# Study of excited $\Xi$ states in
$\psi(3686)\rightarrow{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$

M. Ablikim1, M. N. Achasov13,b, P. Adlarson73, R. Aliberti34, A.
Amoroso72A,72C, M. R. An38, Q. An69,56, Y. Bai55, O. Bakina35, I.
Balossino29A, Y. Ban45,g, V. Batozskaya1,43, K. Begzsuren31, N. Berger34, M.
Berlowski43, M. Bertani28A, D. Bettoni29A, F. Bianchi72A,72C, E.
Bianco72A,72C, J. Bloms66, A. Bortone72A,72C, I. Boyko35, R. A. Briere5, A.
Brueggemann66, H. Cai74, X. Cai1,56, A. Calcaterra28A, G. F. Cao1,61, N.
Cao1,61, S. A. Cetin60A, J. F. Chang1,56, T. T. Chang75, W. L. Chang1,61, G.
R. Che42, G. Chelkov35,a, C. Chen42, Chao Chen53, G. Chen1, H. S. Chen1,61, M.
L. Chen1,56,61, S. J. Chen41, S. M. Chen59, T. Chen1,61, X. R. Chen30,61, X.
T. Chen1,61, Y. B. Chen1,56, Y. Q. Chen33, Z. J. Chen25,h, W. S. Cheng72C, S.
K. Choi10A, X. Chu42, G. Cibinetto29A, S. C. Coen4, F. Cossio72C, J. J. Cui48,
H. L. Dai1,56, J. P. Dai77, A. Dbeyssi19, R. E. de Boer4, D. Dedovich35, Z. Y.
Deng1, A. Denig34, I. Denysenko35, M. Destefanis72A,72C, F. De Mori72A,72C, B.
Ding64,1, X. X. Ding45,g, Y. Ding33, Y. Ding39, J. Dong1,56, L. Y. Dong1,61,
M. Y. Dong1,56,61, X. Dong74, S. X. Du79, Z. H. Duan41, P. Egorov35,a, Y. L.
Fan74, J. Fang1,56, S. S. Fang1,61, W. X. Fang1, Y. Fang1, R. Farinelli29A, L.
Fava72B,72C, F. Feldbauer4, G. Felici28A, C. Q. Feng69,56, J. H. Feng57, K
Fischer67, M. Fritsch4, C. Fritzsch66, C. D. Fu1, Y. W. Fu1, H. Gao61, Y. N.
Gao45,g, Yang Gao69,56, S. Garbolino72C, I. Garzia29A,29B, P. T. Ge74, Z. W.
Ge41, C. Geng57, E. M. Gersabeck65, A Gilman67, K. Goetzen14, L. Gong39, W. X.
Gong1,56, W. Gradl34, S. Gramigna29A,29B, M. Greco72A,72C, M. H. Gu1,56, Y. T.
Gu16, C. Y Guan1,61, Z. L. Guan22, A. Q. Guo30,61, L. B. Guo40, R. P. Guo47,
Y. P. Guo12,f, A. Guskov35,a, X. T. H.1,61, W. Y. Han38, X. Q. Hao20, F. A.
Harris63, K. K. He53, K. L. He1,61, F. H. Heinsius4, C. H. Heinz34, Y. K.
Heng1,56,61, C. Herold58, T. Holtmann4, P. C. Hong12,f, G. Y. Hou1,61, Y. R.
Hou61, Z. L. Hou1, H. M. Hu1,61, J. F. Hu54,i, T. Hu1,56,61, Y. Hu1, G. S.
Huang69,56, K. X. Huang57, L. Q. Huang30,61, X. T. Huang48, Y. P. Huang1, T.
Hussain71, N Hüsken27,34, W. Imoehl27, M. Irshad69,56, J. Jackson27, S.
Jaeger4, S. Janchiv31, J. H. Jeong10A, Q. Ji1, Q. P. Ji20, X. B. Ji1,61, X. L.
Ji1,56, Y. Y. Ji48, Z. K. Jia69,56, P. C. Jiang45,g, S. S. Jiang38, T. J.
Jiang17, X. S. Jiang1,56,61, Y. Jiang61, J. B. Jiao48, Z. Jiao23, S. Jin41, Y.
Jin64, M. Q. Jing1,61, T. Johansson73, X. K.1, S. Kabana32, N. Kalantar-
Nayestanaki62, X. L. Kang9, X. S. Kang39, R. Kappert62, M. Kavatsyuk62, B. C.
Ke79, A. Khoukaz66, R. Kiuchi1, R. Kliemt14, L. Koch36, O. B. Kolcu60A, B.
Kopf4, M. Kuessner4, A. Kupsc43,73, W. Kühn36, J. J. Lane65, J. S. Lange36, P.
Larin19, A. Lavania26, L. Lavezzi72A,72C, T. T. Lei69,k, Z. H. Lei69,56, H.
Leithoff34, M. Lellmann34, T. Lenz34, C. Li46, C. Li42, C. H. Li38, Cheng
Li69,56, D. M. Li79, F. Li1,56, G. Li1, H. Li69,56, H. B. Li1,61, H. J. Li20,
H. N. Li54,i, Hui Li42, J. R. Li59, J. S. Li57, J. W. Li48, Ke Li1, L. J
Li1,61, L. K. Li1, Lei Li3, M. H. Li42, P. R. Li37,j,k, S. X. Li12, T. Li48,
W. D. Li1,61, W. G. Li1, X. H. Li69,56, X. L. Li48, Xiaoyu Li1,61, Y. G.
Li45,g, Z. J. Li57, Z. X. Li16, Z. Y. Li57, C. Liang41, H. Liang69,56, H.
Liang1,61, H. Liang33, Y. F. Liang52, Y. T. Liang30,61, G. R. Liao15, L. Z.
Liao48, J. Libby26, A. Limphirat58, D. X. Lin30,61, T. Lin1, B. J. Liu1, B. X.
Liu74, C. Liu33, C. X. Liu1, D. Liu19,69, F. H. Liu51, Fang Liu1, Feng Liu6,
G. M. Liu54,i, H. Liu37,j,k, H. B. Liu16, H. M. Liu1,61, Huanhuan Liu1, Huihui
Liu21, J. B. Liu69,56, J. L. Liu70, J. Y. Liu1,61, K. Liu1, K. Y. Liu39, Ke
Liu22, L. Liu69,56, L. C. Liu42, Lu Liu42, M. H. Liu12,f, P. L. Liu1, Q.
Liu61, S. B. Liu69,56, T. Liu12,f, W. K. Liu42, W. M. Liu69,56, X. Liu37,j,k,
Y. Liu37,j,k, Y. B. Liu42, Z. A. Liu1,56,61, Z. Q. Liu48, X. C. Lou1,56,61, F.
X. Lu57, H. J. Lu23, J. G. Lu1,56, X. L. Lu1, Y. Lu7, Y. P. Lu1,56, Z. H.
Lu1,61, C. L. Luo40, M. X. Luo78, T. Luo12,f, X. L. Luo1,56, X. R. Lyu61, Y.
F. Lyu42, F. C. Ma39, H. L. Ma1, J. L. Ma1,61, L. L. Ma48, M. M. Ma1,61, Q. M.
Ma1, R. Q. Ma1,61, R. T. Ma61, X. Y. Ma1,56, Y. Ma45,g, F. E. Maas19, M.
Maggiora72A,72C, S. Maldaner4, S. Malde67, A. Mangoni28B, Y. J. Mao45,g, Z. P.
Mao1, S. Marcello72A,72C, Z. X. Meng64, J. G. Messchendorp14,62, G.
Mezzadri29A, H. Miao1,61, T. J. Min41, R. E. Mitchell27, X. H. Mo1,56,61, N.
Yu. Muchnoi13,b, Y. Nefedov35, F. Nerling19,d, I. B. Nikolaev13,b, Z.
Ning1,56, S. Nisar11,l, Y. Niu 48, S. L. Olsen61, Q. Ouyang1,56,61, S.
Pacetti28B,28C, X. Pan53, Y. Pan55, A. Pathak33, P. Patteri28A, Y. P.
Pei69,56, M. Pelizaeus4, H. P. Peng69,56, K. Peters14,d, J. L. Ping40, R. G.
Ping1,61, S. Plura34, S. Pogodin35, V. Prasad32, F. Z. Qi1, H. Qi69,56, H. R.
Qi59, M. Qi41, T. Y. Qi12,f, S. Qian1,56, W. B. Qian61, C. F. Qiao61, J. J.
Qin70, L. Q. Qin15, X. P. Qin12,f, X. S. Qin48, Z. H. Qin1,56, J. F. Qiu1, S.
Q. Qu59, C. F. Redmer34, K. J. Ren38, A. Rivetti72C, V. Rodin62, M. Rolo72C,
G. Rong1,61, Ch. Rosner19, S. N. Ruan42, N. Salone43, A. Sarantsev35,c, Y.
Schelhaas34, K. Schoenning73, M. Scodeggio29A,29B, K. Y. Shan12,f, W. Shan24,
X. Y. Shan69,56, J. F. Shangguan53, L. G. Shao1,61, M. Shao69,56, C. P.
Shen12,f, H. F. Shen1,61, W. H. Shen61, X. Y. Shen1,61, B. A. Shi61, H. C.
Shi69,56, J. L. Shi12, J. Y. Shi1, Q. Q. Shi53, R. S. Shi1,61, X. Shi1,56, J.
J. Song20, T. Z. Song57, W. M. Song33,1, Y. J. Song12, Y. X. Song45,g, S.
Sosio72A,72C, S. Spataro72A,72C, F. Stieler34, Y. J. Su61, G. B. Sun74, G. X.
Sun1, H. Sun61, H. K. Sun1, J. F. Sun20, K. Sun59, L. Sun74, S. S. Sun1,61, T.
Sun1,61, W. Y. Sun33, Y. Sun9, Y. J. Sun69,56, Y. Z. Sun1, Z. T. Sun48, Y. X.
Tan69,56, C. J. Tang52, G. Y. Tang1, J. Tang57, Y. A. Tang74, L. Y Tao70, Q.
T. Tao25,h, M. Tat67, J. X. Teng69,56, V. Thoren73, W. H. Tian57, W. H.
Tian50, Y. Tian30,61, Z. F. Tian74, I. Uman60B, B. Wang1, B. L. Wang61, Bo
Wang69,56, C. W. Wang41, D. Y. Wang45,g, F. Wang70, H. J. Wang37,j,k, H. P.
Wang1,61, K. Wang1,56, L. L. Wang1, M. Wang48, Meng Wang1,61, S. Wang37,j,k,
S. Wang12,f, T. Wang12,f, T. J. Wang42, W. Wang70, W. Wang57, W. H. Wang74, W.
P. Wang69,56, X. Wang45,g, X. F. Wang37,j,k, X. J. Wang38, X. L. Wang12,f, Y.
Wang59, Y. D. Wang44, Y. F. Wang1,56,61, Y. H. Wang46, Y. N. Wang44, Y. Q.
Wang1, Yaqian Wang18,1, Yi Wang59, Z. Wang1,56, Z. L. Wang70, Z. Y. Wang1,61,
Ziyi Wang61, Zongyuan Wang1,61, D. Wei68, D. H. Wei15, F. Weidner66, S. P.
Wen1, C. W. Wenzel4, U. Wiedner4, G. Wilkinson67, M. Wolke73, L. Wollenberg4,
C. Wu38, J. F. Wu1,61, L. H. Wu1, L. J. Wu1,61, X. Wu12,f, X. H. Wu33, Y.
Wu69, Y. J Wu30, Z. Wu1,56, L. Xia69,56, X. M. Xian38, T. Xiang45,g, D.
Xiao37,j,k, G. Y. Xiao41, H. Xiao12,f, S. Y. Xiao1, Y. L. Xiao12,f, Z. J.
Xiao40, C. Xie41, X. H. Xie45,g, Y. Xie48, Y. G. Xie1,56, Y. H. Xie6, Z. P.
Xie69,56, T. Y. Xing1,61, C. F. Xu1,61, C. J. Xu57, G. F. Xu1, H. Y. Xu64, Q.
J. Xu17, W. L. Xu64, X. P. Xu53, Y. C. Xu76, Z. P. Xu41, F. Yan12,f, L.
Yan12,f, W. B. Yan69,56, W. C. Yan79, X. Q Yan1, H. J. Yang49,e, H. L. Yang33,
H. X. Yang1, Tao Yang1, Y. Yang12,f, Y. F. Yang42, Y. X. Yang1,61, Yifan
Yang1,61, Z. W. Yang37,j,k, M. Ye1,56, M. H. Ye8, J. H. Yin1, Z. Y. You57, B.
X. Yu1,56,61, C. X. Yu42, G. Yu1,61, T. Yu70, X. D. Yu45,g, C. Z. Yuan1,61, L.
Yuan2, S. C. Yuan1, X. Q. Yuan1, Y. Yuan1,61, Z. Y. Yuan57, C. X. Yue38, A. A.
Zafar71, F. R. Zeng48, X. Zeng12,f, Y. Zeng25,h, Y. J. Zeng1,61, X. Y. Zhai33,
Y. H. Zhan57, A. Q. Zhang1,61, B. L. Zhang1,61, B. X. Zhang1, D. H. Zhang42,
G. Y. Zhang20, H. Zhang69, H. H. Zhang57, H. H. Zhang33, H. Q. Zhang1,56,61,
H. Y. Zhang1,56, J. J. Zhang50, J. L. Zhang75, J. Q. Zhang40, J. W.
Zhang1,56,61, J. X. Zhang37,j,k, J. Y. Zhang1, J. Z. Zhang1,61, Jianyu
Zhang61, Jiawei Zhang1,61, L. M. Zhang59, L. Q. Zhang57, Lei Zhang41, P.
Zhang1, Q. Y. Zhang38,79, Shuihan Zhang1,61, Shulei Zhang25,h, X. D. Zhang44,
X. M. Zhang1, X. Y. Zhang48, X. Y. Zhang53, Y. Zhang67, Y. T. Zhang79, Y. H.
Zhang1,56, Yan Zhang69,56, Yao Zhang1, Z. H. Zhang1, Z. L. Zhang33, Z. Y.
Zhang74, Z. Y. Zhang42, G. Zhao1, J. Zhao38, J. Y. Zhao1,61, J. Z. Zhao1,56,
Lei Zhao69,56, Ling Zhao1, M. G. Zhao42, S. J. Zhao79, Y. B. Zhao1,56, Y. X.
Zhao30,61, Z. G. Zhao69,56, A. Zhemchugov35,a, B. Zheng70, J. P. Zheng1,56, W.
J. Zheng1,61, Y. H. Zheng61, B. Zhong40, X. Zhong57, H. Zhou48, L. P.
Zhou1,61, X. Zhou74, X. K. Zhou6, X. R. Zhou69,56, X. Y. Zhou38, Y. Z.
Zhou12,f, J. Zhu42, K. Zhu1, K. J. Zhu1,56,61, L. Zhu33, L. X. Zhu61, S. H.
Zhu68, S. Q. Zhu41, T. J. Zhu12,f, W. J. Zhu12,f, Y. C. Zhu69,56, Z. A.
Zhu1,61, J. H. Zou1, J. Zu69,56
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 Chung-Ang University, Seoul, 06974, Republic of Korea
11 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
12 Fudan University, Shanghai 200433, People’s Republic of China
13 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
14 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
15 Guangxi Normal University, Guilin 541004, People’s Republic of China
16 Guangxi University, Nanning 530004, People’s Republic of China
17 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
18 Hebei University, Baoding 071002, People’s Republic of China
19 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
20 Henan Normal University, Xinxiang 453007, People’s Republic of China
21 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
22 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
23 Huangshan College, Huangshan 245000, People’s Republic of China
24 Hunan Normal University, Changsha 410081, People’s Republic of China
25 Hunan University, Changsha 410082, People’s Republic of China
26 Indian Institute of Technology Madras, Chennai 600036, India
27 Indiana University, Bloomington, Indiana 47405, USA
28 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
29 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
30 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
31 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
32 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D,
Arica, Chile
33 Jilin University, Changchun 130012, People’s Republic of China
34 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
35 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
36 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
37 Lanzhou University, Lanzhou 730000, People’s Republic of China
38 Liaoning Normal University, Dalian 116029, People’s Republic of China
39 Liaoning University, Shenyang 110036, People’s Republic of China
40 Nanjing Normal University, Nanjing 210023, People’s Republic of China
41 Nanjing University, Nanjing 210093, People’s Republic of China
42 Nankai University, Tianjin 300071, People’s Republic of China
43 National Centre for Nuclear Research, Warsaw 02-093, Poland
44 North China Electric Power University, Beijing 102206, People’s Republic of
China
45 Peking University, Beijing 100871, People’s Republic of China
46 Qufu Normal University, Qufu 273165, People’s Republic of China
47 Shandong Normal University, Jinan 250014, People’s Republic of China
48 Shandong University, Jinan 250100, People’s Republic of China
49 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
50 Shanxi Normal University, Linfen 041004, People’s Republic of China
51 Shanxi University, Taiyuan 030006, People’s Republic of China
52 Sichuan University, Chengdu 610064, People’s Republic of China
53 Soochow University, Suzhou 215006, People’s Republic of China
54 South China Normal University, Guangzhou 510006, People’s Republic of China
55 Southeast University, Nanjing 211100, People’s Republic of China
56 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
57 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
58 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
59 Tsinghua University, Beijing 100084, People’s Republic of China
60 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
99138, Mersin 10, Turkey
61 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
62 University of Groningen, NL-9747 AA Groningen, The Netherlands
63 University of Hawaii, Honolulu, Hawaii 96822, USA
64 University of Jinan, Jinan 250022, People’s Republic of China
65 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
66 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
67 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
68 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
69 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
70 University of South China, Hengyang 421001, People’s Republic of China
71 University of the Punjab, Lahore-54590, Pakistan
72 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
73 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
74 Wuhan University, Wuhan 430072, People’s Republic of China
75 Xinyang Normal University, Xinyang 464000, People’s Republic of China
76 Yantai University, Yantai 264005, People’s Republic of China
77 Yunnan University, Kunming 650500, People’s Republic of China
78 Zhejiang University, Hangzhou 310027, People’s Republic of China
79 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi 75270,
Pakistan

###### Abstract

Based on a sample of $(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ events
collected with the BESIII detector at BEPCII, the decays of
$\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$ with
$\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+}$,
$\overline{\Lambda}\to\overline{p}\pi^{+}$ are studied. We investigate the two
excited resonances, $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$, which are each
observed with large significance ($\gg 10\sigma$) in the $K^{-}\Lambda$
invariant mass distributions. A partial wave analysis is performed, and the
spin-parities of $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$ are measured to be
$\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$, respectively. The masses, widths, and
product branching fractions of $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$ are also
measured.

###### pacs:

11.80.Et, 13.25.Gv, 14.20.Jn

##  I INTRODUCTION

In the quark model [1], hadrons are viewed as composite objects of constituent
spin-$\frac{1}{2}$ quarks bound by the strong interaction. Mesons are made of
quark-antiquark ($q\overline{q}$) pairs and baryons are made of three quarks
($qqq$). Within this simple quark model, the qualitative properties of hadrons
and the phenomenology of meson and baryon spectroscopy are well-explained. The
accepted full theory of the strong interaction is Quantum Chromodynamics
(QCD), a non-Abelian gauge-field theory that describes the interactions of
quarks and gluons and has the features of asymptotic freedom and confinement
of quarks. The understanding of the quark-gluon structure of baryons is one of
the most important tasks in both particle and nuclear physics. Since baryons
represent the simplest system in which the three colors of QCD neutralize into
colorless objects and the essential non-Abelian character of QCD is
manifested, systematic study of baryon spectroscopy can provide critical
insights into the nature of QCD in the confinement domain.

The mass spectra, together with their production and decay rates, provide the
main sources of information to study their structure. Much experimental work
has been dedicated to the study of baryon spectroscopy. However, the available
experimental information for strange baryons remains very incomplete. In
particular, we are lacking knowledge of the excited baryon states with two
strange quarks, i.e. $\Xi^{*}$ hyperons, due to their small production cross
sections and the complicated topology of the final states. Some
phenomenological QCD-inspired models predict more than thirty such kinds of
hyperons, however, only about a dozen total $\Xi$ states have been observed to
date. Among them, only a few are well established with spin-parity determined.
The spin of $\Xi(1820)$ is determined to be $\frac{3}{2}$ [2] and the
corresponding parity is measured to be negative [3]. Some evidence that the
$\Xi(1690)$ has $J^{P}=\frac{1}{2}^{-}$ was found in a study of
$\Lambda^{+}_{c}\rightarrow\Xi^{-}\pi^{+}K^{+}$ [4].

In recent years, charmonium data samples with unprecedented statistics were
accumulated by the Beijing Spectrometer (BESIII [5]) at the Beijing Electron-
Positron Collider (BEPCII [6]), and these provide great opportunities for
investigating the light baryons produced in charmonium decays. In a previous
analysis using a sample of $106\times 10^{6}$ $\psi(3686)$ events collected
with BESIII, two hyperons, $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$, were observed
in the $K^{-}\Lambda$ invariant mass distribution [7]. Now, with four times
more $\psi(3686)$ events collected at BESIII, we conduct a more extensive
study of the decays $\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$. In
particular, we perform a partial wave analysis (PWA) to study the properties
of intermediate state $\Xi^{*}$ hyperons.

In this paper, we report a PWA analysis of
$\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$, with a sample of
$(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ events collected at BESIII. In the
following, the charge conjugate channel is always implied.

##  II BESIII DETECTOR

The BESIII detector records symmetric $e^{+}e^{-}$ collisions provided by the
BEPCII storage ring, which operates in the center-of-mass energy range from
2.0 to 4.95 GeV. BESIII has collected large data samples in this energy region
[8]. The cylindrical core of the BESIII detector covers 93% of the full solid
angle and consists of a helium-based multilayer drift chamber (MDC), a plastic
scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic
calorimeter (EMC), which are all enclosed in a superconducting solenoidal
magnet providing a 1.0 T (0.9 T in 2012) magnetic field. The solenoid is
supported by an octagonal flux-return yoke with resistive plate counter muon
identification modules interleaved with steel. The charged-particle momentum
resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the dE/dx resolution is $6\%$
for electrons from Bhabha scattering. The EMC measures photon energies with a
resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end cap) region. The
time resolution of the TOF in the barrel region is 68 ps while that in the
endcap region is 110 ps.

##  III DATA SETS AND MONTE CARLO SAMPLES

This study uses $(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ events collected by
the BESIII detector at BEPCII in 2009 ($(107.0\pm 0.8)\times 10^{6}$ events)
and 2012 ($(341.1\pm 2.1)\times 10^{6}$ events, taken with 0.9 T magnetic
field) [9].

A GEANT4-based [10] Monte Carlo (MC) simulation software BOOST [11], which
includes a geometric and material description [12] of the BESIII detector,
detector response and digitization models as well as tracking of the detector
running conditions and performance, is used to generate MC simulated data
samples. An exclusive MC sample for the process
$\psi(3686)\rightarrow{}K^{-}\Lambda\overline{\Xi}^{+}$, is generated to
optimize the selection criteria and estimate the corresponding selection
efficiency. The production of the $\psi(3686)$ is simulated by the generator
KKMC [13], and the subsequent decays are generated with BesEvtGen [14, 15]. An
inclusive MC sample, consisting of $448\times 10^{6}$ $\psi(3686)$ events, is
used to study potential backgrounds. The known decay modes of the $\psi(3686)$
are generated by BesEvtGen with branching fractions set to world average
values [16], and the remaining unknown decay modes are modeled by LUNDCHARM
[17] [18].

##  IV EVENT SELECTION

Considering the full decay chain of
$\psi(3686)\rightarrow{}K^{-}\Lambda\overline{\Xi}^{+}$ as reconstructed from
the decays $\Lambda\rightarrow{}p\pi^{-}$,
$\overline{\Xi}^{+}\rightarrow\overline{\Lambda}\pi^{+}$ and
$\overline{\Lambda}\rightarrow\overline{p}\pi^{+}$, there are six charged
tracks with low momentum in the final state, and the detection efficiency is
very low. Therefore a partial reconstruction method is adopted to obtain
higher statistics by not requiring the prompt $\Lambda$ from the $\psi(3686)$
decay. Following the $K^{-}$ and $\overline{\Xi}^{+}$ reconstruction, the
$\Lambda$ four-momentum is calculated from the recoil of the
$K^{-}\overline{\Xi}^{+}$ system.

With the partial reconstruction method, at least four charged tracks are
required. The polar angles $\theta$, defined with respect to the axis of the
MDC, of all charged tracks is required to satisfy $|\cos\theta|<0.93$. For the
kaon, the point of closest approach to the beam line is required to be within
$\pm 10$ cm in the beam direction and 2 cm in the plane perpendicular to the
beam. Since the $\overline{\Xi}^{+}$ particle has a displaced decay vertex,
looser requirements are imposed on the charged tracks from the
$\overline{\Xi}^{+}$ decay: the point of closest approach to the beam line of
is only required to be within $\pm 15$ cm in the beam direction and 10 cm in
the plane perpendicular to the beam.

Particle identification (PID) for charged tracks combines measurements of the
energy loss in the MDC (d$E$/d$x$) and the flight time in the TOF to form
likelihoods $\mathcal{L}(h)~{}(h=p,K,\pi)$ for each hadron $h$ hypothesis.
Tracks are identified as protons when the proton hypothesis has the highest
likelihood ($\mathcal{L}(p)>\mathcal{L}(K)$ and
$\mathcal{L}(p)>\mathcal{L}(\pi)$), while charged kaons and pions are
identified by comparing the likelihoods for the kaon and pion hypotheses,
$\mathcal{L}(K)>\mathcal{L}(\pi)$ and $\mathcal{L}(\pi)>\mathcal{L}(K)$,
respectively. In this analysis, two negatively charged tracks are required to
be identified as $K^{-}$ and $\overline{p}$, and two positively charged tracks
as pions.

The candidate $\overline{\Xi}^{+}$ baryon is reconstructed in two steps. A
$\overline{p}\pi^{+}$ pair sharing a common vertex is selected to reconstruct
the $\overline{\Lambda}$ candidate via a secondary vertex fit. The
$\overline{\Xi}^{+}$ is then reconstructed with the $\overline{\Lambda}$
candidate and the other $\pi^{+}$ by applying another secondary vertex fit.
For events with more than one $\overline{\Xi}^{+}$ candidate, the
$\overline{p}\pi^{+}$ combination with the minimum
$|M(\overline{p}\pi^{+})-M(\overline{\Lambda})|$ is selected. Here,
$M(\overline{p}\pi^{+})$ is the invariant mass of the $\overline{p}\pi^{+}$
combination, see Fig. 1 (a), and $M(\overline{\Lambda})$ is the known mass of
$\Lambda$ taken from the Particle Data Group (PDG) [16].

A mass window of
$1.110\;\text{GeV}/c^{2}<M(\overline{p}\pi^{+})<1.121\;\text{GeV}/c^{2}$ is
imposed to select $\overline{\Lambda}$ candidates as shown in Fig. 1 (a) by
the blue dashed arrows. The distribution of the $\overline{\Xi}^{+}$ decay
length, $L(\overline{\Xi}^{+})$, is shown in Fig. 1 (b). The distribution of
the $\overline{\Lambda}\pi^{+}$ invariant mass,
$M(\overline{\Lambda}\pi^{+})$, is shown in Fig. 1 (c), after a requirement of
$L(\overline{\Xi}^{+})>0.5~{}\text{cm}$ is applied. The invariant mass
$M(\overline{\Lambda}\pi^{+})$ is required to satisfy
$1.315\;\text{GeV}/c^{2}<M(\overline{\Lambda}\pi^{+})<1.330\;\text{GeV}/c^{2}$.
The distribution of the mass recoiling against $K^{-}$ and the reconstructed
$\overline{\Xi}^{+}$, RM($K^{-}\overline{\Xi}^{+}$), is shown in Fig. 1 (d).
One can see a clear $\Lambda$ baryon signal around $1.115$ GeV/$c^{2}$ and the
main background from $\Sigma^{0}$ decays around $1.193$ GeV/$c^{2}$. The
requirement
$1.080\;\text{GeV}/c^{2}<\text{RM}(K^{-}\overline{\Xi}^{+})<1.140\;\text{GeV}/c^{2}$
is imposed to select prompt $\Lambda$ candidates. In total, 1714 events are
selected.

(a)(b)
(c)(d)

Figure 1:  Distribution of $M(\overline{p}\pi^{+})$ of $\overline{\Lambda}$
candidates (a), decay length of $\overline{\Xi}^{+}$ candidates (b), and
$M(\overline{\Lambda}\pi^{+})$ of $\overline{\Xi}^{+}$ candidates (c). Plot
(d) shows the distribution of RM$(K^{-}\overline{\Xi}^{+})$: the left peak is
$\Lambda$, while the right one is $\Sigma^{0}$. The crosses represent data and
the histograms represent phase space (PHSP) MC. The dashed arrows show the cut
values; for each plot, cuts on the other three quantities are applied.

After the above selection, a one-constraint kinematic fit that constrains the
mass of the missing $\Lambda$ to the PDG value, is performed to improve the
resolution, and no event is rejected. The distributions of
$M(K^{-}\overline{\Xi}^{+})$, $M(\Lambda\overline{\Xi}^{+})$, and
$M(K^{-}\Lambda)$ are shown in Fig. 2 (b), (c) and (d), respectively. Two
diagonal bands on the Dalitz plot as shown in Fig. 2 (a) correspond to the two
structures near 1.7 GeV/$c^{2}$ and 1.8 GeV/$c^{2}$ in the $M(K^{-}\Lambda)$
mass spectrum.

(a)(b)
(c)(d)

Figure 2:  (a) Dalitz plot of
$M^{2}(\Lambda\overline{\Xi}^{+})\;vs\;M^{2}(K^{-}\overline{\Xi}^{+})$ and the
distributions of (b) $M(K^{-}\Lambda)$, (c) $M(K^{-}\overline{\Xi}^{+})$ and
(d) $M(\Lambda\overline{\Xi}^{+})$ after the final selection. The crosses
represent the data and the histograms represent the background events
estimated from the $\Xi$ sidebands.

To investigate possible background events, the same analysis is performed to
the $\psi(3686)$ inclusive MC sample, and nineteen background events are
found. A detailed event type analysis with a generic tool, TopoAna [19], shows
that the main background is from
$\psi(3686)\rightarrow{}\gamma\chi_{c1,c2},\chi_{c1,c2}\rightarrow{}n\overline{K}^{*}\overline{\Lambda}+c.c.$
decays plus some other decays with $K^{-}\pi^{+}\pi^{-}\overline{p}$ in the
final state. A sideband method is used in data to estimate the background
contribution. The sideband regions of $\overline{\Xi}^{+}$ are defined to be
$(1.3025,1.3100)$ and $(1.3350,1.3425)$ GeV/$c^{2}$. A total of 104 background
events, corresponding to a background level of 6%, are obtained. No peaking
background is observed in the recoil mass for the $\overline{\Xi}^{+}$
sideband events. The lower background level in the MC is attributed to the
lack of simulation of the decays of higher-mass excited states and to the
incomplete description of the decays of the excited states that are simulated.
Additionally, the continuum data taken at a center-of-mass energy of 3.65 GeV
with an integrated luminosity of 42.6 pb-1 is used to estimate the background
from Quantum Electrodynamics processes, and no background event remains.

##  V PARTIAL WAVE ANALYSIS

The two-body decay amplitudes in the sequential decay process
$\psi(3686)\rightarrow\overline{\Xi}^{+}\Xi^{*-}_{X},\Xi^{*-}_{X}\to{}K^{-}\Lambda$
are constructed using the relativistic covariant tensor amplitude formalism
[20], and the maximum likelihood method is used in the PWA, with the FDC [21]
package. Here, $\Xi^{*-}_{X}$ denotes an intermediate state such as
$\Xi(1620)^{-}$, $\Xi(1690)^{-}$, $\Xi(1830)^{-}$, etc.

###  V.1 Introduction to PWA

The amplitude $A_{j}$ for the $j$-th possible partial wave in
$\psi(3686)\rightarrow\overline{\Xi}^{+}\Xi^{*-}_{X},\Xi^{*-}_{X}\to{}K^{-}\Lambda$
is described as

$A_{j}=A^{j}_{{\rm prod},X}(BW)_{X}A_{{\rm decay},X},$ (1)

where $A^{j}_{{\rm prod},X}$ is the amplitude describing the production of the
intermediate resonance $\Xi^{*-}_{X}$, $BW_{X}$ is the Breit-Wigner propagator
of $\Xi^{*-}_{X}$, and $A_{{\rm decay},X}$ is the decay amplitude of
$\Xi^{*-}_{X}$.

The total differential cross section $d\sigma/d\Phi$ is

$\frac{d\sigma}{d\Phi}=\left|\sum\limits_{j}c_{j}A_{j}\right|^{2},$ (2)

where $\sigma$ is the total cross section, $\Phi$ is the phase space, and
$c_{j}$ is a complex free parameter to be determined in the fit for each
partial wave $A_{j}$.

The probability to observe the event characterized by the variable $\xi$ is

$P(\xi)=\frac{\omega(\xi)\,\epsilon(\xi)}{\int
d\xi\,\omega(\xi)\,\epsilon(\xi)},$ (3)

where $\xi$ are the four-momenta of the $K^{-}$, $\Lambda$, and
$\overline{\Xi}^{+}$, $\omega(\xi)\equiv d\sigma/d\Phi$ is the probability
density for a single event to populate the PHSP at $\xi$, and $\epsilon(\xi)$
is the detection efficiency to detect one event with $\xi$. The normalization
integral $\int d\xi\,\omega(\xi)\,\epsilon(\xi)$ is calculated using the
exclusive signal MC sample.

The likelihood for observing $N$ events in the data sample is

$\displaystyle\mathcal{L}=P(\xi_{1},\xi_{2},...,\xi_{N})$
$\displaystyle=\prod\limits_{i=1}^{N}P(\xi_{i})$
$\displaystyle=\prod\limits_{i=1}^{N}\frac{\omega(\xi_{i})\,\epsilon(\xi_{i})}{\int
d\xi\,\omega(\xi)\,\epsilon(\xi)}.$ (4)

Rather than maximizing the likelihood function $\mathcal{L}$, the quantity
$-\ln\mathcal{L}$ is minimized to obtain best values of the parameters $c_{j}$
and the masses and widths of the resonances

$-\ln\mathcal{L}=-\sum\limits_{i=1}^{N}\ln\left(\frac{\omega(\xi_{i})}{\int
d\xi\,\omega(\xi)\,\epsilon(\xi)}\right)-\sum\limits_{i=1}^{N}\ln\epsilon(\xi_{i}),$
(5)

For a given data set, the second term is a constant and has no impact on the
determination of the parameters of the amplitudes or on the change of
$-\ln\mathcal{L}$. So, in the fit, the $-\ln\mathcal{L}$ is defined as

$-\ln\mathcal{L}=-\sum\limits_{i=1}^{N}\ln\left(\frac{\omega(\xi_{i})}{\int
d\xi\,\omega(\xi)\,\epsilon(\xi)}\right).$ (6)

The final log-likelihood value minimized for the data, $S$, is the sum of the
log-likelihood values of the events in the $\Xi$ signal region and background
events in the $\Xi$ sideband region with negative weights.

$S=-\ln\mathcal{L}_{\rm{data}}+\ln\mathcal{L}_{\rm{bg}}.$ (7)

The free parameters are optimized by FUMILI [22]. In the minimization
procedure, a change in the log-likelihood of 0.5 represents one standard
deviation for each parameter.

In this analysis, the Breit-Wigner resonance shape used for the $\Xi^{*}$ is

$BW(s)=\frac{1}{M_{\Xi^{*}}^{2}-s_{K\Lambda}-iM_{\Xi^{*}}\Gamma_{\Xi^{*}}},$
(8)

where $s_{K\Lambda}$ is the invariant mass squared of the decay products of
the $\Xi^{*}$.

Since nucleons have structure, form factors modifying the Breit-Wigner shape
are needed to describe them. Different form factors have been discussed in
Refs. [23, 24], and the following ones are used in the fit:

$\displaystyle J$ $\displaystyle=\frac{1}{2}:$ $\displaystyle\quad
F_{N}(s_{K\Lambda})$
$\displaystyle=\frac{\lambda^{4}_{1}}{\lambda^{4}_{1}+(s_{K\Lambda}-M_{\Xi^{*}}^{2})^{2}},$
(9) $\displaystyle J$ $\displaystyle=\frac{3}{2},\frac{5}{2}:$
$\displaystyle\quad F_{N}(s_{K\Lambda})$
$\displaystyle=e^{\frac{-|s_{K\Lambda}-M_{\Xi^{*}}^{2}|}{\lambda^{2}_{2}}},$
(10)

where $J$ is the spin, and the cutoff parameters $\lambda_{1}$ and
$\lambda_{2}$ are set to be 2.0 GeV.

###  V.2 PWA results

According to the $M(K^{-}\overline{\Xi}^{+})$, $M(\Lambda\overline{\Xi}^{+})$,
and $M(K^{-}\Lambda)$ spectra, as shown in Fig. 2, no significant enhancement
is observed except the two structures near 1.7 GeV/$c^{2}$ and 1.8 GeV/$c^{2}$
in the $M(K^{-}\Lambda)$ spectrum. In the PWA, there are seven PDG-listed
candidate $\Xi$ hyperons, $\Xi(1620)^{-}$, $\Xi(1690)^{-}$, $\Xi(1820)^{-}$,
$\Xi(1950)^{-}$, $\Xi(2030)^{-}$, $\Xi(2120)^{-}$ and $\Xi(2250)^{-}$. A
coherent non-resonant contribution is also considered, denoted as non-res,
which is described as a wide intermediate state with certain spin-parity.

####  V.2.1 Nominal fit

In the first step of partial wave analysis, all possible sets of amplitudes
corresponding to the seven PDG-listed candidate $\Xi$ hyperons are evaluated.
The masses and widths of resonances are fixed to the PDG. The significance for
a resonance is calculated based on the improvement in PWA quality, $\Delta S$,
with the change in degrees of freedom considered. Only the $\Xi(1690)^{-}$ and
$\Xi(1820)^{-}$ have significances greater than 5$\sigma$. The other five
$\Xi$ resonances tested ($\Xi(1620)^{-}$, $\Xi(1950)^{-}$, $\Xi(2030)^{-}$,
$\Xi(2120)^{-}$ and $\Xi(2250)^{-}$), each with significance less than
5$\sigma$, are excluded from the nominal fit. Each was tried with a variety of
$J^{P}$ values: $1/2^{\pm},3/2^{\pm},5/2^{\pm},7/2^{\pm}$. Their impact will
be considered as a systematic uncertainty. The $J^{P}$ of $\Xi(1690)$ and
$\Xi(1820)$ are favored to be $\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$,
respectively.

In the next step, the masses and widths of $\Xi(1690)$ and $\Xi(1820)$ are
further optimized. The obtained results are shown in Table 1. The masses and
widths of $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$ are consistent with the PDG
values within $2.6\sigma$.

The projections on the $M(K^{-}\overline{\Xi}^{+})$,
$M(\Lambda\overline{\Xi}^{+})$, $M(K^{-}\Lambda)$ spectra after PWA are shown
in Fig. 3. They agree with those of the data. We observe $464\pm 43$
$\Xi(1690)^{-}$ events with a mass $M=1685^{+3}_{-3}$ MeV/$c^{2}$ and a width
$\Gamma=81^{+10}_{-9}$ MeV, and $776\pm 42$ $\Xi(1820)^{-}$ events with a mass
$M=1821^{+2}_{-3}$ MeV/$c^{2}$ and a width $\Gamma=73^{+6}_{-5}$ MeV. Here,
the uncertainties are statistical only. The statistical significances of both
structures are greater than $10\sigma$. These significances as well as the fit
fractions are given in Table 1.

(a)(b)(c)

Figure 3: Distributions of (a) $M({K^{-}\overline{\Xi}^{+}})$, (b) $M({\Lambda\overline{\Xi}^{+}})$ and (c) $M({K^{-}\Lambda})$. The crosses represent data and red solid histograms represent the projection of the PWA result. The different color histograms represent the intensity of each component in the nominal fit. Here, $\chi^{2}/{\rm nbin}$ demonstrates the goodness of fit in each figure, where nbin is the number of bins in each figure and $\chi^{2}$ is defined as $\chi^{2}=\sum_{i=1}^{\rm nbin}\,(n_{i}-v_{i})^{2}/v_{i}$, where $n_{i}$ and $v_{i}$ are the numbers of events for the data and the fit projections of the nominal fit in the $i^{th}$ interval of each figure, respectively. Table 1: The optimized mass and width for each resonance, along with the fit fraction (FF) for each. Here, the uncertainties are statistical only.  Resonance | $J^{P}$ | M (MeV/$c^{2}$) |  $\Gamma$ (MeV) | $\sigma$ | FF
---|---|---|---|---|---
$\Xi(1690)^{-}$ | $1/2^{-}$ | $1685^{+3}_{-3}$ | $81^{+10}_{-9}$ | 10.8 | 29.0
$\Xi(1820)^{-}$ | $3/2^{-}$ | $1821^{+2}_{-3}$ | $73^{+6}_{-5}$ | 18.3 | 48.0
Non-res | $1/2^{+}$ | - | - |  $>$30 | 23.0

####  V.2.2 Check of the nominal fit

Different $J^{P}$ assignments for the nominal fit have been tested as shown in
Table 2. The likelihood values become worse with respect to that of the
nominal fit.

Table 2: The checks of different $J^{P}$ assignments. $\Delta{}S$ is the change of $S$ compared to the nominal fit. $J^{P}$ | Non-res$\to{}K^{-}\Lambda$ |  $\Xi(1690)^{-}$ | $\Xi(1820)^{-}$
---|---|---|---
$1/2^{-}$ | 53.3 | - | 11.2
$1/2^{+}$ | - | 29.2 | 12.6
$3/2^{-}$ | 44.9 | 110.6 | -
$3/2^{+}$ | 7.7 | 33.6 | 13.9

The other possible non-resonant contributions in the
$\Lambda\overline{\Xi}^{+}$ and $K^{-}\overline{\Xi}^{+}$ systems are
investigated by replacing $1/2^{-}$ non-res$\rightarrow{}K^{-}\Lambda$ in the
nominal fit. As shown in Table 3, the likelihood values also become worse.

Table 3: The checks of other possible non-resonant contributions. Non-res process | $J^{P}$ | $\Delta{}S$
---|---|---
Non-res$\to{}K^{-}\Lambda$ | $1/2^{-}$ | 53.3
Non-res$\to{}\Lambda\overline{\Xi}^{+}$ | $0^{-}$ | 43.9
$1^{-}$ | 14.8
$1^{+}$ | 22.3
Non-res$\to{}K^{-}\overline{\Xi}^{+}$ | $1/2^{-}$ | 8.8
$1/2^{+}$ | 33.9

####  V.2.3 Branching fractions

To determine the detection efficiencies of
$\psi(3686)\rightarrow\Xi(1690)^{-}\overline{\Xi}^{+}$ and
$\psi(3686)\rightarrow\Xi(1820)^{-}\overline{\Xi}^{+}$, signal MC events are
generated using the PWA amplitude for each process. The product branching
fraction of
$\psi(3686)\rightarrow\overline{\Xi}^{+}\Xi^{*-}(\Xi^{*-}\rightarrow{}K^{-}\Lambda)+c.c.$
is calculated with

$\displaystyle\mathcal{B}(\psi(3686)\rightarrow\overline{\Xi}^{+}\Xi^{*-}+c.c.)\cdot\mathcal{B}(\Xi^{*-}\rightarrow{}K^{-}\Lambda)$
(11)
$\displaystyle=\frac{N_{\Xi^{*}}}{N_{\psi(3686)}\cdot\mathcal{B}(\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+})\cdot\mathcal{B}(\overline{\Lambda}\to\overline{p}\pi^{+})\cdot\epsilon_{\Xi^{*}}},$

where $N_{\Xi^{*}}$ is the number of
$\psi(3686)\rightarrow\Xi^{*-}\overline{\Xi}^{+}+c.c.$ events,
$N_{\psi(3686)}=(448.1\pm 2.9)\times 10^{6}$ is the total number of
$\psi(3686)$ events [25], and
$\mathcal{B}(\overline{\Lambda}\to\overline{p}\pi^{+})$ and
$\mathcal{B}(\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+})$ are the
corresponding decay branching fractions [16]. The detection efficiency is
$\epsilon_{\Xi^{*}}=16.0\%$ for $\Xi(1690)^{-}$, and the corresponding product
branching fraction is $(1.06\pm 0.10)\times 10^{-5}$. Similarly, the product
branching fraction for $\Xi(1820)^{-}$ is $(1.78\pm 0.10)\times 10^{-5}$ with
a detection efficiency $\epsilon_{\Xi^{*}}=14.6\%$.

Fitting the RM($K^{-}\overline{\Xi}^{+})$ distribution, shown in Fig. 4,
yields the number of signal events of $\psi(3686)\to
K^{-}\Lambda\overline{\Xi}^{+}$ as $N_{\rm sig}=1572\pm 45$. The signal is
modeled by a signal MC shape of $\Lambda$ convolved with a Gaussian function
while the background components from $\Sigma^{0}$ and $\chi_{cJ}$ are
described by the MC shapes, and other background channels are described with a
third order polynomial function. The branching fraction of this decay is
determined to be

$\displaystyle\mathcal{B}(\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.)$
(12) $\displaystyle=\frac{N_{\rm
sig}}{N_{\psi(3686)}\cdot\mathcal{B}(\overline{\Xi}^{+}\to\overline{\Lambda}\pi^{+})\cdot\mathcal{B}(\overline{\Lambda}\to\overline{p}\pi^{+})\cdot\epsilon}$
$\displaystyle=(3.60\pm 0.10)\times 10^{-5}.$

Here, $\epsilon=15.3\%$ is the detection efficiency for the final state. It is
studied with the exclusive signal MC events which are generated using the PWA
results obtained in this analysis. The uncertainty is statistical only.

Figure 4: Fit to the RM($K^{-}\overline{\Xi}^{+})$ distribution. The crosses
are data and the red curve denotes the best fit. The other curves show the
different fit components listed in the legend.

##  VI SYSTEMATIC UNCERTAINTIES

In this analysis, the sources of the systematic uncertainty are classified
into two categories: the uncertainty from event selection and the uncertainty
from the PWA procedure. The former affects the measurement of branching
fractions, while the latter affects the measurements of masses and widths of
the resonances and the branching fractions of the intermediate resonances. The
different sources of systematic uncertainty are discussed below.

We begin with the systematic uncertainty from event selection:

  1. (i)

The total number of $\psi(3686)$ events is obtained by studying inclusive
hadronic $\psi(3686)$ decays, giving a total uncertainty of 0.65% [25]. This
uncertainty is taken as a systematic uncertainty for this analysis.

  2. (ii)

The uncertainties of kaon tracking and PID efficiencies are estimated using
the control sample of $J/\psi\to K_{S}^{0}K^{*}$. The difference between MC
and data, $1.0\%$, is taken as the systematic uncertainty of the tracking or
PID efficiency.

  3. (iii)

The $\overline{\Xi}^{+}$ reconstruction efficiency is studied with the control
sample of $J/\psi\to\Xi^{-}\overline{\Xi}^{+}$ [26]. The difference between MC
and data, $6.6\%$, is taken as the systematic uncertainty and it includes the
systematic uncertainties of MDC tracking and PID efficiencies for the
$\bar{p}$ and both $\pi^{+}$.

  4. (iv)

In the fit to the RM$(K^{-}\overline{\Xi}^{+})$ distribution, we consider
three sources of uncertainty: the signal model, the background model, and the
fit range. The signal model is changed to a double Gaussian function and the
background model is changed to a second-order polynomial function. In both
cases, the change of the result is taken as the systematic uncertainty. The
fit range is also changed to different values and the maximum change of the
result is taken as the uncertainty. The total systematic uncertainty from
fitting is the sum in quadrature of the three contributions.

The second category is the systematic uncertainty from the PWA procedure:

  1. (i)

In this analysis, the background is estimated from the $\overline{\Xi}^{+}$
sideband events in data. We change the background level by $\pm 1\sigma$ and
redo the PWA fit. We also change the sideband range by $\pm 1\sigma$ and redo
the PWA fit. For each variation, we take the difference as the associated
systematic uncertainty.

  2. (ii)

We replace the non-res component in the nominal fit by the other two processes
(non-res$\rightarrow\Lambda\overline{\Xi}^{+}$ and non-
res$\rightarrow{}K^{-}\overline{\Xi}^{+})$ and redo the PWA fit. The
differences are taken as the systematic uncertainty.

  3. (iii)

Besides $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$ in the nominal fit, each known
possible resonance has been included in the fit. Among them, $\Xi(1620)^{-}$
is the most significant one with a statistical significance of 3$\sigma$. The
difference between the fit results with and without $\Xi(1620)^{-}$ is taken
as the systematic uncertainty due to possible additional resonances.

  4. (iv)

The systematic uncertainty associated with a change of the parameters
$\lambda_{1}$ and $\lambda_{2}$ is evaluated by fixing $\lambda_{2}$ to 2.0
GeV and varying $\lambda_{1}$ between 1.5 GeV and 3.0 GeV. The maximum
difference to the nominal result is taken as the systematic uncertainty.

  5. (v)

In analysis, the branching fractions are obtained with the optimized masses
and widths of $\Xi(1690)$ and $\Xi(1820)$. Alternatively, the branching
fractions are obtained with the masses and widths of $\Xi(1690)$ and
$\Xi(1820)$ fixed to the PDG values. The resulting changes in the measurements
of branching fraction are assigned as systematic uncertainties.

  6. (vi)

The systematic uncertainty from fit bias is evaluated by applying nominal
analysis procedure to signal MC samples generated according to the PWA results
from data. The difference between input and output values is taken as the
systematic uncertainty.

The two categories of systematic uncertainties are listed in Table 4 and Table
5.

Table 4: Relative systematic uncertainties (in $\%$) on the branching fraction measurement of $\psi(3686)\rightarrow{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$. Source | Uncertainty (%)
---|---
Number of $\psi(3686)$ events | 0.7
MDC tracking of $K^{-}$ | 1.0
PID of $K^{-}$ | 1.0
$\overline{\Xi}^{+}$ reconstruction | 6.6
Signal model | 0.8
Background shape | 0.1
Fit range | 0.3
Total | 6.8
Table 5: Systematic uncertainties on the measurements of the $\Xi^{*}$ parameters and branching fractions. Source | $\Xi(1690)$ | $\Xi(1820)$
---|---|---
$\Delta M$(MeV/$c^{2}$) |  $\Delta\Gamma$ (MeV) | $\Delta{\mathcal{B}}/{\mathcal{B}}(\%)$ |  $\Delta M$(MeV/$c^{2}$) |  $\Delta\Gamma$ (MeV) | $\Delta{\mathcal{B}}/{\mathcal{B}}(\%)$
Number of $\psi(3686)$ events | - | - | 0.7 | - | - | 0.7
MDC tracking of $K^{\pm}$ | - | - | 1 | - | - | 1
PID of $K^{\pm}$ | - | - | 1 | - | - | 1
$\overline{\Xi}^{+}$ reconstruction | - | - | 6.6 | - | - | 6.6
Background level | 0 | 3 | 1.0 | 1 | 1 | 0.9
Background sideband | 0 | 4 | 0.4 | 0 | 1 | 1.4
Non-res component | 11 | 9 | 11.3 | 2 | 3 | 15.6
Additional resonances | 5 | 17 | 14.0 | 2 | 2 | 3.7
Different form factors | 0 | 2 | 3.0 | 1 | 8 | 1.3
Fit bias | 1 | 3 | 8.4 | 0 | 1 | 2.4
Resonance parameters of $\Xi(1690)$ and $\Xi(1820)$ | - | - | 19.8 | - | - | 3.2
Total | 12 | 20 | 29.1 | 3 | 9 | 18.1

##  VII SUMMARY AND DISCUSSION

Based on $(448.1\pm 2.9)\times 10^{6}$ $\psi(3686)$ events collected with the
BESIII detector at BEPCII in 2009 and 2012, we report the results of a partial
wave analysis of $\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$ Two
excited hyperons, $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$, are observed in the
$M(K^{-}\Lambda)$ and $M(K^{+}\overline{\Lambda})$ spectra. Their masses,
widths, spin-parities, and product branching fractions are measured. The
results obtained are summarized in Tables 6 and 7. We note that, whereas the
masses of the $\Xi(1690)^{-}$ and $\Xi(1820)^{-}$ are in agreement with
previous measurements, our width values are both larger than those
measurements and only marginally consistent with them. However, our analysis
is the first to use a PWA to include interference effects, and this might help
explain the differences. The spin-parities of $\Xi(1690)^{-}$ and
$\Xi(1820)^{-}$ are measured for the first time, which are consistent with the
quark model. This work improves the knowledge of the excited hyperon spectrum.
To understand the internal structure of baryons and test theoretical
predictions, further investigations with higher statistics are needed.

Table 6: Results obtained for $I(J^{P})$, mass and width for each component. The first (second) uncertainty is statistical (systematic). Resonance | $I(J^{P})$ | M (MeV/$c^{2}$) |  $\Gamma$ (MeV)
---|---|---|---
$\Xi(1690)^{-}$ | $1/2(1/2^{-})$ | $1685^{+3}_{-2}\pm{12}$ | $81^{+10}_{-9}\pm{20}$
$\Xi(1820)^{-}$ | $1/2(3/2^{-})$ | $1821^{+2}_{-3}\pm{3}$ | $73^{+6}_{-5}\pm{9}$
Table 7: Branching fraction results; the first (second) uncertainty is statistical (systematic). Resonance | Branching fraction
---|---
$\mathcal{B}(\psi(3686)\to{}\Xi(1690)^{-}\overline{\Xi}^{+}+c.c.)\times\mathcal{B}(\Xi(1690)^{-}\to{}K^{-}\Lambda)$ | $(1.06\pm 0.10\pm 0.31)\times 10^{-5}$
$\mathcal{B}(\psi(3686)\to{}\Xi(1820)^{-}\overline{\Xi}^{+}+c.c.)\times\mathcal{B}(\Xi(1820)^{-}\to{}K^{-}\Lambda)$ | $(1.78\pm 0.10\pm 0.32)\times 10^{-5}$
$\psi(3686)\to{}K^{-}\Lambda\overline{\Xi}^{+}+c.c.$ | $(3.60\pm 0.10\pm 0.24)\times 10^{-5}$

Acknowledgement

The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key R&D Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400;
National Natural Science Foundation of China (NSFC) under Contracts Nos.
11635010, 11735014, 11835012, 11935015, 11935016, 11935018, 11961141012,
12022510, 12025502, 12035009, 12035013, 12061131003, 12075250, 12075252,
12192260, 12192261, 12192262, 12192263, 12192264, 12192265; the Chinese
Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS
Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale
Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; CAS
Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-
SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; The Institute of Nuclear
and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics
and Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020
research and innovation programme under Marie Sklodowska-Curie grant agreement
under Contract No. 894790; German Research Foundation DFG under Contracts Nos.
443159800, 455635585, Collaborative Research Center CRC 1044, FOR5327, GRK
2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of
Turkey under Contract No. DPT2006K-120470; National Research Foundation of
Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology
fund; National Science Research and Innovation Fund (NSRF) via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation under Contract No. B16F640076; Polish National Science Centre under
Contract No. 2019/35/O/ST2/02907; The Royal Society, UK under Contract No.
DH160214; The Swedish Research Council; U. S. Department of Energy under
Contract No. DE-FG02-05ER41374.

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# Quadrotor Formation Flying Resilient to Abrupt Vehicle
Failures via a Fluid Flow Navigation Function

Matthew Romanoa,†, Harshvardhan Uppalurub,†, Hossein Rastgoftarb, Ella Atkinsa
† Both authors contributed equally to this work.aAuthors are with the Robotics
Department at the University of Michigan {mmroma<EMAIL_ADDRESS>are with the Aerospace and Mechanical Engineering department at the University
of Arizona {huppaluru<EMAIL_ADDRESS>
(January 2022)

###### Abstract

This paper develops and experimentally evaluates a navigation function for
quadrotor formation flight that is resilient to abrupt quadrotor failures and
other obstacles. The navigation function is based on modeling healthy
quadrotors as particles in an ideal fluid flow. We provide three key
contributions: (i) A Containment Exclusion Mode (CEM) safety theorem and proof
which guarantees safety and formally specifies a minimum safe distance between
quadrotors in formation, (ii) A real-time, computationally efficient CEM
navigation algorithm, (iii) Simulation and experimental algorithm validation.
Simulations were first performed with a team of six virtual quadrotors to
demonstrate velocity tracking via dynamic slide speed, maintaining sufficient
inter-agent distances, and operating in real-time. Flight tests with a team of
two custom quadrotors were performed in an indoor motion capture flight
facility, successfully validating that the navigation algorithm can handle
non-trivial bounded tracking errors while guaranteeing safety.

##  I Introduction

Quadrotors are becoming increasingly popular due to advances in sensing,
actuation, and computational power. A quadrotor is cheap and highly
maneuverable with a wide range of useful applications. A single quadrotor has
limited processing and payload carriage capabilities. However, a group of
cooperatively controlled quadrotors can achieve difficult tasks with
considerable advantages in terms of resilience to failure, mission complexity,
and scalability.

Figure 1: Q1 fails and stays in place within the solid red circle. Q2 uses the
CEM navigation algorithm to smoothly avoid the failed vehicle and travels
along the dashed red path. The blue circles indicate five time points for Q2.

Several multi-agent coordination approaches have been investigated with
applications ranging from surveillance [1, 2] to formation flying [3, 4],
rescue missions [5], wildlife monitoring and exploration [6], precision
agriculture [7], cooperative payload transport [8, 9] and hazardous
environment sensing [10]. Virtual structure (VS) is a centralized technique in
which a group of agents, operating as particles of a virtual rigid body,
preserve a strict geometric relationship to one another and a frame of
reference [11, 12]. Desired agent trajectories are calculated in a centralized
manner through rigid body rotations and translations of the virtual body in
$3$-D motion space. However, due to the rigid body requirement and a single
point of failure, the range of applications using VS is limited. Consensus is
one of the most extensively studied cooperative control approaches in which a
team of agents reach agreement/consensus by communicating only with their
neighbours. It is a decentralized coordination approach and is broadly divided
into two categories: consensus without a leader (i.e., leaderless consensus)
[13, 14] and consensus with a leader (i.e., leader-follower consensus) [15,
16]. Containment control [17, 18, 19, 20] is a decentralized leader-follower
method where collective motion of all agents is achieved with multiple
leaders. Defined by geometric constraints i.e., all agents are contained
within a particular area, the follower agents obtain the desired positions
through local communication with in-neighbor agents. Based on the principles
of continuum mechanics, continuum deformation is a recent multi-agent
coordination approach for which agents in a multi-agent system (MAS) are
treated as particles of a body deforming under a homogeneous transformation
[21, 22, 23, 24].

The idea of using potential fields for real-time path planning was originally
proposed for robot arms [25]. An attractive potential is placed at the goal,
repulsive potentials on obstacles, and gradient descent is used to plan a
path. This can be called a navigation function. This is a useful approach,
however it has issues of local minima. The randomized potential field approach
attempts to get around this by executing a random walk whenever the planner
gets stuck in a local minimum, but still provides no guarantee of planning
completeness [26]. For harmonic functions subject to Laplace’s equations, a
navigation function can be constructed that only has saddle points as local
minima such that they can be easily escaped [27]. This model has been applied
to robot formations, but with explicit models of repulsive functions between
robots, one goal sink node, and computation time on the order of one second.
However, the scenario considered in this paper needs an algorithm to quickly
respond to an abrupt failure on the order of milliseconds.

This paper proposes a novel safety recovery algorithm for a multi-quadrotor
system (MQS) and experimentally validates the approach in simulation and
quadrotor flight experiments. We consider each quadrotor in a MQS as a finite
number of particles in an ideal fluid flow pattern. By classifying quadrotors
as healthy and failed agents, we consider failed quadrotors as singularity
points of the fluid flow. We then define desired trajectories of the healthy
agents along flow streamlines so that the failed quadrotors are safely
partitioned outside the motion space (see Figure 1). We enclose the failed
quadrotors with virtual obstacles to define no-fly zones. Because we treat
these obstacles as singularity points that are excluded from the motion space,
our proposed safety-recovery approach is called containment exclusion mode
(CEM) throughout this paper.

Compared to the existing literature, this paper offers the following
contributions:

  1. 1.

We propose a real-time, computationally efficient CEM navigation algorithm
that dynamically modifies quadrotor sliding speed to maintain a desired
maximum speed for all quadrotors.

  2. 2.

We formally specify safety and collision avoidance by providing two CEM safety
theorems. Theorem 1 applies to a large quadrotor system with multiple
obstacles whereas Theorem 2 has been developed for experiments for a single
obstacle.

  3. 3.

We experimentally evaluate the CEM navigation algorithm using a team of
quadrotors in formation while validating our CEM safety theorem.

This paper is organized as follows. A problem statement (Section II) is
followed by a description of our proposed CEM approach (Section III). Section
IV presents our experimental setup, followed by experimental results in
Section V. Section VI concludes the paper.

##  II Problem Statement

Consider a system of quadrotors, flying in formation, defined by set
$\mathcal{S}=\left\\{1,\cdots,N_{q}\right\\}$, where $N_{q}$ is the number of
quadrotors. Let $N_{f}$ be the number of quadrotors that fail to follow the
desired formation and are assumed to hover in this work. The failed quadrotors
are identified by set $\mathcal{F}\subset\mathcal{S}$ and $N_{f}<N_{q}$. The
remaining quadrotors are termed as healthy quadrotors because they follow the
desired formation, and is defined by the set
$\mathcal{H}=\mathcal{S}\setminus\mathcal{F}$.

At reference time $t_{0}$, $N_{f}$ quadrotors fail to follow the desired
formation. The failed quadrotors are enclosed by virtual obstacles as defined
below in Assumption 1. The healthy quadrotors should now follow the safety-
recovery approach called CEM where the quadrotors in $\mathcal{H}$ should
avoid the failed quadrotors and safely pass them before continuing with their
individual trajectories.

Given the problem setup defined above, we first develop a navigation function
where the healthy quadrotors are treated as particles in ideal fluid flow
combining uniform flow and doublet flow in the $x-y$ plane.

##  III Approach

The set $\mathcal{F}$ satisfies the following assumption:

###### Assumption 1

We assume failed quadrotor is enclosed by a virtual obstacle defined as no-fly
zone that is completely contained within an enclosed cylinder of radius
$a_{f}$, centered at $z_{f}=(x_{f},y_{f})$, elongated in the $z$-direction
(e.g. Figure 1).

This ideal fluid flow pattern is generated by the complex function

$\centering
f(\mathbf{z})=\sum_{f\in\mathcal{F}}\left(\mathbf{z}-\mathbf{z}_{f}+\frac{a_{p}^{2}}{\mathbf{z}-\mathbf{z}_{f}}\right),\@add@centering$
(1)

where $\mathbf{z}=x+\mathbf{j}y$ is the complex variable,
$\mathbf{z}_{f}=x_{f}+\mathbf{j}y_{f}$ is the position of singularity point
$f\in\mathcal{F}$ with finite set $\mathcal{F}$ identifying all singularities
in the $x-y$ plane, and $a_{p}=a_{f}+\delta+\epsilon$ is the planned exclusion
radius. $a_{p}$ is larger than the actual exclusion radius ($a_{f}$) by a
bound on healthy agent controller error ($\delta$) and healthy agent radius
($\epsilon$). Eq. (1) can be rewritten as

$\centering f(\mathbf{z})=\phi(x,y)+\mathbf{i}\psi(x,y)\@add@centering$ (2)

where

$\phi\left(x,y\right)=\sum_{f\in\mathcal{F}}\left(\frac{\left(x-x_{f}\right)\left(\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}+a_{p}^{2}\right)}{\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}}\right)$
(3a)
$\psi\left(x,y\right)=\sum_{f\in\mathcal{F}}\left(\frac{\left(y-y_{f}\right)\left(\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}-a_{p}^{2}\right)}{\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}}\right)$
(3b)

are called potential and stream functions, respectively. Note that Eq. (2)
provides a conformal mapping between $x-y$ and $\phi-\psi$ planes, where
Cauchy-Reimann and Laplace equations are satisfied by $\phi(x,y)$ and
$\psi(x,y)$.

$\centering\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial
y}\@add@centering$ $\centering\frac{\partial\phi}{\partial
y}=-\frac{\partial\psi}{\partial x}\@add@centering$
$\centering\nabla^{2}\psi=0,\hskip 85.35826pt\nabla^{2}\phi=0\@add@centering$

Using the ideal fluid flow model on healthy quadrotors $\mathcal{H}$ at any
time $t\geq t_{0}$, $x$ and $y$ components are constrained to slide along the
stream curve

$\centering\psi((x_{i}(t_{0}),y_{i}(t_{0})))=\psi_{i,0},\forall
i\in\mathcal{H}\@add@centering$ (4)

###### Assumption 2

Velocity in the $z$-direction is $0$ i.e., the altitude remains constant for
each healthy quadrotor recovery trajectory.

###### Assumption 3

As the recovery approach is active, we assume that the failed quadrotors do
not leave the enclosing no-fly cylinders of radius $a_{f}$.

Using the approach defined above, the main goal of this paper is to present a
real-time, safe and efficient CEM algorithm that can plan safe recovery
trajectories for all healthy quadrotors. To this end, we first present CEM as
a navigation algorithm in Section III-A. Then, we formally specify safety for
CEM by providing inter-agent collision avoidance conditions (Section III-B).

###  III-A CEM Navigation Algorithm

Algorithm 1 guides agents along constant $\psi$ streamlines in the fluid flow
pattern. This is performed in real-time such that the ”$t$ for loop” (line 12)
is run every $\Delta T$ seconds a total of $m$ times. $\Delta T=10ms$ (100Hz)
in this paper. To minimize computational overhead, this method acts as a
navigation function, calculating the immediate command for each agent to avoid
collisions and then incrementing the trajectory in $\phi$ on each subsequent
loop.

Algorithm 1 CEM Navigation

1:procedure CEM($\mathcal{F}$, $m$, $v_{des}$, $\mathbf{r}$)

2: Inputs: Failed agents $\mathcal{F}$, number of timesteps $m$, desired speed
$v_{des}$, healthy vehicle positions $\mathbf{r}$

3: Output: Trajectory for each vehicle via sendCommands()

4:

5: $\mathbf{r}_{d}\leftarrow$ $\mathbf{r}$

6: $\Delta\phi\leftarrow v_{des}*\Delta T$

7: for i $\leftarrow$ 1, $N_{h}$ do

8: $\phi[i]$ $\leftarrow$ $\phi(\mathbf{r}[i],\mathcal{F})$ $\triangleright$
Eq. 3a

9: $\psi[i]$ $\leftarrow$ $\psi(\mathbf{r}[i],\mathcal{F})$ $\triangleright$
Eq. 3b

10: end for

11:

12: for t $\leftarrow$ 1, m do

13: $\phi_{last}\leftarrow\phi$

14: for k $\leftarrow$ 1, $K$ do

15: for i $\leftarrow$ 1, $N_{h}$ do

16: $\phi[i]\leftarrow\phi_{last}[i]+\Delta\phi$

17: $\mathbf{r}_{d}[i],\dot{\mathbf{r}}_{d}[i]$ $\leftarrow$
CalcXY($\phi[i]$,$\psi[i]$,$\mathbf{r}_{d}[i]$)

18: end for

19: $v_{max}\leftarrow\max_{i\in N_{h}}{||\dot{\mathbf{r}}_{d}[i]||}$

20: $\Delta\phi\leftarrow\Delta\phi*v_{des}/v_{max}$

21: end for

22: sendCommands($\mathbf{r}_{d}$,$\dot{\mathbf{r}}_{d}$)

23: sleep($\Delta T$)

24: end for

25:end procedure

$\phi$ and $\psi$ values for each agent are initialized to their starting
positions. Also, sliding speed $\Delta\phi$ is set to match the desired
vehicle velocities for the non-distorted case. At every time step, $\phi$ is
increased by $\Delta\phi$ and the resulting position and velocity for each
vehicle is calculated using ”CalcXY” (Alg. 2). Maximum velocity $v_{max}$ is
then used to update $\Delta\phi$ for the next iteration to more closely match
$v_{des}$ (Line 20). This update step assumes $v_{max}=C\Delta\phi_{last}$ and
wants to calculate $\Delta\phi_{next}$ such that $v_{des}=C\Delta\phi_{next}$
where $C$ is a constant assumed to encapsulate the gradient for the small step
sizes. Therefore, $\Delta\phi_{next}=\Delta\phi_{last}\frac{v_{des}}{v_{max}}$
should calculate the proper $\Delta\phi$ to obtain $v_{des}$. This is executed
$K$ times. If the assumption that $C$ is constant is reasonable, then $K=2$ is
a good selection which effectively plans twice per time step, first to
calculate $\Delta\phi$ and second to calculate the trajectories using
$\Delta\phi$. $K=2$ was used in this paper for the main results and varied for
additional results to see its effects. After the $K$ loop, commands are output
for that time step and the process repeats for the next time step.

Alg. 2 calculates the position given $\psi$, $\phi$ using a gradient descent
inspired approach. On each iteration ($n=20$ in this paper), the current
estimate is used to calculate the associated $\phi$ and $\psi$ and then the
estimate is updated in the direction of decreasing error via the inverse of
the Jacobian matrix, defined as:

$\mathbf{J}(x,y,\mathcal{F})=\begin{bmatrix}\frac{\partial\phi}{\partial
x}&\frac{\partial\phi}{\partial y}\\\ \frac{\partial\psi}{\partial
x}&\frac{\partial\psi}{\partial y}\end{bmatrix}.$ (5)

$\mathbf{r}_{noise}$ is added to the update step to escape saddle points on
the edge of the exclusion zone as:

$\mathbf{r}_{noise}=\begin{cases}|\mathcal{N}(0,\sigma)|\hat{\mathbf{r}}_{a}+\mathcal{N}(0,\sigma)\hat{\mathbf{r}}_{b}&\text{if
}||\mathbf{r}-\mathbf{r}_{f}||\leq a_{p}\\\
\mathbf{0},&\text{else}\end{cases}$ (6)

where $\mathcal{N}(0,\sigma)$ generates Gaussian noise with zero-mean and
standard deviation $\sigma$,
$\hat{\mathbf{r}}_{a}=\frac{\mathbf{r}-\mathbf{r}_{f}}{||\mathbf{r}-\mathbf{r}_{f}||}$
is the direction away from the exclusion zone, and
$\hat{\mathbf{r}}_{b}=\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}\hat{\mathbf{r}}_{a}$ is perpendicular to
$\hat{\mathbf{r}}_{a}$. $\sigma=1$mm was used in this paper. This procedure
ensures that commanded positions never violate the planned exclusion radius
$a_{p}$. Lastly, the first-order difference equation is used to calculate the
velocity.

Algorithm 2 CalcXY

1:procedure CalcXY($\phi$,$\psi$,$\mathbf{r}$)

2: $\mathbf{r}_{-1}\leftarrow\mathbf{r}$

3: for i $\leftarrow$ 1, $n$ do

4: $\phi_{1}$ $\leftarrow$ $\phi(\mathbf{r},\mathcal{F})$ $\triangleright$ Eq.
3a

5: $\psi_{1}$ $\leftarrow$ $\psi(\mathbf{r},\mathcal{F})$ $\triangleright$ Eq.
3b

6: $J_{h}$ $\leftarrow$ $J_{h}(\mathbf{r},\mathcal{F})$ $\triangleright$ Eq. 5

7: $\mathbf{r}\leftarrow\mathbf{r}-J_{h}^{-1}\begin{bmatrix}\phi_{1}-\phi\\\
\psi_{1}-\psi\end{bmatrix}+\mathbf{r}_{noise}$ $\triangleright$ Eq. 6

8: end for

9: $\dot{\mathbf{r}}\leftarrow(\mathbf{r}-\mathbf{r}_{-1})/\Delta T$

10: return $\mathbf{r},\dot{\mathbf{r}}$

11:end procedure

###  III-B Safety Analysis

Collision avoidance between healthy agents and obstacles is assured via the
construction of the ideal fluid flow pattern. The planned exclusion radius for
obstacles is $\delta+\epsilon$ larger than the actual exclusion radius.
Therefore, if no agents are initialized inside the planned exclusion radius,
the navigation function will never take them closer than $\delta+\epsilon$
which guarantees safety.

Inter-agent collision avoidance is more complex as it involves distortion in
the $x-y$ plane along constant $\psi$ streamlines. Theorem 1 provides a
general, conservative, inter-agent collision avoidance condition while Theorem
2 provides a tighter, more useful condition for the single obstacle case.

###### Theorem 1

Let $r_{i,0}$ be the position of healthy agent $i\in\mathcal{H}$ at the time
of failure and is defined as

$r_{i,0}=[x_{i,0},y_{i,0}]^{T}$

Define $\phi_{i,0}=\phi(x_{i,0},y_{i,0})$, $\psi_{i,0}=\psi(x_{i,0},y_{i,0})$,

$p_{min,0}=\stackunder{min}{\stackunder{i,j}{i\neq
j}}\sqrt{(\phi_{i,0}-\phi_{j,0})^{2}+(\psi_{i,0}-\psi_{j,0})^{2}},$ (7)

with $p_{min,0}$ the minimum separation distance between agents in the
$\phi-\psi$ plane, and

$\lambda_{max}=\max_{x,y}\left(\left(\phi_{x}^{2}+\phi_{y}^{2}\right)\right).$

Assume that the trajectory control error of each individual quadcopter does
not exceed $\delta$ and every quadcopter can be enclosed by a ball of radius
$\epsilon$. Inter-agent collision avoidance during CEM mode is guaranteed, if
the sliding speed $\dot{\phi}_{i}=\dot{\phi}$ is the same for every healthy
quadcopter, and

$\frac{p_{min,0}^{2}}{\lambda_{max}}\geq 4(\delta+\epsilon)^{2}.$ (8)

###### Proof:

Under conformal mapping (1), there exists a one-to-one mapping between
infinitesimal element $dx-dy$ in the $x-y$ plane and infinitesimal element
$d\phi-d\psi$ in the $\phi-\psi$ plane, where $(dx,dy)$ and $(d\phi,d\psi)$
can be related by

$\begin{bmatrix}d\phi\\\ d\psi\end{bmatrix}=\mathbf{J}\begin{bmatrix}dx\\\
dy\end{bmatrix},$ (9)

and $\mathbf{J}$ is the Jacobian matrix in (5). Therefore,

$d\phi^{2}+d\psi^{2}=\begin{bmatrix}dx&&dy\end{bmatrix}\mathbf{J}^{T}\mathbf{J}\begin{bmatrix}dx\\\
dy\end{bmatrix}$ (10)

Using Cauchy-Reimann Relations, we can show that
$\mathbf{J}^{T}\mathbf{J}=\mathrm{diag}\left(\phi_{x}^{2}+\phi_{y}^{2},\phi_{x}^{2}+\phi_{y}^{2}\right)\in\mathbb{R}^{2\times
2}$ is diagonal, and thus, we can write

$\begin{split}d\phi^{2}+d\psi^{2}&=\begin{bmatrix}dx&dy\end{bmatrix}\begin{bmatrix}\phi_{x}^{2}+\phi_{y}^{2}&0\\\
0&\phi_{x}^{2}+\phi_{y}^{2}\end{bmatrix}\begin{bmatrix}dx\\\
dy\end{bmatrix}\\\
=&\left(\phi_{x}^{2}+\phi_{y}^{2}\right)\left(dx^{2}+dy^{2}\right).\end{split}$
(11)

This also implies that

$dx^{2}+dy^{2}\geq\frac{d\phi^{2}+d\psi^{2}}{\lambda_{max}},\qquad x\neq
x_{f},~{}y\neq y_{f}.$ (12)

Eq. (12) implies that

$\left(d_{min}(t)\right)^{2}\geq\frac{\left(p_{min}(t)\right)^{2}}{\lambda_{max}},\qquad\forall
t$ (13)

where

$p_{min}(t)=\stackunder{min}{\stackunder{i,j}{i\neq
j}}\sqrt{{\left(\phi_{i}(t)-\phi_{j}(t)\right)}^{2}+\left(\psi_{i}\left(t\right)-\psi_{j}\left(t\right)\right)^{2}}\qquad\forall
t.$ (14a) $d_{min}(t)=\stackunder{min}{\stackunder{i,j}{i\neq
j}}\sqrt{{\left(x_{i}(t)-x_{j}(t)\right)}^{2}+\left(y_{i}\left(t\right)-y_{j}\left(t\right)\right)^{2}}\qquad\forall
t.$ (14b)

When the sliding speed $\dot{\phi}_{i}=\dot{\phi}$ is the same for every
healthy quadcopter $i$, the healthy agent team moves as particles of a rigid-
body in the $\phi-\psi$ plane, and thus, inter-agent distances in the
$\phi-\psi$ plane are time-invariant. As a result, the minimum separation
distance of the desired formation in the $\phi-\psi$ plane can be assigned at
reference time $t_{0}$, when the failed agent no-fly zone first appears.
Therefore, $p_{min,0}=p_{min}(t)$ and Eq. (15) simplifies to

$\left(d_{min}(t)\right)^{2}\geq\frac{\left(p_{min,0}\right)^{2}}{\lambda_{max}},\qquad\forall
t$ (15)

Since $d_{min}(t)\geq 2(\delta+\epsilon)\forall t$ is the collision avoidance
condition, this implies that the inter-agent collision avoidance is assured,
if Eq. (8) is satisfied.

∎

Theorem 1 provides a sufficient condition for CEM inter-agent collision
avoidance without restricting the number of failures. However, condition (8)
conservatively overestimates the required minimum separation distance and it
does so in the $\phi-\psi$ plane instead of the $x-y$ plane which is harder to
use in practice. To overcome this issue, Theorem 2 provides a tighter bound on
initial formation distances for a collision avoidance guarantee condition for
CEM but it works when there exists a single failed agent in the $x-y$ plane.

###### Theorem 2

Inter-agent collision between every agent is avoided if the minimum separation
distance $d_{min,0}$ at reference time $t_{0}$, when the failed agent appears,
satisfies the following condition:

$d_{min,0}=\min\limits_{i,j,~{}i\neq
j}\sqrt{\left(x_{i,0}-x_{j,0}\right)^{2}+\left(y_{i,0}-y_{j,0}\right)^{2}}\geq
2\left(\delta+\epsilon\right)+a_{p}.$

(16)

###### Proof:

Inter-agent collision between agents is avoided if $d_{min}(t)$, defined by
(14b), satisfies the following condition:

$d_{min}(t)\geq 2\left(\delta+\epsilon\right),\qquad\forall t.$

When a single failure exists in the $x-y$ plane, $\psi(x,y)=0$ wraps the
failed agent by a circle of radius $a_{p}$ with the center positioned at
$\left(x_{f},y_{f}\right)$ (see Eq. (3b), Figure 2). The maximum contraction
of the distance between two arbitrary recovery paths $\psi(x,y)=\psi_{i,0}$
and $\psi(x,y)=\psi_{j,0}$ is less than $a_{p}$, and it occurs at $x=x_{f}$.
So, $d_{min}(t)\geq d_{min,0}-a_{p}=2\left(\epsilon+\delta\right)\forall t$.
Therefore, inter-agent collision between every two agent pair is avoided if
the minimum separation distance $d_{min,0}$, obtained on the configuration of
the healthy agents at reference time $t_{0}$, satisfies inequality (16).

Figure 2: $\phi$, $\psi$ Contours. $a_{p}=1$, $x_{f}=y_{f}=0$. $\phi$ and
$\psi$ integer levels are shown. The $\psi=0$ contour smoothly circumvents the
exclusion zone.

∎

##  IV Experimental Setup

Figure 3 shows a high level view of the experimental setup. The Ground Control
Station (GCS) computer acts as the centralized planner for the system and
sends desired states to each vehicle, where
$\mathbf{X}_{d,i}=\begin{bmatrix}\mathbf{r}_{d,i}^{T}&\dot{\mathbf{r}}_{d,i}^{T}\end{bmatrix}^{T}$.
An Intel core i7-9750H processor laptop was used as the GCS computer. The
real-time commands were computed in C++ and sent over MAVLink using WiFi. The
vehicles were equipped with motion capture markers and tracked via an indoor
3m x 4m, eight Optitrack camera motion capture space at 100Hz. The pose
feedback,
$\mathbf{X}_{i}=\begin{bmatrix}\mathbf{r}_{i}^{T}&\dot{\mathbf{r}}_{i}^{T}\end{bmatrix}^{T}$,
was sent over low-latency 900MHz XBee wireless serial modules to the vehicles.
M330 quadrotors running the rc_pilot_a2sys open source flight controller were
used. A cascaded proportional-integral-derivative (PID) controller tracks the
desired commands via motion capture feedback. The vehicles have an enclosing
circle radius of $\epsilon=28cm$, and it is assumed $\delta=40cm$ bounds
controller tracking error.

Figure 3: High Level Experimental Setup Block Diagram

Figure 4 provides a visual depiction of the radii which keep the vehicles from
colliding with each other per Sec. III-B. The minimum initial formation
distance between any two vehicles is
$d_{min}=2\left(\epsilon+\delta\right)+a_{p}$. Additionally, when a failure
occurs and CEM mode is activated we assume that the failed agent will remain
within $\delta$ of its setpoint so $a_{f}=\epsilon+\delta$. Therefore,
$a_{p}=1.36m$ and $d_{min}=2.72m$.

Figure 4: Formation inter-agent distances to guarantee safety.

##  V Experimental Results

###  V-A CEM Navigation Validation with Six Virtual Vehicles

For this test, the ground control station computer was configured to run
exactly as described in the experimental setup section, except that only the
setpoints were logged and no vehicles were flying. This was done to validate
that the navigation algorithm was able to generate safe trajectories while
also maintaining a desired velocity along constant streamlines and operating
within runtime constraints.

Figure 5 shows the formation and trajectory followed. The formation is similar
to [28] with a leading triangle and interior agents. The initial formation has
agents with a minimum separation of 2.72m. The vehicles begin in a six agent
formation on the bottom before Q1 fails. This activates the CEM navigation
algorithm. An initial ($\phi$, $\psi$) coordinate is calculated for each
vehicle and then on each iteration $\phi$ is advanced in a dynamic fashion to
track a 1.0 m/s desired maximum velocity. Figure 6 shows the commanded
velocities. This shows each component of 2D velocity while the norm at each
step was the quantity being tracked. Figure 7 shows the values of $\phi$
advancing while Figure 8 shows how $\Delta\phi$ is changed to maintain the
specified velocity. Notice how at about 6s and 13s $\Delta\phi$ gets very
small. This coincides with Q2 traversing saddle points at each end of the
exclusion zone. The Jacobian is large here which causes small changes in
$\phi$ to produce a large change in the $x$-$y$ plane (i.e. Q2 moves fast
while all other vehicles move slowly). Additionally, Figure 9 shows the
distance between nearest agents. Every agent pair maintains at least
$2(\delta+\epsilon)$ separation.

Figure 5: Six Vehicle Test 2D Plot Figure 6: Six Vehicle Test Velocities
Figure 7: Six Vehicle Test $\phi$ Figure 8: Six Vehicle Test $\Delta\phi$
Figure 9: Six Vehicle Test Nearest Neighbor Distance

To explore the effectiveness of the velocity tracking aspect of our navigation
algorithm the six vehicle flight was repeated while the value for K was varied
in the range {1,2,5}. Figure 10 shows the maximum velocity of any agent for
the various values of K during one of the two saddle points that cause the
most difficulty. K=1 reached a 2D speed of 2.4 m/s, much higher than the 1.0
m/s setpoint. Speed error with K=2 was significant lower, while K=5 performed
a bit better. However, increasing the value of K essentially replans with a
new $\Delta\phi$ K times each iteration. This has a runtime cost which can be
seen in Figure 11. Runtime is proportional to the value of K and the number of
agents. Runtimes of roughly 0.2ms, 0.4ms, and 1ms were measured for the values
of 1, 2, and 5 respectively. 10ms is the hard upper limit since the navigation
algorithm is running at 100Hz. A value of K=2 follows the velocity profile
well while offering a reasonably low runtime.

Figure 10: How Varying K affects Velocity Tracking Figure 11: How Varying K
affects Runtime

###  V-B Two Vehicle Flight Test

The same formation from the previous section in a reduced configuration (only
Q1 and Q2) was flight tested to validate the navigation algorithm working
safely to avoid collisions with non-zero controller error bounded by
$\delta=40cm$ for each vehicle. Figure 1 shows this test. Figure 12 shows the
trajectories for failed agent Q1 within its exclusion zone (solid red circle)
and Q2 which tracked its desired trajectory computed in real-time (dotted
green) closely by its actual trajectory (solid green). The controller error of
both agents remained within the $\delta$ bound as seen in Figure 13. Figure 14
shows that the actual trajectory of Q2 was a safe distance away from the
exclusion zone. The dynamic value of $\Delta\phi$ can be seen in Figure 15.

Figure 12: Two Vehicle 2D Plot  Figure 13: Two Vehicle Controller Error Figure
14: Two Vehicle Exclusion Zone Distance Figure 15: Two Vehicle $\Delta\phi$

##  VI Conclusion

This paper has presented an algorithm for quadrotor formation flight that
allows healthy quadrotors to safely continue their mission in the presence of
quadrotor failure(s). We presented a novel real-time fluid-flow based CEM
navigation algorithm where recovery trajectories of healthy quadrotors are
calculated as streamlines of an ideal fluid flow. We analyzed two safety
conditions that specify the minimum separation distance required to assure
safety. Experimental results validated the safety condition provided.

Experiments were conducted with quadrotors flying in the $x$ direction of the
$x$-$y$ plane i.e., constant $\psi$ direction of the $\phi$-$\psi$ plane.
Using a rotation matrix, the CEM navigation algorithm can be improved further
to accommodate quadrotor formation flight in any direction of the $x$-$y$
plane. Although this paper demonstrated experiments with a two quadrotor team
and a single failure as well as a six virtual quadrotor team with a single
failure, using Theorem 1, the CEM navigation algorithm III-A can be extended
to large quadrotor groups and multiple quadrotor failures. We will extend
simulation and experiments to larger teams with multiple failed quadrotors in
future work.

## Acknowledgements

This work was supported in part by the National Science Foundation (NSF) under
Award Nos. 2133690, 1914581, and 1739525.

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