# Effect of End-Effector Compliance on Collisions in Robotic Teleoperation
## 3 Effects of Robot Compliance
### Model of Robot Compliance  
The mathematical model of Figure 8 was developed. The slider moves along the approach direction with the motion law imposed by the robot control system (coordinate \(x_{r}\)). It is connected to the motors of the robot by means of ideal transmissions without compliances and clearances. The spring with a stiffness of \(k_{r}\) and the damper with a constant of \(c_{r}\) represent the compliance and the damping of the robot in the approach direction. A share (\(m_{r}\)) of the mass of the robot is added to the tool. The friction force between the object and the plane of motion is not taken into account, but it may cause small (and beneficial) reductions in the final velocity of the object. During the collision, the system is described by the following equations of motion:  
\[(m_{r}+m_{t})\dot{x}_{t}=c_{r}(\dot{x}_{r}-\dot{x}_{t})+k_{r}(x_{r}-x_{t})-k_{c }\delta^{m}-\chi\delta^{n}\dot{\delta} \tag{8}\]  
\[m_{p}\ddot{x}_{p}=k_{c}\delta^{m}+\chi\delta^{n}\dot{\delta} \tag{9}\]  
Equations (8) and (9) hold if \(\delta>0\). Figure 8: Scheme of the impact between the compliant robot and an object. Figure 7: Influence of \(k_{c}\) and \(e\) on contact duration. (**a**) Limit case of Figure 4a; (**b**) limit case of Figure 4b. The main issue is the identification of the robot's parameters, which appear in this model (\(k_{r},c_{r},m_{r}\)). In this research an Omron Adept Viper s650 is considered. The robot is represented in Figure 9.  
It is assumed that robot compliance is due to the compliance of the joints around their rotation axes, while the links and bearings are considered perfectly rigid. With these hypotheses, if the \(i\)-th actuator imposes a rotation \(q_{i}(t)\) on the \(i\)-th joint, the actual rotation \(\theta_{i}(t)\) of the joint does not coincide with \(q_{i}(t)\). In order to consider this behavior, each joint was schematized by means of a spring-damper model, so that the torque produced by the actuator was transmitted to the joint by means of these elements. The motion of a serial six-DOF robot with compliant joints is represented by a system of six second order differential equations:  
\[M_{q}(q)\ddot{\vartheta}+C(q,\dot{q})+D_{q}(\dot{\vartheta}-\dot{q})+K_{q}( \vartheta-q)+G(q)=0 \tag{10}\]  
where  
\(q\) is the vector which contains the joint variables imposed by actuators;  
\(\dot{\vartheta}\) is the vector containing actual joint variables;  
\(M_{q}(q)\) is the robot mass matrix in the joint space;  
\(C(q,\dot{q})\) is the term related to centrifugal and Coriolis forces;  
\(D_{q}\) is the robot viscous damping matrix in the joint space;  
\(K_{q}\) is the robot stiffness matrix in the joint space; and  
\(G(q)\) is the term related to gravity forces. It is possible to neglect centrifugal and Coriolis forces and the effect of gravity when dealing with small oscillations [17]. Therefore, the equations of motion of the robot in the joint space can be simplified as follows:  
\[M_{q}(q)\ddot{\vartheta}+D_{q}(\dot{\vartheta}-\dot{q})+K_{q}(\vartheta-q)=0 \tag{11}\]  
The mass matrix \(M_{q}(q)\) can be estimated using the data provided by the computer-aided design (CAD) model of the robot, retrieved from the website of the manufacturer. The matrices \(D_{q}\) and \(K_{q}\) are diagonal and independent of the robot configuration \(q\). In particular, the terms of these matrices represent the viscous damping coefficient and the stiffness of each joint around its rotation axis, and they can be identified by means of an impulsive modal analysis of the robot [17; 18]. The movement of the tool and its collision with an object in the plane of motion are described in the operational space, so it is necessary to transfer the description of robot dynamics from the joint space to the cartesian space. Figure 9: Robot: Adept Viper s650. Thus, the mass, stiffness and viscous damping matrices in the cartesian space (\(M_{X}\), \(D_{X}\), \(K_{X}\)) have to be calculated.