Program implemented to compute optimal solution for the soft landing problem based on a lossless convexification.
Trajectory generated by the script.
Implemented soft landing powered decent guidance optimization problem based on the paper
Lossless Convexification of Non-Convex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem.
B. Acikmese, J. M. Carson III, and L. Blackmore. IEEE Transactions on Control Systems Technology, Volume 21, Issue 6 (2013).
One of the most critical aspects of a planetary landing mission is the powered-descent phase, which terminates with a soft touchdown on the surface.
However, the problem is a nonconvex finite-horizon optimal control problem in its original form because of the control constraints.
Source of nonconvexity:
- Thrust vectors with a nonzero minimum and maximum on the thrust magnitude
- Thrust vector pointing direction
- Time-varying mass in the dynamics
The planetary soft landing problem can be formulated as a prioritized optimization problem as follows.
The solution of the soft landing problem is obtained by solving Problems 1 and 2. The motivation for this two-step prioritization approach is quite intuitive. The primary goal of planetary landing problem considered in this paper is to land a vehicle as close to a given target as possible, i.e., to minimize the landing error as in Problem 1. There can be multiple optimal solutions for this problem, hence we have a second step where we find the minimum error solution that consumes the least fuel, as in Problem 2.
Its proposed with the following relaxed minimum-error and fuel-optimal control problems that are convex relaxations of Problems 1 and 2.
- Solve the relaxed minimum-landing-error guidance problem (Problem 3) to obtain dP3 (final optimal position). If no solution exists, return infeasible.
- Solve the relaxed minimum-fuel guidance problem to specified range (Problem 4) with dP3 calculated earlier.
The lander is modeled as a lumped mass with a thrust vector for control, and its dynamics are described by,
- ω = (ω1, ω2, ω3) ∈ R3 is the vector of planet’s constant angular velocity
- g ∈ R3 is the constant gravity vector, and
- α > 0 is a constant that describes the fuel consumption (mass depletion) rate.
Euler integration ( w[t+1] = w[t] + dt * f(t,w[t]) ) is used for solving the dynamics equation along each time step, which are added in the relevant constraints for the cvxpy problem.
Python library CVXPY [ https://www.cvxpy.org/ ] used for optimization,
Script is written with python (Version: 3.6) on linux. Additional modules required :
- numpy (tested with Version: 1.18.4 )
- matplotlib (tested with Version: 2.1.1 )
- CVXPY (tested with Version: 1.2)
- Verify and install required modules
- run
python soft_landing_optimizer.py.