**2. Electric series elastic actuator**

### **2.1. Electric series elastic actuator design**

Over the years, digital prototyping has been successfully applied in engineering and manu‐ facturing allowing virtually explore a complete product before it's built (Thurfjell et al., 2002; Soyguder and Alli, 2007; Wang, 2011). As reported by Johansson et al. (2004), the digital prototyping has been important to the development of industrial robots, reducing the time of robot programming phase. According to the authors, the robot task can be simulated in a virtual model of the workcell when still only a digital prototype of the workpiece exists and the risk of technical failure for a transition can be reduced. In this way, before manufacture of the proposed serial elastic actuator, its digital prototyping is presented in this chapter.

### *2.1.1. First design*

The characteristics desired for the actuator are enclosed internal components, to avoid component contamination and accidents risks; size adequate for wearable robots, exoskeletons and prostheses applications, emulating natural muscles; aesthetically suited to applications in wearable robots, exoskeletons and prostheses; and smart enough for applications in unstruc‐ tured environment.

**Figure 10.** First prototype showing internal actuator components: ball screw (1); ball nuts (2); movable arm (3); springs (4, 5); motor & gearbox (6); base tube (7); bearing screw (8); O-ring (9); Allen screws (10); guide ring (11).

The first prototype design is shown in **Figure 10**. There are two ball nuts (2) bolted to plates with a tab and grooved flanges for O-rings (9) installation. The flanges have a central hole and are connected by means of an extender, which is a tube with external male thread at both ends. The springs (4 and 5) slide around the extender and press on both sides a plate fixed to the tube base (7) of the retractable arm (3).

### *2.1.2. The moving sub-assembly*

**Figure 11** shows the exploded view of the moving sub assembly, a set of pieces that behaved as a single device after assembled. Two bolted flanges (4, 3) enclose the ball nut (1). Tubes (2) are used to grip a flange on the nut (1). The flange (4) has threaded holes for Allen screws (5) and flange (3), countersunk holes for these screws and thread. In the central bore of each flange is a threaded guide tube (6), with the pre stressed springs. The circuit board (7) is fixed over the flanges with the sensor for acquiring the deformation of the spring against the linear encoder attached to the flanges (3) and (4).

**Figure 11.** Exploded view of the final design action sub assembly components.

### *2.1.3. DC motor and gearbox*

It was chosen based on the forces for actuator operation and for the spring assembly, according to the planetary gearbox, and the ball screw torque force conversion. The DC motor is a 60 mm, brushless, 400 W; the gearbox is a planetary gearhead 62 mm diameter, 8–50 N.m torque and the ball screw is a 10 mm diameter and 3 mm pitch rolled steel.

### *2.1.4. Final design*

Soyguder and Alli, 2007; Wang, 2011). As reported by Johansson et al. (2004), the digital prototyping has been important to the development of industrial robots, reducing the time of robot programming phase. According to the authors, the robot task can be simulated in a virtual model of the workcell when still only a digital prototype of the workpiece exists and the risk of technical failure for a transition can be reduced. In this way, before manufacture of the proposed serial elastic actuator, its digital prototyping is presented in this chapter.

The characteristics desired for the actuator are enclosed internal components, to avoid component contamination and accidents risks; size adequate for wearable robots, exoskeletons and prostheses applications, emulating natural muscles; aesthetically suited to applications in wearable robots, exoskeletons and prostheses; and smart enough for applications in unstruc‐

**Figure 10.** First prototype showing internal actuator components: ball screw (1); ball nuts (2); movable arm (3); springs

The first prototype design is shown in **Figure 10**. There are two ball nuts (2) bolted to plates with a tab and grooved flanges for O-rings (9) installation. The flanges have a central hole and are connected by means of an extender, which is a tube with external male thread at both ends. The springs (4 and 5) slide around the extender and press on both sides a plate fixed to the

**Figure 11** shows the exploded view of the moving sub assembly, a set of pieces that behaved as a single device after assembled. Two bolted flanges (4, 3) enclose the ball nut (1). Tubes (2) are used to grip a flange on the nut (1). The flange (4) has threaded holes for Allen screws (5) and flange (3), countersunk holes for these screws and thread. In the central bore of each flange is a threaded guide tube (6), with the pre stressed springs. The circuit board (7) is fixed over the flanges with the sensor for acquiring the deformation of the spring against the linear

(4, 5); motor & gearbox (6); base tube (7); bearing screw (8); O-ring (9); Allen screws (10); guide ring (11).

*2.1.1. First design*

210 Recent Advances in Robotic Systems

tured environment.

tube base (7) of the retractable arm (3).

encoder attached to the flanges (3) and (4).

*2.1.2. The moving sub-assembly*

**Figure 12** shows the actuator final design. The iterative digital prototyping design process done resulted in a set of desirable characteristics for the actuator very similar to those originally intended, which are 425 mm long when retracted, with 130 mm range with maximum diameter of 60 mm. For the sake of wider range of operation, the model has interchangeability of springs. The ball screw has 10 mm diameter and 3 mm pitch. Main model structure of carbon steel and aluminum alloy tubes allows a final weight of 3.7 kg. The operating voltage of the ESEA is 48 V. These model's properties allows a 450 N maximum load capacity and 13.4 cm/s continuous speed.

**Figure 12.** Actuator final design with internal components.

### **2.2. Dynamic model**

The dynamic model of the actuator was constructed from the digital prototype data to run a simulation in order to verify its dynamic behavior, bandwidth and validate the project settings, as presented in **Figure 13**.

**Figure 13.** Actuator's mechanical model.

Referring to **Figure 13**, the actuator dynamic model equations are described as follows:

$$\mathbf{L\_a \stackrel{\text{dí}\_a}{\text{dt}} + \mathbf{R\_a \stackrel{\text{i}}{\text{a}}\_a} + \mathbf{K\_b \stackrel{\text{a}}{\text{b}}\_b} \boldsymbol{\phi} = \mathbf{V\_a} \tag{1}$$

In Eq. (1), *La* is the armature inductance, *ia* is the armature current, *Ra* is the armature resistance, *Kb* is the back Electromotive Force (back-EMF) constant, *ω* is the angular velocity, and *Va* is the armature voltage.

$$
\lambda J\_T \ddot{\theta}\_m + C\_T \dot{\theta}\_m = k\_i i\_a \tag{2}
$$

Referring to Eq. (2), *JT* is the total inertia of the actuator, *CT* is the total viscous friction coefficient, *<sup>θ</sup>*˙ *<sup>m</sup>* is the angular displacement velocity of the motor axis, *θ*¨ is the motor angular acceleration, and *kt* is the motor torque proportional constant.

The SEA's total inertia referred in Eq. (2) is the summation of all inertia effects presented by each component as depicted in Eq. (3).

$$J\_T = \left[ J\_m + J\_g + n^2 J\_b + \left(\frac{np}{2\pi}\right)^2 m \right] \tag{3}$$

where *Jm* is the rotor inertia, *Jg* is the gear inertia, n is the gear ratio, *Jb* is the ball screw inertia, *p* is the ball screw pitch, *m* is the action device sub assembly mass.

A similar approach is applied to evaluate the total viscous friction coefficient presented in Eq. (2). The summation of all the friction effects results in the total viscous friction coefficient (Eq. (4))

Series Elastic Actuator: Design, Analysis and Comparison http://dx.doi.org/10.5772/63573 213

$$C\_T = \left[C\_m + C\_g + n^2 C\_b + C \left(\frac{np}{2\pi}\right)^2\right] \tag{4}$$

Referring to Eq. (4), *Cm* is the rotor viscous friction coefficient, *Cg* is the gear reduction viscous friction coefficient, *Cb* is the ball screw viscous friction coefficient, *C* is the action sub assembly viscous friction coefficient.

In Eqs. (5) and (6), Newton's second law is applied on the movable arm and action sub assembly systems to evaluate the action sub assembly and movable arm displacement in order to estimate the spring deflection.

$$m\ddot{\mathbf{x}}\_1 + C\_b \mathbf{x}\_1 = m \left(\frac{np}{2\pi}\right) \ddot{\theta}\_n + C\_b \left(\frac{np}{2\pi}\right) \dot{\theta}\_n = f \tag{5}$$

$$M\ddot{\mathbf{x}}\_2 + C\_L\mathbf{x}\_2 + K\_L\mathbf{x}\_2 = f \tag{6}$$

where *f* is the force between action sub assembly and the movable arm, *M* is the movable arm sub assembly mass, *CL* is the load interface viscous friction coefficient, and *KL* is the load interface elastic constant. The term ( *np* <sup>2</sup>*<sup>π</sup>* ) is the linear displacement of the movable arm reflected to motor axis rotation, and ( *np* <sup>2</sup>*<sup>π</sup>* )<sup>2</sup> is the inertia and viscous friction components of the action device reflected to motor axis.

Once the actuator dynamic model equations are described, the closed loop PID transfer function block diagram in the frequency domain can be made, as shown in **Figure 14**.

**Figure 14.** Actuator's closed loop transfer function block diagram.

#### **2.3. Results and discussion**

**2.2. Dynamic model**

212 Recent Advances in Robotic Systems

as presented in **Figure 13**.

**Figure 13.** Actuator's mechanical model.

armature voltage.

and *kt*

(4))

The dynamic model of the actuator was constructed from the digital prototype data to run a simulation in order to verify its dynamic behavior, bandwidth and validate the project settings,

Referring to **Figure 13**, the actuator dynamic model equations are described as follows:

L Ri K V <sup>a</sup> ++= aa b a

+ = && & *T m T m ta J C ki*

 q

q

is the motor torque proportional constant.

*p* is the ball screw pitch, *m* is the action device sub assembly mass.

each component as depicted in Eq. (3).

w

In Eq. (1), *La* is the armature inductance, *ia* is the armature current, *Ra* is the armature resistance, *Kb* is the back Electromotive Force (back-EMF) constant, *ω* is the angular velocity, and *Va* is the

Referring to Eq. (2), *JT* is the total inertia of the actuator, *CT* is the total viscous friction coefficient, *<sup>θ</sup>*˙ *<sup>m</sup>* is the angular displacement velocity of the motor axis, *θ*¨ is the motor angular acceleration,

The SEA's total inertia referred in Eq. (2) is the summation of all inertia effects presented by

2

é ù æ ö = ++ + ê ú ç ÷ è ø ë û *T mg b np J J J nJ m*

where *Jm* is the rotor inertia, *Jg* is the gear inertia, n is the gear ratio, *Jb* is the ball screw inertia,

A similar approach is applied to evaluate the total viscous friction coefficient presented in Eq. (2). The summation of all the friction effects results in the total viscous friction coefficient (Eq.

2

2

p

(1)

(2)

(3)

*a di dt*

> Once the actuator dynamic model has been described, the parameter values had to be estab‐ lished. The parameters were obtained by the digital prototype design and data sheets of some components, i.e. motor and ball screw (**Table 1**).

Spring constant is the most important parameter in the system. To define the spring constant, we set the minimum impedance level and a maximum bandwidth with respect to the previous selected components of the 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 (Tang et al., 2010). The same PID control tuning strategy is applied in the controller of the linear serial elastic hydraulic actuator (Leal Junior et al., 2015). For this reason, applying the same tuning method in the electric SEA helps on further responses comparison between the hydraulic and the electric SEA.


**Table 1.** Parameters value of the entire ESEA system.

Desired pair of poles is estimated by representing time domain specification, e.g. percent overshoot, settling time. However, in some cases the dominant poles lose its dominant position in the result of the closed loop system. To overcome this problem, a phase margin is employed to guarantee the closed loop system robustness.

Assuming a pair of poles as

$$
\Box p\_{1,2} = -\alpha \pm j\beta \tag{7}
$$

The process G(s) is given and the controller has its conventional configuration with all positive gains as follows.

$$C(\mathbf{s}) = K\_{\rho} + \frac{K\_i}{\mathbf{s}} + K\_d \mathbf{s} \tag{8}$$

Substituting one of the poles into the characteristic equation of the closed loop equation which is the transfer function denominator of the process G(s).

$$(1 + G(p\_1)C(p\_1) = 0) \tag{9}$$

Moreover, the definition of phase margin gives:

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

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 (Tang et al., 2010). The same PID control tuning strategy is applied in the controller of the linear serial elastic hydraulic actuator (Leal Junior et al., 2015). For this reason, applying the same tuning method in the electric SEA helps on further responses comparison between the hydraulic and

)

**Parameter Value Units** Armature inductance (La) 8.2E–04 (H) Armature resistance (Ra) 1.03 (Ω) Total inertia (JT) 9.51E–05 (kg m2

Mass (M) 1.261 (kg) Total viscous friction (CT) 4.594E–05 (Ns/m) Arm and load stiffness (KH) 3.336E+06 (N/m) back-EMF constant (Kb) 1.469E–01 (Vs/rad) Load stiffness (KL) 3.303 +06 (N/m) Load damping (CL) 8.08 (Ns/m) Spring stiffness (K) 3.27E+04 (N/m) Spring viscous friction (C) 8.000E–01 (Ns/m) Torque proportional constant (KT) 1.470E–01 (Nm/A) Gear ratio (n) 0.001 (1) Ball screw pitch (p) 0.003 (m)

Desired pair of poles is estimated by representing time domain specification, e.g. percent overshoot, settling time. However, in some cases the dominant poles lose its dominant position in the result of the closed loop system. To overcome this problem, a phase margin is employed

> *p j* 1,2 =- ± a b

(7)

and lower bounds.

214 Recent Advances in Robotic Systems

the electric SEA.

**Table 1.** Parameters value of the entire ESEA system.

Assuming a pair of poles as

to guarantee the closed loop system robustness.

$$(G(j\alpha\_\g)C(j\alpha\_\g) = -e^{\langle \phi\_\bullet \rangle} \tag{10}$$

Substituting and splitting the complex values and the real ones of Eqs. (9) and (10) gives two real functions of (Eqs. (11) and (12)).

$$f\_1(oo) = \text{Re}\left[\frac{-p\_1}{G(p\_1)}\right] + \frac{\alpha^2 - \beta^2}{2\alpha\beta} \text{Im}\left[\frac{-p\_1}{G(p\_1)}\right] + \frac{\alpha^2 + \beta^2}{2\alpha} \text{Re}\left[\frac{-e^{\beta p\_n}}{G(jo)}\right] \tag{11}$$

$$\,\_{1}f\_{1}(\alpha) = \frac{\alpha^{2}}{2\alpha} \text{Re} \left[ \frac{-e^{\prime \phi\_{n}}}{G(j\alpha)} \right] - \frac{\alpha^{2}}{2\alpha \beta} \text{Im} \left[ \frac{-p\_{1}}{G(p\_{1})} \right] - \alpha \text{Im} \left[ \frac{-e^{\prime \phi\_{n}}}{G(j\alpha)} \right] \tag{12}$$

The positive values of the intersection between *f*1(*ω*) and *f*2(*ω*) are the first approximation for integral gain (*Ki* ). This first value was applied on Eqs. (13) and (14) to evaluate the proportional and derivative gains. Proportional and derivative gains are obtained by substituting the wellknown PID controller equation C(s), presented in Eq. (8), in Eqs. (9) and (10).

$$K\_{\rho} = \frac{2\alpha}{\alpha^2 + \beta^2} \left( K\_i - \text{Re} \left[ \frac{-p\_1}{G(p\_1)} \right] - \frac{\alpha^2 - \beta^2}{2\alpha\beta} \text{Im} \left[ \frac{-p\_1}{G(p\_1)} \right] \right) \tag{13}$$

$$K\_d = \frac{1}{\alpha^2 + \beta^2} \left( K\_i - \text{Re} \left[ \frac{-p\_1}{G(p\_1)} \right] - \frac{\alpha}{\beta} \text{Im} \left[ \frac{-p\_1}{G(p\_1)} \right] \right) \tag{14}$$

The final integral gain is the maximum value between Eqs. (15) and (16).

$$K\_{i,1} = \left(\text{Re}\left[\frac{-p\_1}{G(p\_1)}\right] - \frac{\alpha}{\beta}\text{Im}\left[\frac{-p\_1}{G(p\_1)}\right]\right) \tag{15}$$

$$K\_{i,2} = \left(\operatorname{Re}\left[\frac{-p\_1}{G(p\_1)}\right] - \frac{\alpha^2 - \beta^2}{2a\beta} \operatorname{Im}\left[\frac{-p\_1}{G(p\_1)}\right]\right) \tag{16}$$

To achieve a robust control with large bandwidth it was set an overshoot about 10% and a settling time of 0.02 s. Phase margin was set 60°. **Table 2** shows the integral, proportional and derivative gains of the ESEA's controller.


**Table 2.** Controller parameters.

**Figure 15** shows the Bode diagram of the system, there is a good frequency band of 40 Hz, compatible with the Paine's results (Paine et al., 2013). On the other hand, the phase angle is quite high at 40 Hz, greater than 90°. Yet it is possible to apply the actuator in several systems where force control is necessary, as in some robot arms in industrial plants. Resonance frequency is greater than 200 Hz. This implies that the effects due to vibrations are efficiently filtered into the frequency band of the actuator.

**Figure 15.** Bode diagram of the system.

**Figure 16** shows the time response for a constant input force. The electric SEA shows an 8% overshoot and it settling time is 0.03 s which is very fast. While hydraulic SEA of reference work (Leal Junior et al., 2015) has an overshoot about 12% and a settling time of 0.07 s. Although hydraulic SEA has larger overshoot and settling time than electric SEA of this section, these two parameters are compatible with the reference works results (Paine et al., 2013; Robinson, 2000). Therefore, even in the Hydraulic SEA case the settling time is tolerated which is discussed in Section 4.

### **2.4. Final remarks**

1 1

Re Im

Re Im

**Parameter Value** Proportional gain (Kp) 2.915 Integral gain (Ki) 4.129 Derivative gain (Kd) 0.007

1 1

(15)

(16)

() () æ ö éù éù - - <sup>=</sup> ç ÷ êú êú è ø ëû ëû *<sup>i</sup> p p <sup>K</sup>*

> 2 2 1 1

> 1 1

() 2 () æ ö éù éù - -- <sup>=</sup> ç ÷ êú êú è ø ëû ëû *<sup>i</sup> p p <sup>K</sup>*

ab

*G p G p* a b

To achieve a robust control with large bandwidth it was set an overshoot about 10% and a settling time of 0.02 s. Phase margin was set 60°. **Table 2** shows the integral, proportional and

**Figure 15** shows the Bode diagram of the system, there is a good frequency band of 40 Hz, compatible with the Paine's results (Paine et al., 2013). On the other hand, the phase angle is quite high at 40 Hz, greater than 90°. Yet it is possible to apply the actuator in several systems where force control is necessary, as in some robot arms in industrial plants. Resonance frequency is greater than 200 Hz. This implies that the effects due to vibrations are efficiently

*Gp Gp* a

b

,1

,2

derivative gains of the ESEA's controller.

filtered into the frequency band of the actuator.

**Table 2.** Controller parameters.

216 Recent Advances in Robotic Systems

**Figure 15.** Bode diagram of the system.

This section presented the digital prototyping of a linear electric SEA for exoskeletons, wearable robots and mechatronic devices. It was used as a digital prototyping environment for design and assembly of actuators parts, generation of section views images in perspective and exploded views, and all fabrication drawings of the actuator parts. The prototype is driven by a Maxon EC 60 Ø60 mm, brushless 400W DC motor, with a Planetary Gearhead GP 62 A (Maxon Motor Catalogue) and an R Series 10mm diameter rolled steel ball screw (NSK Catalogue).

This section also presented the actuator's performance, evaluated under a PID impedance controller. **Figure 15** shows the frequency response of the magnitude and phase of Fl/Fd. **Figure 16** shows the step response of unit force input. The step response can also be tuned to final displacement or to contact force. Some improvements to the actuator itself will be needed for the next steps of the project, aiming at the fabrication, assembly and testing of the actuator.

Future works are the construction of a real aluminum alloy based prototype and testing its performance in a real time environment to evaluate the actuator's performance under force,

**Figure 16.** Time response for a 1N step input force.

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 friction have to be analyzed in future works.

A broad comparison between the electric serial elastic actuator and the hydraulic SEA has to be made to define which one is the best for a compliant robot application.
