"Robert Barker in 1643 was a member, with his brother John Barker, of the Marshfield military company under Lieutenant Nathaniel Thomas. He was a surveyor of Marshfield in 1645, 1648, and of Duxbury in 1654, 1672, 1677, 1679; constable of Marshfield 1645, 1648; grandjuryman of Marshfield, 1669, and of Duxbury 1684-85; and was admitted a freeman in 1654. Robert Barker was licensed 7 July, 1646, to keep an inn in Marshfield to retail wine, which was cancelled 5 June 1666. The course at Plymouth on 5 March, 1667-68, granted him nine and one-half acres of meadow at Robinson's Creek, North River, Duxbury. He married Lucy Williams and died between 18 Feb., 1689, when his will, which mentions his children was, made and 15 March, 1691-2, when the inventory of his estate was taken. His estate was valued at 142Â£. His wife died between 7 Mar. 1681-2, when she was fined for selling cider to the Indians, and 18 Feb. 1689. They lived in the old Barker house at Duxbury (in that part which was set off as Pembroke in 1712) which was made a garrison house about 1679; the room and the firstplace at the right of the front door as you entered was said to have been built about 1630."  
Barker gives a further history of the family back to England; I only include his father's generation, because his brother John also immigrated at about the same date Robert did, in 1628-30. Both were early settlers at Plymouth, Massachusetts.
b abt 1616 based on terms of apprenticeship
d Duxbury btw 18 Feb 1689 (date of will) and 14 Mar 1691/2 9date of division of lands to the sons of his son Isaac)
m by 1663 (and by 1642 if she was mother of all his children) Luce (or Lucy) _____; no other record her name, but she was still living 7 Mar 1681/2 when "Lieut. Robert Barker, in behalf os his mother, the wife of Robert Barker, Senior, is fined L2 10s. for that his said mother sold cider to the Indians, contrary to the law of this government." [PCR 6:82]
Most secondary sources identify her as dau of John and Anna Williams of Scituate, but she is not named in the will of john Williams; the confusion may arise because Mary Williams, dau of John, did marry John Barker, brother of Robert... ]
From THE DESCENDANTS OF ROBERT BARKER
The first mention of Robert Barker is found in the Colonial Records under the date Jan. 20, 1632: "Robert Barker, servt. of John Thorp, complayned of his Mr. for want of clothes. The complaint being found just, it was ordered that Thorp should either foorthwith apparell him. or else make over his time to some other that was aide to provide for him." He was later bound to William Palmer as a carpenter's apprentice, and his time was out Apr. 1. 1637. This date probably marks the attainment of his majority, so that we may suppose him to have been born in 1616.
In 1641 he with others, bought from Jonathan, son of Elder Brewster, a ferry and one hundred acres of land at Marshfield, and in 1643 he was a member of the military company in that town under Lieut. Nathaniel Thomas. He held the office of surveyor in Marshfield in 1645 and 1648, constable in 1646, and was admitted freeman in 1654. In 1648 he bought a house and land for 45 shillings, having been licensed in 1646 to keep an inn to retail wine. Although he did not entirely give up his interests in Marshfield, where he kept his liquor license until 1666 and was grandjuryman in 1669, he seems to have removed to Duxbury about 1650-1655, for Aug. 2, 1653, he desired the laying out of lands at Namasskeeset (v. p. 7). There is a tradition that he came with Dolor Davis and a negro, and stayed the first winter in a dug-out. He appears as surveyor of Duxbury from 1654 to 1656, in 1672, 1677, 1679. and grandjuryman 1684-5. The court at Plymouth. Mar. 5, 1667-8. granted him 9 1-2 acres of land at Robinson's Creek, North River, Duxbury, which was perhaps the nucleus of his estate. He prospered in his new home, and left at his deadi. in [691, ±142, is. [id. He married Lucy Williams, who died between Mar. 7. 1681-2, when she was fined for selling cider to the Indian-, and Feb. 18. 1689, the date of her husband's
will. The inventory of his estate was taken Mar. 15, 1 691/2.
Isaac, b say 1642; m Plymouth 28 Dec 1665 Judith Prence, dau of Thomas. 1710 (inv. June 5).
Francis, b abt 1646; m Duxbury 5 Jan 1674 Mary Lincoln, dau of Thomas of Hingham. d. 1720-1 (adm. Feb. 9) 
Rebecca, b abt 1650; m1 by 1670 JosiahSnow (eldest born 6 Dec 1670; her name given incorrectly in some sources as Baker; m2 Marshfield 23 ___ 1694 John Sawyer. d. 1697
Robert b abt 1651 (d Duxbury 25 Sep 1729 a. 78); m1 by 1682 Alice Snow, dau of Anthony and Abigail (Warren) Snow, and g'dau of Richard Warren; m2 Jamestown 7 Oct 17005 Phebe Marsh, who was Phebe (Cook) (Arnold) Marsh, dau of Thos and Thomasin (___) Cook, widow of Oliver Arnold and Jonathan Marsh. Born Feb. 27, 1650-1, d. Sept. 25. 1729 [P. F.] 
Abigail, b abt 1657; m by 1677 Joseph Rogers, son of John Rogers of Marshfield. "... Robert Barker, whose children were Francis, Isaac, Robert, Rebecca and Abigail, in his will d. Feb'y 18, 1689, mentions his daughter, Abigail Rogers...."  d. May, 1718.
There is a Robert Barker (UK) created that shows his parents as Catherine (Acworth) Barker and Robert Barker. It is possible that Robert Barker (US) and Robert Barker (UK) are the same person but the relationship is not proven and the estimates of birth date differ by approximatley 15 years. This note has been added to acknowledge the uncertainty and to prevent further confusion. If it turns out in the future that these two can be merged, sobeit!
↑ Robert Charles Anderson, The Great Migration Directory: Immigrants to New England, 1620–1640 (Boston, Massachusetts. New England Historic and Genealogical Society. 2015)
A History of the Town of Duxbury, Massachusetts, with Genealogical Registers, Justin Windsor, (Boston: Crosby & Nichols, 1849), 223.
New England Historical and Genealogical Register, James Atkins Noyes.
New England Families: Genealogical and Memorial, William Richard Cutter, (New York: Lewis), 512.
New England Historical and Genealogical Register, James Atkins Noyes, 3-4.
Boston Evening Transcript: Genealogy Series
Find A Grave, database and images (https://www.findagrave.com : accessed 27 June 2019), memorial page for Robert Barker (1616–1690), Find A Grave: Memorial #34697547; Maintained by Linda Mac (contributor 47062703) Unknown. (No gravestone photo, no sources.)