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<latex>{\fontsize{16pt}\selectfont \textbf{Push recovery using a stability index}} </latex> /\ \\
<latex>{\fontsize{12pt}\selectfont \textbf{Francisco Giraldez Gamez}} </latex> \\
<latex>{\fontsize{10pt}\selectfont \textit{Semester Project, RSC}} </latex> \\ \\

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

A balance control for an exoskeleton has been developed and tested, to achieve balance recovery from pushes by using a stability
index. The balance controller attempts to maintain stability through stance control,
or by taking a step if necessary. Operational space control in the null space of the
contacts is the proposed method for the stance control. The step uses capture point
based strategies to recover balance, being triggered according to the stability index.
Due to the nature of
the exoskeleton model used for simulation, this balance controller is also applicable for
humanoid robots.

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

This Semester Project is part of the BALANCE EU project, in which an exoskeleton is developed to aid balance-impaired people. ADRL collaborates in the control system design, fulfilling two main tasks: balance and transparency. In this project, the first problem is tackled, by developing a balance controller capable of maintaining stability and rejecting pushes. To monitor stability, an index has been specifically designed by Tecnalia, a BALANCE project partner.

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

{{ :adrl:education:completed_projects:exomodel.png?150|}}


The joint human-exoskeleton system is modelled as the lower half of a humanoid robot, since it is the lower limbs who mainly contribute to stability in biped locomotion. For each leg, the exoskeleton features 4 degrees of freedom: hip adduction-abduction (HAA), hip flexion-extension (HFE), knee flexion-extension (KFE) and ankle flexion-extension (AFE). Additionally, the ankle adduction-abduction (AAA) joint, provided by the human user, is also included in the model due to its key role in lateral stability. Therefore, our robotic system has 10 degrees of freedom from joints, plus 6 from the free-floating base, making a total of 16 degrees of freedom to be controlled. 


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

A control strategy has been designed in order to cover three main aspects, which have been identified as key to fulfill the requirements, namely, to maintain balance during stance while taking a stabilizing step when necessary. \\

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 **Stance control:** posture  has to be kept during the stationary and stepping phases. To this end, operational space control ((M. Mistry and L. Righetti, “Operational space control of constrained and underactuated systems,” Robotics: Science and Systems VII, pp. 225–232,2011))is used to maintain the center of gravity's projection inside the base of support and an upright base orientation, subject to task prioritization ((G. Antonelli, F. Arrichiello, and S. Chiaverini. The null-space-based behavioral
control for mobile robots. In 2005 International Symposium on Computational
Intelligence in Robotics and Automation, pages 15{20, June 2005.)). This provides generalized coordinate torques $\Gamma_{des}$  for control, which are then mapped into the contact constraints null space, . Computing this null-space projection is done through singular value decomposition, to overcome its intrinsic numerical instability. Thus, valid actuation torques $\tau$ are obtained as follows:((L. Sentis, J. Park, and O. Khatib. Compliant control of multicontact and center-ofmass
behaviors in humanoid robots. IEEE Transactions on Robotics, 26(3):483{501,
June 2010.))
$$
\hspace{80}\mathbf{\tau=(N_CS^T)^{+}\Gamma_{des} \hspace{80} \Gamma_{des} = J^T_{pos}F_{cog}+N_{pos}J^T_{or}\Gamma_{cog}+N_{pos}N_{or}\Gamma_0}
$$

**Step control:** the step needs to be taken accurately and on time. To find a stabilizing step goal, the capture point approach is inherited from Matthias Wild's semester project ((http://www.adrl.ethz.ch/doku.php/adrl:education:completed_projects:wild2014f)). A capture point is defined as the point on the ground where a step would bring a falling legged robot to a complete stop. To reach it as effectively as possible, inverse kinematics are used for the swinging leg, as proposed in the hybrid operational space control approach ((M. Hutter et al. Hybrid Operational Space Control for Compliant Legged Systems,
pages 504{. MIT Press, 2013.)). The step trajectories are generated ((Qiang Huang, S. Kajita, N. Koyachi, K. Kaneko, K. Yokoi, H. Arai, K. Komoriya,
and K. Tanie. A high stability, smooth walking pattern for a biped robot. In
Robotics and Automation, 1999. Proceedings. 1999 IEEE International Conference
on, volume 1, pages 65{71 vol.1, 1999.)) and tracked by individual PID controllers in each joint, which requires to decouple the swinging leg from the stance control.


**Step triggering:** deciding when a perturbation is large enough to be recovered by stepping. The stability index (SI), developed ad-hoc by Tecnalia , fulfils this task by considering the current state of the center of gravity and the base of support (BoS). Different distances between the CoG and the BoS are computed, accounting for the CoG position, velocity and acceleration, plus the total area of the BoS. These variables are then weighted with parameters, tuned by Tecnalia so that 0 should represent the stability limit. 

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$$
SI = \alpha D_{min}+\beta D_{tCv}+\gamma D_{tCa}+\delta \sqrt{area(BoS)}  
$$

A finite state machine regulates the behavior of the exoskeleton, by providing each leg's state (e.g. support, step, etc.) as a reference to the stance control and the step control. The respective leg states are triggered by monitoring the stability index and the ground reaction forces, which are computed from rigid-body dynamics for each leg. 

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

This balance controller has been implemented in C++ and tested using SL ((http://www-clmc.usc.edu/publications/S/schaal-TRSL.pdf)). Tests have been devised in order of increasing complexity, including individual tests for each aspect. As opposed to previous theses, the weight of the robot comprises only the exoskeleton's weight (33kg).First, a CoG positioning test validated operational space control, while some push rejection tests proved the capability of the stance control to maintain balance during the support phase. Without stepping, horizontal pushes in any direction can be rejected as far as their magnitude does not exceed 300 N in X direction and 150 N in Y, which is in the order of the robot's weight. 

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


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

During an oblique push rejection experiment, the suitability of the SI as balance metric was also tested. In the graph on the right one can observe how the SI quickly approaches zero as the CoG deviates from its most stable position.

In a step execution test, the exoskeleton was able to take a 0.2 m step to the front in less than 0.5 s. During this experiment, the system was held in the air in order to not rely on the stance control for stability. A test of foremost importance was the static step: the exoskeleton should be able to complete a step without losing balance. It is here where our system finds its limitation, since the stance control cannot achieve balance once the swing leg is lifted from the floor, due to a dynamical instability that causes the AAA joint to twist. \\ 

{{:adrl:education:completed_projects:twistend.png?250 |}}

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

The balance controller provides the desired behaviour of the exoskeleton, allowing for small push rejection and an effective step routine. The stability index shows promising results as a balance metric, especially since it could be tuned/learned to adjust the exoskeleton's sensitivity to disturbances. Nevertheless, the stance control fails to maintain balance on one leg, during the step.

Possible causes for ankle twisting could be the lack of contact force control, an error in the contact constraints or unfeasible robot model parameters. Future work should be focused in solving this problem, as well as further testing the SI during the step phase of balance recovery. It could however be possible that ankle twisting is not an issue when the exoskeleton is attached to a human user.
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