SDSS-V Algorithms: Fast, Collision-free Trajectory Planning for Heavily Overlapping Robotic Fiber Positioners

A rendering of the SDSS-V RFP is shown in the upper inset of Figure 1. The lower panel of this figure shows the relative packing of positioners in the focal plane when viewed edge on. Fiber positioning in the focal plane is achieved through the rotations of two arms about two axes. The alpha arm (lower arm, indicated in gold) rotates about the alpha axis. The alpha axis is collinear with the lower body of the robot. The beta arm (upper arm, indicated in blue) rotates about a beta axis near the edge of the alpha arm. The distance between the alpha axes of neighboring robots we call the pitch. The beta arm carries the optical fiber and risks collisions with the beta arms of neighboring robots. Alpha arms cannot collide with one another. The left two robots in the lower panel of Figure 1 are in an orientation of closest approach between alpha arms, showing the tight clearance between them. Beta arms cannot collide with alpha arms, as they exist in different planes along the optical axis.

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To visualize how a robot positions a fiber, a focal plane view is helpful. Figure 2 shows the focal plane projection of a fiber positioner centered at the robot's xy base position (x_b, y_b). Robot arms and rotation axes are indicated on the figure. The alpha arm length (l_α ) is the distance from the alpha axis to beta axis (7.4 mm for SDSS-V). The beta arm length (l_β ) is the distance from the beta axis to the fiber center (15 mm for SDSS-V). The robot may position a fiber anywhere in the annular patrol zone through the specification of the alpha angle (θ_α ) and beta angle (θ_β ). This coordinate conversion is

$$\begin{eqnarray}&&\begin{array}{c}\left(\begin{array}{c}{x}_{{\rm{f}}}\\ {y}_{{\rm{f}}}\end{array}\right)\,=\,\left(\begin{array}{c}{x}_{{\rm{b}}}\\ {y}_{{\rm{b}}}\end{array}\right)\,+\,\left(\begin{array}{cc}\cos {\theta }_{\alpha } & \cos ({\theta }_{\alpha }+{\theta }_{\beta })\\ \sin {\theta }_{\alpha } & \sin ({\theta }_{\alpha }+{\theta }_{\beta })\end{array}\right)\left(\begin{array}{c}{l}_{\alpha }\\ {l}_{\beta }\end{array}\right),\end{array}\end{eqnarray} \tag{ 1 }$$

where (x_f, y_f) corresponds to the position of the fiber in the xy focal plane.

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The physical limits of travel for the SDSS-V positioner are roughly between −5° and 365° for both the alpha and beta axes. This permits each axis slightly more than one full rotation between hard stops. Operationally, we enforce slightly more restrictive limits of travel during path planning: each axis must remain in the range [0°, 360°).

Notice that the orientation in Figure 2 resembles a human's view of a right arm, and increasing θ_β beyond 180° would resemble a left arm. Every point in a positioner's workspace is physically achievable by both a left and right arm conformation. We will limit targeting to right-armed configurations, though positioners may potentially take on left-armed configurations while maneuvering. This choice does not impact a positioner's ability to cover the available workspace, but it will eliminate an alternative positioning option during target assignment. The elimination of the left arm targeting configuration serves two practical purposes in SDSS-V.

The first purpose is to avoid degenerate solutions. When the FPS system is deployed, a fiber-viewing metrology system is used to determine a robot's orientation by measuring the centroid of a back-lit fiber. If the option of a left arm configuration is eliminated, there is only one possible solution for (θ_α , θ_β ) given a measurement of (x_f , y_f ). This is important when considering a malfunctioning robot that cannot report its absolute position. The robot array will be powered down during science integrations to minimize heat at the focal plane. If we enforce that robots are only powered down in right arm configurations, this will minimize confusion that could arise due to a malfunction after a power cycle. While robots remain powered, their absolute positions are frequently reported.

The second purpose is of mechanical, rather than mathematical, nature. We achieve higher fiber-positioning accuracy when 0° <= θ_β  <= 180°. This is due to an interaction between the beta axis preload spring and the direction of torque imparted by the optical fiber itself. In right arm configurations, the fiber torque and preload work together. In left arm configurations, the fiber torque and preload oppose each other, leading to slightly higher variance in absolute positioning when operating in especially cold weather.

To obtain full focal plane coverage the robots are spaced at a pitch of l_α  + l_β in a hexagonal array (22.4 mm for SDSS-V). This allows a robot's neighbor to patrol its central exclusion zone, leading to heavily overlapping patrol zones between a robot and its neighbors. Figure 3 shows the overlapping patrol zones for an array of 19 SDSS-V positioners, indicating the areas covered by one, two, three, and four fibers in the focal plane. Scaling up the hexagonal array to hundreds of positioners, the majority of the focal surface will be covered by three or more fibers. Only the perimeter of the hexagonal array will have single-fiber coverage with some gaps. The axes in the lower left of the figure indicate the orientation of θ_α with respect to the hexagonal grid.

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The SDSS-V specifications for l_α , l_β , and pitch were chosen to maximize the patrol area for each positioner under the present constraints of the telescope and spectrograph. The BOSS spectrograph has a 500 fiber capacity, dictating the total number of robots. Evenly distributing 500 robots over the telescope's field of view yields a desired pitch of 22.4 mm. Given the pitch, a 7.4 mm alpha arm length grants the largest possible reach while ensuring that alpha arms cannot physically collide with one another. With pitch and alpha arm length selected, a 15 mm beta arm length is required to ensure gap-less coverage of the focal plane. Hörler et al. (2018) provide further discussion on the general topic of RFP design and arrangement, including the densely packed arrangement selected for SDSS-V.

All physical interference between robots in the grid happens between beta arms. We use a relatively simple strategy to represent a beta arm geometry in Cartesian space. The beta arm is modeled by two elements: (1) a line segment and (2) a collision buffer (σ_cb). The beta line segment is constructed between two points (x_f, y_f), and (x_e, y_e). The former point is given in Equation (1). The coordinates of the latter point (elbow point) describe the position of the beta axis:

$$\begin{eqnarray}&&\begin{array}{c}\left(\begin{array}{c}{x}_{{\rm{e}}}\\ {y}_{{\rm{e}}}\end{array}\right)=\left(\begin{array}{c}{x}_{{\rm{b}}}\\ {y}_{{\rm{b}}}\end{array}\right)+\left(\begin{array}{c}{l}_{\alpha }\cos {\theta }_{\alpha }\\ {l}_{\alpha }\sin {\theta }_{\alpha }\end{array}\right).\end{array}\end{eqnarray} \tag{ 2 }$$

The orientation of the beta line segment will depend on a positioner's (θ_α , θ_β ) coordinates, which vary with motion. A positioner's base position (x_b, y_b) and arm lengths (l_α , l_β ) remain fixed.

We measure the proximity between two beta arms by computing the minimum distance between beta arm segments. The procedure for this calculation is provided in Appendix and was adapted from an online source. ⁹ We label the minimum distance between beta arm segments for positioners i and j as D_ij .

The collision buffer (σ_cb) specifies a radius from the beta line segment that safely encloses the physical extent of the beta arm. We refer to this buffered line segment as the collision envelope, which is a cigar-shaped area. We say robots i and j have collided when

$$\begin{eqnarray}&&{D}_{{ij}}\leqslant 2{\sigma }_{\mathrm{cb}}\end{eqnarray} \tag{ 3 }$$

Figure 4 constructs the geometric representation of the beta arm segment (dashed gold line), σ_cb (gold arrow), and collision envelope (blue area) for two colliding RFPs. The red area in the figure indicates the colliding area between the two positioners, where the inequality in Equation (3) is satisfied.

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The two-dimensional collision envelope we have drawn is conservative when considering the three-dimensional shape of the beta arm. Certain robot orientations would allow the head of a beta arm to hover directly above a neighbor's beta axis without suffering a physical collision between beta arms. Consider the view in Figure 1 and imagine the central positioner's beta arm floating rightward. Such an orientation would not be allowed in the focal-plane-projected view of Figure 4, as the area around the elbow joint is encompassed within the collision envelope. The extent of the collision buffer was chosen to completely allocate the space below a beta arm as belonging to the optical fiber, eliminating any chance of physical interaction between a fiber and a neighboring robot body. Admittedly, reserving all space below the beta arm is perhaps unnecessarily greedy, as the fiber is tugged closer to the beta arm body when the arm is extended. The allowance of some amount of safe physical overlap between beta arms could be accomplished by either reducing the length of the beta arm segment to yield space around the elbow or modeling the beta arm as a segment or curve in three-dimensional space. Permitting beta arm overlap would provide additional real estate for both target assignment and path generation, which might prove beneficial. However, we find the collision buffer as described in this section sufficient for SDSS-V survey operations, and the algorithms we present assume these chosen constraints.

We first introduce the parameters, metrics, and general ingredients before describing the procedures of the GC and MC algorithms. We focus our analysis using SDSS-V geometries, although the definitions that follow are generic and can be applied to any layout of RFPs with similar kinematics.

Arm Angles: θ_α , θ_β —The alpha and beta axis angles define the physical orientation of a robot. The range of motion for each axis is [0°, 360°), as described in Section 3.1.

Arm Lengths: l_α , l_β —The lengths of the alpha and beta arms. In this work, we use SDSS-V positioner values of l_α  = 7.4 mm and l_β  = 15 mm.

Collision Buffer: σ_cb—This parameter sets the relative size of the beta arm, a radial distance from the beta arm line segment (see Section 3.3). We vary σ_cb between 1.5 and 3.5 mm. A 1.5 mm σ_cb value exactly encloses the physical envelope of an SDSS-V RFP. A σ_cb between 2 and 2.5 mm maintains a generous amount of positioner separation during motion, providing an extra safety buffer for both (1) real-time RFP positional uncertainties along trajectories and (2) path postprocessing and simplification. We touch on these two topics further in Section 6. In deployment for the SDSS-V FPS, values of σ_cb greater than 2.5 mm are unreasonably large. However, this parameter space will show the potential of these algorithms to achieve high-efficiency solutions in regimes more crowded than our specific application.

Figure 5 shows the minimum, mean, and maximum σ_cb we investigate, emphasizing the crowding effect with increasing σ_cb. A σ_cb of 1.5, 2.5, and 3.5 mm correspond to roughly 6%, 12%, and 20% of the focal plane area occupied by the collision envelopes of positioners when packed into an SDSS-V layout containing 547 positioners.

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Robot Centers:  b _i  = (x_b, y_b)—Base position in the grid for positioner i. For this work, we consider hexagonal grids with a pitch of 22.4 mm, matching the pitch of the SDSS-V FPS. We vary the number of positioners in the grid.

Robot Neighbors: N_i —The set of robot neighbors for positioner i. Two robots are considered neighbors only if they risk collision with one another: when their center-to-center distance is less than twice the sum of their arm lengths and collision buffer σ_cb. If G is the set of all robots in the grid, the set of neighbors for positioner i is

$$\begin{eqnarray}&&{N}_{i}=\{j\in G,j\ne i| \parallel {{\boldsymbol{b}}}_{i}-{{\boldsymbol{b}}}_{j}\parallel \leqslant 2({{\rm{l}}}_{\alpha }+{{\rm{l}}}_{\beta }+{\sigma }_{\mathrm{cb}})\}.\end{eqnarray} \tag{ 4 }$$

Initial Coordinates: ${{\boldsymbol{\theta }}}_{i}^{I}=({\theta }_{\alpha },{\theta }_{\beta })$—The initial alpha and beta angle coordinates for positioner i, a starting point for the routine. Initial coordinates must be noncolliding.

Destination Coordinates: ${{\boldsymbol{\theta }}}_{i}^{D}=({\theta }_{\alpha },{\theta }_{\beta })$—The desired alpha and beta axis angles for a positioner i. The algorithms seek to drive a positioner from its initial coordinates to its destination coordinates while avoiding collisions with neighbors. Destination coordinates must be noncolliding.

Current Coordinates: ${{\boldsymbol{\theta }}}_{i}^{{\rm{C}}}=({\theta }_{\alpha },{\theta }_{\beta })$—The current alpha and beta axis angles for a positioner i. These coordinates evolve with the program step, and their history defines the path followed by a positioner.

Source Coordinates: ${{\boldsymbol{\theta }}}_{i}^{{\rm{S}}}=({\theta }_{\alpha },{\theta }_{\beta })$—The source coordinates specify the alpha and beta axis angles for a positioner to receive light from an astronomical source. In this work, a robot's source coordinates are drawn uniformly from the annular fiber patrol zone. Source coordinates must be noncolliding. All source coordinates are right armed (Section 3.1). If an initial source coordinate assignment creates a collision with a previously assigned robot, source coordinates are redrawn iteratively until the collision vanishes. In a forward-path solution, the source coordinates for positioner i are the destination coordinates, ${{\boldsymbol{\theta }}}_{i}^{D}$. In a reverse-path solution, the source coordinates serve as the initial coordinates, ${{\boldsymbol{\theta }}}_{i}^{I}$.

Angular Step: Δθ—The angular step parameter specifies the step size of the routine: a maximum angular perturbation of a robot's current coordinates θ ^C at each program step. This parameter may be converted into a time step, Δt, by

$$\begin{eqnarray}&&{\rm{\Delta }}t=\displaystyle \frac{{\rm{\Delta }}\theta }{\dot{\theta }},\end{eqnarray} \tag{ 5 }$$

where $\dot{\theta }$ is the angular speed of the positioner's alpha and beta axes (30° s^–1 for SDSS-V RFPs). Smaller angular step values produce more densely sampled paths at the cost of increased program runtime.

Maximum Steps—The maximum number of steps the routine will run for. For this work, we set a limit of 1000°/Δθ steps. This corresponds to 1000 degrees of motion, or roughly 33 s of motion assuming a 30 degree s⁻¹ angular speed for an SDSS-V RFP.

Minimum Approach Distance—This distance designates the closest allowed approach between any two beta arm segments in the routine. For a positioner i with a set of neighbors N_i , we enforce the inequality

$$\begin{eqnarray}&&\min \{j\in {N}_{i}| {D}_{{ij}}\}\gt 2{\sigma }_{\mathrm{cb}}+\mathrm{MD},\end{eqnarray} \tag{ 6 }$$

where MD is a bound on the maximum displacement for a beta arm in a single program step. We choose a conservative value for MD: the displacement of the fiber (x_f, y_f) with the beta arm at full extension (θ_β  = 0), when rotated 2Δθ about the positioner's central axis:

$$\begin{eqnarray}&&\mathrm{MD}=({{\rm{l}}}_{\alpha }+{{\rm{l}}}_{\beta })\sin (2{\rm{\Delta }}\theta ).\end{eqnarray} \tag{ 7 }$$

The inclusion of the maximum displacement term protects against "tunneling collisions," where one positioner may completely jump through a collided orientation in a single step. Smaller values of Δθ permit closer approaches between positioners and thus allow more options for maneuvering.

Cost: C_i —The cost metric is minimized throughout the routine for each positioner. It measures a distance between a positioner's destination coordinates (${{\boldsymbol{\theta }}}_{i}^{D}$) and current coordinates (${{\boldsymbol{\theta }}}_{i}^{{\rm{C}}}$). Note this metric is not equal to the Euclidean distance between the fiber and its destination in the xy focal plane. For a positioner i, we define the cost as

$$\begin{eqnarray}&&{C}_{i}=\parallel {{\boldsymbol{\theta }}}_{i}^{{\rm{C}}}-{{\boldsymbol{\theta }}}_{i}^{D}\parallel .\end{eqnarray} \tag{ 8 }$$

When the cost is zero, the positioner has arrived at its destination.

Energy: E_i —Energy is a measure of local crowding: the sum of inverse square distances to a set of neighbors. We define the energy for a positioner i with neighbors N_i as

$$\begin{eqnarray}&&{E}_{i}=\displaystyle \sum _{j\in {N}_{i}}{\left(\displaystyle \frac{1}{{D}_{{ij}}}\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

This metric is only used in the MC algorithm.

Phobia—Phobia is a scalar between 0 and 1, used only in the MC algorithm. It represents the weighting of relative importance between the energy metric E_i and the cost metric C_i . A weight of 0 produces a positioner insensitive to crowding. A phobia weight of 1 produces a positioner only sensitive to crowding and it will seek to separate from neighbors above all else.

Greed—Greed is a scalar between 0 and 1 used only in the MC algorithm. It represents how much weight a positioner will place on taking the best move when minimizing the E_i or C_i metrics. A greed value of 1 introduces no stochasticity in a positioner's move selection sequence, while a greed value of 0.5 resembles a random, undirected walk. A greed value of 0 produces a robot that will remain stationary.

Neighbor Encroachment—Neighbor encroachment is a proximity threshold that is only considered in the MC algorithm. A positioner i with neighbors N_i will remain stationary only while its cost C_i  = 0 and the following inequality is satisfied:

$$\begin{eqnarray}&&\min \{j\in {N}_{i}| {D}_{{ij}}\}\gt 2{\sigma }_{\mathrm{cb}}+3\mathrm{MD}\end{eqnarray} \tag{ 10 }$$

This provides a similar constraint to the minimum approach distance outlined in Equation (6), but is sensitive over a slightly longer distance. This will cause any robot currently at rest at its destination to be kicked awake by any neighboring robot entering its horizon.

The GC algorithm is a stepwise process. At each iteration, each robot i in a grid of G is visited in turn and presented with a set of nine options from which to select its next state. The options consist of the combination of {−Δθ, 0, Δθ} perturbations applied to the current alpha and beta coordinates ${{\boldsymbol{\theta }}}_{i}^{{\rm{C}}}$. Perturbations are modified if limits of travel are violated or destination coordinates are overshot. The robot will choose the option that both (1) minimizes its cost (Equation (8)), and (2) satisfies the minimum approach criterion (Equation (6)). The routine terminates when either (1) all robots have reached their destination coordinates (∑_i∈G C_i  = 0) or (2) the maximum iteration limit is reached.

The MC algorithm is an extension of the GC algorithm. In contrast to the GC algorithm, the MC algorithm selects subsequent states probabilistically rather than greedily, where the level of stochasticity is controlled by the greed parameter. Additionally, we construct a control policy that jointly minimizes both the cost and energy metrics, where relative weighting between cost and energy is controlled by the phobia parameter. Greed, phobia, cost, and energy are described in Section 4.1.

At each program step, a positioner must first select whether or not to consider a move. If the positioner is at its destination coordinates and the inequality in Equation (10) is satisfied, it will remain stationary until the next program step, otherwise, it will consider a move.

If a positioner considers a move, it must next choose a metric to minimize: either cost or energy. The probability of selecting the energy metric is equal to the phobia parameter. With a metric selected, a positioner will visit each of the nine next-state candidates in random order, where these options are the combinations of {−Δθ, 0, Δθ} perturbations applied to the current alpha and beta coordinates. Perturbations are modified if limits of travel are violated or destination coordinates are overshot. As each state is visited, a positioner will choose to accept or reject the proposed state. A positioner accepts the proposed state with a probability equal to the greed parameter if the following two criteria are met: (1) this state represents the minimum value of the selected metric seen at this step (by the measure of Equations (8) or (9)), and (2) this state satisfies the minimum approach criterion (Equation (6)). If no state is selected, it defaults to remain in its current state until the next program step.

The routine terminates when either (1) all robots have reached their destination coordinates (∑_i∈G C_i  = 0) or (2) the maximum iteration limit is reached.

The GC algorithm represents perhaps the simplest possible control law in the framework we have described. The MC algorithm extends upon the GC algorithm through the inclusion of two additional features: (1) the injection of stochasticity in a robot's decision-making process, and (2) a mechanism for robots to sense and avoid crowded spaces. The greed and phobia parameters provide a tuning knob for these MC features. When greed approaches 1 and phobia approaches 0, the MC algorithm becomes identical to the GC algorithm. In the analysis that follows, we fix greed and phobia parameters to 0.9 and 0.3. This choice of MC parameter setting is not necessarily intended to represent an optimal tuning, but rather to provide a point of comparison between the two approaches.

A complete pseudocode implementation for the GC/MC routine is provided in Appendix.

The performance of the GC and MC algorithms can be highly dependent on the direction of solved motion. Consider two array configuration states between which we want to find collisionless paths for all robots. The first state ${{\boldsymbol{\theta }}}_{i}^{{\rm{F}}}=(0^\circ ,180^\circ )$ represents a "folded" state in which all robot arms are retracted and aligned in a lattice-like orientation. The second state is a randomized source coordinate configuration ${{\boldsymbol{\theta }}}_{i}^{{\rm{S}}}$ that approximates an orientation of RFPs positioned to receive light from astronomical sources in a field. When solving the forward path from initial coordinates θ ^F to destination coordinates θ ^S a vast number of positioners will deadlock and never their destination. However, if we swap the initial and destination coordinates and solve the reverse path θ ^S →  θ ^F, we find our routines yield near-perfect convergence of robots to their desired destinations.

To demonstrate the forward and reverse solutions, we use the GC algorithm, a 19 positioner grid, and a step size Δ_θ  = 0.5°. We set σ_cb = 3.5 mm to simulate an abnormally large level interference between positioners. Noncolliding source coordinates for each positioner are selected randomly from their respective patrol zones. Initial coordinates are set such that the robots begin their journey from a folded position, ${{\boldsymbol{\theta }}}_{i}^{I}={{\boldsymbol{\theta }}}_{i}^{{\rm{F}}}$. Destination coordinates are set to be the source coordinates ${{\boldsymbol{\theta }}}_{i}^{D}={{\boldsymbol{\theta }}}_{i}^{{\rm{S}}}$.

Figure 6 shows three snapshots in robot-travel time when forward motion is propagated according to the GC algorithm. The first panel shows the initial folded state of the positioners, the second panel shows an intermediate state in the routine, and the third panel shows the final state. Stars indicate the source coordinate locations, circles indicate the fiber position at the end of the beta arm. When the fiber is aligned with the source, the positioner has reached its destination. Robots at their destination are colored tan, robots in motion are colored blue. Curved streaks drawn behind the fiber indicate the path the fiber has followed through previous program steps.

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In this example, only 4 of 19 positioners arrive at their destination before the routine's iteration limit is reached. The remaining positioners are in a deadlocked state where all progress is halted due to the inability of robots to navigate beyond their collective blockade.

The next demonstration uses identical configurations to those of the previous example with the following modification: initial coordinates are set to the source coordinates ${{\boldsymbol{\theta }}}_{i}^{I}={{\boldsymbol{\theta }}}_{i}^{{\rm{S}}}$, and destination coordinates are set to the folded configuration ${{\boldsymbol{\theta }}}_{i}^{D}={{\boldsymbol{\theta }}}_{i}^{{\rm{F}}}$. When solved with the GC algorithm, robots will begin from a source coordinate orientation and carve out paths toward a folded configuration. In this case, we are solving exactly the same problem as the previous example (a path between a folded and source orientation) by simply reversing the direction of robot-arm propagation.

Figure 7 shows the results of the reverse GC solve, a sequence analogous to that presented in Figure 6. Positioners begin aligned with source coordinates and step toward a folded destination. In this experiment, we find that all positioners successfully navigate to their destinations after 12 s of motion. This result is striking when contrasted with the forward example in which only a small subset of positioners was able to obtain their desired destination. This pair of examples demonstrates that the reverse-path strategy is a surprisingly good approach.

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The irreversible behavior of the algorithm raises eyebrows (the authors' included), yet it provides unique leverage in solving our path-routing problem. Consider the two following points. First, paths are trivially reversible. If a reverse-path solution is found, we can obtain a forward-path solution by simply reflecting the trajectories of all robots. Second, by building all paths between any source coordinate state and a common folded state, transitioning between any two source coordinate states is also trivial given that reconfigurations always route through the folded state.

The reverse-path approach was discovered somewhat serendipitously through experimentation and motivated by the following thought experiment: is it easier to tie a complex knot or untie a complex knot? Presumably, the latter task is easier, and if one carefully recorded the steps taken while performing the latter task, then the reversal of those steps yields a solution to the former task. Regrettably, we do not have a theoretical justification as to why the reverse process works so well in the context of our problem, though we find the behavior interesting and provide some further discussion in Section 7. In further sections, we rely on statistics gathered from large numbers of simulations to gain confidence in the quality of our solutions.

Here we formulate a statistical measure of algorithmic efficiency for the layout described in Section 5.1. We repeat GC forward/backward pair analysis in a set of 5000 trials, where source coordinates are randomized between trials. Figure 8 shows the resulting histogram of positioner deadlock frequencies for both strategies. The forward strategy will always result in a deadlock that typically involves half of the positioners, whereas the reverse strategy only rarely suffers a deadlock. In these rare cases of reverse-path deadlock, only a few positioners are involved. The chance of deadlock in the reverse solution is small but non-negligible.

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For the reverse solution to be viable, we require a deadlock-free path for every positioner in the grid. The full reconfiguration sequence from source coordinates A to subsequent source coordinates B requires two steps: (step 1) the array moves from A to an intermediate folded state, then (step 2) the array moves from the folded state to B. If any single positioner deadlocks during reverse-path generation, the array has not made it to the transition state, and thus the set of paths cannot be used. Having a strategy for eliminating deadlocks is a necessary feature for the reverse-path solver.

We adopt a brute-force method of deadlock resolution. If a grid is deadlocked, we randomly select a single deadlocked robot, replace its source coordinates, and rerun the path generator. Replacing a source coordinate results in the loss of the original astronomical source. In large grids of positioners, deadlocks are typically isolated in small groups throughout the array. For these cases, we may simultaneously replace one robot from each deadlocked group, which minimizes the number of iterations required of the path generator and improves the total runtime.

This brute-force replacement strategy is illustrated in Figure 9. The top three panels show a sequence of reverse-solved motion that results in a four-positioner deadlock. The lower three panels show the results obtained after replacing a single positioner's source coordinates with a new random position. The array now completely converges to a folded orientation in 12 s of motion, thus resolving four wedged robots with a single replacement.

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We define an efficiency for the reverse GC and MC algorithms in terms of the required number of replacements to solve a grid. For a grid with a total number of positioners n_G requiring source coordinate replacements for a subset of positioners n_R , the efficiency is

$$\begin{eqnarray}&&\mathrm{efficiency}=\displaystyle \frac{{n}_{G}-{n}_{R}}{{n}_{G}}.\end{eqnarray} \tag{ 11 }$$

Note that we count n_R as the number of robots requiring a source coordinate replacement. If a single robot is iteratively tried with multiple sets of source coordinates, we count this as a single replacement. This measure of efficiency is the ratio of astronomical targets obtained to astronomical targets assigned. For the example shown in Figure 9, the efficiency is (19 − 1)/19 ∼ 0.95.

In our trials, we generally (but not necessarily) find that the efficiency is higher than the initial convergence of the grid, where we call convergence the ratio of positioners achieving the destination to the total number of positioners in the grid. In the case shown in Figure 9, the initial convergence (top panel sequence) is (19 − 4)/19 ∼ 0.79. The efficiency increase with respect to convergence is due to the fact that a single replacement will often resolve more than one deadlock. We use efficiency (rather than convergence) to measure algorithmic performance as it is a more relevant statistic in the context of astronomical surveys.

In a cocktail party setting, navigation becomes more challenging as the free space in the room shrinks. The σ_cb parameter is a proxy for either beta arm relative size or collision safety distance, and as σ_cb increases, available space for motion in the grid decreases. Here we investigate the path-routing performance of the reverse-solve GC and MC algorithms for a range of σ_cb values between 1.5 and 3.5 mm to simulate varied levels of crowding.

We use a grid of 547 positioners and set a step size Δ_θ  = 0.1 deg. This step size limits robot-arm perturbations (Equation (7)) to less than 100 μm at each iteration. We run 2000 trials at each σ_cb for each algorithm, amounting to 36,000 trials in total. Initial coordinate assignments are randomized in all trials. We assign folded destination coordinates θ ^D  = (10°, 170°) for all positioners. These destination coordinates leave the positioners with ±10° of free travel in both alpha and beta axes when they arrive at their destination. This choice of parking spot benefits the MC algorithm, in which positioners at rest may be kicked awake when a neighbor encroaches (Equation (10)), allowing for a larger set of evasive options in these situations. The parameter settings for this experiment are summarized in Table 1.

Figure 10 shows the trend of trial-averaged efficiency versus σ_cb for the MC and GC algorithms. The GC algorithm maintains a mean efficiency greater than 0.998 for σ_cb < 2 mm; the MC algorithm maintains a mean efficiency is greater than 0.998 for σ_cb < 3 mm. An efficiency of 0.998 corresponds to a single replacement in a grid of 547 positioners. With either routine, only a fraction of a percent of astronomical targets will be lost in the small σ_cb regimes.

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The MC algorithm outperforms the GC algorithm in terms of efficiency at all σ_cb values. For both algorithms, efficiency declines monotonically with σ_cb. Under the highest crowding conditions (σ_cb = 3.5 mm) mean efficiencies of 0.993 and 0.968 are measured for the MC and GC algorithms. In an environment three times more crowded than the SDSS-V FPS design, we are limiting source replacement to less than a percent using the MC algorithm. This result suggests that very crowded RFP arrays may be feasible from a collision-avoidance standpoint.

Figure 11 shows the minimum observed efficiency as a function of algorithm and σ_cb in the set of 36,000 trials. The trends are similar for both algorithms. For σ_cb ≤ 2 mm, the minimum efficiency remains above 0.98; beyond σ_cb = 2 mm, the minimum efficiency trends steadily downward. For the largest σ_cb, we see minimum efficiencies around 0.92 for the GC algorithm and 0.97 for the MC algorithm.

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Figure 12 presents box plots of measured efficiencies for the MC (upper panel) and GC (lower panel) algorithms across trials. Boxes indicate the interquartile range, and whiskers capture data within 3/2 the interquartile range below and above the low and high quartiles. Outliers are plotted as diamonds. Generally, the variance of efficiency increases with increasing σ_cb. This figure indicates a median target loss on the subpercent level for the MC algorithm over the full range of σ_cb investigated. Subpercent target loss for the GC algorithm is typical when σ_cb < 2.5 mm.

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Reconfiguration times for RFP arrays are usually measured in seconds. However, when integrated over years of a survey, these seconds add up to significant hours of observational overheads. Here we analyze the reconfiguration times from the simulation suite described in Table 1 using SDSS-V positioner velocities. In this section, we measure the "fold time," which is the time required to move between a source (or astronomical target) orientation to a folded state. The total field-to-field reconfiguration time is twice this value, as the positioner array must move first to the folded state before moving to the next desired orientation.

Figure 13 shows the mean fold time as a function of σ_cb and algorithm. The GC outperforms the MC algorithm by a factor of ∼2, depending slightly on σ_cb. The longer path lengths observed in the MC algorithm are due to two effects: (1) the injection of random motion along its path, tuned by the greed parameter, and (2) the additional policy of E_i (energy) minimization, tuned by the phobia parameter. Recall that, as greed approaches 1 and phobia approaches 0, the MC algorithm becomes identical to the GC algorithm. In this sense, the fold time curve for the GC algorithm in Figure 13 represents a lower bound for the MC algorithm. By varying greed and phobia, one might tune the MC algorithm to produce an optimal balance between efficiency and fold time in the context of an astronomical survey. Here we have fixed greed and phobia to 0.9 and 0.3 to provide a point comparison between the two methods.

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Figure 14 shows box plots for fold time across σ_cb and between algorithms. Boxes indicate the interquartile range, and whiskers capture data within 3/2 of the interquartile range below and above the low and high quartiles. Data falling outside whiskers are indicated with diamonds. For the GC algorithm, we see median reconfiguration times less than 30 s for SDSS-V positioners across the full range of σ_cb. For the MC algorithm, we expect reconfiguration times closer to 45 s for the lower half of σ_cb values, and reconfiguration times nearing a minute for the upper half of the σ_cb value range. The GC algorithm generally achieves the SDSS-V benchmark goal of a 30 s RFP reconfiguration time, whereas the MC would require greed and phobia parameter adjustments to reach this benchmark.

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Navigating through a moving crowd is more effective when one takes small steps in the right direction, rather than waiting for the opportunity of a big step to present itself. This analogy holds true for the algorithms we have presented. However, a small step comes at the price of an extended runtime.

A fast-running routine is desirable for at least two reasons. A fast path generator allows for dynamic decision-making throughout a night's observations, as optimal plans are subject to change on short notice. Another benefit of a fast path generator is that it may be incorporated in the optimization stages of survey planning and design. In a large astronomical survey, field assignment for millions of objects is a computationally intensive task, so if path verification is required at this stage, it must have a low computational overhead. Carrying out path planning during field assignment is desirable as it will inform which specific targets ultimately get observed, providing an opportunity for vetting long before a field is visited by the telescope.

We measure two distinct runtimes in our routines. The first runtime is the source replacement runtime τ_sr. This is the computation time accumulated during the source replacement procedure described in Section 5.2. The source replacement procedure requires iterative runs of the path generator to find valid initial coordinates. τ_sr is largely an overhead during target assignment, as once initial coordinates have been assigned and paths shown to converge, they do not need to be recomputed.

The second runtime τ_pg is the computational time required to build paths from a set of initial coordinates that are known to converge. Observing conditions during nightly operations may require that RFP trajectories remain flexible on short notice. For example, the airmass of observation may require slight adjustments to a precomputed set of initial coordinates. Or perhaps a robot malfunctions and must be taken offline during the night while remaining in a position that obstructs neighbors. In cases where we envision small adjustments to initial coordinates, we expect the overall runtime of path generation to be much closer to τ_pg than τ_sr.

For this experiment, we vary Δ_θ , σ_cb, and the algorithm. We run 300 trials at each unique parameter combination, yielding a total of 18000 trials. An analysis of the resulting trends informs a smart decision on step size: a value that produces a reasonable balance between efficiency and runtime. The parameters for this simulation are listed in Table 2.

Figure 15 shows the effects of Δ_θ on efficiency for both GC and MC algorithms. Efficiency declines as both Δ_θ and σ_cb increase. At small σ_cb (1.5–2 mm), there is little variation in efficiency, so large step sizes (0.75–1 deg) seem permissible. However, as σ_cb increases, smaller step sizes are required to remain near the high end of efficiency. This is especially true for σ_cb = 3.5 mm, where efficiency is strongly influenced by the step size. The efficiency degradation with increased step size is due to the maximum displacement (MD) factor entering in Equation (6), which defends against moves during which robots may undetectably jump through a colliding orientation in a single step.

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Figure 16 presents the average observed runtimes across σ_cb , Δ_θ , and algorithm. Here we only consider results for which the trial-averaged efficiency is greater than 0.9. The two left-hand panels show the average runtime τ_pg. Over the full range of Δ_θ , the runtime decreases exponentially, ranging between ∼1–25 s for the GC algorithm. The MC algorithm runtime requires roughly twice the runtime of the GC algorithm, with a slight dependence on σ_cb . The right-hand panels of Figure 16 show the average runtime τ_sr, which includes the iterative process of source coordinate replacement (Section 5.2). When including the source replacement procedure, the runtime scales by roughly an order of magnitude over the full range of σ_cb. This scaling is largely due to the decrease in overall efficiency as σ_cb increases. As σ_cb increases, the chance of deadlock increases, and so the required number of iterations of the path generator increase.

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The trends indicated in Figures 16 and 15 provide the information we need to select decent Δ_θ settings for each algorithm over a range of σ_cb. Table 3 contains a summary of results for each algorithm under some reasonable parameter choices. The table contains runtime, fold time, and efficiency metrics over the full range of σ_cb, providing a concise summary of the performance of the algorithms we have tested over a range of parameter space.

In nightly operations, SDSS-V FPS path computations τ_pg will typically require only a few seconds, even when choosing a conservative σ_cb = 2.5 mm. Should a sudden RFP reconfiguration be desired, the overhead due to path-planning computations will be negligible when compared to the duration of robot motion. The computational speed of these algorithms provides SDSS-V with a nimble observing system that suffers almost no additional overhead when unscheduled reconfigurations are requested at a moment's notice.

In this section, we provide a brief analysis of how algorithmic performance scales with the number of positioners. We use relatively few trials (6000) and choose a midrange crowding level of σ_cb = 2.5 mm. The complete set of simulation parameters are described in Table 4. The results here will give a general feel for behavior in grids more massive than the typical 547 robot array we have focused on thus far.

Figure 17 plots the mean efficiency against grid size for the simulations described in Table 4. The shaded region around each line represents the 95% confidence interval of the mean. For grids larger than the SDSS-V array, the efficiency remains relatively constant. For smaller grids, a direct comparison of efficiency versus grid size should be taken with a grain of salt for two reasons: (1) smaller grids have a larger fraction of positioners without neighbors, and (2) efficiency is computed relative to the total number of positioners in a trial, where smaller grids contain fewer positioners. Despite this, we see that efficiency is not strongly affected by the total number of positioners.

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Figure 18 plots the mean τ_sr and τ_pg runtimes for each algorithm at various grid sizes for the simulation set described in Table 4. For each line in the plot, a 95% confidence interval of the mean is indicated by the shaded region. Runtime τ_pg increases linearly with grid size for both MC and GC algorithms, as is typical for distributed control algorithms. Runtime τ_sr does not show a tight linear response, as larger grids will experience a higher frequency of deadlock events and thus require a higher number of source replacement iterations to solve a grid.

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