Fast Side-Stepping of the Triple Inverted Pendulum via Constrained Nonlinear Feedforward Control Design

Abstract

Abstract-The side-stepping of the triple inverted pendulum serves as a benchmark problem for the presentation of a new feedforward control design technique for finite-time transition problems under consideration of output constraints. The inversion-based design treats the transition task as a twopoint boundary value problem (BVP) defined in the inputoutput coordinates of the pendulum. As a necessary condition for its solvability, a sufficient number of free parameters is provided in a set-up function for the cart acceleration. The new approach allows to directly incorporate output constraints of the pendulum within the BVP, which is solved by a standard MATLAB function. A linear LQR controller is used to stabilize the pendulum along the feedforward side-stepping trajectories. Experimental results for the triple inverted pendulum illustrate the accuracy and robustness of the proposed feedforward/feedback control scheme.

Feedforward Control Design for Finite-Time Transition Problems of Nonlinear Systems With Input and Output Constraints

“…Alternatively, spline functions can be used to parameterize v = Φ(t, p), which allows to condense the "activity" of Φ(t, p) in a desired region, e.g. at the start and stop of the transition (Treuer, 2005;Graichen et al, 2005c). [m]…”

“…With respect to the two-degree-of-freedom control scheme in Figure 1.1, the side-stepping problem can be seen as an extension of the stabilization problem, since the feedforward control Σ F F provides the nominal input trajectory for the side-stepping, while the feedback control Σ F B stabilizes the triple pendulum in the upward position. In this section, the feedforward control design under output constraints is illustrated for the side-stepping of the triple inverted pendulum with constraints on the cart position, velocity, and acceleration (Graichen et al, 2005c;Graichen and Zeitz, 2006b). The feedback control Σ F B is based on an LQ design with the pendulum model linearized around the upward unstable equilibrium.…”

“…The pendulum can also be employed with three links, see Section 4.4 and e.g. (Graichen et al, 2005c) for the side-stepping of the triple inverted pendulum.…”

“…to T = 3 s, which leads to a less aggressive side-stepping maneuver (Treuer, 2005;Graichen et al, 2005c). Moreover, an alternative to the implemented LQ feedback control with the constant gain vector k ∈ R 9 is the reverse-time solution of the Riccati ODE (2.56) in Section 2.4.3.3, in order to compute time-varying gains k(t) ∈ R 9 for the feedback correction ∆u = k…”

Flatness based control of a ball in tube system

“…The main advantage of flatness based control is the two-degree-of-freedom structure of the control loop in terms of a feedforward and a feedback control path. Flatness based control was implemented in many industrial applications in the meantime like chemical reactors in Rothfuss et al (1996) and Graichen et al (2006), hydraulic actuators in Bröcker and Lemmen (2001) and Bröcker (2002) and mechatronic systems in Graichen et al (2005), Kiefer et al (2006), Noda et al (2011) and Henke et al (2014).…”