A branch-and-prune method to solve closure equations in dual quaternions

Journal Article (2021)


Mechanism and Machine Theory





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Using dual quaternions, the closure equations of a kinematic loop can be expressed as a system of multiaffine quations. In this paper, this property is leveraged to introduce a branch-and-prune method specially tailored for solving such systems of equations. The new method is objectively simpler (in the sense that it is easier to understand and to implement) than previous approaches relying on general techniques such as interval Newton methods or methods based on Bernstein polynomials or linear relaxations. Moreover, it relies on two basic operations —linear interpolation and projection onto coordinate planes— that can be efficiently computed. The generality of the proposed method is evaluated on position analysis problems with 0-, 1-, and 2-dimensional solution sets, including the inverse kinematics of serial robots and the forward kinematics of parallel ones. The results obtained on these problems show that the efficiency of the method compares favorably to state-of-the-art alternatives.



Author keywords

Position analysis, multiloop mechanisms, forward kinematics, inverse kinematics, kinematotropic mechanisms, multiaffine systems, branch-and-prune methods

Scientific reference

A. Shabani, J.M. Porta and F. Thomas. A branch-and-prune method to solve closure equations in dual quaternions. Mechanism and Machine Theory, 164: 104424, 2021.