Lagrangian Formulation of Manipulator Dynamics

Euler-Lagrangian Formulation:

It is common to see the equations of motion of manipulators or humanoids reminded in the preamble of research papers. They derived from Newtonian or Lagrangian mechanics. The equations of motion describe the motion of a physical system as a function of time and controls. In their most general form, they are written:

F(q(t), q̇(t), q̇(t), u(t), t) = 0


  • t is the time variable
  • q is the vector of generalized coordinates, for instance the vector of joint-angles for a manipulator.
  • q̇ is the first time derivative (velocity) of q, q.
  • q̇ is the second time derivative (acceleration) of q, q.
  • u is the vector of control inputs.

These equations provide a mapping between the control space and the state space of the robot.


Manipulators are a particular type of articulated system where at least one link is fixed to the environment. We suppose that the fixation is strong enough to withhold any effort exerted on it. Under these assumptions, the equations of motion for a manipulator can written:

M(q)q̇ + q̇yC(q)q̇+g(q) = 𝜏

where denoting by n = dim(q) the degree of freedom of the robot,

  • M(q) is the nxn inertia matrix
  • C(q) is the nxnxn Coriolis tensor
  • g(q) is the n-dimensional gravity vector
  • 𝜏 is the n-dimensional vector of actuated torques.