Theoretical And Applied Geometric Algebra In Symbolic Analyses Of Mechanisms

Mechanism is the skeleton of robots,including analysis of mechanism and synthesis of mechanism.With the operation task becomes more and more diversified and the environment becomes more and more complicated for robots,the mathematical problems in mechanism research are becoming high degree,strong coupling,and nonlinear.Traditional mathematical tool meets some troubles to satisfy solving mathematical problems efficiently and precisely in mechanism research.Furthermore,with the development of science and technology,particularly the development of computer technology,establishing symbolic theory of mechanism,developing computer aided designing program,and realizing the research of mechanism digitally,automatically,visually,intelligently,are time requirement and new trend of modern theory of mechanism development.Geometric algebra is a powerful and efficient mathematical tool,which can solve algebra problems by geometric methods.Geomteric algebra is not only can represent geometry elements without coordinates,but also can represent intersections,distances and angles,and transformations without coordinates.All these advantages make it possible for problems to be geometrical,visible,simple,and efficient.Moreover,geometric algebra can tranfer mathematical problem to be symbolization theoretically,which is good for computer aided problem solving.Therefore,this dissertation focuses on the research of the hot and difficult mathematical problems of mechanism in the framework of geometric algebra.The goal is to simplify,promote the computation efficiency,and gave the symbolic reluts of the mathematical problems of mechanism.The main research contents and innovation points were as follows:(1)The algebra and geometry of Geometric algebras are presented in an easy to understand fashion,requiring only a minimum of mathematical background.The formulas for rigid motions in these algebras are derived in a simple and unified way,emphasizing the essential links between these seemingly disparate algebras and mechanisms.The strengths and weaknesses of the currently competing models of Geometric algebras are also compared and contrasted.The representations for affine geometry are fleshed out inside the Clifford algebra R(4,4).Not only points,lines and planes,but also conic sections and quadric surfaces are represented in this model.The representations for intersections between these geometric elements are also investigated.Formulas for distances and angels are provided.(2)An algorithm for inverse kinematic of industrial robots is proposed based on a geometric algebra model R(3,0,1).The Geometric algebra model R(3,0,1)combines the benefits of dual quaternions and conformal geometric algebra,i.e.,dual quaternions can compute faster while comformal geometric algebra have same translation algorithm for points and planes as well as have algorithm to compute sign distance between points and planes.An algorithm for inverse kinematic of industrial robots based on R(3,0,1)model is disucssed.The unique solution of inverse kinematic of industrial robots is determined by the sign distances between joints and three singular planes,and the sign distances can be computed by R(3,0,1)model.This algorithm can find unique solution without comparing a preferred one which is widely applied in general inverse kinematic solution.This new algorithm has advantages,such as be able to compute the sign distance to the singular planes,simple,high speed to compute unique inverse kinematic solution,visualable when applied to practical robot motion control.This algorithm is numerical verified on PUMA 560.(3)Inverse dynamics of SCARA serial mechanism and 3-RPS parallel mechanism are disscussed based on conformal geometric algebra.The second type of Lagrange Equation for SCARA serial mechanism and the first type of Lagrange Equation for 3-RPS parallel mechanism are rewritten by differential kinematics equation in terms of lines and points in the framework of conformal geometric algebra.After Matrix decomposition,the dynamics equations are produced,and derivation becomes unnecessary.The inverse dynamics equations using conformal geometric algebra not only performs less computations but also take the advantages of the multivectors by computing the dynamics in parallel,which is good for real-time control.Finally,writing code based on Mathmatica software followed by comformal geometric algebra computation rules,the generalized forces for manipultors are computed.The inverse dynamics equations are numerical verified with the help of Adams and Mathmatica software by comparing the relusts to the results computed by Matrix.The inverse dynamics analyses provide fundamental knowledge to the futher development for dynamics discussions based on conformal geometric algebra for mechanisms and robots.(4)Analyzing mobility of parallel mechanisms automatically based on R(3,3)geometric algebra model is proposed.Firstly,since any screw can be represented by a known screw with rigid body transformations,and the rigid body transformations can be derived by the geometric relationship between the two screws,when known the twists of the first joints of the limbs,all twists of limbs of a parallel mechanism can be symbolic expressed and automatically computed by the geometric relationships between the joints of limbs.Then,since the motion of the limb is the union of all twists of joints and the motion of the moving platform is the intersection of all motion of limbs,by taking the advantages that the geometric algebra has the representations of the intersection set and the union set,the motion of all limbs and the moving platform can also be symbolic expressed and automatically computed.Moreover,the symbolic expression of the motion of the moving platform is exactly the moblity of the parallel mechanism.And the degree of a parallel mechanism is given by the blade of the symbolic expression.Finally,some examples are discussed in detail by using the C++ coding to verify the algorithm.The algorithm that analyzing the mobility of parallel mechanisms automatically based on geometric algebra does not need to compute the constraint twists of limbs and does not need to solve linear equations,which makes this algorithm faster and more efficiency.This algorithm sets the stage for automatic type synthesis of parallel mechanism.In conclusion,by combining the methods of the theoretical derivation and computer-aided simulation,the fundenmental problems of mechanism based on geometric algebra have been comprehensive studied and computer-aided verified.The thesis provided a full theory analysis for researching and developing mechanisms and robots based on geometric algebra,and laid a solid foundation for the theory of mechanism to be geometrization,visualization,symbolization,and computer-aided.