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Design and Kinematic Analysis of the Novel Almost Spherical Parallel Mechanism Active Ankle


The novel mechanism Active Ankle features three degrees of freedom that operate in an almost spherical manner. In comparison to spherical devices, its design offers advantages such as high stiffness, a simple and robust construction, and a good stress distribution. In the present paper, a comprehensive study of the design, analysis, and control of the Active Ankle in its almost-spherical work modality is provided. In particular, the kinematic analysis of the mechanism is conducted, solving the full inverse, the rotative inverse, and the forward kinematic problems. In addition, the manipulator’s workspace is characterized and the kinematic control, that has been implemented on a prototype of Active Ankle, is presented together with experimental results that demonstrate the employability as an ankle joint in a full body exoskeleton.

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The work presented in this paper was performed within the project Recupera-Reha, funded by the German Aero-space Center (DLR) with federal funds from the Federal Ministry of Education and Research (BMBF) (Grant 01-IM-14006A).

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Correspondence to Shivesh Kumar.


Appendix A: Reference Configurations

A set of reference configurations of the mechanism Active Ankle is displayed in Table 10.

Table 10 Reference configurations with joint angles, end-effector poses, and point coordinates

Appendix B: Sphere Intersection Problem

The method SphInt in Alg. 9, called from Alg. 5, computes the intersection of three spheres using distance geometry. It is based on the quadrance [42] between two vectors a and b,

$$Q({\boldsymbol{a}, \boldsymbol{b}}) = (\boldsymbol{b} - \boldsymbol{a})\ast (\boldsymbol{b} - \boldsymbol{a}), $$

the squared distance between a and b, and its generalization to vector sets, the Cayley–Menger bideterminants [38].

Appendix C: Representation Conversion Methods

The conversion methods mat2vec and vec2mat, called from Alg. 7 and Alg. 8, are stated in Alg. 12 and Alg. 13.

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Kumar, S., Bongardt, B., Simnofske, M. et al. Design and Kinematic Analysis of the Novel Almost Spherical Parallel Mechanism Active Ankle. J Intell Robot Syst 94, 303–325 (2019).

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