The self-assembled GaSb and InSb nanostructures (quantum dots, QDs) are grown on (001) Ge substrates in Stranski-Krastanow growth mode by molecular beam epitaxy. The structural properties are characterized by ex situ atomic force microscopy (AFM), and the related optical properties are observed by photoluminescence (PL) spectroscopy. Growing of polar GaAs on non-polar Ge creates anti-phase domains (APDs). By careful controlling of growth, APDs surface becomes flat and having large surface area (~?m2) which is sufficient to form QD array in each domain. The effects of APDs on the formation of QDs are discussed. By varying the growth conditions, different QD morphologies for two types of QDs; GaSb and InSb, are observed. At the growth temperature of 400-450?C, GaSb/GaAs QDs are mainly formed on the flat area of each APD. Only a few QDs are also found along the APD boundaries (APBs). The self-assembled GaSb QD shape transforms from circular to rectangular based shape by GaSb depositing over 2.5 monolayer (ML) with the low growth rates ~0.11 ML/s and ~0.09 ML/s, at low growth temperature ~450?C. The free-standing GaSb/GaAs QDs elongate along [110] direction on (001) GaAs surface having the orthogonal nature of GaAs APDs. The PL emission from QDs exhibits a strong blue shift with increasing the excitation power, which is the characteristic of type-II band alignment. Different from the GaSb QD nucleation position, the low-growth-rate InSb QDs are mostly formed at the APBs, where two orthogonal GaAs surfaces meet. The QD size, shape, density and position are varied with the QD growth temperature and In growth rate. At higher growth temperature and low growth rate, larger InSb QDs are observed as qualitatively similar to the GaSb QD system. By increasing growth rate from 0.023 ML/s to 0.14ML/s, the high density (~1010cm-2) InSb QDs array is obtained, and the QDs accumulate in both APDs and APBs. The QD size and shape are very different on APBs and on the flat GaAs APDs. This work enhances our understanding of the relation between molecular beam epitaxial growth conditions and antimony-based QD morphologies and it provides a basis for practical realization of electronic and opto-electronic devices based on QD nanostructures.