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Successive Convexification for 6-DoF Mars Rocket Powered Landing with Free-Final-Time

Michael Szmuk, Behcet Acikmese

arXiv:1802.03827v1math.OC

TL;DR

Minimum-time 6-DoF rocket landing is a nonlinear, non-convex guidance problem complicated by discretization accuracy and reference-trajectory initialization. The paper uses successive convexification to solve a free-final-time formulation through convex SOCP sub-problems. Simulations report convergence with simple, poor initial guesses in fewer than 15 iterations.

  • Problem

    Minimum-time 6-DoF rocket landing is nonlinear and non-convex, with solution validity dependent on discretization accuracy and initialization requiring a suitable reference trajectory.

    Source: §I · p. 2, II.D. Problem Statement · p. 5

  • Method

    The method iteratively linearizes and discretizes the free-final-time landing problem into SOCP sub-problems, initialized with a simple dynamically inconsistent trajectory.

    Source: §I · p. 2, §II · p. 3

  • Results

    Simulations showed convergence in fewer than 15 successive convexification iterations, even with a simple trajectory and poor time-of-flight guess.

    Source: §V · p. 14

  • Takeaways & Limitations

    The authors present the formulation as a promising option for future autonomous landing guidance applications.

    Source: §I · p. 2

Abstract

from arXiv · show

In this paper, we employ successive convexification to solve the minimum-time 6-DoF rocket powered landing problem. The contribution of this paper is the development and demonstration of a free-final-time problem formulation that can be solved iteratively using a successive convexification framework. This paper is an extension of our previous work on the 3-DoF free-final-time and the 6-DoF fixed-final-time minimum-fuel problems. Herein, the vehicle is modeled as a 6-DoF rigid-body controlled by a single gimbaled rocket engine. The trajectory is subject to a variety of convex and non-convex state- and control-constraints, and aerodynamic effects are assumed negligible. The objective of the problem is to determine the optimal thrust commands that will minimize the time-of-flight while satisfying the aforementioned constraints. Solving this problem quickly and reliably is challenging because (a) it is nonlinear and non-convex, (b) the validity of the solution is heavily dependent on the accuracy of the discretization scheme, and (c) it can be difficult to select a suitable reference trajectory to initialize an iterative solution process. To deal with these issues, our algorithm (a) uses successive convexification to eliminate non-convexities, (b) computes the discrete linear-time-variant system matrices to ensure that the converged solution perfectly satisfies the original nonlinear dynamics, and (c) can be initialized with a simple, dynamically inconsistent reference trajectory. Using the proposed convex formulation and successive convexification framework, we are able to convert the original non-convex problem into a sequence of convex second-order cone programming (SOCP) sub-problems. Through the use of Interior Point Method (IPM) solvers, this sequence can be solved quickly and reliably, thus enabling higher fidelity real-time guidance for rocket powered landings on Mars.

Nomenclature

The paper formulates minimum-time 6-DoF rocket landing as a free-final-time problem and solves it through successive convexification. Its model describes rigid-body motion under gravity, with negligible aerodynamics and a single gimbaled engine.

Nomenclature

  • The nomenclature defines the inertial and body frames, vehicle states, commanded thrust, and key constraints used to specify the landing problem.For example, FI denotes the inertial NED frame, FB the body frame, and tf the time-of-flight.Source: Successive Convexification · p. 1

I. Introduction

  • The method initializes with a dynamically inconsistent reference, then repeatedly linearizes and discretizes the problem into SOCP sub-problems around the previous solution.The converged solution is structured to satisfy the original nonlinear dynamics, kinematics, and constraints, without a line search on time-of-flight.Source: §I · p. 2

II. Problem Description

  • The paper develops and demonstrates a 6-DoF free-final-time formulation, solved with successive convexification, for non-convex minimum-time 6-DoF rocket landing, extending prior 3-DoF free-final-time and 6-DoF fixed-final-time minimum-fuel work [1, 2].Source: §I · p. 2, §II · p. 3

II.A. Dynamics and Kinematics

  • The vehicle model assumes constant gravity, negligible aerodynamic forces, and constant inertia and center-of-mass position as mass depletes.A single gimbaled rocket engine provides thrust; the formulation can be adapted to multiple non-gimbaled motors with minor modifications.Source: II.A. Dynamics and Kinematics · p. 3

II.B. State and Control Constraints

The trajectory is constrained by limits on mass, position, attitude, angular rate, thrust magnitude, and gimbal angle. These include both convex constraints and a non-convex lower thrust bound.

II.B. State and Control Constraints

  • The vehicle must remain above its dry mass and within an upward-facing glide-slope cone centered at the landing-site origin.The cone angle is measured from the horizontal and lies in [0°, 90°).Source: II.B. State and Control Constraints · p. 4
  • Vehicle tilt and angular rate are limited by maximum allowable values θmax and ωmax.The tilt angle is defined between the X-axes of the body and inertial frames.Source: II.B. State and Control Constraints · p. 4
  • Commanded thrust magnitude lies in [Tmin, Tmax], while the gimbal angle lies in [0, δmax].The upper thrust bound and gimbal-angle constraint are convex, but the lower thrust bound is non-convex.Source: II.B. State and Control Constraints · p. 4

II.C. Boundary Conditions

The chosen boundary conditions fix the initial mass, position, velocity, and angular rate, while leaving initial attitude free. At the final time, position, velocity, attitude, and angular rate are fixed, but mass remains free.

II.C. Boundary Conditions

  • Initial mass, position, velocity, and angular rate are fixed, while initial attitude and thrust are left unconstrained.Final position, velocity, attitude, and angular rate are fixed, while final mass remains free; final thrust is constrained to point along the body X-axis to null out terminal torque.Source: II.C. Boundary Conditions · p. 5

II.D. Problem Statement

The paper poses a continuous-time, non-convex free-final-time optimal control problem for 6-DoF rocket landing. It minimizes time-of-flight over commanded thrust, subject to the stated dynamics, state and control constraints, and boundary conditions.

II.D. Problem Statement

  • The objective is to minimize time-of-flight in the original non-convex continuous-time free-final-time problem, with commanded thrust and final time as decision variables.The constraints comprise boundary conditions, dynamics and kinematics, state constraints, and control constraints.Source: II.D. Problem Statement · p. 5

III. Convex Formulation

The formulation converts the non-convex free-final-time landing problem into a convex fixed-final-time subproblem, solved repeatedly by successive convexification. Its converged solution satisfies the original dynamics, kinematics, and constraints when initialization is proper and the original problem is feasible.

III. Convex Formulation

  • The free-final-time problem is transformed into a convex fixed-final-time SOCP subproblem for repeated solution.Normalized trajectory time fixes the final normalized time at unity.Source: §III · p. 6

III.A.1. Dynamics and Kinematics

  • The state vector has 14 components and the control vector has 3, with dynamics and kinematics expressed as a nonlinear function of both.The function maps R14 × R3 to R14.Source: III.A.1. Dynamics and Kinematics · p. 6
  • Applying the chain rule expresses the dynamics in normalized trajectory time τ, using σ as the time-dilation coefficient.The normalized-time dynamics are then rewritten in terms of τ.Source: III.A.1. Dynamics and Kinematics · p. 6
  • A first-order Taylor approximation about a reference state, control, and dilation coefficient linearizes the nonlinear dynamics for the convex formulation.The reference trajectory comprises state, control, and dilation coefficient.Source: III.A.1. Dynamics and Kinematics · p. 6

III.A.2. State Constraints

  • The thrust-magnitude lower-bound constraint is the only remaining source of non-convexity and is linearized using a first-order Taylor approximation.The approximation replaces the left side of the constraint equation.Source: III.A.2. State Constraints · p. 7

III.B. Discretization

The continuous-time problem is discretized at evenly distributed points in normalized trajectory time, with first-order-hold controls and discrete dynamics linking successive states.

III.B. Discretization

  • The trajectory is discretized at K evenly distributed points in normalized time to form a finite-dimensional parameter optimization problem.The discretization is performed with respect to τ.Source: III.B. Discretization · p. 7
  • A first-order hold expresses the control over each interval using the controls at its two endpoints.This choice is made to preserve more feasibility.Source: III.B. Discretization · p. 7
  • The discrete-time dynamics relate each state to the next through the state transition and control terms, dilation coefficient, and an offset.The relation is enforced for each interval, while remaining state and control constraints are imposed at every discretization point.Source: III.B. Discretization · p. 7

III.C. Successive Convexification

Successive convexification solves a non-convex problem through a sequence of related convex subproblems. Trust regions and penalized virtual control address boundedness and artificial infeasibility during convergence.

III.C. Successive Convexification

  • Successive convexification solves a non-convex optimization problem indirectly through a sequence of related convex subproblems.The subproblems are solved in succession.Source: III.C. Successive Convexification · p. 8

III.C.1. Trust Regions

  • Soft trust regions around the previous iterate are added to the cost to mitigate unbounded solutions in subproblems.Unboundedness can arise when an iterate's linearization admits an unbounded cost.Source: III.C.1. Trust Regions · p. 8
  • The linearization path is defined by integrating from the previous iterate's state using its control over each discretization interval.This specifies the reference state, control, and dilation coefficient used for linearization.Source: III.C.1. Trust Regions · p. 8

III.C.2. Virtual Control

  • Virtual control is added to the dynamics to mitigate artificial infeasibility, which often occurs in early iterations with a poor initial guess.It is heavily penalized and acts when necessary; a converged solution with negligible virtual-control cost indicates dynamic feasibility.Source: III.C.2. Virtual Control · p. 8

III.C.3. Convex Sub-Problem

Time normalization makes the repeated convex subproblem fixed-final-time: its final normalized time is unity, and the original time-of-flight is replaced by the dilation coefficient.

III.C.3. Convex Sub-Problem

  • Because normalized final time is always unity, the fixed-final-time subproblem uses σi in place of the original time-of-flight tf.Source: III.C.3. Convex Sub-Problem · p. 9
  • The repeated optimization is summarized as Problem 2, a convex discrete-time fixed-final-time problem.Source: III.C.3. Convex Sub-Problem · p. 9
  • Problem 2 includes boundary conditions, state constraints, and trust regions among its listed constraints.The summary also references the earlier discrete dynamics and constraint equations.Source: III.C.3. Convex Sub-Problem · p. 9

III.C.4. Algorithm

The section defines the tolerances used for the algorithm’s exit condition and outlines the successive-convexification loop.

III.C.4. Algorithm

IV. Simulation Results

The simulations test two landing scenarios using the full 6-DoF algorithm and notional non-dimensional quantities. Both use the same discretization and iteration limit, with parameter and boundary-condition information tabulated.

IV. Simulation Results

  • The two scenarios are 2-D in-plane and 3-D out-of-plane maneuvers, both generated with the full 6-DoF algorithm.Notional non-dimensional quantities were used, so the simulations are not intended to match a real-world system.Source: §IV · p. 11
  • Both cases use 50 time discretization points and a maximum of 15 iterations.Source: §IV · p. 11
  • Tables 1 and 2 collect the simulation and algorithm parameters and boundary conditions common to both scenarios.Source: §IV · p. 11, Table 1 · p. 11

IV.A. 2-D In-Plane Example

The simulations demonstrate minimum-time landing trajectories in planar and out-of-plane scenarios, with convergence across varied time-of-flight initializations. The conclusion reports convergence from simple initial trajectories in fewer than 15 iterations and identifies further work on convergence analysis, timing, and aerodynamics.

IV.A. 2-D In-Plane Example

  • In the 2-D case, ten time-of-flight initializations from 1.0 to 10.0 UT yielded converged times within 0.01 UT of each other, by iteration six.The trajectory rides the maximum tilt limit in its first half and the angular-rate limit during most of its second half; both thrust bounds activate throughout.Source: IV.A. 2-D In-Plane Example · p. 11, Figure 2 · p. 12, Figure 1 · p. 12

IV.B. 3-D Out-of-Plane Example

V. Conclusion and Future Work

  • The method initializes with a dynamically inconsistent trajectory, then linearizes and discretizes each iterate into an SOCP subproblem, repeating until changes are small and virtual control subsides.Source: §V · p. 14
  • The algorithm converged in fewer than 15 iterations despite using a simple trajectory and a poor time-of-flight guess.Source: §V · p. 14
  • Future work will examine convergence properties, report real-time timing statistics, and incorporate simple aerodynamic effects.Source: §V · p. 14
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