An efficient explicit dynamic formulation of a Stewart platform parallel robot via new formalism

Parallel manipulators have been proposed to overcome accuracy problem in the end effector positioning, exhibited by serial manipulators(Stewart, 1965)(Reboulet, 1988)(Merlet, 2000). These parallel robots are primarly used in the applications for which the considered processes require a high degree of accuracy, high speeds or accelerations. Aircraft simulator (Stewart, 1965), machining tools (Neugebauer et al., 1998)(Poignet et al., 2002), and various other medical applications (Merlet, 2002)(Leroy et al., 2003)(Plitea et al., 2008) constitute some of the many possible applications of parallel robots. The computation of the inverse dynamic model is essential for an effective robot control. In the field of parallel robots, many approaches have been developed for efficient computation of the inverse dynamics. The formalism of d’Alembert has been used to obtain an analytical expression of the dynamics model (Fichter, 1986)(Nakamura & Ghodoussi, 1989). The principle of virtual works has been applied in (Tsai, 2000) for solving the inverse dynamics of the Gough-Stewart platform and in (Zhu et al., 2005) for a Tau parallel robot. Lagrangian formalism is applied in (Leroy et al., 2003) for the dynamics modeling of a parallel robot used as a haptic interface for a surgical simulator. These various approaches do not seem effective for a robot dynamic control under the real time constraint. A better computational efficiency can be achieved by the development of approaches using recursive schemes, in particular, based on the Newton-Euler formulation. Gosselin (Gosselin, 1996) proposed an approach for the computation of the inverse dynamic model of planar and spatial parallel robots, in which all the masses and inertias are taken into account. This proposed method is difficult to generalize for all the parallel architectures. Dasgupda et al (Dasgupta & Choudhury, 1999) applied this method to several parallel manipulators. Khan (Khan, 2005) has developed a recursive algorithm for the inverse dynamics. This method is applied to a 3R planar parallel robot. Bi et al (Bi & Lang, 2006) use the Newton-Euler recursive scheme for the computation of the articular forces of a tripod system. Khalil et al (Khalil & Guegan, 2004) proposed a general method for the inverse and direct dynamic model computation of parallel robots, which is applied to several parallel manipulators (Khalil & Ibrahim, 2007). Despite the large amount of contributions in this field, there is still a need for improving the computational effeciency of the inverse kinematic and dynamic model clculation for real-time control. In this paper, a parallel robot is considered as a multi robot system with 17

This work presents a new systematic solution to identify the vehicle inertia parameters which are essential inputs for vehicle simulation and vehicle safety research. In conceptual design phase of this work, a virtual three Degree-of-Freedom (DoF) test bench/ parallel manipulator (PM) whose moving platform is used to clamp vehicle under test is developed. In order to realize the kinematic characteristics of the proposed PM, the kinematic analysis consists of inverse kinematic and singularity architecture is carried out. Aiming at obtaining all ten vehicle inertia parameters (i.e., mass, center of gravity and inertia tensor), the observation matrix for parameter identification is derived from the dynamic model of PM. To get the dynamic model, the Euler’s equation and Lagrange approach are applied to implement the dynamic analysis for PM’s moving platform and actuators, respectively. It is beneficial to reduce the complexity of dynamic model and load of numerical computation. In the following section, to minimize the sensitivity of parameter identification to measurement noise, an optimization process of searching for the optimal movement trajectory of PM is proposed. For this purpose, the parameterized finite-Fourier-series are used to definite the general movement trajectory of PM firstly. Subsequently, the parameters of general trajectory are optimized by employing a nonlinear iterative algorithm. Objective of this algorithm is to obtain the minimal condition number of observation matrix and meanwhile to ensure the PM still works in the achievable working space during the test. The results show that the vehicle inertial parameters can be effectively identified by executing the single optimal movement trajectory on the PM. It is expected that the proposed systematic solution could be an important approach to improve the identification efficiency and identification accuracy of vehicle inertial parameters.

Considering the real-time control of a high-speed parallel robot, a concise and precise dynamics model is essential for the design of the dynamics controller. However, the complete rigid-body dynamics model of parallel robots is too complex for online calculation. Therefore, a hierarchical approach for dynamics model simplification, which considers the kinematics performance, is proposed in this paper. Firstly, considering the motion smoothness of the end-effector, trajectory planning based on the workspace discretization is carried out. Then, the effects of the trajectory parameters and acceleration types on the trajectory planning are discussed. But for the fifth-order and seventh-order B-spline acceleration types, the trajectory will generate excessive deformation after trajectory planning. Therefore, a comprehensive index that considers both the motion smoothness and trajectory deformation is proposed. Finally, the dynamics model simplification method based on the combined mass distribution coefficients is studied. Results show that the hierarchical approach can guarantee both the excellent kinematics performance of the parallel robot and the accuracy of the simplified dynamics model under different trajectory parameters and acceleration types. Meanwhile, the method proposed in the paper can be applied to the design of the dynamics controller to enhance the robot's performance.

Despite their well-known advantages in terms of higher intrinsic rigidity, larger payload-to-weight ratio, and higher velocity and acceleration capacities, parallel robots have drawbacks. Among them, the most important one is surely the presence of singularities in the workspace, which divide the workspace into different aspects (each aspect corresponding to one or more assembly modes) and near which the performance is considerably reduced. In order to increase the reachable workspace of parallel robots, a promising solution consists in the definition of optimal trajectories passing through the singularities to change either the leg working modes or the robot assembly modes. Previous works on the field have shown that it is possible to define optimal trajectories that allow the passing through the robot type 2 singularities. Such trajectories must respect a physical criterion that can be obtained through the analysis of the degeneracy conditions of the parallel robot inverse dynamic model. However, the mentioned works were not complete: they lacked a degeneracy condition of the parallel robot inverse dynamic model, which is not due to type 2 singularity anymore, but to a serial singularity. Crossing a serial singularity is appealing as in that case we can change the robot leg working mode and then potentially access to other workspace zones. This absence is due to the fact that the authors used a reduced dynamic model, which was not taking into account all link dynamic parameters. The present paper aims to fill this gap by providing a complete study of the degeneracy conditions of the parallel robot dynamic model and by demonstrating that it is possible to cross the type 2, but also serial singularity, by defining trajectories that respect some given criteria obtained from the analysis of the degeneracy of the robot dynamic model. It also aims to demonstrate that the serial singularities have impacts on the robot effort transmission, which is a point that is usually bypassed in the literature. All theoretical developments are validated through simulations and experiments.

A walking robot consisting of double Stewarts parallel legs was designed by our research team in the past time, which was mainly used for the transportation of the wounded after the disaster. In order to promote stability of control locomotion and ensure invariably horizontal state of the robot platform in the process of motion, the central pattern generator (CPG) based on particle swarm optimization (PSO) is presented to optimize the kinematic model. The purpose of optimization is to solve the hysteresis problem of displacement variation among the electric cylinders. Moreover, the dynamic model of the robot is established, which can provide mechanical basis for the feedback of control signal and make the robot move stably. The simulation results show that the displacement hysteresis problem of the electric cylinders is solved well. Meanwhile, compared with simulation results based on GA-CPG method, it is demonstrated that the robot motion planned using PSO-CPG method has better motion stability and can avoid the impact of legs landing during the transition phase of the motion cycle. The experimental results show that the platform on the robot can maintain an invariably horizontal state, and the locomotion is more stable. It verifies the feasibility of PSO-CPG model and the correctness of the dynamic model of the parallel mobile rescue robot.

Parallel robots must cross Type-II singularities to take full advantage of their workspace. However, since the inverse dynamic model of parallel robots degenerates at these singularities, crossing them requires infinite actuation forces. Previous approaches to crossing singularities with finite forces use Computed Torque Controllers that track a trajectory specifically designed to meet a consistency condition that avoids the degeneracy of the dynamic model. Alternatively, this paper presents an adaptive Computed Torque Controller to cross singularities with finite forces without constraining the trajectory. The proposed controller exploits the modeling errors that prevent an exact vanishing of the tracking errors. These errors multiply the controller gains that intervene in the dynamic model that degenerates at singularities. This allows our controller to adaptively adjust these gains to ensure the consistency condition at all times, avoiding the degeneracy of the dynamic model.

Discussion on: “Comparison of Different Multivariable Control Design Methods Applied on Half Car Test Setup”

This study deals with the dynamic modelling and task space control of the Stewart–Gough platform. The dynamic equations were obtained using the virtual work approach, and include all the 13 presented in the platform, that is, two bodies per each leg and the upper ring. The results show some similarities in the dynamic model of serial and parallel robots, making it possible to extend the control theory of serial robot to the Stewart–Gough platform. To show this, an adaptive control law is proposed to control the parallel platform. The law is obtained from the dynamical equation and its performance is tested using a sinusoidal path for position and orientation of the upper ring of parallel robot. Some additional simulations are carried out to demonstrate a well-behaved controller.

Parallel manipulators with less than six degrees of freedom (DOF) have been increasingly used in high-speed hybrid machine tools. The structural features of parallel manipulators are dynamic, a characteristic that is particularly significant when these manipulators are used in high-speed machine tools. However, normal kinematic control method cannot satisfy the requirements of the control system. Many researchers use model-based dynamic control methods, such as the dynamic feedforward control method. However, these methods are rarely used in hybrid machine tools because of the complex dynamic model of the parallel manipulator. In order to study the dynamic control method of parallel manipulators, the dynamic feedforward control method is used in the dynamic control system of a 3-PSP (prismatic-spherical-prismatic) 3-DOF spatial parallel manipulator used as a spindle head in a high-speed hybrid machine tool. Using kinematic analysis as basis and the Newton-Euler method, we derive the dynamic model of the parallel manipulator. Furthermore, a model-based dynamic feedforward control system consisting of both kinematic control and dynamic control subsystems is established. The dynamic control subsystem consists of two modules. One is used to eliminate the influence of the dynamic characteristics of high-speed movement, and the other is used to eliminate the dynamic disturbances in the milling process. Finally, the simulation model of the dynamic feedforward control system of the 3-PSP parallel manipulator is constructed in Matlab/Simulink. The simulations of the control system eliminating the influence of the dynamic characteristics and dynamic disturbances are conducted. A comparative study between the simulations and the normal kinematic control method is also presented.The simulations prove that the dynamic feedforward control method effectively eliminates the influence of the dynamic disturbances and dynamic characteristics of the parallel manipulator on high-speed machine tools, and significantly improves the trajectory accuracy. This is the first attempt to introduce the dynamic feedfordward control method into the 3-PSP spatial parallel manipulator whose dynamic model is complex and provides a study basis for the real-time dynamic control of the high-speed hybrid machine tools.

Due to the high stiffness, high dynamic performance, the parallel manipulator presents great advantages in the industrial manufacture. However in the machining process, the external low frequency disturbance, e.g. the varying cutting force, has a significant effect on the control system of parallel manipulator, which presents a chatter phenomenon on the end-effector of manipulator. In this paper, a feed forward control strategy is proposed to eliminate the effect of the random external disturbance on the control system of parallel manipulator. By applying the external disturbance force on the inverse dynamic model, the compensation torque is calculated and fed forward into the manipulator driving joints to cancel out the effect of the disturbance acting on the manipulator end-effector. The key issue herein is to be able to establish the accurate dynamic model for the parallel manipulator. Furthermore, in order to guarantee the position precision of the manipulator, a feed forward model-based control strategy combined with the feedback loop PV (position and velocity) control has been developed based on the reference trajectory, which could relatively simplify the highly nonlinear control system of the parallel manipulator and obtain a stable tracking error model. The whole research has been carried out on a parallel manipulator named CaPaMan which has been built in the laboratory of robotics and mechatronics in university of Cassino and South Latium. The results show that the chatter phenomenon could be utterly depressed by the force compensation from the feed forward path of the external disturbance; meanwhile the model-based controller can guarantee the trajectory tracking accuracy within a stable error by choosing the suitable PV gains.

All countries in the world today have different degrees of aging. In order to help the elderly to walk properly, based on the analysis about the common walking stick, this paper have developed a smart walking stick robot which meets the walking habits of the elderly. The walking stick's use process is divided into supporting state and non-support state. By analyzing the variation of force, moment and angle between users and common walking stick under two conditions, the walking stick's mathematical model is established. By analyzing the structure and movement process of the walking stick robot, a dynamic model of the robot is established. At the same time, the admittance control model is used to satisfy the suppleness of the force by controlling the chassis motion of the robot. The final design experiment verified the validity of the walking stick model and the adequacy of admittance control.

A parallel manipulator provides an alternative design to serial manipulators, and can be found in many applications such as mining machines (Arai et al., 1991). Through the design of active joints such that actuators are fixed to the manipulator base, the mass of moving components of the parallel manipulator is greatly reduced, and high speed and high acceleration performance may be achieved. Parallel manipulators, comprised of closed-loop chains due to multiple linkages of the parallel structure, also provide high mechanical rigidity, but adversely exhibit smaller workspace and associated singularities. Considerable research has focused on kinematic analysis and singularity characterization of these devices (Gosselin & Angeles, 1990, Merlet, 1996). Planar parallel manipulators typically consist of three closed chains and a moving platform. According to the arrangement of their joints in a chain, these mechanisms are classified as PRR, RRR etc. where P denotes a prismatic joint and R denotes a revolute joint respectively. An assembly industry, such as the electronic fabrication, demands high-speed, high acceleration placement manipulators, with corresponding lightweight linkages, hence these linkages deform under high inertia forces leading to unwanted vibrations. Moreover, such multiple flexible linkages of a parallel manipulator propagate their oscillatory motions to the moving platform where a working gripper is located. Therefore, such vibration must be damped quickly to reduce settling time of the manipulator platform position and orientation. A number of approaches to develop the dynamic model of parallel manipulators with structural flexibility have been presented in the literature(Fattah et al., 1995, Toyama et al. 2001), but relatively few works related to vibration reduction of a parallel manipulator have been published. Kozak (Kozak et al., 2004) linearized the dynamic equations of a two-degree-of-freedom parallel manipulator locally, and applied an input shaping technique to reduce residual vibrations through modification of the reference command given to the system. Kang (Kang et al., 2002) modeled a planar parallel manipulator using the assumed modes method, and presented a two-time scale controller for linkage vibration attenuation of the planar parallel manipulator. Since both the input shaping technique and the two-time scale control scheme, applied to parallel manipulators,

This thesis explores the fundamentals of a new class of parallel mechanisms called parallel mechanisms with configurable platforms as well as the design and analysis of parallel robots that are based on those mechanisms. Pure parallel robots are formed by two rigid links, the base and the end-effector, connected in parallel by independent serial chains, called legs. The novel concept behind parallel robots with configurable platforms is that the rigid end-effector is replaced by a closed-loop chain, the configurable platform. Some of the links of this closed-loop chain are attached to the legs so its configuration can be fully controlled from the actuators located on the base. The use of a closed-loop chain instead of a rigid end-effector allows the robot to interact with the environment using multiple contact points on the platform. This results in a robot that can combine motion and grasping capabilities into a structure that provides an inherent high structural stiffness. All the actuators are grounded on the base requiring no additional motor at the end-effector location to provide the grasping. High-speed pick-and-place robots and haptic interfaces are examples of devices that can benefit from this new type of architecture.

The parallel robot is a multivariable nonlinear and strongly coupled system, which is difficult to apply to controller design. In order to improve the trajectory tracking accuracy and anti-disturbance ability of the parallel robot, a control method based on improved dynamic integral sliding mode is proposed. Firstly, the dynamic model of the parallel robot is decoupled and simplified. Secondly, considering the errors caused by the decoupling and simplification of the dynamic model and the existence of external disturbances in the actual system, a sliding mode controller is introduced to improve both the trajectory tracking accuracy and the robustness of the system. The traditional sliding mode control method, however, will produce large chattering, which may damage the robot. Therefore, by constructing the second-order dynamic sliding mode surface based on the first-order linear sliding mode, the chattering can be suppressed to a great extent, while maintaining the strong robustness of the sliding mode control. Then, a Lyapunov function is constructed for stability analysis, followed by stability analysis of the closed-loop control system. Finally, an experimental platform is built for comparative experiments to verify the effectiveness and practicability of the proposed controller.

Parallel robots are being applied popular in industry. They usually have many closed-loop kinematic structures, therefore their movement equations have the form of redundant coordinates and are complex. In most cases, parameters of dynamic model of the parallel robots required to be known exactly for control problem using the control laws such as PD, PID, CTC. This study is going to see in which the conditions of the Rostock Delta parallel robot should be used well with the PD control law. The movement equations of Rostock Delta robot in the form of differential equation system with redundant generalized coordinates is used to establish the PD control law. The stableness of the control law is also proved. Tracking control of a specific Rostock Delta robot is then simulated to see whether the PD control law could be used effectively or not in the cases of with and without the error of robot and noise.KeywordsParallel robotPD controlDelta rostock robotModel-based control