Abstract
Recursive matrix relations concerning the kinematics and the dynamics of a constrained robotic system, schematized by several kinematical chains, are established in this paper. Introducing frames and bases, we first analyze the geometrical properties of the mechanism and derive a general set of relations. Kinematics of the vector system of velocities and accelerations for each element of robot are then obtained. Expressed for every independent loop of the robot, useful conditions of connectivity regarding the relative velocities and accelerations are determined for direct or inverse kinematics problem. Based on the general principle of virtual powers, final matrix relations written in a recursive compact form express just the explicit dynamics equations of a constrained robotic system. Establishing active forces or actuator torques in an inverse dynamic problem, these equations are useful in fact for real-time control of a robot.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
Abbreviations
- a k,k−1 :
-
orthogonal relative transformation matrix
- a :
-
general transformation matrix of moving platform
- \(\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}\) :
-
three orthogonal unit vectors
- φ k,k−1 :
-
relative rotation angle of T k rigid body
- \(\vec{\omega}_{k,k-1}\) :
-
relative angular velocity of T k
- \(\vec{\omega}_{k0}\) :
-
absolute angular velocity of T k
- \(\tilde{\omega}_{k,k-1}\) :
-
skew symmetric matrix associated with the angular velocity \(\vec{\omega}_{k,k-1}\)
- \(\vec{\varepsilon}_{k,k-1}\) :
-
relative angular acceleration of T k
- \(\tilde{\varepsilon}_{k0}\) :
-
absolute angular acceleration of T k
- \(\tilde{\varepsilon}_{k,k-1}\) :
-
skew symmetric matrix associated with the angular acceleration \(\vec{\varepsilon}_{k,k-1}\)
- \(\vec{r}_{k,k-1}^{A}\) :
-
relative position vector of the center A k of joint
- \(\vec{v}_{k,k-1}^{A}\) :
-
relative velocity of the center A k
- \(\vec{\gamma}_{k,k-1}^{A}\) :
-
relative acceleration of the center A k
- m k :
-
mass of T k rigid body
- \(\hat{J}_{k}\) :
-
symmetric matrix of tensor of inertia of T k about the link-frame A k x k y k z k
- f q,q−1,m q,q−1 :
-
force or torque of the actuator T q−1
References
Tsai, L.-W.: Robot Analysis: The Mechanics of Serial and Parallel Manipulator. Wiley, New York (1999)
Chablat, D., Wenger, P.: Architecture optimisation of a 3-DOF parallel mechanism for machining applications: the orthoglide. IEEE Trans. Robot. Autom. 19(3), 403–410 (2003)
Liu, X.-J., Wang, J., Gao, F., Wang, L.-P.: On the analysis of a new spatial three degrees of freedom parallel manipulator. IEEE Trans. Robot. Autom. 17(6), 959–968 (2001)
Reboulet, C., Pigeyre, R.: Hybrid control of a 6-DOF in-parallel actuated micro-manipulator mounted on a Scara robot. In: Proceedings of the International Symposium on Robotics and Manufacturing: Research, Education and Applications, Burnaby, Canada, pp. 293–298 (1990)
Valenti, M.: Machine tools get smarter. ASME Mech. Eng. 17, 70–75 (1995)
Cleary, K., Brooks, T.: Kinematics analysis of a novel 6-DOF parallel manipulator. In: Proceedings of the IEEE International Conference on Robotics and Automation, Texas, pp. 708–713 (1993)
Angeles, J.: Fundamentals of Robotic Mechanical Systems: Theory, Methods and Algorithms. Springer, New York (1997)
Carricato, M., Parenti-Castelli, V.: Singularity-free fully-isotropic translational parallel mechanisms. Int. J. Robot. Res. 21(2), 161–164 (2002)
Parenti-Castelli, V., Di Gregorio, R.: A new algorithm based on two extra-sensors for real-time computation of the actual configuration of generalized Stewart–Gough manipulator. J. Mech. Des. 122, 294–298 (2000)
Stewart, D.: A platform with six degrees of freedom. Proc. Inst. Mech. Eng. 1 180(15), 371–386 (1965)
Clavel, R.: Delta: a fast robot with parallel geometry. In: Proceedings of 18th International Symposium on Industrial Robots, Lausanne (1988)
Staicu, S., Carp-Ciocardia, D.C.: Dynamic analysis of Clavel’s delta parallel robot, In: Proceedings of the IEEE International Conference on Robotics & Automation ICRA’2003, Taipei, Taiwan, pp. 4116–4121 (2003)
Tsai, L.-W., Stamper, R.: A parallel manipulator with only translational degrees of freedom. In: ASME Design Engineering Technical Conferences, Irvine, CA (1996)
Hervé, J.-M., Sparacino, F.: Star. A new concept in robotics. In: Proceedings of the Third International Workshop on Advances in Robot Kinematics, Ferrara (1992)
Gosselin, C., Angeles, J.: The optimum kinematics design of spherical three-degrees-of-freedom parallel manipulator. ASME J. Mech. Trans. Autom. Des. 111(2), 202–207 (1989)
Li, Y.-W., Wang, J., Wang, L.-P., Liu, X.-J.: Inverse dynamics and simulation of a 3-DOF spatial parallel manipulator. In: Proceedings of the IEEE International Conference on Robotics & Automation, Taipei, Taiwan, pp. 4092–4097 (2003)
Dasgupta, B., Mruthyunjaya, T.S.: A Newton–Euler formulation for the inverse dynamics of the Stewart platform manipulator. Mech. Mach. Theory 34, 1135–1152 (1998)
Kane, T.R., Levinson, D.A.: Dynamics, Theory and Applications. McGraw-Hill, New York (1985)
Sorli, M., Ferarresi, C., Kolarski, M., Borovac, B., Vucobratovic, M.: Mechanics of Turin parallel robot. Mech. Mach. Theory 32(1), 51–77 (1997)
Geng, Z., Haynes, L.S., Lee, J.D., Carroll, R.L.: On the dynamic model and kinematics analysis of a class of Stewart platforms. Robot. Autonom. Syst. 9, 237–254 (1992)
McCarthy, J.M.: Dual orthogonal matrix in manipulator kinematics. Int. J. Robot. Res. 5(2), 45–51 (1986)
Staicu, S.: Mecanica teoretica. Edit. Didactica & Pedagogica, Bucharest (1998)
Staicu, S.: Inverse dynamics of a planetary gear train for robotics. Mech. Mach. Theory 43, 918–927 (2008)
Staicu, S., Zhang, D., Rugescu, R.: Dynamic modelling of a 3-DOF parallel manipulator using recursive matrix relations. Robotica 24(1), 125–130 (2006)
Guégan, S., Khalil, W., Chablat, D., Wenger, P.: Modélisation dynamique d’un robot parallèle à 3-DDL: l’Orthoglide. In: Conférence Internationale Francophone d’Automatique, Nantes, France, 8–10 Juillet (2002)
Merlet, J.-P.: Parallel Robots. Kluwer Academic, Dordrecht (2000)
Miller, K., Clavel, R.: The Lagrange-based model of delta-4 robot dynamics. Robotersysteme 8, 49–54 (1992)
Staicu, S., Zhang, D.: A novel dynamic modelling approach for parallel mechanisms analysis. Robot. Comput.-Integr. Manuf. 24, 167–172 (2008)
Staicu, S., Liu, X.-J., Wang, J.: Inverse dynamics of the HALF parallel manipulator with revolute actuators. Nonlinear Dyn. 50, 1–12 (2007)
Tsai, L.-W.: Solving the inverse dynamics of Stewart–Gough manipulator by the principle of virtual work. ASME J. Mech. Des. 122, 3–9 (2000)
Zhang, C.-D., Song, S.-M.: An efficient method for inverse dynamics of manipulators based on virtual work principle. J. Robot. Syst. 10(5), 605–627 (1993)
Liu, X.-J., Jeong, J., Kim, J.: A three translational DoFs parallel cube-manipulator. Robotica 21(6), 645–653 (2003)
Staicu, S.: Relations matricielles de récurrence en dynamique des mécanismes. Rev. Roum. Sci. Tech. Sér. Méc. Appl. 50(1–3), 15–28 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Staicu, S., Liu, XJ. & Li, J. Explicit dynamics equations of the constrained robotic systems. Nonlinear Dyn 58, 217–235 (2009). https://doi.org/10.1007/s11071-009-9473-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9473-4