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A non-zero element of the Lie algebra $\mathfrak{se}(3)$ of the specialEuclidean spatial isometry group $SE(3)$ is known as a {\em twist} and thecorresponding element of the projective Lie algebra is termed a {\em screw}.Either can be used to describe a one-degree-of-freedom joint between rigidcomponents in a mechanical device or robot manipulator. This leads to apractical interest in multiple twists or screws, describing the overallinstantaneous motion of such a device. In this paper, invariants of multipletwists under the action induced by the adjoint action of the group aredetermined. The ring of the polynomial invariants for the adjoint action of$SE(3)$ acting on a single twist is well known to be finitely generated by theKlein and Killing forms, while a theorem of Panyushev gives finite generationfor the real invariants of the induced action on two twists. However we are notaware of a corresponding theorem for $k$~twists, where $k\geq3$. FollowingStudy, we use the principle of transference to determine fundamental algebraicinvariants and their syzygies. We prove that the ring of invariants for tripletwists is rationally finitely generated by $13$ of these invariants.

Invariants of the $k$-fold adjoint action of the Euclidean group