# Hybrid leg compliance enables robots to operate with sensorimotor delays and
low control update frequencies

Milad Shafiee Ashtiani ‡, Alborz Aghamaleki Sarvestani ‡ and Alexander Badri-
Spröwitz ∗

###### Abstract

Animals locomote robustly and agile, albeit significant sensorimotor delays of
their nervous system. The sensorimotor control of legged robots is implemented
with much higher frequencies— often in the kilohertz range—and sensor and
actuator delays in the low millisecond range. But especially at harsh impacts
with unknown touch-down timing, legged robots show unstable controller
behaviors, while animals are seemingly not impacted. Here we examine this
discrepancy and suggest a hybrid robotic leg and controller design. We
implemented a physical, parallel joint compliance dimensioned in combination
with an active, virtual leg length controller. We present an extensive set of
systematic experiments both in computer simulation and hardware. Our hybrid
leg and controller design shows previously unseen robustness, in the presence
of sensorimotor delays up to 60 ms, or control frequencies as low as 20 Hz,
for a drop landing task from 1.3 leg lengths high and with a passive
compliance ratio of 0.7. In computer simulations, we report successful drop-
landings of the hybrid compliant leg from 3.8 leg lengths (1.2 m) for a 2 kg
quadruped robot with 100 Hz control frequency and a sensorimotor delay of 35
ms. The results of our presented hybrid leg design and control provide a
further explanation for the performance robustness of animals, and the
resulting discrepancy between animals and legged robots.

##  1 Keywords:

legged robots, parallel and passive compliance, hybrid actuation and leg
design, sensorimotor delay, feedback, latency

##  2 Introduction

Animals make use of their complex networks of mono-articulate and multi-
articulate muscle-tendons to create joint torque and work for bodyweight
support (Biewener, 1989). The time until the animal’s actuator (muscle)
responds to an external stimulus depends on the velocity and other parameters
of nerve conductivity. The delay is 30 ms and higher in a cat-sized animal
(More and Donelan, 2018). In comparison, the entire stance phase at 4 Hz
running with a duty factor of 0.4 lasts only 100 ms. Yet animals are seemingly
unaffected by the significant delay governing their neuromuscular control,
especially at the arguably harshest locomotion event; the touch-down impact.
Instead, evidence shows that running birds traverse unforeseen pot-hole
perturbations with ease (Daley et al., 2006).

On the other end is an emerging system of latest-generation legged robots
driven by ‘proprioceptive’ actuation and control (Park et al., 2017), ‘quasi-
direct-drive’ strategies (Ding and Park, 2017), and alike. These robots are
capable of real dynamic behaviors and agile maneuvers, with high jumps and
landings and fast locomotion (Grimminger et al., 2020; Park et al., 2017).
These systems require high-frequency control loops of 500 Hz and more with
communication and control delays of a few milliseconds (Bledt et al., 2018;
Grimminger et al., 2020; Li et al., 2020). Such fully actuated robots consume
energy even for a simple, standing task. Proprioceptive and other legged robot
designs work well, especially during the stance phase, but they show
limitations in the transition from swing to stance phase. Touch-downs can be
harsh and include rapid leg loading from zero to multiple body weights in a
few milliseconds (Mo et al., 2020), and side-effects from wobbling masses
Günther et al. (2003). Uncertainties in touch-down timing (Daley and Biewener,
2006) are problematic because the robot’s actuators are often gain and
impedance scheduled (Hammoud et al., 2020; Hubicki et al., 2016). Learned
control strategies can mitigate some of the effects of sensor noise and timing
uncertainties and also function in unstructured terrain (Bledt et al., 2018).
To function well, the robot’s legs must rapidly establish secure and steady
ground contact and immediately produce the joint work to support the robot’s
weight.

In teleoperation, as one of the legged robot applications, time-delays are
inherent, and are caused by the communication distance (Varkonyi et al.,
2014). Sensorimotor delays in legged robotics can be defined more broadly for
cases where feedback is transmitted late compared to the expected time-frame
to react, i.e., at step-down or push-like perturbations (Daley, 2018; Daley
and Biewener, 2006). Recent hierarchical, optimization-based controllers
behave robustly and are versatile, but they heavily depend on the sensorimotor
control’s exact timing (Shafiee et al., 2019; Shafiee-Ashtiani et al., 2017).
In general, force feedback works well in minimal-delay systems, and a large
enough feedback delay eventually causes control instabilities. Thus, the
communication delay is a significant challenge for legged robots for real-
world teleoperation applications.

Which leads us back to our initial question: how do animal successfully manage
an in-built neuromuscular control with large sensorimotor delays, yet show no
obvious signs of decline in robustness, responsiveness, or agility. More and
Donelan report an exponential relation between the total neuromuscular delay
and the animal’s mass:

$t_{Delay}=0.031M^{0.21}$ (1)

with $M$ as the animal’s mass in kilogram, and $t_{Delay}$ as the sensorimotor
delay in seconds. For example, for animals with 600 g and 2 kg body weight,
the sensorimotor delays are 27 ms and 35 ms, respectively.

Inspired by the animal locomotion apparatus, legged robots increasingly apply
designs with passive and active compliance (Ambrose and Ames, 2020; Nasiri et
al., 2016). Designs featuring ‘series-elastic actuation’ ((Pratt and
Williamson, 1995)) can improve energy efficiency, robustness, and interaction
safety(Calanca et al., 2015; Hutter et al., 2011) and protect the robot’s
actuators. Compliant and elastic parts applied in-parallel to the actuator
system are labeled ‘parallel-elastic actuation’ (Ambrose and Ames, 2020;
Plooij et al., 2016; Niehues et al., 2015; Roozing, 2018; Ruppert and
Spröwitz, 2019; Spröwitz et al., 2013).

Legged robots with in-series and parallel joint elasticity can locomote purely
driven by feed-forward control, without sensor feedback informing the
controller of the robot’s touch-down status, or the overall robot state
(Spröwitz et al., 2013; Narioka et al., 2012; Ruppert and Spröwitz, 2019;
Spröwitz et al., 2018). Compliance in-parallel to the actuator system can
improve energy efficiency (Roozing et al., 2019; Liu et al., 2018; Yesilevskiy
et al., 2018), and increase the system’s stability (AhmadSharbafi et al.,
2020). Most important for the touch-down event, parallel compliance provides
an _immediate, physical_ response from its spring-equipped leg. The leg’s
spring will charge without delay, sensory feedback or control input, and it
will carry the robot’s weight. Legged robots with in-parallel leg designs have
been shown to mitigate step-down perturbations similarly to running animals
(Spröwitz et al., 2013; Daley and Biewener, 2006). But compliant elements
cause under-actuation and therefore lower the robot’s active control
authority. In parallel-elastic systems, some amount of control authority in
task-space remains, but the robot’s actuators will work against its full-
strength, in-parallel elasticities.

Animals are different in many ways, and it seems they only benefit from
combined passive and active compliant leg actuation (Alexander, 1990).
Physical, compliant structures allow a design where the control task is
partially taken over by the body’s mechanics (Blickhan et al., 2007). Keeping
the animals’ discrepancy between neuromuscular control delays (More and
Donelan, 2018; More et al., 2010) and high locomotion robustness in mind, we
wondered if these aspects are related.

We developed a hybrid between two systems, and merged what we see in legged
animals, to gain the best of two systems; A) Passively, spring-loaded legged
robots that work well with open-loop control, i.e., without control feedback,
and B) High-bandwidth, fully actuated legged robots with full control
authority and rapid response times.

Inspired by biological systems, we show a hybrid robot mechanism and control
design with _complementing levels of passive and active joint compliance_. The
concept successfully overcomes significant sensorimotor delays and works with
lower sensorimotor control update frequencies, than the state-of-the-art fully
actuated legged robots working with delays in the low millisecond range
(Grimminger et al., 2020; Park et al., 2017). Our hybrid robot leg draws
comparatively lower actuator power due to its partial, in-parallel passive
compliance. The proposed hybrid design leads to a balanced level of control
authority, in-between that of robots with passively compliant legs and full
state controlled robots.

In Section 3, we present a theoretical stability analysis of a simplified
joint design with hybrid passive and active stiffness in the presence of
sensorimotor delays. We then present extensive computer simulation and
hardware experiments and investigate the effect of varying actuation update
control frequencies and sensorimotor delays on a robotic leg with varying
ratios of passive and active stiffness and a simulated quadruped robot
equipped with these legs (Section 4). We conclude our work in Section 5.

##  3 Materials and methods

Figure 1: Different source of delay in the robotic leg and biological
counterpart and modeling the leg as a simple pendulum.

We first analyse the poles of a linear and simplified actuated system with
sensorimotor delays. Its mechanics consist of a pendulum mounted with parallel
compliant elements (Figure 1B). We then implement hardware experiments and
simulations to investigate the effect of hybrid active and passive compliance
on a multi-body leg’s control performance. We quantify the total (sum of)
system compliance as the active compliance in-parallel to the passive (spring-
based) compliance, acting at a robotic knee joint (Figure 1C):

$K_{Total}=K_{Active}+K_{Passive}$ (2)

Where $K_{Passive}$($\frac{Nm}{rad}$) is the joint’s passive rotational
stiffness, $K_{Active}$($\frac{Nm}{rad}$) is the joint’s active (virtual)
rotational stiffness ($\frac{Nm}{rad}$) provided by its actuator.
$K_{Total}$($\frac{Nm}{rad}$) is the total rotational stiffness of the joint.
We define the ratio $\lambda_{Passive}$ as the passive stiffness over the
total stiffness.

$\lambda_{Passive}=\frac{K_{Passive}}{K_{Total}}$ (3)

For example, a low passive compliant ratio of $0.1$ indicates that the knee
spring supplies $10\text{\,}\%$ of the knee’s stiffness to carry the robot,
and the active, virtual leg compliance supplies the remaining $90\text{\,}\%$.

###  3.1 Theoretical analysis of a reduced order model

We analysed a simplified system without impacts, with a load and parallel
compliance, to analytically quantify the effects of sensorimotor delays. The
reduced order model consists of a strut-leg mounted as a single degree-of-
freedom pendulum (Figure 1B), and represents a robot’s lower leg. The
equations governing the pendulum motion are:

$I\ddot{\theta}+mgl\cdot
sin(\theta-\theta_{d})+K_{Passive}(\theta-\theta_{d})+B(\dot{\theta}-\dot{\theta}_{d})=\tau_{knee}$
(4)

where $B$ is the physical system’s damping, $K_{Passive}$ is the stiffness of
the parallel compliance element, $I$ is the momentum of inertia, $m$ is the
mass, $l$ is the Center of Mass distance to the pivot point, $g$ is the
standard acceleration. $\theta_{d}$ is the equilibrium knee joint angle that
corresponds to the orientation that a relaxed spring, $\theta$ is a joint
angle and $\tau_{knee}$ is the control torque input to the knee joint. We
implement the input torque as an active compliance:

$I\ddot{\theta}+mgl\cdot
sin(\theta-\theta_{d})+K_{Passive}(\theta-\theta_{d})+B(\dot{\theta}-\dot{\theta}_{d})=-K_{Active}(\theta_{feedback}-\theta_{d})$
(5)

where the $K_{Active}$ is related to the active compliance from motor torque.
$\theta_{feedback}$ is the joint angle read by sensor. We assume a small
enough angular deviation of the pendulum around the equilibrium point:
$sin(\theta-\theta_{d})\simeq(\theta-\theta_{d})$, and can write Equation 5 as
a linear differential equation. We convert Equation 5 to the Laplace domain
and incorporate a fixed time delay $t_{d}$ in the feedback loop of the control
input (active compliance). The resulting closed loop system transfer function
can be presented in the frequency domain as:

$\frac{\Theta_{s}}{\Theta_{ds}}=\frac{K_{Active}e^{-t_{d}s}+mgl+K_{Passive}}{s^{2}I+Bs+K_{Active}e^{-t_{d}s}+K_{Passive}+mgl}$
(6)

The displacement of the system’s poles can be analyzed to understand the
effects that a combination of active and passive compliance have, on the
closed-loop stability in the presence of sensorimotor delay. We linearised the
system’s exponential time delay term with a third-order Padé approximation.
The desired, full joint stiffness $K_{Total}$ is achieved with the combination
of active and passive joint compliance (Equation 3).

Our analysis of the system’s poles is illustrated in Figure 2A. The $"\times"$
mark shows cases where the joint’s compliance is fully active, i.e.,
$\lambda_{Passive}=0$. The $"\circ"$ mark indicates cases where $70\%$ of the
total compliance is caused by the spring, i.e., $\lambda_{Passive}=0.7$.
Figure 2A shows that in a case of $\lambda_{Passive}=0$, and with increasing
feedback loop delay, the dominant system poles move rapidly from their stable
region towards the unstable region at the imaginary axis. For passive
compliance in-parallel with active actuator compliance, the rate of divergence
is much lower. The system’s step response (Figure 2B) indicates that
increasing the sensorimotor delay with a $\lambda_{Passive}=0$ leads to high
oscillations, and a resonance effects eventually destabilizes the joint.
However, in the case of combined passive and active stiffness with an
increased delay of $20ms$, the closed-loop system’s step response is stable
and smooth (Figure 2C, purple line).

Figure 2: A) Pole analysis of a simplified pendulum system with in-parallel,
passive and active joint elasticity. The effects of varying amounts of delay
and passive compliance on the stability of the system are shown. Here, the
system’s mass is $m=0.5\;kg$, the total joint stiffness is
$K_{total}=1.15\;\frac{Nm}{rad}$, and the damping coefficient is
$B=0.14\;\frac{Nms}{rad}$. B) The system’s step response for varying delays
with $\lambda_{Passive}=0$. C) The system’s step response for varying delays
with $\lambda_{Passive}=0.7$.

###  3.2 Computer simulating articulated robot legs

The previous results from the analysis of poles of a simplified, linearised
system provided insights about sensorimotor systems under considerable
sensorimotor delay, and hybrid joint compliance. Here we are extending our
characterization to an articulated leg with hybrid joint compliance,
sensorimotor delays, and impacts from touch-down. The landing task is one of
the most challenging motions due to the large impulsive ground reaction force
applied at the system. The landing load case is nonlinear, hybrid, under-
actuated, and basically a system’s step response. A step response analysis
like the drop-experiment applied here is one conventional way for a system
characterization, in control theory. We implemented our simulation in the
Pybullet simulator (Coumans and Bai, 2016–2019), and we performed extensive
computer simulations for the landing task, for a broad range of sensorimotor
delays, frequencies, and varying $\lambda_{Passive}$ values. We simulated a
single leg and a quadruped robot, both modified from the open-source quadruped
robot ’Solo’ (Grimminger et al., 2020).

In Figure 3, we depict the tested control and sensorimotor strategies. The
black curve is a schematic, desired knee motor torque trajectory, and the
control command frequency (red line) is measured in commands per second. For a
reference; the control frequency of active actuators in proprioceptive legged
robot is often around 1 kHz, i.e., a cycle period
$dt_{control}=\frac{1}{freq}$ of 1 ms. Here we are especially interested in
investigating scenarios with control frequencies much below 1 kHz. We defined
three strategies for applying torque with a duration of $dt_{Activation}$, and
a duty cycle $DC$, which is the fraction of $dt_{control}$ with a non-zero
actuator torque. $dt_{Activation}$ is defined as the time period between
control commands, i.e., $dt_{Activation}=DC\times dt_{control}$, and ranges
from $1\text{\,}\mathrm{m}\mathrm{s}$ to a maximum of $\frac{1}{freq}$ in
$\mathrm{[}\mathrm{m}\mathrm{s}\mathrm{]}$. For $dt_{Activation,min}$, the
control command is applied for $1\text{\,}\mathrm{m}\mathrm{s}$, and then
reset to zero. For $dt_{Activation,max}$, the actuator will send a command
equal to the previous value until the control command is updated (Figure 3,
red line). We further simulate cases where the control command is applied
after a given sensorimotor delay (Figure 3, blue curve).

Figure 3: Knee motor command for different combinations of control frequency,
frequency duty cycle, and sensorimotor delay. A) Is showing a $100\text{\,}\%$
duty cycle control frequency, at a relatively low update frequency. B) Shows a
set sensorimotor delay between the desired knee output torque, and the
commanded output torque. C) Shows a $50\text{\,}\%$ duty cycle at the same
control frequency as A). D) Indicates the portion of parallel, passive joint
compliance, of the joint’s overall joint compliance. Here, a passive
compliance ratio of $\lambda_{Passive}$ 0.5 is shown, such that the knee’s
mechanical spring produces half of the knee torque, and the virtual knee
spring produces the remaining knee torque.

The active compliance controller input is applied at the knee joint as:

$\tau_{knee,motor}=K_{Total}(1-\lambda_{Passive})(\theta_{feedback,Knee}-\theta_{d,knee})$
(7)

To present the physical spring in Pybullet, we implement a spring torque
applied at the knee joint:

$\tau_{knee,spring}=K_{Total}(\lambda_{Passive})(\theta_{knee}-\theta_{d,knee})$
(8)

###  3.3 Setup hardware experiments

We modified a single leg of the 8-DoF open-source, quadruped robot ’Solo’ with
its proprioceptive design and controller architecture (Grimminger et al.,
2020). The leg has two active degrees of freedom, one at the hip and one at
the knee. Both leg segments are $0.16\>m$ long. A brushless motor (Antigravity
MN4004-kv380, T-Motor) drives a two-stage belt transmission with an overall
$9:1$ gear ratio. Two encoders (AEDT-9810-T00, Avago) measure the motor’s
rotor position, which is recalculated into joint angles. We added a physical
spring (SWY 16.5-30, Misumi) in parallel to the knee. The spring is inserted
into the knee joint via a tendon and a pulley with
$18.9\text{\,}\mathrm{m}\mathrm{m}$ radius (Figure 1C). We designed the spring
mount to rapidly exchange springs by softer or harder ones, between
experiments. To simplify the touch-down scenario, the robot leg was dropped
while guided by a vertical rail (Figure 4A). The hip joint is constrained to
follow half of the knee joint angle at all times, controlled by a position
controller, and is meant to create foot contact vertically below the hip
joint. We monitor the robot’s vertical position with a draw-wire sensor (LX-
PA-40, WayCon), we use two draw-wire sensors on top and bottom to cancel
unwanted tension force produce by draw-wire sensors. With the vertical robot
leg position, we quantify the robot’s landing behavior and evaluate the
effectiveness of the proposed hybrid active/passive joint stiffness framework.
The recorded vertical position is sampled by an analog-to-digital (A/D) port
of the brushless motor driver board. The brushless motor driver board sends
motor position and vertical position data via a communication board to a PC,
and it receives the motor control commands, via a SPI Protocol. The PC
communication board is the bridge between the brushless motor driver board and
the PC (Intel® Xeon(R) W-2145 CPU @ 3.70GHz $\times$ 16, 64-bit, RAM 62,5 GB,
Ubuntu 18.04) via EtherCAT. We wrote a custom robot control program in a
Python wrapper, which communicates via the PC communication board with the
robot leg. The Python wrapper adds a time stamp to all input parameters and
sensory data such as joint angles, motor currents, and body height, and saves
all data into a text file for further analysis. A custom Matlab script was
written to post-process and plot the data.

Figure 4: Experimental setup. A) A 2-DoF hybrid compliant leg; a one-
directional spring (‘passive compliance’) extends the knee joint via a knee
tendon and a knee pulley/cam. Knee springs with varying stiffness were mounted
during the experiments, supporting between $0\text{\,}\%100\text{\,}\%$ of the
robot’s weight. A rail guides the robot’s vertical drop, and a set of
potentiometers measures the robot’s height. The leg’s active knee actuation
acts in combination to the in-parallel mounted knee spring. B) Setup details.
C) The URDF model of the hybrid compliant robot leg in the Pybullet
simulation.

##  4 Result and discussion

In this section we present the results from the computer simulation of a
single robot leg with hybrid joint compliance, and from dropping an equivalent
quadruped robot in simulation. We then present the results from our hardware
experiments with a single leg mounted to a vertical slider.

###  4.1 Single leg computer simulation

We study the effects of varying combinations of sensorimotor delay, control
loop update frequency, and the ratio of passive compliance $\lambda_{Passive}$
on the compliance controller performance during landing. We derive a reference
landing hip height trajectory from dropping a parallel compliant leg with
$\lambda_{Passive}=1$. Deviations in settling time and final hip height are
accepted as successful landings, within given limits.

We performed computer simulations to quantify the viability of the landing
task. We varied $\lambda_{Passive}$ from $0$ to $1$ by steps of $0.05$, the
sensorimotor delay from $0\>ms$ to $60\>ms$ in steps $5\>ms$, and the
sensorimotor control loop update frequency in a range of
$[20,\>50,\>250,\>1000]\>Hz$. In Pybullet, we set joint damping values of
$0.01\frac{Nms}{rad}$ and $0.05\frac{Nms}{rad}$ for the hip and knee,
respectively. The weight of the single robotic leg is $0.6\>kg$, the weight of
the quadruped is $2.0\>kg$. We chose the total stiffness for the knee joint
compliance for an approximate leg length deflection of $10\%$ during mid-
stance, after dropping the leg from a height of $42.5\>cm$. We also assumed
that the robot’s weight is concentrated at its hip joint. We realized a
$\lambda_{Passive}=1$ rotational spring as a combination of a pulley with
radius $r=0.0189\>m$, and a linear spring with a stiffness of
$K=4680\>\frac{N}{m}$ acting at the knee joint. The total rotational stiffness
is then $Kr^{2}=1.67\>\frac{Nm}{rad}$ as a combination of passive and active
compliance components. In simulation, the delay and frequency modes do not
affect the $\tau_{knee,spring}$, as it is representing a physical spring.
Instead, delay and frequency variations only affected the active part of the
joint stiffness $\tau_{knee,motor}$. We dropped the robot from $0.425\>m$
height, which is 1.3 times the length of both leg segments. The robot’s body
was constrained to move vertically only, and we monitored the robot’s height
during landing. The vertical hip trajectory represents the system’s step
response. We then assessed the robot’s controller performance by measuring the
settling time and the final hip height. The settling time is the duration to
settle to the hip height within $|\mathrm{Height(t)-Height_{final}}|$ range,
between the hip height trajectory $\mathrm{Height(t)}$ and the steady-state
hip height $\mathrm{Height_{final}}$. The reference final hip height is that
of a $\lambda_{Passive}=1$ robot leg. We chose a critical settling time above
$0.7\>s$, and a critical hip height below $0.3\>m$ as failing cases.

In Figure 5, the results of 315 drop-landing simulations are illustrated, with
varying sensorimotor delays and $\lambda_{Passive}$ settings, and for a single
control frequency of $1\>kHz$. The grey points represent failed simulation
cases with a steady-state hip height of less than $0.3\>m$ or settling times
higher than $0.7\>s$. Our computer simulation results show that in a case of
$\lambda_{Passive}=0.0$, and by increasing the sensorimotor delay to values
above $25\>ms$, all scenarios failed. Yet by setting $\lambda_{Passive}$ high
than $0.7$, the system is stable at the presence of $60\>ms$ delays. The
computer simulation results demonstrate that the hybrid compliant leg has
stable impact control regimes in the presence of large sensorimotor delays,
with an appropriate combination of passive and active compliance.

Figure 5: Results from 315 computer simulations for a control update frequency
of $1000\;Hz$ for the drop-landing task of a simulated robot leg. The passive
compliant ratio $\lambda_{passive}$ was varied between $0$ to $1$ in steps of
$0.05$, and the sensorimotor delay between $0\>ms$ to $60\>ms$ by steps
$5\>ms$. The grey data points and the grey hip height trajectory show a failed
landing task with too much initial deflection and a too large settling time.
The colored data points and trajectories show viable landings. Viable landings
are shown for sensorimotor delays up to $60\text{\,}\mathrm{m}\mathrm{s}$, in
combination with a passive compliant ratio of $\lambda_{Passive}=0.6$.

We then investigated the effect of varying the control update frequency, and
we performed computer simulations for frequencies
$[20,\>50,100\>,\>250,\>1000]\>Hz$, and duty cycles of $[25,\;50,\;100]\>\%$.
The results are shown in Figure 6. Most visible is a decreasing viable area
for all three duty cycles, from reduced the control frequencies. Comparing
duty cycles of $25\%$, $100\;\%$ (Figure 6A, Figure 6C) shows that for a
reduced duty cycle, the viable areas did overall change much. Low amounts of
$\lambda_{Passive}\approx 0.2$ only lead to viable landings in combination
with a DC of $50\text{\,}\%$ or the highest update frequency
($1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$). The Figure 6C shows that a DC of
$100\>\%$ at almost all control frequencies $[\>100,\>250,\>1000]\>Hz$ have a
similar viable area. Only when switching to control frequencies as low as
$20\>Hz$, the viable area largely changes. For a DC of $50\;\%$ (Figure 6B),
the viable area changes slightly when switching between control frequencies
$[\>50,\>100,\>250]\>Hz$. The biggest viable area changes are visible when
changing from $1000\>Hz$ to $250\>Hz$, or from $50\>Hz$ to $20$. At a DC of
$25\;\%$ (Figure 6A) and when switching between control frequencies of
$1000\>Hz$ to $250\>Hz$, the viable area is reduced drastically. The change of
control frequency for the remaining values has seemingly less influence on the
change of viable area. For control frequencies of $20,\>50,\>100,\>250\>Hz$
and high sensorimotor delays around $50,\>60\>ms$, and by reducing the duty
cycle, a larger viable area for large $\lambda_{passive}$ is visible. For all
$\lambda_{Passive}$ above $0.6$ we show stable landing, including critical
combinations of $60\>ms$ delay and $20\>Hz$ updated frequency. These examples
show how capable a hybrid active and passive compliance system can be. In
general, all results indicate stable landing for passive compliant ratios
equal and higher than $\lambda_{Passive}$ $0.7$.

Figure 6: Results showing three different duty cycles (DC). Each DC plot is
made from 1575 simulations, in sum 4725 computer simulations. Top and bottom
plots show the same data. The reference landing performance is the top left
data point in each plot, from dropping a $\lambda_{Passive}=1$ robot leg. A)
$DC=25\>\%$ B) $DC=50\>\%$ C) $DC=100\>\%$.

###  4.2 Quadruped computer simulation

The previous computer simulation results from landing a single leg indicate
that by choosing a high ratio of passive compliance, the robot’s performance
becomes largely independent of the sensorimotor delay, and the control
frequency. But fully passive compliance limits a robot’s agility, and to
maintain control authority we suggest reducing the passive compliance ratio.
In seven drop-landing scenarios, we altered a quadruped robot’s drop height,
and its hybrid passive and active stiffness (Figure 7). The simulation
parameters are provided in Table 1. The robot in case 1 features a passive
compliance ratio of $1$. It was dropped from a height of $0.7\>m$, and landed
successfully. The robot in case 2 uses identical parameters, and it was
dropped from $1\>m$ height, and fails to land properly. This example shows the
drawback of pure, passive and non-adjustable compliance. The robot
configuration in case 3 is a controller with full, bi-directionally active
compliance, no passive compliance, and no sensorimotor motor delay. The
robot’s controller successfully guided the landing. In case 4 a fully-active
compliance with a $17\>ms$ sensorimotor delay failed to land properly, which
shows the vulnerability of active compliance in the presence of sensorimotor
delay. Case 5 shows a successful landing scenario by combining passive and
active compliance, with a $27\>ms$ sensorimotor delay. The control update
frequency was reduced from $1000\>Hz$ to $200\>Hz$. Case 6 uses the same
passive compliance ratio $\lambda_{Passive}$ as case 5, and we reduced the
control update frequency to $100\>Hz$. The robot’s controller did not land the
robot successfully. Finally in case 7, we decreased the passive compliance
ratio $\lambda_{Passive}$ by increasing the active stiffness, and the robot
landed successfully, for a drop height of $1.2\>m$, a sensorimotor delay of
$35\>ms$, and $100\>Hz$ control frequency. The last case shows how an
appropriate combination of hybrid active and passive compliance at low control
frequency increases the control authority, energy efficiency (the spring
partially captures the robot’s load) and robustness in the presence of
sensorimotor delay.

Figure 7: Computer simulated quadruped robots landing, in seven different
scenarios. A) The Robot’s initial state. The initial drop heights are
indicated in red. B) An intermediate robot state at 4 s simulation time. The
panels also provide controller parameters. C) Converged robot state after 10
s. Simulation cases 1, 3, 5, and 7 were landing successfully.

###  4.3 Hardware experiments

We validated previous simulation results with hardware experiments. We
selected passive compliance ratios of
$\lambda_{Passive}=[0\>,0.37\>,0.67\>,1]$ and a total knee stiffness of
$K_{Total}=4680\>Nm$. We then varied the physical spring stiffness, control
frequencies $Frequencies=[1000,\>100,\>10]$ and sensorimotor delays
$\>Delays=[0,\>10,\>20,\>30,\>50]$\mathrm{m}\mathrm{s}$$. In Figure 8 we
assess the difference $e_{sim2real}$ between the computer simulation results
and the hardware experimental results, as the mean-square-error between both
resulting vertical hip trajectory, normalized by the maximum leg length. Grey
color data shows failure cases in both experiments and simulations. Viable
cases with a mean-square-error of less than $6\text{\,}\%$ (Figure 8, deep
red) indicate a good consistency between the hardware experiment, and the
computer simulation. We show four exemplary hip trajectory, with varying
passive compliance ratios $\lambda_{passive}$ (Figure 8, I-IV). The first
three cases are viable cases with a good consistency between simulation and
experiments, and with a $e_{sim2real}$ of less than $6\text{\,}\%$. The fourth
case is a typical failure case. It shows an atypical settling and transition
behavior both in the computer simulation and in the hardware experiment.

Figure 8: Comparing results from computer simulation and hardware experiments,
in form of a mean-square-error of the normalized body height. Left) Good
similarities are shown as colored data patches, grey data patches indicate
unviable landing experiments. I-IV) exemplary hip trajectories for common
control frequency and passive compliant ratio combinations. I-III are
successful landings with short settling times and a good final hip height. IV
shows an unsuccessful landing example.

##  5 Conclusion and summary

We proposed a new, hybrid parallel passive and active compliance architecture
for the design and control of robotic legs, where a portion of the leg loading
is captured by the in-parallel knee spring. The remaining load is captured by
the robot’s active, virtual spring. We show robust landing scenarios with
varying hybrid compliance ratios in the presence of sensorimotor delays and
low control frequencies. The hybrid compliance design remains relatively
energy efficient because the leg’s in-parallel spring stores and releases
significant elastic energy. Yet with an appropriate amount of active
compliance, the robot’s controller maintains its control authority. We
performed extensive computer simulations and systematically characterized the
system’s response to varying sensorimotor delays and control frequencies in a
drop-landing task. The computer simulation results show that drop-landings
were successful for sensorimotor delays up to $60\>ms$, and a control
frequency as low as $20\>Hz$, in combination with a passive compliance ratio
of $\lambda_{Passive}=0.7$. We also simulated drop-landings with a quadruped
robot, with varying total leg stiffness and multiple drop heights. In these
examples, the hybrid compliant actuation showed good robustness in the
presence of sensorimotor delay and low update control frequencies. With a
significant amount of active compliance ratio, we still have good control
authority; we altered the robot’s total compliance and successful drop-landed
the robot from multiple heights. Our results are well consistent with the
locomotion robustness of running animals, which evidently and successfully
deal with neuromuscular sensorimotor delays in a similar range. We verified
our computer simulation results with hardware experiments for a selected range
of control and design parameters and could show a good agreement between both.

## Author Contributions

MSA and AAS contributed to the concept, robot design, experimental setup,
simulation, and experimentation. AB-S developed the original hybrid
active/passive compliant concept. All authors discussed the data, agreed with
the presented results, and contributed to the writing.

## Acknowledgments

The authors thank the International Max Planck Research School for Intelligent
Systems (IMPRS-IS) for supporting AAS. We thank Felix Grimminger and Jad Saud
for support developing the robot leg, the Robotic ZWE for prototyping support
and Julian Viereck for providing the URDF file of the quadruped.

## Supplemental Data

The supplementary material includes a code algorithms snippet driving the
simulations, and our video presenting results from computer simulation and
hardware experiment.

## Data Availability Statement

The datasets recorded and generated for this study are available on request
from the corresponding author.

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| Case
---
number
| Total compliance
---
[$\frac{N.m}{rad}$]
| $\lambda_{Passive}$
---
[%]
| Control frequency
---
[Hz]
| Delay
---
[ms]
1 | 1.6717 | 100 | 1000 | 0
2 | 1.6717 | 100 | 1000 | 0
3 | 1.6717 | 0 | 1000 | 0
4 | 1.6717 | 0 | 1000 | 17
5 | 2.5076 | 67 | 200 | 27
6 | 2.5076 | 67 | 100 | 27
7 | 2.8419 | 59 | 100 | 35

Table 1: Simulation parameter of quadrupedal robot for task of landing with
different control frequency and $\lambda_{Passive}$