Discrete Mechanics and Optimal Control (DMOC)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Discrete mechanics and optimal control (DMOC) is the structure-preserving branch of direct methods, introduced by Junge, Marsden, and Ober-Blöbaum (Junge et al. 2005; Ober-Blöbaum et al. 2011). The key distinction: DMOC discretizes the variational principle, not the ODE. The system Lagrangian $L(\mathbf{q},\dot{\mathbf{q}})$ is replaced by a discrete Lagrangian $L_d(\mathbf{q}_k,\mathbf{q}_{k+1},h)$ ; varying the discrete action yields the forced discrete Euler-Lagrange equations, which serve as NLP constraints; the control cost is the objective. The resulting finite-dimensional constrained optimization is solved by SNOPT or another SQP solver (Ober-Blöbaum et al. 2011; Moore 2011).

DMOC resembles direct collocation in form (discretize-then-optimize), but the objects discretized differ: collocation discretizes $\dot{\mathbf{x}}=\mathbf{f}$ , DMOC discretizes Hamilton's principle $\delta\int L\,\mathrm{d}t = 0$ . The result: DMOC solutions are symplectic and momentum-preserving, with far better long-time energy behavior than non-symplectic schemes (Marsden & West 2001).

Mathematical form

Consider a mechanical system with configuration $\mathbf{q}\in Q$ , Lagrangian $L:TQ\to\mathbb{R}$ , and external force $\mathbf{f}_{\text{ext}}$ . The continuous Lagrange-d'Alembert principle reads

\delta\int_0^T L(\mathbf{q},\dot{\mathbf{q}})\,\mathrm{d}t + \int_0^T \mathbf{f}_{\text{ext}}\cdot\delta\mathbf{q}\,\mathrm{d}t = 0.

Discretization: split $[0,T]$ into $N$ steps of size $h$ , with configuration sequence $\{\mathbf{q}_k\}_{k=0}^N$ . Approximate the discrete Lagrangian by the midpoint rule

L_d(\mathbf{q}_k,\mathbf{q}_{k+1},h)\approx h\,L\bigg(\frac{\mathbf{q}_k+\mathbf{q}_{k+1}}{2},\frac{\mathbf{q}_{k+1}-\mathbf{q}_k}{h}\bigg).

Left and right discrete forces $\mathbf{F}_k^-, \mathbf{F}_k^+$ are approximated similarly. Varying the discrete action $S_d=\sum_k L_d$ yields the forced discrete Euler-Lagrange equations (DEL):

D_2 L_d(\mathbf{q}_{k-1},\mathbf{q}_k) + D_1 L_d(\mathbf{q}_k,\mathbf{q}_{k+1}) + \mathbf{F}_k^- + \mathbf{F}_k^+ = \mathbf{0},\quad k=1,\dots,N-1.

DMOC treats the control force as $\mathbf{f}_{\text{ext}} = \mathbf{u}(t)$ , discretized as node values $\{\mathbf{u}_k\}$ ; DEL equations become NLP equality constraints, initial/terminal conditions are boundary constraints, control energy $\sum_k \|\mathbf{u}_k\|^2$ is the objective, solved by SNOPT or another SQP solver (Ober-Blöbaum et al. 2011; Moore 2011, Ch. 2). Midpoint-rule DMOC is second-order accurate.

Structure-preserving properties

DMOC solutions inherit the key properties of variational integrators:

Symplectic. The symplectic 2-form on phase space is preserved exactly under iteration.
Momentum-preserving. The Noether momentum map associated with any system symmetry is preserved exactly, independent of step size.
Good energy behavior. Symplecticity guarantees no long-time energy drift or dissipation — energy oscillates within a small bound.

These three properties are the core advantages of DMOC over general direct collocation, especially for CR3BP and low-energy Earth-Moon transfers that need long-time integration and are sensitive to energy preservation (Marsden & West 2001; Moore 2011).

Practical implementation

Variable step size. Standard DMOC assumes a uniform $h$ . In CR3BP, dynamics near perilune require very small steps while apoapsis tolerates coarse steps. Moore (2011) introduces time-adaptive DMOC: introduce $t_k$ as variables with constraints $h_k=t_{k+1}-t_k$ on the time-adapted Lagrangian, then apply the time-adapted variational principle. The cost is that time-adaptive DMOC becomes an indirect method (extra costate variables).
Initial guesses. DMOC is a local method and needs a good initial guess. Moore (2011) uses invariant-manifold stitching (Sun-Earth + Earth-Moon manifolds intersecting on a Poincaré section) as the initial guess; DMOC optimization can then eliminate the mid-course $\Delta V$ required by the manifold solution, yielding a fully zero-fuel low-energy channel.
Comparison with collocation.

Feature	DMOC	Direct collocation
Discretization target	Variational principle	ODE
Symplectic	yes	no
Momentum-preserving	yes	no
Large-step behavior	reasonable	error accumulation
Generality	Lagrangian systems only	arbitrary ODE
Tooling maturity	mostly academic	industrial (OTIS, SOCS, …)

Applications

Low-energy Earth-Moon transfers. DMOC + invariant manifolds is a classic pairing: manifolds supply the dynamics-based skeleton as initial guess; DMOC refines in the 4-body model and drives the manifold-stitching mid-course $\Delta V$ to zero (Moore 2011, Ch. 3; Moore et al. 2012).
Formation flight. CubeSat formation reconfiguration benefits from DMOC's momentum preservation — formation geometry remains stable over long simulations.
When not to use. Systems with strong dissipation (atmospheric entry), non-conservative constraints (heat-flux limits), or no Lagrangian in closed form lose DMOC's advantages — direct collocation is then more appropriate.

References

Junge, O., Marsden, J. E., & Ober-Blöbaum, S. (2005). Discrete mechanics and optimal control. IFAC Proceedings Volumes, 38(1), 538–543.
Ober-Blöbaum, S., Junge, O., & Marsden, J. E. (2011). Discrete mechanics and optimal control: an analysis. ESAIM: Control, Optimisation and Calculus of Variations, 17(2), 322–352.
Marsden, J. E., & West, M. (2001). Discrete mechanics and variational integrators. Acta Numerica, 10, 357–514.
Moore, B. E. (2011). Discrete mechanics and optimal control for space trajectory design. PhD thesis, Purdue University.
Moore, B. E., Ober-Blöbaum, S., & Marsden, J. E. (2012). Trajectory design combining invariant manifolds with discrete mechanics and optimal control. Journal of Guidance, Control, and Dynamics, 35(5), 1507–1525.
Leyendecker, S., Ober-Blöbaum, S., Marsden, J. E., & Ortiz, M. (2010). Discrete mechanics and optimal control for constrained systems. Optimal Control Applications and Methods, 31(6), 505–528.