As explained in the previous section, under real conditions the accumulator would gradually empty. In a physical system, this problem would be dealt with by the control system, which would ensure that the accumulator pressure remains periodic (i.e. the final pressure must not drop below the initial pressure value) by modulating the VDA oil flow rate at the expense of reduced push-off power. The control design is beyond the scope of this study and part of future work. However, as a proof of concept, following the working principle of the control system, and to account for the extent of push-off power loss as a result of maintaining a periodic accumulator pressure, this section presents a methodology that was carried out to impose a pseudo control on the results obtained and to observe the extent of reduced push-off power in real conditions.
The control system would ensure that the final accumulator pressure must not drop below the initial pressure value (i.e. pressure is cyclic). To illustrate this functioning, a sample ankle power data was taken, and its positive phases were scaled down. This resulted in a reduced ankle power profile, the eccentric phases of which remain unaffected while the concentric phases were reduced. The original data along with the reduced push-off represent the effect of the control system as a result of maintaining the accumulator pressure by compromising the push-off power, which is presented in Figure 3.12.
Figure 3.12: An ankle power profile sample compared with the effect of the control system.
In order to understand how this could be achieved with the design, the cumulative energy profiles (area under the power curves) corresponding to both the original and reduced power curves, shown in the previous figure is presented in Figure 3.13.
Figure 3.13: Energy profiles for the original ankle power data and for the reduced power data.
Figure 3.13 shows the energy profile obtained by integrating the sample ankle power and reduced power data plotted in Figure 3.12. Comparing the two energy profiles in Figure 3.13, the profile corresponding to the reduced power curve ( in Figure 3.12) has equal initial and final values, which means that the released energy by the ankle was equal to what it stored earlier in gait. In turn, the other profile (obtained from the original ankle power data in Figure 3.12) illustrates that more energy was released than stored earlier.
With the accumulator as the energy storing component, its pressure profile is analogous to an inverted ankle energy profile (during eccentric work phases, cumulative ankle energy is negative, whereas accumulator pressure rises and vice versa). Therefore, the ankle power profile (rate of change of ankle energy) must also resemble the inverted profile of the rate of change of accumulator pressure (dPacc/dt). The accumulator pressure
profile, obtained from simulation with both the ideal and real conditions with the gas- charged accumulator of 245 cc, is shown in Figure 3.14.
Figure 3.14: Accumulator pressures obtained with a gas-charged accumulator for ideal and real conditions.
In Figure 3.14, the final pressure for the profile obtained with the ideal model is higher than the initial pressure, whereas, with real conditions, the final pressure is lower than the initial pressure value. This behaviour is similar to the inverted ankle energy profile in Figure 3.13. Therefore, in order to bring the final pressure back to the initial pressure for real conditions, the time derivative of pressure (dPacc/dt) curve for real conditions must be taken, and the negative phases of which should be scaled down (using factors ranging from 0 to 1). For each of these scaling iterations, the scaled dPacc/dt curve, when integrated, would result in a scaled pressure profile. This process of scaling down the negative phases of dPacc/dt, and subsequently integrating them would result in several scaled pressure profiles, and the one having equal initial and final values (cyclic
pressure profile), would be the desired pressure profile. However, it should be noted that an actual control system would not scaled down the push off power, but would modulate the displacement in order to have equal initial and final accumulator pressures, and the assumption of a perfect displacement control is always assumed in this work. Therefore, the term “scaled” refers to those results where, the push off power has been scaled down and all the other cases (for instance, ideal or real), should not be interpreted as uncontrolled. A perfect control is always assumed for all the conditions.
Figure 3.15 represents the scaled accumulator pressure profile as a result of integrating the scaled-down negative phases of the dPacc/dt curve (which was obtained from simulation with real conditions) for Salford’s gait data. This Figure is plotted along with the ideal and real pressures.
Figure 3.15: Accumulator pressures obtained with a gas-charged accumulator for ideal, real, and real scaled conditions using Salford’s gait data.
To understand the extent of reduced push-off power as a result of reduced oil flow, the scaled pressure profile was used to calculate the oil volume using a re-arranged Eq. (3.44). 𝑉𝐹 =𝑉𝑚𝑖𝑛�𝑃𝑃𝑝𝑟 𝑎𝑐𝑐� 1 𝑘 + 𝑉𝐴�1− �𝑃𝑃𝑝𝑟 𝑎𝑐𝑐� 1 𝑘 �… … …𝐸𝑞. (3.45)
Where, the real scaled pressure is used as Pacc, whereas the rest of the parameters remained unchanged (same values, which were used to calculate the accumulator size). This oil volume was numerically differentiated to get flow rate. The obtained flow rate, when multiplied by the scaled pressure, provided the scaled accumulator hydraulic power (Eq. (3.46).
𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟ℎ𝑦𝑑𝑟𝑎𝑢𝑙𝑖𝑐𝑝𝑜𝑤𝑒𝑟= 𝑑𝑉𝑑𝑡𝐹.𝑃… … …𝐸𝑞. (3.46)
Figure 3.16 and Figure 3.17 show the VDA torque and accumulator hydraulic power obtained using ideal conditions and real scaled conditions, for the Salford’s gait data, and are compared with the original data.
Figure 3.16: Torque profile for ideal and real scaled conditions using Salford’s gait data.
In Figure 3.17, the hydraulic power matches exactly with the in vivo ankle power for ideal conditions. With pseudo control implemented, the system was able to deliver a peak hydraulic power of 137 watts against 159 watts with ideal conditions. In other words, 13.8% of peak power was lost by reducing the push-off power to achieve a periodic accumulator pressure profile.
It should be noted that this comparison is between the two hydraulic powers, one is for the ideal case (which matches exactly with the in-vivo ankle power), and the other is for the real scaled conditions.