Mobility Analysis of Kinematic Chains

There is no dearth of methods to test isomorphism amongst kinematic chains. Search for a computationally easier, logically simple and unique method is still on. Present work is in quest of a reliable test to detect isomorphism among kinematic chains. Work presented here is more versatile as it incorporates more features of the kinematic chain which were not included earlier such as number and type of links, their relative dispositions in the kinematic chain, nature of adjacent links etc. The method proposed is based on the concept of pseudo-probability (pseudo means it appears to be, but not exactly. The approach does not follow in-toto the principles of probability and considerable liberty has been taken in interpreting the word probability hence the word pseudo is used along with the probability schemes). Using the resemblance of different coloured balls in an urn for the number and type of links in a kinematic chain, a matrix (named P-Matrix) representing the kinematic chain in totality is generated. For the sake of comparison a numerical scheme named, pseudo probability scheme, P-Scheme, is developed from the above P-Matrix and is used for testing isomorphism. In fact the method is more powerful in the sense that each row of the proposed P-Matrix is capable of representing the respective kinematic chain distinctly and can be used to compare the kinematic chains with same link assortments, uniquely. The proposed method, besides possessing the potential of testing the isomorphism among simple-joint, single degree of freedom kinematic chains is also capable of multi degrees of freedom and multiple-joint kinematic chains.

Abstract. The similarity recognition of kinematic chains (KCs) is helpful for improving the efficiency of configuration synthesis, which has been paid more and more attention in recent years. The existing recognition methods are divided into the definition method and feature constant method. Among them, the definition method is difficult to adopt in practice because of its long operation time, especially when the number of similar vertices in KCs is large. In this paper, the new concepts of a loop tree (LT) and a loop tree matrix (LTM) have been proposed, which improve the efficiency of similarity recognition. This method is applied on the complete structure of the following: 8-link with 1 DOF (degree of freedom), 9-link with 2 DOF, 10-link with 1 DOF, 12-link with 1 DOF, 13-link with 2 DOF, 14-link with 3 DOF, 15-link with 4 DOF planar single-joint KCs, and contracted graphs with up to six independent loops. All results are verified by the definition method to prove the good applicability, reliability, and efficiency of the proposed method. Simultaneously, the application case of the similarity recognition in a mechanism creation is given to provide a reference for an innovative design.

A number of distinct or non-isomorphic kinematic chains exist for a specified number of links and joints. For example, sixteen distinct chains can be obtained with eight links and two hundred and thirty chains with ten links having a single degree of freedom. Similarly, many space mechanisms can be formed with four links and joints having different degrees of freedom. So far no measure is available to know which of these possesses greater mobility or flexibility. Flexibility is not to be confused with the degree of freedom. Intuitively one feels that a six-link chain has greater flexibility than a four-bar chain both having the same degrees of freedom. Though the mobility of a chain increases with the number of links one is not sure how the structural arrangement, type of links and joints, their numbers and sequence etc. influence the same. Combining graph theory with the concepts of probability, simple formulae are developed to investigate the relative merits of spatial and planar kinematic chains. The greater the flexibility or mobility of the chain, the higher is the ability to meet the motion requirements, i.e., a chain having greater entropy can be expected, say, to reproduce a given function more accurately.

The family of planar kinematic chains with two multiple joints

In a kinematic chain, the links are connected to each other through joints. The connectivity of a joint indicates the number of joints to which it is connected. The connectivity level of a joint indicates the distance by which it is separated from the adjacent joints. The concept of joint connectivity and its application to detect isomorphism among kinematic chains and their inversions has been already reported by authors. The method utilizes the connectivity of joints at different levels to detect isomorphism and inversions among planar kinematic chains. The method is applied to eight-, nine-, and ten-link planar kinematic chains. The results so obtained are in agreement with those available in the literature. In this study, the method is further improved by incorporating the type of joint to make it more effective for the detection of isomorphism and distinct inversions. A joint connectivity table completely representing the kinematic chain is proposed. The application of the method is extended for the determination of additional topological characteristics of chains such as categorization of kinematic chains and selection of preferred frame, input and output links for function and path generation. The concept of ‘Motion Transfer Ability’ is introduced and utilized to develop numerical measures for comparing and categorizing the chains at the synthesis stage of mechanism design for a specific application. The method was successfully tested on planar kinematic chains with single and multiple degrees of freedom and the results for eight- and nine-link kinematic chains are appended.

Recently creation continuously variable transmission (CVT) having property of the adaptation to variable technological loading develops. Traditional CVT contains the hydraulic converter and gear differential with two degrees of freedom (patents of Crockett, Volkov). The hydraulic converter imposes differential constraint on movement of links of the mechanism with two degrees of freedom. It provides definability of movement of transmission. Take place progressive CVT without use of the hydraulic converter (Harries’s patent, Ivanov’s patent). Such transmission contains only gear differential with two degrees of freedom. However the theoretical substantiation of serviceability of such transmission is not convincing for the lack of the proof of static definability of the mechanism with two degrees of freedom, having only one input. Ivanov K.S.’s theoretical researches do not contain enough convincing arguments of presence of definability of such mechanisms. Mechanisms with two degrees of freedom are considered disabled because of their contradiction to a key rule of mechanics that the number of the generalized coordinates should be equal to number of degrees of freedom of the mechanism (that is it should be equal to two, instead of to one). In Ivanov K.S.’s early works it was mentioned presence of paradox of the mechanics reflecting essence of the specified discrepancy. However in these works there was no the elementary demonstrative base of essence of the phenomenon resulting in static definability. The purpose of the present work to describe essence of the phenomenon which in certain conditions result to static definability of the mechanism with two degrees of freedom at presence only of one input, to receive the theoretical description of paradox of mechanics and its authorization using the elementary laws of theoretical mechanics. In the work the presence of paradox of mechanics is proved. The paradox of mechanics takes place in the kinematic chain having the closed contour, and it consist in the following. The structural group with two degrees of freedom having two consistently connected links is statically definable at presence only one entrance link. The paradox of mechanics is authorized by the account of additional constraint which is imposed by kinematic chain with the closed contour and two degrees of freedom on movement of links by a principle of possible works. Use of the equations of a statics and additional constraint allows to receive the equations of interrelation of parameters of transmission and to execute synthesis of transmission on the set conditions of motion. The found laws will allow creating the transfer mechanisms having property of mechanical adaptation to variable technological loading. Mechanical adaptation provides an opportunity of motion of an output link with a speed inversely force of resistance at constant input power.

Parallel-structure mechanisms with four degrees of freedom and four kinematic chains are discussed. The synthesis of these manipulators were performed using a device of closed screw groups. This approach allows avoiding complex equations of synthesis and analysis of singularities. Kinematic chains may impose equal constraints or different constraints, and each may contain a drive. The main result of this article is the synthesis of a new parallel-structure mechanism and a screw analysis of constraints imposed by the kinematic chains.

Force adaptation takes place in a kinematic chain with two degrees of freedom, but only with one input. Such system is considered indefinable. However, if closed contour containing four links is placed between input and output links, then such system has all parameters definability. The closed contour allows making conditions of balance for two links of a contour by a principle of virtual works. These conditions include external and internal forces and are addition in one condition. Work of internal forces for ideal system is equal zero. It leads to receipt of additional constraint in the form of equality to zero of work of external forces. The kinematic chain with two degrees of freedom gets definability of all parameters, and also gets property of force adaptation. - Output link is in motion with a speed, inversely proportional to loading, only at the expense of force properties. Force force adaptation occurs by redistribution of speeds of motion of links in the closed contour under the acting of variable loading. The present work is devoted to a brief description of the theory of force adaptation.

A new atlas for 8-bar kinematic chains with up to 3 prismatic pairs using Joint Sorting Code

It is of great importance in the conceptual creative design of mechanical systems to synthesize as many feasible kinematic structures of mechanisms as possible. However, the methods for the structural synthesis of multiple joint kinematic chains are seldom addressed in literature even though they are widely used in various mechanical products. This paper proposes an automatic method to synthesize planar multiple joint kinematic chains. First, the bicolor topological graph and the bicolor contracted graph are introduced to represent the topological structures of multiple joint kinematic chains. Then, the characteristic number string of bicolor topological graphs is proposed and used to efficiently detect isomorphism in the synthesis progress. Finally, a systematic method for the synthesis of kinematic chains with one multiple joint is proposed, and the whole families of multiple joint kinematic chains with up to 16 links and all possible degrees of freedom are synthesized for the first time.

Addition method with 2 links and 3 pairs of type synthesis to planar closed kinematic chains

An additive approach to find distinct mechanisms of a planar kinematic chain

Dynamic balance has been studied to eliminate the shaking forces and vibration at the base induced by rapid motion of robotic devices. This is done by designing the mass distribution such that the total center of mass of the mechanism is stationary for all motions. However, when the payload changes, for example during pick-and-place action, the dynamic balance cannot be maintained, and vibrations will appear, reducing the accuracy. In this paper, five strategies are described to adapt the dynamic balance under varying payload conditions. Three of these strategies rely on reconfiguration of the mechanism; by changing position the counter weights (I), by changing the joint locations (II) or by altering the amount of counter weight (III). The last two strategies use active control of additional linkages to steer the mechanism in over a reactionless trajectory. These additional linkages can be placed at the base as a reaction mechanism (IV), or within the kinematic chain with redundant joints (V). The implications and differences of these strategies are shown by applying them to a 3 degree of freedom (DOF) planar mechanism. All strategies can provide adaptive dynamic force balance, but have different features, especially added complexity (II & III), reconfiguration force (I & III), or energy consumption (IV & V)

Over the last six decades, kinematicians have devised many tests for the identification of isomorphism among kinematic chains (KCs) with revolute pairs. But when it comes to KCs with prismatic pairs, tests are woefully absent and the age-old method of visual inspection is being resorted to even today. This void is all the more conspicuous because sliders are present in all kinds of machinery like quick-return motion mechanism, Davis steering gear, trench hoe, etc. The reason for this unfortunate avoidance is the difficulty in discriminating between sliding and revolute pairs in the link–link adjacency matrix, a popular starting point for many methods. This paper attempts to overcome this obstacle by (i) using joint–joint adjacency, (ii) labeling the revolute pairs first, followed by the sliding pairs, and (iii) observing whether an element of the adjacency matrix belongs to revolute–revolute (RR), revolute–prismatic (RP) (or PR), or prismatic–prismatic (PP) zone, where R and P stand for revolute and prismatic joints, respectively. A procedure similar to hamming number technique is applied on the adjacency matrix but each hamming number is now split into three components, so as to yield the split hamming string (SHS). It is proposed in this paper that the SHS is a reliable and simple test for isomorphism among KCs with prismatic pairs. Using a computer program in python, this method has been applied successfully on a single degree-of-freedom (DOF) simple-jointed planar six-bar chains (up to all possible seven prismatic pairs) and eight-bar KCs (up to all ten prismatic pairs). For six-bar chains, the total number of distinct chains obtained was 94 with 47 each for Watt and Stephenson lineages. For eight-bar chains, the total number is 7167 with the distinct chain count and the corresponding link assortment in parenthesis as 3780(0-4-4), 3037(1-2-5), and 350(2-0-6). Placing all these distinct KCs in a descending order based on SHS can substantially simplify communication during referencing, storing, and retrieving.

The structural synthesis of Baranov trusses is an open problem even though it finds its wide usage in design and analysis of mechanisms and robotics. A Baranov truss is a planar zero DOF kinematic chain, which contains no rigid subchains. The present work aims at synthesising the complete collection of N - link Baranov trusses by adding a binary link to (N – 1) - link, 1 degree of freedom (DoF) simple jointed kinematic chains. From this collection, chains which contain degenerate sub-chains are identified and removed as they are not Baranov truss by definition. Next, an isomorphism test is carried out to identify distinct chains and resulting complete collection of N-link Baranov trusses is reported.