Model Based Control of a Parallel Robot – A Comparison of Control Algorithms

In this contribution the control behavior of a special construction of a parallel robot, called multi-axes test facility, is investigated. After a brief discussion of the different tasks of the robot the construction of the robot is briefly presented. To solve the tasks, different control algorithms are derived based on model equations of different complexity of the robot. Depending on the task to be performed by the robot, the controllers compensate the kinematic and/or kinetic coupling of the degrees of freedom of the robot, stabilize the system and achieve the desired spatial motion of each degree of freedom as well as sufficient robustness with respect to parameter uncertainties and load variations. A few results obtained in computer simulations and laboratory experiments are presented and judged with respect to the quality of control, the closeness to reality of the computer simulations, and the amount of costs and work needed to realize the different solutions.     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

Comparative study on control concepts of a robot manipulator with multiple-link/joint flexibilities

If one is dealing with active vibration suppression on a highly nonlinear flexible system, various techniques are needed. On the one hand a suitable dynamic model of the system is required. And on the other hand intelligent model based control concepts are necessary for active vibration damping. We deal with a basic model, where the flexibilities are approximated with linear springs and dampers, a so called lumped element model (LEM). For the control design we propose a control structure with two degrees of freedom (2DoF) for solving the tracking problem, based on the LEM. Such an approach allows designing the feedforward part independently of the feedback part. Hereby the feedforward control is based on the flatness approach, while for the feedback control several strategies are studied using acceleration- and gyrosensor-measurements. The contribution is completed with a validation by measurements from a very fast trajectory on an articulated robot with two flexible links and three elastic joints. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control Strategies for a Hybrid Articulated Robot

Due to higher requirements in productivity and cost efficiency of production lines, robots and other manipulators have to move faster. One possibility to fulfill the mentioned goals is to build lightweight constructions having elastic deformations in joints and links. The elastic components tend to vibrations and static deflections. Methods that compensate or minimize these drawbacks are the focus of this paper. An articulated robot with 6 joints and flexibility in joints and links is under consideration. The joints are actuated by DC motors combined with Harmonic Drive gears which offer high gear ratios but undergo elastic deformations. The links are flexible in two bending directions and in torsional sense. To achieve ordinary differential equations, a Ritz approach together with the projection equation is used. The obtained model is used for feedforward and feedback control design. Based on reference trajectories and on a rigid body model, estimations for the elastic deflections are calculated. These deflections are used to alter the reference trajectory in order to minimize the error of the tool center point. For basic active damping, non-local curvature feedback is used. Together with PD joint control and the feedforward control, satisfying results are obtained. Additionally, a sliding control approach is presented. The stiffness of the tool center point is enhanced with the drawback of less active damping. Simulation results and measured data are presented and compared. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computed torque control of an under-actuated service robot platform modeled by natural coordinates

The paper investigates the motion planning of a suspended service robot platform equipped with ducted fan actuators. The platform consists of an RRT robot and a cable suspended swinging actuator that form a subsequent parallel kinematic chain and it is equipped with ducted fan actuators. In spite of the complementary ducted fan actuators, the system is under-actuated. The method of computed torques is applied to control the motion of the robot.The under-actuated systems have less control inputs than degrees of freedom. We assume that the investigated under-actuated system has desired outputs of the same number as inputs. In spite of the fact that the inverse dynamical calculation leads to the solution of a system of differential–algebraic equations (DAE), the desired control inputs can be determined uniquely by the method of computed torques.We use natural (Cartesian) coordinates to describe the configuration of the robot, while a set of algebraic equations represents the geometric constraints. In this modeling approach the mathematical model of the dynamical system itself is also a DAE.The paper discusses the inverse dynamics problem of the complex hybrid robotic system. The results include the desired actuator forces as well as the nominal coordinates corresponding to the desired motion of the carried payload. The method of computed torque control with a PD controller is applied to under-actuated systems described by natural coordinates, while the inverse dynamics is solved via the backward Euler discretization of the DAE system for which a general formalism is proposed. The results are compared with the closed form results obtained by simplified models of the system. Numerical simulation and experiments demonstrate the applicability of the presented concepts.     

Directional fields algebraic non-linear solution equations for mobile robot planning

In this study, a novel approach to robot navigation/planning by using half-cell electrochemical potentials is presented. The half-cell electrode’s potential is modelled by the Nernst equation to yield automatic search/detection of pipeline flaws by using the direct current voltage gradient (DCVG) technique. We introduce a theory of spherical volumetric electric density in the soil to sustain our postulates for navigational potential fields. The Nernst potential is correlated with the distance to a pipe’s flaw by proposing a fitted theoretical-empirical nonlinear regression model. From this, volumetric derivatives are solved as gradient-based fields to control wheeled robot’s motion. A nonlinear system for trajectory planning is proposed, and analytically solved by an algebraic solution. This solution directly adjust robot’s speed kinematic values to lead it toward the flaw. The inverse/forward kinematic constraints are non-holonomic, and are recursively integrated into the general potential equation. Analytical modelling is reported, and a set of numerical simulations are presented to prove the feasibility of the proposed formulations.     

Role of Pectoral Fin Flexibility in Robotic Fish Performance

Pectoral fins play a vital role in the maneuvering and locomotion of fish, and they have become an important actuation mechanism for robotic fish. In this paper, we explore the effect of flexibility of robotic fish pectoral fins on the robot locomotion performance and mechanical efficiency. A dynamic model for the robotic fish is presented, where the flexible fin is modeled as multiple rigid elements connected via torsional springs and dampers. Blade element theory is used to capture the hydrodynamic force on the fin. The model is validated with experimental results obtained on a robotic fish prototype, equipped with 3D-printed fins of different flexibility. The model is then used to analyze the impacts of fin flexibility and power/recovery stroke speed ratio on the robot swimming speed and mechanical efficiency. It is found that, in general, flexible fins demonstrate advantages over rigid fins in speed and efficiency at relatively low fin-beat frequencies, while rigid fins outperform flexible fins at higher frequencies. For a given fin flexibility, the optimal frequency for speed performance differs from the optimal frequency for mechanical efficiency. In addition, for any given fin, there is an optimal power/recovery stroke speed ratio, typically in the range of 2–3, that maximizes the speed performance. Overall, the presented model offers a promising tool for fin flexibility and gait design, to achieve speed and efficiency objectives for robotic fish actuated with pectoral fins.     

Towards Washout Filter Concepts for Motion Simulators on the Base of a Stewart Platform

In this paper we present a pneumatically driven Stewart platform as a basis for motion simulators. Motion platforms, that simulate perceived situations in aeroplanes, cars or ships, have workspace constraints in every degree of freedom. Therefore it is necessary to adapt the accelerations and angular rates in order to stay within the physical restrictions. For realizing a flight or ride simulator on the basis of the Stewart platform, “ Washout Filters” are used to change the signals of the open source software FlightGearand to minimize the sensation error between simulator and aircraft. Different filter concepts are implemented and evaluated. The platform presented in this document is a parallel kinematic robot, driven by fluidic muscles. The pneumatically actuated muscles are only able to produce tensile forces. Therefore a spiral spring from a passenger car is used to apply the compressive forces and torques. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Active Vibration Damping and Model-Based Control for an Adaptronic Actuator

The static and dynamic flexibility of a machine tool is often the major limiting factor for the machining quality, especially if high processing speeds as well as high accuracies are desired. Particularly parallel kinematic machines, which have been developed for high processing speeds and accuracy and which often use lightweight structures, can suffer from severe problems with vibrations of system components and the loss of accuracy due to deformations of machine parts. Introducing so–called adaptronic or smart components into the machine tool structure opens up new and interesting possibilities to actively control the deformations and vibrations of flexible machine parts. They allow to specifically target such undesired properties and to significantly reduce or even eliminate their influence on the system performance. Concepts of active vibration damping and model–based control are presented here for such an adaptronic actuator implemented in a machine tool with parallel kinematics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Dynamic Model for a Hybrid Articulated Robot

The dynamic modeling of hybrid systems, consisting of flexible and rigid parts results in large partial differential equation systems (PDE). With the assumption of small deflections and the Ritz expansion the PDE can be approximated by an ordinary differential equation system (ODE) but the number of degrees of freedom is generally high. In this paper a hybrid articulated robot with 2 flexible links and 6 joints is under consideration. The joints are equipped with Harmonic Drive gears with high gear ratio but relative low stiffness. Therefore additionally degrees of freedom are introduced for the elastic deflection of the gears. The links are modeled with flexibility in two bending directions and in torsional sense. To be able to achieve structured equations the projection equation in subsystem representation is used. The projection equation is based on the momentum and the angular momentum equations of each single finite or infinitesimal body which are projected into the space of minimal coordinates and subsequently are summed up. Groups of bodies are collected to the so called subsystems with separated describing velocities. These subsystems are linked together with the kinematical chain. Because the robot is tree structured it is possible to obtain an explicit expression for the second derivatives of the minimal coordinates with a recursive scheme (O(n) efficiency). The robot is controlled with a feed forward controller and a linear PD joint controller. Simulation results and measured data are presented and compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic analysis and system identification of an LCD glass-handling robot driven by a PMSM

In this paper, Hamilton’s principle is employed to derive Lagrange’s equations of an liquid crystal display (LCD) glass-handling robot driven by a permanent magnet synchronous motor (PMSM). The robot has three arms driven by two timing belts. The dynamic formulations can be expressed by one and four independent variables, which are named as the rigid and flexible models, respectively. In order to verify the dynamic formulation is correct, we reduce the flexible model to the rigid one under some assumptions. In this paper, we adopt the real-coded genetic algorithm (RGA) to identify all the parameters of the robot and PMSM simultaneously. It is found that the RGA can identify system parameters which are difficult to be measured in practical problems, for examples, the inductance, stator resistance, motor torque constant, damping coefficient of the motor and timing belts. In numerical simulations, vibrations due to flexibility of the timing belts are investigated for the angular displacements, speeds, accelerations of arms, and the horizontal and vertical displacements of the robot. The angular displacements of the robot arm and the translational positions of the robot end are obtained in the numerical simulations and experimental results. From their comparisons, it is demonstrated that identification results of the dynamic model with four independent variables present the better matching with experimental results of the system.     

Sensitivity Analysis of the Design Parameters for the Calibration of Cable-driven Parallel Robots

Uncertainties in the kinematic parameters like the pulley positions take a major influence onto the force capability of a cable-driven parallel robot. For that purpose this paper describes a calibration method to estimate exactly the underlying kinematic parameters. As the kinematic is influenced by a variety of different parameter, the calibration can be very complex and time consuming. In this approach, a sensitivity analysis of a cable-driven parallel robot is presented to simplify and enhance the calibration. The results are discussed and the further steps are introduced. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot

This paper deals with the dynamics and motion planning for a spherical rolling robot with a pendulum actuated by two motors. First, kinematic and dynamic models for the rolling robot are introduced. In general, not all feasible kinematic trajectories of the rolling carrier are dynamically realizable. A notable exception is when the contact trajectories on the sphere and on the plane are geodesic lines. Based on this consideration, a motion planning strategy for complete reconfiguration of the rolling robot is proposed. The strategy consists of two trivial movements and a nontrivial maneuver that is based on tracing multiple spherical triangles. To compute the sizes and the number of triangles, a reachability diagram is constructed. To define the control torques realizing the rest-to-rest motion along the geodesic lines, a geometric phase-based approach has been employed and tested under simulation. Compared with the minimum effort optimal control, the proposed technique is less computationally expensive while providing similar system performance, and thus it is more suitable for real-time applications.     

Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

In the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors

The paper deals with the dynamics of a spherical rolling robot actuated by internal rotors that are placed on orthogonal axes. The driving principle for such a robot exploits nonholonomic constraints to propel the rolling carrier. A full mathematical model as well as its reduced version are derived, and the inverse dynamics are addressed. It is shown that if the rotors are mounted on three orthogonal axes, any feasible kinematic trajectory of the rolling robot is dynamically realizable. For the case of only two rotors the conditions of controllability and dynamic realizability are established. It is shown that in moving the robot by tracing straight lines and circles in the contact plane the dynamically realizable trajectories are not represented by the circles on the sphere, which is a feature of the kinematic model of pure rolling. The implication of this fact to motion planning is explored under a case study. It is shown there that in maneuvering the robot by tracing circles on the sphere the dynamically realizable trajectories are essentially different from those resulted from kinematic models. The dynamic motion planning problem is then formulated in the optimal control settings, and properties of the optimal trajectories are illustrated under simulation.     

Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach

This paper proposes a modular and control oriented model of a double flexible-link manipulator that stems from the modelling of a spatial flexible robot. The model consists of the power preserving interconnection between two infinite dimensional systems describing the beam’s motion and deformation with a finite dimensional nonlinear system describing the dynamics of the actuated rotating joints. To derive the model, Timoshenko’s assumptions are made for the flexible beams. Using Hamilton’s principle, the dynamic equations of the system are derived and then written in the Port-Hamiltonian (PH) framework through a proper choice of the state variables. These so called energy variables allow to write the total energy as a quadratic form with respect to a state dependent energy matrix. The resulting model is shown to be a passive system, a convenient property for control design purposes.     

Design an interactive virtual physics environment with uncertainties control to support motion tracking

The development of robot or character motion tracking algorithms is inherently a challenging task. This is more than ever true when the latest trends in motion tracking are considered. Some researchers can deal with kinematic and dynamic constraints induced by the mechanical structure. Another class of researchers fulfills various types of optimality conditions, yet others include means of dealing with uncertainties about the robot or character and its environment. In order to deal with the complexity of developing motion tracking algorithms, it is proposed in this paper to design an interactive virtual physics environment with uncertainties for motion tracking based on sliding mode control. The advantages of doing so are outlined and a virtual environment presented which is well suited to support motion tracking development. The environment makes full use of multi-body system dynamics and a robust sliding mode controller independent of model as simulation kernel. So the environment is capable of simulating setups which fulfill the requirements posed by state-of-the-art motion tracking algorithm development. The demonstration results verified the validity of the environment.     

Switched control of a nonholonomic mobile robot

We present a switched control algorithm to stabilize a car-like mobile robot which possesses velocity level nonholonomic constraint. The control approach rests on splitting the system into several second-order subsystems and then stabilizing the system sequentially using finite-time controllers, finally resulting in the mobile robot being moved from one point to another point. State dependent switching control is employed in which the controllers switches on a thin surface in the state-space. Robustness analysis is presented by redefining the switching signal using relaxed switching surface. Both, non-robust and robust controllers are validated through numerical simulation.     

Robust Optimal Trajectory Planning for Robots by Stochastic Optimization

In the optimal control of industrial, field or service robots, the standard procedure is to determine first offline a reference trajectory and a feedforward control, based on some selected nominal values of the unknown stochastic model parameters, and to correct then the inevitable and increasing deviation of the state or performance of the robot from the prescribed state or performance of the system by online measurement and control actions. Due to the stochastic variations of the model parameters, increasing measurement and correction actions are needed during the process. By optimal stochastic trajectory planning (OSTP), based on stochastic optimization methods, the available a priori and sample information about the robot and its working environment is incorporated into the control process. Consequently, more robust reference trajectories and feedforward controls are obtained which cause much less online control actions. In order to maintain a high quality of the guiding functions, the reference trajectory and the feedforward control can be updated at some later time points such that additional information about the control process is available. After the presentation of the Adaptive Optimal Stochastic Trajectory Planning (AOSTP) procedure based on stochastic optimization methods, several numerical techniques for the computation of robust reference trajectories and feedforward controls under real-time conditions are presented. Additionally, numerical examples for a Manutec r3 industrial robot are discussed. The first one demonstrates real-time solutions of (OSTP) based on a sensitivity analysis of a before-hand calculated reference trajectory. The second shows the differences between reference trajectories based on deterministic methods and the stochastic methods introduced in this paper. Based on simulations of the robots behavior, the increased robustness of stochastic reference trajectories is demonstrated.