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Dynamics Modeling of Gear Transmissions with Asymmetric Load-Dependent Friction

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div>This paper proposes a new concise mathematical model of gear transmission dynamics with asymmetric load-dependent friction. It is of the form of a differential-algebraic inclusion (DAI) characterized by some parameters, including input-side and output-side asymmetry coefficients. The presented model properly captures the static friction and even the non-backdrivability. It is applicable to different classes of transmissions, such as leadscrew transmissions, worm gear transmissions, and spur gear transmissions. The DAI representation is extended into a multi-dimensional representation for articulated rigid-body systems driven through joint transmissions. Moreover, simulation algorithms are derived through the implicit Euler discretization. Some simulation examples illustrate the validity of the presented representations. </div
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Dynamics Modeling of Gear Transmissions with AsymmetricDynamics Modeling of Gear Transmissions with Asymmetric
Load-Dependent FrictionLoad-Dependent Friction
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Kikuuwe, Ryo (2022): Dynamics Modeling of Gear Transmissions with Asymmetric Load-Dependent Friction.
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Dynamics Modeling of Gear
Transmissions with Asymmetric
Load-Dependent Friction
Ryo Kikuuwe1
This paper proposes a new concise mathematical model of gear transmission dynamics with asymmetric load-
dependent friction. It is of the form of a differential-algebraic inclusion (DAI) characterized by some parameters,
including input-side and output-side asymmetry coefficients. The presented model properly captures the static friction
and even the non-backdrivability. It is applicable to different classes of transmissions, such as leadscrew transmissions,
worm gear transmissions, and spur gear transmissions. The DAI representation is extended into a multi-dimensional
representation for articulated rigid-body systems driven through joint transmissions. Moreover, simulation algorithms
are derived through the implicit Euler discretization. Some simulation examples illustrate the capability of the presented
simulation algorithm to reproduce load-dependent asymmetric frictional behaviors, which cannot be reproduced by
conventional load-independent joint friction models.
Nonsmooth mechanics, Gear, Friction, Backdrivability
1 Introduction
Gear transmissions are extensively used in robotic mech-
anisms to extract low speeds and large forces from com-
mercial servo actuators. The modeling of friction in gear
transmissions is an important issue for the control and
simulation of various robotic systems. Controller design
considering the effects of friction is necessary for realizing
highly efficient and dynamic motion (Wensing et al. 2017;
Sim and Ramos 2021). Appropriate friction models are also
beneficial to force control (Aung and Kikuuwe 2017), of
which the stability is enhanced by friction compensation
(Aung et al. 2015;Iwatani and Kikuuwe 2017).
An important feature of the internal friction of gear
transmissions is the asymmetry (Wang and Kim 2015;Wang
2012). Specifically, the effect of internal friction is usually
smaller when the torque is applied to the input shaft than
when it is applied to the output shaft. In an extreme case
where the torque applied to the output shaft is blocked by
the internal friction, the transmission is said to be non-
backdrivable or self-locking (Plooji et al. 2005). Most high-
ratio gear transmissions do not have high backdrivability,
which means that there is a large frictional loss when it is
driven from the output shaft. The asymmetric frictional loss
is also understood in terms of the load dependency (Dohring
et al. 1993) because the effect of friction depends on the
balance between the torques on the input and output shafts.
Previous modeling approaches of the internal friction
considering the asymmetry are based on the Coulomb
friction model, with which the friction force between
components is proportional to the normal force on the contact
surface. Researchers investigated the friction of leadscrew
transmissions (Mablekos-Alexiou et al. 2021;Dupont 1990),
worm gear transmissions (Dohring et al. 1993;Mablekos-
Alexiou et al. 2021;Yeh and Wu 2009;May et al. 2000),
and spur gear transmissions (Wang and Kim 2015;Wang
2012;Diez-Ibarbia et al. 2018). In many of these previous
approaches, different expressions must be switched between
the forward-driving and backdriving cases, which are defined
by the direction of the power flow. Some researchers
(Matsuki et al. 2019;Yeh and Wu 2009;Wang and Kim
2015;Mablekos-Alexiou et al. 2021;Sim and Ramos 2021)
use different inertia values between the forward-driving and
backdriving cases. Most previous methods cannot handle the
static friction state, where the direction of the power flow
cannot be defined.
This paper proposes a new mathematical representation
of the dynamics of gear transmissions with asymmetric
frictional loss. The proposed representation is of the
form of a differential-algebraic inclusion (DAI)*with a
Lagrange multiplier representing the internal force. The
frictional characteristics of the transmission are represented
by two parameters, which are named as input-side and
output-side asymmetry coefficients. The DAI representation
is then extended into a multi-dimensional representation
for articulated rigid-body systems driven through joint
transmissions. This paper also derives simulation algorithms
based on the DAI through the implicit Euler discretization.
Some numerical examples illustrate its application to robotic
1Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8527, Japan.
In this paper, a DAI stands for a system of equations that involves at least
one differential inclusion (DI) and at least one variable that is algebraically
determined. A DI stands for a pair of mathematical expressions connected
by “” instead of “=.”
The remainder of this paper is organized as follows.
Section 2provides some mathematical preliminaries.
Section 3presents the main result of the work, which is
the new representation of the dynamics of frictional gear
transmissions. Section 4investigates the dynamic and static
properties of the presented representations. Section 5shows
the extension of the model to robotic systems having multiple
degrees of freedom. Section 6shows some numerical
examples. Section 7provides concluding remarks.
2 Mathematical Preliminaries
Throughout this paper, Rstands for the set of all real
numbers, B[1,1] Ris the closed unit ball in R,
0denotes the zero vector or zero matrix of appropriate
dimensions, convXstands for the convex hull of X, and
2Xis the set of all subsets of X. The notation f:X0→ X1
means that fis a single-valued function from the set X0to
the set X1, and f:X0X1means that fis a set-valued
function from the set X0to the set X1, i.e., f:X0X1
stands for f:X02X1. When a function f:X0→ X1or
f:X0X1is called with a set Xs⊂ X0as its argument, it
should be read as follows:
f(Xs) = [
f(x) = {f(x)∈ X1|x∈ Xs}⊂X1.(1)
This implies that, with a continuous and monotone function
f:RR, one has f(B)=[f(1), f (1)].
We define the set-valued signum function sgn : RB
and the projection function proj : 2R×RRas follows:
sgn(x)x/|x|if x̸= 0
Bif x= 0 (2)
proj(X, x)argmin
ξ∈X |ξx|.(3)
Note that, for ab,
proj([a, b], x) = max(a, min(x, b)) (4)
holds. With a continuous and monotone function f:B → R,
the following statement holds true:
yx∈ −f(sgn(y)) y=xproj(f(B), x).(5)
It can be proven by investigating all possible signs of yas
yx∈ −f(sgn(y))
(0 < y =xf(1)) (0 > y =xf(1))
(0 = yx[f(1), f (1)])
(y=xf(1) x>f(1))
(y=xf(1) x<f(1))
(y= 0 x[f(1), f (1)])
y=xproj(f(B), x).(6)
The relation (5) is convenient to convert an algebraic
inclusion involving the set-valuedness into an ordinary
equation involving only single-valued functions.
With a vector xRn,xstands for the 2-norm of x
(i.e., x=xTxR) and |x|stands for the element-wise
Figure 1. A model of a gear transmission with asymmetric
absolute-value operator (i.e., |x|= [|x1|,·· · ,|xn|]TRn).
We also use the element-wise signum function sgn :Rn
Bndefined as follows:
sgn(x)[s1,··· , sn]T∈ Bn|s1sgn(x1)∧ · · ·
Note that Bnis an n-dimensional hypercube, not a unit
ball. For brevity, this paper uses [x]diag(x)Rn×nfor
xRn. Note that the following relation holds:
With a matrix XRm×n,σmax(X)stands for the
maximum singular value of X. The following statement
σmax(X) = max
It should also be noted that the infinity norm of a vector x
satisfies x=σmax([x]).
3 DAI Representation of a Gear
3.1 Main Result
As a model of a gear transmission having asymmetric load-
dependent friction, let us consider the system illustrated in
Fig. 1. In this system, two carts, representing the input and
output shafts of a transmission, slide against each other. The
contact between the carts is assumed to be bilateral, i.e., the
normal force can be either positive or negative. The moving
directions of the carts are at the angles of θuand θvfrom the
tangential direction of the contact surface. Similar models
have also been investigated in previous work, e.g., (Brogliato
2016;Dohring et al. 1993;Matsuki et al. 2019;Sim and
Ramos 2021;Mablekos-Alexiou et al. 2021), often being
intended as a simplified model of general gear transmissions.
It should be noted that this paper does not consider the
backlash in contrast to (Matsuki et al. 2019).
As illustrated in Fig. 1, let ˆmbe the inertia of the input
cart, Mbe the inertia of the output cart, uRbe the input
velocity, vRbe the output velocity, ˆ
fuRbe the force
acting on the input cart, and fvRbe the force from the
output cart to the external load. Without loss of generality, we
assume that θu(0, π/2] and θv(0, π/2]. The geometric
Kikuuwe 3
constraint between the input and output carts can be written
as follows:
v=u/κ, ˙v= ˙u/κ (10)
where κsin θv/sin θu. The equations of motion of the
carts can be written as follows:
fuλtcos θuλnsin θu(11a)
M˙v=fvλtcos θv+λnsin θv.(11b)
Here, [λt, λn]Tis the contact force acting from the input cart
to the output cart. The normal component λnis determined
to enforce the constraint (10). The tangential component λt
is determined by the dry friction at the contact surface as
where µis the friction coefficient at the contact surface.
Throughout this paper, it is assumed that there is no
difference between the maximum static friction coefficient
and the kinetic friction coefficient. The expression (12)
states that the tangential component λtis within the interval
[µ|λn|, µ|λn|]in the static friction and is on the boundary
of the interval in the kinetic friction.
Solving (11) with respect to λtand λnconsidering the
constraint (10) yields the following expression:
λt=sin θu
sin(θu+θv)(κ2ˆm+M) ˙vκˆ
λn=sin θu
tan θvM
tan θu˙vκˆ
tan θvfv
tan θ!.(13b)
By setting mκ2ˆm,fuκˆ
fu,γuµ/ tan θv,γv
µ/ tan θu, and λµλnsin(θu+θv)/sin θuand considering
(12), one can rewrite (13) as follows:
(m+M) ˙vfufv− |λ|sgn(v)(14a)
(γumγvM) ˙v=γuκfu+γvfvλ. (14b)
Note that (14) is a DAI and it is the equation of motion of the
system of Fig. 1.
The DAI representation (14) is the main result of this
paper. Although (14) is derived from the specific model in
Fig. 1, it is general enough to describe different types of
gear transmissions as will be shown in subsequent sections.
The system (14) can be seen as a nonsmooth differential-
algebraic system with the state v, the inputs {fu, fv}, and the
Lagrange multiplier λ, and is characterized by the following
four parameters:
γuR: the input-side “asymmetry coefficient,
γvR: the output-side “asymmetry coefficient,
m(0,): the input-side inertia, and
M(0,): the output-side inertia.
It should be noted that the system (14) is invariant to the
simultaneous sign changes of the pair {γu, γv}because it
only reverses the sign of λand does not affect ˙v. Thus,
without loss of generality, we can assume that γu+γv0.
Figure 2. Leadscrew transmission.
By taking a closer look on the DAI (14), one may call
(14a) as a visible dynamics that determines the acceleration
˙vand (14b) as a hidden dynamics that determines the
internal force λ. When γu=γv= 0, i.e., when there is
no friction between the carts, the hidden dynamics (14b)
vanishes and the whole system (14) reduces to an ordinary
differential equation (m+M) ˙v=fufv, which is quite
a conventional representation. When γu̸= 0 or γv̸= 0,
the hidden dynamics (14b) determines the force λand
it influences the visible dynamics (14a). Further physical
intuitions regarding the coefficients γuand γvand the overall
dynamics is not easy to draw directly from the DAI (14).
Section 4will present more in-depth analyses by converting
(14) into several different expressions.
Expressions similar to (14) have also been presented in
(Dupont 1993,1992;Dohring et al. 1993), but it has not
been clarified how the inertia-like coefficient in the hidden
dynamics is determined and how the external force acts
in the hidden dynamics. Moreover, they are written as an
ordinary differential equation (ODE) describing only the
kinetic friction state (i.e., only the case of v̸= 0). The
treatment of the static friction state in (Dupont 1993,1992;
Dohring et al. 1993) is based on some heuristics, not on
differential equations.
It is not new to employ DIs and DAIs to describe
mechanical systems involving the static friction (Acary and
Brogliato 2008;Brogliato 2016;Blumentals et al. 2016;
Kikuuwe and Brogliato 2017;Kikuuwe 2019). In fact,
by setting θu+θv=π/2, the DAI (14) reduces to the
representation provided in (Brogliato 2016, Section 5.5.5)
except that the constraint between the carts is bilateral
instead of unilateral. To the best of the author’s knowledge,
however, representations of the form of the DAI (14) for
general geared transmissions are not found in the literature.
3.2 Example 1: Leadscrew Transmission
The proposed DAI representation (14) describes some
different types of transmissions. The following sections
present some examples.
A leadscrew transmission such as those illustrated in Fig. 2
can be seen as one of the simplest examples. Let us assume
that its lead angle is θu, the screw radius is r, and the friction
coefficient is µ. Let ˆmbe the moment of inertia of the input
screw and Mbe the inertia of the output slider. Let ube
the angular velocity of the input screw and vbe the linear
velocity of the output slider. Then, the system of Fig. 2
becomes equivalent to the system of Fig. 1with uand θv
being replaced by ru and π/2θu, respectively. (See also
(Mablekos-Alexiou et al. 2021, Fig. 2).) Therefore, one can
Figure 3. Worm gear transmission, adapted from (Yeh and Wu
see that the dynamics of a leadscrew transmission in Fig. 2
can be described by the proposed DAI representation (14)
with m=κ2ˆm,fu=κˆ
fu,κr/ tan θ,γuµtan θ, and
γvµ/ tan θ.
3.3 Example 2: Worm Gear Transmission
As another example, let us consider a worm gear
transmission illustrated in Fig. 3, which has been investigated
also in (Yeh and Wu 2009;May et al. 2000). Partially
borrowing Yeh and Wu’s (2009) notations, let w>0
and g>0be the pitch radii of the worm and the gear,
respectively, ϕn(0, π/2) be the pressure angle, and λl
(0, π/4] be the lead angle. Other symbols, u,v,ˆ
and M, are defined in the same manner as in Section 3.1 and
Fig. 1.
Yeh and Wu (2009) have derived two sets of equations
of motion of this system; one is for the “left engagement”
case and the other is for the “right engagement” case. These
cases correspond to the cases where the force from the worm
to the gear in Fig. 3is rightward and leftward, respectively.
Borrowing their notations, let Wn>0be the magnitude of
the normal component of the contact force. Then, by setting
λn=Wnin the “left engagement” and λn=Wnin the
“right engagement,” Eqs. (2) and (3) in (Yeh and Wu 2009)
can be unified into the following form:
fu+ (λncos ϕnsin λlWfcos λl)w(15a)
M˙v=fv(λncos ϕncos λl+Wfsin λl)g(15b)
where Wfis the tangential component of the contact force.
The Coulomb friction law results in Wfµ|λn|sgn(u), and
the geometric constraint can be written as v=u/κ where
κ=g/(wtan λl). Thus, solving (15) results in
λt= ˆm˙u+M˙vtan λlˆ
futan λlfv(16a)
λn= ˆm˙utan λlM˙vˆ
futan λlfv.(16b)
One can see that (16) reduces to the proposed DAI
representation (14) by setting
wtan λl
, γu=µtan λl
cos ϕn
, γv=µ
tan λlcos ϕn
Figure 4. Spur-gear transmission.
The model derived by Yeh and Wu (2009) consists of
four different equations of motion that must be appropriately
switched according to the engagement (left or right) of the
gears and the direction (positive or negative) of the velocity
u. In contrast, the presented formalism (14) is a single unified
expression that always holds, even in the case of static
3.4 Example 3: Spur-Gear Transmission
We can also see that the proposed DAI representation (14)
can be applied to describe spur-gear transmissions. Let us
consider a spur gear transmission of Fig. 4, where Ruand Rv
are the pitch radii of the input and output gears, respectively,
and αis the pressure angle. When the origin is placed at the
pitch point and the x-axis is aligned to the line connecting the
rotation centers of the gears, the coordinates of the input shaft
and the output shaft are [Ru,0]Tand [Rv,0]T, respectively.
As a property of the spur gear geometry, the contact points
are located on the two lines making the angle αwith the y
axis at the origin, i.e., their coordinates are ξ[sin α, cos α]T
or ξ[sin α, cos α]Twhere ξR. The unit vectors along
the tangential and normal directions at a contact point are
[cos α, sin α]Tand [sin α, cos α]T, respectively.
Let us consider the case where there is only one contact
point ξ[sin α, cos α]T. Let the contact force from the input
gear to the output gear be denoted by [λx, λy]T. Then, the
equations of motion of the gears are written as follows:
fu[ξsin α+Ru, ξ cos α]T,[λx, λy]T(18a)
M˙v=fv+[ξsin αRv, ξ cos α]T,[λx, λy]T(18b)
where [a, b]T,[c, d]T=ad bc. The positive directions of
uand vare clockwise and counter-clockwise, respectively.
The tangential and normal components {λt, λn}of the
contact force is given as follows:
λn=cos αsin α
sin αcos αλx
Eliminating {λx, λy}from (18) and (19) results in the
λt=(m+M) ˙vκfu+fv
ξ(1 + κ)(20a)
λn=(γumγvM) ˙vγufuγvfv
µ|ξ|(1 + κ)(20b)
Kikuuwe 5
where mκ2ˆm,fuκˆ
fu,κRv/Ru, and
γusgn(ξ)µtan α+µ|ξ|
Rvcos α(21a)
γvsgn(ξ)µtan α+µ|ξ|
Rucos α.(21b)
Due to the Coulomb friction law, λtand λnare related by
the following:
λt∈ −µ|λn|sgn(ξ)sgn(u).(22)
Considering (20) and (22), we can obtain the proposed DAI
representation (14) with λ=µ|ξ|(1 + κ)λnand {γu, γv}
defined in (21).
At every given instant, the equation of motion of this
system is the DAI (14) but the coefficients {γu, γv}in (21)
depend on the location of the contact point, which varies as
the gears rotate. The number of contact points also varies
between one and two as the gears rotate. By integrating
{γu, γv}in (21) with respect to ξover a full contact cycle,
one may obtain average values of the coefficients {γu, γv}
with some assumptions on the gear teeth geometry in a
similar manner to those in (Wang and Kim 2015;Wang 2012;
Diez-Ibarbia et al. 2018;Yada 1997). Another approach to
obtain {γu, γv}may be from empirical measurements of the
torques in both shafts in the forward-driving and backdriving
cases, as discussed in the upcoming Section 4.2.
4 Properties and Use of DAI (14)
This section discusses some properties of the nonsmooth
differential-algebraic system (14).
4.1 Statics: Backdrivability and
Let us consider the case where the system (14) is in the static
equilibrium, i.e., v= 0 and ˙v= 0. This situation can also be
said to be the static friction state. In this case, the DAI (14)
reduces to the following:
fufv∈ |γufu+γvfv|B.(23)
If one set fu= 0 with (23), it results in fv∈ |γvfv|B, which
holds true with any fvRif |γv| ≥ 1. This means that,
when fu= 0, any large force fvapplied to the output shaft
does not produce any movements at the static friction state
if |γv| ≥ 1. That is, the transmission is non-backdrivable
(or self-locking) if |γv| ≥ 1. In the same light, one may
say that the transmission is non-forward-drivable if |γu| ≥
1, although it is not the case with most practical gear
transmissions. Fig. 5shows the map of the backdrivability
and the forward-drivability in the γu-γvplane.
From some straightforward derivations, (23) can be
rewritten as follows:
((1 + γv)fv(1 γu)fu)×
((1 + γu)fu(1 γv)fv)0.(24)
Neglecting the singular case of 1 + γv= 0 1 + γu= 0,
one can rewrite (24) as follows:
(1 + γu)(1 + γv)(fvηffu)(fuηbfv)0(25)
Figure 5. Backdrivability and forward-drivability in relation to
the asymmetry coefficients γuand γv.
Figure 6. The set of [fu, fv]T(represented by gray regions)
that satisfies (25). When [fu, fv]Tbelongs to the gray regions,
the system (14) can be in the static friction state. (BD =
backdrivable, nBD = non-backdrivable, FD = forward-drivable,
nFD = non-forward-drivable.)
1 + γv
, ηb1γv
1 + γu
Fig. 6illustrates the set of [fu, fv]Tthat satisfy the condition
(25) of the static friction with different values of γuand
γv. Note that the system is non-backdrivable when the set
(the gray region) includes the fvaxis, and is non-forward-
drivable when the gray region includes the fuaxis.
We can also see from Fig. 6that being both non-
forward-drivable and non-backdrivable does not mean that
the transmission is non-drivable, but means that it can be
driven only if the input and output torques are applied in
appropriate ratios. For example, in the cases of “nBD &
nFD” in Fig. 6, the system is always in the static friction
when either fuor fvis zero but the pairs {fu, fv}within the
white regions result in the kinetic friction.
4.2 Statics: Forward/Backward Efficiencies
In the case where |γu|<1∧ |γv|<1, i.e., where the
transmission is both backdrivable and forward-drivable, we
have ηf(0,1) and ηb(0,1) from (26). In this case, the
condition fv=ηffu, which is one of the critical conditions
of (25), takes place in the marginally forward-driving
situation where the input shaft is driven with the torque fu
and the output shaft applies the torque fvto the external
load. Thus, ηfcan be interpreted as the forward efficiency
of the transmission. Conversely, the other critical condition
fu=ηbfvof (25) takes place in the marginally backdriving
situation where the transmission is driven from the output
shaft. Thus, ηbcan be interpreted as the backward efficiency
of the transmission. In most practical cases, backdrivable
and forward-drivable geared transmissions satisfy 0< ηb<
ηf<1, which results in |γu|<|γv|<1considering (26).
In practice, the values of the forward and backward
efficiencies {ηf, ηb}can be obtained by some experiments or
are provided by the manufacturers. Because the efficiencies
{ηf, ηb}are related to the asymmetry coefficients {γu, γv}
through (26), one can obtain the values of {γu, γv}from the
values of {ηf, ηb}by calculating the following equations:
, γv=12ηb+ηbηf
which is the solution of (26) with respect to {γu, γv}.
In case where the transmission is non-backdrivable or
non-forward-drivable, i.e., where |γu|>1∨ |γv|>1, the
quantities ηfand ηbcan have values outside the interval
(0,1). In such a case, we do not have simple interpretations
of these coefficients, except that they are the slopes of the
lines in Fig. 6.
4.3 Dynamics: DI Representation
The dynamics of the nonsmooth differential-algebraic
system (14) is now investigated. We can see that (14) can
be equivalently rewritten as follows:
(m+M) ˙v=fufvs|λ|(28a)
Lemma 1in Appendix Asuggests that (28) is equivalently
rewritten as follows:
(m+M) ˙v=fufvs|λ|(30a)
λλ(ϕ(fu, fv,ˆρ),ˆρs)(30b)
Figure 7. Set-valued function λdefined in (31). The first and
fourth lines of (31) correspond to the thick black lines, the
second line of (31) corresponds to the green surface, and the
third line of (31) corresponds to the blue and red surfaces. If
ϕ= 0 ∨ |ρ|<1,λ(ϕ, ρ)is single-valued. If ϕ̸= 0 ∧ |ρ|>1,
λ(ϕ, ρ)is either empty-valued or dual-valued. The single-valued
function λsdefined in (35) consists only of the green and blue
surfaces in this figure.
where λ:R×RRand ϕ:R×R×RRare defined
as follows:
λ(ϕ, ρ)
0if ϕ= 0
ϕ/(1 sgn(ϕ)ρ)if ϕ̸= 0 ∧ |ρ|<1
1 + ρif ϕρ < 0∧ |ρ|>1
ϕ/2if ϕρ < 0∧ |ρ|= 1
if ϕρ > 0∧ |ρ| ≥ 1
ϕ(fu, fv, ρ)γufu+γvfvρ(fufv).(32)
The function λis illustrated in Fig. 7.
Considering the definition (31) of λand the relations (30b)
and (30c), one can see that there exists a unique solution
λof (30b) if |ˆρ|<1. Note that |ˆρ|<1is satisfied if the
transmission is both backdrivable and forward-drivable, i.e.,
|γu|<1∧ |γv|<1. Meanwhile, if |ˆρ|>1, the solution λof
(30b) may be non-existent or non-unique.
Eliminating λand sfrom (30) yields the following DI:
(m+M) ˙vfufvΦ(sgn(v),ϕ(fu, fv,ˆρ),ˆρ)(33)
where Φ : B × R×RRis defined as follows:
Φ(s, ϕ, ρ)sλ(ϕ, ρs).(34)
Note that the DI representation (33) is an equivalent form
of the proposed DAI representation (14). The DI (33) can
be seen as less suited for physical interpretation but more
suited for further mathematical analysis than the DAI (14).
If |ˆρ|<1, the right-hand side of (33) is single-valued as
long as v̸= 0. If |ˆρ|<1and v= 0,Φ(sgn(v), ϕ, ˆρ)is the
closed interval [Φ(1, ϕ, ˆρ),Φ(1, ϕ, ˆρ)] because Φ(s, ϕ, ˆρ)is
a continuous, monotone, and bounded function of s∈ B, as
stated in Lemma 2in Appendix A.
Remark 1 (Existence of solution of the DI (33)). If
|ˆρ|<1, the right-hand side of (33) is outer semicontinuous
The outer semicontinuity is often referred to as upper semicontinuity
(Smirnov 2002, p. 32) in the set-valued sense.
Kikuuwe 7
(cf. (Hiriart-Urruty and Lemar´
echal 2001, p. 14)) with
respect to v, and is compact and convex for all {v, fu, fv}.
Therefore, according to Theorem 4.7 of (Smirnov 2002), the
DI (33) has an absolutely continuous solution with respect to
vif fuand fvare measurable functions of time t.
Remark 2 (Uniqueness of solution of the DI (33)). If |ˆρ|<
1, the right-hand side of (33) is a locally bounded maximal
monotone (cf. (Acary and Brogliato 2008, Section 2.1.2))
function of v. Therefore, according to Theorem 1 in (Cellina
1995), the DI (33) has unique solutions from almost all initial
values of v.
In the ill-posed case, i.e., in the case of |ˆρ|>1, the right-
hand side of the DI (33) is empty-valued if ˆρvϕ(fu, fv,ˆρ)>
0and dual-valued otherwise. These empty-valuedness and
dual-valuedness have also been discussed in previous work
(Dupont 1993,1992). Such cases are often referred to as
dynamic jamming or dynamic wedging (see, e.g., Section 3.2
of (Blumentals et al. 2016)). One may also say that the rigid-
body assumption or the Coulomb-friction assumption fails to
hold in this case. Some relevant discussions can be found in
(Brogliato 2016, Section 5.5.5).
Although a thorough mathematical analysis of such ill-
posed cases is left outside the scope of the paper, one
may interpret the empty-valued case as the situation where
the triplet {fu, fv, v}in the DI (33) is not permitted and
thus vshould instantaneously reach zero with the infinite
acceleration ˙v=±∞. The dual-valued case may be more
debatable, but seeing the structure of λin Fig. 7, it would be
natural to neglect the red parts, which are not continuously
connected to other portions of the graph. In this line of
consideration, one may replace λby the following single-
valued function λs:R×RR∪ {±∞}:
λs(ϕ, ρ)
0if ϕ= 0
ϕ/(1 ρ)if ϕ > 0ρ < 1
ϕ/(1 + ρ)if ϕ < 0ρ > 1
if ϕ > 0ρ1
−∞ if ϕ < 0ρ≤ −1,
which corresponds to the green and blue surfaces in Fig. 7,
with the empty-valuedness being replaced by +or −∞.
4.4 Dynamics: Apparent Inertia
Some researchers (Matsuki et al. 2019;Wang and Kim
2015;Mablekos-Alexiou et al. 2021;Sim and Ramos 2021)
describe dynamical systems like Fig. 1with expressions in
which the inertia varies between the forward-driving and
backdriving cases. Here, it is shown that their results are
consistent with the DAI (14) and DI (33).
Under the condition |ˆρ|<1, if (fu, fv,ˆρ)>0, the DI
(33) reduces to
(m+M) ˙v=fufvϕ(fu, fv,ˆρ)/(1 ˆρ).(36)
If (fu, fv,ˆρ)<0, (33) reduces to
(m+M) ˙v=fufv+ϕ(fu, fv,ˆρ)/(1 + ˆρ).(37)
Through some straightforward derivations considering the
definitions of ϕ,ˆρ,ηf, and ηb, one obtains the following
(ηfm+M) ˙v=ηffufvif (fu, fv,ˆρ)>0
(m/ηb+M) ˙v=fubfvif (fu, fv,ˆρ)<0.(38)
The ‘if’ condition in the first line can be seen as the forward-
driving case, and the other as the backdriving case. These
expressions coincide with those in previous work, e.g., (Yeh
and Wu 2009, Eqs. (18) and (19)), (Matsuki et al. 2019,
Section II.A), (Sim and Ramos 2021, Eqs. (5) and (6)),
(Mablekos-Alexiou et al. 2021, Eq. (20)), and (Wang and
Kim 2015, Eqs. (5) and (6)).
It might be possible to say that the presented DI (33) is
more natural than (38) in that the internal friction appears
as a single term, instead of influencing the inertial term.
A more important point compared to (38) is that the DI
(33), as well as the DAI (14), is a unified representation
that stands for all the cases of backdriving, forward-driving,
and the static friction, without involving switching between
different expressions. This property makes (14) and (33)
suited for computation, especially with a fixed timestep, and
for rigorous mathematical analysis.
4.5 Discrete-time Representation for
For simulation purposes, the continuous-time DAI repre-
sentation (14) or its equivalent DI form (33) needs to be
approximated by a discrete-time representation. The explicit
discretization, such as the forward Euler scheme, is not
applicable because, with such a scheme, the set-valuedness
in the DAI (14) or the DI (33) acts only as a discontinu-
ity, which results in chattering around v= 0 and fails to
realize the static friction. To deal with such problems, the
implicit (backward) Euler discretization method has been
known to be useful, especially for the time integration of
systems involving Coulomb friction (Acary and Brogliato
2008;Acary et al. 2010;Anitescu and Potra 1997;Stewart
and Trinkle 1996;Kikuuwe et al. 2006).
The implicit-Euler discretization of the DI (33) can be
given as follows:
hfu,k fv,k
Φ(sgn(vk), ϕ(fu,k, fv ,k,ˆρ))(39)
where his the timestep size and the subscript kis the
discrete-time index. For simulation purposes, (39) must be
solved to obtain vkaccording to given {vk1, fu,k, fv,k }.
The algebraic inclusion (39) can be rewritten as follows:
k∈ −hΦ(sgn(vk), ϕk,ˆρ)
kvk1+h(fu,k fv,k )
ϕkϕ(fu,k, fv ,k,ˆρ).(42)
Lemma 2suggests that the function Φ(s, ϕk)is continuous
and monotone with respect to s∈ B. Therefore, by using the
relation (5), one can equivalently rewrite (40) as follows:
kproj hΦ(B, ϕk,ˆρ)
m+M, v
Therefore, an algorithm to solve (39) is written as follows:
ϕk:= ϕ(fu,k, fv ,k,ˆρ)(44a)
k:= vk1+h(fu,k fv,k )
vk:= v
kproj hΦ(B, ϕk,ˆρ)
m+M, v
The expression (44) is an algorithm of the time integration
of the DAI (14) or the DI (33), which updates the velocity vk
in accordance with the input forces fu,k and fv,k . With the
algorithm (44), vkbecomes exact zero if
k/h [Φ(1, ϕk,ˆρ),Φ(1, ϕk,ˆρ)],(45)
which means that the algorithm (44) properly captures the
static friction. One can use the algorithm (44) even in the ill-
posed case by replacing λby λsin (35) in which replaced
by a very large floating-point number.
5 Multi-DOF Extension
5.1 DAI Representation
This section considers the equation of motion of an
articulated rigid-body system that has njoints driven by
actuators through transmissions. First, let us write the
equation of motion of an articulated rigid-body system as
q) = ˆ
where qRnis the joint angle vector, Mh(q)Rn×nis
the inertia matrix of the system, h(q,˙
q)Rnis the force
including the centrifugal, Coriolis and gravitational forces,
J(q)Rn×6is the Jacobian matrix that translates the joint
velocity to the end-effector velocity, fR6is the external
force acting on the end-effector, and ˆ
τvRnis the vector of
torques applied to the joints.
Next, let us consider the transmissions and the actuators
that are to be connected to the system (46). Let γuRn
and γvRnbe the vectors of the input- and output-side
asymmetry coefficients, respectively, mRnand mv
Rnbe the vectors including the input- and output-side
moments of inertia of the transmissions, respectively, and
τuRnbe the vector of the actuator torques. Note that m
and τuare evaluated in the output shafts, i.e, the real moment
of inertia of the motors are mdivided by the squares of the
reduction ratios, and the real actuator torques are τudivided
by the reduction ratios. Then, the equation of motion of the
transmission part of the system can be written as follows:
([mr]+[mv]) ¨
([γu][m][γv][mv]) ¨
q= [γu]τu+ [γv]ˆ
Here, λRnserves as a Lagrange multiplier. Note that (47)
is only a collection of nindependent systems each of which
is in the form of the DAI (14).
By combining the collection (47) of the transmissions with
the main part of the articulated rigid-body system (46) and
eliminating ˆ
τv, one obtains the following DAI:
([m] + M(q)) ¨
([γu][m][γv]M(q)) ¨
q= [γu]τu+ [γv]τvλ(48b)
Note that the DAI (48) can be considered as the multi-DOF
version of the DAI (14). A mathematical difficulty is raised
by the cross-coupling of accelerations resulting from the
non-diagonal elements of the matrix M(q). Also note that
this representation (48) is applicable not only to serial-link
manipulators but also to parallel-link manipulators.
5.2 DI Representation
Let us investigate dynamic properties of the DAI (48) in a
similar approach to that in Section 4.3. It can be equivalently
rewritten as follows:
([m] + M(q)) ¨
P(q)([γu][m][γv]M(q))([m] + M(q))1(52)
τλ[γu]τu+ [γv]τv.(54)
Note that P(q)can be seen as a multi-dimensional version
of the coefficient ˆρdefined in (29). Let us define a function
λ:Rn×Rn×nRnas follows:
Then, (51) can be rewritten as follows:
([m] + M(q)) ¨
λλ(τλP(q)τα,P(q)[s]) (56b)
Note that λin (55) reduces to λin (31) with n=
1. Considering Lemma 3in Appendix Aand the fact
σmax([s]) 1, one can see that
σmax(P(q)) <1(57)
is sufficient for the uniqueness of λin (56b). The fact that the
solution does not always uniquely exist has been pointed out
in (Dupont 1992)(Dupont 1993, Section 3), but its sufficient
or necessary condition had not been clarified.
Unfortunately, the condition (57) may be violated even
if all joints are both backdrivable and forward-drivable.
One can see that all joints are backdrivable (resp., forward-
drivable) if and only if all elements of γv(resp., γu)
are smaller than 1. Therefore, all joints of the system are
both backdrivable and forward-drivable if γu<1
γv<1. This condition is not sufficient for P(q)defined
by (52) to satisfy the condition (57). A clear physical
interpretation for the condition (57) is left as an open
By eliminating λand sfrom (56), one can obtain the
following DI representation:
([m] + M(q)) ¨
Kikuuwe 9
q),τλP(q)τα,P(q)) (58)
where Φ:Bn×Rn×Rn×nRnis defined as follows:
The DI representation (58) is a multi-dimensional version of
the DI (33), and Φin (59) is a multi-dimensional version of
Φin (34). As long as σmax(P)<1,Φ(s,ϕ,P)is single-
valued because of the property of λstated in Lemma 3.
Under this condition, the right-hand side of (58) is set-valued
only if ˙
qhas one or more zero elements.
Remark 3 (Numerical computation of λ). The proof of
Lemma 3, which uses the fixed point theorem, suggests that
the numerical computation λ(ϕ,P)can be performed by
several iterations of the computation λ:= ϕ+P|λ|with
the initial value setting λ:= ϕ. This function needs to
be computed in the algorithm presented in the upcoming
Section 5.3.
Remark 4 (Solution of the DI (58)). For the existence
of the uniqueness of solution of the DI (58), the continuity
and the monotonicity of the map from sto λ(ϕ,P[s]) are
important. Under the condition σmax(P)<1, its continuity
is implied by Lemma 4, but its monotonicity, unfortunately,
has not been proven yet. The continuity implies that the
right-hand side of (58) is outer semicontinuous (cf. (Hiriart-
Urruty and Lemar´
echal 2001, p. 14)) with respect to ˙
q. If the
monotonicity of λ(ϕ,P[s]) is proven, the right-hand side
can be shown to be compact and convex for all {q,˙
Then, Theorem 4.7 of (Smirnov 2002) suggests that the
DI (58) always has an absolutely continuous solution with
respect to {q,˙
q}if τuand fare measurable functions of
time t. In addition, if the monotonicity of λ(ϕ,P[s]) is
proven, the right-hand side of (58) can be shown to be a
locally bounded maximal monotone function of {q,˙
q}. In
this case, Theorem 1 in (Cellina 1995) suggests that the DI
(58) has unique solutions from almost all initial values of
5.3 Discrete-time Representation for
A simulation algorithm for the time integration of the DAI
(48) or the DI (58) is now derived. An implicit-Euler
discretization of the DI (58) can be given as follows:
hτα,k Φ(sgn(vk),ϕk,Pk1)(60)
ϕkτλ,k Pk1τα,k (62)
Mk1[m] + M(qk1)(63)
The algebraic inclusion (60) is rewritten as follows:
Figure 8. An illustration of the operator axproj defined in (67)
with n= 2. The vertices A, B, C, and D are AΦ(s,ϕ,P)where
s= [1,1]T,[1,1]T,[1,1]T, and [1,1]T, respectively.
When vis in the pink regions, y= axproj(v,A,ϕ,P)is the
closest vertex. When vis in the green regions, yis the
projection of vonto the closest edge along either axis. When v
is within the quadrilateral ABCD, y=v.
One needs to obtain vkby solving (65). To this end, let us
define an operator axproj : Rn×Rn×n×Rn×Rn×n
Rnas follows:
Then, the solution vkof (65) is written as follows:
k, hM1
Provided that an algorirhm for the operator axproj is
available, one can obtain an algorithm for the time
integration of the DAI (48) or the DI (58) as follows:
Mk1:= [m] + M(qk1)(69a)
Pk1:= ([γu][m][γv]M(qk1)) ˆ
τv,k := h(qk1,vk1)J(qk1)Tfk(69c)
ϕk:= [γu]τu,k + [γv]τv,k Pk1(τu,k τv,k )(69d)
k:= vk1+hˆ
k1(τu,k τv,k )(69e)
vk:= v
k, h ˆ
qk:= qk1+hvk.(69g)
Fig. 8illustrates how the operator axproj should work.
Here, AΦ(Bn,ϕ,P)is a polyhedron spanned by 2nvertices
in Rn, and y= axproj(v,A,ϕ,P)is the projection of von
the polyhedron in a particular axis-aligned manner. Note that
(68) is a multi-dimensional version of (43). As a special case,
if both Aand Pare diagonal matrices, the computation of
axproj reduces to the following:
axproj(v,A,ϕ,P) =
proj(A11 Φ(B, ϕ1, P11), v1)
proj(Ann Φ(B, ϕn, Pnn), vn)
where the symbols with subscripts denote elements of
correspondent vectors or matrices.
There are several ways to perform the computation of
axproj. An important fact is that it is classified as a
variational inequality (VI) problem (Facchinei and Pang
2003, Section 1.1). By borrowing the notation of (Facchinei
and Pang 2003, Section 1.1), a VI problem is a problem
to obtain xRnfor a convex set KRnand a map F:
RnRnso that
xSOL(K, F ){xRn|0F(x) + N(x, K )}(71)
where Nis an operator called the normal cone (see
(Facchinei and Pang 2003, Section 1.1) for its definition).
One can see that, if one defines a function F:RnRnas
the operator axproj can be rewritten as follows:
axproj(v,A,ϕ,X) = F(SOL(Bn, F )) + v.(73)
One simple way to solve a VI is, as stated in (Acary and
Brogliato 2008, Section 12.6.7), to perform an iteration of
s:= proj(K, scF (s)) with a small positive number cand
an appropriate initial value of s. After obtaining the solution
sof the VI, the output yof axproj can be obtained by y:=
F(s) + v. More computationally efficient schemes may be
found in the literature, e.g., (Acary and Brogliato 2008;
Facchinei and Pang 2003), or can be constructed considering
the particular structure of the function Fin (72), but it is left
outside the scope of this paper. It should also be noted that,
if P= 0, it reduces to an affine variational inequality (AVI)
problem, for which more sophisticated solvers are available.
Remark 5 The algebraic problem of axproj can also
be found in (Dupont 1993, Section 3.2.1), in which it is
suggested to check all possible combinations of the joint
velocity signs until a consistent solution is found. Such an
approach would require a careful algorithm design to deal
with the case where some joints are in the static friction. As
far as the author is aware, this paper is the first where this
problem is formulated as a tractable VI problem.
6 Numerical Examples
This section presents illustrative simulation results of the
proposed algorithms (44) and (69). The purpose is to
illustrate the capability of the proposed method to properly
simulate scenarios involving the load-dependency of the
friction and the static friction. Such scenarios cannot be
simulated with load-independent, constant friction models,
which are adopted by most open-source robotics simulators.
The paper does not attempt to compare them with previous
methods because, as far as the author is aware, there are
no existing models that capture asymmetric friction with the
static friction.
6.1 1-DOF System
Some simulations of a 1-DOF system are performed to
validate the algorithm (44). The system is illustrated in
Fig. 9(a), in which a rod is driven by a rotary actuator through
a transmission. The timestep size is set as h= 0.001 s.
The parameters are set as m= 1 kg·m2,M= 1 kg·m2,
γu= 0.2, and γv∈ {0.2,0.5,0.95,1.05}. The parameter M
includes the moment of inertia of the rod. As illustrated in
Fig. 9(a), the rod is always subject to the viscosity Be=
Figure 9. One-DOF simulation; (a) the system, (b) results of
position p, and (c) results of force fv.
10 Nms/rad, and gains contact with an external object with
the stiffness Ke= 10000 Nm/rad at the location p= 0 rad.
Thus, the force from the rod to the external environment
is fv= min(Kep, 0) + Be˙p. The actuator force fuis set as
fu=40 Nm in the beginning, switched to fu= 0 Nm at
t= 0.9s, and to fu= 20 Nm at t= 1.4s.
Note that γv>1means that the transmission is non-
backdrivable. The value of ˆρ, defined in (29), for the
aforementioned four values of γvsatisfy |ˆρ|<1and thus the
system is well-posed for all cases. It should also be noted that
the reduction ratio κdoes not appear in this system because
the aforementioned values of mand fuare those evaluated
at the output shaft; i.e., being evaluated at the input shaft,
the actuator’s moment of inertia is m/κ2and the torque
generated by the actuator is fu.
This simulation scenario is for illustrating the asymmetric
frictional behaviors, with which the rod can be moved only
by a small actuator torque in the free space but can keep
applying a large force during the contact with the external
object. Such features cannot be produced by conventional
load-independent friction models.
Results are shown in Figs. 9(b) and (c). In the beginning,
the rod is moved in the negative direction due to fu=
40 Nm until the collision with the spring. The rod is then
pushed against the spring until t= 0.9s with the force fu=
40 Nm. During this time, the rod bounces on the spring
with smaller γvvalues, but maintains a large contact force
fv, much larger than the actuator force fu, with larger γv
values. These results properly reflect the asymmetric friction
of the transmission.
Once fuis set zero at t= 0.9s, the rod is pushed back
from the spring with |γv|<1, but it is still stuck with
|γv|>1. In this period, the contact force fvis still large
with |γv|>1even though the actuator force fuis zero.
It is consistent with the fact that the transmission is non-
backdrivable with |γv|>1. After the actuator force fuis
switched to be positive at t= 1.4s to pull the rod in
the positive direction, the rod is detached also with |γv|>
1. These results properly reflect the asymmetric friction
behaviors of transmissions, both those backdrivable and non-
backdrivable, which cannot be reproduced by conventional
load-independent friction models.
Kikuuwe 11
Figure 10. Two-DOF simulation. The actuator force
τu=J(q)[fux(t),0]Tand the wall position wx(t)are given as
functions of time t.
Figure 11. Results of two-DOF simulations with
γv= [0.1,0.1]T,[0.6,0.6]T, and [1.1,1.1]T; (a)(b)(c) snapshots
at 0.3s intervals for t < 2.0s, (e) the position pxof the robot’s
end-effector, and (e) the force fxfrom the wall to the robot.
6.2 2-DOF System
Some simulations are also performed with a 2-DOF system
to validate the algorithm (69). The timestep size is set as
h= 0.001 s. The system is a two-link robot driven by rotary
actuators through transmissions as in Fig. 10. Each link has a
length of 1m and a mass of 1kg lumped at its distal end.
The rotor inertias of the actuators evaluated at the output
shafts are both m= 1 kg·m2. Each joint is subject to the
viscosity of 5Nms/rad. The gravity is assumed not to act.
As illustrated in Fig. 10, an elastic wall is placed in parallel
to the yaxis and its x-coordinate wxis moved from the
initial position wx01.37 m to the terminal position wx1
1.12 m. The wall applies the force f= [Kmax(px
wx),0]Tto the end-effector of the robot.
The actuators’ torques τuR2are determined by an
x-direction force command fux through the statics τu:=
J(q)T[fux,0]Twhere J(q)R2×2is the Jacobian matrix
translating the joint velocities to the end-effector velocity. At
t= 0, the end-effector’s x-coordinate is set to be the same as
the initial position of the wall wx0. The simulation scenario
is defined as follows:
t[0 s,0.5s];fux := 0 N, wx:= wx0; The robot is
in contact with the wall.
Figure 12. Results of two-DOF simulations with
γv= [0.1,1.1]Tand [1.1,0.1]T; (a)(b) snapshots at 0.3s
intervals for t < 2.0s, (c) the position pxof the robot’s
end-effector, and (d) the force fxfrom the wall to the robot.
t[0.5s,1s];fux := 500 N, wx:= wx0; The robot
pushes the wall.
t[1 s,1.5s];fux := 500 N, wx:= ((1.5t)wx0+
(t1)wx1)/0.5; The wall moves to the left, pushing
back the robot.
t[1.5s,2s];fux := 500 N, wx:= wx1The wall
stops, while the robot keeps pushing.
t[2 s,2.5s];fux := 0 N, wx:= wx1; The robot
stops pushing.
t[2.5s,3s];fux := 500 N, wx:= wx1; The robot
pulls itself apart from the wall.
This scenario is for illustrating the characteristics of
asymmetric joint friction, with which the robot resists large
external force but can be driven with small actuator torques.
Note that such behaviors cannot be produced by load-
independent joint friction models provided by common
robotics simulators.
Simulations are performed with γu= [0.1,0.1]Tand
three different settings for γv, which are γv= [0.1,0.1]T,
[0.6,0.6]T, and [1.1,1.1]T. The results are shown in Fig. 11.
It shows that a larger value of γvresults in a large contact
force fxwhen the wall pushes the robot to the left (t
[1 s,2s]). In addition, in the case of |γv|>1, the robot is
not moved by the wall (t[1 s,1.5s]) and the contact force
is high even after the robot stops pushing (t[2.0s,2.5s]),
exhibiting the non-backdrivability of the mechanism. Even in
this case, the end-effector is pulled apart from the wall when
fux is set negative, exhibiting the forward-drivability of the
system. These results indicate that the presented algorithm
(69) is capable of properly simulating the transmission
dynamics, including backdrivability and forward-drivability.
Another set of simulations with the same scenario is
performed with different joint properties; γv= [0.1,1.1]T
and γv= [1.1,0.1]T. The results are shown in Fig. 12.
It shows that, when the wall pushes the robot to the left
(t[1 s,1.5s]), the joint with |γv|>1is fixed while the
other joint is moved. This result is physically plausible,
exhibiting the capability of the presented algorithm (69) to
simulate transmissions of different friction properties in a
single system.
7 Concluding Remarks
This paper has presented a dynamics modeling scheme for
gear transmissions having asymmetric and load-dependent
friction. The representation is of the form of a DAI
composed of a ‘visible’ dynamics and a ‘hidden’ dynamics
combined through a Lagrange multiplier. The presented
model properly captures the static friction and even the
non-backdrivability. It is characterized by two coefficients
that have simple relations with the forward and backward
efficiencies of the transmission. The model is extended into a
multi-dimensional model for robotic systems equipped with
joint transmissions. Discrete-time simulation algorithms are
derived through the implicit-Euler discretization of the
presented DAIs. Some illustrative simulation results are
presented, in which transmissions with asymmetric, load-
dependent, and non-backdrivable properties are properly
The presented model would be further improved by
including the load-independent component of Coulomb-like
friction and the Stribeck-like effect caused by the lubricants
in the transmission. An efficient calibration method for
the parameters γuand γvwould also need to be sought.
Moreover, the control applications of the method, such as
friction compensation, would demand some clarifications
and improvements on the computational efficiency of the
algorithm axproj.
A Some Lemmas
Lemma 1 (About λ). For any sets of real numbers
{ρ, ϕ, λ}, the following statement holds true:
λρ|λ|=ϕλλ(ρ, ϕ)(74)
where λ:R×RRis defined in (31).
Proof. If λ= 0, the statement (74) reduces to
ϕ= 0 0λ(ρ, ϕ),(75)
which can be seen to be true by definition (31) of λ.
If λ̸= 0, one has the following:
λ=ϕ/(1 ρsgn(λ))
(λ=ϕ/(1 ρ)>0) (λ=ϕ/(1 + ρ)<0)
(λ=ϕ/(1 ρ)ϕ < 0ρ > 1)
(λ=ϕ/(1 ρ)ϕ > 0ρ < 1)
(λ=ϕ/(1 + ρ)ϕ > 0ρ < 1)
(λ=ϕ/(1 + ρ)ϕ < 0ρ > 1)
(λ∈ {ϕ/(1 ρ), ϕ/(1 + ρ)} ∧ ϕ > 0ρ < 1)
(λ∈ {ϕ/(1 ρ), ϕ/(1 + ρ)} ∧ ϕ < 0ρ > 1)
(λ=ϕ/(1 ρ)ϕ > 0ρ[1,1))
(λ=ϕ/(1 + ρ)ϕ < 0ρ(1,1])
λλ(ρ, ϕ),(76)
which demonstrates (74).
Lemma 2 (Continuity, boundedness, and monotonicity of
Φ). For all {s, ϕ, ρ} ∈ B × R×Rsatisfying |ρ|<1, the
function Φ(s, ϕ, ρ)of sdefined in (34) is continuous,
bounded, and monotone.
Proof. The continuity and boundedness are trivial because
of the definition (34) of Φ, which depends on the definition
(31) of λ. The monotonicity is proven as follows. First, if ρ=
0,λ(ϕ, ρs) = ϕand Φ(s, ϕ, ρ) = s|ϕ|, which is monotone
with respect to s. Next, if ρ̸= 0, Lemma 1suggests that
λ(ϕ, ρs)ρs|λ(ϕ, ρs)|=ϕ(77)
and, because of the definition (34) of Φ, one has
Φ(s, ϕ, ρ)=(λ(ϕ, ρs)ϕ)/ρ. (78)
The definition (31) implies that the right-hand side of (78) is
monotone with respect to sin both cases of ρ > 0and ρ < 0.
Lemma 3 (Single-valuedness of λ). For all {ϕ,P} ∈
Rn×Rn×nsatisfying σmax(P)<1, the set λ(ϕ,P)is a
singleton where λ:Rn×Rn×nRnis defined in (55).
Proof. Let X(λ)P|λ|+ϕand assume that σmax(P)<
1. Then, the map Xis contracting, i.e., ∀{x,y} ∈ Rn×Rn,
∥X(x)− X (y)=P(|x|−|y|)
σmax(P)∥|x|−|y|∥ <∥|x|−|y|∥ ≤ ∥xy.(79)
Thus, due to Banach fixed-point theorem, there exists
a unique solution for X(λ) = λ, and it means that the
inclusion λλ(ϕ,P)has a unique solution.
Remark 6 The proof of Lemma 3is along the same line
as is the proof of Proposition 9 of (Blumentals et al. 2016),
which deals with a system that is almost the same as (48).
Lemma 4 (Continuity and boundedness of λ). For all
{ϕ,P,s} ∈ Rn×Rn×n× Bnsatisfying σmax (P)<1, the
single-valued function λ(ϕ,P[s]) of sis continuous and
Proof. For brevity, let us use σP=σmax(P)<1. With s
Bnand λ=λ(ϕ,P[s]), one has the following,
ϕ=λP[s]|λ|∥ ≥ (1 σPs)λ,(80)
which results in the following:
λ≤ ∥λ∥ ≤ ϕ
This means that λ(ϕ,P[s]) is a bounded function of s.
With s1∈ Bnand s2∈ Bn, let us define λ1=
λ(ϕ,P[s1]) and λ2=λ(ϕ,P[s2]). From the definition
(55), the following relations are satisfied:
Eliminating ϕfrom the above results in the following:
λ1λ2P[s1](|λ1|−|λ2|) = P[|λ2|](s1s2).(83)
Kikuuwe 13
Considering that ∥|λ1|−|λ2|∥ ≤ ∥λ1λ2and σP<1,
one obtains the following:
(1 σP)λ1λ2∥≤∥L.H.S. of (83)
=R.H.S. of (83)∥ ≤ σPλ2s1s2.(84)
Considering (81) and (84), one has the following:
λ1λ2∥ ≤ σPλ2s1s2
(1 σP)2s1s2.(85)
Therefore, for any ε > 0, choosing δso that
0< δ < (1 σP)2
results in that s1s2< δ implies λ1λ2< ε. This
means that λ(ϕ,P[s]) is a continuous function of s.
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