Trajectory Planning and Second-Order Sliding Mode Motion/Interaction Control for Robot Manipulators in Unknown Environments

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 8, AUGUST 2012 3189

Trajectory Planning and Second-Order SlidingMode Motion/Interaction Control for Robot

Manipulators in Unknown EnvironmentsLuca Massimiliano Capisani, Member, IEEE, and Antonella Ferrara, Senior Member, IEEE

Abstract—The problem of determining an interaction controlstrategy, allowing a manipulator to reach a goal point even in thepresence of unknown obstacles, is faced in this paper. To this end,on the basis of position/orientation and force measurements, first,a path planning strategy is proposed. The path planning is basedon an a priori trajectory, which is determined without the priorknowledge of the obstacle presence in the workspace, and on areal-time approach to generate auxiliary temporary trajectorieson the basis of the properties of the obstacle surface in a vicinityof the contact point, estimated through force measurements. Todetermine the input laws of the manipulator, a robust hybridposition/force control scheme is adopted. First- and second-ordersliding mode controllers are considered to generate the robot inputlaws, and the obtained performances are experimentally com-pared with those of classical PD control. Experiments are made ona COMAU SMART3-S2 anthropomorphic industrial manipulator.

Index Terms—Force control, path planning, robust control.

I. INTRODUCTION

FOR robotic manipulators, a typical critical task is the man-ufacturing in the vicinity of objects or with interaction with

these latter [1]. Depending on the features of the sensors whichare employed, different approaches can be used to control therobotic system [2]–[4]. When a force sensor is available, thecontrol strategies can take into account the local properties ofthe obstacle surface in the vicinity of the contact point [5]–[13]and can be effective even if the position and shape of theobstacles are not a priori known.

In this paper, we assume that a force sensor is mountedat the tip of the end-effector of the manipulator (the case ofa three degrees of freedom manipulator is considered for thesake of simplicity, but the proposed approach can be appliedto any n degrees of freedom manipulator). Assuming that therobot manipulator operates in an environment with unknownobstacles, the goal is to develop a trajectory planning strategycombined with a hybrid position/force control scheme to reduce

Manuscript received April 23, 2010; revised September 8, 2010,January 18, 2011, and April 20, 2011; accepted June 2, 2011. Date of publi-cation June 23, 2011; date of current version March 30, 2012.

The authors are with the Department of Computer Engineering and Sys-tems Science, University of Pavia, 27100 Pavia, Italy (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2011.2160510

the risk of harming the robot, the sensor, or the obstacle, whencontact between robot and obstacle occurs, while guaranteeingrobust position/force tracking. In particular, a major feature ofthe proposed approach is to decouple the position control andthe force control so as to reduce the complexity of the controllerdesign.

A first problem which needs to be solved is that the forcesmeasured by the sensor are not only the interaction forces, butalso dynamical forces due to the motion of the tip [14]–[18].So, in analogy with the preliminary work [16], a suitable modelof the sensor is proposed, along with an approach to correctlydetermine the contact force and the local properties of theunknown obstacle surface in the vicinity of the contact point.The problem of determining the surface properties on the basisof the contact force is analyzed also in [12]. In that case, a 2-Dforce vector is considered to obtain the local approximation ofthe surface of the planar obstacle. Particular solutions are thenadopted to reduce the effects of the presence of uncertaintiesin friction phenomena. In our case, we consider not only theplanar force vector, but also the measure of the correspondingtorque revealed by the sensor (see also [19]). In analogy tothis approach, we consider an experimental setup in which thecontact friction between the tip of the robot and the surfaceof the obstacle is suitably reduced, as will be explained in thepaper.

On the basis of previous works on interaction control, aneffective approach to take into account the presence of obstaclesinside the workspace and, even, the interaction with them isthe parallel approach, see [5]. However, differently from thehybrid control approach, this method requires the knowledge ofinformation on the task geometry at the planning level. Then, inour case, to reach the pre-defined goal point, a particular pathplanning strategy is proposed on the basis of hybrid control.This path planning strategy does not require information on theobstacles shape before the contact between robot and obstacleis established. First, an a priori trajectory is designed withoutconsidering the obstacles’ presence. This trajectory is modifiedin real time when the manipulator interacts with the obstacles.The proposed hybrid position/force control scheme makes thetip of the robot follow the a priori planned trajectory if nocontact with an obstacle occurs, then it makes the tip of therobot follow the shape of the obstacle until the goal point isreached (if this lies on the obstacle) or the a priori trajectory iscrossed again so that this can be followed until the goal point,or until a next contact takes place.

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3190 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 8, AUGUST 2012

The scheme aims at linearizing and decoupling the roboticdynamical system, by relying on the inverse dynamics and in-verse kinematic approaches. Moreover, particular feedforwardsare introduced to decouple and linearize the interaction betweenthe manipulator and the obstacle. Finally, two feedback loopsare present in the system. The first loop is devoted to make thetip position/orientation track the reference trajectory, while thesecond loop is devoted to control the magnitude of the contactforce. In this scheme, any controller applicable to a second-order system can be used. Three particular combinations, two ofwhich are based on the sliding mode control methodology (see[20]–[26]) are here considered, analyzed, and experimentallytested.

This paper is organized as follows. Section II describes thesystem kinematic model. Section III describes the dynamicalmodel of the sensor and of the tip. How the contact force can beestimated is discussed in Section IV. Section V describes thetrajectory planning approach. Section VI details the proposedhybrid control scheme and the proposed controllers, the stabil-ity properties of which are illustrated in Section VII. The ex-periments made on a COMAU SMART3-S2 anthropomorphicrigid robot manipulator with an ATI Gamma force sensor aredescribed in Section VIII, so as to make a comparison betweenthree different combinations of the position/force controllers.Finally, Section IX gathers some conclusions.

II. KINEMATIC MODEL OF THE ROBOT

Consider a typical industrial anthropomorphic manipulator.Only vertical planar motions can be obtained by locking threeof the six joints of the robot (note that the extension to thespatial motions is possible, but in this case, the dynamic modelof the tip is more complicated). The kinematic model of theresulting planar manipulator is the relation between the config-uration q = [q1, q2, q3]T of the three considered joints and theend-effector position and orientation P = [Px, Py, φ]T in thevertical plane {x, y}, which is the workspace. The angular termq1 is the orientation of the first link with respect to the y axisclockwise positive, while qj , j = 2, 3, define the displacementof the jth link with respect to the (j − 1)th, clockwise positive(see Fig. 1). The position/orientation of the extreme point of thekinematical chain is

P =3∑

k=1

⎡⎢⎣ lk sin

(∑kz=1 qz

)lk cos

(∑kz=1 qz

)qk

⎤⎥⎦+ lt

⎡⎢⎣

sin(∑3

i=1 qi

)cos(∑3

i=1 qi

)0

⎤⎥⎦=C(q)

(1)

where qi and li are the angular displacement and the lengthof the ith link, respectively, while lt is the length of the tip(P and qi are obviously functions of time). As indicated inFig. 1, the first two components of the vector P , denoted byP12, represent the position of the extremal point of the tip of theend-effector, while the orientation of the tip with respect the yaxis is given by φ = q1 + q2 + q3, i.e., the third component ofvector P . Note that, in (1), C(q) is used to denote the kinematicrelation.

Fig. 1. Three link planar manipulator.

Now, consider the center of gravity G of the rigid body givenby the sensor and its tip (see Fig. 1). The position of the centerof mass G and its velocity vG are given by

G = Ps + lg

[sin(φ)cos(φ)

], vG = Ps + lgφ

[cos(φ)− sin(φ)

](2)

where lg is the distance between G and the position Ps of theforce sensor, which is given by

Ps =3∑

k=1

⎡⎢⎣ lk sin

(∑kz=1 qz

)lk cos

(∑kz=1 qz

)qk

⎤⎥⎦

m being the total mass of the tip and the sensor, which causesthe forces that are measured by the sensor itself. Note that lgand m are unknown. The acceleration aG is

aG = Ps + lg

[−φ2 sin(φ) + φ cos(φ)−φ2 cos(φ) − φ sin(φ)

]=[aGx

aGy

]. (3)

Note that the angular velocity of the tip is given by φez ,where ez is the unit vector in the z direction, normal to {x, y}.The angular acceleration is given by φez .

III. SENSOR AND TIP DYNAMICAL MODEL

A fundamental part of a force control loop is the determina-tion of the contact force between the tip of the manipulator andthe environment [10]. The considered force sensor measuresthe force f acting on its tip. This force is described in theO − xy vertical plane indicated in Fig. 1, which represents themanipulator workspace. The force f is composed by two terms

f = [fx, fy, τz]T = f0 + fc, fc = [fcx, fcy, τcz]T (4)

where f0 refers to the forces related to the tip dynamics and fc

refers to the forces related to the contact with the environment.Vector f contains the force and the torque generated becauseof the contact, and the dynamical effects on the sensor tip,where fx and fy are the components of the force and τz isthe corresponding torque revealed by the sensor. The torque τcz

on Ps is generated by the forces fcx and fcy. The objective isto determine a suitable model in order to eliminate the effectof f0 in (4), so that the remnant force is actually the contactforce. This implies that if the tip is not in contact with theenvironment, the considered force has to be zero.

CAPISANI AND FERRARA: TRAJECTORY PLANNING AND MOTION/INTERACTION CONTROL FOR ROBOT MANIPULATORS 3191

Fig. 2. Proposed hybrid position/force control architecture.

The sensor is composed by two parts: the first part is fastenedwith the robot, and its mass cannot produce forces measurableby the sensor since it can be viewed as a part of the link; thesecond part is fastened with the tip, and its mass, jointly with themass of the tip, can produce significant dynamical effects whichcan be revealed by the sensor. For this reason, the robot and thefirst part of the sensor are not considered in the formulation ofthe dynamical model.

By relying on (3) and on the transport theorem [27], it ispossible to express f ′

0, which is obtained by describing f0 inthe rotated {x′, y′} reference, as follows:

f ′0 =

⎡⎣ f0x′

f0y′

τ0z

⎤⎦ =

⎡⎣ f0x′ − maGx′ − mg sin φ

f0y′ − maGy′ − mg cos φ

τ0z − Iφ + mlg(aGx′ + g sin φ)

⎤⎦ (5)

where g = 9.806 m/s2, I is the inertia of the sensor tip, whichis unknown, and the terms f0x′ , f0y′ , τ0z take into accountthe unknown constant biases always present on the acquiredgeneralized force. The unknown terms of the model (5) needto be estimated, to allow the estimation of the forces measuredby the sensor but not related to contact effects. The identifi-cation procedure here proposed follows the guidelines of [28]and [16].

IV. ESTIMATION OF THE CONTACT FORCE

Once the parameters of the model for the f0 term in (4) areestimated, it is possible to determine a better estimation of thecontact force fc by evaluating the term

fc = f − f0 = f − Φi(Ps, φ, φ, φ)θLS =[f ∗

c , τcz

]T(6)

where fc is the estimate of fc, f0 is the estimate of f0 [see (5)],f ∗

c = [fcx, fcy]T , while the term Φi(Ps, φ, φ, φ) is the nonlineartransformation of the inputs at the time instant i, θLS is theestimated parameters vector.

Note that the objective is to estimate the magnitude and thedirection of the contact force fc, while in (6), three equationsare present, hence the problem of estimating the contact forceis overdetermined. Let us denote with R the magnitude of thecontact force (ϕ and ϕ′ represent its direction with respect to

the y axis and to the y′ axis, respectively). In the absence ofnoise and unmodeled effects, fc is given by

fc = fc =

⎡⎣ fcx

fcy

τcz

⎤⎦ =

⎡⎣ −R sin ϕ

−R cos ϕ−Rd sin ϕ′

⎤⎦ (7)

where d = ‖P − PS‖ = lt. In the presence of noise and un-modeled effects, fc differs from fc. Then, (7) can be rewrittenas fc = J (ϑo) + ε, ϑo = [R,ϕ]T , in which J (·) represents thenonlinear model (7) and ε ∈ R

3 is unknown. By minimizing theterm εT ε with respect to ϑo, one obtains

ϕ = arctan

[fcx − τczd

fcy(1 + d2)

](8)

R =

∣∣∣∣∣ (τczd − fcx) sin ϕ − fcy cos ϕ

1 + d2 sin2 ϕ

∣∣∣∣∣ (9)

(see the preliminary works [15] and [19] for a more detailedexplanation), which are the estimates of R and ϕ, respectively.

V. TRAJECTORY PLANNING

Fig. 2 shows the entire system, which includes the controllerlogic and the controlled plant. The trajectory Pr(t) is thereference trajectory to be tracked by the manipulator whenthe experiments are performed. Note that the a priori deter-mination of a collision-free trajectory is not possible becausethe obstacles positions and shapes are unknown. In fact, as itwill be explained in Section V-A, Pr(t) can coincide with twopossible trajectories during the control phase. On the basis ofsome conditions on the obstacle geometry, it can coincide withPd(t) which is an a priori trajectory generated on the basisof the manipulator geometry and performances neglecting theobstacles, or it can coincide with P ′

d(t), which is a temporaryauxiliary trajectory. In our case, the a priori trajectory Pd(t) =Pd(s(t)) = [x(s(t)), y(s(t)), φ(s(t))]T is given by

Pd (s(t)) =

⎡⎣ a0 + a1s(t) + a2s

2(t) + a3s3(t)

b0 + b1s(t) + b2s2(t) + b3s

3(t)c0 + c1s(t) + c2s

2(t) + c3s3(t)

⎤⎦ (10)


where s(t) is a scalar function of the time t which is definedon-line during the movement on the basis of a required velocityshape vd(t) for the trajectory, as will be specified. A particularset of vectors

[a, b, c]T = [[a0, . . . , a3], [b0, . . . , b3], [c0, . . . , c3]]T

(11)

represents a suitable parametrization of the trajectory Pd(s(t)).These vectors are obtained by solving the systems⎧⎨⎩

a0 + a1si + a2s2i + a3s

3i = x(si)

b0 + b1si + b2s2i + b3s

3i = y(si)

c0 + c1si + c2s2i + c3s

3i = φ(si)

i = 0, . . . , 3 (12)

where the set {si, x(si), y(si), φ(si)}, i = 0, . . . , 3, containsthe four points of the manipulator workspace to be interpolatedby the trajectory Pd(s(t)). When the manipulator enters incontact with an obstacle, the a priori trajectory Pd(s(t)) cannotbe followed. In this case, the actual trajectory P (t) of the robotturns out to be modified by virtue of the application of thehybrid control, so as to follow the shape of the obstacle. Notethat, in Fig. 2, Σp is the position control projector operator,Σp = r⊥rT

⊥ , with r⊥ being a unit vector normal to the surfaceof the obstacle in the contact point, as it will be clarified later.

To impose an arbitrary velocity function vd(t) to the tra-jectory Pd(s(t)), the function s(t) has to be selected so thatthe velocity of Pd(s(t)) is equal to vd(t). To this end, ascalar valued function v(Pd(s(t))) is introduced: it representsa criterion to evaluate the velocity of Pd(s(t)). Let the functionv(Pd(s(t))) be linear with respect to a scalar factor r, i.e.,

v(rPd (s(t))

)= rv

(Pd (s(t))

), r ∈ R. (13)

For instance, in our case, we choose

v(Pd (s(t))

)=√

x2(t) + y2(t). (14)

Then, the objective is to determine a suitable function s(t)such that, for any time instant t, the following relation holds

vd(t) = v(Pd (s(t))

). (15)

To solve this problem, one can note that because of (13),since s(t) is scalar

v(Pd (s(t))

)= v

(dPd(s)

dss(t))

= v

(dPd(s)

ds

)s(t). (16)

Hence, once selected the velocity vd(t), the goal is to finds(t) such that (15) holds. From (15) and (16), it yields

s(t) =vd(t)

v(

dPd(s)ds

) (17)

which in turn, given the initial condition s(0) = s0, allows oneto find the function s(t), satisfying the velocity requirementsfor Pd(s(t)). Note that any Pd(t) trajectory defined in theworkspace of the robot manipulator, having limited velocitiesand accelerations, can be acceptable for the hybrid control

Fig. 3. (Left) Example of a situation in which the contact force betweenthe tip and the obstacle vanishes. The tip starts at Pstart. At P1, the contactbecomes evident. The dotted shape is followed. After, the tip reaches the pointP (T ) and the contact force vanishes. Finally, the dash-dot temporary trajectoryis determined and followed until Pd(s(T )) is reached. (Right) Example ofalignment between the contact force direction −ef and the trajectory directionP ′

d(t) (dash-dot in the figure): the manipulator cannot move.

scheme (see, for instance, [10] and [29]) for other planningstrategies.

A. Real-Time Modification of the A Priori Trajectory

To control the robot even in the presence of obstacles whichare a priori unknown, it is necessary to take into account ofsome events which require to switch the reference trajectoryPr(t) between the a priory trajectory Pd(·) and the auxiliarytrajectory P ′

d(·). Such events are: vanishing of the estimatedcontact force between the tip of the sensor and the obstacle(magnitude of the contact force below the limit Rth for somesampling instants) and discontinuous or fast change of thedirection of the contact force. Note that, to deal with theseissues, the following error signals have been taken into account{

ep(t) = Σp (Pd (s(t)) − P (t))ef (t) = (I − Σp) [fcd(t) − Ks (P (t) − P ∗(t))] (18)

where (P (t) − P ∗(t)) takes into account the tip bending. A par-ticular solution is also presented in case of alignment betweenthe contact force direction and the ep direction occurs, to avoidtoo slow velocities during the interaction with the obstacle.

1) Vanishing of the Contact Force (Fig. 3 Left): Supposethat, at a certain time instant t = T , the contact force magnitudeR becomes lower than threshold Rth: in this case, the hybridforce/position control is switched to the position/orientationcontrol by setting Σp = I and p = 0. Then, in this time in-stant, it is necessary to switch the trajectory Pr(t) to avoid asudden change of the control action. So, Pr(t) coincides to anew temporary trajectory P ′

d(sr(t)) which is generated on thetime instant t = T . It is a straight line defined between thetip position/orientation P (T ) and the a priori trajectory posi-tion/orientation Pd(s(T )). Then, the function sr(t) is generatedaccording to (17) but with dP ′

d(sr)/dsr at the denominator,to allow the possibility of keeping the desired velocity vd(t)also along the temporary trajectory P ′

d(sr(t)). The temporarytrajectory P ′

d(sr(t)) is kept until one of the following twoconditions is verified: the contact force becomes again evident,i.e., greater than a threshold RTH , with RTH > Rth, or P (t)is in a small vicinity of Pd(s(T )). In both cases, the trajectory


Fig. 4. Example of a situation in which the contact direction ϕ has an abruptchange to ϕ′. Then, the direction in which the position/orientation control actsep (dotted arrow) suddenly changes to e′p (dash-dot arrow).

Pd(s(t)) is newly considered as reference trajectory (Pr(t) =Pd(s(t))). Of course, in the first case, the position/orientationcontroller is again switched to the hybrid control. Note that,considering the a priori Pd(s(t)) trajectory in the first case,i.e., when the contact force becomes again evident, can gen-erate some discontinuities on the reference trajectory Pr(t).However, our experimental experience allows us to say that thissituation is usually generated because of the presence of noiseon the force measurements. This implies that then the timeintervals in which a temporary trajectory P ′

d(t) is consideredas reference trajectory Pr(t) are short, and, as a consequence,the discontinuities which may be generated by adopting thisapproach do not affect too much the controlled system behavior.Obviously, a possible solution to this problem would be tochange the trajectory in a smoother way.

2) Alignment Between the Contact Force Direction and thePosition/Orientation Trajectory Direction (Fig. 3 Right): Sup-pose that, at a certain time instant, the temporary trajectoryP ′

d be aligned with the force error direction −ef . In this case,it is simple to show that the manipulator cannot move in thedirections in which the position/orientation control is possiblebecause the position/orientation controller output is orthogonalto the projector Σp. To solve this problem, when this case isdetected by the controller, another temporary trajectory has tobe planned, such that it is not aligned with the contact force.It is sufficient to evaluate the trajectory Pd(s(t)) not exactlyin the point Pd(s(T )), where T is the time instant in whichthe alignment is detected, but in the point Pd(s(T + δ)), withδ > 0.

3) Fast Change of the Measured Contact Direction (Fig. 4):When the estimated direction ϕ of the contact force has anabrupt change, stability of the controlled system may be lostbecause of peaking of the control signals. The influence of thisproblem on the performances can be reduced by consideringslow trajectories Pd(s(t)) and P ′

d(t). However, this solution isnot efficient in the presence of noise on sensor measurementsand in the presence on edges of the obstacle manifold. Theproposed solution consists in determining and checking themagnitude of the first derivative of ϕ(t), i.e., ϕ(t). Note that thisquantity has to be estimated from the discrete time evaluationϕ(kh) of the angle ϕ(t) in (8). Consider the discrete signalϕk = ϕ(kh), k ∈ N, where h is the sampling time.

Then, a suitable estimation of the term |ϕk| is obtainedrelying on the formula∣∣∣∣dϕ

dt

∣∣∣∣ =∥∥∥∥dv

dt

∥∥∥∥ �∥∥∥∥Δv

Δt

∥∥∥∥ =‖Δv‖

h=

‖vk − vk−1‖h

(19)

where v is the unit vector correspondent to ϕk

vk = [vx, vy, vz]TΔ= [sin(ϕk), cos(ϕk), 0]T . (20)

Thus, from (19), one has

ˆϕk =

√[sin(ϕk) − sin(ϕk−1)]

2 + [cos(ϕk) − cos(ϕk−1)]2

h.

(21)

Finally, the condition ˆϕk > Vmax is evaluated where Vmax

is the maximum allowed angular velocity.Now, suppose that the condition on ˆϕk is verified at time

instant T . Then, the hybrid position/force control is switched toposition/orientation control, and a new temporary trajectory isdetermined, which starts at the actual position/orientation P (T )of the end-effector and ends to the current position/orientationof the a priori planned trajectory Pd(s(T )) as explained inSection V-A.1 (see also Fig. 3 Left). In this way, the trackingerror becomes zero. Of course, in this case, discontinuitiescan occur on velocity. Yet, during the interaction phase, itis advisable that the velocity is low, so that the effects ofthe discontinuities are not so remarkable, as confirmed byexperimental experience.

VI. HYBRID POSITION/FORCE CONTROL SCHEME

The proposed control scheme (see Fig. 2) takes explicitlyinto account the force measurements by the force control loop,as is proposed in [5], [6]. However, in these approaches, anintegral action in the force control loop is introduced to reducethe effects of possible steady state force errors which may arisefrom reduced information on the obstacle surface geometry orfriction effects in the contact point. In our case, we deal withthese problems both by integrating two sliding mode controllers(to cope with position/orientation and force control) and by es-timating the geometric properties of the surface of the obstacle,as it will be described in the paper. The loop at the top of Fig. 2is specifically designed for the position/orientation trackingcontrol. The controller C1 determines the position/orientationcontrol signal yc1(t). Then, the operator Σp selects the directionin which it is possible to perform this kind of control. Note that,in the absence of a contact force, Σp is the identity matrix. Afterthe selection, the signal yp(t) is generated. This signal is thefirst of the components of the auxiliary input signal y(t) in (24).An acceleration feedforward is present along the directions inwhich the position/orientation control is performed, as can beseen in Fig. 2.

The loop at the bottom of the figure is specifically designedfor the interaction control. The force f(t) detected by the sensoris transformed according to the relationship f ∗

c = [f − f0]1,2

indicated in the block present in the loop (the notation [·]1,2

indicates the first two components of the vector). Note that


f0 ∈ R3 contains the generalized forces related to the tip dy-

namics. Such a vector can be determined on the basis of thedynamical model of the sensor tip as illustrated in [15]. Thedesired contact force fcd(t) is obtained by selecting a con-stant contact force Rd ∈ R in the f(t) direction, i.e., fcd(t) =Rd[sin ϕ(t), cos ϕ(t)]T being ϕ(t) the angle of the direction ofthe contact force in the contact point with respect to the positivey-axis counterclockwise positive, and ϕ(t) a suitable estimateof ϕ(t). The controller C2 computes the control input signalyc2(t) which is the second component of the auxiliary inputsignal y(t) in (24).

The force control problem is dealt with as a position/orientation control problem due to the introduction of a virtualspring model (Ks in Fig. 2 is the spring constant). Thus, in theproposed control scheme, one has, as position/orientation andforce controllers, two position controllers.

A. System Kinematics, Dynamics, and Feedback Linearization

To design efficient decoupling control schemes, the dynamicsof the system has to be considered. To this end, the inversedynamics control described in [20] has been implemented. Forthe reader’s convenience, a brief description of the proposedcontrol strategy is hereafter reported. The kinematic and dy-namic models of the system are given by

P (t) = C (q(t)) ,

τ(t) = B (q(t)) q(t) + n (q(t), q(t)) ,

n (q(t), q(t)) = C (q(t), q(t)) q(t) + g (q(t)) + Fssign (q(t))

+ Fv q(t) + JT (q(t)) f(t)

= n′ (q(t), q(t)) + JT (q(t)) f(t) (22)

where P (t) and C(q(t)) as in (1), the matrix B(q) is theinertia matrix, C(q, q)q is the vector of the centripetal andCoriolis torques, g(q) is the vector which takes into accountthe gravitational effects acting on the robot, Fssign(q) + Fv qtakes into account of the friction effects, and the additionalterm JT (q)f is added to the torque τ (see also Fig. 2) tocompensate the effect of the external interaction force. Theinverse dynamics control consists in defining an auxiliary inputsignal y(t) such that

τ(t) = B (q(t)) y(t) + n′ (q(t), q(t)) + pJT (q(t)) fcd(t)(23)

where n′(·) is the identified version of n′(·) appearing in (22),B(q) is assumed to be known, see [28], [30], and pJT (q)fcd,p being 1 if there is contact, 0 otherwise, is an additionalterm to compensate the external interaction force as previouslyexplained. The auxiliary input signal y(t) in (23) is determinedin the following way:

y(t) = J−1 (q(t))(y(t) − J (q(t), q(t)) q(t)

)(24)

where y(t) ∈ R3 is another auxiliary input. By differentiating

twice the first of (22), one has

P (t) = y(t) + η(t) (25)

where η(t) is an uncertain term which, on the basis of physicalconsiderations, can be assumed to be bounded.

B. Determination of the Workspace Projectors Σp and I − Σp

In Fig. 2, the workspace projector Σp, to decouple theposition/orientation and force control tasks, is given by

Σp = r⊥rT⊥ , r = [cos(ϕ),− sin(ϕ)]T (26)

where r⊥ is a unit vector orthogonal to the contact force [15].By relying on Σp, it is possible to obtain yp(t) and yf (t) as

yp(t) = Σpyc1(t), yf (t) = (I − Σp)yc2(t) (27)

yc1(t), yc2(t) being the outputs of the position/orientationcontroller and of the force controller, respectively. The auxiliaryinput y(t) is then determined as

y(t) = yp(t) + yf (t) + ΣpPd(·) + ˆϕ[sin(ϕ)cos(ϕ)

]rT Pd(·) (28)

where ˆϕ is an estimate of ϕ, ΣpPd is the feedforward ofthe acceleration tangential to the surface in the contact point,and ˆϕ[sin(ϕ), cos(ϕ)]T rT Pd(s(t)) is the feedforward of theacceleration normal to the surface in the contact point. Withthis choice of the auxiliary input, it is possible to decoupleposition/orientation control and force control when the tipinteracts with the obstacle profile.

C. Dynamics of the Decoupled System

Starting from the tracking errors in (18), where P (t) as in(1) would be the tip position if the tip were rigid, while P ∗(t)is the point in which the tip actually touches the surface of theobstacle, it is possible to derive the equations of the roboticsystem when this latter interacts with an object. By computingthe derivatives of these two error signals, it is possible to obtainthe error dynamics (time dependency is omitted for simplicity){

ep = Σpep + 2Σpep + Σp(Pd − P )

ef = −Σpef + 2Σpef − (I − Σp)[ϕ2fcd + Ks(P − P ∗)

](29)

where Σp can be written as Σp = (ϕ/ϕ)Σp + 2ϕ2(I − 2Σp).Relying on (25) and (28), (29) becomes{

ep = 2Σpep + Σpep − yp − Σpη

ef = 2Σpef − Σpef − yf − (I − Σp)η.(30)

In (30), it is possible to note that the errors ep(t) and ef (t)have a decoupled second-order dynamics with respect to theauxiliary inputs yp(t) and yf (t).

D. Proposed Controllers

Here follows a description of three particular solutionswhich have been analyzed and experimentally tested. Each


position/force controller has, as input signal, the position/forcetracking error, i.e.,{

e′p(t) = Pd (s(t)) − P (t)

e′f (t) =[fcd(t) − f ∗

c

].

(31)

1) PD Position/Force Controller: In the first configuration,a Proportional-Derivative (PD) controller is implemented forboth C1 and C2, i.e.,

C1 =Kppe′p(t) + Kp

d e′p(t), C2 =Kfp e′f (t) + Kf

d e′f (t) (32)

where Kpp ,Kp

d ,Kfp ,Kf

d ∈ R3×3 are diagonal matrices. Note

that, due to the feedback linearization, it is possible to find asuitable choice of such matrices that stabilizes the closed loopsystem [10].

2) First-Order Sliding Mode Position/Force Controller: Inthe second configuration, the position/orientation and the forcecontrollers are of conventional (first-order) sliding mode type.The objective is to steer to zero in finite time the slidingvariables si(t) and sfi(t), ∀i = 1, 2, 3, defined as⎧⎪⎪⎨

⎪⎪⎩si(t) = x2i(t) + βx1i(t), β > 0sfi(t) = χ2i(t) + βχ1i(t)x1(t) = e′p(t), x2(t) = x1(t)χ1(t) = e′f (t), χ2(t) = χ1(t)

(33)

where the subscript i denotes the joint of the manipulator. Theinput control laws produced by C1 and C2 are designed as{

yc1i(t) = UMisign [si(t)]yc2i(t) = UMisign [sfi(t)]

(34)

where UMi > |ηi(t)|, ηi(t) being the ith component of theuncertain term η in (25), [31]. Finally, the terms yp(t) and yf (t)of the auxiliary input y(t) are determined as

yp(t) = Σpyc1(t), yf (t) = (I − Σp)yc2(t). (35)

3) Second-Order Sliding Mode Position/Force Controller:In the third configuration, a second-order sliding mode con-troller of Sub-Optimal type (see [32]) has been implementedfor both position/orientation and force controllers. As for po-sition/orientation control, the second-order sliding mode con-troller has the objective to steer to zero the error state variablesx1(t), x2(t). One has⎧⎨

⎩ySMi(t) = αiGisign {si(t) − 0.5siM}x1(t) = e′p(t), x2(t) = x1(t)si(t) = x2i + βx1i, β > 0

(36)

where αiGi, and β are design parameters (see [32]), and siM isthe last extremal value of si(t).

Note that it is possible that a sudden change of the contactforce direction occurs. In this case, the integrated input law isnot adequate because the past values (referred to the previouscontact directions) still affect the integrated output. So, theactual control component yp(t) needs to be determined byselecting the current direction, i.e.,

yc1(t) = ySM (t), yp(t) = Σpyc1(t). (37)

In this way, it is guaranteed that the position/orientationcontroller always generates an input in the direction in whichit is possible to move the tip without the risk of damages.

A second-order sliding mode controller of Sub-Optimal typehas been implemented also as force controller. In this case, thesecond-order sliding mode controller has the objective to steerto zero the error state variables χ1(t), χ2(t). One has⎧⎨

⎩yfSMi(t) = αfiGfisign {sfi(t) − 0.5sfiM}χ1(t) = e′f , χ2(t) = χ1(t)sfi(t) = χ2i + βfχ1i, βf > 0

(38)

where αfiGfi, and βf are design parameters (see [32]), andsfiM is the last extremal value of sfi(t). Finally, as before,the possible sudden change of the contact force direction isconsidered. So, the force control input yf (t) is computed as

yc2(t) = yfSM (t), yf (t) = (I − Σp)yc2(t). (39)

In this way, it is guaranteed that the position/orientationcontroller always generates an input in the direction in whichit is possible to move the tip without the risk of damages.

VII. STABILITY OF THE CONTROLLED SYSTEM

The following assumptions need to be considered.Assumption 1: Given a C2 trajectory Pd(s(t)), the in-

verse kinematics relation C−1(Pd(s(t))) exists for all the timeinstants t. Moreover, velocities Pd(s(t)) and accelerationsPd(s(t)) are compatible with the manipulator performances.

Assumption 2: The trajectory Pd(s(t)) avoids kinematicsingularities, i.e., ∀ t ∃ J−1(q(t)) with q(t) such thatPd(s(t)) = C(q(t)).

Assumption 3: The maximum absolute noise f(t) affect-ing the contact force measurements is bounded, and

|f | < Fm ∧ |f | < (FM − Fm). (40)

Assumption 4: No friction is generated when the tiptouches the obstacles. In other words, [fcx, fcy]T is normal tothe obstacle surface in the contact point, and ϕ depends only onthe direction normal to the surface.

Assumption 5: The mechanical structure of the manipula-tor and of the obstacle is rigid, and the elastic properties of thetip of the sensor are the same in any possible direction of thecontact force. Moreover, the estimate ϕ(t) of the angle ϕ(t) isaccurate enough to assume that ϕ(t) = ϕ(t), ∀t.

Note that these assumptions are quite realistic (of course,friction may be important as the end-effector moves around theobstacle, but, in our case, to minimize its effects, a small wheelhas been placed at the end-effector of the tip, see Fig. 5. Byadopting this solution, we have achieved satisfactory experi-mental results). Note also that, by suitably selecting Fm andFM in (40), it is possible to avoid the situation in which thehybrid control takes place without a contact between the tipand the obstacle, as well as it is possible to avoid that switchingbetween the hybrid control and the position/orientation controltakes places only because of the noise presence on the measure-ments. The choice of these parameters can be made on the basis


Fig. 5. COMAU SMART3-S2 robot and the force sensor with its tip.

of the noise and uncertainties which are present in the system.In our case, we have obtained an estimation of Fm and FM

after an experimental tuning procedure. The last statement ofAssumption 5, in practice, is not easily satisfied, but we haveseen that, in experiments we can obtain satisfactory results alsoin case of non exact estimation of ϕ(t).

The stability properties of the PD controller are quite obvi-ous. The following theorems can be proved for the sliding modecontrollers.

Theorem VII.1 (First-Order Sliding Mode Control Sta-bility): Given system (30) where yp(t), yf (t), and y(t) aredetermined as in (35), and (28), respectively, then, starting fromthe initial condition ep(0) = ep(0) = ef (0) = ef (0) = 0, thesliding manifold [si, sfi]T = 0, for i = 1, . . . , 3, si, and sfi asin (33) is a positively invariant set for the system, and the originof the system state space is an asymptotically stable equilibriumpoint.

Proof: Since the decoupled system dynamics (25) can bereformulated as in (30), the control objective is to steer to zerothe two signals ep(t) and ef (t). The two subsystems in (30)are second-order systems. By determining si, sfi from (33),one has that the structure of the obtained auxiliary systems isthe same as that of the system considered in [20]. By virtue ofthe boundedness of the uncertain term η(t) in (30), the proofof the positively invariance of the set [si, sfi] = 0, and of theasymptotic stability of the origin of the system state space isanalogous to that provided in [20] (see Theorem 1). �

Theorem VII.2 (Second-Order Sliding Mode Control Sta-bility): Given system (30) where yp(t), yf (t), and y(t) aredetermined as in (37), (39), and (28), respectively, then, startingfrom the initial condition ep(0) = ep(0) = ef (0) = ef (0) =0, the second-order sliding manifold [si, si, sfi, sfi]T = 0 fori = 1, . . . , 3, si and sfi as in (36) and (38), respectively, isa positively invariant set for the system, and the origin ofthe system state space is an asymptotically stable equilibriumpoint.

Proof: Since the decoupled system dynamics (25) can bereformulated as in (30), the control objective is to steer to zerothe two signals ep(t) and ef (t). The two subsystems in (30)are second-order systems. By determining si, si from (36), andsfi, sfi from (38), one has that the structure of the obtainedauxiliary systems is the same as that of the system consideredin [20]. By virtue of the boundedness of the uncertain termη(t) in (30), the proof of the positively invariance of the set[si, si, sfi, sfi] = 0, and of the asymptotic stability of the originof the system state space is analogous to that provided in [20](see Theorem 2). �

TABLE IPARAMETERS OF THE TWO TRAJECTORIES

CONSIDERED IN THE EXPERIMENTS

TABLE IISECOND, THIRD, AND FOURTH COLUMNS: ROOT MEAN SQUARE (RMS)

ERRORS VALUES OBTAINED WHILE EXECUTING THE EXPERIMENTS

WITH THE PROPORTIONAL-DERIVATIVE (PD) CONTROLLER,FIRST-ORDER (FO) SLIDING MODE AND SECOND-ORDER (SO)

SLIDING MODE. THE LAST COLUMN REPORTS THE RATIOS BETWEEN

THE ERROR VALUES OBTAINED WITH SUB-OPTIMAL SECOND-ORDER SLIDING MODE AND PROPORTIONAL-DERIVATIVE

VIII. EXPERIMENTAL TESTS

A. Details of the Experimental Setup

The COMAU SMART3-S2 industrial anthropomorphic rigidmanipulator, located at the Department of Electrical Engineer-ing of the University of Pavia, is shown in Fig. 5. It consistsof six-DOF actuated by six brushless electric motors. 12-bitresolvers supply accurate angular position/orientation measure-ments. Torque transmission is provided by reducers. For ourpurposes, joints 1, 4, and 6 of the robot have been locked sothat only joints 2, 3, and 5 are used. Yet, the proposed approachcan be easily extended to a n-joints robot. The three consideredjoints are numbered as {1, 2, 3}.

The COMAU SMART3-S2 robot is equipped with the ATIGAMMA force sensor. The analog output of the sensor isacquired and sampled through a FLEX dsPIC micro-controller.The FLEX board and the robot controller are connected througha serial line. The sampling time of the whole real time systemis 1 ms, while force values are acquired every 2 ms, due to thetime needed to convert the analog signal.

B. Experimental Results for the Interaction Control

Table I describes the parametrization of the two trajectoriesconsidered in the experiments, while Table II reports the out-comes of the experiments. As it can be seen, considering theSub-Optimal second-order sliding mode controller significantlyimproves the capability of the controlled system of followingthe trajectories even in contact with the environment. To this


Fig. 6. Actual and desired trajectories of the tip when the second trajectory isfollowed on the first obstacle by relying on the Sub-Optimal SOSM controller(left). Actual and desired trajectories of the tip when the first trajectoryis followed on the second obstacle by relying on the Sub-Optimal SOSMcontroller (right).

Fig. 7. First obstacle shape: the SMART3-S2 robot interacts with a planar ob-ject. Second obstacle shape: the SMART3-S2 robot interacts with a nonplanarobject.

end, the last column of Table II reports the ratio between theRoot Mean Square (RMS) error defined as

eRMSi =

√√√√ N∑j=1

(ypij − Pij)2

N − 1(41)

N being the number of sampled data, and obtained by consid-ering as controllers second-order sliding mode control versusproportional-derivative control. Note that the shapes of theactual trajectories P are very similar to our initial purposePd(s(t)), as illustrated in Fig. 6. In the two plots of Fig. 6,one can see that the actual trajectory P coincides with theshape of the obstacles which have been touched by the tip ofthe manipulator (see Fig. 7). The actual force magnitude anddirection which have been measured during two experimentsare reported in Fig. 8. In the first and third plots, the actualforce magnitude and the respective control status (identified by“Cont.” in the legend) are depicted. The control status is 0 onlyif position/orientation control is activated, while it is equal to 1when hybrid control is activated.

IX. CONCLUSION

A hybrid control scheme is proposed which allows one toimplement different combinations of position/orientation andforce controllers. Before starting the control phase, an a prioritrajectory is determined, assuming that no obstacle is present inthe environment. The trajectory is then modified in real time toaccount for the obstacle presence. The hybrid control schemerelies on the inverse dynamics and on the inverse kinematicsof the robot and exploits a combination of position/orientationand force controllers. Experiments are made on a COMAU

Fig. 8. First plot: actual and desired force values on the tip, value of theflag cont (= 1 if there is contact); second plot: direction of the actual contactforce when the second trajectory is followed interacting with the first obstacle.Third and fourth plots: the same quantities when the first trajectory is followedinteracting with the second obstacle.

SMART3-S2 anthropomorphic rigid manipulator, allowing oneto make a comparison among three possible choices of thecontrollers themselves. The control algorithms based on thetheory of second-order sliding modes demonstrate to be goodsolutions for this application.

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Luca Massimiliano Capisani (M’06) gained in2001 the “diploma in Commercial and ComputerScience.” His diploma has been awarded by theyoung Industrial Association of Mantua. He receivedthe laurea (M.Sc.) (cum laude) degree in computerscience engineering from the University of Pavia,Pavia, Italy, in 2006 and the Ph.D. degree in elec-tronics, computer science, and electrical engineeringfrom the Engineering Faculty, University of Pavia,in 2010.

Actually, he is Post-Doc Researcher at the Uni-versity of Pavia. He has been Lecturer for the course of Mechanics of theUniversity of Pavia. His research activities are mainly in the area of modeling,identification and robust position and force control of anthropomorphic roboticmanipulators. In particular, the problems of robust tracking using sliding modecontrollers, planar and spatial path planning and interaction and vision- basedobstacle avoidance techniques, fault detection and identification and robust dis-tributed networked control systems for robot manipulators are coped with. Onthese subjects, he has authored and coauthored twenty-four papers, includingfive journal papers, three international scientific book chapters, and sixteenIEEE conference papers, which have been presented during the sessions.During the Ph.D. period, he has worked for three private companies in the pickand place robotics, industrial PCs, remote monitoring fields.

Dr. Capisani is reviewer of the IEEE TRANSACTIONS ON AUTOMATIC

CONTROL, the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOL-OGY, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, Robotica,International Journal of Control, and of the International Journal of AdaptiveControl and Signal Processing journals. Moreover, he is reviewer for the mainconferences in the field of automatic control and robotics.

Antonella Ferrara (S’86–M’88–SM’03) was bornin Genova, Italy. She received the Laurea degreein electronic engineering and the Ph.D. degree incomputer science and electronics from the Universityof Genova, Genova, in 1987 and 1992, respectively.

She has been Assistant Professor in the Depart-ment of Communication, Computer and System Sci-ences of the University of Genova, since 1992. InNovember 1998, she became Associate Professor ofAutomatic Control at the University of Pavia, Pavia,Italy. Since January 2005, she is Full Professor of

Automatic Control in the Department of Computer Engineering and SystemsScience of the University of Pavia. Her research activities are mainly in thearea of sliding mode control, with application to robotics, automotive control,and process. She has authored or coauthored more than 250 papers, including75 international journal papers.

Dr. Ferrara got the “IEEE North Italy Section Electrical Engineering StudentAward” as a student at the Faculty of Engineering of the University of Genovain 1986. She has been Associate Editor of the IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY from 2000 to 2004. She is AssociateEditor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL since 2006.She currently is and has been member of the International Program Committeeof numerous international conferences and events. She is Senior Member of theIEEE Control Systems Society, member of the IEEE Technical Committee onVariable Structure and Sliding Mode Control, member of the IEEE Roboticsand Automations Technical Committee on Autonomous Ground Vehicles andIntelligent Transportation Systems, and member of the IFAC Technical Com-mittee on Transportation Systems.

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Trajectory Planning and Second-Order Sliding Mode Motion/Interaction Control for Robot Manipulators in Unknown Environments