A New Optimal Method for Calculating the Null Space of a Robot using NOC Algorithm; Application on Parallel 3PRS Robot

Document Type : Original Article


Department of Mechanical Engineering, Faculty of Engineering, University of Kharazmi, Tehran, Iran


In this paper, a new optimal method for modelling of a 3PRS robot is proposed according to NOC algorithm. An optimal method of selecting the generalized coordinate is presented and a new algorithm of extracting the null space of over and under constrained robots is proposed through which a lower amount of mathematical calculations is required. In this method, using the principal of derivatives of implicit functions, the null space of constraint matrix will be extracted. Afterwards the null space matrix is calculated with orthogonal columns. The proposed method is implemented on a 3PRS robot which is an under constrained robot. This robot is a kind of parallel spatial robot with 6 DOFs which can be controlled using 3 active prismatic joints and 3 passive rotary ones. This robot similar to other parallel robots has heavy, complicated and nonlinear model which needs heavy and time consuming mathematical calculations. The proposed strategy of extracting the null space of the robot, extremely and heavily decreases the volume of required mathematical calculations for modelling the robot and consequently decreases the inevitable consumed time of processing and numerical errors and increases the accuracy of simulations.


[1]     Stewart, D., A Platform with Six Degrees of Freedom, Proceedings of the Institution of Mechanical Engineers, Vol. 180, No. 1, 1965, pp. 371-386.
[2]     Ruiz, F., Campa, C., Roldán-Paraponiaris, and Altuzarra, O., Dynamic Model of a Compliant 3PRS Parallel Mechanism for Micromilling, in Microactuators and Micromechanisms: Springer, 2017, pp. 153-164.
[3]     Li, Y., Xu, Q., Kinematics and Inverse Dynamics Analysis for a General 3-PRS Spatial Parallel Mechanism, Robotica, Vol. 23, No. 02, 2005, pp. 219-229.
[4]     Li, Y., Staicu, S., Inverse Dynamics of a 3-PRC Parallel Kinematic Machine, Nonlinear Dynamics, Vol. 67, No. 2, 2012, pp. 1031-1041.
[5]     Altuzarra, O., Gomez, F. C., Roldan-Paraponiaris, C., and Pinto, C., Dynamic Simulation of a Tripod Based in Boltzmann-Hamel Equations, in Proceedings of ASME International Design Engineering Technical Conferences, Vol. 5, 2015, pp. 8-22.
[6]     Nikravesh, P., Haug, E., Generalized Coordinate Partitioning for Analysis of Mechanical Systems with Nonholonomic Constraints, ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 105, 1983, pp. 379-384.
[7]     Singh, R., Likins, P., Singular Value Decomposition for Constrained Dynamical Systems, Journal of Applied Mechanics, Vol. 52, No. 4, 1985, pp. 943-948.
[8]     Kim, S., Vanderploeg, M., A General and Efficient Method for Dynamic Analysis of Mechanical Systems using Velocity Transformations, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 108, No. 2, 1986, pp. 176-182.
[9]     Liang, Lance, G. M., A Differentiable Null Space Method for Constrained Dynamic Analysis, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, No. 3, 1987, pp. 405-411.
[10]  Angeles, J., Lee, S. K., The Formulation of Dynamical Equations of Holonomic Mechanical Systems using a Natural Orthogonal Complement, Journal of Applied Mechanics, Vol. 55, 1988, pp. 243-249.
[11]  Angeles, J., Ma, O., Dynamic Simulation of N-Axis Serial Robotic Manipulators using a Natural Orthogonal Complement, The International Journal of Robotics Research, Vol. 7, No. 5, 1988, pp. 32-47.
[12]  Saha, S. K., Angeles, J., Dynamics of No Holonomic Mechanical Systems using a Natural Orthogonal Complement, Journal of Applied Mechanics, Vol. 58, No. 1, 1991, pp. 238-243.
[13]  Terze, Z., Lefeber, D., and Muftić, O., Null Space Integration Method for Constrained Multibody Systems with no Constraint Violation, Multibody System Dynamics, Vol. 6, No. 3, 2001, pp. 229-243.
[14]  Pendar, H., Vakil, M., and Zohoor, H., Efficient Dynamic Equations of 3-RPS Parallel Mechanism through Lagrange Method, in Proceedings of the Robotics, Automation and Mechatronics, IEEE Conference, Vol. 2, 2004, pp. 1152-1157.
[15]  Rao, K., Saha, S., and Rao, P., Dynamics Modelling of Hexaslides using the Decoupled Natural Orthogonal Complement Matrices, Multibody System Dynamics, Vol. 15, No. 2, 2006, pp. 159-180.
[16]  Phong, V., Hoang, N. Q., Singularity-Free Simulation of Closed Loop Multibody Systems by using Null Space of Jacobian Matrix, Multibody System Dynamics, Vol. 27, No. 4, 2012, pp. 487-503.
[17]  Marino, Parker, L. E., Antonelli, G., and Caccavale, F., A Decentralized Architecture for Multi-Robot Systems Based on the Null-Space-Behavioral Control with Application to Multi-Robot Border Patrolling, Journal of Intelligent & Robotic Systems, Vol. 71, No. 3-4, 2013, pp. 423-444.
[18]  Raoofian, Kamali, A., and Taghvaeipour, A., Forward Dynamic Analysis of Parallel Robots using Modified Decoupled Natural Orthogonal Complement Method, Mechanism and Machine Theory, Vol. 115, 2017, pp. 197-217.
[19]  Coleman, T. F., Sorensen, D. C., A Note on the Computation of an Orthonormal Basis for the Null Space of a Matrix, Mathematical Programming, Vol. 29, No. 2, 1984, pp. 234-242.
[20]  Berry, M., Heath, M., Kaneko, I., Lawo, M., Plemmons, R., and Ward, R., An Algorithm to Compute a Sparse Basis of the Null Space, Numerische Mathematik, Vol. 47, No. 4, 1985, pp. 483-504.
[21]  Coleman, T. F., Pothen, A., The Null Space Problem I. Complexity, SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 4, 1986, pp. 527-537.
[22]  Dai, J. S., Jones, J. R., Null–Space Construction using Cofactors from a Screw–Algebra Context, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 458, No. 2024, 2002, pp. 1845-1866.
[23]  Bourdoux, Khaled, N., Joint TX-RX Optimisation for MIMO-SDMA Based on a Null-Space Constraint, Proceedings of the Vehicular Technology Conference, Vol. 1, 2002, pp. 171-174.
[24]  Nie, P. Y., A Null Space Method for Solving System of Equations, Applied Mathematics and Computation, Vol. 149, No. 1, 2004, pp. 215-226.
[25]  Ye, J., Xiong, T., Computational and Theoretical Analysis of Null Space and Orthogonal Linear Discriminant Analysis, Journal of Machine Learning Research, Vol. 7, No. Jul, 2006, pp. 1183-1204.
[26]  Betsch, P., The Discrete Null Space Method for the Energy Consistent Integration of Constrained Mechanical Systems: Part I: Holonomic Constraints, Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 50, 2005, pp. 5159-5190.
[27]  Betsch, P., Leyendecker, S., The Discrete Null Space Method for the Energy Consistent Integration of Constrained Mechanical Systems, Part II: Multibody Dynamics, International Journal for Numerical Methods in Engineering, Vol. 67, No. 4, 2006, pp. 499-552.
[28]  Leyendecker, S., Betsch, P., and Steinmann, P., The Discrete Null Space Method for the Energy-Consistent Integration of Constrained Mechanical Systems. Part III: Flexible Multibody Dynamics, Multibody System Dynamics, Vol. 19, No. 1, 2008, pp. 45-72.
[29]  Leyendecker, S., Ober‐Blöbaum, S., Marsden, J. E., and Ortiz, M., Discrete Mechanics and Optimal Control for Constrained Systems, Optimal Control Applications and Methods, Vol. 31, No. 6, 2010, pp. 505-528.
[30]  Spivak, M., Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus. CRC Press, 2018.
[31]  Spong, M. W., Hutchinson, S., and Vidyasagar, M., Robot Modeling and Control, Jon Wiley & Sons, 2005, ISBN-100-471-649.