^{2}BIA ZA Les Boutriers, 8 rue de l’Hautil, 78000 Conflans fin d’Oise, France.
^{3}School of Engineering, Department of Mechanical Engineering, P. O. Box 28282, Dubai, United Arab Emirates.
Keywords: Hydraulic Actuation; Mathematical Modeling; System Identification Technique; Virtual Modeling;
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
$$\text{}{H}_{p1}\text{}=\text{}{R}_{b}-E-\text{}{l}_{p}-d\text{}\left(1\right)$$ $$\text{}{H}_{p2}\text{}=\text{}{R}_{b}+E-\text{}{l}_{p}-d\text{}\left(2\right)$$
By activating the voice-coil of the micro-valve in positive or negative direction, CHeB is connected to high pressure fluid line Ps, while CHeA is connected to the return line, and vice versa. In order for the eccentricity to follow the value of input displacement X, a closed loop system is needed. This closed loop circuit has been carried out mechanically, by connecting the micro-valve external fixed part to the pump housing. Therefore, when X changes, the micro-valve is opened and E changes until it reaches the value of X, where the micro-valve closes and eccentricity remains constant at this value. An external force applied to the actuator increases the pressure in the output piston, and consequently the micro-pump pistons. This increase in pressure, increases the force applied by the micro-pistons on the housing, and can change the eccentricity. In this case the micro valve opens which will correct the eccentricity and bring it back to the same value as X.
Each micro-piston in the radial pump is connected to the intake line during half of each rotation, where it sucks oil. During the second half, it pushes oil to one of the two cylinder chambers. The micro-pistons are pushed to the carriage wall by centrifugal forces. The microscopic model of flow is:
$$\text{}{Q}_{mic}=\text{}{\displaystyle \sum}_{i=1}^{n}{Q}_{pi}\left(3\right)$$
$${L}_{pi}=\text{}\sqrt{{R}_{b}^{2}-\text{}{E}^{2}si{n}^{2}(\omega t+\text{}{\phi}_{i})}-E\mathrm{cos}\left(\omega t+\text{}{\phi}_{i}\right)\text{}(5)$$
$$\stackrel{\u0307}{{H}_{{p}_{i}}}\text{}=\text{}\stackrel{\u0307}{{L}_{{p}_{i}}\text{}}=\text{}-\dot{E}\text{}\mathrm{cos}\left(\omega t+\text{}{\phi}_{i}\right)+E\text{}\omega \text{}\mathrm{sin}\left(\omega t+{\phi}_{i}\text{}\right)-\text{}\frac{E\text{}\dot{E}\text{}si{n}^{2}\left(\omega t+\text{}{\phi}_{i}\right)+\text{}{E}^{2}\omega \text{}\mathrm{sin}\left(\omega t+\text{}{\phi}_{i}\right)\mathrm{cos}\left(\omega t+\text{}{\phi}_{i}\right)\text{}}{\sqrt{{R}_{b}^{2}-\text{}{E}^{2}si{n}^{2}(\omega t+\text{}{\phi}_{i})}}\text{(6)}$$
$$({H}_{p2}-\text{}{H}_{p1}){S}_{pi}\text{}=\text{}2E{S}_{pi}\text{}(7)\text{}$$
$${Q}_{mac}\text{}=\text{}2N{S}_{p}E\omega \text{}(8)$$
$${Q}_{e}\text{}=\text{}{C}_{d}\text{}2\pi \text{}{r}_{tig}\left(X-E\right)\text{}\sqrt{\frac{2}{\rho}}\text{}\sqrt{\frac{{P}_{s}}{2}\text{}-\text{}\zeta \text{}}\text{}(9)$$
The oil compressibility is defined as the relative change in oil volume per unit change in pressure. Oil compressibility should be taken into account when response time and high-precision control of hydraulic actuators are important. The resistance of a fluid to being compressed is defined by the variable, Bulk Modulus (β), which is the inverse of compressibility as shown in Equation 10.
Coefficients respectively. These two values are a function of the surface of the leakage section.
$$\Delta {p}_{i}=\{\begin{array}{c}{P}_{rp}-{P}_{pi}\theta \u03f5[0,\pi ]\\ {P}_{pi}-{P}_{c\lambda}\theta \u03f5[\pi ,2\pi ]\end{array}\text{}(15)$$
For this purpose, MATLAB-Simulink was used to model all the dynamic equations of the IEHA actuator, as well as fluid properties such as bulk modulus and internal leakage as shown in Appendix I. The blocks on the left-hand side of the figure represent the dynamics of the micro-valve, which takes signal X as an input, the resulting eccentricity E is applied to the micropump which is simulated as 15 individual micro-pistons (middle blocks). The flow produced by the micro-pistons is sent to the passive distributor which directs it to side A or B of the linear actuator (right hand side blocks). The electric motor has been modeled as constant input rotating at a fixed speed. The used parameters values are given in Table 1.
The following simulation results show the variation of internal parameters against a step input of 0.04 [cm] of the micro valve position X. Generally, the input command X decides how much the micro-valve should be opened and as the pressure in the chamber increases, the carriage moves and E changes. Figure 5 shows the response of the carriage eccentricity, E, while it follows the micro-valve displacement X. As seen in Figure 5, the time response of E is around 0.007 [s] and it follows the input X with a negligible delay.
In order to illustrate the pressure variation in the micropistons, the variations of the output cylinder pressures are linked to the oscillation of flow. By looking at pump flow curve (Figure 6), it is clear that the oscillates with a frequency of 10 [Hz]. This oscillation is due to the fluid rippling from the micro-radial pump. There is also a slight peak in the beginning, the faster the eccentricity the bigger this peak of flow is.
In the beginning of the cylinder’s movement as shown in Figure 7, the pressure across the cylinder increases due to fluid compressibility. And as the load starts to move, the pressure difference across the cylinder drops and the velocity reaches the stable value of 2.2 [cm/s] as shown Figure 8. The pressure difference across the output stays positive (0.8 bar). This is
Parameter |
Physical Quantity |
Value |
Ps |
Supply pressure of micro-valve |
10 bar |
rtig |
Micro-valve radius |
0.25 cm |
me |
Carriage mass |
0.091 kg |
Se |
Carriage active surface area |
1.644 cm2 |
Emax |
Maximum eccentricity |
0.05 cm |
ve |
Chamber volume of carriage |
0.0822 cm3 |
β |
Bulk modulus |
800 MPa |
Cd |
Vena contracta coefficient |
0.62 |
ρ |
Fluid density |
840 kg/m3 |
ω |
Electric motor rotational speed |
3000 rpm |
Prp |
Supply pressure of micro-pump |
10 bar |
N |
Number of micro-pistons |
15 |
rrp |
Micro-pump in-out radius |
0.2 cm |
mp |
Micro-piston mass |
0.45 g |
Sp |
Micro-piston surface area |
0.197 cm2 |
Lp |
Micro-piston stroke |
0.27 cm |
Rb |
Interior ring radius |
0.3 cm |
d |
Distance between chamber bottom & shaft center |
0.4 cm |
m |
Load mass |
10 kg |
vc |
Chamber volume of output cylinder |
23.562 cm3 |
Sc |
Output cylinder surface area |
2.356 cm2 |
In these results the output flow of the pump seems to immediately increase the pressure force in the output cylinder and make the piston velocity increase to a stable value. This is because the dynamics of the passive distributor are not taken into account in this model in Simulink.
Returning to the equations in subsection 5-5.3, if Pc is derived and replaced from Equation 17 into Equation 18, and by considering the macroscopic value of flow according to Equation 8, we will have the eccentricity E expression in terms of Fext, Y, and their derivatives:
On the other hand, the force of the micro-pump pistons exerted on the carriage chambers is given by Equation 16. By replacing Pc, the micro pump’s force can also be written in terms of the output position, the external force, and their derivatives:
$A=\text{}{K}_{10}\dot{Y}+{K}_{11m}{Y}^{(3)}+{K}_{11}\stackrel{\u0307}{{F}_{ext}}$
$\begin{array}{l}B=\text{}{K}_{8}\left({K}_{10}\ddot{Y}+{K}_{11m}{Y}^{\left(4\right)}+{K}_{11}\stackrel{\xb7\xb7}{{F}_{ext}}\right)+{K}_{9}\left({K}_{13}{Y}^{\left(4\right)}+{K}_{14m}{Y}^{\left(6\right)}+{K}_{14}{F}_{ext}^{\left(4\right)}\right)\\ -\left({K}_{16}+{K}_{17m}\ddot{Y}+{K}_{17}{F}_{ext}\right)\mathrm{sin}\left(\omega t\right)+{K}_{18m}\mathrm{cos}\left(\omega t\right){Y}^{\left(3\right)}+{K}_{18}cos(\omega t){\dot{F}}_{ext}\end{array}$
$${K}_{1}=\frac{{m}_{e}{S}_{c}}{4{P}_{s}{S}_{e}{S}_{p}N\omega}{K}_{2m}=\frac{{m}_{e}{v}_{c}m}{8{P}_{s}{S}_{e}\beta {S}_{c}{S}_{p}N\omega}{K}_{2}=\frac{{m}_{e}{S}_{c}}{8{P}_{s}{S}_{e}\beta {S}_{c}{S}_{p}N\omega}$$
$${K}_{3}=\frac{{S}_{p}}{2{S}_{e}}{K}_{4m}=-\frac{{S}_{p}m}{4{S}_{e}{S}_{c}{P}_{s}}{K}_{4}=-\frac{{S}_{p}}{4{S}_{e}{S}_{c}{P}_{s}}$$
$${K}_{7}=2\Pi {C}_{d}{r}_{tig}\sqrt{\frac{{P}_{s}}{\varrho}}\beta \text{}{K}_{8}={S}_{e}\beta \text{}{K}_{9}=\frac{{v}_{e}}{2{S}_{e}}$$
$${K}_{10}=\frac{{S}_{c}}{2{S}_{p}N\omega}\text{}{K}_{11m}=\frac{{v}_{c}m}{4\beta {S}_{c}{S}_{p}N\omega}\text{}{K}_{11}=\frac{{v}_{c}}{4\beta {S}_{c}{S}_{p}N\omega}$$
$${K}_{13}=\frac{{S}_{c}{m}_{e}}{2{S}_{p}N\omega}\text{}{K}_{14}=\frac{{v}_{c}{m}_{e}}{4\beta {S}_{c}{S}_{p}N\omega}\text{}{K}_{14}=\frac{{v}_{c}{m}_{e}m}{4\beta {S}_{c}{S}_{p}N\omega}$$
$${K}_{15}=\frac{{v}_{c}{m}_{e}}{4\beta {S}_{c}{S}_{p}N\omega}\text{}{K}_{16}=-{P}_{rp}{S}_{p}\omega \text{}{K}_{17m}=\frac{{S}_{p}\omega m}{2{S}_{c}}$$
$${K}_{17}=\frac{{S}_{p}\omega}{2{S}_{c}}\text{}{K}_{18m}=\frac{{S}_{p}m}{2{S}_{c}}\text{}{K}_{18}=\frac{{S}_{p}}{2{S}_{c}}$$
By comparing the coefficients K(1…4), we see that the terms K2 and K2m will always be very small compared to the others, because they are divided by the bulk modulus, which is of an
$${C}_{s}={K}_{7}\sqrt{\begin{array}{c}1+{K}_{1}{Y}^{\left(3\right)}+{K}_{3}\mathrm{cos}\left(\omega t\right)+\\ {K}_{4m}\ddot{Y}\mathrm{cos}\left(\omega t\right)+{K}_{4}{F}_{ext}cos(\omega t)\end{array}}\text{}$$
The system input-output linear function has been identified with variable load (can also be interpreted as being in contact with the environment). Unlike simple micro-valves, dynamics of IEHA depends on the load (this is also the property that makes it back-drivable). Increase in the external force, causes increase of the cylinder pressure, which will increase the pressure in the micro-pistons. Consequently, the oil pressure on the carriage goes up and this can displace the carriage, (i.e. modify the eccentricity E). Hence, the output flow of the micro-pump is changed. This effect is also seen in the analytical relation between Y and X. Therefore, we consider Fext as an input to our system. Different external forces were applied to the output stroke. These forces included a step and a chirp signal of maximum frequency of 10 Hz, with amplitude of 100 ~ 1000N. Inserting X and Fext as two separate inputs to the system, a new linear model has been estimated as follows:
$${G}_{1}\left(s\right)=\frac{K}{s(1+{\lambda}_{1}s)(1+\text{}{\lambda}_{2}s)(1+\text{}{\lambda}_{3}s)}\text{(27)}$$
$${G}_{2}\left(s\right)=\frac{{K}^{\text{'}}}{s(1+{\lambda}_{1}^{\text{'}}\text{s})}\text{}(28)$$ where
$$\text{\lambda}=\text{}{10}^{-3}\left[1.14,\text{}1.14,\text{}1.11\right]$$
$${\text{\lambda}}_{1}^{\text{'}}=0.43\text{X}{10}^{-6}$$
$$\text{K}=115.4$$
$${\text{K}}^{\text{'}}=6.3\text{X}{10}^{-4}$$
To validate the piston response, another simulation is conducted. A step input of 0.04 [cm] is given to the micro-valve, then after 0.5 [s], the micro-valve is closed and the output is suspended at a certain position. Then a varying sinusoidal force is applied on the cylinder, which will pull and push the piston. The results are shown in Figure 13.
From those two previous simulation results, the virtual model was compared with the linear model while responding
Since, we have already a linear based model for the IEHA in addition to the virtual model; a model predictive control algorithm seems adequate. The model predictive control is a strategy that is based on the explicit use of some kind of linear system model to predict the controlled variables over a certain time period [30]. In our case the linear model of the IEHA can be the predicting model. This way we avoid the undesirable dynamics of the actuator in the feedback. Another advantage of this method is that this feedback is faster than the one coming from the nonlinear model. Therefore, the control signal is adjusted by “predicting” the future error. The control block diagram is shown in Figure 12.
For future work, the internal parameters of the hydraulic mathematical model such as leakage and compressibility are not constant and vary with time and temperature. Hence, a further work must consider a better solution to identify the internal states of the system. A better use of an adaptive controller accordingly can be pursued. On the other hand, the passive distributor model has to be included in order to get a complete description of the IEHA model. The new virtual model presented can be used for manufacturing other actuator prototypes with all of its inner parameters identified. Finally, the hardware use of the IEHA to actuate an active compliant hydraulic arm is under development. This will enable us to have an integrated hydraulic actuator for the underdevelopment humanoid robot HYDROïD.
- Alfayad S, Benouedou F, Namoun F. Convertor for converting mechanical energy into hydraulic energy and robot implementing said convertor. US Patent WO/2009/118366. 2009.
- Semini C, Tsagarakis NG, Guglielmino E, Focchi M, Cannella F, Caldwell DG. Design of HyQ - a hydraulically and electrically actuated quadruped robot. Proc. Inst Mech Eng Part I, J Syst Control Eng. 2011;225:831– 849. doi: 10.1177/0959651811402275.
- Sakagami Y, Watanabe R, Aoyama C, Matsunaga S, Higaki N, Fujimura K. The intelligent ASIMO: system overview and integration. In Proc. of IROS. 2002:2478-2483. doi: 10.1109/IRDS.2002.1041641.
- Konno A, Sellaouti R, BEN Amar F, Ouezdou FB. Design and Development of the Biped prototype ROBIAN. ICRA. 2002:1384 - 1389.
- Kaneko K, Kanehiro F, Morisawa M, Akachi K, Miyamori G, Hayashi A, et al. Humanoid robot HRP-4 - Humanoid robotics platform with lightweight and slim body. IEEE International Conference for Intelligent Robotic Systems. 2011:4400–4407.
- Buschmann T, Lohmeier S, Ulbrich H. Humanoid robot Lola: Design and walking control. J Physiol-Paris. 2009;103(3–5):141–148.
- Tellez R, Ferro F, Garcia S, Gomez E, Jorge E, Mora D, et al. Reem-B an autonomous lightweight human-size humanoid robot. 8th IEEE International Conference on Humanoid Robots (Humanoids08). 2008. doi: 10.1109/ICHR.2008.4755995
- Zhou X, Bi S. A survey of bio-inspired compliant legged robot designs. Bioinspir Biomim. 2012;7(4):041001. doi: 10.1088/1748- 3182/7/4/041001.
- Hurst JW. The role and implementation of compliance in legged locomotion. Dissertation. Carnegie Mellon University. 2008.
- Pratt JE. Exploiting inherent robustness and natural dynamics inthe control of bipedal walking robots. Dissertation, Massachusetts Institute of Technology. 2000.
- Brooks RA, Breazeal C, Marjanovic M, Scassellati B, Williamson MM.The Cog Project: Building a Humanoid Robot Lect Notes Comput Sci. 1999;1562:52–87.
- Robinson DW, Pratt JE, Paluska DJ, Pratt GA. Series elastic actuator development for a biomimetic walking robot. AdvancedIntelligent Mechatronics. Proceedings. IEEE/ASME International Conference.1999. 561-568.
- Edsinger-Gonzales A, Weber J. Domo: A Force Sensing Humanoid Robot for Manipulation Research. IEEE International Conference on Humanoid Robots. 2004.
- Albers A, Brudniok S, Ottnad J, Sauter C, Sedchaicharn K. ARMARIII design of the upper body. Technical report. Institute of Product Development University of Karlsruhe (TH). 2006.
- Raibert M, Blankespoor K, Nelson G, Playter R. Bigdog, the Rough-Terrain Quadruped Robot. In Proceedings of the 17th World Congress. 2008;41(2):10822-10825.
- Bentivegna DC, Atkeson CG, Kim JY. Compliant control of a hydraulichumanoid joint. Proc. 2007 7th IEEE-RAS Int Conf Humanoid Robot. HUMANOIDS 2007. 2008:483–489.
- Semini C. HyQ - Design and Development of a Hydraulically Actuated Quadruped Robot. Darwin. 2010..
- Hyon SH, Suewaka D, Torii Y, Oku N, Ishida H. Development of afast torque-controlled hydraulic humanoid robot that can balancecompliantly. Humanoid Robots (Humanoids). 2015 IEEE-RAS15th International Conference on, 2015:576-581.doi: 10.1109/HUMANOIDS.2015.7363420.
- Alfayad S, Ouezdou FB, Namoun F, Cheng G. Highly Integrated Electro- Hydraulic Actuator for Robotics - Part II: Theoretical modelling, Simulation, Control & Comparison with real measurements. Sensors and Actuators A: Physical. 2011;169(1):124-132.
- Bobrow J E, Desai J. Modeling and Analysis of a High-Torque, Hydrostatic Actuator for Robotic Applications. Experimental Robotics I. 1990:215-228.
- Habibi S, Goldenberg A. Design of a New High PerformanceElectrohydraulic Actuator. Proc 1999 IEEE/ASME. 1999;5:158–164..
- Gnesi E, Maré JC, Bordet JL. Modeling of EHA Module Equipped with Fixed-Displacement Vane Pump. 13th Scand Int Conf Fluid Power. SICFP. 2013:141–153.
- N. Takahashi, T. Kondo, and M. Takada, Masutani K, Okano S, TsujitaM. Development of prototype electro-hydrostatic actuator for landing gear extension and retraction system. Sumitomo Precision Products Co Ltd. 2008 :165–168.
- Altare G, Vacca A, Richter C. A Novel Pump Design for an Efficient andCompact Electro-Hydraulic Actuator. In IEEE Aerospace Conference. 2014..
- Altare G, Vacca A. A Design Solution for Efficient and Compact Electrohydraulic Actuators. Procedia Engineering. 2015;106:8–16.
- Alfayad S, Ouezdou FB, Namoun F, Cheng G. Lightweight High Performance Integrated Actuator for Humanoid Robotic Applications: Modeling, Design & Realization. In Proc IEEE Int Conf on Robotics and Automation (ICRA). 2009.
- Kazerooni H. Dynamics and control of instrumented harmonic drives.J Dyn Syst Meas Control. 1995;117:15-19.
- Kaminaga H, Yamamoto T, Ono J, Nakamura Y. Backdrivableminiature hydrostatic transmission for actuation of anthropomorphicrobot hands. Humanoid Robots. 2007 7th IEEE-RAS InternationalConference on. 2007:36-41.
- Zhu Q, Mao Y, Xiong R, Wu J. Adaptive Torque and Position Controlfor a Legged Robot Based on a Series Elastic Actuator. Int Journal of Advanced Robotics Systems. 2016:1.
- Chalupa P, Novák J. Modeling and model predictive control ofa nonlinear hydraulic system. Computers & Mathematics with Applications Journal. 2013;66(2):155–164.
- Alfayad S, Ouezdou FB, Namoun F, Cheng G. New Highly Integrated Electro-Hydraulic Actuator for Robotics - Part I : Principle, PrototypeDesign & First Experiments. Sensors and Actuators A: Physical.2011;169(1):115-123.