**4. Comparison between series elastic actuators**

### **4.1. Comparison parameters**

As mentioned in Section 1, the analysis of an actuator comprises the investigation of force bandwidth, mechanical output impedance, dynamic range, force density and the time response for the system without load. Each of these parameters has its own importance for the system and its own method to be determined. Robinson (2000) first introduced the comparison methodology adopted in this chapter. In these analyses, the load stiffness and viscous damping presented for ESEA and HSEA presented in **Tables 1** and **3** respectively is not considered on the following analyses. Therefore, it is expected that the systems responses have deviations between the ones presented in Sections 2 and 3.

### *4.1.1. Force bandwidth*

The force bandwidth of an actuator is how quickly it can generate the desired forces. In order to be able to perform the desired task, the bandwidth must be sufficiently high to transmit the forces through the machine structure. To control a human joint, only a few Hertz of bandwidth is necessary. Moreover, the bandwidth needed for a four-legged dynamic robot for weight carrying is small too. Reference authors estimate that only 20 Hz is more than necessary for the given application. The bandwidth analysis is divided in the closed loop force bandwidth and the large force bandwidth. Closed loop bandwidth is evaluated with a step input of a small force in the actuator closed loop model (**Figure 23**). The large force bandwidth is obtained with an oscillation of the full steady-state output force of the actuator as an input on the closed loop model. HSEA has 4.9 kN of output force and ESEA has only 450 N. Due to this difference between the actuators capabilities, only the closed loop bandwidth is compared in this chapter. Referring to **Figure 23**, *Xm* is the motor displacement, *Fd* is the desired force, *Fl* is the output force, *ks* is the spring stiffness, *K* and *Vc* are model characteristics.

**Figure 23.** Simplified block diagram for closed loop bandwidth evaluation (Robinson, 2000).

#### *4.1.2. Mechanical output impedance*

The mechanical output impedance is defined as the minimum force needed to move a certain load. An actuator with low impedance can be considered a pure force source since its internal dynamics are negligible. Low impedance actuators can also be referred as back-drivable (Robinson, 2000). The analysis of the output impedance is made by applying to each actuator model the case of a load, which is free to move, attached to the actuator's output. The impe‐ dance is evaluated around a constant desired output force as an analogy of the concept of impedance on electrical circuits (**Figure 24**). Referring to **Figure 24**, the *xl* is the load displace‐ ment, *xs* is the spring displacement and *Fl* is the force output.

**Figure 24.** Simplified block diagram for mechanical output impedance evaluation (Robinson, 2000).

### *4.1.3. Dynamic range*

*4.1.1. Force bandwidth*

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The force bandwidth of an actuator is how quickly it can generate the desired forces. In order to be able to perform the desired task, the bandwidth must be sufficiently high to transmit the forces through the machine structure. To control a human joint, only a few Hertz of bandwidth is necessary. Moreover, the bandwidth needed for a four-legged dynamic robot for weight carrying is small too. Reference authors estimate that only 20 Hz is more than necessary for the given application. The bandwidth analysis is divided in the closed loop force bandwidth and the large force bandwidth. Closed loop bandwidth is evaluated with a step input of a small force in the actuator closed loop model (**Figure 23**). The large force bandwidth is obtained with an oscillation of the full steady-state output force of the actuator as an input on the closed loop model. HSEA has 4.9 kN of output force and ESEA has only 450 N. Due to this difference between the actuators capabilities, only the closed loop bandwidth is compared in this chapter. Referring to **Figure 23**, *Xm* is the motor displacement, *Fd* is the desired force, *Fl* is the output

force, *ks* is the spring stiffness, *K* and *Vc* are model characteristics.

**Figure 23.** Simplified block diagram for closed loop bandwidth evaluation (Robinson, 2000).

impedance on electrical circuits (**Figure 24**). Referring to **Figure 24**, the *xl*

**Figure 24.** Simplified block diagram for mechanical output impedance evaluation (Robinson, 2000).

The mechanical output impedance is defined as the minimum force needed to move a certain load. An actuator with low impedance can be considered a pure force source since its internal dynamics are negligible. Low impedance actuators can also be referred as back-drivable (Robinson, 2000). The analysis of the output impedance is made by applying to each actuator model the case of a load, which is free to move, attached to the actuator's output. The impe‐ dance is evaluated around a constant desired output force as an analogy of the concept of

is the force output.

is the load displace‐

*4.1.2. Mechanical output impedance*

ment, *xs* is the spring displacement and *Fl*

The dynamic range is the ratio between the maximum output force of the actuator and the minimum resolvable force of this device. The dynamic range analysis on an actuator is very important since it gives a measure of how sensitive the actuator is to small force compared to the actuator maximum force. For example, if an actuator has high maximum force and high minimum resolvable force, its dynamic range is low. Therefore, this actuator can only make tasks of high force and it is not able to manipulate fragile objects or have direct contact with humans. Moreover, a large dynamic range is desirable for almost all robotic applications, since it allows the robot to be versatile to perform high sensitive operations and large force de‐ manding operations. Since it can sense or grasp almost all fragile objects and manipulate it in precise tasks and, on the same time, can perform high force demanding tasks such as weight lift, one of the largest dynamic range actuator is the human muscle.

### *4.1.4. Force density*

Force density is defined as the ratio between the output force and the mass or the volume of an actuator. An actuator with large force density allows the application in lighter robots with good force output characteristics since it not overburden the robot structure and allows the quick response of the mechanism.

### **4.2. Results and discussion**

The two SEA presented in previous sections are compared based on the key parameters of the series elastic actuators aforementioned which are bandwidth, output impedance, time response, power density, dynamic range.

The spring stiffness affects all parameters presented above. In order to evaluate the differences between the two systems with the same parameter for the compliant sensor, the spring stiffness is 3.336E+03 kN/m for both actuators.

### *4.2.1. Force bandwidth results*

**Figure 25** shows the HSEA and ESEA Bode diagram which allows evaluating the two systems closed loop force bandwidth.

**Figure 25.** Closed loop bode diagram.

The ESEA frequency band is about 40Hz with a phase angle of 90° which is quite high. On the other hand, the HSEA has 30Hz but the phase angle is about 55° which is more appropriate for practical applications. Since a four-legged robot works on small bandwidths, the smaller bandwidth range of the HSEA is not an issue for this application. In addition, the two actuators present resonance frequency above 100Hz which is far enough to the required frequency.

### *4.2.2. Time response results*

The time response for a Unit step input of the system without the load is given in **Figure 26**. In the ESEA and HSEA design made in Sections 2 and 3, respectively, the system dynamic has a load damping of 8 Ns/m and a load stiffness of 3E+06 N/m. ESEA shows an 8% overshoot and settling time of 0.03 s which is very fast. While HSEA has an overshoot about 12% and a settling time of 0.07 s. Although HSEA has larger overshoot and settling time than ESEA, these two parameters are compatible with the reference works results (Paine et al., 2013; Robinson, 2000). Therefore, even in the HSEA case the settling time is tolerated.

**Figure 26.** Time response of the actuators.

### *4.2.3. Mechanical output impedance results*

Since output impedance is the force at an actuator output given a moving load position, the impedance analysis was made by setting a load position with a constant desired force. An ideal actuator has zero impedance. In other words, the actuator is a pure force source. If this happens, the dynamics of the actuator are completely decoupled from the dynamics of the load motion (Robinson, 2000). Although the compliant element of the series elastic actuator limits the bandwidth, it reduces the output impedance which is desirable for many actuator applications (Pratt et al., 2002).

Nevertheless, these two actuation systems have the same spring stiffness. Therefore, the differences between the systems output impedance are due to the actuator's moving system which are the motor with gear reduction for the ESEA and servo valve with hydraulic piston for the HSEA (Robinson, 2000).

**Figure 27** shows the output impedance for both hydraulic and electric sources. The output impedance of the series elastic actuator is small at low frequency and it achieves the spring stiffness magnitude (which is −130dB for the spring previous selected) when the frequency rises. This behavior was proved by simulations and physical SEA experiments (Robinson, 2000).

**Figure 27.** HSEA and ESEA output impedance.

Since the both systems have the same compliant element, they achieve the same value with frequency increase. Therefore, the difference on the impedance is given by the curve slope. Examining the slope of the both curves, the HSEA has the smallest value of the impedance when the frequency increases. It can be seen two peaks of the magnitude, the first one due to controller gains and the second occurs in the systems resonance frequency.

### *4.2.4. Power density results*

The ESEA frequency band is about 40Hz with a phase angle of 90° which is quite high. On the other hand, the HSEA has 30Hz but the phase angle is about 55° which is more appropriate for practical applications. Since a four-legged robot works on small bandwidths, the smaller bandwidth range of the HSEA is not an issue for this application. In addition, the two actuators present resonance frequency above 100Hz which is far enough to the required frequency.

The time response for a Unit step input of the system without the load is given in **Figure 26**. In the ESEA and HSEA design made in Sections 2 and 3, respectively, the system dynamic has a load damping of 8 Ns/m and a load stiffness of 3E+06 N/m. ESEA shows an 8% overshoot and settling time of 0.03 s which is very fast. While HSEA has an overshoot about 12% and a settling time of 0.07 s. Although HSEA has larger overshoot and settling time than ESEA, these two parameters are compatible with the reference works results (Paine et al., 2013; Robinson,

Since output impedance is the force at an actuator output given a moving load position, the impedance analysis was made by setting a load position with a constant desired force. An ideal actuator has zero impedance. In other words, the actuator is a pure force source. If this happens, the dynamics of the actuator are completely decoupled from the dynamics of the load motion (Robinson, 2000). Although the compliant element of the series elastic actuator limits the bandwidth, it reduces the output impedance which is desirable for many actuator applications

Nevertheless, these two actuation systems have the same spring stiffness. Therefore, the differences between the systems output impedance are due to the actuator's moving system which are the motor with gear reduction for the ESEA and servo valve with hydraulic piston

**Figure 27** shows the output impedance for both hydraulic and electric sources. The output impedance of the series elastic actuator is small at low frequency and it achieves the spring stiffness magnitude (which is −130dB for the spring previous selected) when the frequency

2000). Therefore, even in the HSEA case the settling time is tolerated.

*4.2.2. Time response results*

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**Figure 26.** Time response of the actuators.

(Pratt et al., 2002).

for the HSEA (Robinson, 2000).

*4.2.3. Mechanical output impedance results*

Power density is the actuator power per mass unit. High power density actuators can delivery great power with a small package. It prevents over-burdening the actuator structure with excessive mass allowing a responsive robot performance. The lightweight structure with a quick performance is very important for a weight carrying dynamic robot because it provides a compact robot which is suited to a non-structured environment and has a great output force.

As shown in previous section, the hydraulic series elastic actuator has more output force than the electric SEA. Moreover, the ESEA is heavier than the HSEA.

The output force of the HSEA is 4.9kN and it weighs 1.5kg given the velocity of 0.5m/s, the HSEA has a power density about 1600W/kg which is a great performance. On the other hand, the ESEA has 450N of output force and weighs 3.7kg. The ESEA is also slower than HSEA, the electric SEA velocity is 13.4cm/s which gives a power density of 16W/kg. This result shows a poor performance of ESEA for weight carrying dynamic robot.

Even though the HSEA mass applied on the power density calculation did not include the mass of servo valve, reservoir, pump, hoses and pipes which are heavier than the ESEA batteries that also are not included on the electric SEA mass. Even if these accessories mass is considered, the HSEA still has way more power density even with these items included.

### *4.2.5. Dynamic range results*

Dynamic range is the ratio between the maximum output force and the minimum resolvable output force. It gives quantitative information of how sensitive the actuator is with respect of its maximum output force capability. It is a crucial parameter for dynamic robots because it allows the actuator to be used in sensitive activities such as human contact, equipment operation, and non-structured environment exploration. In addition, the actuator with great dynamic range can do hard work activities such carrying, push or pull weights. In summary, high dynamic range is an important characteristic for a four-legged dynamic robot application.

The minimum resolvable output force is evaluated by the resolution of the spring deflection sensor. Resolution gives the minimum spring deflection that can be detected. This deflection times the spring stiffness gives the actuator minimum resolvable force.

Both the SEAs have the same spring stiffness in this comparison. Also, both actuators have the same spring deflection sensor which is a linear encoder. Therefore the minimum output force is the same for both which is 3N.

The HSEA dynamic range is 1500:1 while the ESEA range is 150:1. It shows that the hydraulic SEA has a range 10 times bigger than electric one.
