Real-time optimal powered descent guidance algorithms that enable precision landing are key to maximizing the probability of landing success. Such algorithms allow the vehicle to cope with a more adverse set of model uncertainties, environmental disturbances, and unexpected obstacles as the vehicle approaches the landing site. This increase in robustness can be used to attempt landings in more challenging and scientifically interesting areas, or instead can be reallocated by exchanging excessive propellant mass for an increased payload mass fraction. In recent years, optimal powered descent guidance technology has played a key role in the robust recovery of commercial vertical-takeoff-vertical-landing reusable launch vehicles [1], and is at the forefront of the drive to reduce launch costs for the foreseeable future. Solving the powered descent guidance problem quickly and reliably is necessary because the vehicle has a limited amount of propellant, and must react quickly to deviations and conditions observed close to the ground. Doing so is challenging due to the nonlinear nature of the dynamics, the non-convex nature of the state and control constraints, and the free-final-time nature of the problem. Work on powered descent guidance began during the Apollo program, with [2, 3, 4] approaching the problem using optimal control theory and calculus of variations. However, these methods were not incorporated into the Apollo flight code since the polynomial-based guidance methods in use at the time were deemed sufficiently optimal, and were far simpler to design and implement [5]. After the Apollo program, research continued in search of analytical solutions to the 3-degree-of-freedom (DoF) landing problem [6, 7, 8, 9]. In the early 2000s, interest in the powered descent landing problem renewed, this time with a focus on robotic missions to Mars. A number of works using direct methods were published, with [10, 11] using numerical simulations to demonstrate theoretical results on the 3-DoF problem, and with [12, 13, 14] culminating in the use of Pontryagin’s maximum principle to losslessly convexify the 3-DoF problem. These works were later generalized in [15, 16, 17, 18, 19, 20, 21], and were demonstrated in a sequence of flight experiments in the early 2010s [22, 23, 24, 25].

In 2015, a dual-quaternion-based approach was proposed in which a 6-DoF line-of-sight constraint was convexified [26, 27]. This method was inherently equipped to handle 6-DoF motion, but relied on piecewiseaffine approximations to deal with the nonlinear dynamics. As such, the solution degraded in accuracy with coarser temporal discretizations.