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.

Absshact-The general problem of control of state-space constrained dynamic systems is considered. Based on the derivation of the comlmint form as explicit functions of the state and the inpot, a general model is presented that encompasses both the constrained case and the corresponding unconstrained case. S t a b i i and point-to-point motion of these systems in the vicinity of an operating point are considered under operating conditiom which either maintain or deliberately violate the constraints Algorithms for computation of the neasary feedback gains in the vicinity of the operating point are discmsd For a threelink biped model, several motions in the vicinity of the vertical stance are considered, and the neressQTy feedback gains are derived. Digital computer simulation of some biped motions are carried out to serve as examples and to demonstrate use of the themy. 'Ibis work is to be regarded as an elementary step in better understanding of human motor control and in the design of robots and prostttetic devices.

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A computational method for simulation of 3-D movement of the trunk under the control of 48 anatomically oriented muscle actions was developed. Neural excitation of muscles was set based on inverse dynamics approach along with the stability-based optimization. The effect of muscle spindle reflex response on the trunk movement stability was evaluated upon the application of a perturbation moment. The method was used to simulate the trunk movement from the upright standing to 60 degrees of flexion. Incorporation of the stability condition as an additional constraint in the optimization resulted in an increase in antagonistic activities demonstrating that the antagonistic co-activation acts to increase the trunk stability in response to self-induced postural internal perturbation. In presence of a 30 Nm flexion perturbation moment, muscle spindles decreased the induced deviation of the position and velocity profiles from the desired ones. The stability-generated co-activation decreased the reflexive response of muscle spindles to the perturbation demonstrating that the rise in muscle co-activation can ameliorate the corruption of afferent neural sensory system at the expense of higher loading of the spine.

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