Reza Fotouhi

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 and temporal planning of a wheeled mobile robot on an uneven surface

Computing a realistic velocity profile for a mobile robot is a challenging task due to the large number of kinematic and dynamic constraints involved. In order for a mobile robot to complete its task it must be able to plan and follow a trajectory. It may also be necessary to follow a given velocity profile, depending on the environment. Temporal planning, or following a given velocity profile, can be used to minimize time of motion and to avoid moving obstacles. For example, assuming the mobile robot is a smart wheelchair, it must follow a prescribed path while following a strict speed limit. This paper presents a temporal planning algorithm that is implemented on a wheeled mobile robot to be used in an indoor setting, such as a hospital ward. The path planning stage is accomplished by using cubic spline functions. A trajectory is created by assigning an arbitrary time of 1 s 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 non-linear time scaling technique is used to fit the mobile robots 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. The main contribution of this paper is in developing a temporal planning algorithm, which is capable of moving on an uneven surface (graded non-flat), and its implementation on the mobile robot at the robotics lab in the University of Saskatchewan. This algorithm is computationally very efficient as it requires low computation cost and does not involve major iterations.

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.

A manoeuvre control strategy for flexible-joint manipulators with joint dry friction

A new control strategy based on the singular perturbation method and integral manifold concept is introduced for flexible-joint manipulators with joint friction. In controllers so far developed based on the singular perturbation theory, the dynamics of actuators of flexible-joint manipulators are partially modelled, and the coupling between actuators and links is ignored. This assumption leads to inaccuracy in control performance and error in trajectory tracking which is crucial in high-precision manipulation tasks. In this paper, a comprehensive dynamic model which takes into account the coupling between actuators and links is developed and a composite controller is then designed based on the singular perturbation theorem and integral manifold concept. To overcome the joint friction, a novel method is introduced in which a linear feed-forward torque is designed using the principle of work and energy. Finally, the experimental set-up of a single rigid-link flexible-joint manipulator in the Robotics Laboratory at the University of Saskatchewan is used to verify the proposed controller. Experimental results employing the new controller show that the trajectory tracking error during and at the end of the motion of the robot manipulator is significantly reduced.

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.

On the Stability of a Friction Compensation Strategy for Flexible-Joint Manipulators

A new control strategy for flexible-joint manipulators with joint friction is proposed. The proposed controller includes two main components: a friction compensating torque, and a composite controller torque to compensate for the friction in the joints and the flexibility of the joints, respectively. The novel approach that is used to compensate for the friction in the joints includes a nonlinear one-state LuGre model and a PD control. The second method of Lyapunov is utilized to find sufficient conditions for the global asymptotic stability of the friction compensating torque and it is shown that the tracking error is bounded as long as appropriate PD gains are chosen. The performance of the proposed controller is experimentally verified using a setup of a two-rigid-link flexible-joint manipulator. Experimental results are presented for different flexibilities of the joints and also for different desired trajectories. These results demonstrate the high performance of the controller under different situations while the tracking error remains very small.

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.

Welded T-joint connections of square thin-walled tubes under a multi-axial state of stress

T-joint connections are used extensively in industry as parts of machine components and structures. The T-joint connection is typically constructed through the welding of its tubular members, with significant stress and strain concentrations occurring at the toe of the weld under loadings. In this paper, a welded T-joint connection of square hollow-section (SHS) tubes subjected to a multi-axial state of stress is examined both numerically and experimentally. The hot spot strains and stresses in the connection are determined through a detailed finite element (FE) analysis of the joint. The weld geometry is accurately modelled using FE. To model the weld, several full-scale welded T-joints were cut at the connection to obtain the size and depth of penetration of the weld. For the experimental study, a test rig with a hydraulic actuator capable of applying both static and cyclic loadings is designed and used. Strain gauges are installed at several locations on the joint to validate the FE model. The verified FE model is then used to study the through-the-thickness stress distributions of the tubes. It is shown that the membrane stresses which occur at the mid-surface of the tubes remain similar regardless of the weld geometry. The weld geometry only affects the bending stresses. It is also shown that the stress concentrations are highly localized at the vicinity of the weld toe. At a distance of about half of the weld thickness from the weld toe, the effect of the weld geometry on the bending stresses becomes insignificant as well. To reduce the stress concentrations at the T-joint, plate reinforcements are used in a number of different arrangements and dimensions to increase the load-carrying capacity of the connection.

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.