## 3 Results
### Trajectory Execution  
The first experiment consists of the execution of a straight line trajectory along the horizontal direction, the \(x\) axis. While for the direct force control, the plane is placed beneath the trajectory, for admittance control, a steel ramp intersects the trajectory causing contact with the end-effector.  
If the objective is to be compliant with the environment, thus to adapt to obstacles, the desired force is set to zero along the axis perpendicular to the plane, i.e., \(\mathcal{F}_{e}\)'s \(z\) axis, the compliant direction.  
Figure 7: Reference frames used in the interaction controllers.  
The impedance matrices for the real UR10 are: \(\mathbf{M}_{d}=\text{blockdiag}\{0.1,0.1,0.1\}\), \(\mathbf{K}_{D}=\text{blockdiag}\{300,300,500\}\), \(\mathbf{K}_{P}=\text{blockdiag}\{900,1300,10\}\). In order to achieve comparable performances with the simulated robot, because of the different dynamics of the manipulator and the higher stiffness of the objects in the virtual environment, the matrices are: \(\mathbf{M}_{d}=\text{blockdiag}\{1.0,1.0,0.001\}\), \(\mathbf{K}_{D}=\text{blockdiag}\{900,900,5000\}\), \(\mathbf{K}_{P}=\text{blockdiag}\{1300,1300,90\}\). In both cases, the \(z\) component has a higher damping and a lower stiffness since the robot must be highly compliant and moderately damped in that direction, while along \(x\) and \(y\) axes, it must be stiffer, as the position must be kept with accuracy.  
As it is clear from Figure 8, as soon as the robot is in contact with the ramp (around 20 s), it slides over it, causing a non-negligible position error and higher contact forces along \(z\). Since the trajectory is longer than the ramp, when the interaction ends (around 140 s), the robot recovers the desired position and force, with zero error.  
With the direct force controller, since we want to track a specific force reference, we set the desired force to a ramp from 0 to 50 N along \(z\), which is the only force-controlled direction. PI gains for the real UR10 are: \(K_{P_{z}}=0.8\cdot 10^{-5}\) and \(K_{I_{z}}=6\cdot 10^{-5}\). In the simulation, where both the robot and the environment have slightly different characteristics, the proportional gain is lower: \(K_{P_{z}}=0.025\cdot 10^{-7}\). In Figure 9, we can see that, along position-controlled directions, the position error is zero, while along the force-controlled direction, the force error is below 2 N.  
The same experiments are performed with the Smart-Six by only changing configuration parameters, i.e., reference frames, impedance matrices, and PI gains in configuration  
Figure 8: Command end-effector positions (**left**) and measured end-effector forces (**right**) of the UR10 during the execution of the workspace trajectory with the admittance controller. Quantities are compared to the desired positions and forces.  
Figure 9: Commanded end-effector positions along the position-controlled axes (**left**) and end-effector forces along the force-controlled axis (**right**) of the UR10 during the execution of the workspace trajectory with the direct force controller. Quantities are compared to the desired positions and forces.  
files. Similar observations made for the UR10 hold, as can be seen from Figure 10, demonstrating the flexibility of the proposed architecture. It is worth noting that the controllers' performance can be improved through better knowledge of the interaction dynamics and an informed tuning of the controllers' parameters, but this is addressed elsewhere in the literature [31, 48, 49], and goes beyond the scope of this communication.