Kinematic Analysis And Synthesis Of Spatial Mechanisms And Compliant Mechanisms

The kinematic analysis and synthesis of the mechanism is one of the most fundamental and most important topics in the research of robot mechanisms, which does not only lay a foundation for the design of mechanisms, but also provides theoretical support to the engineering application of robots. The dissertation develops the research on the hot and difficult kinematic problems of the spatial mechanism and the compliant mechanism in order to realize the mathematics mechanization of the kinematics of mechanisms. The main contents and contributions can be summarized as follows:(1) For the forward displacement analysis of a general 6-4A Stewart in-parallel platform mechanism, firstly, a 6-4A structure is changed into an equivalent structure; secondly, a mathematical modeling is established based on the trilateration formula containing a few number of Cayley-Menger determinants derived by using barycentric coordinates; and then five basic constraint equations with five variables are obtained from eight equations with eight variables by introducing vector loop relationships and substitution variables; and then the equation containing two variables is derived from the four equations containing other three variables by the vector elimination; finally, a 32th-degree polynomial equation in a univariate is derived from a 10 by 10 Sylvester resultant which is constructed from the equation by vector elimination and the last of the five and the mathematics mechanization of this problem is implemented. The problem is modeled based on the geometric invariant, as a result, the solution procedure is simpler and more efficient and readily to program.(2) The new closed-form solution of the forward displacement analysis of a general 5-5B Stewart in-parallel platform mechanismis presented. First of all, a new equivalent mechanism is transformed from the 5-5B Stewart in-parallel platform mechanism; secondly, the basic closure equations are derived based on the geometric invariant; and then three kinematic constraint equations in two variables are derived from the basic closure equations by the vector algebraic elimination technique. Then, the greatest common divisor (GCD) of two constraint equations is symbolically factored out by using computer algebra system. Finally, a 24th-degree univariate polynomial equation is reduced from this GCD polynomial together with the third constraint equation by constructing a 10 by 10 Sylvester resultant matrix. Its novelty lies in elimination strategy of the basic closure equations. The solution procedure is all symbolic, and it is feasible to program for the forward displacement analysis of the general 5-5B Stewart in-parallel platform mechanisms. As a result, the paper completes the mathematics mechanization of the problem.(3) The algebraic solution method is improved for the forward displacement analysis of a general 6-6 Stewart in-parallel platform mechanism. By using Cay ley’s formula, the basic constraint equations of the forward kinematics of the general 6-6 Stewart in-parallel platform mechanism are established. Four polynomial equations in four variables are reduced from the six basic constraint equations by substituting variables and linearly elimination. Before Grobner basis theory is used to the four polynomial equations, the degree of one of the four variables is elevated by variable substitution in order to eliminate it priority and under the graded reverse lexicographical order,16 reduced Grobner bases are obtained and then a 10 by 10 Sylvester resultant is constructed from the chosen 10 ones of the aforementioned 16 reduced bases. It will lead to a 40th-degree univariate polynomial equation. The advantage of this new algorithm lies in that the size of the resultant is relatively small and therefore the computational speed is improved.(4) The complete real solution of the five-orientation rigid body guidance problem for a spherical four-bar linkage is solved. Firstly, a 6th-degree univariate polynomial equation is deduced from the constraint equations of the spherical RR dyad by using Dixon resultant elimination method. Secondly, based on Sturm’s theorem, the relationships between the design parameters are derived in order to obtain six real roots. Moreover, the Grashof condition and the circuit defect condition are taken into account. Given the relationships between the design parameters and the aforementioned two conditions, two objective functions are constructed and optimized by the adaptive genetic algorithm. The contribution of this study proposes a new method by which for any kinematic problem, a univariate high-degree polynomial equation is derived by algebraic elimination and the complete real solutions are obtained based on Sturm’s theorem, and it provides a new research idea for many other similar kinematic problems.(5) In this dissertation, a new closed-form solution to the inverse static force analysis of a spatial three-spring system is presented. A system of three polynomial equations in three variables is derived based on the geometric constraint and static force balancing. A 20 by 20 Dixon resultant matrix is firstly derived from these three polynomials after removing the linearly dependent rows and columns and then reduced to an 18 by 18 matrix according to removing the linearly dependent rows and columns. A 46th-degree univariate polynomial equation is yielded from the above 18 by 18 matrix. By further analysis, we found that 24 roots were degenerated and only the remaining 22 roots are the ones for the three-spring system. The presented algebraic elimination procedure is following the one solved by the method of analytical geometry and reveals some intrinsic geometry nature of this challenging problem.(6) The mobility criteria of the compliant mechanisms are proposed based on the decomposition of compliance matrix. In this dissertation, we investigate the use of the eigentwist and eigenwrench decomposition of compliance matrices to identify the mobility of spatial compliant mechanisms. We firstly prove that the eigencompliances are invariant to the coordinate transformation. We then introduce the characteristic length to scale the eigencompliances and compliance matrices to compare translational compliances with rotational ones. We propose two mobility criteria for the compliance matrix of any given compliant mechanisms. We have also proposed two guidelines for choosing the characteristic length for serial open chains and closed loop mechanisms respectively. The robustness of the chosen characteristic length is discussed. A general procedure for determining the mobility for any compliant mechanism is presented. Finally, two case studies are provided to verify our method. According to the proposed mobility criteria, we do not only identify the number of mobility of compliant mechanisms, but also recognize the direction of the mobility. The result is similar to the mobility of the rigid-body mechanisms analyzed by screw theory.In conclusion, this dissertation studies some unsolved hot and difficult kinematic problems for the spatial mechanisms and compliant mechanisms and makes some ground breaking research results in the following four aspects. On one aspect, the dissertation presents a new modeling method for the kinematics analysis of the mechanisms based on geometric invariant; on the second aspect, the dissertation firstly accomplishes the mathematics mechanization of the general 5-5B Stewart in-parallel platform mechanism; on the third aspect, the dissertation firstly proposes a method that combines the algebraic elimination method and Sturm’s theorem to solve the kinematic analysis or synthesis problem for the complete real roots; on the last aspect, this dissertation firstly presents the mobility criteria for compliant mechanisms based on the decomposition of the compliance matrix. Moreover, the dissertation proposes the algebraic elimination procedure by following the one by the method of analytical geometry for the nonlinear polynomial equations and it provides a new idea for the solution of the nonlinear polynomial equations.