Journal of Intelligent & Robotic Systems

, Volume 77, Issue 2, pp 251–264 | Cite as

Complete Analytical Forward and Inverse Kinematics for the NAO Humanoid Robot

  • Nikolaos Kofinas
  • Emmanouil Orfanoudakis
  • Michail G. Lagoudakis


The design of complex dynamic motions for humanoid robots is achievable only through the use of robot kinematics. In this paper, we study the problems of forward and inverse kinematics for the Aldebaran NAO humanoid robot and present a complete, exact, analytical solution to both problems, including a software library implementation for real-time on-board execution. The forward kinematics allow NAO developers to map any configuration of the robot from its own joint space to the three-dimensional physical space, whereas the inverse kinematics provide closed-form solutions to finding joint configurations that drive the end effectors of the robot to desired target positions in the three-dimensional physical space. The proposed solution was made feasible through a decomposition into five independent problems (head, two arms, two legs), the use of the Denavit-Hartenberg method, the analytical solution of a non-linear system of equations, and the exploitation of body and joint symmetries. The main advantage of the proposed inverse kinematics solution compared to existing approaches is its accuracy, its efficiency, and the elimination of singularities. In addition, we suggest a generic guideline for solving the inverse kinematics problem for other humanoid robots. The implemented, freely-available, NAO kinematics library, which additionally offers center-of-mass calculations and Jacobian inverse kinematics, is demonstrated in three motion design tasks: basic center-of-mass balancing, pointing to a moving ball, and human-guided balancing on two legs.


Robot kinematics Humanoid robots Aldebaran NAO robot 

Mathematics Subject Classification (2010)

68T40 53A17 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Nikolaos Kofinas
    • 1
  • Emmanouil Orfanoudakis
    • 1
  • Michail G. Lagoudakis
    • 1
  1. 1.Intelligent Systems Laboratory, School of Electronic and Computer EngineeringTechnical University of CreteChaniaGreece

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