Space-filling curves have been found useful for many applications in diverse fields. A space-filling curve is a path in a 2(r)×2(r) raster domain, which visits each location exactly once. In mathematical terms, space-filling curves linearize a 2D integer space, bijectively mapping the space to the integer line. An algorithm is presented, which generates a large number of space-filling curves/spatial orders. Functions are derived such that the code of each location can be calculated from its coordinates and, conversely, a location code can be decoded to yield the coordinates. The algorithm first generates generate 4×4 spatial orders; they subsequently may be scaled up to any desired domain of size 2(r)×2(r) . The underlying theory of the algorithm, the processes for scaling up, encoding, and decoding are described in detail. The curves are generated as a set of incongruent curves, followed, if required, by the sets of associated congruent curves. A number of space-filling curves are illustrated.

Spatial orders such as the Morton (Z) order, U-order, or X-order have applications in matrix manipulation, graphic rendering and data encryption. It is shown that these spatial orders are single examples of entire classes of spatial orders which can be defined in arbitrary numbers of dimensions and base values. Secondly, an algorithm is proposed which can be used to transform between these spatial orders and Cartesian coordinates. It is shown that the efficiency of the algorithm improves with a larger base value. By choosing a base value that corresponds to the available memory page size, the computational effort required to perform operations such as matrix multiplication can be optimized.

In the well-known electromechanical analogy that converts between electrical and mechanical system representations, mass is the dual of a grounded capacitor. Consequently, any electrical circuit that contains ungrounded capacitors, such as a filter, does not have a mechanical equivalent. A new mechanical system element representing a pulley-based differential transmission is proposed, which, when connected to a mass, is shown to simulate a capacitor in the general case. This new differential mass model provides an additional conceptual framework to model complex mechanical systems such as robotic manipulators.

A haptic interface controlled in impedance mode should present its user with a uniform force capability matching the human hand. This paper reviews a mechanism design methodology that maximizes a workspace-inclusive isotropy index. Force maximization in coreless or voice-coil motors is also considered in order to maximize haptic interface acceleration. Example designs of a planar 3-DOF haptic mouse, a 5-DOF haptic pen and a MagLev joystick are presented.

Good robot performance often relies upon the selection of design parameters that lead to a well conditioned Jacobian or impedance "design" matrix. In this paper, a new design matrix normalization technique is presented to handle the problem of nonhomogeneous physical units and to provide a means of specifying a performance based design goal. The technique pre- and post-multiplies a design matrix by scaling matrices corresponding to a range of joint and task space variables. The task-space scale factors are used to set the relative required strength or speed along any axes of end point motion while the joint-space scale factors are treated as free design parameters to improve isotropy through nonhomogeneous actuation. The effect of scaling on actual designs is illustrated by a number of design examples using a global search method previously developed by the authors.

Good robot performance often relies upon the selection of design parameters that lead to a well conditioned Jacobian or impedance "design" matrix. In this paper a new design matrix normalization technique is presented to cope with the problem of nonhomogeneous physical units. The technique pre and post-multiplies a design matrix by diagonal scaling matrices corresponding to the range of joint and task space variables. In the case of the Jacobian, normalization leads to a practical interpretation of a robot's "characteristic length" as the desired ratio between maximum linear and angular force or velocity. The scale factors can also be used to set relative required strength or speed along any axes of end-point motion and/or can be treated as free design parameters to improve isotropy through asymmetric actuation. The effect of scaling on actual designs is illustrated by a number of design examples using a global search method previously developed by the authors.

A coarse-fine approach to the design of high fidelity haptic interfaces is proposed based on prior work and new psychophysics studies. The approach involves a fine-motion six-degree-of-freedom parallel Lorentz actuator mounted on a series/parallel coarse-motion stage and coupled through a compliant transmission. The frequency response of a simplified model is used to illustrate the advantages of the approach. The workspaces of three coarse-motion platforms are compared. A novel twin-elbow manipulator with all but one of the drive motors in the base is proposed for its simplicity and large workspace size.