multi-input second-order sliding-mode hybrid control of constrained manipulators

Dynamics and Control, 10, 277–296, 2000c© 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Multi-Input Second-Order Sliding-Mode HybridControl of Constrained Manipulators

G. BARTOLINI* [email protected], [email protected] of Electrical and Electronic Engineering, University of Cagliari, Italy

A. FERRARA [email protected] of Computer and Systems Engineering, University of Pavia, Italy

E. PUNTA [email protected] of Communication, Computer and System Sciences, University of Genova, Italy

Editor: M. J. Corless

Received June 25, 1999; Revised May 26, 2000; Accepted July 26, 2000

Abstract. This paper deals with the hybrid position/force control of constrained manipulators subjected to uncer-tainties. A solution is proposed that is based on sliding-mode control theory, which proved to be highly effectivein counteracting uncertainties for some classes of nonlinear systems. Specific problems involved in this techniqueare chattering elimination and the algebraic coupling between constraint forces and possibly discontinuous controlsignals. Both the problems are addressed in this paper by exploiting the robustness properties of a second-ordersliding-mode control algorithm. This algorithm, recently proposed by the authors for solving the single-inputhybrid control problem, is generalized in this paper to deal with the class of multi-input differential algebraicsystems describing the behaviour of constrained mechanical systems.

Keywords: second-order sliding-mode control, multi-input second-order sliding-mode control, mechanical sys-tems, constrained manipulators

1. Introduction

The problem of the hybrid position/force control of robot manipulators has been extensivelyinvestigated during the last decade to improve the performances of both simulation andcontrol. The treatment of this control problem needs to start with the representation of theconstrained mechanism with holonomic constraints in terms of a redundant set of differentialalgebraic equations (DAE),n differential equations withm< n algebraic constraints [11].

Since the hybrid control of a constrained robot is the primary objective, the nature ofthe dependence between the control vector and the constraint forces or the free trajectoriesmust be explicitly defined, that is, what is usually called the “reduction problem” has tobe faced. This problem consists in identifying a projection of the original DAE throughwhich a separation between the algebraic and differential parts of the system equation canbe formally accomplished [13], [18], [26]. The involved nature of such a projection requiresa perfect knowledge of the model of the manipulator and of the object to be manipulated. Inthat case, the feedback linearization control approach can be applied to the reduced system

* This work was partially supported by contract MAS3CT 950024 - AMADEUS of the European Community.

278 BARTOLINI, FERRARA AND PUNTA

according to [19], but both the complexity and the sensitivity of the related transformations touncertainties and disturbances prevent the use of this technique in some practical situations.

When uncertainties affect the knowledge of the model, the control problem turns out to bedifficult to solve. Reference [20] presents a particular projection technique for constrainedmanipulators for which uncertainties are expressed in a suitably parametrized form to whichthe classical model reference adaptive approach applies. Unfortunately, this approachbecomes very cumbersome when the number of degrees of freedom (d.o.f.) increases.

A procedure based on the variable structure control (VSC) technique, which exploits somestructural properties of Lagrangian systems, reveals to be, at least theoretically, effectiveand easy to use [9], [23], [25]. But, this approach suffers from the so-called chatteringphenomenon, that is, the high frequency oscillation caused by the discontinuity of thecontrol signals. Various techniques to eliminate the chattering effect have been proposedrecently [7], [10], [4]. An effective one consists in representing the system, through timedifferentiation, in an augmented state space in which the time derivative of the controlacts as the actual control signal. The derivative of the control is the input to an integratorwhose output is the true control. This technique relies on the possibilities of either takinginto account the actuator dynamics or introducing an integrator into the control loop, ifsuch dynamics is sufficiently fast. The new control is discontinuous on a suitable manifoldbut its integral (for instance, a force or a torque) is continuous. Note that this philosophyfor chattering elimination is analogous to the one that underlies the use of pulse widthmodulated (PWM) amplifiers [21]; therefore, it is suitably exploitable from a practicalpoint of view.

This paper is aimed at extending the proposed [6] second-order sliding-mode controlapproach to the case in which the evolution of the system, a constrained manipulator, isrepresented by a class of DAE with anyn − m. In Section 2, the case of the controlof a system represented by a DAE withn − m = 1, already discussed in [6], is brieflyrecalled for the readers’ convenience. The main contribution of this paper consists in ageneralization of the procedure to the case of ann d.o.f. manipulator constrained to anm-dimensional surface in the presence of various kinds of uncertainties. The treatment isdeveloped in Sections 3 and 4. In Section 3, the control of a system that can be representedby a multi-dimensional DAE is described as a generalization of the methodology presentedin Section 2. In Section 4, the hybrid control strategy is described in more details. Twosituations are explicitly addressed: in the first, the system is assumed to be uncertain butthe Jacobian matrix relevant to the constraint equation is known. This assumption allows atransformation of the control matrix into a triangular form that enables one to decouple theposition control and the force control problem. The second situation is relevant to the caseof an uncertain Jacobian matrix. The particular structures of the matrices appearing in thereduced equation are exploited just to design a procedure that first hierarchically fulfills theposition control objective, by forcing the state trajectory on a suitable second-order slidingmanifold in a finite time, and then satisfies the force reference tracking problem by forcingthe resulting equivalent system to a first-order sliding-mode. Simulations relevant to afour-degree-of-freedom manipulator constrained to the horizontal plane are presented inSection 5 to illustrate the effectiveness of the proposed methodology. Finally, conclusionsare drawn in Section 6.

MULTI-INPUT SECOND-ORDER SLIDING-MODE HYBRID CONTROL 279

2. Chattering Elimination

The aim of this section is to briefly recall the control problem which is presented and solvedin [6], the generalization of which is the main contribution of this paper.

Consider a constrained mechanical system described by the following differential alge-braic equations{

x = A(x)+ B11(x)u1+ B12(x)u2

y = D(x)+ B21(x)u1+ B22(x)u2(1)

wherex ∈ R, y ∈ R, u1 ∈ R, andu2 ∈ R. The system is uncertain and it is assumed thatthe uncertain terms are bounded by known bounds

A(x) < A ∂∂x A(x) < A

D(x) < D ∂∂x D(x) < D

Bi jm ≤ Bi j (x) ≤ Bi j M Bi jm ≤ ∂∂x Bi j (x) ≤ Bi j M

The problem consists in steeringx andy to the desired valuesx∗ andy∗, for instancex∗ = 0andy∗ = 0, in a finite time. Moreover, the control signals must be continuous functions oftime.

This problem cannot be solved by standard robust control techniques due to the uncer-tainties and the algebraic relation betweeny and the control vectoru. In fact the standardVSC approach makes it possible to steerx to zero but noty which, since it is algebraicallycoupled to a discontinuous control, chatters at a theoretically infinite frequency.

A possible solution is instead provided by the following procedure: differentiate (1) andsetx1 = x, x2 = x, to obtain

x1 = x2

x2 = F(x1, x2,u)+ B11(x1)v1+ B12(x1)v2

y = G(x1, x2,u)+ B21(x1)v1+ B22(x1)v2

u1 = v1

u2 = v2

(2)

F(x1, x2,u) = ∂

∂x1A(x1)x2+ ∂

∂x1B11(x1)x2u1+ ∂

∂x1B12(x1)x2u2

G(x1, x2,u) = ∂

∂x1D(x1)x2+ ∂

∂x1B21(x1)x2u1+ ∂

∂x1B22(x1)x2u2

wherex2 is not available, since the first equation of (1) is uncertain. System (2) is anuncertain system for which the control vector isv. The control task is that of steering boththe system outputsx1 andy to zero. Since the relative degree [12] betweeny andv is equalto one and the relative degree betweenx1 andv is equal to two, the control objective can beregarded as the attainment in a finite time of a first-order sliding motion ony = 0 and of a


Figure 1. The trajectories of the double integrator under the action of the suboptimal strategy.

second-order sliding motion onx1 = 0. The second-order sliding motion onx1 = x2 = 0can be enforced by the controlv1 applying previously introduced [3], [4] second-ordersliding-mode control strategies. Once the second-order sliding motion onx1 = 0 occurs,the first-order sliding motion ony = 0 is enforced by the controlv2 according to the controlhierarchy concept.

For the readers’ convenience it is provided a synthetic and qualitative presentation ofsecond-order sliding-mode control approaches.

Consider a double integratorz1 = z2, z2 = w, and assume that only the sign ofz2 isavailable, there are algorithms capable of causing the finite time reaching of the origin.One, in particular, can be derived from the well-known time optimal bang-bang controlapproach and can be expressed as follows.

When t= 0, set z1M = z1(0), i = 0, tMi = 0. While t ∈ [0,∞), if z2(t) = 0 then setz1M = z1(t), i = i + 1, and tMi = t , then apply

w(t) = −WM sign

[z1(t)− 1

2z1M

]WM > 0 (3)

The application of the control strategy (3) steers the state trajectory of the double inte-grator to the origin of the state plane with at most two commutations, instead of the singlecommutation characterizing the optimal bang-bang control, Figure 1. In this sense, thecontrol law (3) gives rise to a suboptimal strategy.

Now consider the double integrator systemz1 = z2, z2 = h(z) + d(z)w perturbedby uncertain terms|h(z)| < H , 0 < d1 < d(z) < d2. The solution to the perturbedcase proposed in [5] is based on the previously outlined suboptimal algorithm. It canbe proved that, despite the uncertainties, if the control amplitude is sufficiently high, theapplication of the control strategy (3) to the perturbed double integrator generates a sequence


Figure 2. The worst case trajectory is not contractive, the possible control counteractions are: to increase thecontrol amplitudeWM (a), to use an asymmetric commutation logic (b), to anticipate the commutation (c).

of successive singular points{z1(tMi ), tMi : z2(tMi ) = 0}, and this sequence is strictly

contractive, that is|z1(tMi+1)||z1(tMi )| ≤ q < 1. Moreover, the reaching time is a series of positive

elements upper-bounded by a geometric series with ratio strictly less than one. Therefore,∑∞i=1(tMi − tMi−1) = T <∞.The analysis can be carried out considering the worst case, that is when the uncertainties

always act against the attainment of the contraction. Assume that, for a certain choice ofWM ,the worst case behaviour, sketched in Figure 2 with the dashed line, is not characterized bythe contraction effect, there are at least three ways to achieve the prefixed control objective:

1. to increase the control amplitudeWM , that is to reduced2WM+Hd1WM−H , Figure 2 case (a);

2. to use an asymmetric commutation logic by suitably choosing a parameterα, that is, ifz1M [z1(t)− 1

2z1M ] > 0 then the control amplitude isWM elseαWM , Figure 2 case (b);

3. to anticipate the commutation at the moment wheny1 = βy1(tMi ), with β ∈ [ 12, γ ],

γ < 1, Figure 2 case (c).

Each of the above actions makes the worst case turn out to be characterized by a finite timeconvergent response.

On the basis of the previous considerations a rather general control algorithm which canbe successfully applied to the perturbed double integrator, has the following form:

Algorithm 1

• When t= 0, set z1M = z1(0), i = 0, tMi = 0

• While t ∈ [0,∞)1. If z2(t) = 0 then set z1M = z1(t), i = i + 1, and tMi = t ;


2. If z1M [z1(t)− 12z1M ] > 0 then W(t) = WM else W(t) = αWM

3. Apply

w(t) = −W(t) sign

[z1(t)− 1

2z1M

]WM > max

(H

d1,

4H

3αd1− d2

)α > 1 α 6= d2

3d1(4)

Under the control action (4), the sequence{z1(tMi ), tMi : z2(tMi ) = 0} is convergent tozero and the series

∑∞i=1(tMi − tMi−1) converges to a finite valueT which is the reaching

time.In the above control algorithm it is assumed|h(z)| < H , this assumption which appears

to give local validity to the treatment, is made for simplicity sake. Indeed in [5] and in veryrecent work [8] it is proved that it is possible to compute at the first singular pointt = tM1

a constant upperboundH [z1(tM1)] which is proved to remain valid∀t ∈ [tM1,∞), while toreach the first singular pointz1(tM1) it is sufficient to know a constant upperboundZ2 ofthe modulus of the unavailable quantityz2(0).

The above control algorithm requires the knowledge of the sign ofz2(t), therefore it isassumed the availability of ideal peak detectors, with infinite bandwidth to identify the sin-gular points. Practical aspects such as time delays, discretization, and measurement errorshave been addressed in [4], [16], and are the object of theoretical as well as experimentalinvestigation.

Now let us go back to our original treatment and observe that, as it was shown in [6], a con-trol problem which can be faced via the recalled procedure is just the hybrid force/positioncontrol of a planar manipulator withn = 2 degrees of freedom the end effector of which isconstrained to a line.

3. A Multi-Input Version of the Control Problem

It is presented a possible generalization to multi-input cases of the problem considered inthe previous section.

In contrast with the standard multi-input first-order sliding-mode control a Lyapunov likeapproach is not available for the second-order sliding-mode control of uncertain systems.Hereafter it is presented a solution for the considered control problem in presence of asufficiently wide class of uncertainties.

Consider a system characterized by the following differential algebraic equations{x = A(x)+ B11(x)u1+ B12(x)u2

y = D(x)+ B21(x)u1+ B22(x)u2(5)

wherex ∈ Rn−m, y ∈ Rm, u1 ∈ Rn−m, andu2 ∈ Rm.The vectorsu1 andu2 collect the control signals devoted to steeringx and y to zero,

respectively. The system is assumed to be uncertain with some structural assumptions the


reasonableness of which will be discussed in the next sections. The procedure follows thesame steps previously performed for the single input case.

Differentiate (5) obtainingx = F(x, x,u)+ B11(x)v1+ B12(x)v2

y = G(x, x,u)+ B21(x)v1+ B22(x)v2

u1 = v1

u2 = v2

(6)

F(x, x,u) = A(x)+ B11(x)u1+ B12(x)u2

G(x, x,u) = D(x)+ B21(x)u1+ B22(x)u2

x is assumed measurable, whilex is not available due to uncertainties in system dynamics.Assume that:

• The vector fieldF(x, x,u) = [F1(x, x,u) · · · Fn−m(x, x,u)]T is uncertain and, in a suf-ficiently large open set containing the origin ofRn−m, the following holds|Fi (x, x,u)| <Fi , whereFi are known constants.

• The vector fieldG(x, x,u) = [G1(x, x,u) · · ·Gm(x, x,u)]T is uncertain and, in a suf-ficiently large open set containing the origin ofRm, the following holds|Gi (x, x,u)| <Gi , whereGi are known constants.

• The control matrix

B(x) =[

B11(x) B12(x)

B21(x) B22(x)

](7)

is assumed uncertain and invertible and such that bounds of the modula of its componentsare known. The submatrixB11(x) is positive definite.

A crucial role in the generalization process under investigation, is played by the ma-trix B11(x), which has been assumed to be positive definite. In the first-order sliding-mode case, a Lyapunov approach can be applied and the positive definiteness of the matrixB11(x) is sufficient to ensure the finite-time reaching of the chosen sliding manifold. Ourcase would need a “second-order” Lyapunov-like criterion, that is, one for which, givena positive scalarV(x) for which the first time derivativeV(x) is neither measurable normanipulable, one can define how to modifyV(x) through the control vector in order tosteerV(x) to zero asymptotically or in a finite time. Due to the lack of a synthetic cri-terion, the only possible way is that of trying to decouple the multi-input problem intoa set of single-input problems. This can be done only for a particular class of matri-ces.


Proposition 1 If B11(x) is not only positive definite but also dominant diagonal, that is

0<n−m∑

j=1, j 6=i

|B11i j (x)| < B11i i (x) i = 1, . . . ,n−m

then it is possible to choose each component of the control vectorv1 in accordance to thesecond-order sliding-mode control Algorithm 1, such that the vector x is steered to zero ina finite time.

Proof: The firstn−m equations in (6) can be rewritten as

xi = Fi (x, x,u, v2)+n−m∑

j=1, j 6=i

B11i j (x)v1j + B11i i (x)v1i i = 1, . . . ,n−m (8)

Fi (x, x,u, v2) = Fi (x, x,u)+m∑

j=1

B12i j v2j

Assume that any control signalv1i has the form

v1i = −VM sign

(xi − 1

2xi M

)i = 1, . . . ,n−m

and letF∗i be an upperbound of the term

|Fi (x, x,u, v2)| < F∗i

Equation (8) can be rewritten as

xi = Fi (x, x,u, v2)− gi (x)VM sign

(xi − 1

2xi M

)i = 1, . . . ,n−m (9)

where

g1i (x) < gi (x) < g2i (x)

g1i (x) = B11i i (x)−n−m∑

j=1, j 6=i

|B11i j (x)|

g2i (x) = B11i i (x)+n−m∑

j=1, j 6=i

|B11i j (x)|

A value ofVM valid for any equation can be derived, according to the control Algorithm 1,by considering the worst case, that is the one which gives the highest value ofVM .

F∗M = max1≤i≤n−m

F∗i

g∗1 = min1≤i≤n−m

g1i


g∗2 = max1≤i≤n−m

g2i

V∗M = max

(F∗Mg∗1,

4F∗M3g∗1 − g∗2

)VM ≥ V∗M (10)

The control lawv1i = −VM sign(xi − 12xi M ) with VM satisfying inequality (10) is sufficient

to steer to zero the vectorsx andx.

Note that the condition (10) whichVM must satisfy has been found according to theAlgorithm 1, but the scaling factorα appearing in the algorithm has not been used. Thismeans thatB11i i must be assumed sufficiently dominant diagonal so that|3g∗1 − g∗2| turnsout to be greater than some positive value. This assumption is reasonable according to theprocedure which follows, in fact, as it will be soon realized, the diagonal elements of therelevant control matrices can be made arbitrarily dominant.

Now it will be proved that it is still possible to solve the control problem whenB11(x) ispositive definite but not diagonal dominant (e.g., the inertia matrix of a Lagrangian system).

The proposed procedure consists of the following steps.Consider again equation (6) and an observer

z= −Pv1+ w (11)

P = pI is a diagonal matrix with positive diagonal elements the amplitude of which,p,can be chosen asp ≥ maxi=1,...,n−m

∑n−mj=1, j 6=i |B11i j |, so thatP+B11(x) is positive definite

and dominant diagonal.Define the observation error ase= x − z, its dynamics is described by

e= [F(x, x,u, v2)− w] + [ P + B11(x)]v1 (12)

with e not available. For equation (12), since the control matrix is diagonal dominant, asecond-order sliding-mode control vector can be found as above, capable to steere andeto zero in a finite time.

By applying the equivalent control method, one= e= 0, v1 is equivalent to

v1eq = −[ P + B11(x)]−1[F(x, x,u, v2)− w]

This expression can be substituted in equation (11)

z= Q(x, x,u, v2)+ {I − P[ P + B11(x)]−1}w (13)

with Q(x, x,u, v2) = P[ P + B11(x)]−1F(x, x,u). System (13) contrarily to system (12)is characterized by a complete availability of the state since the observer state is involved.On the other hand ifp is sufficiently high{I − P[ P + B11(x)]−1} is positive definite, sowell known Lyapunov like procedures can be adopted as in [24], [22] to steerz, z to zeroasymptotically.


Thus, setsz = z+ czand chooseV = 12sT

z [ I − P(P + B11(x))−1]−1sz

V = sTz {[ I − P(P + B11(x))

−1]−1[Q(x, x,u, v2)+ cz] + w}w = −0 sign(sz)

sz tends to zero in a finite time andz andz tend to zero asymptotically.It is possible to claim that by this procedure the controlled system tracks the observed

state in a finite time by a second-order sliding-mode, whereas the observer statez is steeredto zero by a classical sliding-mode control of the first-order.

Indeed, att = t , the manifoldx = x = 0 is reached. The distance of this manifold toany manifoldsx = x + cx = 0, for which a set of well-defined regularization proceduresexist (Filippov’s continuous solution concept, equivalent control and approximability) iszero. This means that the effect of the controlv1 on any other equation of the systemcan be expressed by replacingv1 with v1eq. The equivalent control signalv1eq, except forexponentially vanishing terms, is

v1eq = −B−111 (x)[B12(x)v2+ F(x, x,u)]

The effect ofv1 on the second equation in (6) is therefore

y = G(x, x,u)− B21(x)B−111 (x)F(x, x,u)+ [B22(x)− B21(x)B

−111 (x)B12(x)]v2

= G∗(x, x,u)+ B∗22(x)v2

This equation is not affected by the amplitude of the controlv1 and defines a multi-inputuncertain system with control matrixB∗22(x).

If B∗22(x) though uncertain, belongs to the class of control matrices for which a multi-inputfirst-order sliding-mode control strategy exists [24], then the vectorv2 forcing y to zero in afinite time can be synthetized independently of the boundsVM of the discontinuous controlv1. The set of possibilities for the matrixB∗22(x) is sufficiently wide to cover a large numberof practical situations.

To sum up, we have assumed thatB11(x) is positive definite to derive, under someboundness conditions, the convergence ofx to zero in a finite time, and then, by invokingthe approximability properties of sliding-modes, to obtain the convergence of the vectoryto zero in a finite time.

The convergence of both x and y to zero can be guaranteed by a combined procedureinvolving a single-input second-order sliding-mode algorithm and standard multi-inputfirst-order sliding-mode control algorithm.

4. Hybrid Control of Constrained Manipulators

In this section, the previously obtained results, are applied to deal with the force/positioncontrol of constrained manipulators. The development is based on a Lagrangian formulationof the dynamic equations for a robot with holonomic constraints.

Consider a manipulator withn degrees of freedom. Assuming that the end effectoris constrained on a smoothm-dimensional manifold, and that the frictionless interaction


occurs with an infinitely stiff rigid environment, the dynamics of the constrained robot isexpressed by a redundant set ofn+m differential algebraic equations{

M(q)q + C(q, q)q + g(q) = τ + f

8(q) = 0(14)

whereq ∈ Rn is the vector of the joint generalized coordinates,f ∈ Rn is the vector ofthe generalized constraint (reaction) forces,τ ∈ Rn is the vector of the generalized inputforces,M(q) is the inertia matrix (symmetric and positive definite), the vectorC(q, q)qincludes Coriolis and centrifugal torques, andg(q) is the vector of the gravitational torques.The constraint smoothm-dimensional manifold is defined by the second equation of (14),where the function8(q): Rn → Rm is at least twice continuously differentiable. Thegeneralized constraint forcesf can be expressed as

f = JT (q)λ

whereλ ∈ Rm is the vector of the Lagrange multipliers.According to standard projection techniques [18], [20], [26], it can be considered a

partition of the joint-coordinate vectorq = [qT1 qT

2 ]T whereq1 ∈ Rn−m andq2 ∈ Rm,it is assumed the existence of a convex subsetÄ1 ⊂ Rn−m and of a functionk ∈ C2

(k: Ä1 → Ä2, with Ä2 open subset ofRm) such that8(q1, k(q1)) = 0 for all q1 ∈ Ä1.The vectorq2 is uniquely defined by the vectorq1: q2 = k(q1) for all q1 ∈ Ä1. Thevectorq2 is uniquely defined by the vectorq1: q2 = k(q1) for all q1 ∈ Ä1. Moreover,the regionÄ = Ä1 × Ä2 ⊂ Rn is defined. Within the regionÄ, the vector of thegeneralized coordinatesq can be parametrized, even if the parameterization is not uniquelydefined. The end effector motion is constrained on the smooth manifoldS = {(q, q) :8(q) = 0, J(q)q = 0}, whereJ(q) is the Jacobian matrixJ(q) = ∂8(q)/∂q. Then,only (n−m) coordinates inq are independent. The vector of the generalized coordinatesq can be expressed in terms ofq1, the independent sub-vector, asq = [qT

1 kT (q1)]T andthe vector of the velocitiesq in terms of the independent velocities asq = H(q)q1, beingH(q) = [ In−m (−J−1

2 (q)J1(q))T ]T = [H T1 (q) H T

2 (q)]T whereJ1(q) = ∂8(q)/∂q1 and

J2(q) = ∂8(q)/∂q2. According to the previous assumptions, the Jacobian matrixJ2(q)never degenerates in the setÄ.

The first equation of the model (14) can be multiplied, as in [20], by the matrixT(q)defined asT(q) = [H(q) (J(q)M−1(q))T ]T .

As a result, it is obtained an equation of the form{M∗(q)q1+ C∗(q, q)q1+ g∗(q) = H T (q)τ

λ = Z(q)[Cλ(q, q)q1+ g(q)− τ ](15)

M∗(q) = H T (q)M(q)H(q)

Cλ(q, q) = M(q)H(q)+ C(q, q)H(q)

C∗(q, q) = H T (q)Cλ(q, q)


g∗(q) = H T (q)g(q)

Z(q) = [ J(q)M−1(q)JT (q)]−1J(q)M−1(q)

Z(q) = [Z1(q) Z2(q)]

Z1(q) ∈ Rm×(n−m) Z2(q) ∈ Rm×m

whereq2 is expressed asq2 = k(q1).If it is chosen the decoupling control transformation

τ = H+(q)τ1+ JT (q)τ2 (16)

whereH+(q) = H(q)(H T (q)H(q))−1, the reduced model is{M∗(q)q1 = θ∗1 (q, q)+ τ1

λ = θ∗2 (q, q)− Z(q)H+(q)τ1− τ2(17)

Now differentiate (17) and obtain the following equation{M∗(q)q

(3)1 = ψ∗1 (q, q, q)+ τ1

λ = ψ∗2 (q, q, q, τ1)− Z(q)H+(q)τ1− τ2(18)

Note that, in this equation,M∗(q) is a positive definite matrix, sinceM(q) is a positivematrix andH(q) is full rank∀q ∈ Ä. The vectorsψ∗1 (q.q, q) andψ∗2 (q, q, q, τ1) representthe unknown terms resulting from the time differentiations ofθ∗1 (q, q) andθ∗2 (q, q) andfrom [ d

dt M∗(q)]q1 and [ddt Z(q)H+(q)]τ1.

Equation (18) can be interpreted asn−m equations defining the dynamics of the inde-pendent coordinatesq1 with the control vectorτ1. The second block of equations describesthe dynamics of the vectorλ, which is affected by bothτ1 and τ2. Due to the triangularstructure of the relevant control matrix, the terms dependent onτ1 can be considered asa disturbance, whereasτ2 constitutes the true control signal. The procedure outlined inthe previous section can be applied to the present case by choosing the sliding manifoldss1 = q1 + cq1 ands2 = λ − λd and settingx = s1, z = s2, u1 = τ1, u2 = τ2. Equation(18), can easily be expressed in the form of equation (6) withB11 = M−1

∗ (q), which hasbeen proved to be positive definite,B22 = Im, B12 = 0, andB21 = Z(q). By adopting theprocedure described in the previous section, the control objective can be achieved.

4.1. The Case of Uncertain J(q)

If J(q) is uncertain, the previous triangularization of the overall control matrix cannotbe performed. To deal with this case, we do not follow the traditional robust controltheory based on the sensitivity analysis of a nominalJ0(q) with respect to a variation1J(q) bounded in norm. Such a treatment is very conservative and often results in strictconditions on‖1J(q)‖ hard to verify in practice. In our treatment, we exploit the structureof the uncertain matrixH(q) and the flexibility of the method based on a control hierarchy


[24] together with the results discussed in Section 3. Let us show that the previous methodcan be used to reachs1 = 0 in a finite time. After that, the resulting dynamics describingthe evolution ofs2 is characterized by a control matrix that is the invertible matrixJ2(q).Uncertainties in this square matrix can be managed as in traditional first-order sliding-modecontrol by exploiting the structural properties of such a matrix or by using a componentwisecontrol hierarchy.

Consider equation (15) and differentiate it. Due to the uncertainties it is now impossibleto perform the transformation (16), thus letτ = [τ T

1 τ T2 ]T , it is obtained{

M∗(q)q(3)1 = φ1(q, q, q)− H T

1 (q)τ1− H T2 (q)τ2

λ = φ2(q, q, q)− Z1(q)τ1− Z2(q)τ2(19)

It follows from definition of the matrixZ(q) that the following condition holdsZ(q)JT (q) =Im. This fact can be exploited in order to derive a relation between the two submatricesZ1(q) andZ2(q). In fact, sinceZ1(q)JT

1 (q)+ Z2(q)JT2 (q) = Im it results that

Z2 = [ Im − Z1(q)JT1 (q)] J−T

2 (q) (20)

whereJ−T2 (q) = [ JT

2 (q)]−1.

It follows that, sinceH1 = Im, H2 = −J2(q)−1J1(q), in equation (19) the matrixB11

turns out to be still equal to the positive definite matrixM−1∗ (q), as in the previous case,

so that it is possible to apply the second-order multi-input sliding-mode procedure to steers1 = q1 + cq1 to zero in a finite time. It is possible to claim that, when the second-ordersliding-mode on the manifolds1 = 0 is reached, the representation of the system on such amanifold can be attained by replacingτ1 with τ1eq, which, except for exceptional decayingterms, can be expressed as

τ1eq = JT1 (q)J

−T2 (q)τq − φ1(q, q, q) (21)

Substituting the expression forτ1eq into the second equation of (19) and considering thepreviously obtained expression forZ2(q), one can show that, when a sliding-mode occurson the intersection of the discontinuity surfacess1 = 0, it results

λ = φ2(q, q, q)+ Z1(q)φ1(q, q, q)− Z1(q)JT1 (q)J

−T2 (q)τ2− Z2(q)τ2 (22)

and, from (20),

λ = φ3(q, q, q)− J−T2 (q)τq (23)

so that now we have a set ofm first-order differential equations coupled through the matrixJ−T

2 (q). The situation is already improved over the previous one, as onlyJ2(q) is involved.Moreover, the last control problem to be faced being a classical first-order sliding-modecontrol, the list of possible uncertain control matrices that can be dealt with by existingtheory is sufficient to cover significant application cases.

In particular ifm = 1, J2(q) is a scalar, and the above methodology requires the knowl-edge of the sign and a lower bound to the modulus of this quantity. A single constraint


Figure 3. Physical manipulator model.

equation corresponds to a surface in<3; therefore, a wide class of hybrid control prob-lems can be dealt with in an uncertain context requiring only this information. For morecomplex situations, the approach presented in [1], and based on the choice of a simplex ofconstant vectors, appears very promising, as it seems to be characterized by good propertiesof robustness to the uncertainties in the control matrix [2].

5. Simulations

We have considered a four degree-of-freedom manipulator of the type sketched in Figure 3and the end-effector of which is constrained to the horizontal plane. The simulation hasbeen carried out by assuming to know only the sign ofJ2(q), which, in this case(m =1), is a scalar quantity. Due to the great complexity of the reduced order model, thesimulation has been based on a DAE-type model for which solution variable-structurecontrol concepts have been used, minimizing the well-known [11], [15] numerical problemsfor DAE systems of index more than two, typical of numerical integration kernels availablein the literature. Simulation results, which are comparable with those presented in [6] fora two link manipulator, show the effectiveness of the proposed methodology, even in thenumerically approximate context of a simulation experiment.

The convergence of thes1 components is shown in Figures 4 and 5, and the convergenceof thes2 component is plotted in Figure 6. The behavior of the Lagrange multiplierλ ispresented in Figure 7; moreover, the temporal evolutions of all the joint coordinates areshown in Figures 8 and 9. The application of the proposed control procedure causes the timederivatives of the actual control signals, namelyτ10, τ11, τ12, andτ2, to be discontinuous, asshown in Figures 12 and 13. For such signals, we have obtained continuous, chattering-free,control signalsτ10, τ11, τ12, andτ2 (Figures 10 and 11).


Figure 4. Convergence of thes1 vector components.

Figure 5. Convergence of thes1 vector component.

6. Conclusions

In this paper, a multidimensional control problem concerning differential algebraic systemscharacterized byn−mdifferential equations coupled tomalgebraic ones has been dealt with.The system has been assumed to be uncertain in both its components. The proposed solutionhas been applied to the case of hybrid force/position control of constrained manipulators.The result is that this problem can be solved despite a high degree of “ignorance” of therobot and the constraint equations.

We are aware that our work is only a first step toward a feasible solution of the practicalhybrid force/position control of constrained mechanism. Various directions need to beinvestigated to give our approach the necessary robustness with respect to real phenomena(unmodelled dynamics of the sensors and actuators, Coulomb friction, collisions, etc.), not


Figure 6. Convergence of thes2 vector component.

Figure 7. Behavior of the Lagrange multiplierλ.


Figure 8. Behavior of the joint coordinates.

Figure 9. Behavior of the joint coordinates.

Figure 10.The continuous control signals.


Figure 11.The continuous control signals.

Figure 12.The chattering time derivatives of the control signals.

Figure 13.The chattering time derivatives of the control signals.


considered in this analysis, even if works about this topics are in progress. The practicalimplementation of the proposed algorithm has been successfully carried out in our lab, inorder to control a flexible finger of an underwater hand, during the first phase of the projectAMADEUS (C40. 96 AMADEUS2-A. C.) of the European Community.

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multi-input second-order sliding-mode hybrid control of constrained manipulators

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