# Screw theory-based mobility analysis and projection-based kinematic modeling of a 3-CRRR parallel manipulator

- 70 Downloads

## Abstract

Forward kinematics analysis of parallel manipulators requires solving highly complicated nonlinear equations, which deriving a closed-form solution is often a real challenge. Being used in closed loop position control of mechanisms, the forward kinematics solution of parallel manipulators is of great importance. Here, we investigate the mobility, forward kinematics, and inverse kinematics of a previously introduced three-degree-of-freedom spatial parallel manipulator from a new perspective. The manipulator is a 3-CRRR parallel mechanism proposed for object manipulation tasks. The mobility of the mechanism is, first, discussed using screw theory, showing that the robot has only three translational degrees of freedom. Next, the forward kinematics of the robot is analyzed based on a geometric approach. Using this method, which is the main novelty of our article, the spatial representation of the manipulator is transformed to a simpler planar representation by a projection-based interpretation, to reduce the complexity of kinematic equations. Afterward, the position of the end-effector is extracted by some algebraic expressions written based on geometrical properties of the robot. Then, the inverse kinematics of the mechanism is analyzed through the same approach. Finally, the kinematic modeling is verified using numerical and analytical methods. The results show that the obtained kinematic model has high accuracy.

## Keywords

Parallel manipulator Forward kinematics Inverse kinematics Mobility analysis Screw theory## List of symbols

- \(O_{{\rm inertia}}\)
origin of the reference Cartesian coordinate frame

*XYZ*- \(O_i\)
origin of the coordinate frame corresponding to the

*i*-th limb- \(O_p\)
origin of the coordinate frame attached to the end-effector

*Pr*unit vector denoting the screw of a prismatic joint

- Rev\(_i\)
unit vector denoting the screw of the

*i*-th revolute joint*a*distance between two adjacent revolute joints in a limb

*b*horizontal distance between the upper and the lower revolute joints of a limb

*h*vertical distance between the upper and the lower revolute joints of a limb

- \(S_i\)
*i*-th screw- \(S_{{\mathrm{lmss}}_i}\)
*i*-th screw associated with the limb motion screw system- \(S_{{\mathrm{lcss}}_i}\)
*i*-th screw associated with the limb constraint screw system- \(S_{{\mathrm{pcss}}_i}\)
*i*-th screw associated with the platform constraint screw system- \(S_{{\mathrm{pmss}}_i}\)
*i*-th screw associated with the platform motion screw system- \(\alpha _i, \beta _i, \gamma _i\)
direction cosines of the i-th screw associated with the platform constraint screw system in the reference coordinate frame

- \(A_i\)
maximum reach point of the

*i*-th limb on the ground plane- \(B_i\)
instantaneous contact point of the

*i*-th limb to the ground plane- \(C_i\)
upper end of the

*i*-th limb- \(D_i\)
projection of \(C_i\) on the ground plane

- \(L_i\)
length of the

*i*-th limb- \(x_p, y_p, z_p\)
components of the position of the end-effector with respect to the reference coordinate frame

- \(proj_i\)
projection of the

*i*-th limb on the ground plane- \(x_{A_i}, y_{A_i}, z_{A_i}\)
components of the position of \(A_i\) with respect to the reference coordinate frame

- \(x_{B_i}, y_{B_i}, z_{B_i}\)
components of the position of \(B_i\) with respect to the reference coordinate frame

- \(x_{C_i}, y_{C_i}, z_{C_i}\)
components of the position of \(C_i\) with respect to the reference coordinate frame

- \(x_{D_i}, y_{D_i}, z_{D_i}\)
components of the position of \(D_i\) with respect to the reference coordinate frame

*dl*distance between \(D_1\) and \(D_2\) or \(D_3\) along the

*x*-axis of the reference coordinate frame*dw*distance between \(D_1\) and \(D_2\) or \(D_3\) along the

*y*-axis of the reference coordinate frame- \(\theta _i\)
angle between the motion direction of the

*i*-th limb on the ground plane and the positive direction of the*x*-axis of the reference coordinate frame*e*vertical distance between \(C_i\) and the moving platform

- proj\(_p\)
projection of the end-effector position on the ground plane

- \(R_i\)
joint variable corresponding to the

*i*-th limb

## References

- 1.Chiu Y-J, Perng M-H (2001) Forward kinematics of a general fully parallel manipulator with auxiliary sensors. Int J Robot Res 20(5):401–414CrossRefGoogle Scholar
- 2.Cui G, Sun M, Meng W, Zhang H, Sun C (2015) The research of kinematic performances of 3-UPU-UPU parallel mechanism for automobile assembly line. In: 2015 IEEE international conference on mechatronics and automation (ICMA). IEEE, pp 2514–2520Google Scholar
- 3.Dhingra A, Almadi A, Kohli D (2000) A grobner-sylvester hybrid method for closed-form displacement analysis of mechanisms. J Mech Des 122(4):431–438CrossRefGoogle Scholar
- 4.Dogangil G, Davies B, Rodriguez y Baena F (2010) A review of medical robotics for minimally invasive soft tissue surgery. Proc Inst Mech Eng Part H J Eng Med 224(5):653–679CrossRefGoogle Scholar
- 5.Dongsu W, Hongbin G (2007) Adaptive sliding control of six-DOF flight simulator motion platform. Chin J Aeronaut 20(5):425–433CrossRefGoogle Scholar
- 6.Gallardo-Alvarado J (2016) Kinematic analysis of parallel manipulators by algebraic screw theory. Springer, BerlinCrossRefGoogle Scholar
- 7.Gallardo-Alvarado J, Rico-Martínez JM, Alici G (2006) Kinematics and singularity analyses of a 4-DOF parallel manipulator using screw theory. Mech Mach Theory 41(9):1048–1061MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Gallardo-Alvarado J, Rodríguez-Castro R, Islam MN (2008) Analytical solution of the forward position analysis of parallel manipulators that generate 3-RS structures. Adv Robot 22(2–3):215–234CrossRefGoogle Scholar
- 9.Gan D, Liao Q, Dai JS, Wei S, Seneviratne L (2009) Forward displacement analysis of the general 6–6 Stewart mechanism using Gröbner bases. Mech Mach Theory 44(9):1640–1647CrossRefzbMATHGoogle Scholar
- 10.Huang X, Liao Q, Wei S (2010) Closed-form forward kinematics for a symmetrical 6–6 Stewart platform using algebraic elimination. Mech Mach Theory 45(2):327–334CrossRefzbMATHGoogle Scholar
- 11.Innocenti C, Parenti-Castelli V (1990) Direct position analysis of the Stewart platform mechanism. Mech Mach Theory 25(6):611–621CrossRefGoogle Scholar
- 12.Liu J, Li Y, Zhang Y, Gao Q, Zuo B (2014) Dynamics and control of a parallel mechanism for active vibration isolation in space station. Nonlinear Dyn 76(3):1737–1751MathSciNetCrossRefGoogle Scholar
- 13.Mahmoodi A, Sayadi A, Menhaj MB (2014) Solution of forward kinematics in Stewart platform using six rotary sensors on joints of three legs. Adv Robot 28(1):27–37CrossRefGoogle Scholar
- 14.Merlet J-P (1992) Direct kinematics and assembly modes of parallel manipulators. Int J Robot Res 11(2):150–162CrossRefGoogle Scholar
- 15.Merlet J-P (2004) Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int J Robot Res 23(3):221–235CrossRefGoogle Scholar
- 16.Merlet J-P (2006) Parallel robots, vol 128. Springer, BerlinzbMATHGoogle Scholar
- 17.Moosavian SAA, Nazari AA, Hasani A (2011) Kinematics and workspace analysis of a novel 3-DOF spatial parallel robot. In: 2011 19th Iranian conference on electrical engineering (ICEE). IEEE, pp 1–6Google Scholar
- 18.Morell A, Tarokh M, Acosta L (2013) Solving the forward kinematics problem in parallel robots using support vector regression. Eng Appl Artif Intell 26(7):1698–1706CrossRefGoogle Scholar
- 19.Nanua P, Waldron KJ, Murthy V (1990) Direct kinematic solution of a Stewart platform. IEEE Trans Robot Autom 6(4):438–444CrossRefGoogle Scholar
- 20.Omran A, Bayoumi M, Kassem A, El-Bayoumi G (2009) Optimal forward kinematics modeling of Stewart manipulator using genetic algorithms. Jordan J Mech Ind Eng 3(4):280–293Google Scholar
- 21.Patel Y, George P et al (2012) Parallel manipulators applications—a survey. Modern Mech Eng 2(3):57–64CrossRefGoogle Scholar
- 22.Pierrot F, Reynaud C, Fournier A (1990) Delta: a simple and efficient parallel robot. Robotica 8(2):105–109CrossRefGoogle Scholar
- 23.Sadjadian H, Taghirad HD (2005) Comparison of different methods for computing the forward kinematics of a redundant parallel manipulator. J Intell Robot Syst 44(3):225–246CrossRefGoogle Scholar
- 24.Shafiee-Ashtiani M, Yousefi-Koma A, Iravanimanesh S, Bashardoust AS (2016) Kinematic analysis of a 3-UPU parallel robot using the Ostrowski-homotopy continuation. In: 2016 24th Iranian conference on electrical engineering (ICEE). IEEE, pp 1306–1311Google Scholar
- 25.Sheng L, Wan-long L, Yan-chun D, Liang F (2006) Forward kinematics of the Stewart platform using hybrid immune genetic algorithm. In: Proceedings of the 2006 IEEE international conference on mechatronics and automation. IEEE, pp 2330–2335Google Scholar
- 26.Stewart D (1965) A platform with six degrees of freedom. Proc Inst Mech Eng 180(1):371–386CrossRefGoogle Scholar
- 27.Tsai L-W (1999) Robot analysis: the mechanics of serial and parallel manipulators. Wiley, HobokenGoogle Scholar
- 28.Varedi S, Daniali H, Ganji D (2009) Kinematics of an offset 3-UPU translational parallel manipulator by the homotopy continuation method. Nonlinear Anal Real World Appl 10(3):1767–1774MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Xie F, Liu X-J, You Z, Wang J (2014) Type synthesis of 2T1R-type parallel kinematic mechanisms and the application in manufacturing. Robot Comput Integr Manuf 30(1):1–10CrossRefGoogle Scholar
- 30.Yang C-F, Zheng S-T, Jin J, Zhu S-B, Han J-W (2010) Forward kinematics analysis of parallel manipulator using modified global Newton–Raphson method. J Central South Univ Technol 17(6):1264–1270CrossRefGoogle Scholar
- 31.Yang X, Wu H, Li Y, Chen B (2017) A dual quaternion solution to the forward kinematics of a class of six-DOF parallel robots with full or reductant actuation. Mech Mach Theory 107:27–36CrossRefGoogle Scholar
- 32.Yu J, Dai JS, Zhao T, Bi S, Zong G (2009) Mobility analysis of complex joints by means of screw theory. Robotica 27(6):915–927CrossRefGoogle Scholar
- 33.Yurt S, Anli E, Ozkol I (2007) Forward kinematics analysis of the 6–3 SPM by using neural networks. Meccanica 42(2):187–196CrossRefzbMATHGoogle Scholar
- 34.Zhang D, Lei J (2011) Kinematic analysis of a novel 3-DOF actuation redundant parallel manipulator using artificial intelligence approach. Robot Comput Integr Manuf 27(1):157–163CrossRefGoogle Scholar
- 35.Zhao J, Li B, Yang X, Yu H (2009) Geometrical method to determine the reciprocal screws and applications to parallel manipulators. Robotica 27(6):929–940CrossRefGoogle Scholar
- 36.Zhou W, Chen W, Liu H, Li X (2015) A new forward kinematic algorithm for a general Stewart platform. Mech Mach Theory 87:177–190CrossRefGoogle Scholar