A Bimodal Hydrostatic Actuator for Robotic Legs with Compliant Fast Motion and High Lifting Force Alex Lecavalier1 Jeff Denis1 Jean-S ebastien Plante1 Alexandre Girard1

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A Bimodal Hydrostatic Actuator for Robotic Legs with Compliant Fast

Motion and High Lifting Force

Alex Lecavalier1, Jeff Denis1, Jean-S´

ebastien Plante1, Alexandre Girard1

Abstract— Robotic legs have bimodal operations: swing

phases when the leg needs to move quickly in the air (high-

speed, low-force) and stance phases when the leg bears the

weight of the system (low-speed, high-force). Sizing a traditional

single-ratio actuation system for such extremum operations

leads to oversized heavy electric motor and poor energy

efﬁciency, which hinder the capability of legged systems that

bear the mass of their actuators and energy source. This paper

explores an actuation concept where a hydrostatic transmission

is dynamically reconﬁgured using valves to suit the require-

ments of each phase of a robotic leg. An analysis of the mass-

delay-ﬂow trade-off for the switching valve is presented. Then, a

custom actuation system is built and integrated on a robotic leg

test bench to evaluate the concept. Experimental results show

that 1) small motorized ball valves can make fast transitions

between operating modes when designed for this task, 2) the

proposed operating principle and control schemes allow for

seamless transitions, even during an impact with the ground

and 3) the actuator characteristics address the needs of a leg

bimodal operation in terms of force, speed and compliance.

I. INTRODUCTION

A robotic leg needs to quickly move through the air to

reposition its foot, for instance, when stabilization or for

fast gaits. Also, an ideal robotic leg have a small reﬂected

inertia to limit the effects of the impact when the foot hits

the ground. Lightly geared and direct-drive electric motors

(EM) are thus well suited for those requirements and have

been used for creating highly dynamic legged robots [1]

[2]. On the other hand, in the stance phase, the leg must

apply large forces to bear the weight of the robot and its

payload. Without large reduction ratios, EM actuators exhibit

poor torque density and efﬁciency at low-speed [1] [3].

Thus, they are not well suited to stance phase requirements,

especially if the robot needs to lift and carry heavy payloads.

Alternatively, increasing the reduction ratios to meet the

stance phase requirements will limit the maximum velocity

and increase the inertia, thus penalizing the performance of

the swing phase. These conﬂicting requirements for legs lead

designers to compromise between multiple characteristics,

illustrated in Figure 1, when using a ﬁxed reduction ratio.

Fig. 1. Trade-offs of geared motors with a ﬁxed reduction ratio.

This work was supported by the Fonds qu´

eb´

ecois de la recherche sur

la nature et les technologies (FRQNT) and the Natural Sciences and

Engineering Research Council of Canada (NSERC).

1All authors are with the Department of Mechanical Engineering, Uni-

versit´

e de Sherbrooke, Qc, Canada.

Fig. 2. Bimodal demonstration on a robotic knee: swing phase (high-speed),

stance phase (high-force).

Dynamically changing the reduction ratio, like most car

powertrains, would allow a designer to avoid this perfor-

mance compromise. If a robot leg actuator can downshift to

a large reduction ratio during the stance phase, and upshift to

a small reduction ratio for the swing phase, then the electric

motor does not need to be oversized and would always

work in an efﬁcient operating range. Leveraging variable

transmission actuators in this way have been sporadically

explored by researchers in the ﬁeld of robotics in the last

decades. In this sense, Hirose relied on two parallel motors

of different reduction ratio and an electromagnetic clutch to

create a dual-mode transmission mechanism for an articulate

prismatic leg [4]. Bell proposed a dual-motor design for

which the geared motor is electrically disconnected for high-

speed motions to prevent back-emf power dissipation, but the

geared motor inertia stay coupled to the output which limits

the possible reduction ratios [5]. Jeong et al. presented a

single motor two-speed transmission based on twisted string

actuation (TSA) and a dog clutch that is light and compact,

but with many limitations in the operating conditions [6].

Lee et al. proposed a compact dual reduction actuator with

a latching mechanism for a knee joint exoskeleton adapted

for the walking phase and sit-to-stand phase, but without

dynamic switching capabilities [7]. For seamless transitions,

Jang et al. developed a continuously variable transmission

(CVT) based on TSA, two motors and a differential gearbox,

arXiv:2210.05765v1 [cs.RO] 11 Oct 2022

but is limited by the range of reduction ratios. Other serial

dual-motor architectures were investigated too, requiring a

differential and a brake that can be used to conduct seamless

transitions, and shown the mass and energy advantages over

single-ratio actuators [3] [8] [9]. All in all, despite promising

results, many challenges remain such as the trade-off be-

tween the complexity (and size) of the variable transmission

and its ability to change the ratio in terms of range, speed,

seamlessness and operating conditions in which it is possible

to change the reduction ratio [10]. A compact device that

allows a fast and seamless switch between a small and a

large reduction ratio in any operating conditions would be

a breakthrough for many applications, especially for robotic

legs.

This paper explores a novel two-speed hydrostatic ar-

chitecture proposed in [11] which shown mass and energy

advantages over a single-motor design. The concept lever-

ages a hydrostatic transmission which is also beneﬁcial to

delocalize the motor of the moving linkages [12] [13] [14]

[15]. As illustrated in Fig. 3, motorized ball valves are

used to dynamically reconﬁgure the system between two

operating modes tailored to the swing phase and stance

phase. The concept is similar to the two-speed architecture

explored by Verstraten [9] and Girard [3], but in the ﬂuidic

domain. The salient feature is that compact ball valves

replace the high-force brake and the differential that were

required in the mechanical domain, reducing the number of

cumbersome components. Furthermore, the concept is more

ﬂexible in terms of conditions in which the system can

downshift. This paper presents an experimental assessment

of the performance of this concept for actuating a robotic

knee as shown in Figure 2. The novel contributions are: 1) a

mass-delay-ﬂow tradeoff analysis for motorized ball valves

used in the concept and 2) an experimental assessment of the

ability of the concept to switch seamlessly underload, and

3) an experimental demonstration that the actuator prototype

exhibit capabilities (force, speed and compliance) addressing

the needs of robotic legs. Section II presents the working

principle and model of the proposed bimodal hydrostatic

system. Section III discusses the use of hydraulic valves for

reconﬁguring the circuit and presents a motorized ball valve

prototype. Section IV presents the actuator prototype and the

leg test bench. Section V presents control scheme used for

coordinating the motor and valves, and experimental results

with the prototype.

II. WORKING PRINCIPLE AND MODEL

The proposed two-speed architecture consists of a lightly

geared electric motor (EM1) and a highly geared electric

motor (EM2) which are respectively coupled to a high

pitch and low pitch ball screw that actuates two hydraulic

cylinders. Those two cylinders called master cylinder 1 (M1)

and master cylinder 2 (M2) are connected to a slave cylinder

on the leg, through a ﬂexible hydraulic line. This results

in a kinematically redundant system: the displacement of

both master cylinders adds up to create a displacement at

the slave cylinder (neglecting compressibility of the ﬂuid)

but the pressure is shared in the circuit. Additionally, two

hydraulic valves can close the path to the lightly geared M1.

(a) High-speed mode; (b) High-force mode;

Fig. 3. Bimodal actuation principle of the proposed hydrostatic architecture.

Hence, this architecture permits two main modes of op-

eration: a high-speed mode (HS) for fast movement and

backdrivability when the valves are opened and a high force

mode (HF) when the valves are closed. When valves are

open (Fig. 3a), both M1 and M2 can contribute to the output

motion (ﬂow adds up, pressure is shared). This results in

high-speed capability and a low reﬂected inertia at the output,

but with force limited by EM1. When valves are closed (Fig.

3b), only M2 contributes to the output (M1 is connected

to the reservoir and can move freely). This results in high-

force low-speed capabilities, as the conﬁguration leads to

a direct coupling of the highly geared piston to the slave

piston. Furthermore, it is possible to generate large forces at

high speeds in quadrants two and four (Fig. 4) using partial

opening of the valves to restrict the ﬂow in order to brake the

system. The operating modes capabilities in terms of force-

speed are illustrated in the four quadrants at Figure 4, using

the speciﬁcations of the built actuator prototype.

Fig. 4. Bimodal system operating regions in terms of force and speed.

A. Equations of motion

Here we present simpliﬁed equations of motions (EoM)

describing the system behaviour, for all operating modes,

based on lumped-parameter approach illustrated at Fig. 5.

Note that a single-action system is illustrated for simplicity.

If we consider that the ﬂuid in the circuit is incompressible,

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ABimodalHydrostaticActuatorforRoboticLegswithCompliantFastMotionandHighLiftingForceAlexLecavalier1,JeffDenis1,Jean-S´ebastienPlante1,AlexandreGirard1AbstractRoboticlegshavebimodaloperations:swingphaseswhenthelegneedstomovequicklyintheair(high-speed,low-force)andstancephaseswhenthelegbearstheweighto...

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A Bimodal Hydrostatic Actuator for Robotic Legs with Compliant Fast Motion and High Lifting Force Alex Lecavalier1 Jeff Denis1 Jean-S ebastien Plante1 Alexandre Girard1

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