Kinematics and design of a class of parallel manipulators

This dissertation is concerned with the kinematic analysis and design of a class of three degree-of-freedom, spatial parallel manipulators. The class of manipulators is characterized by two platforms, between which are three legs, each possessing a succession of revolute, spherical, and revolute joints. The class is termed the "revolute-spherical-revolute" class of parallel manipulators. Two members of this class are examined. The first mechanism is a double-octahedral variable-geometry truss, and the second is termed a double tripod. The history the mechanisms is explored--the variable-geometry truss dates back to 1984, while predecessors of the double tripod mechanism date back to 1869. This work centers on the displacement analysis of these three-degree-of-freedom mechanisms. Two types of problem are solved: the forward displacement analysis (forward kinematics) and the inverse displacement analysis (inverse kinematics). The kinematic model of the class of mechanism is general in nature. A classification scheme for the revolute-spherical-revolute class of mechanism is introduced, which uses dominant geometric features to group designs into 8 different sub-classes. The 'forward' kinematics problem is discussed: given a set of independently controllable input variables, solve for the relative position and orientation between the two platforms. For the variable-geometry truss, the controllable input variables are assumed to be the linear (prismatic) joints. For the double tripod, the controllable input variables are the three revolute joints adjacent to the base (proximal) platform. Multiple solutions are presented to the forward kinematics problem, indicating that there are many different positions (assemblies) that the manipulator can assume with equivalent inputs. For the double tripod these solutions can be expressed as a 16th degree polynomial in one unknown, while for the variable-geometry truss there exist two 16th degree polynomials, giving rise to 256 solutions. For special cases of the double tripod, the forward kinematics problem is shown to have a closed-form solution. Numerical examples are presented for the solution to the forward kinematics. A double tripod is presented that admits 16 unique and real forward kinematics solutions. Another example for a variable geometry truss is given that possesses 64 real solutions: 8 for each 16th order polynomial. The 'inverse' kinematics problem is also discussed: given the relative position of the hand (end-effector), which is rigidly attached to one platform, solve for the independently controlled joint variables. Iterative solutions are proposed for both the variable-geometry truss and the double tripod. For special cases of both mechanisms, closed-form solutions are given. The practical problems of designing, building, and controlling a double-tripod manipulator are addressed. The resulting manipulator is a first-of-its kind prototype of a tapered (asymmetric) double-tripod manipulator. Real-time forward and inverse kinematics algorithms on an industrial robot controller is presented. The resulting performance of the prototype is impressive, since it was to achieve a maximum tool-tip speed of 4064 mm/s, maximum acceleration of 5 g, and a cycle time of 1.2 seconds for a typical pick-and-place pattern.