## 2 Problem Statement  
Figure 1 presents the robot in contact with the uneven surface. The robot has to execute the following task: to exert a desired force \(f_{d}\) in the normal direction (\(x\) direction) to the surface while a desired velocity \(v_{d}\) is maintained along the surface (\(y\) direction). During the execution of the task (see Figure 1), the following assumptions are considered. First, the end-effector of the robot is always in contact with the surface; the methods provided in this paper are not valid when the robot loses contact with the surface. Second, it is assumed that no forces are produced along the \(z\)-axis since the end-effector is moving along the \(y\)-axis. Third, the end-effector is in compliance with the surface in the \(x\)-direction. The compliance in the \(x\)-direction helps to direct most of the force produced by the curved surface to the \(x\)-direction; hence, the force along the \(y\)-axis is minimum and one can consider that the normal force is mainly defined by the force in the \(x\)-direction. Then, the interaction force is studied using the one-degree-of-freedom (1 DOF) model presented in Figure 2, considering only the movement in the \(x\) direction [6]. The force exerted by the robot on the surface is described by the following elastic model:  
\[f=K(x-x_{e}) \tag{1}\]  
where \(K>0\) is the accumulative stiffness of the tool plus the environment, \(x\) is the end-effector position, and the location of the surface in \(x_{e}\). The control objective is to design the robot's input \(v\) that ensures the desired force \(f_{d}\) is applied on the surface. Considering the models in Figure 2 and Equation (1), the force control system presented in Figure 3 is used to study and to design the control \(v\). The force control system is composed of the following blocks: \(G_{c}(s)\) is the controller, \(G_{LP}(s)\) describes the dynamics of a filter used to attenuate noise from force sensor measurements, and \(G_{T}(s)\) corresponds to the delay produced by sensor-hardware communication. The block \(K\) corresponds to the stiffness. The block named \(Robot\) is the single-input single-output model of the robot, and the time-constant \(\tau\) defines how fast the robot's position \(x\) responds to the control input \(v\). The signals \(f_{d}\), \(f\), \(e=f_{d}-f\), \(v\), and \(x_{d}\) represent the desired force, the measured force, the force error, the control signal, and a disturbance emerging in the position \(x\), respectively. Figure 1: The contact task. \(v\) is the velocity in the normal direction, and \(v_{y}\) is the velocity along the surface. Figure 2: The one-degree-of-freedom interaction force model. The transfer functions of the force control system in Figure 3 are as follows. \[G_{c}(s) = K_{p}+K_{d}s+\frac{K_{j}}{s} \tag{2}\] \[G_{LP}(s) = \frac{1}{\tau_{LPS}+1}\] (3) \[G_{T}(s) = e^{-Ts} \tag{4}\]  
where \(s\) is the Laplace variable, \(K_{p},K_{d},K_{j}>0\) are control gains, \(\tau_{LP}\) is the time constant of the filter, and \(T\) is the time-delay value. The transfer functions and its parameters were already identified and presented in [17] by our research group. Note that the controller \(G_{c}(s)\) is a direct force control, and it is similar to an admittance control since its input is a force and its output is a velocity. Direct force control and admittance control have been studied and tested in industrial robots; however, during the adjustment of the gains, there exists an unavoidable compromise between performance and stability [8; 9]. Furthermore, when delays and filters are included in the force control system (see Figure 3), these additional dynamics deteriorate the stability of the force control system [3]. Additionally, when disturbances emerge, the design of the gains should consider disturbance rejection as well as the stability of the system. In this paper, we propose a method for the analysis and design of force control systems, such as the one in Figure 3, considering performance, stability margins, and robustness against disturbances.