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.

State-space intermittent feedback stabilization of a dual balancing task

“…Figure 4 shows typical combined CIP/body balancing patterns, during the dual task, in the time and frequency domains and Table 2 reports characteristic parameters of the different subjects. As regards the CIP-like device, generally speaking, as one may expect, such patterns are similar to those recorded in studies where the CIP was stabilized by a sitting subject 23,24 . The range of motion of the stick, measured by the standard deviation of the θ s angle, is close to 2 deg; the spectrum of such oscillations has a well-marked peak, below 1 Hz (0.39 Hz on average); the range of hand motion, measured by the standard deviation of x s , is about 0.1 m. However, what is more interesting is the interaction/coordination between the two balancing motions.…”

“…However, the goal of the control process is more limited: it consists of keeping one variable (θ s ) in a small neighborhood of equilibrium, while allowing the other variable (x s ) to oscillate semi-freely in a large range compatible with arm-length. The model, that can be derived by the Lagrange equations and has been already employed in a previous work 24 , is expressed by the following equation: x are the controller gain parameters: the function of the first two is to constrain the stick oscillations near a limit cycle in the phase plane around the vertical, thus avoiding the fall, while the function of the other two elements of the control action is simply to limit the CIP motion to a physiological range, compatible with the arm length. The two groups of gains must be tuned considering a trade-off between stick motion and hand motion: increasing the hand gains will reduce the range of hand motion but increase the range of stick motion, with a greater risk of fall; in contrast, reducing the hand gains will also reduce the stick risk of fall but may force the hand beyond arm reachable positions.…”

“…The missing control action must be provided by an active feedback mechanism driven by the more or less accurate estimate of the oscillations that need to be balanced: different possible alternatives of such feedback control actions have been investigated. The basic choice is between continuous-time 9-13 and discontinuous-time or intermittent control action [14][15][16][17][18][19][20][21][22][23][24] , with or without an observer and a predictor in the control structure. The main challenge, for the family of balancing tasks we are considering, is the strong delay of the feedback information about the ongoing sway, of the order of 0.2 s, and the fact that this delay is comparable to the potential falling time constant of the oscillating body.…”

“…In previous studies, it was demonstrated that the state-space intermittent feedback stabilization paradigm can explain in a detailed manner the oscillatory patterns of two very different balancing tasks: quiet upright standing [20][21][22] and stabilization of a CIP 23,24 . In both cases, the intermittent control paradigm matches the rationale of the minimum intervention principle 25 and in any case minimization of energetic costs is a relevant feature at stability's edge 26 .…”

Centre of pressure versus centre of mass stabilization strategies: the tightrope balancing case

“…The oscillatory regimes in the two planes are characterized by different spectral features ( figure 6 ). The oscillations in the coronal plane are characterized by a clear resonant peak between 0.3 and 0.4 Hz, namely a behaviour which is similar to the stabilization of a cart inverted pendulum [ 18 – 20 ]: in spite of the obvious differences between the two cases, they share the fact that the oscillating body has the CoP fixed to the lower end of the pendulum, inhibiting any possible contribution of the CoP to its stabilization. By contrast, the sway in the sagittal plane is characterized by a typical f − β power spectral density (PSD) well known in the standard posturographic literature, where it is often modelled as a stochastic motion of the CoP acting on the foot soles during stance [ 32 , 33 ].…”