Limits of MPCC Formulations in Direct Optimal Control with Nonsmooth Differential Equations

Nurkanović, Armin; Albrecht, Sebastian; Diehl, Moritz

doi:10.23919/ecc51009.2020.9143593

Cited by 22 publications

(36 citation statements)

References 23 publications

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“…In the penalty reformulation an 1 or ∞ norm of the complementarity residual (i.e., the bilinear terms) is added to the objective and penalized by a positive weight ρ mpcc [6,18,23]. Moreover, homotopy approaches help to avoid convergence to spurious local solutions [21,29]. An appealing property of such reformulation is: if ρ mpcc is larger than a critical value, then the complementarity constraints are satisfied exactly at the solution [6,23].…”

Section: Solving Discretized Ocp Via Homotopies For Mpccsmentioning

confidence: 99%

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

Nurkanović

Sartor

Albrecht

et al. 2021

IEEE Control Syst. Lett.

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This paper introduces a novel reformulation and numerical methods for optimal control of complementarity Lagrangian systems with state jumps. The solutions of the reformulated system have jump discontinuities in the first time derivative instead of the trajectory itself, which is easier to handle theoretically and numerically. We cover not only the easier case of elastic impacts, but also the difficult case, when after the state jump the system evolves on the boundary of the dynamic's feasible set. In nonsmooth mechanics this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamic system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. We detail how to recover the solution of the original system, show how to select appropriate auxiliary dynamics and give practical numerical methods to handle discontinuous ODEs with nonunique sliding motions. Moreover, we introduce a novel auxiliary ODE for time-freezing for elastic impacts and overcome some drawbacks of [22]. The theoretical findings are illustrated on the nontrivial numerical optimal control example of a hopping one-legged robot.

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Section: Solving Discretized Ocp Via Homotopies For Mpccsmentioning

confidence: 99%

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

Nurkanović

Sartor

Albrecht

et al. 2021

IEEE Control Syst. Lett.

Self Cite

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“…Therefore, we have equations (14a), (15a), and (15b) for stage i in the same form as equations (11) for stage i + 1.…”

Section: A Backward Recursionmentioning

confidence: 99%

“…In this stage, the phase of interest changes from k to k−1 when k > 1, and the backward recursion terminates when k = 1. The equations (11) until i k are expressed as:…”

Section: A Backward Recursionmentioning

confidence: 99%

“…Therefore, together with an equation (7e) for k − 1, we obtain equations (14a), (15a), and (15b) for stage i k in the same form as equations (11) for stage i + 1 with…”

Section: A Backward Recursionmentioning

confidence: 99%

“…However, this case has the same LICQ problem as the vanishing constraints, requiring a great deal of computational time to alleviate the numerical ill-conditioning. Furthermore, it often suffers from undesirable stationary points [11].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Structure-Exploiting Newton-Type Method for Optimal Control of Switched Systems

Katayama¹,

Ohtsuka²

2021

Preprint

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This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilized to discretize continuous-time optimal control problems (OCPs) and formulate a nonlinear program (NLP), which guarantees the local convergence of a Newton-type method. A dedicated structure-exploiting algorithm (Riccati recursion) is proposed to perform a Newton-type method for the NLP efficiently because its sparsity structure is different from a standard OCP. The proposed method computes each Newton step with linear time-complexity for the total number of discretization grids as the standard Riccati recursion algorithm. Additionally, the computation is always successful if the solution is sufficiently close to a local minimum. Conversely, general quadratic programming (QP) solvers cannot accomplish this because the Hessian matrix is inherently indefinite. Moreover, a modification on the reduced Hessian matrix is proposed using the nature of the Riccati recursion algorithm as the dynamic programming for a QP subproblem to enhance the convergence. A numerical comparison is conducted with off-the-shelf NLP solvers, which demonstrates that the proposed method is up to two orders of magnitude faster. Whole-body optimal control of quadrupedal gaits is also demonstrated and shows that the proposed method can achieve the whole-body model predictive control (MPC) of robotic systems with rigid contacts.

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Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

Nurkanović,

Sperl,

Albrecht

et al. 2024

Numer. Math.

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This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.

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Limits of MPCC Formulations in Direct Optimal Control with Nonsmooth Differential Equations

Cited by 22 publications

References 23 publications

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

Structure-Exploiting Newton-Type Method for Optimal Control of Switched Systems

Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

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