1 A Step Towards Testing of Foot Prostheses Using Real-Time Substructuring (RTS) 5 there is no delay in the transfer system (orange). In the real-time experiments, the force offset is set when the experimental part is already mounted on the test bench. Hence, the measured force is Fm =0Nif themass mE is not in contact. However, the force that acts at the interface of the reference system is Fint =mE·g. Since the interface forceFint acts on the numerical part and is thus required for a correct RTS simulation, the measured force must be corrected by Fint =Fm+mE · g. (1.1) In other words, the difference between the measured forces and the interface forces in the reference system is 2mEg, which results from the different orientation of the gravitational force. However, due to setting the offset of the force sensor when the mass mE is already mounted, we only have to compensate mEg according to Eq. (1.1). 1.3.3 Delay Compensation The delay and dynamics of the transfer system can lead to instability and inaccuracy of the RTS simulation, as they bring negative damping into the system. To compensate, there are a lot of different delay compensation techniques. For the experiments presented in Sect. 1.4, we used a polynomial extrapolation as published in [10]. The formulation is z (t) = 2 i=0 ai · z(t −i · τ) with a0 =3, a1 =−1 and a2 =3, if a polynomial of degree n = 2 is used and the time delay of the transfer system is τ. Here, the position value that is calculated by the numerical simulation at the current time t is z and the command that is sent to the actuator is z . z is an extrapolated value that depends on the value of z in the past. 1.4 Experiment This section presents the experiments performed and results obtained with the mechanical system presented in Sect. 1.3. Firstly, the load case is described with the corresponding simulation parameters in Sect. 1.4.1 and then the results are shown in Sect. 1.4.2. 1.4.1 Loading Condition and Simulation Parameters The aim of our research is to carry out a RTS simulation of a human walking with a prosthesis. A major question is how a human being stabilizes its body so that it does not fall, even if there are disturbances. One approach used in robotics for stabilizing bipeds is to use a planned trajectory and track this trajectory with a controller [17]. Using this approach, we prescribed a desired trajectory zd (and ˙zd) for the mass mV in the numerical substructure. A PD controller (parameters Kp and Kd) attempts to follow this trajectory by adding an external force, if the desired motion is not tracked: Fext =(mV +mE) · g +Kp · (zd −z (V)) +Kd · (˙zd − ˙z (V)). The first term corresponds to the static external force for holding the system in the air. Hence, the external force Fext that acts on the system as displayed in Fig. 1.4 is the force that is required to keep mass mV on the desired trajectory. The higher the values Kp andKd, the faster the controller that tries to hold mass mV on the desired trajectoryzd. Even small deviations from the trajectory are adjusted in a short period. If the parameters of the PD controller are reduced, the ability of the mass to follow its desired trajectory is reduced as well. Returning to the aim of testing a prosthesis using the RTS approach, this is similar to the case where the forces coming from a badly designed prosthesis are so large compared to the equilibrating forces, that the patient needs for example to tension a muscle much more than usual or tilt the hip. The dynamic behavior of the mechanical system is more relevant in this case and not dominated by the properties of the PD controller.
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