Closed-form time derivatives of the equations of motion of rigid body systems

Derivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

GRAVITY–OPERATED IMPACT FEEDER DYNAMICS

This paper deals with the dynamics of rigid body that collide with a rigid surface; such motion with impact of bodies takes place in the gravity operated impact feeders. Feeders are mechanisms of the single- peace feeding for the forced moving of the oriented workpieces. In this work the vertical gravity-impact feeder for moving of the prismatic or plane details is presented. The parts move on inclined guiding plates, free fall and collide with the down plates, the impact phenomena may be used both for decreasing of velocity and for the orientation of the tracking workpieces (lateral reversing). System of equations of plane motion of detail, including stages of sliding on the slope guideway, free flight, impact and motion to the next guideway, are written down. System of equations is solved numerically with help of MathCAD program.

Redundancy-Free Integration of Rotational Quaternions in Minimal Form

Redundancy-free computational procedure for solving dynamics of rigid body by using quaternions as the rotational kinematic parameters will be presented in the paper. On the contrary to the standard algorithm that is based on redundant DAE-formulation of rotational dynamics of rigid body that includes algebraic equation of quaternions’ unit-length that has to be solved during marching-in-time, the proposed method will be based on the integration of a local rotational vector in the minimal form at the Lie-algebra level of the SO(3) rotational group during every integration step. After local rotational vector for the current step is determined by using standard (possibly higher-order) integration ODE routine, the rotational integration point is projected to Sp(1) quaternion-group via pertinent exponential map. The result of the procedure is redundancy-free integration algorithm for rigid body rotational motion based on the rotational quaternions that allows for straightforward minimal-form-ODE integration of the rotational dynamics.