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This convexity condition is relaxed in this paper, as we show that it is enough that the vector field belongs to a strict tangent cone based on a local supporting hyperrectangle. The new condition has the natural physical interpretation of requiring shared reference directions in addition to the available local relative states. Such shared reference directions can be further interpreted as if each agent holds a magnetic compass indicating the orientations of a global frame. It is proved that the cooperative multiagent system achieves exponential state agreement if and only if the time-varying interaction graph is uniformly jointly quasi-strongly connected. Cooperative-antagonistic multiagent systems are also considered. For these systems, the relation has a negative sign for arcs corresponding to antagonistic interactions. State agreement may not be achieved, but instead it is shown that all the agents' states asymptotically converge, and their limits agree componentwise in absolute values if and in general only if the time-varying interaction graph is uniformly jointly strongly connected.<\/jats:p>","DOI":"10.1137\/140982283","type":"journal-article","created":{"date-parts":[[2015,9,22]],"date-time":"2015-09-22T12:19:04Z","timestamp":1442924344000},"page":"3057-3080","source":"Crossref","is-referenced-by-count":12,"title":["Multiagent Systems with Compasses"],"prefix":"10.1137","volume":"53","author":[{"given":"Ziyang","family":"Meng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guodong","family":"Shi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Karl Henrik","family":"Johansson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,9,22]]},"reference":[{"key":"atypb1","first-page":"7","author":"Altafini C.","year":"2012","journal-title":"PloS One"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2012.2224251"},{"key":"atypb3","unstructured":"J.P. 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