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The problem of stability of a robot in a dynamic environment is solved. Sufficient stability conditions for a given programmed path of the robot are established. The effect of the dynamic environment is illustrated by three examples Keywords: dynamic environment, robot, asymptotic stability Introduction. In recent decades, the problem of stability of robots in dynamic environment has attracted great interest. The reason is that the environment may destabilize the programmed motion of the robot's effector. The control problem for a robot interacting with its environment is formulated as follows: given a force of interaction F t r ( ) between the robot and its environment, find a control law that would steer the robot along a programmed path x t r ( ). We will discuss results that allow us to establish conditions for the stability of a robot moving along a programmed path, depending on the parameters of the environment.1. Problem Formulation. Auxiliary Results. Let us examine the special case where the force of interaction between the robot and its environment asymptotically tends to the expected one. In this case, it is important to establish the conditions for the stability of the expected programmed path x t r ( )of the robot depending on the parameters of the environment. According to [6,9], the motion of a robot is described by the following system of differential equations:

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