Stabilization of a Cart Inverted Pendulum: Improving the Intermittent Feedback Strategy to Match the Limits of Human Performance

Abstract

Stabilization of the CIP (Cart Inverted Pendulum) is an analogy to stick balancing on a finger and is an example of unstable tasks that humans face in everyday life. The difficulty of the task grows exponentially with the decrease of the length of the stick and a stick length of 32 cm is considered as a human limit even for well-trained subjects. Moreover, there is a cybernetic limit related to the delay of the multimodal sensory feedback (about 230 ms) that supports a feedback stabilization strategy. We previously demonstrated that an intermittent-feedback control paradigm, originally developed for modeling the stabilization of upright standing, can be applied with success also to the CIP system, but with values of the critical parameters far from the limiting ones (stick length 50 cm and feedback delay 100 ms). The intermittent control paradigm is based on the alternation of on-phases, driven by a proportional/derivative delayed feedback controller, and off-phases, where the feedback is switched off and the motion evolves according to the intrinsic dynamics of the CIP. In its standard formulation, the switching mechanism consists of a simple threshold operator: the feedback control is switched off if the current (delayed) state vector is closer to the stable than to the unstable manifold of the off-phase and is switched on in the opposite case. Although this simple formulation is effective for explaining upright standing as well as CIP balancing, it fails in the most challenging configuration of the CIP. In this work we propose a modification of the standard intermittent control policy that focuses on the explicit selection of switching times and is based on the phase reset of the estimated state vector at each switching time and on the simulation of an approximated internal model of CIP dynamics. We demonstrate, by simulating the modified intermittent control policy, that it can match the limits of human performance, while operating near the edge of instability.

Keywords: Cart Inverted Pendulum; intermittent feedback control; internal model simulation; phase reset; saddle-like instability.

Figure 1

Schematic representation of the CIP model. θ and x are the two DoFs; f (the control variable) is the force applied by the user to the cart. The stick is a thin uniform rod. The hinge is frictionless as well as the virtual rail that allows the motion of the cart. The x-axis is aligned with the antero-posterior direction of the subject's body.

Figure 2

Off-phase trajectories of the CIP model with the same initial angular tilt (−2 deg) but different distance from the stable manifold: the green line (the red line is the corresponding unstable manifold). The blue part of each trajectory runs from the initial position (at t = t_off) until the intersection with one of the two coordinate axes (at t = t_c: the duration of such segment is determined by Equation 9); the second part is red-colored and has a fixed duration, equal to the sensory feedback delay δ (t_on = t_c+δ). (A) Refers to a CIP model with the following parameters: L = 50 cm; δ = 100 ms). (B) Refers to a CIP model with much more challenging parameters: L = 32 cm; δ = 230 ms. In both cases the cart mass is 250 g and the stick mass is 125 g.

Figure 3

Characteristic timing of the hyperbolic trajectories in the off-phases. Δt_cross is the time taken by an hyperbolic trajectory to cross the border, between the off-area and the on-area, as a function of the distance of the starting point $\left[{\theta }_0,{\dot{\theta }}_0\right]$ from the stable manifold ($\dot{\theta =-\sqrt{A_{11}}\theta }$). Such distance is measured by the parameter ${\gamma }_{off}=\left|\frac{{\dot{\theta }}_{off}}{{\theta }_{off}\sqrt{A_{11}}}\right|.{\gamma }_{off}=1$ means that the starting point of an off-phase is exactly on top of the manifold and in this case the crossing time diverges, whereas it quickly decreases with the increase of |γ_off − 1|. (A) Displays Δt_cross as a function of γ_off for different values of the stick length and feedback delay δ = 230 ms. Since the stability condition of the off-phase for the standard intermittent control policy is Δt_cross > δ, the graph clearly shows that the interval of values of γ_off that support such condition strongly decreases with the shortening of the stick length. (B) Focuses on the most challenging configuration of the CIP balancing task (L = 32 cm, δ = 230 ms) and compares the range of values of γ_off that support stability in the standard and in the new intermittent control policy (Δt_cross > δ vs. Δt_cross > δ/(1+ρ), respectively). ρ = 0.8 is the “contraction factor”.

Figure 4

(A) Time series of the stick angle θ(t) during a 2 min balancing exercise. (B) Corresponding PSD. (C) time series of the cart displacement during the same time interval. CIP parameters: stick length L = 32 cm; cart mass M = 0.25 kg; Stick mass m = 0.125 kg; feedback delay δ = 230 ms. Controller parameters: P_θ = 5 N/rad; D_ω = 0.1826 Ns/rad; P_x = 0.01 N/m; D_v = 0.1 Ns/m.

Figure 5

Phase portrait of a 2 min CIP balancing exercise with the modified intermittent control policy. CIP parameters: stick length L = 32 cm; cart mass M = 0.25 kg; stick mass m = 0.125 kg; feedback delay δ = 230 ms. Controller parameters: P_θ = 5 N/rad; D_ω = 0.1826 Ns/rad; P_x = 0.01 N/m; D_v = 0.1 Ns/m. The green and red lines correspond to the stable and unstable manifolds, respectively. Measurement units: deg vs. deg/s.

Figure 6

(A) Histogram of γ_off, i.e., of the distance from the stable manifold of the state vector at t = t_off. (B) Histogram of the duration of the on-phase. (C) Histogram of the duration of the off-phase.

Figure 7

(A) Probability of falling over 100 repetitions; (B) Standard deviation of the stick oscillations; (C) Standard deviation of the cart motion, computed for the successful trials. The control parameter P_θ is varied between 4 and 20.