Conventional Jacobian Matrix Research Articles

Optimal design of a six-axis vibration isolator via Stewart platform by using homogeneous Jacobian matrix formulation based on dual quaternions

A six-axis vibration isolation system is essential to high-precision space systems for attenuating vibrations on precise instruments. The kinematic optimal design is researched for the space six-axis vibration isolator via Stewart mechanism. Jacobian matrix is the basis of the kinematic performance index. However, the conventional Jacobian matrix is not usually dimensionally homogeneous due to the inhomogeneous physical units, caused by the different mathematical representations of the rotation and translation. In this paper, we propose a dual quaternion approach to derive the dimensionally homogeneous Jacobian matrix of a general six-axis parallel mechanism. Two quaternions are used to parameterize the rotation and translation of the platform. The dimensionally scaling factor for the generalized Jacobian matrix is defined as the ratio of the norms of the two quaternions. The dimensionally homogenous Jacobian matrix is then obtained and applied to the optimal design of the six-axis vibration isolator. The performance index of isotropy is considered to make the isolator minimum kinematic coupling in its working configuration.

Formulating an Invariant Manipulability Index of Gough-Stewart Platform

Manipulability indexes have been extensively used as objective functions or constraints in the optimal design of robots, especially for the robots with pure active joints and the end-effectors only involving positioning or orientation. The moving platform of Gough-Stewart platform can involve both positioning and orientation, which leads to the conventional Jacobian matrix with entries bearing disparate physical units, then the established manipulability index based on the conventional Jacobian matrix has no physical meaning and is variable under a change of units. Based on a dimensionless Jacobian matrix, this paper established a new manipulability index, which has physical meaning and is invariant under a change of units. In order to design a good performance motion simulator, the extreme values of the new and old manipulability indexes of 2 commercial flight simulators in their reachable workspaces were found by random search method, respectively. From the results, the extreme values of the new manipulability index are in regularity, and can be used as references in the optimal design of the Gough-Stewart platform as motion simulator.

Dexterity measures and their use in quantitative dexterity comparisons

The majority of proposed dexterity measures rely on the use of the condition number of the manipulator’s Jacobian matrix mapping actuator velocities to end effector velocities. Unfortunately, for the vast majority of manipulators, the conventional Jacobian matrix has inconsistent units making the aforementioned measures dependent on units and scale. Recently, a Jacobian matrix mapping the joint velocities to independent Cartesian velocity components of three points on the end effector was proposed. In most cases, this Jacobian matrix has consistent units and in many cases it is dimensionless. This dimensionally-homogeneous Jacobian matrix yields meaningful dexterity measures that allow quantitative dexterity comparisons between manipulators with different architectures. Although these new measures were originally proposed within the context of parallel manipulator design and analysis, they can also be used for serial architectures. In this paper, the analysis of the Tricept manipulator’s dexterous workspace, as a function of architectural variables, is provided through the use of a quantitative metric for dexterity. Furthermore, the generality of this metric is also demonstrated by first employing it to analyze a serial manipulator and then comparing it to a parallel manipulator having the same degrees-of-freedom. The paper also proposes a strategy to deal with singularities introduced mathematically by the novel Jacobian formulation.

Dimensional synthesis of a 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix

A study of dimensional synthesis of a 3-DOF parallel manipulator with coupling of translation and rotation is carried out. The architecture of the manipulator is composed of a moving platform attached to a fixed base through three identical PRS (prismatic-revolute-spherical) serial limbs, whose unique topology leads to the physical unit inconsistency of the conventional Jacobian matrix and the emergence of the parasitic motion. Then this paper introduces a kinetostatic performance index of the manipulator based on the condition of a dimensionally homogeneous Jacobian matrix, later, the workspace of the aforementioned manipulator is searched and the influence of the crucial design variables on the workspace is analyzed. Finally, a dimensional synthesis method of the manipulator is proposed, which may be regarded as a nonlinear programming problem with subject to the parasitic motion and other several engineering constraints.

New Jacobian Matrix and Equations of Motion for a 6 d.o.f Cable-Driven Robot

In this paper, we introduce a new method and new motion variables to study kinematics and dynamics of a 6 d.o.f cable-driven robot. Using these new variables and Lagrange equations, we achieve new equations of motion which are different in appearance and several aspects from conventional equations usually used to study 6 d.o.f cable robots. Then, we introduce a new Jacobian matrix which expresses kinematical relations of the robot via a new approach and is basically different from the conventional Jacobian matrix. One of the important characteristics of the new method is computational efficiency in comparison with the conventional method. It is demonstrated that using the new method instead of the conventional one, significantly reduces the computation time required to determine workspace of the robot as well as the time required to solve the equations of motion.

Force transmission analyses with dimensionally homogeneous Jacobian matrices for parallel manipulators

To avoid the unit inconsistency problem in the conventional Jacobian matrix, new formulation of a dimensionally homogeneous inverse Jacobian matrix for parallel manipulators with a planar mobile platform by using three end-effector points was presented (Kim and Ryu, 2003). This paper presents force relationships between joint forces and Cartesian forces at the three End-Effector points. The derived force relationships can then be used for analyses of the input/output force transmission. These analyses, forward and inverse force transmission analyses, depend on the singular values of the derived unit consistent Jacobian matrix. Using the proposed force relationship, a numerical example is presented for actuator size design of a 3-RRR planar parallel manipulator.

New dimensionally homogeneous jacobian matrix formulation by three end-effector points for optimal design of parallel manipulators

Development of optimal design methods for parallel manipulators is important in obtaining an optimal architecture or pose for the best kinetostatic performance. The use of performance indexes such as the condition number of the conventional Jacobian matrix that is composed of nonhomogeneous physical units, however, may lack in physical significance. In order to avoid the unit inconsistency problem in the conventional Jacobian matrix, we present a new formulation of a dimensionally homogeneous Jacobian matrix for parallel manipulators with a planar mobile platform by using three end-effector points that are coplanar with the mobile platform joints. The condition number of the new Jacobian matrix is then used to design an optimal architecture or pose of parallel manipulators for the best dexterity. An illustrative design example with a six-degree-of-freedom Gough-Stewart platform parallel manipulator by using the proposed formulation is shown to generate the same optimal configurations as those from using the other existing dimensionally homogenous Jacobian formulation methods.