Kinematic and Dynamic Simulation of 6 DOF P-U-S Type Manipulators


A parallel manipulator typically consists of a moving platform and a fixed base that are connected together by several limbs. Because of the closed-loop architecture, not all of the joints can be independently actuated and usually the number of actuated joints is selected to be equal to the number of degrees of freedom of the manipulator. Parallel manipulators for which the number of chains is strictly equal to the number of d.o.f. of the end-effector are called fully parallel kinematic manipulators (FPKMs).

Architecturally, there are many choices for type of legs, joints and numbers of attached legs to the platform which can be significant factors for determining the workspace and actuation requirements. Considerable efforts have focused on enumerating and classifying various types of parallel architecture manipulators. Further significant efforts have also been expended for dimensional optimization to enhance kinematic, kinetostatic and dynamic performance of such systems.

It has been studied earlier that it is advantageous to locate the actuations adjacent to the fixed base (rather than attaching it midway in an articulated leg like the traditional Stewart-Gough platform). Such architectures proved to be beneficial based on the following factors: (1) absorption of major portion of reaction forces by the ground resulting in almost vibration-free operation with light-weight mobile components, (2) reduced effect of inertia due to the elimination of actuator’s weight, (3) absence of interference of actuators and routing cables due to base location of actuators. Further, by selecting the base actuated joint to be prismatic, the proximal links are not subjected to the bending moments and the corresponding stresses. The resulting class of 6-DOF P-U-S (active prismatic– universal– spherical, shortly referred to as 6-P-U-S hereafter) FPKMs is generally composed of six sliding actuators, six fixed length links and a mobile platform (as in Figure 1). The sliders are active prismatics that move along linear motion guides and are fixed on the ground. Hence, such systems are also referred to as “HexaSlides” or “Hexaglides” in the past. The axes of the prismatic joints along which the centers of the universal joints are being translated will be referred to as the rail axes. The links are of constant length and are connected to the sliders through universal joints (U). Finally, the links are connected to the mobile platform through spherical joints (S).

 In our research, we analyzed a general 6-P-U-S architecture and conducted a kinematic analysis for workspace as well as singularities. Using the symbolic kinematic modeling, we then specialized this general system to a specific architecture that would not posses any form of workspace interior singularities. We even analyzed the workspace aspect for such parallel-architecture manipulators – with the primary goal of optimizing the link geometries and parameters to enhance overall workspace and other selected geometric workspace-based performance-measures. While multiplicity of supporting serial-chains engenders challenges for control, we will seek to exploit it for workspace-design by noting that the parallel manipulator workspace can be determined as the intersection of the individual serial-supporting chains. In other words, the workspace can be determined by geometric combination – union and/or intersection – of swept serial-chain workspace volumes. Based on the comprehensive kinematic and workspace analysis of the platform, we were able to obtain the optimal geometrical parameters for 6-P-U-S system and based on these parameters Quanser Inc. fabricated the final device.

Nevertheless, there are many challenges to successful deployment of such a method. Two facets contribute to the increased computation cost and complexity engendered in workspace design and analysis. First, while the process is conceptually simple and that can be trivially implemented for planar manipulators, the complexity increases considerably when spatial manipulators are considered. Further, effective visualization is a key to workspace-design and analysis and can be difficult for 6-DOF workspaces. To overcome this issue, such workspaces are classified into various groups out of which two types are commonly considered important and studied: (i) a constant orientation workspace, taken to be the 3D space of points where the manipulator can reach while keeping its orientation fixed; (ii) a constant position workspace –set of orientations possible for the manipulator while keeping the position of the platform center fixed.

The efficient geometric-based workspace calculation techniques were proposed for this purpose. The efficiency and performance of this improvised method was compared against the conventional (parametric sweeping) method. Finally, this scheme was extended to reap the benefits of automatic volume calculation and a subsequent kinematic design optimization using a commercial CAD package. While our focus is the 6-P-U-S manipulator, the method is benchmarked against published results for the Stewart-Gough platform as well.

A Gaussian divergence theorem based framework was also implemented that could be used to compute exact workspaces for parallel manipulators. Presently, we have implemented it to compute the entire workspace of a 3-PRR planar manipulator using a 2D Gaussian divergence theorem. In the near future, we also intend to extend this work to spatial class of parallel manipulators especially to 6-P-U-S to enable 3D volume computation.


Despite advances in state-of-the-art and considerable efforts over past few decades, the unified method to develop dynamic models of parallel architecture systems still remains a challenge. The applicability of many standard approaches including Lagrangian modeling, principle of virtual work etc has been restricted to standard manipulators due to the complexities and nonlinearities— introduced mainly by the multiplicity of chains of such systems under consideration. This is particularly the case in spatial robots, whose nature as systems requires integration of physical and information infrastructures and system-based thinking. Our guiding vision however is to create an overall framework for simulation of inverse dynamic analysis and implement simultaneous force and trajectory control on this complicated manipulator. We also intend to explore the advantage and effectiveness of a model-based control performance using symbolic computation available in commercial packages like Maple, MapleSim etc. 

*This project is funded by the National Science Foundation CAREER Award under Grant IIS-0347653 and CNS-0751132.



 Students Involved:

- Madusudanan Sathia Narayanan, PhD Candidate, University at Buffalo

- Xiaobo Zhou, PhD Candidate, University at Buffalo

- Luca Carbonari, Visiting Scholar

- Leng-Feng Lee, PhD Candidate, University at Buffalo

- Hrishi L Shah, M. S., University at Buffalo [Graduated]

- Srikanth Kannan, M. S., University at Buffalo [Graduated]

- Yao Wang, M. S., University at Buffalo [Graduated]



 Related Publications - Conference Proceedings:



H., Shah, M.S., Narayanan, and V.N., Krovi, “CAD-Enhanced Workspace Optimization for Parallel Manipulators: A Case Study”, IEEE 2010 Conference on Automation Science and Engineering, Toronto, Ontario, Canada, August 21-24, 2010. (Best Conference Poster Award) [IEEEXplore]




M.S. Narayanan, Sourish Chakravarty, Hrishi Shah, and V.N. Krovi, "Kinematic- Static- and Workspace Analysis of a 6- P-U-S Parallel Manipulator", Proceedings of ASME 2010 International Design Engineering Technical Conferences, Montreal, Quebec, Canada, August 15-18, 2010.






 Related Publications - Theses:


Hrishi L Shah, "Role of Automated Symbolic Generation of Plant Models in Robotics Education", M.S. Thesis, Dept. of Mechanical & Aerospace Engineering, SUNY at Buffalo, Sep. 2010.



Madusudanan Sathia Narayanan, "Analysis of Parallel Manipulator Architectures for Use in Masticatory Studies", M.S. Thesis, Dept. of Mechanical & Aerospace Engineering, SUNY at Buffalo, Sep. 2008.



Srikanth Kannan, "Quantitative Analysis Of Masticatory Performance in Vertebrates", M.S. Thesis, Dept. of Mechanical & Aerospace Engineering, SUNY at Buffalo, Sep. 2008.



Yao Wang, "Symbolic Kinematics and Dynamics Analysis and Control of a General Stewart Parallel Manipulator", M.S. Thesis, Dept. of Mechanical & Aerospace Engineering, SUNY at Buffalo, Sep. 2008.



by Automation, Robotics & Mechatronics Laboratory, Mechanical and Aerospace Engineering, University at Buffalo
Last Updated: March 07, 2011