Jacobian-based stiffness analysis method for parallel manipulators with non-redundant legs

Abstract

Stiffness is an important element in the model of a parallel manipulator. A complete stiffness analysis includes the contributions of joints as well as structural elements. Parallel manipulators potentially include both actuated joints, passive compliant joints, and zero stiffness joints, while a leg may impose constraints on the end-effector in the case of lower mobility parallel manipulators. Additionally, parallel manipulators are often designed to interact with an environment, which means that an external wrench may be applied to the end-effector. This paper presents a Jacobian-based stiffness analysis method, based on screw theory, that effectively considers all above aspects and which also applies to parallel manipulators with non-redundant legs.

Consistent modeling resolves asymmetry in stiffness matrices

“…[29] are replicated using the Jacobian-based stiffness analysis method introduced in Ref. [12]. The general formulation of this analysis method is repeated here for convenience, namely…”

“…The vector t e is the joint torque vector where all entries associated to zero stiffness joints were removed (see Ref. [12] for more details), J −1 is the 6N × 6 full inverse Jacobian, and J −1 e is the inverse Jacobian of elasticity that maps an end-effector displacement twist on dq e , which is the joint displacement vector in which all entries associated to zero stiffness joints are removed. Finally, matrix K q,e is the stiffness matrix of the parallel manipulator expressed in the space spanned by the elastic joint coordinates, which is a diagonal matrix whose entries are the spring stiffness values if structural stiffness is not considered.…”

“…For serial manipulators, this dependency on loading was expressed by Chen and Kao [26] in a term that is a function of the derivative of the Jacobian, and which represents the change in the transmission of the already applied joint torques as a function of a displacement. The effect of loading on the Cartesian stiffness matrix has also been expressed for parallel manipulators [10,12,27]. The effect of loading on the static equilibrium conditions was investigated by Pashkevich et al [27] and Klimchik et al [28].…”

“…The popularity of these methods is explained by their compact, symbolic formulation of the Cartesian stiffness matrix as a function of a manipulator's configuration and joint stiffness values. Jacobian-based stiffness analysis methods have been developed both for serial manipulators [6,7] and parallel manipulators [8][9][10][11][12]. These stiffness analysis methods have been used for the analysis of a wide range of parallel manipulators, such as machining robots [1,13,14], haptic devices [15,16], and precision manipulators [17,18].…”

“…[29] are replicated using the analysis method introduced in Ref. [12], which results in symmetric stiffness matrices. Next, it is shown that the asymmetry in Ref.…”