# Solving an Ordinary Differential Inequality

In his 2008 paper Viability and Predictive Control for Safe Locomotion, Pierre-Brice Wieber discusses what happens when the center-of-mass of a humanoid reaches the boundary of its supporting contact area. Mathematically, this gives rise to the Ordinary Differential Inequality (ODI):

Under the boundary condition:

The paper provides an analytical solution to the ODI as follows:

Let us now detail how to get there.

## Bounding by an ODE solution

The method to derive a lower-bound on ODI solutions relies on a "parallel" solution to the related Ordinary Differential Equation (ODE):

Theorem (Petrovitsch, 1901):let \(\dot{\gamma} \geq f(\gamma, t)\) denote an ODI, and let \(\eta\) denote a solution to the ODE \(\dot{\eta} = f(\eta, t)\) subject to the boundary condition \(\eta(t_0) = \gamma(t_0)\). Then,

In our case, a solution to the ODE associated with our second-order ODI satisfies:

It is well known that the set of solutions to such second-order differential equations, whose characteristic polynomial has real roots, are linear combinations of hyperbolic functions, plus the non-homogeneous part of the solution:

We now choose the boundary conditions such that \(y(t_0) = x(t_0) = -b\) and \(\yd(t_0) = \xd(t_0)\), resulting in:

We cannot apply Petrovitsch's theorem directly as our variable \(\gamma = (x, \xd)\) is two-dimensional, but the underlying idea is the same. Consider the difference \(\delta\) between \(x\) (ODI solution) and \(y\) (ODE solution). Then:

Due to the initial condition, either \(\delta\) is uniformly \(0\) for all \(t \geq t_0\) (in which case the bound is tight), or \(\delta\) is increasing for \(t \geq t_0\) and thus positive. In both cases, this difference ends up being positive, so that \(\forall t \geq t_0, x(t) \geq y(t)\) and \(y\) is indeed an analytical lower-bound to all solutions of the ODI.