In this study, an approach to solve the kinematics of the Stewart platform is introduced with the purpose to enhance its originality. Contrary to the Gough platform, the forward displacement analysis of the Stewart platform is based on closure equations that allow to obtain nearly closed-form solutions. These equations lead to decoupled functions which are solved by means of the well-known Sylvester's dialytic method of elimination. The input–output equations of velocity and acceleration as well as the singularities of the robot are obtained systematically by resorting to reciprocal-screw theory. Numerical examples are provided in order to demonstrate the application of the proposed method. Furthermore, the numerical results obtained via screw theory are successfully compared with simulations generated with the aid of commercially available software. Finally, the contribution may be useful to recognize that the flight simulator proposed by Stewart almost five decades ago and the tire testing machine attributed to Gough are in reality robot manipulators with quite different characteristics.
StewartD. A platform with six degrees of freedom. Proc IMechE, Part A: J. Power and Energy1965–66; 180(15): 371–386.
2.
StewartD. A Platform with six degrees of freedom: a new form of mechanical linkage which enables a platform to move simultaneously in all six degrees of freedom developed by Elliott-automation. Aircr Eng Aerosp Tec1966; 38(4): 30–35.
3.
GoughVE. Contribution to discussion of papers on research in automobile stability. Proceedings of the automobile division of the institution of mechanical engineersNew York, NY, 1957, 392–394.
4.
GoughVEWhitehallSGEleyG. Universal tyre testing machine. In Proceedings of the 9th international automobile technical congress, discussion pp.250ff; Federation Internationale des Societes d'Ingenieurs des Techniques de l'Automobile (FISITA). IMechE 1: London, UK, 1962, 117–137
5.
FichterEFKerrDRRees-JonesJ. The Gough-Stewart platform parallel manipulator: a retrospective appreciation. Proc. IMechE Part C: J. Mechanical Engineering Science, JMES 50TH ANNIVERSARY ISSUE PAPER2009; 223: 243–281.
6.
7.
Gallardo-AlvaradoJOrozco-MendozaHRico-MartinezJ.M. A novel five-degrees-of-freedom decoupled robot. Robotica2010; 28(6): 909–917.
8.
MerkleRC. A new family of six degree of freedom positional devices. Nanotechnology1997; 8(2): 47–52.
9.
ZuoAWuQMJGruverWA. Stereo vision guided control of a Stewart Platform. Proceedings of the 2002 IEEE international symposium on intelligent control. Vancouver: Canada, 27–30 October 2002, 125–130.
10.
LeeS.-HSongJ.-BChoiWC. Position control of a Stewart platform using inverse dynamics control with approximate dynamics. Mechatronics2003; 13: 605–619.
11.
KorobeynikovAVTurlapovVE. Modeling and evaluating of the Stewart platform. Proceedings of the international conference graphicon 2005. Novosibirsk Akademgorodok: Russia, 20–24 June 2005.
12.
CappelKL. Letter to the editor: six degrees of freedom. Mechanical Engineering2002New YorkASME
13.
HustyML. An algorithm for solving the direct kinematic of general Stewart-Gough platforms. Mech Mach Theory1996; 31(4): 365–380.
14.
NanuaPWaldronKJMurthV. Direct kinematic solution of a Stewart platform. IEEE Trans Robot Autom1990; 6(4): 438–444.
15.
InnocentiCParenti-CastelliV. A new kinematic model for the closure equations of the generalized Stewart platform mechanism. Meccanica1991; 26(4): 247–252.
16.
RaghavanM. The Stewart platform of general geometry has 40 configurations. ASME J Mech Des1993; 115(2): 277–282.
17.
WamplerCW. Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates. Mech Mach Theory1995; 31(3): 331–337.
18.
InnocentiC. Forward kinematics in polynomial form of the general Stewart platform. ASME J Mech Des2001; 123(2): 254–260.
19.
RollandL. Certified solving of the forward kinematics problem with an exact algebraic method for the general parallel manipulator. Adv Rob2005; 19(9): 995–1025.
20.
WangY. A direct numerical solution to forward kinematics of general Stewart-Gough platforms. Robotica2007; 25(1): 121–128.
21.
TarokhM. Real time forward kinematics solutions for general Stewart platforms. Proceedings of the 2007 IEEE international conference on robotics and automation. Roma: Italy, 10–14 April 2007, 901–906.
22.
RicoJMDuffyJ. An application of screw algebra to the acceleration analysis of serial chains. Mech Mach Theory1996; 31(5): 445–457.
23.
RicoJMDuffyJ. Forward and inverse acceleration analyses of in-parallel manipulators. ASME J Mech Des2000; 122(3): 299–303.
24.
GallardoJRicoJMFrisoliA. Dynamics of parallel manipulators by means of screw theory. Mech Mach Theory2003; 38(11): 1113–1131.
25.
Gallardo-AlvaradoJOrozco-MendozaHRodríguez-CastroR. Finding the jerk properties of multibody systems using helicoidal vector fields. Proc. IMechE, Part C: J. Mechanical Engineering Science2008; 222(11): 2217–2229.
26.
BonevIAZlatanovDGosselinC.M. Singularity analysis of 3-DOF planar parallel mechanisms via screw theory. ASME J Mech Des2003; 125(3): 573–581.
27.
St-OngeBMGosselinC. Singularity analysis and representation of the general Gough-Stewart platform. Int J Robot Res2000; 19(3): 271–288.
28.
LiHDGosselinCMRichardMJ. Analytic form of the six-dimensional singularity locus of the general Gough-Stewart platform. ASME J Mech Des2006; 128(1): 279–287.
29.
BandyopadhyaySGhosalA. Geometric characterization and parametric representation of the singularity manifold of a 6-6 Stewart platform manipulator. Mech Mach Theory2006; 41(11): 1377–1400.
30.
LiHGosselinC.MRichardM.J. Determination of the maximal singularity-free zones in the six-dimensional workspace of the general Gough-Stewart platform. Mech Mach Theory2007; 42(4): 497–511.
