This study discusses kinematic and dynamic precession models for a rolling ball with a finite contact area and a point contact respectively.In literature, both conventions have been applied.In this paper, we discuss in detail the kinematic and dynamic models to describe the ball precession and the radius of a circular rolling path.The kinematic model can be used if the contact area and friction coefficient are sufficient to prevent slippage.The dynamic precession model has significance in multi-body simulation environments handling rolling balls with ideal point contacts.We have applied both the kinematic and dynamic precession model to evaluate the no-slip condition of the existing GimBall-robot.According to the result, the necessity of an external precession torque may cause slipping at lower velocities than expected if ignoring this torque., ̇ Precession rate, spin rate (rad/s) ̈ Precession acceleration of the ball (rad/s 2 ) ω Forward rolling rate of the ball (rad/s) ̇ Forward rolling acceleration (rad/s 2 ) μ Friction coefficient at contact μ a Friction coefficient share for acceleration force μ T Friction coefficient share for vertical torque

Dynamic models describing the ball-robot motion form the basis for developments in ball-robot mechanics and motion control systems. For this paper, we have conducted a literature review of decoupled forward-motion models for pendulum-driven ball-shaped robots. The existing models in the literature apply several different conventions in system definition and parameter notation. Even if describing the same mechanical system, the diversity in conventions leads into dynamic models with different forms. As a result, it is difficult to compare, reproduce and apply the models available in the literature. Based on the literature review, we reformulate all common variations of decoupled dynamic forward-motion models using a unified notation and formulation. We have verified all reformulated models through simulations, and present the simulation results for a selected model. In addition, we demonstrate the different system behavior resulting from different ways to apply the pendulum reaction torque, a variation that can be found in the literature. For anyone working with the ball-robots, the unified compilation of the reformulated dynamic models provides an easy access to the models, as well as to the related work.

This work presents an assessment of different control techniques based on Limit Cycle Walking. The study is performed on a two-dimensional kneed bipedal simulator developed with Open Dynamics Engine, which includes realistic configuration of weight allocation, link length, inertia distribution and impacts. The controllers are model-based driven from an analytic 2D kneed biped. Several controllers from the Computed Torque (CT) family together with a Passivity Based controller (PBC) are compared based on their energy efficiency and robustness. The previous controllers are used in their original form, with minor adjustment to handle hybrid systems such as the kneed biped case. PBC proved not suitable for real implementation however a combined strategy between PBC and a soft PD control has shown promising results. PD Computed Torque control also offered high tolerance to disturbances with reasonable energy consumption. Furthermore, the performance of all the controllers showed high dependency on the parameters of the robot.