International Journal of Advanced Robotics
and Automation

The work presented in this paper is an important step toward a better understanding of a compact hydraulic robotic actuator, based on the Integrated Electro-Hydraulic Actuator (IEHA) developed by Alfayad and Ouezdou [1]. The novel advantage of this actuator is being highly compact and autonomous (no need for central hydraulic source), while keeping a good power to weight ratio. In order to present and develop the working dynamics of this actuator, a highly detailed mathematical model for the system is presented. The proposed model is simulated using MATLAB-Simulink software to identify the effect of the internal system parameters on system dynamics and prepare an input-output test-bed model. Such test-bed model is used to obtain the transfer function of the system and its order. Analysis of the effects of the main parameters was carried out and a lower order of the system was identified. A linear model of the system is derived and validated using system identification technique. Finally, a robust motion controller is applied on the proposed linear model and the simulation results are presented.

Keywords: Hydraulic Actuation; Mathematical Modeling; System Identification Technique; Virtual Modeling;

Cd: The venta contracta coefficient [−]CHeA & CHeB The two sides of the carriage moving w.r.t the fixed frame [−]; d: Distance between the bottom of chamber and The center of the shaft [mm]; E: Eccentricity of the integrated actuator [mm]; Fp: The force exerted by the oil pressure in the Radial pump on the carriage [N]; Fe: The hydraulic force in the carriage chambers [N]; Fext: The external force exerted on the cylinder [N]; Fc: The hydraulic force exerted on the cylinder [N]; Hp1: The distance between the pistons of length l_p at the dead bottom position and the bottom of its chamber [mm]; Hp2: The distance between the pistons at the high Dead point and the bottom of its chamber [mm]; Hpi: The height of oil in piston chamber (i) at each Instant [mm]; Kin: Internal leakage coefficient [−]; Kout: Internal leakage coefficient [−]; Lp: The stroke length of the piston [mm]; Lpi: The distance between the shaft center + Piston contact point on the surface of the Housing [mm]; m: The end effectors mass [kg]; me: Mass of the carriage of the IEHA [kg]; N: Number of micropistons [−]; Prp: The pressure of the oil at the intake channel of the radial pump [bar]; Pc: The pressure difference between the two Cylinder chambers A and B [bar]; Ps: High pressure line [bar]; PA: Pressure in chamber A of the micro-pistons [bar]; PB: Pressure in chamber B of the micro-pistons [bar]; Pi: The micropistons of the IEHA [-]; Qmac: The average macroscopic flow of the N Micro-pistons [m3/s]; Qmic: The average microscopic flow of the N Micro-pistons [m3/s]; Qe: The flow from microvalve into the carriage Chambers [m3/s]; Qeleak: Leakage flow from the carriage micro-Pistons to the body of the actuator [m3/s]; Q: The flow from the micro-pump [m3/s]; QleakA: The leakage between the carriage and output Cylinder + the one between the hydraulic Chamber to the micro pumps body [m3/s]; Qpileak: The leakage flow of the micro-pistons in the Radial pump [m3/s]; Qleak: The internal leakage between the two Chambers of the cylinder [m3/s]; Rb: The radius of the carriage [mm]; rtig: The micro-valve radius [mm]; rrp: The radius of the in-out opening section of the micro-pump [mm]; Sc: The surface area of the linear hydraulic Cylinder’s piston [cm2]; Se: The surface area of the carriage chambers [cm2]; Spi: The active area of a single piston [mm2]; V: Fluid volume [m3]; ve: The volume of the chamber of the carriage [m3]; vpi: The volume of the IEHA micro-pistons [m3]; vc: The volume of the chamber of the output Cylinder [m3]; X: Micro-valve input displacement [mm]; Y: The end effector (output load cylinder) Position [mm]; β: Bulk modulus of elasticity [MPa]; Ɵ: The angle between the piston and the Reference axe [degrees]; ω: Rotational speed of the shaft [rad/sec]; ρ: The density of the oil [kg/m3]; ζ: The actual pressure in the piston chamber [bar]; φi: Phase angle of the piston pi [deg]

The development of bio-inspired robots; an important and active area of research; has been going on for the past few years. These robots outperform the mobile robots in terms of mobility and versatility [2]. Moreover, they combine many desirable features such as human-like locomotion capabilities and humanfriendly design and behavior. However, the performances exhibited by bio-inspired robots are more or less limited. This is mainly due to their actuation. Indeed, in our opinion, the first still an open question that should be answered while designing a bio-inspired robot concerns its actuation.

Research in the actuation of bio-inspired robots that aims to mimic the performances of biological systems, has been investigated by several research teams. Nevertheless, as far as we know, no actuator able to reproduce the biological muscle capabilities in term of producing force and speed already exists. Focusing on humanoid robots, a continuous need for enhancing their performances leads to identifying the desired actuator properties. These properties are: i) high power to mass ratio; ii) high integration within the robot body; iii) safe-interaction of the humanoid with the surrounding environment while performing human-like behavior.

Basically, the actuation for humanoid robots can be either electrically or hydraulically. Most robotic applications are electrically driven. Generally, electric motors with high gear ratio drives are popular because of their small size and cheap price. In addition, electric motors are proven to be easy to use and control. Significant examples of electrically actuated robots are: ASIMO [3], ROBIAN [4], HRP biped series [5], Johnnie and LOLA [6], REEM [7].

However, electric actuation has several drawbacks. Indeed, electric motors normally produce small torques relative to their size and weight, thereby making reduction sub-systems with high ratios essential to convert velocity into torque. These reduction components are limited and cannot increase indefinitely, which resulted in having reduced dynamic capabilities systems. Moreover, the presence of high reduction ratios causes limited passive back-drivability, which may lead to unsafe interaction with humans as well as troubles for walking in unforeseen terrains [8].

Nevertheless, several research works to enhance the performance of electric actuators for humanoid robots were proposed. For instance, flexible elements were added between the motor and the load, as can be seen in ECD leg [9], Tekken [10] in order to have locomotion activities over unforeseen terrain. Other robots, like COG [11] and DOMO use series elastic actuators developed by Robinson et al [12]. In this kind of robots, robustness of the system is an issue because of complex electric connection [13]. On the other hand, harmonic drives were also utilized for example in ARMAR III [14]. In this robot, motors and the harmonic drives are located in the thorax of the robot. This design intended to decrease the weight of the arm. However, this inherently leads to complex transmission system through wires, which also reduces the robustness.

In general, electric actuators coupled with gearbox reduction systems currently didn’t fulfill all the needs of humanoid robots. Indeed, this actuation solution neither has a high power-toweight ratio nor is able to simultaneously provide the speed and forces required for highly dynamic robots. Therefore, other methods of actuation are commonly sought after.

The other concurrent actuation solution is hydraulic technology based. It has several advantages, which mainly include: 1) high power to mass ratio; 2) ability to produce high torque at low speed; 3) the high stiffness compared to electric ones; 4) the ability to perform continuous, intermittent, reversing and stalled motions without damage; 5) the ability to emulate human musclo-skeletal systems by means of high-bandwidth force control. Examples of robots using hydraulic actuation include: Bigdog [15], Sarcos [16], HyQ [17], Tae-Mu [18] and the underdevelopment humanoid robot HYDROïD [19].

However, this type of actuators has its main drawbacks. The major one is due to the necessity of a central Hydraulic Power Unit (HPU) to supply high-pressure fluid to all the robot joints. . This HPU is always bulky and the leakage from the hydraulic tube connections can cause safety issues, especially in human-robot interaction. Hence, it was necessary to have integrated hydraulic actuators inside the robot and near the joints.

Research for hydraulic actuator for robotics has been started with Bobrow et al [20], in which a closed loop hydrostatic actuator was introduced. This actuator was driven directly from an electric motor without a gear train enabling large speed reductions and corresponding torque amplification. However, achieving high torques is not possible without the use of large electric motors and power amplifiers, which leads to an increased overall dimensions and large mass for the system. Moreover, this actuator suffered from dead band caused by the inversion of electric motor rotation. S. Habibi et al. followed [21], with an improvised electrohydraulic actuator that tackled the effect of dead band by using a high gain cascaded control strategy with motor speed feedback. Nevertheless, this actuator had several design constraints to achieve high performance. These constraints include the usage of a symmetrical actuator in addition to appropriate sizing of hydraulic components to minimize the pressure drop. These constraints questioned the supposed compactness of an electrohydraulic actuator dedicated for robotic applications.

Recent research was done on the EHA compactness by Gnesi et al [22] and Takahashi et al [23], in which both presented two EHAs made for aircraft applications. The first presented an EHA with a vane pump and double acting cylinder, while the latter used a similar design but with a piston pump. The most recent contribution was attributed to Altare et al [24] and [25], in which a miniature gear pump is presented along with its actuator. The volumetric displacement of this EHA was found to be 0.13 cc/rev and it was also, made solely for aerospace applications.

In 2011, an IEHA (Integrated Electro-Hydraulic Actuator) was developed by Alfayad et al. [26] and [31]. Its main objective was eliminating the need for a central pressure source and to be implemented for each joint of the humanoid hydraulic robot HYDROïD. Due to the different pressures needed by the robot joints, each single IEAH actuator can be considered autonomous. Thanks to its compactness, the IEHA can be placed as near as possible to the hydraulic actuator (rotary or linear). Hence, the pressure drop is reduced and leakage may only exist inside the actuator. Moreover, the IEHA contains a passive distributor connecting the pump with the actuator. This passive distributor delivers the oil between the pump and the actuator without the need of flipping the direction of the electric motor driving the pump. Hence, the effect of dead band is avoided enhancing the position control of the actuator. In addition to the dead band elimination, the presence of the passive distributor enables the IEHA to handle any asymmetrical actuator. Therefore, we believe that this type of actuation is a one of the best choices for enhancing the capabilities of robotic systems.

The main objective for the IEHA is to be implemented on the humanoid robot HYDROïD. Hence, the human robot interaction and compliance are of high priority. There are different ways to ensure that the robot does not risk injuring the user. One approach to soft human robot contact is back-drivability. This ability enables the mechanical system to move the input axis from the output axis. In other words, the force applied from output axis of the actuator must be greater than force lost in the actuator due to static friction [27, 28]. A more advanced approach is to apply an active compliance, which can be reached through accurate force/position control [29]. Both of these approaches need a complete dynamic model of the actuation system, including the IEHA actuators and the several transmission mechanisms.

Consequently, the goal of the work presented in this paper concerns the study the internal parameters of the IEHA which highly complex and therefore identify their influence on the behavior of the whole system. The ultimate goal is deducing a practical dynamic model that approximates the IEHA behavior for active compliance control targeting. Indeed, the dynamic model will be used to choose the best control strategy to ensure the back-drivability. Moreover, the identification of the internal parameters in the literature was done in most cases through empirical assumptions and experimental results. Due to the high compactness of this kind of actuators. In the presented contribution, the approach is based on the identification of the IEHA nonlinear behavior based on the approach detailed on Figure 1. A combination of virtual modeling and mathematical analysis has been used to identify the linear model of the system. Based on the fundamental hydraulic and dynamic equations of the system, a virtual model has been developed. This model is used to provide a test bench for identifying the order of the system.

Meanwhile, an analytical study of the equations is carried out to define an enhanced input-output relation of the system. Finally, a system identification technique is used to validate the linearized model.

This paper is organized as follows; section 4 briefly describes the architecture of the IEHA electro-hydraulic actuator, IEHA. Section 5 introduces the fundamental hydraulic and mechanical equations used in the development of the virtual model of the

actuator. In Section 6, the virtual model of the system under MATLAB-Simulink is developed and the influence of the internal parameters is studied. A mathematical analysis of the inputoutput relation of the system is presented in Section 7. In Section 8, a linearized model of the system is achieved through black-box system identification. This linear model is used in Section 9 to implement a predictive position controller for the actuator. The conclusion and the future work are shown in Section 10.

The integrated hydraulic actuator (IEHA) is based on the power transmission from an electric motor to a hydraulic actuator. The basic idea consists of converting the electric power to mechanical one by using a highly integrated micropump producing pressure and flow (shown in Figure 2). This energy converter contains an in-built micro-valve that controls the eccentricity. The radial pump is connected to the output piston (linear or rotary) through a passive distributor. In order to simplify the figure, only two of the pump’s micro-pistons are shown. Since the passive distributor response is very fast compared to the rest of the components, its dynamics can be neglected in the present study. In this section, the functioning principle of the micro-pump and the micro valve are detailed.

To produce hydraulic energy in the authors proposed solution [1], a micro radial pump is used to deliver the hydraulic power required. The flow discharge of such pump can be controlled by modifying the eccentricity of the main shaft driven by the electric motor with respect to the housing. For a given direction of rotation of shaft, fluid enters the pump, and the centrifugal forces and hydraulic pressure push the micro-pistons to the walls of the housing during half of the rotation cycle, [0, π], where the corresponding micro-pistons are connected to the intake port. As the rotor continues around, the vanes sweep the fluid to the opposite side. For the other half of the cycle [π, 2π], the micropiston volume decreases and fluid exits the discharge port. Figure 3 presents a simplified IEHA model with two micro-pistons. The distance between the bottom of chamber and the center of the shaft is denoted d, while Rb is the radius of the carriage. Hp1 is the distance between the micro-piston of length lp at the dead bottom position and the bottom of its chamber. In the same way, Hp2 is the distance between the micro-piston at the high dead point and the bottom of its chamber as given in Equations 1 and 2. The piston stroke is defined by the volume of fluid produced during a rotation for a given eccentricity (E). In order to calculate

the variation of the micro-pump stroke, the distance that the piston travels during half-rotation corresponding to either the intake or the discharge is determined.
$$H_{p1}=R_b-E-l_p-d\left(1\right)$$ $$H_{p2}=R_b+E-l_p-d\left(2\right)$$ $$Q_{pi}=\{\begin{array}{c}{S}_{pi} \dot{H_{p_i}} \theta ϵ [0, \pi ]\\ - S_{pi} \dot{H_{p_i}} \theta ϵ [\pi , 2\pi ]\end{array}$$

$$Q_{mic}={\displaystyle \sum }_{i=1}^n{Q}_{pi} \left(3\right)$$ $$L_{pi}=d+H_{pi}+l_p(4)$$

$$L_{pi}=\sqrt{R_b^2-E^{2}si{n}^2(\omega t+{\phi }_i)}-E\cos \left(\omega t+{\phi }_i\right)(5)$$
$$\dot{H_{p_i}}=\dot{L_{p_i}}=-\dot{E}\cos \left(\omega t+{\phi }_i\right)+E\omega \sin \left(\omega t+{\phi }_i\right)-\frac{E\dot{E}si{n}^2\left(\omega t+{\phi }_i\right)+E^2\omega \sin \left(\omega t+{\phi }_i\right)\cos \left(\omega t+{\phi }_i\right)}{\sqrt{R_b^2-E^{2}si{n}^2(\omega t+{\phi }_i)}}\text{ (6)}$$
$$(H_{p2}-H_{p1})S_{pi}=2E{S}_{pi}(7)$$
$$Q_{mac}=2N{S}_{p}E\omega (8)$$
$$Q_e=C_{d}2\pi r_{tig}\left(X-E\right)\sqrt{\frac{2}{\rho }}\sqrt{\frac{P_s}{2}-\zeta }(9)$$ $$\dot{p}=-\beta \frac{\dot{V}}{V}(10)$$ $$Q_{leakA}=K_{in}\left(P_A-P_B\right)+K_{ex}(P_A)(11)$$ $$F_e+F_p=m_e\stackrel{··}{E}(12)$$ $$Q_e=S_e\dot{E}+\frac{v_e}{\beta }\dot{\zeta }-Q_{eleak}(13)$$ $$C_d\Pi r_{rp}^2\sqrt{\frac{2}{\rho }}\sqrt{\Delta P_i}=Q_{pi}+\frac{v_{pi}}{\beta }-Q_{p{i}_{leak}}(14)$$ with
$$\Delta p_i=\{\begin{array}{c}{P}_{rp}- P_{pi} \theta ϵ [0, \pi ]\\ P_{pi}- P_{c\lambda } \theta ϵ [ \pi , 2\pi ]\end{array}(15)$$ $$F_p=(P_{rp}-P_c)S_p\cos (\omega t)(16)$$ $$F_c-F_{ext}=m\stackrel{··}{Y}(17)$$ $$Q=S_c\dot{Y}+\frac{v_c}{\beta }\dot{P_c}-Q_{leak}\left(18\right)$$ $$E=\frac{S_c}{2N{S}_p\omega }\dot{Y}+\frac{1}{2N{S}_p\omega }\frac{v_c}{\beta }\frac{1}{2S_c}m{Y}^{(3)}+\frac{1}{2N{S}_p\omega }\frac{v_c}{\beta }\frac{1}{2S_c}{\dot{F}}_{ext}(19)$$ $$F_p=\left(P_{rp}-\frac{1}{2S_c}\left(m\ddot{Y}+F_{ext}\right)\right)S_p\cos \left(wt\right)(20)$$ $$\zeta =\frac{1}{2S_e}\left(m_e\ddot{E}-F_p\right)(21)$$ $$\zeta = \frac{1}{2S_e}(\frac{m_e{S}_c}{2N{S}_p\omega }{Y}^{\left(3\right)}+\frac{m_e}{2N{S}_p\omega } \frac{v_c}{\beta }\frac{1}{2S_c}m{Y}^{\left(5\right)}+\frac{m_e}{2N{S}_p\omega }\frac{v_c}{\beta }\frac{1}{2S_c}{F}_{ext}^{\left(3\right)}-\left(P_{rp}- \frac{1}{2S_c}\left(m\ddot{Y}+F_{ext}\right)\right)S_{p}cos\left(\omega t\right)) (22)$$ $$2\Pi C_d{r}_{tig}\sqrt{\frac{2}{\rho }}(X-E)\sqrt{\frac{P_s}{2}-\zeta }=S_e\dot{E}+\frac{v_e}{\beta }\dot{\zeta }\text{ (23)}$$ $$X=A+\frac{B}{C}(24)$$ Where,
$A=K_{10}\dot{Y}+K_{11m}{Y}^{(3)}+K_{11}\dot{F_{ext}}$

$\begin{array}{l}B=K_8\left(K_{10}\ddot{Y}+K_{11m}{Y}^{\left(4\right)}+K_{11}\stackrel{··}{F_{ext}}\right)+K_9\left(K_{13}{Y}^{\left(4\right)}+K_{14m}{Y}^{\left(6\right)}+K_{14}{F}_{ext}^{\left(4\right)}\right)\\ -\left(K_{16}+K_{17m}\ddot{Y}+K_{17}{F}_{ext}\right)\sin \left(\omega t\right)+K_{18m}\cos \left(\omega t\right)Y^{\left(3\right)}+K_{18}cos(\omega t){\dot{F}}_{ext}\end{array}$
$$K_1=\frac{m_e{S}_c}{4P_s{S}_e{S}_{p}N\omega }{K}_{2m}=\frac{m_e{v}_{c}m}{8P_s{S}_e\beta S_c{S}_{p}N\omega }{K}_2=\frac{m_e{S}_c}{8P_s{S}_e\beta S_c{S}_{p}N\omega }$$
$$K_3=\frac{S_p}{2S_e}{K}_{4m}=-\frac{S_{p}m}{4S_e{S}_c{P}_s}{K}_4=-\frac{S_p}{4S_e{S}_c{P}_s}$$
$$K_7=2\Pi C_d{r}_{tig}\sqrt{\frac{P_s}{\varrho }}\beta K_8=S_e\beta K_9=\frac{v_e}{2S_e}$$
$$K_{10}=\frac{S_c}{2S_{p}N\omega }{K}_{11m}=\frac{v_{c}m}{4\beta S_c{S}_{p}N\omega }{K}_{11}=\frac{v_c}{4\beta S_c{S}_{p}N\omega }$$
$$K_{13}=\frac{S_c{m}_e}{2S_{p}N\omega }{K}_{14}=\frac{v_c{m}_e}{4\beta S_c{S}_{p}N\omega }{K}_{14}=\frac{v_c{m}_{e}m}{4\beta S_c{S}_{p}N\omega }$$
$$K_{15}=\frac{v_c{m}_e}{4\beta S_c{S}_{p}N\omega }{K}_{16}=-P_{rp}{S}_p\omega K_{17m}=\frac{S_p\omega m}{2S_c}$$
$$K_{17}=\frac{S_p\omega }{2S_c}{K}_{18m}=\frac{S_{p}m}{2S_c}{K}_{18}=\frac{S_p}{2S_c}$$ $$X=A+\frac{B_s}{C_s}(25)$$ $$\begin{array}{l}{B}_s=K_8(K_{10}\ddot{Y}+K_{11m}{Y}^{\left(4\right)}+K_{11}\stackrel{··}{F_{ext})}+K_9\left(K_{13}{Y}^{\left(4\right)}\right)-\\ \left(K_{16}+K_{17m}\ddot{Y}+K_{17}{F}_{ext}\right)\sin \left(\omega t\right)+K_{18m}\cos \left(\omega t\right)Y^{\left(3\right)}+K_{18}\cos \left(\omega t\right)\dot{F_{ext}}\end{array}$$
$$C_s=K_7\sqrt{\begin{array}{c}1+K_1{Y}^{\left(3\right)}+K_3\cos \left(\omega t\right)+\\ K_{4m}\ddot{Y}\cos \left(\omega t\right)+K_4{F}_{ext}cos(\omega t)\end{array}}$$
$$Y\left(s\right)=G_1\left(s\right)X+G_2\left(s\right)F_{ext}\left(26\right)$$
$$G_1\left(s\right)=\frac{K}{s(1+{\lambda }_{1}s)(1+{\lambda }_{2}s)(1+{\lambda }_{3}s)}\text{ (27)}$$
$$G_2\left(s\right)=\frac{K^{\text{'}}}{s(1+{\lambda }_1^{\text{'}}\text{s})}(28)$$ where
$$\text{λ}={10}^{-3}\left[1.14,1.14,1.11\right]$$
$${\text{λ}}_1^{\text{'}}=0.43\text{ X }{10}^{-6}$$
$$\text{K}=115.4$$
$${\text{K}}^{\text{'}}=6.3\text{ X }{10}^{-4}$$