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On-Orbit Optimal Kinodynamic Planning for Low-Thrust Trajectory Maneuvers

Ibrahima Sory Sow, Juan Alvarez Padilla,

Nayana Suvarna, Fausto Vega

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Problem Statement

Challenges

Nonlinear dynamics
High-dimensionality
Numerical conditioning
Memory issues

Provide a low-thrust trajectory for orbital maneuvers

Optimal sampling-based planner

Optimal cost-to-go ~ LQR

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Planning Representation

Nonlinear Relative Keplerian Dynamics

where

Chaser spacecraft at starting state

Target Spacecraft (final state)

Circular Restricted 3 body problem

\ddot{p_x} = \frac{-\mu}{r_c^3}(r_t+p_x)+\dot{\theta}^2p_x+2\dot{\theta}(\dot{p_y} - p_y \frac{\dot{r}_t}{r_t}) + \frac{\mu}{r_t^2} + \frac{u_x}{m}

\ddot{p_y} = \frac{-\mu}{r_c^3}p_y + \dot{\theta}^2p_y - 2\dot{\theta}(\dot{p_x}-p_x \frac{\dot{r_t}}{r_t}) + \frac{u_y}{m}

\ddot{p_z} = \frac{-\mu}{r_c^3}p_z + \frac{u_z}{m}

Equations representing the motion of a satellite in the presence of two bodies. Assuming the satellite does not have a gravitational effect on the two bodies and that the two bodies follow circular

Rotating frame with normalized coordinates to improve the conditioning of the problem. Equilibrium points are the Lagrange points. Essentially want to plan a trajectory between the two bodies in this system.

6 dimensional state

U is the effective potential which is a function of the potential energy and kinetic energy

\ddot{x} -2\dot{y} = -\bar{U_x}

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Kinodynamic Planning Procedure

Distance

Algebraic Riccati

Optimal policy

Cost

Perez et al. "LQR-RRT*: Optimal sampling-based motion planning with automatically derived extension heuristics.”

Maybe a simple block diagram?

Kinodynamic planning ( run sim forward ⇒ new sample satisfying dynamics)

Do a quick schematic with arrows

LQR cost-to-go (put value function V(x) = x.T S x) + say where we use it + Linearization around current sample for rewiring

Motivation: shown that RRT’s efficiently explore the state space when the metric reflects the true cost-to-go
Optimal policy: pi*(x) = - K x

We plan to compare this approach with other flavor of RRT* and optimal sampling-based planners (state-of-the-art)
Using Euclidean distance as a distance metric results to poor exploration and slow convergence to the optimal solution

x_{\text{nearest}} = \arg\min_{v \in V} (v - x)^T S(v - x)

\pi^*(x) = -Kx

K = R^{-1}B(x_0, u_0)^T S

V(x) = x^T S x

J = \int_0^{\infty} x^T Q x + u^T R u

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Experiment Setup

Results

	R = 1	R = 10	R = 100
Sample size	486 samples	1353 samples	2738 samples
Tree size	357 nodes	1044 nodes	1913 nodes
Path size	42 nodes	60 nodes	175 nodes
Solve time	18.539 sec	88.453 sec	282.093 sec

Initial State

Goal State

Asteroid Center

Asteroid Radius

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Controls with a TVLQR controller

Time (s)

Thrust vector magnitude per Second (N/s)

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Live Demo