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In this paper semirings with an idempotent addition are considered. These algebraic structures are endowed with a partial order. This allows to consider residuated maps to solve systems of inequalities $A \otimes X \preceq B$. The purpose of this paper is to consider a dual product, denoted $\odot$, and the dual residuation of matrices, in order to solve the following inequality $ A \otimes X \preceq X \preceq B \odot X$. Sufficient conditions ensuring the existence of a non-linear projector in the solution set are proposed. The results are extended to semirings of intervals

The rehabilitation robot Lokomat allows automated treadmill training for patients with neurological gait disorders. The basic position control approach for the robot has been extended to patient-cooperative strategies. These strategies provide more freedom and allow patients to actively influence their training. However, patients are likely to need additional support during patient-cooperative training. In this paper, we propose an algorithm based on iterative learning control that shapes a supportive torque field. The torque field is supposed to assist the patient as much as needed in performing the desired task. We evaluated the algorithm in a proof-of-concept experiment with 3 healthy subjects. Results showed that the amount of support was automatically adapted to the activity and the individual needs of the subjects. Furthermore, the support improved the performance of the subjects. Adaptive Support for Patient-Cooperative Gait Rehabilitation with the LokomatAlexander Duschau-Wicke, Thomas Brunsch, Lars Lünenburger, and Robert RienerAbstract-The rehabilitation robot Lokomat allows automated treadmill training for patients with neurological gait disorders. The basic position control approach for the robot has been extended to patient-cooperative strategies. These strategies provide more freedom and allow patients to actively influence their training. However, patients are likely to need additional support during patient-cooperative training. In this paper, we propose an algorithm based on iterative learning control that shapes a supportive torque field. The torque field is supposed to assist the patient as much as needed in performing the desired task. We evaluated the algorithm in a proof-of-concept experiment with 3 healthy subjects. Results showed that the amount of support was automatically adapted to the activity and the individual needs of the subjects. Furthermore, the support improved the performance of the subjects.