Solution of the direct and inverse kinematic problems by a common algorithm based on the mean of multiple computations

Abstract

Abstract. For the control of the movement of a multijoint manipulator a "mental model" which represents the geometrical properties of the arm may prove helpful. Using this model the direct and the inverse kinematic problem could be solved. Here we propose such a model which is based on a recurrent network. It is realized for the example of a three-joint manipulator working in a two-dimensional plane, i.e., for a manipulator with one extra degree of freedom. The system computes the complete set of variables, in our example the three joint angles and the two work-space coordinates of the endpoint of the manipulator. The system finds a stable state and a geometrically correct solution even if only a part of these state variables is given. Thus, the direct and the inverse kinematic problem as well as any mixed problem, including the underconstrained case, can be solved by the network.

Universally manipulable body models—dual quaternion representations in layered and dynamic MMCs

“…In contrast, Cruse and Steinkühler (1993) proposed a holistic network, forming a unified inverse-forward model, which can represent any configuration that is geometrically possible and may therefore be termed a universally manipulable body model. While this network, termed mean of multiple computation (MMC) network, can be used as a forward model, as an inverse model and as any mixed model (Cruse and Steinkühler 1993;Steinkühler and Cruse 1998), it was originally applied only to 2D structures and the extension to 3D structures was restricted to Cartesian coordinates.…”

“…While this network, termed mean of multiple computation (MMC) network, can be used as a forward model, as an inverse model and as any mixed model (Cruse and Steinkühler 1993;Steinkühler and Cruse 1998), it was originally applied only to 2D structures and the extension to 3D structures was restricted to Cartesian coordinates. An extension to general coordinates including joint angles, for example, requires additional processing of constraints which introduce non-linearities.…”

“…The classical MMC network (Cruse and Steinkühler 1993;Steinkühler and Cruse 1998) is an autoassociator, a type of self-organising map that is constrained by geometric relationships representing the kinematics of, to take a simple example, a serial chain manipulator, or, as a more complex case, of a six-legged walker with 18 degrees of freedom. By means of the geometric constraints, the network is forced into its attractor states, which always represent valid, geometrically correct solutions.…”

“…Traditional MMC networks can be easily represented in neural networks (Cruse and Steinkühler 1993) because the equations used to represent the geometric structure are based on summation and the resulting equations can be directly transferred into a weight matrix for a neural network. In contrast, for the structure proposed here, the equations describing geometrical relations use quaternion multiplication for the concatenation of transformations.…”

The evolution of cognition—a hypothesis

“…This principle is reflected by the name MMC which stands for mean of multiple computation. In general terms, this model consists of a recurrent network which relaxes to adopt a stable state corresponding to a geometrically correct solution, even when the input does not fully constrain the solution (Cruse & Steinkühler, 1993). Any special constraints of the variables (e.g., joint limits) can be introduced.…”