Finite Thrust Maneuver

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A finite thrust maneuver is an orbital maneuver that accounts for the finite magnitude of engine thrust and non-instantaneous burn duration. Unlike the impulsive thrust assumption, finite thrust maneuvers require solving a two-point boundary value problem (TPBVP) to obtain the optimal control law, using Pontryagin's minimum principle to determine thrust direction and switching strategy.

Core Elements

Optimal Control Model

System dynamics:

X˙=f(X,u,t)\dot{\boldsymbol{X}} = f(\boldsymbol{X}, \boldsymbol{u}, t)

Performance index:

J=ϕ[X(tf),tf]+∫t0tfL(X,u,t) dtJ = \phi[\boldsymbol{X}(t_f), t_f] + \int_{t_0}^{t_f} L(\boldsymbol{X}, \boldsymbol{u}, t)\,dt

Hamiltonian Function

H=L(X,u,t)+λTf(X,u,t)H = L(\boldsymbol{X}, \boldsymbol{u}, t) + \boldsymbol{\lambda}^T f(\boldsymbol{X}, \boldsymbol{u}, t)

State equation: X˙=∂H/∂λ\dot{\boldsymbol{X}} = \partial H / \partial \boldsymbol{\lambda}; co-state equation: λ˙=−∂H/∂X\dot{\boldsymbol{\lambda}} = -\partial H / \partial \boldsymbol{X}.

Pontryagin's Minimum Principle

Optimality condition for constrained control:

H(X∗,u∗,λ∗,t)≤H(X∗,u∗+δu,λ∗,t)H(\boldsymbol{X}^*, \boldsymbol{u}^*, \boldsymbol{\lambda}^*, t) \leq H(\boldsymbol{X}^*, \boldsymbol{u}^* + \delta\boldsymbol{u}, \boldsymbol{\lambda}^*, t)

Optimal Control Law

Optimal thrust direction in Cartesian coordinates:

α∗=−λv∥λv∥\boldsymbol{\alpha}^* = -\frac{\boldsymbol{\lambda}_v}{\|\boldsymbol{\lambda}_v\|}

The thrust direction is opposite to the velocity co-state direction.

Switching Function

HT=−∥λv∥m−λm1gIspH_T = -\frac{\|\boldsymbol{\lambda}_v\|}{m} - \lambda_m \frac{1}{gI_{sp}}

  • HT>0H_T > 0: T=0T = 0 (engine off)
  • HT<0H_T < 0: T=Tmax⁡T = T_{\max} (full thrust)
  • HT=0H_T = 0: 0<T<Tmax⁡0 < T < T_{\max} (arc thrust)

Mass Equation

m˙=−TgIsp\dot{m} = -\frac{T}{gI_{sp}}

Application Value

Finite thrust maneuver methods are applicable to orbital design for solar electric propulsion (SEP) and other low-thrust propulsion systems. By solving the TPBVP for the optimal thrust strategy, fuel-optimal or time-optimal orbital transfers can be achieved. With advances in low-thrust propulsion technology, finite thrust maneuver methods are increasingly used in deep-space exploration and long-duration on-orbit missions.

References

  • Zheng W, An X, Zhou X, He R. Spaceflight Mechanics [M]. National University of Defense Technology, 2026.
  • Jia P, Chen K, et al. Long-Range Rocket Ballistics [M]. National University of Defense Technology Press.