Kinematic-Mapping Based Solution to Various Mixed-Exact-and-Approximated Problem in Planar Motion Synthesis

2016 ◽  
Author(s):  
Ping Zhao ◽  
Xiangyun Li ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Constraint Equation ◽  
Parallel Manipulators ◽  
Linear Constraint ◽  
Surface Fitting ◽  
Motion Synthesis ◽  
Analysis Technique ◽  
Additional Task ◽  
Kinematic Mapping ◽  
Fitting Problem

The design of mechanisms that lead a rigid-body through a set of prescribed discrete poses is usually referred to as “motion synthesis”. In practical motion synthesis cases, aside of realizing a set of given poses, various types of geometric constraint conditions could also require to be satisfied, e.g. defining the coordinates of the center/circle points of dyad linkages, setting the ground line/coupler line for four-bar linkages, realization of additional task positions, etc. Some of these constraint conditions require to be realized exactly while others might allow approximation. To solve this mixed-exact-and-approximated problem, this paper proposed a kinematic-mapping-based approach, which builds on the previous work of the realization of an arbitrary number of approximated poses as well as up to four exact poses. We now have found that the aforementioned various types of constraint conditions could be converted to each other through a general linear constraint equation. Thus, those “approximated conditions” could be uniformly converted to several prescribed discrete poses so as to be formulated as a general approximated motion synthesis problem, which is actually a general quadratic surface fitting problem in kinematic-mapping space, while up to four “exact conditions” could be imposed as linear constraint equations to this surface fitting system such that they could be exactly realized. Through null-space analysis technique, both type and dimensions of the resulting optimal dyad linkages could be determined by the solution of this surface-fitting problem with constraints. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators.

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping

2016 ◽  
Vol 8 (5) ◽  
Author(s):  
Ping Zhao ◽  
Xin Ge ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Dimensional Synthesis ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
Mapping Theory ◽  
Discrete Motion ◽  
Fitting Problem

It has been well established that kinematic mapping theory could be applied to mechanism synthesis, where discrete motion approximation problem could be converted to a surface fitting problem for a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach is presented, aiming at the problem of mixed exact and approximate motion realization with three types of planar dyad chains (RR, RP, and PR). A two-step unified strategy is established: first N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, then up to four exact poses could be imposed as pose-constraint equations to this surface fitting system such that they could be strictly satisfied. The former step, the surface fitting problem, is converted to a linear system with two quadratic constraint equations, which could be solved by a null-space analysis technique. On the other hand, the given exact poses in the latter step are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

On the Complete Synthesis of Finite Positions With Constraint Decomposition via Kinematic Mapping

2014 ◽  
Author(s):  
Ping Zhao ◽  
Q. J. Ge ◽  
Anurag Purwar
Keyword(s):  
Null Space ◽  
Constraint Equation ◽  
Linear Constraint ◽  
Quadratic Equations ◽  
Additional Task ◽  
Prismatic Joints ◽  
Kinematic Mapping ◽  
Position Synthesis ◽  
Additional Constraints ◽  
Novel Algorithm

In this paper, we revisit the classical Burmester problem of the exact synthesis of a planar four-bar mechanism with up to five task positions. A novel algorithm is presented that uses prescribed task positions to obtain “candidate” manifolds and then find feasible constraint manifolds among them. The first part is solved by null space analysis, and the second part is reduced to finding the solution of two quadratic equations. Five-position synthesis could be solved exactly with up to four resulting dyads. For four-position synthesis, a limited number of solutions could be selected from the ∞1 many through adding an additional linear constraint equation without increasing the computational complexity. This linear constraint equation could be obtained either by defining one of the coordinates of the center/circle points, by picking the ground line/coupler line, or by adding one additional task position, all of which are proved to be able to convert into the same form as in (23). For three-position synthesis, two additional constraints could be imposed in the same way to select from the ∞2 many solutions. The result is a novel algorithm that is simple and efficient, which allows for task driven design of four-bar linkages with both revolute and prismatic joints, as well as handling of different kinds of additional constraint conditions in the same way.

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization via Kinematic Mapping

2015 ◽  
Author(s):  
Ping Zhao ◽  
Xin Ge ◽  
Bin Zi ◽  
Q. J. Ge
Keyword(s):  
Null Space ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Realization Problem ◽  
Constraint Equations ◽  
Kinematic Mapping ◽  
Motion Approximation ◽  
Mapping Theory ◽  
Discrete Motion ◽  
Fitting Problem

It has been well established that kinematic mapping theory could be applied in mechanism synthesis area, where discrete motion approximation problem could be converted to surface fitting problem of a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach for mixed exact and approximate motion realization problem for three types of planar dyad chains (RR, RP, PR) is presented. For all three types of dyads, N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, while up to four prescribed poses could be imposed as pose-constraint equations to this surface fitting system such that they could be exactly realized. The surface fitting problem is converted to a linear system with two quadratic constraint equations, which could be solved by null space analysis technique. On the other hand, the given exact poses are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

A Task Driven Approach to Simultaneous Type Synthesis and Dimensional Optimization of Planar Parallel Manipulator Using Algebraic Fitting of a Family of Quadrics

2013 ◽  
Author(s):  
Xiangyun Li ◽  
Ping Zhao ◽  
Q. J. Ge ◽  
Anurag Purwar
Keyword(s):  
Least Squares ◽  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Surface Fitting ◽  
Image Space ◽  
Synthesis Process ◽  
Dimensional Synthesis ◽  
Planar Parallel Manipulator ◽  
Kinematic Mapping ◽  
The Given

This paper studies the rigid body guidance problem for 3-DOF planar parallel manipulators (PPM) with three-triad assembly. We present a novel, unified, and simultaneous type and dimensional synthesis approach to planar parallel manipulator synthesis by using kinematic mapping, surface fitting, and least squares techniques. Novelty of our approach lies in linearization of a highly non-linear problem and the fact that the nature of the given motion or displacement drives the synthesis process without assuming triad topology or their geometry. It has been well established that by using planar quaternions and kinematic mapping, workspace related constraints of planar dyads or triads can be represented as algebraic constraint manifolds in the image space of planar displacements. The constraints associated with planar RR-, PR- and RP-dyads correspond to a single quadric in the image space, while that of each of the six planar triads (RRR, RPR, PRR, PPR, RRP and RPP) map to a pair of quadrics and the space between them. Moreover, the quadrics associated with RRR- and RPR-triads are of the same type as that of RR dyads, of PRR- and PPR-triads as that of PR-, and RRP- and RPP-triads as that of RP-dyad. This simplification nicely extends a dyad synthesis problem to a triad synthesis one. The problem is formulated as the least-squares error minimization problem to find a trinity of quadrics that best fit the image points of task displacements. The fitting error corresponding to each single quadric of the trinity is regarded as variation (thickness) of that quadric, which turns that quadric into a pair of quadrics. Hence, three dyads with minimal surface fitting errors can be converted to three triads in the Cartesian space.

Time-Differentiable Null Space Method for Constrained Mechanical Systems (Application to a Rheonomic System)

2013 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Constraint Equation ◽  
Coefficient Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the coefficient matrix which appears in the constraint equation in velocity level. In a previous report, the author presented a method to obtain a time differentiable null space matrix for scleronomic systems, whose constraint does not depend on time explicitly. In this report, the method is generalized to rheonomic systems, whose constraint depends on time explicitly. Finally, the presented method is applied to four-bar linkages.

Analysis of Active Joint Failure in Parallel Robot Manipulators

2004 ◽  
Vol 126 (6) ◽  
pp. 959-968 ◽  
Author(s):  
Mahir Hassan ◽  
Leila Notash
Keyword(s):  
Jacobian Matrix ◽  
Null Space ◽  
Robot Manipulators ◽  
Parallel Manipulators ◽  
Parallel Robot ◽  
Task Space ◽  
Active Joint ◽  
Joint Failure ◽  
Space Vectors ◽  
Six Degree Of Freedom

In this study, the effect of active joint failure on the mobility, velocity, and static force of parallel robot manipulators is investigated. Two catastrophic active joint failure types are considered: joint jam and actuator force loss. To investigate the effect of failure on mobility, the Gru¨bler’s mobility equation is modified to take into account the kinematic constraints imposed by various branches in the manipulator. In the case of joint jam, the manipulator loses the ability to move and apply force in a specific portion of its task space; while in the case of actuator force loss, the manipulator gains an unconstrained motion in a specific portion of the task space in which an externally applied force cannot be resisted by the actuator forces. The effect of joint jam and actuator force loss on the velocity and on the force capabilities of parallel manipulators is investigated by examining the change in the Jacobian matrix, its inverse, and transposes. It is shown that the reduced velocity and force capabilities after joint jam and loss of actuator force could be determined using the null space vectors of the transpose of the Jacobian matrix and its inverse. Computer simulation is conducted to demonstrate the application of the developed methodology in determining the post-failure trajectory of a 3-3 six-degree-of-freedom Stewart-Gough manipulator, when encountering active joint jam and actuator force loss.

Design of Planar 1-DOF Cam-Linkages for Lower-Limb Rehabilitation Via Kinematic-Mapping Motion Synthesis Framework

2019 ◽  
Vol 11 (4) ◽  
Author(s):  
Ping Zhao ◽  
Lihong Zhu ◽  
Bin Zi ◽  
Xiangyun Li
Keyword(s):  
Lower Limb ◽  
Large Deviation ◽  
Motion Synthesis ◽  
Kinematic Constraint ◽  
Walking Motion ◽  
Kinematic Mapping ◽  
Limb Rehabilitation ◽  
The Given ◽  
Additional Constraints ◽  
Lower Limb Rehabilitation

When designing linkage mechanisms for motion synthesis, many examples have shown that the optimal kinematic constraint on the task motion contains too large deviation to be approximately viewed as a single rotational or translational pair. In this paper, we seek to adopt our previously established motion synthesis framework for the design of cam-linkages for a more accurate realization, while still maintaining a 1-degree-of-freedom (DOF) mechanism. To determine a feasible cam to lead through the task motion, first a kinematic constraint is identified such that a moving point on the given motion traces a curve that is algebraically closest to a circle or a line. This leads to a cam with low-harmonic contour curve that is simple and smooth to avoid the drawbacks of cam mechanisms. Additional constraints could also be imposed to specify the location and/or size of the cam linkages. An example of the design of a lower-limb rehabilitation device has been presented at the end of this paper to illustrate the feasibility of our approach. It is shown that our design could lead the user through a normal walking motion.

Motion Synthesis of Mechanisms Using Constraint Manifolds

1994 ◽  
Author(s):  
Y. K. Wu ◽  
Ian S. Fischer
Keyword(s):  
Curve Fitting ◽  
Optimization Algorithms ◽  
Image Space ◽  
Motion Synthesis ◽  
Numerical Examples ◽  
Synthesis Of Mechanisms ◽  
Normal Distance ◽  
Kinematic Mapping

Abstract The motion synthesis of a mechanism is obtained by curve fitting the intersection of the constraint manifolds representing each leg of the linkage in an image space to points obtained by the kinematic mapping of the prescribed positions and orientations. The dimensions of the mechanism can be found by using optimization algorithms to minimize the normal distance between all the desired image points and image curve of the tracer frame. The theory is illustrated by numerical examples.

The DOMAIN family of propagation operations for intervals on simultaneous linear equations

1995 ◽  
Vol 9 (3) ◽  
pp. 197-210 ◽  
Author(s):  
R. Chen ◽  
A.C. Ward
Keyword(s):  
Linear Equations ◽  
Constraint Equation ◽  
Linear Constraint ◽  
The Other ◽  
Interval Matrix ◽  
Domain Family ◽  
Output Vector ◽  
Constraint Equations ◽  
Simultaneous Linear Equations ◽  
Mathematical Operations

AbstractThis paper defines, develops algorithms for, and illustrates the utility in design of a class of mathematical operations. These accept as inputs a system of linear constraint equations, Ax = b, an interval matrix of values for the coefficients, A, and an interval vector of values for either x or b. They return a set of values for the “domain” of the other vector, in the sense that all combinations of the output vector values set and values for A, when inserted into the constraint equation, correspond to values for the input vector that lie within the input interval. These operations have been mostly overlooked by the interval matrix arithmetic community, but are mathematically interesting and useful in the design, for example, of structures.

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