High-Accuracy Rocket Landing via Lossless Convexification

Abstract

With the development of rocket technology, achieving high-precision landing has become a key technical challenge in the field of aerospace. To cope with this challenge, we propose a lossless convexification algorithm based on the integral pseudospectral method in this paper. Firstly, for the fuel optimization problem, the continuous dynamic equations and constraints are discretized with high accuracy using an integral-type pseudospectral method. By constructing a global integration matrix at Legendre–Gauss nodes, the original complex continuous problem is effectively transformed into a discrete form that is more tractable for numerical optimization. Secondly, the non-convex constraints are transformed using the lossless convexification technique, thereby reformulating the original problem as a second-order cone programming (SOCP) problem. The effectiveness of the proposed algorithm is validated through numerical simulations, which demonstrate high landing accuracy, robustness, and fuel efficiency. These results highlight the algorithm’s high performance and strong potential for practical application in space missions.

1. Introduction

In recent years, with the development of space exploration missions such as lunar exploration and precise Mars landing, the importance of powered descent guidance (PDG) technology has become increasingly prominent [1,2,3,4]. PDG, as the final phase of a flight mission, must guide the spacecraft precisely and safely to the target location while satisfying physical constraints and ensuring fuel optimality [5]. The history of PDG dates back to the Apollo era, where the task was to ensure that the vehicle landed in a designated area on the Moon at nearly zero velocity [6]. Common analytic methods included the gravity turn [7] and the Apollo polynomial guidance algorithms [8,9]. However, these methods suffered from insufficient precision or difficulty in simultaneously satisfying process constraints and fuel optimality. Subsequently, indirect guidance methods based on Pontryagin’s minimum principle were proposed [10,11,12,13], but these methods encountered difficulties such as sensitivity to the initial guess of the costate variables.

Recently, the application of convex optimization theory in trajectory optimization has brought new breakthroughs to PDG technology [14,15,16,17]. Acikmese and Blackmore et al. proposed lossless convexification algorithms that transform the original problem into a convex problem, achieving both fuel optimality and good convergence [14]. Liu and Lu proposed an algorithm that successively transforms a nonlinear problem into convex subproblems, broadening the application scope [15,16]. Currently, successive convexification (SCP) still faces risks such as slow convergence, and the choice of discretization points and method affects computational complexity and accuracy. Therefore, balancing computation accuracy and efficiency is a key issue in applying convex optimization to PDG. The emergence of pseudospectral methods offers a new solution to this problem [18,19,20,21,22]. For example, Sagliano et al. combined pseudospectral methods with convex optimization to effectively improve solution accuracy, and on this basis introduced an hp-pseudospectral method to further improve computational efficiency [18,19]; reference [20] proposed an adaptive pseudospectral method whose discretization density can automatically adjust, thereby improving terminal precision and computational efficiency. However, traditional guidance algorithms adopt a differential form of pseudospectral discretization. When the number of nodes is large, this can lead to poor numerical stability and an imprecise differentiation matrix, making it difficult to meet the demands of precision landing.

Therefore, this paper proposes a lossless convex optimization algorithm based on the integral pseudospectral method. Unlike traditional differential discretization methods, the integral discretization uses a global integration matrix to perform a high-precision algebraic approximation of the differential equations, which significantly enhances numerical stability and ensures high accuracy and efficiency in the computation. At the same time, by using lossless convexification technology to convert the non-convex constraints of the fuel-optimal problem into convex constraints, the solution efficiency and precision of the trajectory optimization problem are improved, and the feasibility and reliability of the algorithm under complex constraints are enhanced.

This paper is organized as follows: Section 2 models the rocket landing problem; Section 3 applies the pseudospectral discretization to the constructed problem and convexifies the nonconvex constraints to form an SOCP problem; Section 4 provides numerical examples to demonstrate the high performance of the proposed algorithm framework; and Section 5 concludes the paper.

2. Problem Statement

In this section, we first establish the dynamics model for rocket landing. We then describe the various constraints acting on the spacecraft, and finally formulate a fuel-optimal rocket landing problem with free terminal time. During the rocket’s powered descent, the motion is primarily driven by thrust provided by the engine. Since the aerodynamic forces are much smaller compared to thrust, we can neglect atmospheric drag and the non-inertial forces due to Earth’s rotation. Under these assumptions, the equations of motion can be expressed as follows: [14]:

$$\begin{array}{cc}\hfill \dot{\mathbf{r}}\left(t\right)& =\mathbf{v}\left(t\right)\hfill \\ \hfill \dot{\mathbf{v}}\left(t\right)& ={\displaystyle \frac{\mathbf{T}\left(t\right)}{m\left(t\right)}}-\mathbf{g}\hfill \\ \hfill \dot{m}\left(t\right)& =-\alpha \parallel \mathbf{T}\left(t\right)\parallel \hfill \end{array}$$

(1)

where $\mathbf{r}\left(t\right)$, $\mathbf{v}\left(t\right)$, and $m\left(t\right)$ denote the position vector, velocity vector, and the rocket mass, respectively; $\mathbf{T}\left(t\right)$ is the thrust vector; $\alpha \triangleq {\displaystyle \frac{1}{I_{\mathrm{sp}}{g}_0}}$ is the mass consumption coefficient; $I_{sp}$ is the specific impulse of the rocket engine; $g_0$ is the gravitational constant; and $\mathbf{g}$ is the gravitational acceleration vector (assumed constant).

The thrust vector $\mathbf{T}\left(t\right)$ is subject to the following constraint:

$$T_{min}\le \parallel \mathbf{T}\left(t\right)\parallel \le T_{max}$$

(2)

where $T_{min}$ and $T_{max}$ are the lower and upper bounds of the thrust magnitude, respectively.

The initial and terminal state constraints can be expressed as follows:

$$\begin{array}{c}\mathbf{r}\left(t_0\right)={\mathbf{r}}_0,\mathbf{v}\left(t_0\right)={\mathbf{v}}_0,m\left(t_0\right)=m_0,\mathbf{r}\left(t_f\right)={\mathbf{r}}_f,\mathbf{v}\left(t_f\right)={\mathbf{v}}_f,m\left(t_f\right)\ge m_{\mathrm{dry}}\end{array}$$

(3)

where $t_0$ and $t_f$ are the initial and final times; ${\mathbf{r}}_0$, ${\mathbf{v}}_0$, and $m_0$ represent the initial position, velocity, and mass; and $m_{\mathrm{dry}}$ is the dry mass of the rocket.

Based on the above equations of motion and constraints, the fuel-optimal powered descent problem with free terminal time can be formulated as follows:

$$\begin{array}{c}\mathrm{P}0:\underset{t_f,T\left(t\right)}{\min }{\int }_0^{t_f}\parallel T\left(t\right)\parallel dt\\ \dot{r}\left(t\right)=v\left(t\right)\\ \dot{v}\left(t\right)=-g+{\displaystyle \frac{T\left(t\right)}{m\left(t\right)}}\\ \dot{m}\left(t\right)=-\alpha \parallel T\left(t\right)\parallel \\ 0<T_{min}\le \parallel T\left(t\right)\parallel \le T_{max}\\ r\left(t_0\right)=r_0,v\left(t_0\right)=v_0,m\left(t_0\right)=m_0\\ r\left(t_f\right)=0,v\left(t_f\right)=0,m\left(t_f\right)\ge m_{\mathrm{dry}}\end{array}$$

(4)

3. Methodology

In this section, we propose a lossless convex trajectory optimization method for rocket landing based on an integral Gaussian pseudospectral method. First, a global integration matrix is constructed using Legendre–Gauss (LG) nodes, converting the differential equations into algebraic equations. Second, for non-convex constraints such as those on thrust, lossless convexification theory is applied to effectively transform the original problem into an SOCP problem.

3.1. Integral-Type Discretization of Gaussian Pseudospectral Method

To solve the problem, the continuous optimal control formulation must be discretized. In this work, an integral form of the Gaussian pseudospectral method is employed. This approach leverages the roots of orthogonal Legendre polynomials as collocation points, enabling high-order approximation of state and control trajectories on a non-uniform grid. Compared with the evenly spaced nodes of classical Euler discretization, the pseudospectral method achieves superior accuracy with significantly fewer nodes. The Gaussian pseudospectral methods all use normalized time variables, and the original problem time domain needs to be transformed by the following time-domain transformation relation [23]:

$$\begin{array}{c}t={\displaystyle \frac{t_f-t_0}{2}}\tau +{\displaystyle \frac{t_f+t_0}{2}}\end{array}$$

(5)

The distribution point consists of N LG nodes, we use the N-point Gauss–Legendre (LG) quadrature on the normalized domain $\tau \in (-1,1)$. The collocation nodes ${\left\{{\tau }_k\right\}}_{k=1}^N$ are the roots of the Legendre polynomial $P_N$, which can be defined in the following:

$$\begin{array}{c}{P}_N\left({\tau }_k\right)={\displaystyle \frac{1}{2^{N}N!}}{\displaystyle \frac{d^N}{d{\tau }_k^N}}\left[{\left({\tau }_k^2-1\right)}^N\right],k=0,1,\dots ,N\end{array}$$

(6)

The associated Gaussian quadrature weights are given by the following:

$$\begin{array}{c}{w}_k={\displaystyle \frac{2}{1-{\tau }_k^2}}{\left[{\dot{P}}_N\left({\tau }_k\right)\right]}^{-2},k=0,1,\dots ,N\end{array}$$

(7)

where ${\dot{P}}_N$ denotes the derivative of the Legendre polynomial. The Lagrange interpolating basis $L_i\left(\tau \right)$ is defined as follows:

$$\begin{array}{c}{L}_k\left(\tau \right)={\displaystyle \prod _{j=0j\ne k}^N}{\displaystyle \frac{\tau -{\tau }_j}{{\tau }_k-{\tau }_j}}\end{array}$$

(8)

In the Gaussian pseudospectral method, the state and control quantities are approximated by the Lagrange interpolating polynomial basis $L_k\left(\tau \right)$ approximation of the state vector differential $\dot{\mathbf{x}}\left(\tau \right)$:

$$\dot{\mathbf{x}}\left(\tau \right)\approx \sum _{k=0}^N{\dot{\mathbf{x}}}_k\left({\tau }_k\right)L_k\left(\tau \right)$$

(9)

Substituting Equation (1) into Equation (9) and integrating both sides yields the following:

$$\begin{array}{c}x\left(\tau \right)\approx x(-1)+{\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}f\left(x_k,{\tau }_k,u_k\right){\int }_{-1}^{\tau }{L}_k\left(s\right)ds\end{array}$$

(10)

The integral matrix of the Lagrange polynomial $A_{ik}\left(\tau \right)$ at the collocation points is defined as follows:

$$\begin{array}{c}{A}_{ik}\left({\tau }_i\right)={\int }_{-1}^{{\tau }_i}{L}_k\left(\tau \right)d\tau \end{array}$$

(11)

Therefore, combining the dynamics in Equation (10), the trajectory differential equation at the collocation points can be transformed into an algebraic constraint:

$$\begin{array}{c}{x}_i\approx x_0+{\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}f\left(x_k,{\tau }_k,u_k\right)A_{ik},i=1,\dots ,N\end{array}$$

(12)

It should be noted that the algebraic constraints in Equation (12) do not include the terminal state variable. Thus, the terminal state is estimated using Gaussian quadrature as follows:

$$\begin{array}{c}{x}_f\approx x_0+{\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}{w}_{k}f\left(x_k,{\tau }_k,u_k\right)\end{array}$$

(13)

For the performance index, the cost functional is reconstructed at the collocation points using Gaussian quadrature in the following form:

$$\begin{array}{c}minJ={\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}{w}_k∥T\left(x_k,{\tau }_k,u_k\right)∥\end{array}$$

(14)

Through this discretization, the continuous optimal control problem is transformed into a finite-dimensional nonlinear programming problem.

3.2. Lossless Convexification of Thrust Constraints

To address the intrinsic non-convexity of the thrust magnitude constraint, we introduce a slack variable $\mathrm{\Gamma }\left(t\right)$ to replace the original thrust magnitude $\parallel \mathbf{T}\left(t\right)\parallel$ and substitute it into Equation (1). Accordingly, the mass dynamics, thrust magnitude constraints are rewritten as follows:

$$\dot{m}\left(t\right)=-\alpha \mathrm{\Gamma }\left(t\right)$$

(15)

$$0<T_{min}\le \mathrm{\Gamma }\left(t\right)\le T_{max}$$

(16)

This reformulation introduces the following additional constraint linking the slack variable to actual thrust magnitude:

$$∥\mathit{T}\left(t\right)∥\le \mathrm{\Gamma }\left(t\right)$$

(17)

As proven in [14], by introducing the slack variable $\mathrm{\Gamma }\left(t\right)$ together with Equation (17), the solution to the relaxed problem remains optimal for the original problem. This constitutes the core step of the lossless convexification approach. Due to the nonlinear terms in the system dynamics, we further introduce a set of equivalent transformations to linearize the problem formulation, defined as follows:

$$\mathrm{\sigma }\left(t\right)\triangleq {\displaystyle \frac{\mathrm{\Gamma }\left(t\right)}{m\left(t\right)}},\mathit{u}\left(t\right)\triangleq {\displaystyle \frac{\mathit{T}\left(t\right)}{m\left(t\right)}},z\left(t\right)\triangleq lnm\left(t\right)$$

(18)

Under this transformation, the dynamics and performance index can be rewritten in terms of $(u,\sigma )$, yielding linear dynamics and a convex cost. Therefore, substituting Equation (18) into Equation (1), the dynamics constraints of the problem can be expressed as follows:

$$\dot{x}=\left[\begin{array}{c}\dot{r}\\ \dot{v}\\ \dot{z}\end{array}\right]=\left[\begin{array}{c}v\\ \mathit{u}-\mathit{g}\\ -\alpha \sigma \end{array}\right]=f\left(x,\mathit{u},\sigma \right)$$

(19)

Substituting Equation (19) into Equation (12) yields the dynamics in integral form as a set of linear equality constraints:

$${\mathbf{x}}_i={\mathbf{x}}_0+{\displaystyle \frac{t_f-t_0}{2}}\sum _{k=1}^N{A}_{ik}f\left(\mathit{x},\mathit{u},\sigma \right),i=1,\dots ,N.$$

(20)

Fuel optimality is defined as maximizing the final mass $m_f$. Thus, integrating both sides of the mass dynamics equation (Equation (15)) yields

$$m_f=m_0-\alpha {\int }_{t_0}^{t_f}\mathrm{\Gamma }\left(t\right)dt$$

(21)

After the change of variables $\sigma =\mathrm{\Gamma }/m$ and $z=lnm$, we integrated the mass dynamics equation (Equation (19)); hence,

$$m_f=m_{0}exp\left(-\alpha {\int }_{t_0}^{t_f}\sigma \left(t\right)dt\right)$$

(22)

Since $\alpha >0$, the function $exp\left(-\alpha {\int }_{t_0}^{t_f}\sigma \left(t\right)dt\right)$ is strictly decreasing in the integral value. Therefore, minimizing ${\int }_{t_0}^{t_f}\sigma \left(t\right)dt$ yields the same optimal solution as minimizing ${\int }_{t_0}^{t_f}\mathrm{\Gamma }\left(t\right)dt$, and both formulations are fuel-optimal. To obtain a linear (and thus convex) performance index with respect to the decision variable, we adopt ${\int }_{t_0}^{t_f}\sigma \left(t\right)dt$ as the continuous-time cost functional.For discretization, we apply Gaussian quadrature on the Legendre–Gauss nodes ${\left\{t_k\right\}}_{k=1}^N$ with corresponding weights ${\left\{w_k\right\}}_{k=1}^N$. Denoting ${\sigma }_k=\sigma \left(t_k\right)$, the cost is approximated by

$$J={\displaystyle \frac{t_f-t_0}{2}}\sum _{k=1}^N{w}_k{\sigma }_k.$$

(23)

The thrust constraints together with Equation (17) can be accordingly reformulated at each discretization node as follows:

$$∥\mathit{u}\left(t_i\right)∥\le \sigma \left(t_i\right)$$

(24)

$$\begin{array}{c}0<T_{min}{e}^{-z_i}\left(t_i\right)\le {\sigma }_i\le T_{max}{e}^{-z_i}\end{array}$$

(25)

Since the constraint in Equation (22) is nonconvex, it needs to be transformed into a convex form. Specifically, the lower bound can be reformulated as a convex second-order cone constraint, while the upper bound remains nonconvex. For the upper bound, we approximate the exponential term using a first-order Taylor expansion around a reference point $z_0\left(t\right)$. This yields the following convex relaxation:

$$\begin{array}{c}{T}_{min}{e}^{-z_0^i}\left[1-(z_i-z_0^i)+\frac{{(z_i-z_0^i)}^2}{2}\right]\le {\sigma }_i\le T_{max}{e}^{-z_0^i}\left[1-(z_i-z_0^i)\right]\end{array}$$

(26)

where $z_0^i=ln\left(m_{\mathrm{wet}}-\alpha T_{max}{t}_i\right)$. Since Equation (26) is an approximation of Inequality (25), it introduces approximation errors. However, as shown in [14], this approximation preserves feasibility with respect to the original constraints and satisfies engineering-level accuracy requirements, thereby effectively transforming the original nonconvex constraint into a convex form. Here, $m_{\mathrm{wet}}$ denotes the initial mass, and $z_0^i$ is the reference value at time $t_i$. To ensure consistency of $z\left(t\right)$, the following additional constraints are imposed:

$$ln\left(m_{\mathrm{wet}}-\alpha T_{max}{t}_i\right)\le z\left(t_i\right)\le ln\left(m_{\mathrm{wet}}-\alpha T_{min}{t}_i\right)$$

(27)

Through convexification, applying the change of variables in (18) and discretizing on Legendre–Gauss nodes via the integral pseudospectral scheme recasts the continuous dynamics as linear (affine) equalities in $(x_i,u_i,{\sigma }_i,z_i)$ (see (19)). On the constraint side, the $\parallel u_i\parallel \le {\sigma }_i$ together with a first-order Taylor approximation of the exponential term at a fixed reference $z_0$ for the thrust bound (see (26) and (27)) yields convex conic constraints. The performance index is linear in $\sigma$ after Gaussian quadrature (see (20)). Consequently, Problem $P0$ is reformulated as an SOCP, denoted $P1$:

$$\begin{array}{c}P1:\underset{t_f,T\left(t\right)}{\min }{\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}{w}_k{\sigma }_k\\ \mathrm{s}.\mathrm{t}.x_i\approx x_0+{\displaystyle \frac{t_f-t_0}{2}}{\displaystyle \sum _{k=1}^N}f(x_k,u_k,{\sigma }_k)A_{ik},i=1,\dots ,N\\ T_{min}{e}^{-z_0^i}\left[1-(z_i-z_0^i)+\frac{{(z_i-z_0^i)}^2}{2}\right]\le {\sigma }_i\le T_{max}{e}^{-z_0^i}\left[1-(z_i-z_0^i)\right]\\ \parallel u_i\parallel \le {\sigma }_i,ln\left(m_{\mathrm{wet}}-\alpha T_{max}{t}_i\right)\le z_i\le ln\left(m_{\mathrm{wet}}-\alpha T_{min}{t}_i\right)\\ r\left(t_0\right)=r_0,v\left(t_0\right)=v_0,m\left(t_0\right)=m_0,r\left(t_f\right)=0,v\left(t_f\right)=0,m\left(t_f\right)\ge m_{\mathrm{dry}}\end{array}$$

(28)

4. Numerical Demonstration

In this section, the proposed algorithm is applied to solve Problem P1. The simulation parameters are provided in

Table 1. Parameters of the rocket in 3DOF simulation.

Parameter	Value	Units
$m_0$	15,000	kg
$m_{dry}$	10,000	kg
${\mathbf{r}}_{\mathbf{0}}$	[0, 500, 500]	m
${\mathbf{v}}_{\mathbf{0}}$	[50, 0, 50]	m/s
${\mathbf{r}}_{\mathbf{f}}$	[0, 0, 0]	m
${\mathbf{v}}_{\mathbf{f}}$	[0, 0, 0]	m/s
$I_{sp}$	300	s
$T_{min}$	100	kN
$T_{max}$	250	kN
$g_0$	9.807	m/s2

. To accurately discretize the dynamics, Problem P2 is formulated using the integral Gaussian pseudospectral method on Legendre–Gauss (LG) nodes with N = 50 interior collocation points. The simulation environment is constructed in MATLAB R2023a using CVX and solved by the MOSEK solver with default primal/dual feasibility tolerances and iteration limits. Owing to the lossless convexification, no user-supplied initial guess is required. The terminal time is free and is determined via an outer-loop bisection method that iteratively solves the inner SOCP.

Under the aforementioned configuration, the algorithm successfully generates a fuel-optimal landing trajectory that satisfies all physical and boundary constraints (see Figure 1 and Figure 2). Figure 1 shows the rocket of position, velocity, and mass, while Figure 2 presents the rocket of the thrust components along each coordinate axis and its magnitude. As illustrated in both figures, the state variables (position, velocity, and mass) and the control input (thrust) evolve smoothly and remain strictly within their prescribed constraint bounds throughout the entire descent phase. This result strongly validates the effectiveness and numerical accuracy of the proposed method for high-precision powered descent guidance of rockets.

To verify the stability of the algorithm described in Section 3, a Monte Carlo simulation was performed. The rocket’s initial state, including position, velocity, and mass, was randomly generated as follows (within specified ranges to simulate real mission uncertainties)

$$\left\{\begin{array}{c}{r}_{x,0}={\epsilon }_1\times 100,\hfill \\ r_{y,0}={\epsilon }_2\times 100+500,\hfill \\ r_{z,0}={\epsilon }_3\times 100+500.\hfill \\ v_{x,0}={\epsilon }_4\times 25+50,\hfill \\ v_{y,0}={\epsilon }_5\times 25,\hfill \\ v_{z,0}={\epsilon }_6\times 25-50.\hfill \\ m_0={\epsilon }_7\times 500+15000,\hfill \end{array}\right.$$

(29)

where ${\epsilon }_1\sim {\epsilon }_7$ are random numbers generated in $[-1,1]$, $r_{x,0}$, $r_{y,0}$, and $r_{z,0}$ are the three coordinates of the initial position, $v_{x,0}$, $v_{y,0}$, and $v_{z,0}$ constitute the initial velocity, and $m_0$ denotes the initial mass of the rocket. Other parameters remain consistent with the previous numerical examples.

To evaluate the robustness and adaptability of the algorithm under varying initial conditions, we conducted 100 Monte Carlo trajectory simulations, with results shown in Figure 3, Figure 4 and Figure 5. Specifically, Figure 3 displays the terminal landing errors and fuel consumption across all trials; Figure 4 presents the time histories of the state variables—position, velocity, and mass—over the Monte Carlo simulations; and Figure 5 illustrates the thrust components along each coordinate axis and their magnitude throughout the descent.

In these simulations, the initial states—including position and other state components—were randomly perturbed according to a uniform distribution within prescribed bounds to emulate uncertainties present in real missions. All 100 trials successfully converged to feasible solutions and achieved high-precision landings, demonstrating that the proposed algorithm maintains excellent robustness and reliability even under significant initial state perturbations.

Furthermore,

Table 2. Statistics of the Landing conditions for 100 Monte Carlo Trials.

Parameter	Mean Value
Terminal position error	1.01 m
Terminal velocity	0.16 m/s
Fuel consumption	1263.81 kg

summarizes the terminal performance statistics from these 100 Monte Carlo runs: the average terminal position error is 1.01 m, the average terminal velocity is 0.16 m/s, and the average fuel consumption is 1263.81 kg. These results further confirm that the proposed algorithm not only achieves sub-meter landing accuracy but also maintains high fuel efficiency.

5. Conclusions

This paper addresses the fuel-optimal trajectory optimization problem for the rocket-powered descent phase and proposes a lossless-convexification framework based on an integral pseudospectral method. By constructing a global integration matrix, the method achieves high-accuracy integral discretization. Combined with lossless convexification, it reformulates the original nonlinear dynamics and nonconvex thrust constraints as an SOCP. The approach requires no initial guess and ensures trajectory optimality, numerical stability, and reliable solution quality. Extensive numerical simulations—including a Monte Carlo study with 100 randomized perturbations of the initial state—show that the algorithm consistently generates smooth, feasible trajectories with physically reasonable thrust profiles under significant uncertainty, demonstrating strong robustness and accuracy.

We note that the present work focuses on offline solver performance; real-time capability on flight-qualified hardware has not yet been verified. End-to-end online deployment will require dedicated solver development. Future work will extend the framework to multi-constraint landing missions (e.g., with glide-slope enforcement and thrust-pointing limits), provide comparisons with differential pseudospectral methods and SCP-based methods, and explore applications to multi-phase trajectory optimization and real-time guidance.