Balance control in humans has long been an active area of research due to its important implications in physiology, sports medicine, medical diagnostics, and humanoid robotics. Numerous models have been proposed for this purpose and analysed both from experimental and theoretical perspectives. Still, few results exist that pertain to quantitative evaluation of control limitations in balance control especially during dexterous body motions as demanded in dance or acrobatics, but also during ordinary stance. This status quo is explained by the absence of computationally feasible approaches to global stability analysis of under-actuated multi-body systems that interact with the environment through contact points while remaining under the action of gravity. Admittedly, the stabilization problem of a multi-body system to its upright position in the presence of gravity is far from simple as the dynamical model of the system is not only highly dimensional, but also heavily non-linear.
The aim of this talk is to elucidate the theoretical issues underlying semi-global stabilization of multi-body systems under the action of gravity and to present ideas for the development of a unified approach to analysis and stabilizing feedback control design for such systems. Semi-global stabilization refers to control designs characterized by large basins of attraction to a system equilibrium with computable bounds on the size of such basins. Highly demanding balancing tasks, including that of regaining the upright posture while standing on the tip of one foot, will be of special interest as the solution of such problems subsumes simpler control situations. The simplest model permitting to explain the difficulties of balancing a body “sur les pointes” is a multi-link inverted pendulum deprived of actuation at the first joint located at the contact point with the ground. As it turns out, the pendulum system is controllable to the unstable equilibrium corresponding to the upright body position provided that the remaining joints of the pendulum are fully actuated.
The system is classified as a second order non-holonomic system as the acceleration constraints at the joints that hold parts of the body together are not integrable. Systems of this type cannot be stabilized globally by continuous state feedback and necessitate application of hybrid or else time-varying control methods.
It will be shown that the planar version of the balancing “sur les pointes” problem can be solved semi-globally by partial input-output linearization and linear feedback with a non-linear correction term that employs the integral of the generalized momentum in the model of the dynamical system. Somewhat surprisingly, a similar construction is impossible if the balancing problem is considered in three dimensional space because in the 3D case the generalized momentum fails to be integrable. Passivity based energy shaping approaches, although theoretically promising, cannot deliver feedback laws either, as their derivation is inhibited by the overwhelming complexity of symbolic calculations.
An alternative approach to balancing rigid body structures in gravity fields is hence proposed which relies on two components, well recognized as critical in the physics of human balance control: the angular momentum of the body with respect to the foot and the reaction force from the ground. Related system variables – the two horizontal components of the linear momentum of the system as a whole, are then identified as output variables permitting partial linearization of the system. The latter permits for an immediate construction of a partial feedback control law that insures partial stabilization of the system to the set of its relative equilibria. The last phase of the balancing control design involves a modification of the partial feedback control by way of a sub-optimal solution of an infinite horizon minimum-energy-related optimal control problem which serves as an instrument for simultaneous stabilization and energy-injection-shaping towards achieving the desired final energy balance corresponding to the upright equilibrium of the system.
The proposed approach is not conceived to deliver closed form state feedback laws.
It should rather be regarded as a form of on-line robust feedback stabilization strategy much akin to the model predictive control concept. The obvious implication is that the approach is highly tunable by optimal cost adjustment, and can, in principal, accommodate for the satisfaction of actuation constraints. It also generalizes easily to apply to more complex balancing problems involving multi-link rigid bodies. There is just one caveat: Full state estimation procedures must be in place for the feedback control to be computed.
As the proposed stabilizing control approach invokes intuitively familiar, real physical concepts of momentum, energy, and reaction forces, it can perhaps shed some light on how the balancing control processes are effectuated by the brain.