**3. Hydraulic series elastic actuator**

### **3.1. Hydraulic series elastic actuator design**

### *3.1.1. Methodology*

A basic model of a series elastic hydraulic actuator has the serial element, i.e. a rod with the spring's arrangement and a hydraulic circuit which basically comprises of the hydraulic cylinder and the servo valve. The actuator may be understood as a combination of two parts. The first one is the serial elastic element presented. The other one is the hydraulic circuit which has some construction alternatives as well.

Once the hydraulic circuit and serial elastic element of the actuator are defined, the next step is to select the hydraulic components and design the spring based on the application. This is done by defining a condition for this system which, in this case, is the actuator output force to be up to 5000 N. To achieve this output force, the springs are selected based on consolidate knowledge presented by mechanical engineering design bibliography. Moreover, the rod had to be able to support the load without buckling. A special attention must be given to the actuator's springs because they affect system's impedance and bandwidth. After setting these two parameters, iteration could be necessary to evaluate the desirable spring constant.

After the selection of the rod and the spring, it is possible to use the force caused by the spring under allowable extension-compression limits as the force output of the hydraulic cylinder. This force and the maximum speed of 0.5 m/s, as defined by Robinson and Pratt (2000), led to the selection of the correct cylinder by a commercial data sheet; otherwise it would be possible to manufacture a custom cylinder. After the parts design, they are assembled in Siemens Solid Edge (Siemens PLM Software, Germany) digital prototyping environment which allows obtaining inertial properties of the assembly.

### *3.1.2. Digital prototyping design*

The design was done by the methodology mentioned on the previous section. Following these steps, the actuator achieved a good volume-output force ratio for the available components combination. Also all the internal components had been enclosed to avoid accidents, compo‐ nents contamination and make the actuator suitable to human interaction (Bento Filho et al., 2014).

**Figure 17** shows the prototype of the actuator. There is a hydraulic cylinder (1) that moves forward and dislocates the movable rod (2) fixed in it with a screwed connection. The model has two low friction sliding cylinders (6) and (8) fixed on the base tube (3) and one guide (7) fixed on the movable rod (2). There are two springs (4) and (5) fixed between the low friction sliding cylinders (6) and (8). A connection guide (10) is used to connect the hydraulic cylinder rod (1) to the base tube (3). The load terminal (9) is also connected directly on the base tube (3). The encapsulation tube (12) is the last part of the assembly. It is connected on the hydraulic cylinder (1) and base tube (3). A guide bolt (11) was applied to make the connection between the base tube (3) and the encapsulation tube (12), but without interfere on the base tube's (3) linear movement.

position, velocity and mixed controllers. It is also important to verify the effect of spring stiffness and gear ratio in load bandwidth. Another further investigation is the development of tuning strategies between the spring stiffness – actuator bandwidth and elastic energy store relationship. Since these effects were neglected in this section, actuator's backslash and viscous

A broad comparison between the electric serial elastic actuator and the hydraulic SEA has to

A basic model of a series elastic hydraulic actuator has the serial element, i.e. a rod with the spring's arrangement and a hydraulic circuit which basically comprises of the hydraulic cylinder and the servo valve. The actuator may be understood as a combination of two parts. The first one is the serial elastic element presented. The other one is the hydraulic circuit which

Once the hydraulic circuit and serial elastic element of the actuator are defined, the next step is to select the hydraulic components and design the spring based on the application. This is done by defining a condition for this system which, in this case, is the actuator output force to be up to 5000 N. To achieve this output force, the springs are selected based on consolidate knowledge presented by mechanical engineering design bibliography. Moreover, the rod had to be able to support the load without buckling. A special attention must be given to the actuator's springs because they affect system's impedance and bandwidth. After setting these two parameters, iteration could be necessary to evaluate the desirable spring constant.

After the selection of the rod and the spring, it is possible to use the force caused by the spring under allowable extension-compression limits as the force output of the hydraulic cylinder. This force and the maximum speed of 0.5 m/s, as defined by Robinson and Pratt (2000), led to the selection of the correct cylinder by a commercial data sheet; otherwise it would be possible to manufacture a custom cylinder. After the parts design, they are assembled in Siemens Solid Edge (Siemens PLM Software, Germany) digital prototyping environment which allows

The design was done by the methodology mentioned on the previous section. Following these steps, the actuator achieved a good volume-output force ratio for the available components combination. Also all the internal components had been enclosed to avoid accidents, compo‐ nents contamination and make the actuator suitable to human interaction (Bento Filho et al.,

be made to define which one is the best for a compliant robot application.

friction have to be analyzed in future works.

**3. Hydraulic series elastic actuator**

**3.1. Hydraulic series elastic actuator design**

has some construction alternatives as well.

obtaining inertial properties of the assembly.

*3.1.2. Digital prototyping design*

2014).

*3.1.1. Methodology*

218 Recent Advances in Robotic Systems

**Figure 17.** Actuator's components: (1) hydraulic cylinder; (2) movable rod; (3) base tube; (4) and (5) springs; (6), (7) and (8) low friction sliding cylinders; (9) load terminal; (10) connection guide; (11) guide bolt; (12) encapsulation tube.

There are four low friction sliding cylinders (6), (7), (8), and (10) in this actuator model. The application of the four sliding cylinders is to give to the system the necessary stability. Without these components, the movable rod (2) would oscillate creating additional forces on the springs instead of only an axial force. Moreover, the load terminal is connected with the base tube only.

The springs are responsible to the hydraulic SEA stiffness reduction. However, this component has to be carefully assembled since it will stretch and compress depending on the movement of the actuator and the load attached to it. Choosing the spring is always challenging, because it has to be a balance between a large bandwidth requiring a high stiffness and impedance that needs a low spring constant. Therefore, after choosing all system characteristics and compo‐ nents, a minimum acceptable point must be defined between the largest force bandwidth and the minimum tolerance impedance level. After setting these two bounds, iteration is required to estimate the spring constant.

The movement of the actuator can be analyzed in three different cases:

Case 1: Movement without load

This is the simplest case, because it is a free movement. The hydraulic cylinder moves forward or backward and the movable rod and all the others components attached to it such as base tube, guides and load terminal follow the movement of the hydraulic cylinder. In this case, there is only a little deflection in the springs due to system's inertia.

### Case 2: Movement with load

When a load is attached in the load terminal, the movement of the actuator depends on the springs. For a forward movement, the spring (5) deflects and spring (4) stretches due to springs preload. These deflections and stretching depends on the load mass. The actuator moves forward only when the spring's deformation resulting force equals the applied load. This system characteristic can reduce the frequency range and makes a limit of bandwidth of the system, which is the major drawback of a series elastic actuator.

For the backward movement the same characteristics can be observed, but the spring (4) deflects and spring (5) stretches.

Case 3: Contact of the load terminal with the load

When the actuator reaches an obstacle or the load during its free movement, the springs deform and the value of the resulting force makes the actuator continues its forward movement or stop.

The prototype shows outstanding characteristics of output force and compactness. The actuator has 200 mm length when retracted with 100 mm of range. The maximum diameter is 26 mm. Moreover, with 100 bar of working pressure, this SEA produces output force up to 4.9 kN which is a great power to weight ratio since the estimated weight of the actuator is 1.5 kg. In order to ease the manufacturing, model structure material chosen is the carbon steel. Actuator's parameters are measured for modeling and control purposes. Important parame‐ ters are the deflection of the springs and the displacement of the movable rod, which are due the movement of the hydraulic cylinder.

A linear potentiometer was chosen to evaluate the movable rod's displacement. Since the potentiometer did not have the resolution to evaluate the deflection of the springs, an optical sensor used on a printer was chosen to evaluate this deflection. This sensor has a scaled tape with black lines, with distance between the lines known, which does not allow light beam passage. The tape movement produces pulses, which are proportional to spring deflection.

The assembly of the sensors was shown in **Figure 18**. The linear potentiometer (3) was connected directly on the hydraulic cylinder rod (2) and the hydraulic cylinder bore (1). The optical sensor has a circuit board (7) with a sensor (9). This circuit board was connected on the sliding cylinder (6) by a bolt connection. The optical tape (8) was connected on the sliding cylinders (4) and (5) by a bolt connection and positioned on the optical sensor (9).

**Figure 18.** Sensors assembly: (1) hydraulic cylinder bore; (2) hydraulic cylinder rod; (3) linear potentiometer; (4), (5) and (6) guides; (7) circuit board; (8) optical sensor tape; (9) optical sensor.

### **3.2. Dynamic model**

The movement of the actuator can be analyzed in three different cases:

there is only a little deflection in the springs due to system's inertia.

system, which is the major drawback of a series elastic actuator.

This is the simplest case, because it is a free movement. The hydraulic cylinder moves forward or backward and the movable rod and all the others components attached to it such as base tube, guides and load terminal follow the movement of the hydraulic cylinder. In this case,

When a load is attached in the load terminal, the movement of the actuator depends on the springs. For a forward movement, the spring (5) deflects and spring (4) stretches due to springs preload. These deflections and stretching depends on the load mass. The actuator moves forward only when the spring's deformation resulting force equals the applied load. This system characteristic can reduce the frequency range and makes a limit of bandwidth of the

For the backward movement the same characteristics can be observed, but the spring (4)

When the actuator reaches an obstacle or the load during its free movement, the springs deform and the value of the resulting force makes the actuator continues its forward movement or

The prototype shows outstanding characteristics of output force and compactness. The actuator has 200 mm length when retracted with 100 mm of range. The maximum diameter is 26 mm. Moreover, with 100 bar of working pressure, this SEA produces output force up to 4.9 kN which is a great power to weight ratio since the estimated weight of the actuator is 1.5 kg. In order to ease the manufacturing, model structure material chosen is the carbon steel. Actuator's parameters are measured for modeling and control purposes. Important parame‐ ters are the deflection of the springs and the displacement of the movable rod, which are due

A linear potentiometer was chosen to evaluate the movable rod's displacement. Since the potentiometer did not have the resolution to evaluate the deflection of the springs, an optical sensor used on a printer was chosen to evaluate this deflection. This sensor has a scaled tape with black lines, with distance between the lines known, which does not allow light beam passage. The tape movement produces pulses, which are proportional to spring deflection.

The assembly of the sensors was shown in **Figure 18**. The linear potentiometer (3) was connected directly on the hydraulic cylinder rod (2) and the hydraulic cylinder bore (1). The optical sensor has a circuit board (7) with a sensor (9). This circuit board was connected on the sliding cylinder (6) by a bolt connection. The optical tape (8) was connected on the sliding

cylinders (4) and (5) by a bolt connection and positioned on the optical sensor (9).

Case 1: Movement without load

220 Recent Advances in Robotic Systems

Case 2: Movement with load

deflects and spring (5) stretches.

stop.

Case 3: Contact of the load terminal with the load

the movement of the hydraulic cylinder.

The system consists of a servo-hydraulic system with a serial elastic element. The overall system has two degrees of freedom. One degree is the hydraulic cylinder displacement and the other is the serial elastic element displacement. This displacement and the spring deflection were used to define the output force of the system and to control it. **Figure 19** illustrates the mechanical model of this system discussed above.

```
Figure 19. Actuator's mechanical model.
```
Although most of the servo-hydraulic system is nonlinear, each component of this model has been linearized (Qian et al., 2014). The linearization occurs at an operating point and also can be made at multiple operating points. The controller is tuned based on the linearized model. Although an actuator does not work at a fixed operation point, the controller based on a linearized system has good response even when it is off the operation point.

The servo valve's pilot stage is assumed to be of first order. Since high order dynamics in the valve are more than an order of magnitude above the frequency range of 40 Hz which is the frequency of interest (Robinson and Pratt, 2000), the higher order dynamics can be neglected. Eq. (17) represents the servo valve pilot stage equation as seen in Qian et al. (2014).

$$\frac{\underline{Q}\_s(s)}{\nu\_i(s)} = \frac{K\_{vp}}{\pi s + 1} \times K\_p \times K\_c \times K\_v \tag{17}$$

where *Kvp* is the flow gain of the pilot-stage servo valve and *τ* pilot-stage's equivalent time constant, *vi* valve control signal transmitted from the hydraulic system internal proportional controller, *Kp* represents inner loop proportional gain, *KC* is the valve pressure gain which relates flow from the pilot stage with spool position. The coefficient *Kv* is the flow gain of the servo valve and *Qs* is the supply flow. Rearranging the flow terms and applying the wellknown cylinder force equation gives Eq. (18) (Qian et al., 2014).

$$\frac{F\_A(\mathbf{s})}{Q\_s(\mathbf{s})} = \frac{\frac{4}{\beta V \mathbf{s} + C\_l} \times A}{1 + \frac{4}{\beta V \mathbf{s} + C\_l} \times A^2 \times X\_1(\mathbf{s})\mathbf{s}}\tag{18}$$

This β is the fluid bulk modulus, *V* is the fluid volume, *A* is the cross sectional area, *X*1(*s*) is the action sub-assembly displacement.

The serial elastic element has to be modeled by Newton's second law (Eq. (19)). The properties of the movable rod and the springs are considered on the stiffness, damping and mass terms. The load characteristics are also considered in the actuator's dynamic model (Eq. (20)).

$$X\_1(\mathbf{s})(m\mathbf{s}^2 + \mathbf{C}\mathbf{s} + K) = F(\mathbf{s})\tag{19}$$

where *X*1(*s*) is the action sub-assembly displacement, *m* is the action sub-assembly mass, *C* is the damping coefficient, *K* is the stiffness coefficient and *F* is the output force.

$$(X\_2(\mathbf{s})(M\mathbf{s}^2 + C\_L\mathbf{s} + K\_L) = F(\mathbf{s})\tag{20}$$

Referring to Eq. (20), *M* is the movable arm mass, *CL* and *KL* are the load interface viscous friction coefficient and load interface elastic constant respectively. *X2(s)* is the movable arm displacement. The simplified closed loop transfer function block diagram for the dynamic model is presented in **Figure 20**. Each subsystem presented in the figure has a transfer function according the equation discussed on this topic. The actuator was modeled to have a natural velocity feedback (Qian et al., 2014). The leakage of the servo valve stages was neglected. The assumptions made in this model are constant fluid properties, servo valves are not saturated, supply pressure is much greater than the load pressure, friction force can be modeled as viscous damping, main stage spool opening is proportional to pilot stage flow (Qian et al., 2014) **Figure 20** shows the hydraulic SEA block diagram.

**Figure 20.** Closed loop transfer function block diagram.

### **3.3. Results and discussion**

The servo valve's pilot stage is assumed to be of first order. Since high order dynamics in the valve are more than an order of magnitude above the frequency range of 40 Hz which is the frequency of interest (Robinson and Pratt, 2000), the higher order dynamics can be neglected.

*pCv*

2

*A*

1

<sup>1</sup> *X s ms Cs K F s* ( )( ++ =) () (19)

<sup>2</sup> ( )( ++ =) () *X s Ms C s K F s L L* (20)

(17)

(18)

Eq. (17) represents the servo valve pilot stage equation as seen in Qian et al. (2014).

() 1 = ´´´ <sup>+</sup>

*KKK vs s*

where *Kvp* is the flow gain of the pilot-stage servo valve and *τ* pilot-stage's equivalent time constant, *vi* valve control signal transmitted from the hydraulic system internal proportional controller, *Kp* represents inner loop proportional gain, *KC* is the valve pressure gain which relates flow from the pilot stage with spool position. The coefficient *Kv* is the flow gain of the servo valve and *Qs* is the supply flow. Rearranging the flow terms and applying the well-

4

( ) <sup>4</sup> <sup>1</sup> ( ) ´ <sup>+</sup> <sup>=</sup>

*Q s A X ss Vs C*

b

*A l*

b

2

the damping coefficient, *K* is the stiffness coefficient and *F* is the output force.

2

*F s Vs C*

+ ´´ +

*l*

This β is the fluid bulk modulus, *V* is the fluid volume, *A* is the cross sectional area, *X*1(*s*) is the

The serial elastic element has to be modeled by Newton's second law (Eq. (19)). The properties of the movable rod and the springs are considered on the stiffness, damping and mass terms. The load characteristics are also considered in the actuator's dynamic model (Eq. (20)).

where *X*1(*s*) is the action sub-assembly displacement, *m* is the action sub-assembly mass, *C* is

Referring to Eq. (20), *M* is the movable arm mass, *CL* and *KL* are the load interface viscous friction coefficient and load interface elastic constant respectively. *X2(s)* is the movable arm displacement. The simplified closed loop transfer function block diagram for the dynamic model is presented in **Figure 20**. Each subsystem presented in the figure has a transfer function according the equation discussed on this topic. The actuator was modeled to have a natural velocity feedback (Qian et al., 2014). The leakage of the servo valve stages was neglected. The

( )

*Q s K*

*i*

known cylinder force equation gives Eq. (18) (Qian et al., 2014).

( )

*s*

action sub-assembly displacement.

222 Recent Advances in Robotic Systems

*s vp*

t

Once the actuator dynamic model is described, the parameter values had to be established. The parameters were obtained by the digital prototype and data sheets of some components i.e. servo valve and hydraulic actuator. **Table 3** summarizes the HSEA parameters applied on dynamic model.


**Table 3.** HSEA parameters.

Spring constant is the most important parameter in the system. To define the spring constant, the minimum impedance level and a maximum bandwidth were set with respect to the previously selected components of the hydraulic system. Iteration was made to choose the best value with upper and lower bounds.

After knowing all the actuator's parameters, the PID controller could be designed. The controller was tuned by applying the pole placement and phase margin method presented in Section 2. A PID controller is able to produce good responses. This controller is widely applied on industrial robots and processes due to its robustness and simple implementation.

To achieve a robust control with large bandwidth, an overshoot of about 10% and a settling time of 0.02 s were set. Phase margin was set to 60°. **Table 4** shows the controller parameters for the system's characteristics aforementioned.


**Table 4.** Controller parameters.

The bode diagram of the system (**Figure 21**) shows an excellent frequency band of about 100 Hz which was greater than all references works presented (Bento Filho et al., 2014; Paine et al., 2013; Robinson and Pratt, 2000). The phase angle for this frequency band was around 45°. The resonance frequency of the system was 1100 Hz which is far enough to the system's operation frequency. This implies that the effects due to vibrations are efficiently filtered into the frequency band of the actuator.

**Figure 21.** Bode diagram of the system.

To show the system's response, a Unit step input was applied. **Figure 22** shows the system's response where it could be seen that the settling time is around 0.02 s which was a very fast convergence to this system. The maximum overshoot was a little higher than 10% which was an acceptable value.

**Figure 22.** Time response of the system.

### **3.4. Final remarks**

Spring constant is the most important parameter in the system. To define the spring constant, the minimum impedance level and a maximum bandwidth were set with respect to the previously selected components of the hydraulic system. Iteration was made to choose the best

After knowing all the actuator's parameters, the PID controller could be designed. The controller was tuned by applying the pole placement and phase margin method presented in Section 2. A PID controller is able to produce good responses. This controller is widely applied

To achieve a robust control with large bandwidth, an overshoot of about 10% and a settling time of 0.02 s were set. Phase margin was set to 60°. **Table 4** shows the controller parameters

The bode diagram of the system (**Figure 21**) shows an excellent frequency band of about 100 Hz which was greater than all references works presented (Bento Filho et al., 2014; Paine et al., 2013; Robinson and Pratt, 2000). The phase angle for this frequency band was around 45°. The resonance frequency of the system was 1100 Hz which is far enough to the system's operation frequency. This implies that the effects due to vibrations are efficiently filtered into the

To show the system's response, a Unit step input was applied. **Figure 22** shows the system's response where it could be seen that the settling time is around 0.02 s which was a very fast convergence to this system. The maximum overshoot was a little higher than 10% which was

on industrial robots and processes due to its robustness and simple implementation.

**Parameter Value** Proportional gain (Kp) 54.9946 Integral gain (Ki) 4412.1219 Derivative gain (Kd) 0.0575

value with upper and lower bounds.

224 Recent Advances in Robotic Systems

**Table 4.** Controller parameters.

frequency band of the actuator.

**Figure 21.** Bode diagram of the system.

an acceptable value.

for the system's characteristics aforementioned.

This section presented a digital prototyping of a linear series elastic hydraulic actuator for compliant robots. It was used in digital prototyping environment for design and assembly of actuators parts, generation of images of section views in perspective and exploded views, and all fabrication drawings of the parts of the actuator. The model obtained was very compact with a high output force which is desirable for all applications. A dynamic model with a PID controller was also presented which described the linear characteristics of the actuator. The system showed good time and frequency responses. The results obtained help the develop‐ ment of robots and manipulators for non-structured environments with another level of compactness and force capacity.

Future work is the construction of the actuator and investigation of benefits and limitations of this actuator on a compliant robot with human interaction.
