Optimal Control

Optimal Control of Mechatronic Systems

The optimization of the dynamical behavior of a mechanical system is crucial for technical applications. Therefore, optimal control te is usually applied to determine trajectories of the system, which satisfy some optimality criteria. By designing the input of the system, the optimal behavior is achieved [ST14], [TKOBT11].

The hardware implementation of the optimal control method is demonstrated using a double pendulum on a cart system [TKOBT11]. We consider the swing-up and the stabilization of the double pendulum from its stable lower equilibrium to its unstable upper equilibrium. Using the method Discrete Mechanics and Optimal Control, a set of pareto optimal trajectories can be determined, which represents a compromise between the time and energy consumption. The hardware realization of the optimal control is supported by an additional trajectory tracking controller.

Pareto front of the optimal trajectories for the swing-up of the double pendulum. Each cross indicates a solution, which is a compromise between the time consumption and the energy consumption.


A dynamical system may possess natural dynamical structures, in which the uncontrolled motion of the system reaches certain desired equilibrium points “for free”. The search and utilization of such structures are crucial for designing energy efficient optimal control maneuvers.

Specifically, we search and study the stable and unstable manifolds of the upper equilibrium of a double pendulum system, in which the uncontrolled motion of the double pendulum reaches the upper equilibrium without additional energy support.

In our approach, we design energy efficient open loop optimal control trajectories by concatenating uncontrolled trajectories on strong stable manifolds with short controlled trajectory towards these manifolds. The results are evaluated using numerical simulation and applied to a real test rig. The improvement of the energy consumption is significant [FTOBT14].

Illustration of the 4-dimensional manifold.
Stable (left) and unstable (right) manifold containing the upper equilibrium of the double pendulum. The angles (φ1 and φ2) and the corresponding angular velocities (dφ1/dt und d
φ2/dt) of the double pendulum are shown.



Related publications:

  • [ST14] Zeeshan Shareef, Ansgar Trächtler; Simultaneous Path Planning and Optimization for Robotic Manipulators using Discrete Mechanics and Optimal Control;2014; Robotica
  • [FTOBT14] Kathrin Flaßkamp, Julia Timmermann, Sina Ober-Blöbaum, Ansgar Trächtler; Control Strategies on Stable Manifolds for Energy-Efficient Swing-Ups of Double Pendula; 2014; International Journal of Control
  • [TKOBT11] Julia Timmermann, Shaady Khatab, Sina Ober-Blöbaum, Ansgar Trächtler; Discrete Mechanics and Optimal Control and its Applica-tion to a Double Pendulum on a Cart; IFAC World Congress 2011