Euler-Lagrange Equations

A way of deriving a robot's equations of motion from kinetic and potential energy, without analyzing constraint forces one by one.

Lagrangian mechanics was developed by Joseph-Louis Lagrange, laid out fully in his 1788 book Mécanique analytique. The method starts by writing the Lagrangian L = T − V (T is kinetic energy, V is potential energy), then, for each generalized coordinate q_i (such as a joint angle), writes an equation d/dt(∂L/∂q̇_i) − ∂L/∂q_i = τ_i, where τ_i is the generalized force acting on that coordinate (such as a motor torque). The advantage is that it only requires computing energy — the constraint forces between joints cancel out automatically — and the number of equations equals the number of degrees of freedom. For a robot arm, this works out to the standard form M(q)q̈ + C(q,q̇)q̇ + g(q) = τ: M is the mass matrix, the C term captures Coriolis and centrifugal forces, and g is the gravity term. This set of equations underlies torque control, gravity compensation, model predictive control, and physics simulation; deriving it by hand suits a two- or three-joint textbook example, while real robots compute it numerically with algorithms like the recursive Newton-Euler algorithm.

ExampleA simple pendulum: length l, mass m, angle θ, kinetic energy T = ½ml²θ̇², potential energy V = −mgl·cosθ. Substituting into the equation gives ml²θ̈ + mgl·sinθ = τ — the familiar pendulum equation.

Also called

Lagrangian Dynamics, Lagrangian Mechanics

Related

Newton-Euler Equations · Generalized Coordinates · Mass Matrix · Coriolis and Centrifugal Terms · Forward Dynamics · Inverse Dynamics

Sources

Lagrangian mechanics - Wikipedia
Underactuated Robotics (MIT) - Multi-Body Dynamics