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<latex>{\fontsize{16pt}\selectfont \textbf{Improving the Transparent Behavior of a Robotic Gait Orthosis}} </latex> \\ \\
<latex>{\fontsize{12pt}\selectfont \textbf{Nina Sauthoff}} </latex> \\
<latex>{\fontsize{10pt}\selectfont \textit{Master Project RSC}} </latex> \\ \\


<latex> {\fontsize{12pt}\selectfont \textbf{Abstract} </latex> \\
{{ :adrl:education:completed_projects:lokomat.png?200|}}
The Lokomat ((Riener, R., Lunenburger, L., Jezernik, S., Anderschitz, M., Colombo, G., and Dietz, V.
(2005). Patient-cooperative strategies for robot-aided treadmill training: first experimental
results. IEEE Transactions on Neural Systems and Rehabilitation Engineering,
13(3):380–394.)) (Figure 1) is a robotic gait orthosis for neurorehabilitation. In order to increase the scope of possible training tasks, the control of interaction torques between robot and patient has to be improved. In particular, transparent behavior of the device is crucial to introduce new tasks as it allows to remove robotic support from selected joints.


<latex> {\fontsize{12pt}\selectfont \textbf{Impedance Control} </latex> \\

A common control concept in robot-aided neurorehabilitation is impedance control ((Hogan, N. (1985). Impedance control - an approach to manipulation. i - theory. ii -
implementation. iii - applications. ASME Transactions Journal of Dynamic Systems
and Measurement Control B, 107:1–24.)). In this control scheme, the robot joints are coupled to their reference position by virtual impedances. An impedance $Z$ describes the relation between input velocity $v$ and output force $F$. In the linear case, the transfer function is given as\\

<latex> 
\begin{equation}
Z(s) = \frac{F(s)}{v(s)}.
\end{equation}
</latex>

In a typical implementation for control, $Z$ includes a virtual spring-damper element with spring stiffness $K$ and damping coefficient $B$ that is used to calculate joint torques $\tau$. The virtual element is deflected by the length $e$ and with the velocity $\dot{e}$ :\\


<latex> 
\begin{equation}
\setcounter{equation}{2}
\frac{\tau(s)}{\dot{e}(s)} = \frac{K}{s}+B.
\end{equation}
</latex>


<latex> {\fontsize{12pt}\selectfont \textbf{Control Modifications for Better Transparency} </latex> \\

High performance impedance control is characterized by transparency. A device is described as transparent if its control is able to display the properties of virtual elements realistically with minimal influence of its real-world characteristics inertia, friction and gravity.\\
To reduce or completely remove robotic support from selected joints of the Lokomat, transparency at low desired interaction torques (K→0, B→0) is needed. To reach this goal, the torque control performance has to be improved.\\
Hardware and control performance were evaluated in initial system identification experiments. Based on their results, several controller modifications were implemented and evaluated:
  * Changes in the controller structure
  * New method for velocity estimation from position measurement
  * Load velocity compensation
  * Backlash compensation
  * Loop shaping to remove phase offset 

<latex> {\fontsize{12pt}\selectfont \textbf{Focus: Load Velocity Compensation} </latex> \\

In physical systems, forces are applied to objects with a finite stiffness and damping. Therefore, the applied force is does not only depend on the input velocity but also on their own velocity. The transfer function that describes a simple system which includes the intrinsic load
velocity feedback (Figure 2) is given by\\

{{ :adrl:education:completed_projects:lvfb.png?400|}}

<latex>
\begin{equation}
\setcounter{equation}{3}
\frac{F(s)}{v(s)} = \frac{\left( K_t+B_t s\right)\left(J_L s +B_L\right)}{J_L s^2 + \left(B_t+B_L\right)s + K_t},
\label{eq:lvfb}
\end{equation}
</latex>\\

where $K_t$ is the transmission stiffness, $B_L$ and $B_t$ are the load and transmission damping coefficients, $J_L$ the load inertia, $F$ the applied force and $v$ the input velocity. 

During force control, one bandwidth limitation is therefore given by the transfer function's zero (Equation 3) originating from the load dynamics. Higher control performance can be expected when the influences of the load on the force dynamics are removed. The basic solution for this problem to add an additional velocity $v^*=v_L$ that compensates for the load velocity. Without the feedback loop, the dynamics reduce to
\begin{equation}
\setcounter{equation}{4}
\frac{F(s)}{v(s)} = \frac{K_t}{s}+B_t, 
\end{equation}
now only depending on the transmission dynamics and the controlled input velocity and not on the load dynamics anymore.

This approach was shown to lead to crucial torque control performance improvements of hydraulically and electrically actuated robots ((Boaventura, T., Focchi, M., Frigerio, M., Buchli, J., Semini, C., Medrano-Cerda, G., and
Caldwell, D. (2012). On the role of load motion compensation in high-performance
force control. Proceedings of the IEEE/RSJ International Conference on Intelligent
Robots and Systems, pages 4066–4071.)) ((Focchi, M. (2013). Strategies To Improve the Impedance Control Performance of a
Quadruped Robot. PhD thesis, University of Genoa Istituto Italiano di Tecnologia.)).


<latex> {\fontsize{12pt}\selectfont \textbf{Results} </latex> \\

The combination of several control modifications results in a significant reduction of undesired interaction torques at low desired impedances (K→0, B→0). Figure 3 shows the corresponding Bode plot of the transfer function 

\begin{equation}
\setcounter{equation}{5}
H(s) = \frac{\tau_{measured}(s)}{\tau_{desired}(s)} ,
\end{equation}

{{ :adrl:education:completed_projects:bode.png?450|}}

where $\tau$ is the interaction torque at one joint. Ideal behavior would result in a magnitude of 0 db and a phase of 0°.
While the controller improves the magnitude behavior as desired, the phase offset increases. However, the phase behavior is of minor importance if the torque magnitude approaches zero.


<latex> {\fontsize{12pt}\selectfont \textbf{Conclusion} </latex> \\

The torque control performance at low desired interaction torques is improved. Within this thesis project, the new control has already been shown to be a crucial componentent in the design of new rehabilitation tasks.

<latex> {\fontsize{12pt}\selectfont \textbf{Future Work} </latex> \\
The reliability of the new control inside the new task design has only been verified with healthy subjects walking in the device. Both the control and the task design have to be tested and tuned in a realistic clinical setup.
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