Decoupling force and position control in constrained

motion with friction

Giorgio Bartolini ¦ and Elisabetta Punta ?

¦ DIEE - University of CagliariPiazza d’Armi - 09123 Cagliari, Italy

e-mail: giob@dist.unige.it, giob@elettro1.unica.it

? ISSIA CNR - National Research Council of ItalyVia De Marini, 6 - 16149 Genova, Italye-mail: punta@ge.issia.cnr.it

Abstract

This paper deals with the force/position control of con-strained mechanical systems subjected to uncertain-ties and disturbances of various nature, including fric-tion. The hybrid control problem with continuous ac-tive forces and discontinuous friction disturbance isaddressed in the paper by exploiting the robustnessproperties of the second-order sliding-mode control al-gorithms. An algorithm of this kind is proved to beeffective to avoid complex stick-slip phenomena and toattain, by suitably choosing the sliding surfaces, a dy-namic decoupling of the force and position control tasksduring constrained motion with friction.

1 Introduction

In this paper the hybrid control of simple constrainedmechanisms with frictional contact is considered. Inprevious works [1], [2], the second order sliding modecontrol has been exploited as a method which is robustagainst uncertainties in modelling exogenous distur-bances in both actuation and sensing devices and whichis chattering free in the sense that the force/torque in-put is continuous. The aim of this paper is to show,starting from elementary mechanical systems, that,when there is friction between the mechanical systemand the constraint surface, this procedure, decouplingforce and position control, can no longer be adopted.Even if appropriate and precise models of the frictionhave been developed [3], [4], the adopted friction modelis discontinuous on velocity and therefore the motioncan be forced by friction to undesired sliding motion,called trapping point. The point to point motion un-der the action of the suboptimal second order slidingmode control, is proved to be characterized by no inter-mediate trapping point, if the modulus of the system’s

uncertainties can be bounded by known positive func-tion of available quantities.In general cases, as far as the force control is concerned,there is an algebraic relation relating control, normalforce (to be controlled), and friction. The applicationof the procedure described in [1], [2], to track a desirednormal force profile would imply the time differentia-tion of the friction term. This fact makes the mathe-matical description of the force control extremely deli-cate during any rest phase, that is whenever a trappingpoint occurs, and in particular if the adopted frictionmodel is discontinuous. To avoid this fact, which takesplace in any case except the trivial one, it is necessaryto slightly modify the force control objective.In the paper first it is proved that, in presence of fric-tion, the decoupling between force and position controlcannot be performed in general, but just in some triv-ial and particular cases, then it is shown that, start-ing from these cases, it is possible to find suitablesliding surfaces so that, when a motion is forced onthem, asymptotically the desired decoupled behaviouris achieved.

2 Position control of mechanical systemswith friction

Consider a mechanical system affected by a boundedand discontinuous disturbance modeling the frictionphenomena

x1 = x2

x2 = x3 = f(x1, x2) + g(x1)u− C(x2) sign (x2)(1)

C(x2) is the modulus of the friction term which in gen-eral depends on system states and external actions [5],[6], but it is assumed, together with its derivatives,bounded in the bounded domain of interest. For ex-

ample, according to models proposed in literature [7],[3], C(x2) could be expressed as C(x2) = Cc + (Cs −Cc) exp−( x2

vs)2

, where Cc is the Coulomb friction level,Cs is the level of stiction, and vs is the Stribeck veloc-ity. The viscous friction effect is incorporated in theterm f(x1, x2), therefore C(x2) ≤ Cs. Moreover as-sume to know a function F (‖x‖) non decreasing withits argument ‖x‖ and two constants g1 and g2 suchthat the following holds |f(x1, x2)| < F (‖x‖), and0 < g1 ≤ g(x1) ≤ g2. The control aim is that of steer-ing exponentially to zero both x1(t) and x2(t) by meansof a continuous control u(t).An undesired sliding motion on x2 = 0 (“trappingpoint”) can occur if |f(x1, x2) + g(x1)u(t)| ≤ C(x2)and the point is maintained until the sign of the in-equality changed.Pathological phenomena such as stick-slip limit cyclesassociated with the occurrence of a sequence of trap-ping points, can arise when standard control algorithmsare used [5]. The solution proposed in [8] avoids anyproblem of this kind since it is ensured that during themotion from the initial position to the desired final oneat most a single trapping point is encountered.The continuous control u(t) is the output of a dynami-cal system u = h(u, v), where v(t) is the actual controlsignal. In particular let u = v.System (1) with friction and continuous control can beinterpreted as a discontinuous system, due to the pres-ence of friction, in which the control objective is thatof avoiding sliding motion on x2 = 0.If a sliding motion on x2 = 0 occurs, system (1) is in a“trapping point” x1 = x1, x2 = 0, and according to theFilippov’s solution concept, the system representationis

x1 = x1 6= 0x1 = x2 = 0x2 = x3 = f(x1, 0) + g(x1)u + (1− 2α)C(0)u = v

α = f(x1,0)+g(x1)u+C(0)2C(0)

(2)

where α ∈ [0, 1] according to the trapping conditionstated below.Definition 1: System (1) represented by (2) duringthe sliding motion on x2 = 0, is leaving the trappingpoint (x1, 0) at t = t0, if

α[x(t0), u(t0)] =

1 ddt [α(x, u)]t=t0

> 00 d

dt [α(x, u)]t=t0< 0

After t = t0 and until a new trapping point isreached, system (1), under mild regularity conditionson f(x1, x2) and g(x1), can be represented by

x1 = x2

x2 = x3 = f(x1, x2) + g(x1)u± C(x2)x3 = x2 = d

dtf(x1, x2) + ddtg(x1)u± d

dtC(x2) + g(x1) v

If s(x1, x2) = x2 + c x1 is steered to zero in a finitetime by a possibly discontinuous signal v(t), the samecontrol objective as in the standard first order slidingmode control case is achieved by means of a continuouscontrol action u(t) = u(0) +

∫ t

0v(τ)dτ , therefore the

chattering phenomenon is eliminated. To obtain theseresults differentiate s(x1, x2) two times

s = f(x1, x2) + g(x1)u± C(x2) + c x2

s = m1(x1, x2) + m2(x1, x2)u + g(x1)v(3)

where m1(x1, x2) and m2(x1, x2) collect all the termsdepending on x1 and x2.By setting s(x1, x2) = x2 + c x1 = y1, y1 = y2, then,since relations (3) hold, the whole system can be rep-resented as

x1 = x2 = y1 − c x1

y1 = y2 = f(x1, x2) + g(x1)u + c x2

y2 = m1(x1, x2) + m2(x1, x2)u + g(x1)vu = v

(4)

Assume that there exist two functions m1(‖x‖) andm2(‖x‖) non decreasing with their argument ‖x‖ suchthat the following inequalities hold |m1(x1, x2)| <m1(‖x‖) and |m2(x1, x2)| < m2(‖x‖). It was proved in[8] that the regulation problem for system (1) is solvedby applying a discontinuous control law v(t), designedaccording to the 2-SM algorithm in [9], and expressedby

v(t) = −V (t)sign[y1(t)− 1

2y1M

](5)

V (t) =

VM y1M

[y1(t)− 1

2y1M

]> 0

α∗VM y1M

[y1(t)− 1

2y1M

]< 0

where VM > 0, α∗ ≥ 1, and, ∀t ∈ [0,∞), if y2(t) = 0then y1M = y1(t), i = i + 1, and tMi = t.With a suitable choice of the control parameters α∗ andVM the control law v(t) guarantees that: a trappingpoint once reached is left in a finite time; starting froma trapping point the sliding manifold can be reached ina finite time without encountering any further trappingpoint; on the sliding manifold the origin of the phaseplane is reached exponentially.The proposed solution [8] relies on the fact that, if asliding mode occurs on the sliding surface s(x1, x2) =x2 + c x1 = 0, the only reachable trapping point isx1 = x2 = 0, in fact, on s(x1, x2) = 0, the frictionterm turns out to be continuous until the state originis reached and starting from any point of the slidingmanifold the state trajectories converge exponentiallyto the origin according to the zero dynamics of thesystem constrained to the surface.

3 Constrained mechanical systems: theforce/position control in presence of friction

Consider a constrained mechanical system

M(q)q + C(q, q)q + g(q) = τ + τc

Γ(q) = 0 (6)

with q ∈ Rn, τ ∈ Rn, τc ∈ Rn and Γ(q) : Rn → Rm.According to standard projection techniques [10], [11],it can be considered a partition of the lagrangian co-ordinate vector q′ = [ q′1 q′2 ] where q1 ∈ Rn−m andq2 ∈ Rm, it is assumed the existence of a convex subsetΩ1 ⊂ Rn−m and of a function k ∈ C2 ( k : Ω1 → Ω2,with Ω2 open subset of Rm) such that Γ(q1, k(q1)) = 0for all q1 ∈ Ω1. The vector q2 is uniquely defined bythe vector q1: q2 = k(q1) for all q1 ∈ Ω1. Moreover,the region Ω = Ω1 × Ω2 ⊂ Rn is defined. Withinthe region Ω, the vector of the generalized coordi-nates q can be parameterized, even if the parameter-ization is not uniquely defined. The system’s motionis constrained on the smooth manifold S = (q, q) :Γ(q) = 0, J(q)q = 0, where J(q) is the Jacobian ma-trix J(q) = ∂Γ(q)/∂q. Then, only (n−m) coordinatesin q are independent. The vector of the generalizedcoordinates q can be expressed in terms of q1, the inde-pendent sub-vector, as q′ = [q′1 k′(q1)] and the vector ofthe velocities q in terms of the independent velocities asq = H(q)q1, being H ′(q) =

[In−m

(−J−12 (q)J1(q)

)′]

where J1(q) = ∂Γ(q)/∂q1 and J2(q) = ∂Γ(q)/∂q2. Ac-cording to the previous assumptions, the Jacobian ma-trix J2(q) never degenerates in the set Ω.The presence of the constraint surface introduces theaction of the generalized constraint forces τc. When thecontact with the constraint is frictional, τc = f + ff ; fand ff can be expressed as

f = J ′(q)λ and ff = H(q)fr(µ, λ, q) (7)

where λ ∈ Rm is the vector of the Lagrange multi-pliers and fr ∈ Rn−m is the friction which depends onthe modulus of the contact forces during the constraintmotion, the friction coefficients µ ∈ Rn−m, and the sys-tem’s velocities.The first equation of the model (6) can be multiplied,as in [10], by the matrix T (q), defined as T ′(q) =[H(q)

(J(q)M−1(q)

)′].As a result, it is obtained an equation of the form

M∗(q)q1 + C∗(q, q)q1 + g∗(q) = H ′(q)τ + H ′(q)ff (µ, λ, q)λ = Z(q) [Cλ(q, q)q1 + g(q)− ff (µ, λ, q)− τ ] .

(8)M∗(q) = H ′(q)M(q)H(q)Cλ(q, q) = M(q)H(q) + C(q, q)H(q)C∗(q, q) = H ′(q)Cλ(q, q)g∗(q) = H ′(q)g(q)Z(q) =

[J(q)M−1(q)J ′(q)

]−1J(q)M−1(q)

Z(q) = [Z1(q) Z2(q)]Z1(q) ∈ Rm×(n−m) Z2(q) ∈ Rm×m.

Either, by multiplying the first equation of the model(6) by the matrix T (q), defined as T ′(q) = [H(q) J ′(q)],it is obtained an equation of the form

M∗(q)q1 + C∗(q, q)q1 + g∗(q) = H ′(q)τ + H ′(q)ff (µ, λ, q)λ = Z(q) [M(q)H(q)q1 + Cλ(q, q)q1 + g(q)− τ ] ,

(9)Z(q) = [J(q)J ′(q)]−1

J(q)Z(q) = [Z1(q) Z2(q)]Z1(q) ∈ Rm×(n−m) Z2(q) ∈ Rm×m.

It appears from(8) and (9) that, when the friction ef-fect is taken into account, a decoupled model can nolonger be exploited. Then the plain application of thepreviously introduced control procedure for the regu-lation of a mechanical system in presence of frictionto the case when both position and force have to beregulated, is not straightforward. In fact we have toconsider an uncertain algebraic equation coupled to anuncertain dynamical system by means of a term, thefriction effect, discontinuous on the system’s velocities.The first case we are going to analyze is that of a massconstrained to a surface. For this system, even in pres-ence of uncertainties, it is still possible to obtain a de-coupled model, thanks to the particular structure ofthe inertia matrix M(q), which is constant and, muchmore, diagonal, being all the elements on the diagonalthe same.

3.1 Force/position control of a massconstrained to a surfaceConsider a mass constrained to a surface, its dynamicscan be expressed as

Mq + g = τ + τc

Γ(q) = 1√1+γ2

( q2 − γ q1 ) = 0 (10)

where q = [ x y ]′ is the vector of the Cartesian coor-dinates of the mass, Γ(q) = 0 is the constraint surface,M = mI is the inertia matrix, g is the vector of thegravitational forces, τ is the vector of the control forces,and τc = f + ff is the vector of the constraint forces,including both the reaction and the friction term.It can be defined the Jacobian matrix J = ∂Γ(q)

∂q =1√

1+γ2[−γ 1 ]. Due to the constraint, the coordi-

nate q2 can be expressed as a function of q1, the in-dependent coordinate, and it results q2 = γ q1, whilethe vector of the velocities q is function of the in-dependent velocity q1 as follows, q = H q1, beingH =

[1 − J−1

2 J1

]′= [ 1 γ ]′.

The constraint force is given by f = J ′λ and, follow-ing the Coulomb model, the friction term is written asff = −H 1√

1+γ2µ |λ| sign q1, being µ the friction co-

efficient of the constraint surface.Let the control vector τ be chosen as follows

τ = H1√

1 + γ2u1 + J ′ u2. (11)

Finally, for system (10), the reduced order model (8),which is not only equivalent, but even formally thesame of (9), results to be

m∗ q1 = −g∗ + nH (u1 − µ |λ| sign q1 )λ = g − u2,

(12)

where m∗ = m(1 + γ2

), g∗ = ( g1 + γ g2 ), g =

(−γ g1+g2 )1+γ2 , and nH =

√1 + γ2 , and it has been ex-

ploited the particular structure of the inertia matrix,M = mI, to get the decoupled equations (12).The final expression (12) is suitable to be dealt withby a combination of first order and second order slid-ing modes. If the problem consists in steering both q1

and λ to their desired values q1dand λd(t), the follow-

ing procedure can be applied.Choose two sliding outputs

s1 = ( q1 − q1d) + c ( q1 − q1d

) ,

s2 =∫ t

0[ λ(θ)− λd(θ) ] dθ − m ( q1 − q1d

) ,

and consider the two control signals v1(t) and v2(t)

u1 = v1 = −V1(t) sign[s1(t)− 1

2s1M

],

u2 = v2 = V2 sign s2(t)

the choice of V1 > 0 can be made on the basis ofcomputable upper-bounds of the system which willbe specified in the following, while it is chosen V2 =(λdM

+ h2)

with λdM> maxt∈[0,∞) |λd|.

Due to the decoupled structure of the system, if v2 isapplied, λ → λd in a finite time interval during which|λ(t)| < |λ(0)|+ λdM

, being λdM> maxt∈[0,∞) |λd|.

Assume that the system in t = 0 is in a trappingpoint, while it remains in the trapping point, that is∀t ≥ 0 : µ |λ(t)| >

∣∣∣ u1(t)− g∗nH

∣∣∣, the Filippov’s repre-sentation of the system is

q1 = 0q1 = 0λ = g − u2

u = v.

According to (5), s1(0) = s1M= c[q1(0) − q1d

] is anextremal value, therefore, while the system is in thetrapping point, the applied control signal v1 is

v1 = −V1(t) sign s1(0)

Since u1(t) increases linearly with time there exists atime instant t = t0 at which µ |λ(t0)| >

∣∣∣ u1(t0)− g∗nH

∣∣∣,this means that the system at t = t+0 is leaving the trap-ping point. After t = t+0 it makes sense differentiatingequation (12),

q(3)1 = nH

m∗

(u1 − µ d

dt |λ| sign q1

)

λ = −u2

where ddt |λ| = ∂

∂λ |λ| λ and, according to the chain rulefor non smooth functions [12], ∂

∂λ |λ| is given by

∂

∂λ|λ| =

sign λ λ 6= 0,p p ∈ [−1, 1] λ = 0.

Let y1 = s1, z1 = s2, and φ(t) = −q1d(3) − c q1d

, then

y1 = y2

y2 = φ(t) + c q1 + nH

m∗

(v1 − µ λ sign λ sign q1

)

z1 = z2 = −v2 − λd.

The first two equations can be expressed as

y1 = y2

y2 = m1(q1, λ, λ) + m2 u1 + nH

m∗v1

(13)

where, according to the first equation in (12), it hasbeen expressed q1 as a function of q1, u1, λ, and thevarious components have been collected into the twoterms m1(q1, λ, λ) and m2. Bounds of the two termscan be easily found: m2 of the term m2 and an upper-bound of the modulus of m1(q1, λ, λ) is m1.The proposed control procedure can be applied to sys-tem (12), and it is obtained the asymptotic convergenceof x and x to zero. This result is obtained by meansof a continuous control u1(t), which is the integral of adiscontinuous signal v1(t)designed following the secondorder sliding mode Algorithm 1 with the parameterschosen according to the conditions found in the con-sidered case of a mass constrained to a surface. TheLagrange multiplier λ converges to the desired valueand even the force control objective is guaranteed bymeans of a continuous control u2(t) which is the inte-gral of a discontinuous signal v2(t).

3.2 Force/position control of a constrainedCartesian robotWe consider a planar Cartesian robot with two degreesof freedom with the end effector constrained on an sur-face. With respect to the previously considered caseof the mass (10), the difference in the system’s equa-tions is given by the fact that now the inertia matrixM is still diagonal, but with different elements, that isM = diagmi, i = 1, 2, m1 6= m2. According to thesecond projection method (9), we attain the followingconstrained model for the Cartesian robot

m∗ q1 = −g∗ +√

1 + γ2 (u1 − µ |λ| sign q1 )λ = m q1 + g − u2,

(14)

where m∗ =(m1 + γ2 m2

), g∗ = ( g1 + γ g2 ), m =

γ (−m1+m2 )1+γ2 , and g = (−γ g1+g2 )

1+γ2 . To decouple the twoequations (14), we choose a sliding output s2 differentwith respect to the previous case, then let

s1 = ( q1 − q1d) + c ( q1 − q1d

)s2 =

∫ t

0[λ(θ)− λd(θ) ] dθ − m ( q1 − q1d

) ,(15)

the first and second time derivative of s2 do not dependon ff since

s2 = −u2 − λd + m q1d,

s2 = −v2 − λd + m q(3)1d

,

while s1 is identical to the previous case. It is possi-ble to force both s1 and s2 in (15) to zero, indepen-dently, by using a couple of second order sliding modecontrollers. As a result we have that s1 = s1 = 0 ands2 = s2 = 0 in finite time. The condition s1 = 0 impliesthat q1 → q1d

arbitrarily exponentially fast, thereforethe reaching of the condition s2 = 0 means that λ → λd

arbitrarily exponentially fast.The problem we are considering is a point to point mo-tion control which is, in this case, the motion betweentwo different trapping points. Regardless the simplifiedmodel adopted for the friction term, that is the abruptsign model, it is guaranteed that no discontinuity takesplace in s2. The proposed control procedure is appliedto system (14), and it is obtained the convergence ofq1 and λ to their desired values (Figure 1). This re-sult is obtained by means of continuous control signalsu1(t) and u2(t), Figure 2, which are the integral of adiscontinuous signals v1(t) and v2(t), designed accord-ing to the proposed control procedure. In Figure 3 itis shown the convergence of both s1 and s2, given by(15), to zero.Continuing our analysis, let us consider a further casewhen both m1 and m2 are unknown but constant. Theonly assumption we make is that u2(t) is measurable.From the second equation in (14), we have that u2(t) =−λ + m q1 + g. Let us introduce the auxiliary signals

u2(t) = −λd + m(t) q1d+ g,

v2(t) = −λd + ˙m q1d+ m q

(3)1d

and choose the sliding outputs s1 and s2 as follows

s1 = ( q1 − q1d) + c ( q1 − q1d

)s2 =

∫ t

0[ u2(θ)− u2(θ) ] dθ;

(16)

setting m(t) = m(t)− m, s2 can be rewritten as

s2 =∫ t

0[λ(θ)− λd(θ) ] dθ − m ( q1 − q1d

)+

∫ t

0m(θ) q1d

(θ) dθ.

Differentiating s2 with respect to time, we obtain

s2 = ( λ− λd )− m ( q1 − q1d) + m q1d

,

where m(t) and ˙m(t) depend on the adopted adapta-tion law.Assuming to know bounds for the desired trajectoriesλd, λd, q1d

, q(3)1d

, and for the uncertain system’s term m,the uncertain drift term v2 can be bounded, thereforedesigning v2 according to a second order sliding modecontrol algorithm, it is possible to steer to zero both s2

and s2. Due to the decoupling between s1 and s2, also

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

0

1

2

3

4

5

q 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

8

λ

t [sec]

Figure 1: The state q1 and the force λ converge to theirdesired values.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

u 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

u 2

t [sec]

Figure 2: The continuous control forces u1 and u2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.01

−0.005

0

0.005

0.01

s 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t [sec]

s 2

Figure 3: the convergence of s1 and s2 to zero.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

q 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−2

0

2

4

6

8

t [sec]

λ

Figure 4: The state q1 and the force λ converge to theirdesired values.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

u 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

u 2

t [sec]

Figure 5: The continuous control forces u1 and u2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.01

−0.005

0

0.005

0.01

s 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

0

0.2

0.4

0.6

0.8

t [sec]

s 2

Figure 6: the convergence of s1 and s2 to zero.

s1 and s1 can be steered to zero, as a result q1 → q1d

arbitrarily exponentially fast.On s2 = 0 the following identity holds ( λ− λd ) =−m q1d

+ η, being η an exponentially vanishing quan-tity. This fact allows to consider ( λ− λd ) as a pre-diction error corrupted by an exponentially vanishingterm

(λ− λd ) = Θ X + η

where Θ = m(t) and X = −q1d.

Choosing as adaptation law

˙Θ = −ΓX (λ− λd ) ,

since η → 0, it holds that ( λ− λd ) → Θ X; by stan-dard argumentation, based on Barbalat’s Lemma, wehave that Θ X → 0. This in turn implies, since s2 = 0,λ → λd.When the system (14) is uncertain, the application ofthe control procedure obtains the convergence of q1 andλ to their desired values (Figure 4). This result is at-tained by means of continuous control signals u1 andu2, Figure 5, which are the integral of a discontinuoussignals v1 and v2, designed according to the proposedcontrol procedure. In Figure 6 it is shown the conver-gence of both s1 and s2, given by (16), to zero.

4 Conclusions

In this paper it is shown that a second order slidingmode control algorithm conceived as a solution to thechattering elimination problem is also effective againstdiscontinuous disturbances like friction. The efficacy ofthe presented method to avoid complex stick-slip phe-nomena is related to the fact that, during transients,no oscillations or overshoot take place if the control pa-rameters are suitably tuned. Furthermore, by suitablychoosing the sliding surfaces, the proposed methodol-ogy results to be effectual even to attain a dynamicdecoupling of the force and position tasks during con-strained motion with friction.

AcknowledgementsWork partially supported by MURST, P.RI.N. “Guid-ance and Control of Underwater Vehicles” and by CNR,

“Progetto Giovani”, codice no. CNRG004E90, linea diricerca: Fluidodinamica Industriale.

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