Robotic Systems · Lecture 2

Euler, semi-implicit Euler, and RK4

Simeon Nedelchev

Explicit Euler

x_{k+1}=x_k+\Delta t f(x_k)

First-order global accuracy on smooth fixed-time problems.

Oscillator energy grows.

v_{k+1}=v_k+\Delta t a(q_k)

q_{k+1}=q_k+\Delta t v_{k+1}

The update uses the new velocity.

\omega\Delta t<2

This bound matters for symplectic Euler.

Formal order alone does not determine stability.

Four slope evaluations per step.

Fourth-order global accuracy for smooth problems.

Energy conservation remains approximate.

Halve the timestep and keep final time fixed.

Expected error ratios: about 2 for Euler, 16 for RK4.

Exact solution: $(q,v)=(\cos t,-\sin t)$ .

Compare trajectory error and energy drift.