Interconnection and damping assignment passivity-based control of a class of underactuated...

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control 2011; 21:738–751Published online 30 July 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1622

Interconnection and damping assignment passivity-based control ofa class of underactuated mechanical systems with dynamic friction

Jesus Sandoval1,∗,†, Rafael Kelly2 and Víctor Santibanez3

1Facultad de Ingeniería, Universidad Autonoma de Baja California, Carretera Tijuana-Ensenada km. 107,

Ensenada, B. C. 22800, Mexico2Division de Física Aplicada, CICESE, Carretera Tijuana-Ensenada, B. C. 22800, Mexico

3Instituto Tecnologico de La Laguna, Apdo. Postal 49, Adm. 1, Torreon, Coahuila 27001, Mexico

SUMMARY

This paper considers the extension of the interconnection and damping assignment passivity-based controlmethodology for a class of underactuated mechanical systems with dynamic friction. We present a newdamping assignment approach to compensate friction by means of a nonlinear observer. Friction at theactuated joints is assumed to be captured by a bristle deflection model: the Dahl model. Based on theLyapunov direct method we show that, under some conditions, the overall closed-loop system is stableand, by invoking the theorem of Barbashin–Krasovskii, we arrive to asymptotic stability conditions.Experiments with an underactuated mechanical system, the Furuta pendulum, show the effectiveness ofthe proposed scheme when friction is compensated. Copyright � 2010 John Wiley & Sons, Ltd.

Received 3 December 2008; Revised 3 March 2010; Accepted 22 May 2010

KEY WORDS: underactuated mechanical systems; Hamiltonian systems; stabilization; Dahl friction model

1. INTRODUCTION

Friction is a complex physical phenomenon that appears in all mechanical systems, and a lotof strategies to vanish its harmful effects in control systems have been extensively investigatedfor a long time. Steady-state errors and oscillations are typical effects of friction in the motioncontrol of mechanical systems. There are many schemes of model-based friction compensationthat attempt to reduce these undesirable behaviors caused by friction, see [1–3]. These schemesare based on friction models which are able to predict an extensive variety of friction phenomena(e.g. presliding displacement, frictional lag and varying break-away force), all of them observedin practice. The nature of the models is quite different, but all approximate the real friction.Accuracy of approximation is mostly given by the complexity of the model, either static ordynamic. Dynamic friction models, such as the Dahl model [4], and LuGre model [5], havebeen used to compensate friction in control systems (see [6] and references therein). The Dahlmodel is one of the simplest friction dynamic models, which is inspired on a bristle deflec-tion interpretation, and it is able to predict important phenomena that appear at low velocities,such as presliding displacement. Essentially, the Dahl model is a Coulomb friction model witha lag in the change of the friction force, when the direction of motion is changed [1]. Some

∗Correspondence to: Jesús Sandoval, Av. Conchalito 4758 Fracc. Esperanza III, 23090 La Paz, B. C. S., Mexico.†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

IDA-PBC OF A CLASS OF UNDERACTUATED MECHANICAL SYSTEMS 739

remarkable properties of this friction model are: it is extensively used in simulation studies, frictiondepends only on displacement, it does not capture Stribeck effect and it does not describe stick-slipphenomenon [2].

On the other hand, the interconnection and damping assignment passivity-based control (IDA-PBC) method, introduced by Ortega et al. [7], is a control design methodology that assigns a desireddynamics to the closed loop. The stabilization of a class of underactuated mechanical systems—those having more degrees of freedom than control inputs—has been successfully solved via theIDA-PBC method. This method utilizes the Hamiltonian formulation as a natural framework. In theLagrangian formalism, a related technique, called Controlled Lagrangian, shares a similar reasoningas used in the IDA-PBC method, and some equivalence conditions between both techniques arepresented in [8, 9]. Further details on the Controlled Lagrangian method are described in Bloch’sbook [10]. In the original formulation of the IDA-PBC method the friction is absent, but recently adetailed analysis of the IDA-PBC method applied to the stabilization of underactuated mechanicalsystems with viscous friction, has been shown in Lagrangian formulation [11] as well as inthe Hamiltonian formulation [12]. These works show how the natural damping can affect thesystem’s stability when such a damping is not considered in the IDA-PBC methodology. Moreover,in [11], experimental evidence has revealed undesirable behavior due to friction phenomena fora pendulum–cart system in closed loop with a control law based on the Controlled Lagrangiantechnique. In that work, it was pointed out that a compensatory force was used to minimize theeffect of dynamic friction, but this force is not shown in the control law and it is not clear how itwas considered.

We have carried out experiments applying IDA-PBC control to a Furuta pendulum where frictionis not considered; those experiments revealed an oscillatory behavior which remained in spiteof modifying the parameters of the control system. The evidence exhibited in our experimentsmotivated us to analyze friction compensation in the framework of the IDA-PBC methodology.As a consequence, we have extended the original formulation of this methodology to the case whenfriction is present only in the actuated joints of a class of underactuated mechanical systems. Thekey issue in our proposal is the incorporation of the Dahl model in the formulation of the IDA-PBCmethod, preserving its design philosophy. The latter means that the Dahl model can be embeddedin an IDA-PBC control law with friction compensation. An advantage of the inclusion of the Dahlmodel in the IDA-PBC method is the simplicity in the design of the observer that deals with theestimation of the unmeasurable internal state of the friction model. Unfortunately, this proceduremay fail with other dynamic friction models, and a novel strategy must be considered if we wish toincorporate them in the IDA-PBC methodology. We believe that the results of this extension will beuseful to achieve better performance in experiments of a wide class of IDA-PBC control systemsfor underactuated mechanisms reported in the literature (see Reference [13] for a complete survey).In sum, the main contribution of this paper is an extension of the IDA-PBC method applied to thecontrol of a class of underactuated mechanical systems, with dynamic friction characterized by theDahl model.

The remainder of this paper is organized as follows. In Section 2, we introduce the newformulation of the IDA-PBC method to control underactuated mechanical systems with friction, thedesign of a nonlinear observer, and the stability analysis using Lyapunov’s direct method. Section 3shows the application of the IDA-PBC methodology using friction compensation to the control ofa Furuta pendulum, including the experimental results. Finally, we offer some concluding remarksin Section 4.

2. IDA-PBC CONTROL OF UNDERACTUATED MECHANICALSYSTEMS WITH DYNAMIC FRICTION

In this section, we extend the original formulation of the IDA-PBC method to control a class ofunderactuated mechanical systems when friction is considered. The total energy function, given

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 2011; 21:738–751DOI: 10.1002/rnc

740 J. SANDOVAL, R. KELLY AND V. SANTIBÁNEZ

by the sum of the kinetic plus potential energies of the mechanical system is‡

H (q,p)= 12 pT M−1(q)p+V (q), (1)

where q∈Rn and p∈Rn are the vectors of generalized positions and momenta, respectively, M =MT>0 is the so-called inertia matrix and V is the potential energy. Moreover, it is assumed thatthe first m of the n joints are actuated; if the system is underactuated then m<n. We will call thesem joints as the actuated coordinates. On the other hand, the Dahl model applied to underactuatedsystems is [6]:

z = −G�(q)GT�oz+[GGT]Tq, (2)

f(z, q) = GGT�oz+G FvGTq, (3)

where z∈Rn is a vector of unmeasurable internal states, G =[

Im0

]∈Rn×m , such that rank(G)=

m, �=diag{(1/ fc1 )|q1|, . . . , (1/ fcm )|qm |} is a diagonal positive-semidefinite matrix, where fci

(i =1,2, . . . ,m) denotes the Coulomb parameter for each of the m actuated coordinates, �o =diag{�01, . . . ,�0n } is a diagonal positive-definite matrix which contains the ‘stiffness’ parametersof the n coordinates, f∈Rn is the vector of forces due to the friction, Fv =diag{ fv1, . . . , fvm } is adiagonal positive-definite matrix which contains the viscous friction coefficients of the m actuatedcoordinates, and

q= M−1p= [q1 · · · qn]T (4)

is the vector of generalized velocities. Although the state z is not measurable, we assume that allthe parameters of the model are known. For the analysis, we consider that friction is present onlyin the actuated joints, which is equivalent to say that z= [z1, . . . , zm, zm+1, . . . , zn]T with zk =0, fork =m+1, . . . ,n. If we take the Dahl friction model (2)–(3) into account, the equations of motionof the whole system can be written as

d

dt

⎡⎢⎣

q

p

z

⎤⎥⎦=

⎡⎢⎢⎣

0 In 0

−In −G FvGT −GGT

0 [GGT]T −G�GT

⎤⎥⎥⎦

⎡⎢⎣

∇q H

∇p H

∇zW

⎤⎥⎦+

⎡⎢⎣

0

Gu

0

⎤⎥⎦ (5)

where u∈Rm is the vector of control inputs, ∇(·) =�/�(·),W (z)= 1

2 zT�oz,

and the right-hand side of (3) is included in the second line of (5), where the minus sign is due tothe fact that friction force is in opposite direction as the control input.

The rationale behind the IDA-PBC method is to assign a desired structure in closed loop,preserving the structure in (5), with a desired energy function given by

Hd (q,p)= 12 pT M−1

d (q)p+Vd (q), (6)

where Md = MTd >0 and Vd are the desired inertia matrix and potential energy function, respectively.

The desired closed-loop system is

d

dt

⎡⎢⎣

q

p

z

⎤⎥⎦=

⎡⎢⎢⎣

0 M−1 Md 0

−Md M−1 J2(q,p)−G KvGT −GGT

0 [GGT]T −G�GT

⎤⎥⎥⎦

⎡⎢⎣

∇q Hd

∇p Hd

∇zWd

⎤⎥⎦ , (7)

‡To simplify the expressions, the arguments of all functions will be omitted, and will be explicitly written only thefirst time that the function is defined.



where J2 =−J T2 and Kv = K T

v >0 are free matrices [7], z=z− z, being z the estimation of thestate z, and we introduce in our formulation a positive-semidefinite function

Wd (z)= 12 zT�oz. (8)

Now, for this new class of systems (7) the main challenge of the IDA-PBC method still consistsof solving the following set of PDEs, called matching equations [7]:

G⊥{∇q(pT M−1p)− Md M−1∇q(pT M−1d p)+2J2 M−1

d p} = 0, (9)

G⊥{∇qV − Md M−1∇qVd} = 0, (10)

where G⊥ ∈R(n−m)×n is a full-rank left annihilator of G, so that G⊥G =0. Given the Md and Vdsolutions of (9)–(10), we define the control law

u = [GTG]−1GT [∇q H − Md M−1∇q Hd + J2 M−1d p]︸︷︷︸

ues

−KvGT M−1d p︸︷︷︸

udi

+FvGT M−1p+GT�oz︸︷︷︸ufric

(11)

being ues the energy shaping control and udi the damping injection [7], while the new terms givenby ufric define the friction compensation.

Remark 1We underscore the fact that friction compensation is absent in the original formulation as shownin [7]. Furthermore, it is worth to remark that, in our formulation, the matching Equations (9) and(10) do not change with respect to the original formulation. This is because friction compensationdefines a new damping injection given by unew

di =udi +ufric.

2.1. Friction observer

Since the Dahl model is dynamic and has an unmeasurable state z, some kind of observer mustbe established. It is known that there are several well-established methodologies for carrying outobserver designs, and also there are several nonlinear observers proposed in a particular way[5, 6, 14, 15]. Nevertheless, inspired by Kelly et al. [16], in this paper we present the followingfriction observer

˙z= [GGT]T[M−1p− M−1d p]−G�GT�oz, (12)

which provides an estimate of the unmeasurable state z. An advantage from the inclusion of theDahl model (2)–(3) in (5) is the simplicity to design the observer. To clarify this procedure, noticethat, from the definition z=z− z, the time derivative can be written as

˙z= z−˙z. (13)

Thus, the proposed observer (12) is achieved by substituting z and ˙z from (5) and (7), respectively,in (13).

Remark 2An interesting connection between our proposal and the work by Rocha-Cozatl and Moreno [17] isthe nonlinear observer defined by (12), which could be considered as an unknown input observer inaccordance with [17], where friction force—captured by the Dahl model—would be the unknowninput (see Section 2.1 in [17]). Therefore, we believe that a further study could be addressed toanalyze our proposal in the theoretical framework given by [17].



2.2. Analysis

In this subsection, we extend the analysis shown in [7] to the case when friction is considered.First, we proceed to define a necessary condition to ensure that [qT pT zT]T = [q∗T 0T 0T]T is anisolated equilibrium point of (7), where

q∗ ∈argmin{Vd} (14)

is the isolated minimum of interest, and arg min{Vd} is the set of all local minimal of Vd . Towardthis end, we introduce the following:

Assumption A1The following set of algebraic equations

G⊥∇qV =0 (15)

has as unique solution q=qa , at least locally. Moreover, substituting q=qa into (10), we have that

G⊥Md M−1∇qVd =0 (16)

only has a unique solution q=q∗ locally.

Remark 3Assumption A1 includes a class of underactuated mechanical systems where the vector of desiredpositions q∗ in the closed-loop system belongs to the set of critical points of the open-loop system,such that ∇qV (q)|q=q∗=0. Some examples are the inertial wheel pendulum and the ball and beamsystems shown in [7], the cart–pole system [18], the acrobot [19], the Furuta pendulum [20], andthe pendubot [21].

We now present our main result on stability in the following proposition.

Proposition 1Under Assumption A1, system (7) has a stable isolated equilibrium point [qT pT zT]T =[q∗T 0T 0T]T. This equilibrium point is asymptotically stable if inside

�={[qT pT zT]T ∈ D :Hd (q,p, z)=0}the maximum invariant set is [q∗T 0T 0T]T, where D ⊆R3n and

Hd = Hd +Wd (17)

qualifies as a Lyapunov function. An estimate of the domain of attraction is given by LV (c), where

LV (c)�

⎧⎪⎨⎪⎩

⎡⎢⎣

q

p

z

⎤⎥⎦∈ D :Hd <c

⎫⎪⎬⎪⎭

and

c=sup{c>Hd (q∗,0,0)|LV (c) is bounded}. (18)

ProofNote that the set of equilibria from (7) is given by

E={[qT pT z]T ∈ R3n :p=0; Md M−1∇qVd +GGT�oz=0}.The equation shown in the set E can be written as[

GT

G⊥

][Md M−1∇qVd +GGT�oz]=0



such that

GT[Md M−1∇qVd +GGT�oz] = 0, (19)

G⊥Md M−1∇qVd = 0. (20)

Now, by design, (20) implies that G⊥∇qV =0, in accordance with (10). Then, by usingAssumption A1 we ensure that (20) holds uniquely in q=q∗, at least locally. Finally, substitutingthis previous result in (19) yields z=0, recalling that we have assumed that the friction is onlypresent in the actuated joints. Thus, we conclude that [q∗T 0T 0T]T is an isolated equilibriumof (7).

Next, taking into account (6), (8) and (14), it is easy to verify that (17) is a positive-definitefunction in a neighborhood of the point [q∗T 0T 0T]T, where an appropriate constant could beaggregated to Vd such that it vanishes at q=q∗. The time derivative of (17) along the trajectoriesof system (7) is given by

Hd = −pT M−1d G KvGT M−1

d p− zT�oG�GT�oz

= −[

p

z

]T [M−1

d G KvGT M−1d 0

0 �oG�GT�o

][p

z

], (21)

which is a nonpositive function. Based on Lyapunov’s theory, these results ensure that the equi-librium point [q∗T 0T 0T]T is stable.

It only remains to establish asymptotic stability conditions. To this end, in accordance with thetheorem of Barbashin–Krasovskii [22] we define the set � as:

�=

⎧⎪⎨⎪⎩

⎡⎢⎣

q

p

z

⎤⎥⎦∈ D :pT M−1

d G KvGT M−1d p+ zT�oG�GT�oz=0

⎫⎪⎬⎪⎭ .

The next step is to determine the maximum invariant set in �. Therefore, we look for all trajectoriesthat, starting in �, remain therein forever. To this end, notice that in � the following must besatisfied:

pT M−1d G KvGT M−1

d p+ zT�oG�GT�oz=0. (22)

There are two ways of fulfilling the previous equation. The first choice is p=0 and z∈Rn due to that�(0)=0 in accordance with the definition q= M−1p. Then, p=0 implies p=0 and, substitutingthis result into (7), under Assumption A1, we obtain q=q∗ and z=0. Thus, we conclude that aninvariant set in � is the equilibrium point [q∗T 0T 0T]T. The second choice is z=0 and

GT M−1d p=0. (23)

The former implies that ˙z=0 and substituting this result, together with (23), in (7), led us to

p=−Md M−1∇q Hd + J2 M−1d p. (24)

Moreover, (23) implies that its time derivative holds, i.e.

GT M−1d p−GT M−1∇q Hd +GT M−1

d J2 M−1d p=0, (25)

where we have considered (24). Thus, the problem to solve reduces to prove that [q∗T 0T 0T]T isthe unique solution that simultaneously satisfies (23), (24) and (25), which means that there areno other invariant sets in �, besides the equilibrium point [q∗T 0T 0T]T. Although there is nota general procedure to solve this problem, particular solutions exist for a class of underactuatedmechanical systems where the friction is absent (see [7, 18, 19]). This latter is equivalent to considerz=0 in our analysis, such that the results shown in the works [7, 18, 19] ensure, for a class



of underactuated mechanisms, that the set of Equations (23)–(25) only have a unique solution[qT pT]T = [q∗T 0T]T locally. Hence, by following the steps given in those works it is possible toprove that the equilibrium point [q∗T 0T 0T]T will be the maximum invariant set in �.

Finally, the estimate of the domain of attraction follows from the fact that LV (c) is the largestbounded sublevel set of Hd . This completes the proof. �

In the following section, we apply the new formulation of the IDA-PBC method to a Furutapendulum.

3. APPLICATION: THE FURUTA PENDULUM

The Furuta pendulum is an underactuated mechanical system consisting of two links with tworevolute joints [23]. The first joint is endowed with an actuator that applies a torque to it, while theabsence of actuation in the second joint determines the underactuated nature of the mechanism.A schematic picture of the Furuta pendulum is shown in Figure 1, where q1 and q2 are the jointpositions. The first link turns around the vertical axis while the second link, located at the extremeof the first link, turns in a vertical plane. The parameters of the two links are the length of the linkli , the distance to the center of mass lci , the moment of inertia Ii and the mass mi , with i =1,2.

The Hamiltonian model of the Furuta pendulum shown in Figure 1 can be described by (5) withthe matrices

M(q2)=[

a1 +a2 sin2(q2) a3 cos(q2)

a3 cos(q2) a2

], G =

[1

0

](26)

and the potential energy function

V (q2)=a4 cos(q2), (27)

where the constants ai are defined as

a1 = I1 +m1l2c1

+m2l21, a2 = I2 +m2l2

c2,

a3 = m2l1lc2, a4 =m2glc2 .(28)

The control objective is to drive the first link of the Furuta pendulum to a desired position with thesecond link in the upward vertical direction, starting from a neighborhood of this configuration.

Figure 1. Furuta pendulum and characteristic parameters.



Formally, the control objective can be established as

limt→∞

[q1(t)

q2(t)

]=

[qd1

�

], (29)

where qd1 is the desired position.Following the procedure shown in [21], together with the friction compensation, according to

our scheme, we designed an IDA-PBC control law to stabilize the Furuta pendulum, which is

u = −[�1∇1Vd +�2∇2 Hd ]+ j2[−d2 p1 +d1 p2]

�d︸︷︷︸ues

−kv[d3 p1 −d2 p2]

�d︸︷︷︸udi

+ fv1 q1 +�o1 z︸︷︷︸ufric

, (30)

˙z = −d3 p1 +d2 p2

�d+ q1 − �o1

fc1

|q1|z, (31)

where kv is a positive constant; j2 and di (i =1,2,3) are the elements of the matrices J2 and Md ,respectively, such that

Md =[

d1 d2

d2 d3

]=

⎡⎢⎢⎢⎣

k3r3 cos(q2)

�

r3 cos(q2)

�

−r1 +r2 cos2(q2)

�

⎤⎥⎥⎥⎦ , (32)

J2 =[

0 j2

− j2 0

](33)

and

Vd = a4

k1[1+cos(q2)]+ kp

2

⎡⎢⎢⎣q1 −qd1 +

a3arctan

(a2 sin(q2)

r4

)r4

⎤⎥⎥⎦

2

, (34)

where kp>0, �=k1k2a3 +a2 sin2(q2),

r1 = k1a22 +k2

1k2a2a3, r2 =k1a22 +k1a2

3,

r3 = k1a1a3 −k21k2a2

3, r4 =√

k1k2a2a3

and

j2 = p1[d3�1 −d2�2]+ p2[−d2�1 +d1�2]

�d(35)

being k1, k2, k3 positive constants satisfying some conditions, and �d =d1d3 −d22>0 is the deter-

minant of Md . The remaining terms of the control law (30)–(31), �1, �2 and the conditions on thegains k1,k2 and k3 are described in the Appendix. Finally, in accordance with Assumption A1,considering (27) with G⊥ = [0 1], we find out that (15) has as unique solution qa = [q1 �]T∀q1 ∈R

and �/2<q2<3�/2. Substituting qa into (16) with M , Md and Vd given in (26), (32) and (34),respectively, we verify that q∗ = [qd1 �] is a unique solution of (16) locally. Hence, the Furutapendulum satisfies Assumption A1 such that the proposed method is applicable.



Figure 2. Experimental Furuta pendulum.

Table I. Parameters.

Link 1 Link 2 Gains

l1 =0.252 l2 =0.366 k1 =k2 =0.7m1 =1.636 m2 =0.336 k3 =0.2lc1 =0.019 lc2 =0.177 k p =0.2I1 =0.0075 I2 =0.004 kv =15

3.1. Experimental results

We have built a Furuta pendulum in order to do real-time control experiments (see Figure 2). Themotor used in the experimental Furuta pendulum is the DM1004-C model from Parker Compumotor.The motor is operated in a torque mode, so it acts as a torque source and accepts an analogvoltage as a reference torque signal. The motor DM1004-C is capable of delivering a maximumtorque of 4 N m. The control algorithm is executed at a 2.5 ms-sampling period in a PC hostcomputer equipped with the MultiQ-PCI data acquisition board from Quanser Consulting Inc. Froma practical viewpoint, we have found that the Dahl friction model closely matches the experimentscarried out in our direct-drive actuator [6].

Following the procedures in [24, 25], we have estimated the parameters of the Dahl model offriction for the only actuator of the experimental mechanism:

�o1 = 2764 Nm/rad,

fc1 = 0.745 Nm,

fv1 = 0.144 Nms/rad.

The values of the parameters of the Furuta pendulum are given in the first two columns ofTable I.

We carried out two different experiments to compare the performance of the IDA–PBC controlsystem when friction compensation is present and when it is absent. We considered in bothexperiments the gains given in the third column of Table I (which satisfy the required conditionsgiven in Appendix), and an initial configuration with q2 ∈ (154.22◦,205.78◦) (see details also in



Figure 3. Time evolution of q1(t) without friction compensation.

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Figure 4. Time evolution of q2(t) without friction compensation.

Figure 5. Time evolution of q1(t) using friction compensation.

Appendix). Moreover, we took care that the control input did not exceed the maximum torque ofthe motor.

The desired position in the experiments was qd1 =−90◦. The plots depicted in Figures 3 and 4show the time evolution of joint positions when friction compensation is not present, while Figures 5and 6 show the results when it is considered. Note that the oscillatory behavior in the jointpositions vanishes toward the desired position, with a small offset, when we incorporate frictioncompensation in the control law. This offset can be associated to the discretization process of the



Figure 6. Time evolution of q2(t) using friction compensation.

control algorithm, where velocities are estimated through a backward difference algorithm, insteadof being directly measured.

4. CONCLUSIONS

In this paper, we have presented an extension of the IDA-PBC methodology to control a classof underactuated mechanical systems with friction, where we have modeled friction using theDahl model. We have carried out some experiments to compare the performance of the IDA-PBC control system on a Furuta pendulum, when friction compensation is present and when itis absent. The exhibited experimental evidence showed how the oscillatory behavior of the jointpositions—mainly due to the high friction in the actuator—vanished when friction compensationwas incorporated in the control law. The price to be paid for having the possibility of including theDahl model into the IDA-PBC methodology is the limitation to add new free terms—such as theStribeck and stick-slip effects—in the control law, to achieve a better performance in the presenceof dynamic friction.

APPENDIX A: TERMS IN THE IDA-PBC CONTROL LAW OF THE FURUTA PENDULUM

Consider the control law (30)–(31) and the definitions of Md , J2 and Vd in (32)–(34). The followingconditions ensure that Md is a positive-definite matrix:

d1>0, d1d3 −d22>0.

Both conditions are satisfied if k1, k2 and k3 are choosen such that

k1k2<a3

a2(A1)

and

k3>r2

3 cos2(q2)

[k1k2a3 +a2 sin2(q2)][−r1 +r2 cos2(q2)]. (A2)

The element d3 of matrix Md is positive for q2 ∈ (�−ε,�+ε), where

ε=arccos

(√r1

r2

)



such that |ε|<�/2 with (r1/r2)<1, where

r1

r2= k1a2

2 +k21k2a2a3

k1a22 +k1a2

3

thereby the condition (A1).On the other hand, some algebraic manipulations allow to verify that the desired energy function

Vd is a nonnegative function concerning [q∗1 q∗

2 ]T = [qd1 �]T, ∀qd1 ∈R, q2 ∈ (�/2,3�/2). There-fore, both conditions—positiveness of Md and Vd—simultaneously hold with �−ε<q2<�+ε.Furthermore, it is true that

�−(ε−�)�q2��+(ε−�)

with ��1 and, considering (A2), we have that

k3>r2

3 cos2(ε−�)

[k1k2a3 +a2 sin2(ε−�)][−r1 +r2 cos2(ε−�)](A3)

ensures the positiveness of Md . Here, we have utilized the following trigonometric identities:

cos(a±b) = cosa cosb∓sina sinb

sin(a±b) = sina cosb±cosa sinb

being a =� and b=ε−�.Note that the gain k3 shown in Table I has been chosen taking into account (A3), where

k3>0.1288, with �=0.15 and ε defined by

ε=arccos

(√r1

r2

)=0.45(rad)=25.78◦

such that the initial configuration for q2 is bounded by

q2 ∈ (154.22◦,205.78◦).

Furthermore, substituting the values of Table I in (28), we obtain a3/a2 =1.03; therefore, wechoose k1 and k2 such that k1k2 =0.49 satisfies (A1).

The �i parameters in (35) are

�1 = −sin(q2)[a2�21 cos(q2)−a3�1�2],

�2 = −sin(q2)[2a2�1�3 cos(q2)−a3�1�4 −a3�2�3]+ k1r3 sin(q2)[�+2a2 cos2(q2)]

�2

and

�1 = k3a2�−r3a3 cos2(q2)

��, �2 = d2 −�1a3 cos(q2)

a2,

�3 = k1a3 cos(q2)

�, �4 =−k1,

where �=a1a3 −a22 is the determinant of the inertia matrix M . The rest of terms in (30) are

∇1Vd = kp

⎡⎣q1 −qd1 +

a3arctan(

a2 sin(q2)r4

)r4

⎤⎦ ,

∇2 Hd = �d [ f1 p21 −2 f2 p1 p2]− f3[d3 p2

1 −2d2 p1 p2 +d1 p22]

2�2d

+∇2Vd



being

∇2Vd = −a4 sin(q2)

k1+ kp[a2a3 cos(q2)]

r24 +a2

2 sin2(q2)

⎡⎣q1 −qd1 +

a3arctan(

a2 sin(q2)r4

)r4

⎤⎦ ,

f1 = −2r2 cos(q2)sin(q2)

�− [−r1 +r2 cos2(q2)][2a2 sin(q2)cos(q2)]

�2,

f2 = �[−r3 sin(q2)−r3 cos(q2)[2a2 sin(q2)cos(q2)]]

�2,

f3 = k3 f1 −2d2 f2.

ACKNOWLEDGEMENTS

The authors thank the anonymous reviewers for their valuable comments, and also Dann De la Torre, fromCICESE, for his valuable support during the experimental tests. The first author thanks Ricardo Campafor many helpful remarks. The first and third authors acknowledge Instituto Tecnologico de La Paz andDGEST/CONACyT for their financial support, respectively.

REFERENCES

1. Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A survey of analysis tools and compensation methods forthe control of machines with friction. Automatica 1994; 30(7):1083–1138.

2. Åstrom K. Control of systems with friction. Proceedings of the Fourth International Conference on Motion andVibration Control. Zurich, Switzerland, August 1998; 25–32.

3. Hensen RH, van de Molengraft HJ. Friction induced hunting limit cicle: an event mapping approach. Proceedingsof the American Control Conference, Alaska, U.S.A., 2002; 2267–2272.

4. Dahl PR. Solid friction damping of mechanical vibrations. AIAA Journal 1976; 40(12):1675–1682.5. Canudas de Wit C, Olsson H, Åstrom K, Lischinsky P. A new model for control of systems with friction. IEEE

Transactions on Automatic Control 1995; 40(3):419–425.6. Moreno J, Kelly R, Campa R. Manipulator velocity control using friction compensation. IEE Proceedings Control

Theory Applications 2003; 150(2):119–126.7. Ortega R, Spong MW, Gomez-Estern F, Blankenstein G. Stabilization of a class of underactuated mechanical

systems via interconnection and damping assignment. IEEE Transactions on Automatic Control 2002;47(8):1213–1233.

8. Blankenstein G, Ortega R, van der Schaft A. The matching conditions of controlled Lagrangians and interconnectionand damping assigment passivity based control. International Journal of Control 2002; 75(9):645–665.

9. Chang DE, Bloch AM, Leonard NE, Marsden JE, Woolsey CA. The equivalence of controlled Lagrangian andcontrolled Hamiltonian systems. ESAIM: Control, Optimisation and Calculus of Variations 2002; 8:393–422.

10. Bloch AM. Nonholonomic Mechanics and Control. Springer: New York, 2003.11. Reddy KC, Whitacre WW, Woolsey CA. Controlled Lagrangians with gyroscopic forcing: an experimental

application. Proceedings of the American Control Conference, Boston, MA, U.S.A., 2004; 511–516.12. Gomez-Estern F, van der Schaft A. Physical damping in IDA–PBC controlled underactuated mechanical systems.

European Journal of Control 2004; 10:451–468.13. Ortega R, García-Canseco E. Interconnection and damping assignment passivity-based control: a survey. European

Journal of Control 2004; 10:432–450.14. Canudas de Wit C, Lischinsky P. Adaptive friction compensation with partially known dynamic friction model.

International Journal of Adaptive Control and Signal Processing 1997; 11:65–80.15. Olsson H, Åstrom KJ, Canudas de Wit C, Gafvert M, Lischinsky P. Friction models and friction compensation.

European Journal of Control 1998; 4(3):176–195.16. Kelly R, De La Torre D, Campa R. Energy control of the Furuta pendulum with friction: speed gradient method

in Hamiltonian formulation. Proceedings of the VI Congreso Mexicano de Robotica, Torreon, Mexico, 2004. (inSpanish).

17. Rocha-Cozatl E, Moreno J. Passivity and unknown input observers for nonlinear systems. The 15th WorldCongress of the International Federation of Automatic Control (IFAC ), Barcelona, Spain, 2002.

18. Acosta JA, Ortega R, Astolfi A, Mahindrakar A. Interconnection and damping assignment passivity-basedcontrol of mechanical systems with underactuation degree one. IEEE Transactions on Automatic Control 2005;50(12):1936–1955.



19. Mahindrakar A, Acosta JA, Ortega R, Astolfi A, Viola G. Further constructive results on interconnection anddamping assignment control of mechanical systems: the acrobot example. International Journal of Robust andNonlinear Control 2006; 16(14):671–685.

20. Viola G, Ortega R, Banavar J, Acosta JA, Astolfi A. Total energy shaping control of mechanical systems:simplifying the matching equations via coordinate changes. IEEE Transactions on Automatic Control 2007;52(6):1093–1099.

21. Sandoval J, Ortega R, Kelly R. Interconnection and damping passivity-based control of the pendubot. The 17thWorld Congress of the International Federation of Automatic Control (IFAC ), Seoul, Korea, 2008; 7700–7704.

22. Khalil HK. Nonlinear Systems. Prentice-Hall: Upper Saddle River, NJ, U.S.A., 1996.23. Åstrom K, Furuta K. Swinging up a pendulum by energy control. Automatica 2000; 36:287–295.24. Kelly R, Llamas J, Campa R. A measurement procedure for viscous and Coulomb friction. IEEE Transactions

on Instrumentation and Measurement 2000; 49(4):857–861.25. Olsson H. Control systems with friction. Ph.D. Dissertation, Department of Automatic Control, Lund Institute

of Technology, Lund, Sweden, 1996.


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