Probabilistic Guidance of a Swarm Deployed from the Back ...publish.illinois.edu/saptarshibandyopadhyay/files/...swarm. From the science perspective, the key requirement on the swarm

70th International Astronautical Congress, 21–25 October 2019 Washington, D.C., USA.

IAC–19–C1.7.5

Probabilistic Guidance of a Swarm Deployed from the Back Shell of aMars Spacecraft

Saptarshi BandyopadhyayJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

[email protected]

Jean-Pierre de la CroixJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

[email protected]

Issa NesnasJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

[email protected]

David BayardJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

[email protected]

Soon-Jo ChungCalifornia Institute of Technology, Pasadena, CA, USA

[email protected]

Fred HadaeghJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA

[email protected]

This paper presents a probabilistic guidance algorithm for a notional swarm of assets deployed from the backshell of the Mars spacecraft. Such a swarm could provide valuable science data, with large spatiotemporalvariation, from the Martian surface. Our probabilistic swarm guidance algorithm maximizes the coveragearea of the conceptual swarm while uniformly distributing the assets on the Martian surface and guaranteeingstrong-connectivity of the swarm’s communication network topology. Numerical simulations demonstrate theeffectiveness and versatility of our probabilistic swarm guidance algorithm.keywords: swarm, guidance, robots, Mars, sensor network,

1. Introduction

The National Aeronautics and Space Administra-tion (NASA) is considering multiple missions to Marsfor a potential Mars Sample Return (MSR) cam-paign.1 At the same time, a number of mission con-cepts (e.g., Prandtl-M glider,2 Tensegrity lander3) arebeing developed to fit in the empty space in the back-shell of Mars spacecraft as secondary payloads. Theseassets could be deployed as secondary payloads afterthe primary payload has separated from the back-shell of Mars spacecraft. We envisage that a swarmof such secondary payloads (100–1000 assets) couldbe deployed from the backshell of Mars spacecraft fordistributed science on Mars (see Fig. 1). In this paperwe present a guidance algorithm for such a notionalswarm.

From the science perspective, the key requirementon the swarm distribution would be to maximizeits coverage area while staying away from some re-gions. In addition, the swarm has to maintain astrongly connected communication network topologyso that the science data collected by each agent canbe sent to Earth via the backshell of Mars spacecraft.These requirements and constraints are captured bya desired swarm distribution on the Martian surface.The objective of our probabilistic swarm algorithm isto determine the release time, angle of deployment,and initial velocity of each asset so that the swarmachieves the desired distribution on the Martian sur-face, despite the assets being subject to significantenvironmental disturbances. Moreover, the assets donot have position or attitude sensors and would fall

IAC–19–C1.7.5 Page 1 of 8


Fig. 1: An artist’s concept of the deployment of aswarm of assets from the backshell of Mars space-craft

towards the Martian surface in an open-loop mannerafter being deployed from the Mars spacecraft. Thislack of feedback control makes this swarm guidanceproblem fundamentally different from existing prob-abilistic swarm guidance algorithms in the literature.For example, probabilistic swarm guidance using ho-mogeneous or inhomogeneous Markov chains4–6 oroptimal transport7 requires position information ofthe assets. Our probabilistic swarm guidance algo-rithm captures the transition probabilities from dif-ferent release times, angles of deployment, and initialvelocities to locations on the Martian surface, andthen maximizes the probability of the swarm achiev-ing the desired distribution. This novel probabilisticswarm guidance algorithm is the main contributionof this paper.

This paper is organized as follows. The concep-tual science objectives of the Martian swarm are dis-cussed in Section 2. Section 3 presents the possiblehardware assets that could be used for this swarm.The desired swarm distribution is designed in Sec-tion 4. Our probabilistic swarm guidance algorithmand numerical simulations are presented in Section 5.More numerical simulations are presented in Section6 and the paper is concluded in Section 7. Here Nand R represent the sets of natural numbers and realnumbers respectively.

2. Possible Science Objectives

A number of interesting science observations, withlarge spatiotemporal variation, could be carried outby a swarm of 1000 assets on the Martian surface.Here we present some possible science objectives ofsuch a swarm:

• Temperature and pressure profiles

• Wind speed and wind cycle

• Distributed Methane measurement

• Water cycle between soil and atmosphere

• Interior properties using miniature seismographs

• Soil properties using miniature spectrometer

• Gradiometry

• Dynamics of polar ice caps

• Sensor localization network for future missions

The key requirements on the spatial distributionof the conceptual swarm from the science perspectiveare:

• The coverage area of the swarm should be max-imized in order to maximize the spatiotemporalrange of the data collected by the swarm.

• The communication network topology of theswarm should be strongly connected so that thescience data collected by each agent can be sentto Earth via the backshell of Mars spacecraft.

• The swarm must stay away from the primarypayload (the Martian rover) and other sensitiveareas on the Martian surface for planetary pro-tection purposes.

These requirements and constraints are capturedwhile designing the desired swarm distribution on theMartian surface.

Fig. 2: Possible swarm assets, with increasing rangeafter deployment



3. Possible Hardware Assets

A number of concepts are being developed to fit inthe empty space in the backshell of Mars spacecraftas possible secondary payloads, as shown in Fig. 2.These potential hardware concepts include:

• Deep Space 2 microprobe8

• Printable spacecraft9

• Pop-Up Flat Folding Explorer Robot (PUFFER)robot10

• Tensegrity lander3

• Prandtl-M glider2

• MARSdrop microlander11

In order to simplify the dynamics of these swarmassets, we use the ballistic trajectory dynamics inthis paper. Let xi(t) = {xi(t), yi(t), zi(t)} representthe position of the ith asset in an inertial coordinateframe fixed on the Martian surface. For given initialconditions in position xi(0) and velocity xi(0), thedynamics of the assets is given by:

xi(t) = {0, 0,−gM} , [1]

where gM = 3.7 ms−2 represents the gravitational ac-celeration on Mars. We assume that the maximuminitial velocities that can be imparted to an asset atthe time of deployment by the deployment mecha-nism in the Mars spacecraft is ±40 ms−1 in X and Yaxes. We also assume the the assets experience a ve-locity disturbance at the time of deployment, whichis bounded by ±5 ms−1 in all three axes.

4. Desired Swarm Distribution

We assume that N ∈ N assets will be deployed assecondary payloads. In this section, we generate thedesired distribution of the swarm.

Let rcomm represent the maximum communicationradius of each asset, i.e., each asset can communicatewith any other asset that less than rcomm distanceaway from it. In this paper, we assume that rcomm =60 m.12

The theoretical bound on the area density of a un-firmly distributed swarm that grantees that the com-munication network topology is strongly connected isgiven by:13

limN→∞

P(πNr2comm − logN ≤ c) = e−e−c

, [2]

where c ∈ R. Since this bound is not tight for finitenumber of agents, comparison of Monte Carlo sim-ulation results and theoretical bounds for differentnumber of assets N is shown in Fig. 4. These re-sults show that the swarm distribution must have anarea density of

√Nrcomm ≥ 1.7 in order to guaran-

tee strong connectivity of its communication networktopology.

During the Entry Descent Landing (EDL) se-quence, the Mars rover separates from the backshellof Mars spacecraft at 1.6 − 2 km above the Martiansurface and the Mars spacecraft is falling downwardsat a speed of approximately 100 ms−1.14 A nominaltrajectory of the backshell of Mars spacecraft, afterseparation, is shown in Fig. 3(a).

Assume that a swarm of 1000 assets are randomlyreleased, with uniform randomness in release timesand initial conditions. The trajectories of the swarmassets are shown in Fig. 3(b). Note that the assets areclustered around the central region where the mainspacecraft landed, as shown in Fig. 3(c)–(e). More-over, such a swarm non-uniformly samples the cover-age area of approximately 2.3×105 m2 and a numberof peripheral assets are not connected to the domi-nant communication network. The red line in Fig.3(e) shows the boundary of the strongly connectedsub-group of the swarm. Clearly, a well-designedswarm guidance algorithm should be able to achievea larger coverage area with better distribution of as-sets. This problem is the main focus of this paper.

The desired distribution of the swarm (µd) on theMartian surface that satisfies the area density re-quirement of

√Nrcomm ≥ 1.7 is given by Fig. 3(f). It

covers an area of approximately 1.3 × 106 m2 on theMartian surface.

5. Probabilistic Swarm Guidance Algorithm

Motivated by the probabilistic swarm guidance al-gorithm using inhomogeneous Markov chains,5 we de-sign a Markov matrix to capture the transition prob-abilities from different release times, angles of deploy-ment, and initial velocities to locations on the Mar-tian surface, and then maximize the probability of theswarm achieving the desired distribution. Our prob-abilistic swarm guidance algorithm consists of threesteps:

• Step 1: Designing bins with appropriate initialconditions.

• Step 2: Computing transition probabilities foreach bin.



(a) Main Spacecraft Trajectory (b) Trajectories of the swarm as-sets

(c) Swarm on Martian surface

(d) Connected assets on Martiansurface

(e) Swarm distribution on Mar-tian surface

(f) Desired swarm distribution(µd)

Fig. 3: Swarm assets assumed to be randomly released after backshell separation

Fig. 4: Monte Carlo simulation results and theoreti-cal bounds show the fraction of the swarm that isstrongly connected, for different number of assetsN

• Step 3: Maximizing probability of achieving thedesired distribution.

5.1 Step 1: Designing Bins:

The workspace on the Martian (i.e., approximately±1000 m on both axes from the main spacecraft land-ing site) is partitioned into cells as shown in Fig.3(e)–(f). Each cell is of the size rcomm × rcomm andthere are a total of ncell = 1089 cells on the Mar-tian surface. Let C[j] represent the jth cell, for all

j ∈ {1, . . . , ncell}.The trajectory of the main spacecraft is discretized

into time steps of 0.5 sec as shown in Fig. 5(a).There are a total of ntime = 32 discrete releasetime. Let T [k] represent the kth release time, forall k ∈ {1, . . . , ntime}.

Let us define ncell × nbin bins, where the binB[k, j] encodes the initial velocities for an asset, re-leased at time T [k], to reach the center of the cellC[j] in the absence of disturbances. During thisstep, the initial conditions for all bins B[k, j], wherek ∈ {1, . . . , ntime} and j ∈ {1, . . . , ncell}, are com-puted. If the initial conditions in a bin exceed themaximum initial velocity threshold, then those binsare considered invalid. The trajectories from validbins for a few release times are shown in Fig. 5(b)–(f).

5.2 Step 2: Computing Transition Probabilities:

Let the Markov matrix M ∈ R(ntime×ncell)×(ncell)

capture the transition probabilities from bins (encod-ing release times and initial conditions) to cells on theMartian surface in the presence of disturbances. Forexample, the element M [(k − 1)× ncell + j, `] in theMarkov matrix encodes the probability that an assetreleased from bin B[k, j] (i.e., at time T [k] with ini-tial conditions to reach cell C[j]) actually reaches cell



(a) Release Times (b) Trajectories from bins at 0 sec (c) Trajectories from bins at 2 sec

(d) Trajectories from bins at 4.5sec

(e) Trajectories from bins at 7 sec (f) Trajectories from bins at 9.5sec

Fig. 5: Trajectories from valid bins for a few release times

(a) k = 1, j = 305 (b) k = 2, j = 711

(c) k = 3, j = 834 (d) k = 4, j = 353

Fig. 6: Visualization of the Markov row M [(k− 1)×ncell + j, :] for a few bins, where the red boundarydenotes cell C[j]

C[`] in the presence of disturbances, i.e.,

M [(k − 1)× ncell + j, `] = P (C[`]|B[k, j]) . [3]

During this step, all the elements in the Markov ma-trixM for valid bins are computed using Monte Carlosimulations. Thus the bins encode deterministic ini-tial conditions for the assets, but the probabilisticfinal location of the assets on the Martian surface is

captured by the Markov matrix. The Markov rowsfor a few valid bins are shown in Fig. 6.

5.3 Step 3: Maximizing Probability of Achieving theDesired Distribution:

Let the variable A ∈ Z1×(ntime×ncell)≥ represent the

assignment of the N assets to the ntime × ncell bins,where Z≥ represents the set of non-negative integers

and A1(ntime×ncell)×1 = N . The probabilistic swarmdistribution, if agents are released as per the assign-ment A is given by µs = AM . The desired swarmdistribution µd ∈ R1×ncell is given in Fig. 3(f). Ourobjective is to minimize the difference between µs

and µd using the following optimization problem:

minA∈Z1×(ntime×ncell)

≥

DL1 (µs,µd) [4]

subject to µs = AM , [5]

A1(ntime×ncell)×1 = N . [6]

Here DL1 represents the L1 distance between the twodistributions. This optimization problem is solvedduring this step.

The trajectories of the swarm assets deployed us-ing our probabilistic swarm guidance algorithm areshown in Fig. 7. Note that the assets are uni-formly distributed over a large region of approxi-mately 1.3 × 106 m2 on the Martian surface, whichis ≈ 5 times larger than the coverage area obtained



(a) Desired swarm distribution(µd)

(b) 3D View (c) Top View (Swarm on Martiansurface)

(d) Connected Assets on MartianSurface

(e) Probability Distribution onMartian Surface

Fig. 7: Swarm assets, deployed using our probabilistic swarm guidance algorithm, achieve the desired distri-bution

from random deployment in Fig. 3. Moreover, theswarm’s communication network topology is stronglyconnected. Thus, our probabilistic swarm guidancealgorithm achieves the desired distribution of assetson the Martian surface.

6. Numerical Simulations

Here we show that any desired distribution canbe achieved using our probabilistic swarm guidancealgorithm, if the desired location on the Martian sur-face can be reached by the swarm assets. Fig. 8 andFig. 9 show that the swarm can easily incorporatecomplex shapes and no-go zones.

7. Conclusion

In this paper, we presented a probabilistic swarmguidance algorithm for potential assets deployed fromthe back shell of a Mars spacecraft. Our probabilis-tic swarm guidance algorithm maximizes the coveragearea of the swarm while uniformly distributing the as-sets on the Martian surface and guaranteeing strong-connectivity of the communication network topology.Numerical simulations with 1000 assets and 60 mcommunication radius show that the swarm couldcover a large region of approximately 1.3× 106 m2 on

the Martian surface. Numerical simulations also showthat the probabilistic swarm guidance algorithms canbe used to achieve complex shapes with no-go zones.Such a swarm on the Martian surface could yield valu-able science data over a large spatiotemporal range.

Acknowledgments

This research was carried out at the Jet PropulsionLaboratory, California Institute of Technology, undera contract with the National Aeronautics and SpaceAdministration. The information in this document ispre-decisional and is provided for planning and dis-cussion purposes only. c©2019 California Institute ofTechnology. All rights reserved.

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(b) 3D View of Swarm Trajecto-ries

(c) Top View (Swarm on Martiansurface)



Fig. 8: Swarm assets achieve the new desired distribution of shape H


(b) 3D View of Swarm Trajecto-ries

(c) Top View (Swarm on Martiansurface)



Fig. 9: Swarm assets achieve another new desired distribution of shape Θ, with no-go zones



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Probabilistic Guidance of a Swarm Deployed from the Back ...publish.illinois.edu/saptarshibandyopadhyay/files/...swarm. From the science perspective, the key requirement on the swarm