A new serial approach of the forward kinematic model of spherical parallel manipulators for real-time applications

Parallel architectures are increasingly used as haptic devices to provide low inertia, high stiffness and compactness. Thus, spherical parallel manipulators have been developed to generate the three rotational movements in a sufficient workspace. However, these parallel structures have complex kinematic models and can suffer from critical singularity issues. This paper proposes a serial approach to solve the forward kinematic model of a spherical parallel manipulator, which is used as a haptic device in minimally invasive surgery. The new forward kinematic model is based on the serial positioning of the three sensors on one leg rather than placing the three sensors on the three actuated base joints. The forward kinematic model calculation is thus simplified to be suitable for real-time applications (computing time around 5 µs) without cost increase. Parallel singularity effects are removed using this approach and the accuracy of the forward kinematic model is highly enhanced. Simulations were carried out to show the benefits of this approach. The resulting errors of the forward kinematic model calculation due to measurement noises do not exceed 0.2° along the workspace. Experiments were carried out to demonstrate the control of a surgical robot.

Prestack traveltime approximations

Many of the explicit prestack traveltime relations used in practice are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multifocusing, based on the double square-root (DSR) equation, and the common reflection stack (CRS) approaches. Using the DSR equation, I constructed the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I recasted the eikonal in terms of the reflection angle, and thus, derived expansion based solutions of this eikonal in terms of the difference between the source and receiver velocities in a generally inhomogenous background medium. The zero-order term solution, corresponding to ignoring the lateral velocity variation in estimating the prestack part, is free of singularities and can be used to estimate traveltimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. The higher-order terms include limitations for horizontally traveling waves, however, we can readily enforce stability constraints to avoid such singularities. In fact, another expansion over reflection angle can help us avoid these singularities by requiring the source and receiver velocities to be different. On the other hand, expansions in terms of reflection angles result in singularity free equations. For a homogenous background medium, as a test, the solutions are reasonably accurate to large reflection and dip angles. A Marmousi example demonstrated the usefulness and versatility of the formulation.