Stability and Performance Limits of Latency-Prone Distributed Feedback Controllers

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by  Ye Zhao

1
Stability and Performance Limits of Latency-Prone
Distributed Feedback Controllers
Y. Zhao, N. Paine, K.S. Kim, and L. Sentis∗
Abstract—Robotic control systems are increasingly relying on
distributed feedback controllers to tackle complex sensing and
decision problems such as those found in highly articulated
human-centered robots. These demands come at the cost of a
growingcomputationalburdenand,asaresult,largercontroller
5 latencies.Tomaximizerobustnesstomechanicaldisturbancesby
1 maximizing control feedback gains, this paper emphasizes the
0 necessity for compromise between high- and low-level feedback
2 control effort in distributed controllers. Specifically, the effect
of distributed impedance controllers is studied where damping
n
feedback effort is executed in close proximity to the control
a
plant and stiffness feedback effort is executed in a latency-
J
prone centralized control process. A central observation is

3 that the stability of high impedance distributed controllers is
1 very sensitive to damping feedback delay but much less to
stiffness feedback delay. This study pursues a detailed analysis

] of this observation that leads to a physical understanding of
Y
the disparity. Then a practical controller breakdown gain rule
S is derived to aim at enabling control designers to consider
s. the benefits of implementing their control applications in a Fig. 1: Depiction of Various Control Architectures. Many
distributed fashion. These considerations are further validated
c control systems today employ one of the control architectures
[ throughtheanalysis,simulationandexperimentaltestingonhigh
performance actuators and on an omnidirectional mobile base. above: a) Centralized control with only high-level feedback

1 controllers (HLCs); b) Decentralized control with only low-
IndexTerms—DistributedFeedbackControl,HighImpedance
v level feedback controllers (LLCs); c) Distributed control with
Control, Feedback Delays, Mobile Robotics.
4
both HLCs and LLCs, which is the focus of this paper.
5
8 I. INTRODUCTION
2
0 As a result of the increasing complexity of robotic control stable than those with small delays, such as locally embed-
. systems, such as human-centered robots [1, 2] and industrial ded controllers. Without the fast servo rates of embedded
1
surgical machines [3], new system architectures, especially controllers, the gains in centralized controllers can only be
0
5 distributed control architectures [4, 5], are often being sought raisedtolimitedvalues,decreasingtheirrobustnesstoexternal
1 for communicating with and controlling the numerous de- disturbances [6] and unmodelled dynamics [7, 8].
v: vice subsystems. Often, these distributed control architectures As such, why not removing centralized controllers alto-
i manifest themselves in a hierarchical control fashion where a gether and implementing all feedback processes at the low-
X centralized controller can delegate tasks to subordinate local level? Such operation might not always be possible. For
r controllers(Figure1).Asitisknown,communicationbetween instance, consider controlling the behavior of human-centered
a
actuatorsandtheirlow-levelcontrollerscanoccurathighrates robots(i.e.highlyarticulatedrobotsthatinteractwithhumans).
while communication between low- and high-level controllers Normally this operation is achieved by specifying the goals
occurs more slowly. The latter is further slowed down by of some task frames such as the end effector coordinates.
the fact that centralized controllers tend to implement larger One established option is to create impedance controllers on
computational operations, for instance to compute system those frames and transform the resulting control references to
models or coordinate transformations online. actuator commands via operational space transformations [9].
Such a strategy requires the implementation of a centralized
A. Control architectures with feedback delays feedback controller which can utilize global sensing data,
One concern is that feedback controllers with large delays, access the state of the entire system model, and compute
such as the centralized controllers mentioned above, are less thenecessarymodelsandtransformationsforcontrol.Because
of the aforementioned larger delays on high-level controllers,
Y. Zhao, K.S. Kim and L. Sentis (* is the corresponding author) are
does this imply that high gain control cannot be achieved in
with the Department of Mechanical Engineering, The University of Texas,
Austin, TX 78712 USA (e-mail: yezhao@utexas,edu, [email protected], some human centered robot control due to stability problems?
[email protected]). Itwillbeshownthatthismaynotneedtobethecase.Butfor
N. Paine is with the Department of Electrical and Computer En-
now, this delay issue is one of the reasons why various cur-
gineering, The University of Texas, Austin, TX 78712 USA (e-mail:
[email protected]). rentlyexistinghuman-centeredrobotscannotachievethesame
2
level of control accuracy that it is found in high performance
industrial manipulators. More concretely, this study proposes
a distributed impedance controller where only proportional
(i.e., stiffness) position feedback is implemented in the high-
level control process with slow servo updates. This process
will experience the long latencies found in many modern
centralized controllers of complex human-centered robots. At
Fig. 2: Actuator and Control Plant Model. This diagram
the same time, it contains global information of the model
represents a generalization of rigid actuators considered in
andtheexternalsensorsthatcanbeusedforoperationalspace
this paper. F is the applied motor force, x is the load
control.Forstabilityreasons,ourstudyproposestoimplement M
displacement output, m is effective output inertia, b is the
the derivative (i.e., damping) position feedback part of the
actuator’s passive damping, and F is an external disturbance
controllerinlow-levelembeddedactuatorprocesseswhichcan d
force.
therefore achieve the desired high update rates.
B. Analysis of sensitivity to delay
Third, the proposed distributed controllers is leveraged an
To focus the study on the physical performance of the implementationintoanomnidirectionalbase.Theresultsshow
proposed distributed control approach, our study first focuses a substantial increase in closed-loop impedance capabilities,
on a single actuator system with separate stiffness and damp- which results in higher tracking accuracy with respect to the
ing servos and under multiple controller delays. Then the monolithic centralized controller counterpart approach.
physical insights gained are used as a basis for achieving
highimpedancebehaviorsinsingleactuatorsystemsandinan II. RELATEDWORK
omnidirectional mobile base. Let us pose some key questions
Advances in distributed control technologies [5, 10] have
regarding distributed stiffness-damping feedback controllers
enabled the development of decentralized multi input and
considered in this paper: (A) Does controller stability have
multi output systems such as humanoid systems and highly
differentsensitivitytostiffnessanddampingfeedbackdelays?
articulated robots [1, 2]. However the effect on controller
(B)Ifthatisthecase,whatarethephysicalreasonsforsucha
performance due the ever growing computational demand on
difference? (C) In applications where load uncertainty exists,
feedback servos and latency-prone serial communications in
how robust is the distributed controller to these uncertainties?
human-centered robots have been largely overlooked on these
To answer these questions, this paper studies the physical
studies.
behavior of the proposed realtime distributed system using
Distributed control architectures in robotics combine cen-
control analysis tools applied to the system’s plant, including
tralized processes with self-contained control units in close
the phase margin stability criterion and time-based trajectory
proximity to actuators and sensors. Two inherent advantages
tracking response. Using these tools our study reveals that
ofhavinglowlevelprocessesare:1)toincreasecomputational
system stability and performance are much more sensitive to
resources by offloading the workload of the central computer
damping feedback delays than to stiffness feedback delays.
onto distributed processors [4] and 2) to increase control
system stability and tracking performance due to reduced
C. Benefits of the proposed distributed control architecture
feedback delays. This study focuses on the latter advantage,
As it will be empirically demonstrated, the benefit of the i.e., how a distributed controller improves control system sta-
proposed split control approach over a monolithic controller bilityandperformanceovermonolithiccentralizedcontrolap-
implemented at the high-level is to increase control stability proaches.Adetailedanalysis,exploration,andimplementation
duetothereduceddampingfeedbackdelay.Asadirectresult, of the high impedance capabilities of distributed controllers
closed-loop actuator impedance may be increased beyond for a mobile base with latency-prone centralized processors
the levels possible with a monolithic high-level impedance has not previously been performed.
controller.Thisconclusionmaybeleveragedonmanypractical Robustnessandtheeffectsofdelayhaveoftenbeenstudied
systems to improve disturbance rejection by increasing gains in work regarding Proportional-Integral-Derivative (PID) con-
without compromising overall controller stability. As such, troller tuning. A survey of PID controllers including system
these findings are expected to be immediately useful on many plants using phase margin techniques with linear approxima-
complex mechatronic and control systems. tions is conducted in [12]. The works [13, 14] study auto-
To demonstrate the effectiveness of the proposed methods, tuning and adaption of PID controllers while the work [15]
thisstudyimplementstwotypesoftestsonahighperformance furthers these techniques by developing optimal design tools
actuator followed by experiments on a mobile base. First, a applied to various types of plants including delays. The study
position step response is tested on an actuator under various in [16] proposed an optimal gain scheduling method for DC
combinations of stiffness and damping feedback delays. The motor speed control with a PI controller. In [17], a back-
experimentalresultsshowhighcorrelationtotheircorrespond- stepping controller with time-delay estimation and nonlinear
ingsimulationresults.Second,trajectorytrackingperformance damping is developed for variable PID gain tuning under
of a complex joint trajectory with load uncertainty is tested disturbances. The high volume of studies on PID tuning
both in simulation and in hardware on the same actuator. methods highlight the importance of this topic for robust
3
Fig. 3: Single Input / Single Output Controller with Distributed Structure. A simple proportional-derivative control law is
used to control an actuator. P denotes the actuator plant with motor current input, i , and position output, x. ν−1 represents
M
a scaling constant mapping the desired force, F , to the motor current, i . K is the stiffness feedback gain while B is
D M
the damping feedback gain. The damping feedback loop is labeled as embedded to emphasize that it is meant to be locally
implemented to take advantage of high servo rates. On the other hand the stiffness loop is implemented in a high-level
computational process close to external sensors and centralized models, for operational space control purposes. Operational
spacecontrolisnormallyusedinhuman-centeredroboticapplicationswherecontrollersusetaskcoordinatesandglobalmodels
for their operation. An external disturbance is denoted as F inserted between the controller and plant block as suggested by
d
[11]. This simple controller is used to illustrate the discrepancies to latencies between the servo loops. It does not correspond
to a practical robot controller as it contains only a single degree of freedom. After studying the physical advantages of these
type of structure, we leverage it to a multi-axis robotic base shown in Figure 10 demonstrating the ability to decouple the
damping and stiffness servos to simultaneously achieve system stability and operational space control.
control under disturbances. However, none of those studies stability and performance is more sensitive to damping servo
considers the sensitivity discrepancy to latencies between the latencies than stiffness servo latencies. Then a novel servo
stiffness and damping servos as separate entities nor do they breakdown rule is proposed to evaluate the benefits of using a
considerthedecouplingofthoseservosintoseparateprocesses distributed control architecture. As a conclusion, this paper
for stability purposes as it is done in this paper. proposes to use stiffness servos for centralized operational
The field of haptics [18], networked control [19, 20] and space control while realizing embedded-level damping servos
teleoperation [16] have also thoroughly studied delays and as joint space damping processes for stability and tracking
filtering effects. Specifically, haptics is more related to our accuracy.
work because it is a special case of distributed control in
III. BASICDISTRIBUTEDCONTROLSTRUCTURE
which the master and slave devices require separate feedback
This section describes the actuator model used to analyze
controllers. Due to the destabilizing effects of time delays,
closed-loop system stability, propose a basic distributed con-
significant effort has been put forth to ensure systems are
trol architecture that delocalizes stiffness and damping servo
stable by enforcing passivity criteria [18]. Other works [21–
loops, and analyzes the sensitivity of these control processes
23] further relax this constraint and focus on how delay and
to loop delays.
filtering affect stability. Related work has been performed
considering additional real-world effects such as quantization
A. Actuator plant model
and coulomb friction on system stability [24]. Once more,
Manyrigidelectricalactuatorsliketheonesusedinmodern
theses studies do not analyze nor exploit the large sensitivity
robots can be approximately modeled as a second-order plant
discrepancy between stiffness and damping feedback loops
with a force acting on an inertia-damper pair (Figure 2).
nor propose solutions to increase performance based on this
Consideringacurrent-controlledmotor,thecontrolplantfrom
discrepancy.
current, i , to position, x, is
M
Consequently, our main contribution is to analyze, provide
x(s) x(s) F (s) ν
control system solutions, implement and evaluate actuators P(s)= = M = , (1)
i (s) F (s) i (s) ms2+bs
and mobile robotic systems with latency-prone distributed M M M
architectures to significantly enhance their trajectory tracking where F is the applied motor force, ν (cid:44)F /i =ηNk ,
M M M τ
capabilities under disturbances and system uncertainties. In η is the drivetrain efficiency, N is the gear speed reduction
particular, a new study is performed to reveal that system and k is the motor torque constant.
τ
4
B. Closed-loop distributed controller model
Figure 3 shows our proposed distributed controller built
using a proportional-derivative feedback mechanism. It in-
cludes velocity feedback filtering (Q s), stiffness feedback
v
delay (T ), damping feedback delay (T ), with T (cid:54)= T ,
s d s d
stiffness feedback gain (K) and damping feedback gain (B).
Excluding the unknown load (F ), the desired motor force
d
(F ) in the Laplace domain associated with the proposed
D
distributed controller is
FD(s)=K(xD−e−Tssx)+B(xDs−e−TdsQvxs), (2)
where s is the Laplace variable, x and x˙ (i.e., x s in the
D D D
Laplace domain) are the desired output position and velocity,
and e−Tss and e−Tds represent Laplace transforms of the
time delays in the stiffness and damping feedback loops,
respectively. Using the above equation and Equation (1), one
can derive the closed-loop transfer function from desired to
output positions as
x Bs+K
P (s)= = , (3)
CL xD ms2+(b+e−TdsBQv)s+e−TssK
where Q is chosen to be a first order low pass filter with a
v
cutoff frequency f
v
2πf
Q (s)= v . (4)
v s+2πf
v
To derive the open-loop transfer function [11] of the dis-
tributed controller, one can re-write Equation (3) as Fig.4:PhaseMarginSensitivitytoLoopDelays.Thisfigure
shows phase margin simulations of the open loop transfer
Bs+K
function shown in (6) as a function of the natural frequency
P (s)= ms2+bs , (5)
CL 1+P (s) defined in (10) and the servo delays shown in Figure 3. A
OL
phase margin of 0◦ is considered unstable, however, from
wherePOL(s)(cid:44)P(s)H(s)istheopen-looptransferfunction, simulations of various types of actuators [25], oscillatory
POL(s)= e−TdsBmQsv2s++bes−TssK, (6) basehaavhioorribzeognitnasl blienleo.wDtheleaythsrersahnoglidngofb5e0tw◦,eednisp1laymesdaanbdov5e
ms are simulated for both the stiffness and damping servos.
P(s)istheactuator’splant,andH(s)istheso-calledfeedback The simulations are performed based on identical actuator
transfer function. parametersthanthoseusedintheexperimentalsection,Section
Thepresenceofdelaysandfilteringcausestheaboveclosed V,i.e.passiveoutputinertiam=256kgandpassivedamping
loopplanttobehaveasahighorderdynamicsystemforwhich b = 1250 Ns/m. Equations (9) and (10) can subsequently be
typicalgainselectionmethodsdonotapply.However,tomake usedtoderivedthestiffnessanddampingfeedbackgains.The
the problem tractable, one can define a dependency between key observation here is that variations of the phase margin
the stiffness and damping gains using an idealized second curves above are much more pronounced from damping servo
order characteristic polynomial [11] delays than from stiffness servo delays.
s2+2ζω s+ω2, (7)
n n
and the natural frequency,
where ω is the so-called natural frequency and ζ is the so-
n …
calleddampingfactor.Insuchcase,theidealizedcharacteristic
ω 1 K
polynomial (i.e. ignoring delays, T = T = 0, and filtering, f (cid:44) n = . (10)
s d n 2π 2π m
Q =1)associatedwithourclosedloopplantofEquation(3)
v
would be ThesecondorderdependencyofEquation(9)willbeusedfor
s2+(B+b)/m·s+K/m. (8) the rest of this paper for deriving new gain selection methods
throughthethoroughanalysisoftheoscillatorybehaviorofthe
Choosing the second order critically damped rule, ζ = 1 closed loop plant of Equation (3). In particular our study will
and comparing Equations (7) and (8) one can get the gain use the phase margin criterion and other visualizations tools
dependency √ to study how the complete system reacts to feedback delays
B =2 mK−b, (9) andsignalfiltering.Phasemarginistheadditionalphasevalue
5
above −180◦ when the magnitude plot crosses the 0 dB line with
(i.e., the gain crossover frequency). It is common to quantify
A (ω)(cid:44)Bωcos(T ω)−Ksin(T ω)+Kτ ωcos(T ω),
system stability by its phase margin. 1 d s v s
A (ω)(cid:44)Bωsin(T ω)+Kcos(T ω)+Kτ ωsin(T ω).
2 d s v s
(12)
C. Phase margin sensitivity comparison
Note that Euler’s Formula (e−jx = cosx−jsinx) has been
This subsection focuses on utilizing frequency domain
used to obtain the above results.
control method to analyze the phase margin sensitivity to The phase margin, PM (cid:44)180◦+∠P (ω ), of the plant
OL g
time delays on the distributed control architecture shown on (11), where ∠. is the angle of the argument, is
Figure 3. Different delay range scales are considered: (1) a ï ò
A (cid:104)mω (cid:105)
small range scale (1 − 5 ms) to show detailed variations, PM =atan 1g +90◦−atan g −atan[τ ω ], (13)
and (2) a larger range scale (5−25 ms) to cover practical A2g b v g
control ranges. These scales roughly correspond to embedded with ω being the gain crossover frequency [27] and A (cid:44)
g ig
and centralized computational and communication processes
A (ω ),i={1,2}.
i g
found in highly articulated robots such as [26]. Phase margin
FollowingthederivationsofAppendixA,onecanobtainthe
plots, are subsequently obtained for the controller of Equa-
sensitivity equations below expressing variations of the phase
tion (3) and shown on Figures 4 and 5 as a function of the
margin with respect to stiffness and damping delays,
natural frequency given in Equation (10) and using the gain
relationship of Equation (9). ∂PM = (cid:2)−K2(τv2ωg2+1)+KBωg M(cid:3) ωg, (14)
From Figure 4, it is noticeable that reducing either stiffness ∂Ts A12g+A22g
or damping feedback delays will increase the stability of
the controller. But more importantly, it is clearly visible that ∂PM = (cid:2)−B2ωg2+KBωg M(cid:3) ωg. (15)
phase margin behavior is much more sensitive to damping ∂T A2 +A2
d 1g 2g
servo delays (T ) than to stiffness servo delays (T ). Not
d s (cid:112) (cid:16) (cid:17)
only there is a disparity on the behavior with respect to where M = (τ ω )2+1·sin (T −T ) ω +φ is also
v g s d g
the delays, but phase margin is fairly insensitive to stiffness derived in that Appendix A.
servo delays in the observed time scales. Such disparity and
behavior is the central observation that motivates this paper B. Gain crossover sensitivity condition
and the proposed distributed control architecture. Figure 5
Fromthecontrolanalysisofthedistributedplantperformed
simulates step position response of the controller for a range
in previous sections, increasing damping delays decreases the
of relatively large stiffness delays and for two choices of
phase margin. This observation means that the sensitivity of
damping delays, a short and a long one. It becomes clear
the phase margin to damping delays must be negative, i.e.
thatreducingdampingdelaysignificantlyboostsstabilityeven
in the presence of fairly large stiffness delays. These results ∂PM
<0. (16)
emphasize the significance of implementing damping terms at ∂Td
the fastest possible level (e.g. at the embedded level) while Also from those analysis, it is observed that the phase margin
proportional (i.e. stiffness) servos can run in latency prone is more sensitive to damping than to stiffness delays. This
centralized processes. observation can be formulated as
∂PM ∂PM
< . (17)
IV. BASISFORSENSITIVITYDISCREPANCY ∂Td ∂Ts
In the previous section it was observed different behavior Let us re-organize the numerator of Equation (15) to be
of the controller’s phase margin depending on the nature of written in the alternate form
delay.Dampingdelayseemstoaffectmuchmorethesystem’s ∂PM = [−Bωg+KM]Bωg2. (18)
phase margin than stiffness delay. This section will analyze ∂T A2 +A2
d 1g 2g
this physical phenomenon in much more detail and reveal the
An upper bound of the above equation occurs when the
conditions under which this disparity occurs. (cid:16) (cid:17)
maximal condition sin (T −T )ω +φ =1 is met, i.e.
s d g
î ó
A. Equations expressing phase margin sensitivity to delays ∂PM −Bωg+K(cid:112)(τvωg)2+1 Bωg2
≤ . (19)
Detailed mathematical analysis is developed to find further ∂T A2 +A2
d 1g 2g
physical structure for the causes of stability discrepancies
Based on the above inequality, (16) is met if the following
betweendampingandstiffnessdelays.Letusre-visittheopen
gain crossover sensitivity condition is met,
loop transfer function of Equation (6). The resulting open
K
loop transfer function, including the low pass velocity filter
ω > . (20)
g (cid:112)
of Equation (4), in the frequency domain (s=jω) is B2−K2τ2
v
jA (ω)+A (ω) The above equation is only a sufficient condition for fulfilling
P (ω)= 1 2 , (11)
OL jω(jmω+b)(jτ ω+1) Condition (16). Obtaining a closed form solution for that
v
6
Fig. 5: Comparison of Step Response with Slow and Fast Damping Servos. The figures above compare the effects of
implementing damping feedback on slow or fast servo processes. The simulations are performed on the same actuator used
in the experimental section, Section V. The top row depicts damping feedback implemented with delays of 15 ms while the
bottom row depicts a faster damping servo with delay of only 1 ms. For both rows, various stiffness delays are identically
used ranging from 5 ms to 25 ms. Subfigures (a) and (d) show simulations of the phase margin as a function of the natural
frequency, which in turn is a function of the feedback stiffness gain. The first point to notice here is that the phase margin
values for subfigure (a) are significantly lower than for (d) due to the larger damping delay. Secondly, both (a) and (d) show
small variations between the curves, corroborating the small sensitivity to stiffness delays that will be studied in Section IV.
Corresponding step responses are shown along for various natural frequencies. The key observation here is that embedded
damping greatly improves the stability of the controller despite large stiffness delays.
condition would be very complex due to the presence of Using the plant (11), it can be shown that the above equation
trigonometric terms. Therefore, the remainder of this section resultsintheequality(seetheAppendixBforthederivations),
is to study under what circumstances Condition (20) holds.
At the same time, Inequality (17) can be re-written in the (Bωg)2+K2(τv2ωg2+1)−2KBωgM
form =ω2(cid:16)(ω m)2+b2(cid:17)(cid:16)τ2ω2+1(cid:17). (23)
g g v g
∂PM −∂PM = (cid:2)−B2ωg2+K2(τv2ωg2+1)(cid:3)ωg <0, (21)
∂T ∂T A2 +A2 The above equation is intractable in terms of deriving a
d s 1g 2g
closed loop expression of the gain crossover frequency. To
where it has been subtracted the right hand sides of Equa- tackle a solution our study introduces transformations of
tions (14) and (15) for the derivation. Notice that in that the parameters and numerically derives parameter ranges for
subtraction the sine functions cancel out. Coincidentally, the which Condition (20) holds. Let us start by creating a new
above inequality is also met if the gain crossover sensitivity variable that allows to write (20) as an equality,
condition (20) is fulfilled. In other words, that condition is
sufficient to meet both Inequalities (16) and (17). K
δ ∈[−1,∞) s.t. ω =(1+δ) . (24)
g (cid:112)
B2−K2τ2
v
C. Servo breakdown gain rule
Thus, demonstrating the gain crossover sensitivity condi-
To validate the gain crossover condition (20), our study tion (20) is equivalent to demonstrating that δ >0. Rewriting
solves for the gain crossover frequency, which consists of the Equation (9) as K =(B+b)2/4m and substituting K in the
frequency at which the magnitude of the open loop transfer above equation, (24) can be further expressed as
function is equal to unity, i.e.
(B+b)2
ω =(1+δ) . (25)
|POL(ωg)|=1. (22) g (cid:112)16B2m2−(B+b)4τ2
v
7
Dividing Equation (23) by a new term K2UV, with U (cid:44)
τ2ω2+1, and V (cid:44) B2ω2/K2, while substituting ω on the
v g g g
right hand side of Equation (23) by Equation (25), and using
M as shown in Equation (37), Equation (23) becomes
(cid:16) (cid:17)
2sin (T −T )ω +φ
1 1 s d g
+ − √
U V U ·V
Å ã
(cid:0)1+δ(cid:1)2(cid:0)B+b(cid:1)4 b 2
= + . (26)
16B4−B2(B+b)4τ2/m2 B
v
Using Equation (24) it can be further demonstrated that V =
(τ ω )2+(1+δ)2.ThusU andV areonlyexpressedinterms
v g
of (τ ω )2. To further facilitate the analysis, let us introduce
v g
three more variables
Å ã
α(cid:44)sin (T −T )ω +φ ∈[−1,1], (27) Fig. 6: Controller Values Meeting the Gain Crossover
s d g
Sensitivity Condition. The surfaces above show the range of
β ∈(0,∞) s.t. B =βm, (28) feedback parameters that meet the gain crossover sensitivity
condition of Equation (20). δ > 0 represents the excess gain
γ ∈(0,∞) s.t. B =γb. (29)
ratio by which the condition is met. γ > 0 represents the
Notice that α can be interpreted as an uncertainty, β is the ratio between damping feedback gain and passive damping.
ratio between damping gain and motor drive inertia and γ is β ∈ [10,400] is chosen to cover a wide range of actuator
theratiobetweendampinggainandmotordrivefriction.Using parameters.Thesurfacesabovedemonstratethatawiderange
these variables, (26) simplifies to of practical gains, γ, meet the aforementioned gain crossover
√ sensitivity condition. The values of the above surfaces are
U +V −2α U ·V (cid:0)1+δ(cid:1)2(cid:0)1+γ)4 1
solved by numerically identifying the smallest real root of
= + . (30)
U ·V 16γ4−(1+γ)4β2τ2 γ2 Equation (30). In the bottom right surface, it can be seen that
v
δ > 0 for γ > 2, meaning that the gain crossover sensitivity
Using Equations (25), (28) and (29), the term (τ ω )2 ap-
v g
condition is met if the ratio between damping feedback gain
pearing in the variables U and V on Equation (30) can be
and passive damping is larger than two.
expressed as
(cid:0)1+δ(cid:1)2(cid:0)1+γ)4
(τvωg)2 =β2τv216γ4−(1+γ)4β2τ2 (31) causes the phase margin to be more sensitive to damping
v delays than to stiffness delays. The threshold above can
Thus,Equation(30)doesnotcontaindirectdependencieswith therefore be interpreted as a breakdown gain rule which is
ωg and therefore can be represented as the nonlinear function sufficienttomeetthegaincrossoversensitivitycondition(20),
and from which the aforementioned phase margin sensitivity
f(α,β,γ,δ,τ )=0 (32)
v discrepancy follows.
Let us demonstrate under which conditions δ > 0, which This threshold hints towards a general rule for breaking
willimplythatEquation(20)holds.Inourlab,velocityfilters controllers down into distributed servos, as was illustrated
with τ = 0.0032s are commonly used for achieving high in Figure 3. Namely, if the maximum allowable feedback
v
performance control [25], and therefore Equation (30) will be damping gain for a given servo rate is significantly larger
solved for only that filter. Notice that it is not difficult to try than twice the passive actuator damping, then the controller’s
new values of τ when needed. Additionally, when sampling stiffness servo can be decoupled from the damping servo to a
v
Equation (30) for the values of α shown in Equation (27) we slower computational process without hurting the controller’s
observed that not only δ is fairly invariant to α but the lowest stability.
value of δ occurs for α = 1. These behaviors are omitted
here for space purposes. Therefore, as a particular solution, D. An example: analyzing real-world actuators by the servo
Equation (30) is solved for the values breakdown rule
As a means of demonstrating the utility of the breakdown
f(α=1,β,γ,δ,τ =0.0032)=0. (33)
v
gain rule of Equation (34), here we analyze several real-
The above function is solved numerically and the solution world actuation systems. Our goal is to determine whether
surface is plotted in Figure 6. As it can be seen, δ > 0 for the properties of each system make them good candidates
γ >2,allowingustostatethatusingadistributedPDfeedback for distributed control schemes with decoupled stiffness and
control law like the one in Figure 3 with the particular choice damping feedback loops.
of the filter τ = 0.0032s and with damping gains greater Table I shows actuator parameters for the Valkyrie hu-
v
than manoid and the UT-SEA actuator [28], as well as the max-
B > 2b, (34) imum feedback damping gains that are implemented in those
8
TABLE I: UT-SEA/Valkyrie Actuator Parameters
Actuator outputinertia passivedamping dampinggain ratio
Type m b B γ
UT-SEA 360kg 2200N·s/m 50434N·s/m 22.92
Valkyrie1 270kg 10000N·s/m 46632N·s/m 4.66
Valkyrie2 0.4kg·m2 15Nm·s/rad 68Nm·s/rad 4.55
Valkyrie3 1.2kg·m2 35Nm·s/rad 196Nm·s/rad 5.60
Valkyrie4 0.8kg·m2 40Nm·s/rad 145Nm·s/rad 3.61
Valkyrie5 2.3kg·m2 50Nm·s/rad 360Nm·s/rad 7.20
Valkyrie6 1.5kg·m2 60Nm·s/rad 259Nm·s/rad 4.32
actuatorstoachievemaximumimpedancecontrol.Ourlabhas
been involved in developing these two sets of actuators. In all
instances, the embedded servos had effective delays of 0.5
ms. In order to compute the maximum feedback damping
gains as a function of the previous servo rate, our recent
work [25] is used. In that work, a new rule is provided to
compute maximum feedback gains for a phase margin of 50◦
given the actuator parameters and the servo rate. Figure 8
shows,(1)picturesofvariousValkyrieactuators,(2)asurface
depictingthemaximumallowabledampinggainsasafunction
of actuator parameters, and (3) Valkyrie’s actuators mapped
into the surface. The surface is computed for effective delays
of 0.5 ms. Within the surface, it shows the line corresponding
to the breakdown gain rule of Equation (34). As can be seen,
all actuators implement feedback damping gains that were
abovethebreakdowngainboundary.Itfollowsthatthosegains
would be highly sensitive to damping servo delays.
To maintain these maximum actuator gains, servo latency
for the damping process must not be increased. However,
accordingtotheservobreakdownrule,thestiffnessservopro-
Fig. 8: Various Actuators Meeting Servo Breakdown Rule.
cesses shall be fairly insensitive to delays and therefore could
The top image shows various actuators from NASA that our
bedecoupledandimplementedinaslowercentralizedprocess.
group helped to build. The bottom image shows the surface
Such decoupling is advantageous in multi-axis robots where
of maximum allowable damping gains (computed according
centralized processes contain sensor and model information
to the gain rule described in [25]) as a function of actuator
needed for operational space control to coordinate the robot’s
parameters. γ is the feedback damping gain ratio described in
movement. In Subsection V-C we discuss such an application
Equation (29), and m and b are the output inertia and passive
for an omnidirectional mobile robotic base.
dampingofthevariousactuators.Asitisshown,themaximum
allowablegainsareabovetheservobreakdownboundary.This
V. EXPERIMENTALEVALUATION entails that stiffness servos are fairly insensitive to delays and
The proposed controller of Figure 3 is implemented in our therefore can be decoupled from the damping servos when
UTlinearrigidactuatorshowninFigure7.Thislinearpushrod needed.
actuator has an effective output inertia of m = 256 kg and
an approximate passive damping of b = 1250 Ns/m. The
and small delays are used for either or both the stiffness and
sampling period is 0.5 ms, i.e., with a 2 kHz servo rate. At
dampingloops.Thefourcombinationsofresultsareshownin
the same time, the controller is simulated by using the closed
the figure with delay values of 1 ms or 15 ms.
loop plant given in Equation (5). Identical parameters to the
real actuator are used for the simulation, thus allowing us to The first thing to notice is that there is a good correlation
compare both side by side. between the real and the simulated results both for smooth
and oscillatory behaviors. Small discrepancies are attributed
to unmodelled static friction and the effect of unmodelled
A. Step response implementation
dynamics. More importantly, the experiment confirms the an-
First, a test is performed on the actuator evaluating the ticipateddiscrepancyindelaysensitivitybetweenthestiffness
response to a step input on its position. The results are shown and damping loops. Large servo delays on the stiffness servo,
in the bottom part of Figure 7 which shows and compares corresponding to subfigures (a) and (b) have small effects on
the performance of the real actuator versus the simulated the step response. On the other hand, large servo delays on
closedloopcontroller.Varioustestsareperformedforthesame the damping servo, corresponding to subfigures (c) and (d),
reference input with varying time delays. In particular large stronglyaffectthestabilityofthecontroller.Infact,for(c)and
9
Fig.7:Step Response Experiment with Distributed Controller.Subfigures(a)through(d)showvariousimplementationson
the UT linear rigid actuator corresponding to the simulations depicted on Figure 5. Overlapped with the data plots, simulated
replicas of the experiments are also shown to validate our models. The experiments not only confirm the higher sensitivity
of the actuator to damping than to stiffness delays but also indicate a good correlation between the real actuator and the
simulations.
TABLE II: Root Mean Square Tracking Errors
(d) the results corresponding to f = 12 Hz are omitted due
n
to the actuator quickly becoming out of control. In contrast,
Experiment Simulation
StiffnessDelay
the experiment in (b) can tolerate such high gains despite the Position Velocity Position Velocity
large stiffness delay. Ts (ms) Err(rad) Err(rad/s) Err(rad) Err(rad/s)
2 0.0182 0.3866 0.0204 0.3970
20 0.0247 0.6366 0.0289 0.6332
B. Trajectory tracking with an uncertain load 30 0.0360 0.8753 0.0386 0.8178
Performance limits are explored at their fullest in the test Once more, the test confirms the predicted robustness to
shown in Figure 9. Here, an unmodelled weight of 4.5kg stiffness delays that was studied in previous sections. Increas-
is attached to the load arm which is also connected to the ing 10 times T from 2 ms to 20 ms only increases the RMS
s
pushrod actuator through a pivot. The weight is unmodelled joint position and angular velocity errors by less than two
and therefore constitutes a disturbance. By estimation, the fold as shown in Table II. By using high gains, the controller
total disturbance torque that the controller has to deal with ensures that joint tracking is accurate despite the large load
is Fd = 16.84 Nm. A trajectory with output angle variations disturbances. As shown in the previous table, the maximum
within [86◦,126◦] is designed to test the controller’s perfor- root mean square (RMS) error for the tracked joint position is
mance under the load disturbance. This trajectory is inspired 0.0182 rad ≈1◦ for stiffness delays of T =2 ms and 0.036
s
by that of a biped locomotion knee joint motion [29] during rad ≈2◦ for delays T =30 ms.
s
fast walking, with angular velocities varying between ±2.5
rad/s.
C. Distributed operational space control of a mobile base
This experiment tests the tracking performance under the
load disturbance on both the real actuator and also on a Asaconceptproofofourdistributedarchitectureonamulti-
numerical simulation of the controller model depicted in axis mobile platform, a Cartesian space feedback operational
Figure 3. Disturbance forces for the numerical simulation are space controller (OSC) [30] is implemented on an omnidi-
applied based on the position of the arm and considering rectionalmobilebase.Theoriginalfeedbackcontrollercanbe
only gravitational effects. Judging from the visualization of foundin[31]whichwasimplementedasacentralizedprocess
theerrorsinthatfigureandtherootmeansquareoftheerrors with no distributed topology at the time. The mobile base
depicted in Table II, there is a good correlation between the is equipped with a centralized PC computer running Linux
real experiment and the simulation values for both the joint with the RTAI realtime kernel. The PC connects with three
positions and angular velocities. actuator processors embedded next to the wheel drivetrains
10
Fig. 9: Trajectory Tracking Experiment under Load Disturbances with Distributed Controller. The figures on the top
row show snapshots of the testbed with an unmodelled load of 4.5 kg. The plots show trajectory tracking performance for a
small damping delay, T = 2 ms and various stiffness delays ranging from T = 2 ms to T = 30 ms. Trajectory tracking
d s s
errors on the bottom left remain relatively small despite the large stiffness delays, confirming the advantages of implementing
damping feedback in a fast computational processes. A simulated experiment is also shown on the bottom right confirming
good correlation between the real and simulated performance.
viaEtherCatserialcommunications.Theembeddedprocessors error.ToimplementtheCartesianstiffnessfeedbackprocesses
do not talk to each other. The high level centralize PC has a in both architectures, the Cartesian positions and orientations
roundtrip latency to the actuators of 7ms due to process and of the mobile base on the ground are computed using wheel
buscommunications,whilethelowlevelembeddedprocessors odometry and according to the methods discussed in [31].
have a servo rate of 0.5ms. Notice that 7ms is considered too To achieve the highest stable stiffness gains, the following
slow for stiff feedback control. To accentuate even further the procedure is followed: (1) first, Cartesian stiffness gains are
effect of feedback delay on the centralized PC, an additional adjusted to zero while the damping gains in either Cartesian
15ms delay is artificially introduced by using a data buffer. space (COSC) or joint space (DOSC) - depending on the
Thus, the high level controller has a total of 22 ms feedback controller architecture - are increased until the base starts vi-
delay. brating;(2)theCartesianstiffnessgains,oneitherarchitecture,
are increased until the base starts vibrating or oscillating; (3)
An operational space controller (OSC) is implemented in a desired Cartesian circular trajectory is commanded to the
the mobile base using two different architectures. First, the base and the position and velocity tracking performance are
controller is implemented as a centralized process, which will recorded.
be called COSC, with all feedback processes taking place Based on these experiments, DOSC was able to attain a
in the slow centralized processor and none in the embedded maximum Cartesian stiffness gain of 140N/(m kg) compared
processors.Thisimplementationisthesameasitwasdonefor to30N/(mkg)forCOSC.Thisresultmeansthattheproposed
ourpreviousworkin[31].Inthiscase,themaximumstiffness distributedcontrolarchitectureallowedtheCartesianfeedback
gains should be severely limited due to the effect of the large process to increase the stiffness gains by 4.7 times with
latencies.Secondadistributedcontrollerarchitectureisimple- respect to the centralized controller implementation. In terms
mentedinspiredbytheoneproposedinFigure3butadaptedto of tracking performance, the results are shown in Figure 10.
our desired operational space controller, which will be called BothCartesianpositionandvelocitytrackinginDOSCaresig-
DOSC. In this version, the Cartesian stiffness feedback servo nificantlymoreaccurate.Theproposeddistributedarchitecture
is implemented in the centralized PC in the same way than in reduces Cartesian position root mean error between 62% and
COSC, but the Cartesian damping feedback servo is removed 65% while the Cartesian velocity root mean error decreases
from the centralized process. Instead, our study implements between 45% and 67%.
dampingfeedbackinjointspace(i.e.proportionaltothewheel
velocities)ontheembeddedprocessors.Aconceptualdrawing
VI. CONCLUSIONSANDDISCUSSION
of these architectures is shown in Figure 10. The metric
used for performance comparison is based on the maximum Themotivationforthispaperhasbeentostudythestability
achievable Cartesian stiffness feedback gains, the Cartesian and performance of distributed controllers where stiffness and
position tracking error, and the Cartesian velocity tracking damping servos are implemented in distinct processors. These