Rapidly Exploring Random Trees with Physics-Informed Neural Networks for Constrained Energy-Optimal Rendezvous Problems

Abstract

This article introduces physics-informed neural networks (PINNs) to the field of motion planning by utilizing a PINN framework as the steering function in the kinodynamic rapidly-exploring random tree (RRT*) algorithm. The goal of this paper is to show that PINN-based methods can be used successfully for aerospace motion planning applications. We test the RRT* algorithm coupled with PINN steering, what we call PINN-RRT*, by solving spacecraft energy-optimal motion planning problems governed by the Hill–Clohessy–Wiltshire (HCW) equations of motion and nonlinear equations of relative motion (NERM), where a deputy satellite must rendezvous with a chief satellite while avoiding spherical keep-out-zones and complying with an approach corridor. The particular PINN framework we employ approximates the solution of nonlinear two-point boundary value problems (TPBVPs), which must be solved to form connections between waypoints in the RRT* tree, via the Theory of Functional Connections (TFC). TFC enables the PINN to analytically satisfy the boundary conditions (BCs) of the TPBVP. Thus, the admissible solution search space of each nonlinear TPBVP is reduced to just the trajectories that already satisfy the BCs. Using our proposed approach, each energy-optimal TPBVP solution during the run-time of the PINN-RRT* algorithm was computed in centiseconds and with an average error on the order of machine epsilon for both the HCW and NERM dynamics.

Appendices

Appendix A: RRT* Example

An example of an iteration of the RRT* algorithm is shown by the collection of subfigures in Fig. 13. In order to describe the algorithm the example is walked through by pointing out the lines in Algorithms 1–3 where each step takes place. Once the tree is initialized, a for-loop that extends the tree for N iterations begins. Figure 13 displays the RRT* algorithm in action past the tenth iteration. At the start of an iteration a node, denoted as $x_\text {rand}$, is first sampled from the the C-space (line 3 of Algorithm 1). Its nearest neighbor $x_\text {nearest}$ is then found and an edge that is steered from $x_\text {rand}$ towards $x_\text {nearest}$ is constructed (lines 4 and 5 of Algorithm 1). The reader should be aware that the steering threshold is $\bar{c}=2$ for the example shown in Fig. 13, i.e., $\texttt {Cost}(\{ x_{10},x_\text {new} \})=2$. A set of vertices $X_\text {near}$ that can reach $x_\text {new}$ without exceeding the metric shown in Eq (5) is then computed using the Near function (line 6 of Algorithm 1). Note that the metric shown as Eq (5) is larger than $\bar{c}=2$ in our example. This explains why many edges have a cost greater than 2. Next, A parent node of $x_\text {new}$ is then selected by calling ChooseParent (line 7 of Algorithm 1).

The ChooseParent procedure (Algorithm 2) searches through the vertices in the set $X_\text {near}$ and determines the vertex that reaches $x_\text {new}$ while incurring the least amount of accumulated cost. This minimum cost incurring vertex is denoted as $\varvec{x}_\text {min}$ and the edge that connects $x_\text {min}$ to $x_\text {new}$ is $\{x_\text {min},x_\text {new}\}$. From subfigure e it is clear $x_\text {min}=x_2$ because the accumulated cost of edges $\{x_\text {init},x_2\}$ and $\{x_2,x_\text {new}\}$ equals 10, which is less than the accumulated cost of any other path towards $x_\text {new}$. After the minimum vertex is found, $\{x_\text {min},x_\text {new}\}$ is checked for collisions with obstacles. If $\{x_\text {min},x_\text {new}\}$ is collision-free then $x_\text {new}$ and $\{x_\text {min},x_\text {new}\}$ are added to the tree (lines 9 and 10 of Algorithm 1). Note that this is why we stated earlier that Fig. 13 is an example of some iteration past the tenth iteration and not definitively that it shows iteration eleven. Many other nodes sampled at previous iterations could have resulted in building edges that collided with the obstacles, which would not have been added to the tree. Thus, the number of iterations the RRT* algorithm has performed is always greater or equal to the number of nodes in the tree. RRT* then attempts to rewire the nodes in $X_\text {near}$ (Algorithm 3). If the edge $\{x_\text {new},x_\text {near}\}$ has less total incurred cost than the edge that connects $x_\text {near}$ with its parent, $\{x_\text {parent},x_\text {near}\}$, then $\{x_\text {parent},x_\text {near}\}$ is removed from the tree and $\{x_\text {new},x_\text {near}\}$ takes its place. Lastly, the best solution is found by connecting the goal region, an individual state in this case, with the path in the tree that accumulates the least cost (not shown in Algorithm 1).

Fig. 13

Simple RRT* example

Appendix: B NERM TPBVP

Table 8 Free final time energy-optimal TPBVP governed by NERM dynamics

Table 8 shows the free final time energy-optimal TPBVP governed by the NERM dynamics. Time appears implicitly through the true anomaly rate $\dot{f}$, true anomaly acceleration $\ddot{f}$, and chief radius $r_c$. Time can be related to the mean anomaly following Kepler’s laws of planetary motion, given the orbital elements of the chief’s orbit. With the mean anomaly at a particular time instance, the true anomaly f at that time instance can be found using the equation of center. Again, using Kepler’s laws of planetary motion, $r_c$ can be found once f is known at a particular time instance. Then, transforming the chief orbital elements with f into state vectors in the geocentric equatorial frame ($\varvec{r}_{\text {gef}_c}$ and $\varvec{v}_{\text {gef}_c}$) allows $\dot{f}$ and $\ddot{f}$ to be found:

$$\begin{aligned} \begin{aligned} \dot{f}&= \frac{\left( \varvec{r}_{\text {gef}_c} \times \varvec{v}_{\text {gef}_c} \right) }{\vert \varvec{r}_{\text {gef}_c} \vert ^2} \\ \ddot{f}&= \frac{-2 \left( \varvec{r}_{\text {gef}_c} \times \varvec{v}_{\text {gef}_c} \right) \vert \varvec{v}_{\text {gef}_c} \vert }{\vert \varvec{r}_{\text {gef}_c} \vert ^3}. \end{aligned} \end{aligned}$$

Appendix C: NERM TPBVP Constrained Expressions

The position and velocity in Table 8 can be coupled together because velocity is simply just the derivative of position. Thus, only the constrained expressions for position elements are needed, and its derivative is the constrained expression for velocity elements. Since the constraints in Table 8 are identical to those in Table 2, the constrained expression for the position and velocity for the TPBVP in Table 8 are Eqs (25) and (26). Unlike, the TPBVP in Table 1, the costates cannot be coupled with the state. Hence, they need their own expression. Luckily, they are not constrained in Table 8. Thus, their constrained expressions are just their respective free functions, i.e., linear combinations of Chebyshev orthogonal polynomials:

$$\begin{aligned} \lambda _{r_i}(t) = \lambda _{r_i}(z)&= \varvec{P}^\intercal (z) \varvec{\beta }_{r_i} \\ \dot{\lambda }_{r_i}(t) = b^2 \lambda '_{r_i}(z)&= b^2 \varvec{P}^\intercal (z) \varvec{\beta }_{r_i} \\ \lambda _{v_i}(t) = \lambda _{v_i}(z)&= \varvec{P}^\intercal (z) \varvec{\beta }_{v_i} \\ \dot{\lambda }_{v_i}(t) = b^2 \lambda '_{v_i}(z)&= b^2 \varvec{P}^\intercal (z) \varvec{\beta }_{v_i}. \end{aligned}$$

Plugging the constrained expressions into the system ODEs shown in Table 8 and following the steps outlined in Sect. 3, then allows the TFC solution to be obtained.

Appendix D: LERM TPBVP

Table 9 shows the fixed final time energy-optimal TPBVP governed by the LERM dynamics. Solving this TPBVP with TFC can be accomplished using the constrained expressions shown and referenced in the previous Appendix subsection.

Table 9 Fixed final time energy-optimal TPBVP governed by LERM dynamics