Planar Multibranch Open-Loop Robotic Manipulators Subjected to Ground Collision

In this article, a recursive approach is used to dynamically model a tree-type robotic system with floating base. Two solution procedures are developed to obtain the time responses of the mentioned system. A set of highly nonlinear differential equations is employed to obtain the dynamic behavior of the system when it has no contact with the ground or any object in its environment (flying phase); and a set of algebraic equations is exploited when this tree-type robotic system collides with the ground (impact phase). The Gibbs–Appell (G–A) formulation in recursive form and the Newton’s impact law are applied to derive the governing equations of the aforementioned robotic system for the flying and impact phases, respectively. The main goal of this article is a systematic algorithm that is used to divide any tree-type robotic system into a specific number of open kinematic chains and derive the forward dynamic equations of each chain, including its inertia matrix and right-hand side vector. Then, the inertia matrices and the right-hand side vectors of all these chains are automatically integrated to construct the global inertia matrix and the global right-hand side vector of the whole system. In fact, this work is an extension of Shafei and Shafei (2016, “A Systematic Method for the Hybrid Dynamic Modeling of Open Kinematic Chains Confined in a Closed Environment,” Multibody Syst. Dyn., 38(1), pp. 21–42.), which was restricted to a single open kinematic chain. So, to show the effectiveness of the suggested algorithm in deriving the motion equations of multichain robotic systems, a ten-link tree-type robotic system with floating base is simulated.