## 1 Introduction  
Robots executing contact tasks are essential to automate plenty of manufacturing processes. Regulating the force produced during the interaction between the robot and the environment is critical. There are principally two approaches to regulate the force; one is called indirect force control, since the force is regulated through motion control, i.e., changes in the position error at the end-effector, and the second one is called direct force control, since force feedback is directly compared with a desired force to calculate the robot's control input [1]. Direct force control is preferred when the application requires a precise regulation of the force. Additionally, direct force control is capable of accomplishing the contact task without damaging the environment and the robot itself [2]. However, the advantages of direct force control come at a price, since preserving stability is challenging, mainly because of the presence of unavoidable dynamics such as sensor dynamics, filters, and delays [3]. On the other hand, when direct force control techniques are implemented in industrial robots, one should design controllers that generate velocity or position inputs, since these are the standard inputs of industrial robots [4, 5]. Admittance controllers are the ones having velocity/position as an output and force as an input [6]. Despite the fact that the implementation of admittance-type controllers has shown efficiency and efficacy [7], there exists a compromise between performance and stability during its design [8, 9]. Recent research on force control has been focused in the design of Proportional-Integral-Derivative (PID) controllers that reach quickly the desired force with limited overshoot. For example, the authors in [10] present a force control system based on PID that ensures asymptotic convergence of the force error to zero with small overshoot and short settling time. In [11], a force control system is presented based on PID that keeps the force within the desired value despite uncertainty in the surface's model. In [12], the authors analyze the effect of the surface's stiffness in the force control, and they present a PID controller that reaches the desired force without overshoot. Furthermore, advanced control techniques have been recently applied to regulate force. An application to medical robotics in [13] presents a force controller based on sliding mode control that ensures convergence of the force error to zero in finite time. Data-driven control is used in [14] to present a data-driven force control that ensures global convergence of the error to a steady state. Notwithstanding the prominent results presented in the mentioned references, the velocity of the robot in the tangent direction (along the surface) has not been studied, although this velocity is important since it is related to the velocity at which the task can be executed. Furthermore, a quantitative approach to design the control gains that produces a specified tolerance error is hardly discussed. The mentioned methods (estimation of the velocity of the task and a quantitative design) are relevant to practical applications required to execute the task as fast as possible and to keep the force error within acceptable limits. In this paper, we propose a methodology to analyze and to design the force control of a robot in contact with an uneven surface. The proposed methods are presented considering the general contact task of maintaining a desired velocity along the surface (in the tangent direction to the surface) while a desired force is applied on the normal direction to the surface. This contact task properly describes applications when the priority is to regulate the force in one direction, such as polishing and assembly tasks, as well as medical applications (see [15]). We study admittance direct force controllers with Proportional-Integral-Derivative (PID) structure to have methods suitable for industrial robots allowing velocity/position inputs and to fit our methods with the industrially accepted PID controller.