2020 # Mixed‐integer quadratic programming for automatic walking footstep placement, duration, and rotation

**Abstract:** Summary
This paper presents a mixed‐integer model predictive controller for walking. In the proposed scheme, mixed‐integer quadratic programs (MIQP) are solved online to simultaneously decide center of mass jerks, footsteps positions, durations, and rotations while respecting actuation, geometry, and contact constraints. Most walking controllers require preplanned footstep rotations to avoid dealing with the nonlinearity introduced by foot rotation decision. The main contribution of this work is an optimizatio…

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“…When all the flows ϕ ij are strictly positive, the objective (8a) is equal to (7) by definition, and this bilinear program is exactly the problem we derived above. On the other hand, when a flow ϕ ij is zero, problem (8) does not raise any issue: the (i, j)th addend in (8a) is well defined and correctly evaluates to zero, even when ij (x i , x j ) = ∞. To see this, notice that ϕ ij = 0 implies y ij = z ij = 0 by (8b) and (8c), and Definition 1 reads…”

confidence: 99%

“…When all the flows ϕ ij are strictly positive, the objective (8a) is equal to (7) by definition, and this bilinear program is exactly the problem we derived above. On the other hand, when a flow ϕ ij is zero, problem (8) does not raise any issue: the (i, j)th addend in (8a) is well defined and correctly evaluates to zero, even when ij (x i , x j ) = ∞. To see this, notice that ϕ ij = 0 implies y ij = z ij = 0 by (8b) and (8c), and Definition 1 reads…”

confidence: 99%

“…Proposition 1. The addition of the integrality constraints ϕ ij ∈ {0, 1} for all (i, j) ∈ E does not affect the optimal value of problem (8).…”

confidence: 99%

“…Therefore, the solution problem of MISO nonlinear system in each fermentation process can be transformed into a mixed integer quadratic programming (MIQP) problem [40]:…”

confidence: 99%

“…New stability conditions and control designs based on numerical methods, including linear matrix inequalities and gradient based optimization methods, are introduced to address time delays, parametric uncertainties, and the nonlinearities involved in the switching dynamics with stability and performance guarantees. The second group of articles, 5‐14 constituting the major category of this issue, applies adaptive control, optimization methods, and hybrid system theories to various practical fields, ranging from UAVs, AGVs, robotics, and water systems to batch processes. The last group 15,16 considers distributed cyber physical systems, which deals with parametric uncertainties, communication constraints, and signal estimations by resorting to adaptive control and fast Kalman filtering.…”

mentioning

confidence: 99%