Our paper is devoted to a specific class of Optimal Control Problems (OCPs) in theoretical mechanics. We consider a minimax-Type optimal control processes governed by dynamic systems with randomly varying time delays. In particular we deals with the minimax-Type OCPs associated with a family of delayed Lagrange differential equations for the robot dynamics. The mathematical abstractions under consideration provide an adequate approach to many real-world robotic systems. Moreover, the proposed minimax dynamic optimization approach has a fundamental interpretation as a system robustness with respect to the unavoidable delays in robot control. The obtained convex structure of a linearized robot dynamics makes it possible to reduce the originally given delayed OCP to an auxiliary convex program in a suitable Euclidean space. The equivalent transformation we propose involves the wide range of effective algorithms for an effective computational treatment of the resulting convex OCP. We finally propose a concrete gradient based computational approach for the optimal control design of the controlled Lagrange-Type robot dynamics.