Mathematical modeling of a robot collision with its environment

Abstract

In this article the collision of a robot with its environment is studied. In normal applications of a robot arm, a collision takes place because of the velocity of the end effector relative to the object at the time of contract. The collision has effects on the velocities and internal forces of the robotic system. Firstly, the generalized velocities representing joint rates have abrupt changes at the moment of collison with the environment. The mathematical model is derived to establish the quantitative relationship between this abrupt change and the severity of the collision. The latter is represented by either an external impulsive force or the instantaneous change of the linear velocity of the contact point. Secondly, internal to the system, large impulsive forces and torques of constraint may develop at each joint because of the collision. These impulses cause possible damages to the system. The mathematical model is also derived to establish a quantitative relation between the impulsive forces and torquest of constraint and the collision. These two models are applied to a Stanford Arm designed to pick up an object by its end effector, and the consequences of the collision are analyzed. Journal ofRobotic Systems, 2(3 , 289-307 (1 985) 0 1985 by John Wiley & Sons, 1 nc.

Exploiting redundancy to reduce impact force

“…From this diagram, by the use of Path 6 = Path 3→4→5, we can derive the velocity change in joint space in terms of the velocity change of the end-effector in operational space: (13) where the subscripts i and f mean before and after impact, respectively. The same expression of (13) in [1] and [12] proves the correctness of the result found by the use of the Impact Premultiplier Diagram. The matrix J -in (13) equals the inertia-weighted g-inverse J H + in (6).…”

“…The relationship between and is the same as (3) for dynamics, since the Jacobian matrix can be assumed as constant over the infinitesimal time period. The relationship between and is obtained by integrating (4) as: (12) These relationships are substituted into the Impact Premultiplier Diagram in Figure 2, which is the integrated version of the Dynamic Premultiplier Diagram in Figure 1. The diagram helps understand the impact physics and also provides a natural solution of the joint velocity after impact.…”

“…Recently, research has been performed on how redundancy can be used to minimize the force of impact between the end-effector of a manipulator and an object [1][5] [12][10] [3]. Minimization of the impact force is important, since expensive and hard-to-replace end-effectors can be damaged by excessive force.…”

“…Walker [10] showed that reducing the instantaneous force of impact can be accomplished by minimizing the effective mass of the redundant manipulator. Zheng and Hemami [12] explored the relationship between abrupt changes in generalized joint rate velocities and the intensity of collision during an instantaneous impact event. Cai [1] provided a method for determining the impulse of a perfectly elastic collision having a duration that is less than the controller's sampling period.…”

Drum Beating and a Martial Art Bojutsu Performed by a Humanoid Robot

“…Zheng and Hemami discussed a mathematical model of a robot that collides with an environment (Zheng & Hemami, 1985). When a robot arm fixed on the ground collides with a hard environment, the transition from the free space to constrained space may bring instability in the control system.…”