### Computational Kinematic Design of Robotic Systems

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** Eric Lee,
Rutgers University **

In recent years, complex, spatial robotic manipulators and mechanisms have attracted the interest of machine designers because they can combine high dexterity with complex three dimensional trajectories, relatively large workspaces and high accuracy. Examples of such devices include prosthetic devices, high speed assembly machines and walking robots. The kinematics analysis and synthesis are two important steps in the design of these mechanical devices by studying the geometrical properties of them. The analysis problem studies the mechanism capabilities and performance when its type and geometric parameters are given. The synthesis problem is the process of designing a mechanism to accomplish a specific task. The mathematical formulation of both problems lead to complex systems of multivariable polynomial equations. Due to their nonlinearity, multiple solutions exist. It is not uncommon that a kinematics problem possesses hundred or even thousand of solutions. These multivariate polynomial systems are, therefore, not easy to solve and many important design problems remain unsolved to this date.

The methods used in solving polynomial systems can be classified as either analytical or purely numerical methods. Analytical methods are those that, by carefully creating and subtracting new equations from old ones, eliminate all but one variable in the multivariate polynomial system and reduce it to a single univariate polynomial, which is then solved by a numerical method. These methods involve the manipulation of large amount of intermediate symbolic information and large integer coefficients. Examples of these methods are the classical elimination method and the Gröbner Basis Method which are based on modern theory of commutative algebra. Purely numerical methods rely on iterative algorithms which compute the numerical solutions directly from the polynomial systems. One of the most popular numerical methods is the polynomial continuation method which computes all the possible solutions by tracking all the possible solution paths. Thus, both classes of method involve large amount of computation and storage of information. Solving such problems is not possible without the careful use of modern computing technology. The main objective of this research is to apply existing computational methods as well as develop generic methodologies for the kinematic analysis and design of robotics systems and mechanisms. Using these computational methods, analysis and synthesis algorithms for different types of mechanisms will be developed and used in design automation and control of them.

**Abstract Author(s):** Eric Lee