controller is designed according to the gain-scheduling technique. Due to the application of the observer-based controller there is no need for the measurement of the time derivative of the flexible link’s lateral deflection, in which its measurement is difficult if not impossible in practice. This feature of the new controller is an advantage for it. To facilitate the derivation and implementation of this controller, several properties of the matrices in the dynamic model of the CRFE are introduced and used which are other contributions of this research. The effectiveness and feasibility of the new controller are shown by simulation and experimental studies.

End-effector trajectory tracking of a flexible link manipulator using integral manifold concept

A new controller for the end-effector trajectory tracking of a single flexible link manipulator is introduced. The linear dynamic model of the single flexible link manipulator is expressed in the singularly perturbed form. To reduce the end-effector trajectory tracking error, a corrective torque is added to the computed torque command of the rigid link counterpart of the single flexible link manipulator. The corrective torque is derived based on the concept of the integral manifold of the singularly perturbed differential equations. This corrective torque is of order ε 2 where , and f is the fundamental natural frequency of the single flexible link manipulator. The implementation of the introduced technique does not require the full-state measurements since by designing an observer, the time derivative of the links lateral deflection is estimated. The stability proof of the new controller, which is based on the Lyapunov criterion, is presented. The results of the simulation and experimental studies are also included to, respectively, show the effectiveness and feasibility of the new controller.

On the Zeros of the Transfer Function of Flexible Link Manipulators and Their Non-Minimum Phase Behaviour

In this article, a study of the zeros of the transfer function, between the base torque and the end-effector displacement, for flexible link manipulators is performed. The analysis is carried out on a single flexible link manipulator with the initial part of the link being rigid. This type of manipulator is referred to as a slewing single rigid—flexible link manipulator (SRFLM). A new method for finding the zeros of the transfer function of an SRFLM without using the corresponding transfer function is introduced. The changes of the locations of the zeros of an SRFLM owing to the changes in all the physical parameters (PPs) are investigated. It is shown that there are PPs where the increase in their values moves the zeros further from the imaginary axis; while by increasing the values of some other PPs the zeros become closer to the imaginary axis. Finally, there are PPs where the locations of the zeros are independent of their values. These findings will be beneficial in the design as well as control of flexible link manipulators and are among the main contributions of this work.

Causal inversion of a single-link flexible-link manipulator via output planning

Recently, a method was introduced by Bensoman and Le Vey (July 2003) for the stable inversion of a single-input single-output (SISO) non-minimum phase system through the employment of output planning. The use of a causal input via output planning permits the initial conditions to be freely selected. In the application of this method to a single-link flexible-link manipulator (SFLM), for any initial conditions a polynomial is sought for the end-effectors trajectory that cancels the effects of the internal instability of the inverse system. A drawback of this method, however, is that allows for the development of only one polynomial for any initial conditions imposed on the input and output. An extension of this method is proposed in this paper which does not suffer from this drawback. This extension employs exponential functions to define the end-effectors trajectory, which leads not to just one solution, but to a family of solutions. The results of some simulation studies that verify the proposed extension are included.

Vibration analysis of a Timoshenko beam on a moving base

In this paper free vibration of a Timoshenko beam with a tip payload, which is mounted on a cart (referred to as TBC) is studied. The cart (base) can only have lateral displacement and the tip payload has both mass and mass moment of inertia. The center of mass of the payload does not coincide with the point where the beam connects to the payload. Therefore, the tip of the beam is exposed to an extra bending moment due to the inertial force of the payload. By employing Hamilton’s principle, the governing equations of motion and the associated boundary conditions for the TBC are first derived and then transferred into dimensionless forms. By using these governing equations and their associated boundary conditions, the closed-form frequency equation (characteristic equation) of the TBC is derived. This closed-form frequency equation is validated both analytically and numerically. The closed-form expressions for the mode shapes of the TBC and their orthogonality are also presented. By using the closed-form characteristic equation, a sensitivity study is performed and the changes in the natural frequencies versus changes in the physical parameters are investigated. The results presented in this paper are valuable for precise dynamic modeling and model-based control of flexible mobile manipulators; a flexible mobile manipulator is a flexible link manipulator with a moving base.

Kinematic Analysis of the Spherically Actuated Platform Manipulator

New methods for the inverse and forward kinematic analysis of the novel six degrees of freedom (6DOF) parallel manipulator which has only two legs are presented. The actuation of the new mechanism is through two base-mounted spherical actuators. In the inverse pose kinematic, active joint variables are directly calculated with no need for the evaluation of passive joint variables. In the forward pose kinematic, closed form solution adopting a new approach is presented. It is shown that the inverse and forward pose kinematic have sixteen and four different solutions, respectively. Moreover, closed form equations for the rate kinematic analysis are proposed. Finally, two different categories of the singularity points for the new mechanism with their geometrical interpretation are introduced. In one category the mechanism loses one or more DOF while in the other one it gains one or more DOF.

Trajectory Planning for a Wheeled Mobile Robot and its Robotic Arm

A temporal planning algorithm that is implemented on a wheeled mobile robot is presented. This algorithm has two parts: the first part is developed to control the motion of the mobile robot, and the second part is designed to control a 6-degrees-of-freedom (DOF) arm attached to the robot. In most robotic applications, it is necessary for the mobile robot to plan and follow a desired path. It may also be necessary for the robot to follow a given velocity profile, which is known as temporal planning. The advantage of temporal planning method in this paper is in its simplicity and its computational efficiency. A rudimentary trajectory is first created by assigning an arbitrary time to each segment of the path. This trajectory is made feasible by applying a number of constraints and using a linear scaling technique. When a velocity profile is given, a nonlinear time scaling technique is used to fit the mobile robot’s linear velocity to the given velocity profile. A method for avoiding moving obstacles is also implemented. Simulation and experimental results showed good agreement with each other. A novelty of this paper is in developing and implementing a new method for control of a 6-DOF arm attached to the mobile robot. Two methods have been proposed and tested for position control of the robot arm; i) linear end-effector increment (LEI), and ii) linear joints increment (LJI). It is shown that LEI is more precise than LJI in trajectory tracking of robot arm; however, the singularity of Denavit-Hartenberg (DH) transformations matrix limits the application of this method for specific trajectories. The LJI is developed to avoid the singularity in DH transformation matrix. The experimental results for four different paths show the effectiveness of the LJI approach. The successful experimental results of path and temporal planning of a wheeled mobile robot and motion control of its industrial arm is reported.Copyright

Parameter Identification of a Friction Model for Robotic Joints

In this paper a new identification method to obtain the friction parameters in the joints of robotic manipulators is presented. These parameters are coulomb friction, static friction, Stribeck velocity constant and viscous damping coefficient. The available methods to find these parameters either require the design of a controller or the precise value of system parameters such as mass moment of inertia. In contrast, the new method proposed here finds these parameters by a nonlinear optimization approach which requires neither any knowledge of system’s parameters nor any controller design. The corresponding nonlinear optimization problem is solved using an efficient technique which does not require iteration or any initial estimate of optimization parameters. The new method proposed in this paper was experimentally verified on a robotic manipulator available in the robotics laboratory at the University of Saskatchewan.Copyright

Truncation errors of assumed shape modeling for flexible-link manipulators

The assumed mode shape method has been widely used to derive finite degree-of-freedom dynamic models for flexible-link manipulators, which theoretically have infinite degree-of-freedom dynamics. For a single flexible manipulator, this approximation changes locations of the zeros of transfer functions between base torque and end-effector displacement. The change in locations of zeros considerably affects accuracy of the model and therefore the performance of model-based controllers. This article presents a comprehensive study on the change in locations of zeros due to the truncation associated with assumed mode shape method. It is shown that the locations of approximate zeros depend on four non-dimensional parameters, whereas the locations of analytical zeros depend on only two non-dimensional parameters. Approximate zeros are obtained from assumed mode shape method models, whereas analytical zeros are derived from infinite order models. A thorough study of the differences between approximate zeros and analytical zeros versus the number of mode shapes as well as all the physical parameters is performed. Moreover, guidelines are provided to select the numbers of mode shapes such that the approximate zeros become close to the analytical zeros. These guidelines can easily be used by control and modeling engineers, making them valuable for modeling and control of flexible robot manipulators.