What is this war machine shown on Trajan's column?

It seems to be not entirely clear. But it also seems to represent an unusual depiction of an otherwise well known instrument:

Scene of a siege from Trajan's Column (Rome). The defenders roll barrels and tree trunks downhill (top right), but these are intercepted by three peculiar machines. The German scholar Otto Lendle made the astute observation that these are likely, to represent Apollodorus' 'ship's prow tortoises'. (C. Cichorius, Die Reliefs der Traianssaule, Berlin 1900).
_{Duncan B Campbell: "Besieged. Siege Warfare in the Ancient World", Osprey: Oxford, New York, 2006, p197.}

Another view of the relevant picture:

_{File:085 Conrad Cichorius, Die Reliefs der Traianssäule, Tafel LXXXV (Ausschnitt 01).jpg}

In the original called the Schiffsbugschildkröten of Apollodorus. If true then it would be one of his Poliorketika, another form of testudo or (chelone).

This work of Apollodorus seems to be this one:

or earlier:

Source for last two pictures: British Library, Burney MS 69, ff 33v-52r

Apollodorus of Damascus, Poliorcetica (TLG 1165.001), imperfect, lacking the final paragraph (193.1). Twenty-eight coloured drawings. Tit. ᾽Εκ τῶν ᾽Απολλοδώρου πολιορκητικὰ.

Personally, I'd been lead to believe this looks most likely like a crossbreed between an early trebuchet and some kind a wrecking ball, although both are not recognised at such an early date.

The explanatory text given on the site where the picture in the question was found says:

TRAJAN'S COLUMN LXXXV/CXIV (Scene 114): War Machines
Scene 114/CXIV: Close view of odd machines that have been interpreted variously as serving offensive or defensive purposes: "anxious to show these things to the best advantage, the Maestro has raised the line of the wall almost to the top of the scroll, leaving him just enough room to display their component parts, but no room to show how they were operated and by which side" (Lepper/Free 1988: 165).
From cast 308, now in the Museo della Civiltà Romana, Rome. Compare Cichorius Pl. LXXXV, scene 114 and Coarelli Pl. 140. Ref: RBU2011.8203

_{Source trajan's Column.org Quoting: Lepper, F. and S. Frere. 1988, (rev.) 2015. Trajan ‘s Column : A New Edition of the Cichorius Plates. Gloucester, UK and Wolfeboro, NH. Revised edition 2015: Fonthill media.}

But the more widespread conjecture is that:

The curious wheeled machine in scene CIV is stylistically very similar to a surviving drawing from Apollodorus' Poliorcetica and may represent another direct input from this architect, *albeit one misunderstood by the sculptor.[…]
The strange device depicted in front of the polygonal masonry wall in scene CXIV has attracted a great deal of scholarly speculation, non of which has solveed the problem of its function. It corresponds with no known form of ancient artillery, wall-batterer, pile-driver or defensive engine. Detached examination reveals no logical explanation of its working parts. Its main compositional role on the frieze seems to have been to fill the empty space above the group of Roman soldiers in the scene and the sculptor himself evidently did not have much idea of what he was depicting. The triangular frameworks of beams with associated circular objects are most reminiscent of the tread-mill cranes which would have been visible around the forum-basilica building site, but this observation only makes the device's obscurity more surprising, given that such cranes were intelligibly depicted elsewhere in Roman sculpture. A surviving manuscript illustration from Apollodorus of Damascus's Poliorcetica pictures a siege tower on little wheels like those in scene CXIV, and employs the same overall perspective style. This again suggests Apollodorus' direct involvement with the frieze and further reveals architect's general knowledge of military engineering. It does not, however, provide enlightenment as to the function of the device in scene CXIV.

_{J. C. Coulston: "Trajan's Column: The Sculpting and Relief Content of a Roman Propaganda Monument", Thesis, Newcastle University Library: Newcastle upon Tyne, 1988, p 161–162. (PDF)}

This actually came up in discussion when I was studying the archaeology of Roman Britain.

In fact, as far as I'm aware, nobody knows for certain, but several archaeologists that I have spoken to suggested that they were meant to be three-wheeled plutei, which were screens intended to protect the attacking force besieging a city.

The objects can be seen here in context against the walls of the city:

• Image source Wikimedia

Plutei and other forms of protective shield are described in Vegetius' Epitome of Military Science (pp128-129). In fact, in the linked edition, in the footnotes on page 129 the editor, N.P Milner, notes that:

The triangular chassis with three wheels which are illustrated on Trajan's column, Cichorius Pl. LXXXV scene cxiv, may be intended to represent plutei.

It is worth noting that the archaeologist Ian Richmond, then at the British School at Rome, was of a different opinion. He did not consider that the objects were part of the Roman siege equipment at all.

In his paper Trajan's Army on Trajan's Column, published in Papers of the British School at Rome Vol. 13 (1935), pp. 1-40, he discusses the features of the objects and concludes that it more likely that the objects represented festucae, or 'pounders' used in the construction of the Murus Gallicus, and as such, would have been used by the Dacian defenders, rather than the Romans attacking the city of Sarmisegetusa.

Such devices are mentioned by Vitruvius in Book 3 of his De architectura where he describes the construction of temple foundations:

"Intervalla autem concamaranda aut solidanda festucationibus, uti distineantur"

translation:

"The spaces between the columns are to be arched over, or made solid by being rammed down, so that the columns may be held apart."

Professor Richmond went on to argue that:

"... their presence here will indicate that this part of the fortification was newly completed, shortly before the arrival of the Romans."

This interpretation has the advantage that it does at least explain all the visible features of the objects but, as I said earlier, it is not possible to be certain.