Collocated Adaptive Control of Underactuated Mechanical Systems
CCollocated Adaptive Control ofUnderactuated Mechanical Systems
Francesco Romano and Daniele Pucci and Francesco Nori Abstract — Collocated adaptive control of underactuated sys-tems is still a main concern for the control community, all themore so because the collocated dynamics is no longer linear withrespect to the constant base parameters. This work extends andencompasses the well known adaptive control result for fullyactuated mechanical systems to the underactuated case. The keypoint is the introduction of a fictitious control input that allowsus to consider the complete system dynamics, which is assumedto be linear with respect to the base parameters. Local stabilityand convergence of time varying reference trajectories for thecollocated dynamics are demonstrated by using Lyapunov andBarbalat arguments. Simulation and experimental results ona two-link manipulator verify the soundness of the proposedapproach.
I. I
NTRODUCTION
Nonlinear feedback control of underactuated mechanicalsystems is not new to the scientific community [1], [2], [3].Aircraft, underwater vehicles, and humanoid robots are onlya few examples where the number of control inputs is fewerthan the system’s degrees of freedom, which characterizes thenature of an underactuated system [4]. Clearly, the lack ofactuation along with model uncertainties significantly com-plexify the control problem associated with these systems.Given an open-chain mechanical system, this work proposescontrol strategies for a subset of the system’s degrees offreedom by using estimates of its dynamical model. In thelanguage of automatic control, the laws presented in thispaper fall into the category of adaptive control schemes [5].Underactuated mechanical systems arise specific issueswhen attempting the control of the complete set of degrees offreedom. Assuming that the system’s desired configurationis feasible, the nature of a controller that asymptoticallystabilizes this configuration is intimately related to the natureof the system itself. For instance, systems without potentialterms in general forbid the existence of time-invariant feed-back continuous stabilizers [6]. This claim, which followsfrom an application of Brockett’s Theorem [7], motivated thedevelopment of discontinuous and/or time-varying feedbackstabilizers for specific classes of systems [8], [9], [10].Clearly, the complexity of the control problem reduces whenattempting to stabilize only a subset of the system’s degreesof freedom.In the specialized literature, several methods have beenproposed to control a subset of the system’s degrees of *This paper was supported by the FP7 EU projects CoDyCo (No. 600716ICT 2011.2.1 Cognitive Systems and Robotics), and Koroibot (No. 611909ICT-2013.2.1 Cognitive Systems and Robotics). All authors belong to the Department of Robotics, Brain and CognitiveSciences, Italian Institute of Technology, Via Morego 30, Genoa, Italy [email protected] . freedom. Inverse dynamics [11], [4], sliding mode [12],and energy based techniques [13] are among the maintools exploited by these works. The common denominatorof these approaches is to partition the set of degrees offreedom into two subsets, usually referred to as collocated and noncollocated . The former, whose cardinality equals thenumber of control inputs, contains the actuated degrees offreedom. The latter accounts for the remaining nonactuated degrees of freedom – see [4] for additional details. Then,the control objective is usually defined as the asymptoticstabilization of either set to desired values.To cope with model uncertainties, which may impair theeffectiveness of the aforementioned approaches, adaptivecontrol schemes have also been proposed. Adaptive stabi-lizations of the collocated and noncollocated set is achievedin [14]. The main drawback of this approach is that themeasurement of the system’s acceleration is required by thefeedback control action. Leaving aside causality issues, thismeasurement may not be always available.In the case of fully actuated mechanical systems, adaptivestabilization of time varying reference trajectories can beachieved [15], [16]. The key assumption is that the system’sdynamics can be expressed linearly with respect to a set ofconstant base parameters . The extension of these works tothe underactuated case is not straightforward. As a matter offact, the collocated dynamics is no longer linear with respectto the base parameters when expressed independently of thenoncollocated accelerations.Assuming that the control objective is the asymptoticstabilization of the collocated degrees of freedom, the presentpaper basically extends [15] to the underactuated case. Thekey point is the introduction of a fictitious control input thatallows us to consider the complete system’s dynamics, whichis linear with respect to the base parameters. No accelerationmeasurement is required by the proposed control laws.The paper is organized as follows. Section II provides no-tation and background, and also proposes a slightly differentformulation of [15] that simplifies both the presentation andthe proof of the paper’s contribution. The main control resultsare presented in Section III. Validations of the approach arepresented in Section IV, first through simulations carried outwith the simulator Gazebo, and then through experimentsperformed with a two-link underactuated robot. Remarks andperspectives conclude the paper.II. B ACKGROUND
A. Notation
The following notation is used throughout the paper. a r X i v : . [ c s . S Y ] M a y The set of real numbers is denoted by R . • Let u and v be two n -dimensional column vectors ofreal numbers, i.e. u, v ∈ R n , the inner product betweenthem is x (cid:62) y , where “ (cid:62) ” stands for the transposeoperator. • Given a function of time f ( t ) , its time derivative isdenoted by ˙ f ( t ) . Given a function f of several variables,its gradient w.r.t. some of them, say x , is denoted as ∂ x f . • The euclidian norm of either a vector or a matrix of realnumbers is denoted by | · | . • I n ∈ R n × n denotes the identity matrix of dimension n ; n ∈ R n denotes the zero vector of dimension n ; n × m ∈ R n × m denotes the zero matrix of dimension n × m . • The generalized coordinates characterizing the mechan-ical system are given by an n -dimensional vector of realnumbers denoted as q ∈ R n ; its first and second ordertime derivatives are indicated as ˙ q and ¨ q , respectively. • M ( · ) ∈ R n × n , C ( · ) ∈ R n × n , and g ( · ) ∈ R n denotethe inertia matrix, the Coriolis matrix, and the grav-ity torques obtained by applying Lagrange’s formal-ism [17]. B. System modeling and properties
In light of the above notation, we assume that the appli-cation of Lagrange formulation yields a system’s model ofthe following form: M ( q, π )¨ q + C ( q, ˙ q, π ) ˙ q + g ( q, π )+ F v ( π ) ˙ q + F ( q, ˙ q, π ) = τ , (1)where π ∈ R p is the vector of the (constant) system’s baseparameters [18], F v ∈ R n × n and F ( · ) ∈ R n model viscousand nonlinear friction torques (i.e. F v is a positive definitematrix), and τ is the vector of control inputs (i.e. desiredactuators’ torques) to be designed for achieving specificcontrol objectives. Furthermore, the following properties onthe model (1) hold true [17]: Property 1.
The inertia matrix M is a symmetric positivedefinite matrix, which implies λ ( π ) I n ≤ M ( q, π ) ≤ λ ( π ) I n , with λ and λ two strictly positive constants. Property 2.
The matrix ˙ M − C is skew-symmetric, i.e. x (cid:62) ( ˙ M − C ) x = 0 , ∀ x ∈ R n . Property 3.
The Coriolis matrix C ( q, ˙ q, π ) satisfies | C ( q, ˙ q, π ) | ≤ λ ( π ) | ˙ q | , for some bounded constant λ . Property 4.
The gravity vector g ( q, π ) satisfies | g ( q, π ) | ≤ γ ( π ) , for some bounded constant γ . Property 5.
The model (1) can be expressed linearly withrespect to the system’s base parameters π . Also, there existsa regressor matrix Y ( · ) ∈ R n × p such that M ( q, π )¨ q + C ( q, ˙ q, π ) ξ + g ( q, π )+ F v ( π ) ξ + F ( q, ˙ q, π ) = Y ( q, ˙ q, ξ, ¨ q ) π, for any vector ξ ∈ R n . The matrix Y ( · ) is the so-called Slotine-Li regressor. As foran example, all above assumptions are satisfied in the caseof rigid robot manipulators. C. A known adaptive control result
Let r ( t ) denote a time-varying reference trajectory for thejoint variables q . Throughout the paper, we assume that: Assumption 1.
The reference trajectory r ( t ) is bounded innorm on R + , and its first and second order derivatives arewell-defined and bounded on this set. We present below a revisited version of an adaptivecontrol scheme that ensures the asymptotic stabilization ofthe tracking error e := q − r (2)to zero without the knowledge of the inertial parameters π .The benefits of this new formulation will be clear in the nextsection. First, define: s := ˙ q − ξ, (3a) ˜ π := ˆ π − π, (3b)where ˜ π is the inertial parameters estimation error. Byconsidering the dynamics ˙ˆ π and ˙ ξ as auxiliary control inputs,fusing and reformulating the results [15] [16] lead to thefollowing lemma. Lemma 1.
Assume that Properties 1-5 and Assumption 1hold true. Apply the following control laws to System (1) τ = Y ( q, ˙ q, ξ, ˙ ξ )ˆ π − Ks, (4a) ˙ˆ π = − Γ Y (cid:62) ( q, ˙ q, ξ, ˙ ξ ) s, (4b) ˙ ξ = ¨ r − Λ ˙ e − Λ e, (4c) with K, Λ , Λ ∈ R n × n diagonal, constant positive definitematrices, and Γ ∈ R p × p a constant positive definite matrix.Then, there exists a constant vector β ∈ R n such that theequilibrium point ( (cid:82) t e ( s ) ds, e, s, ˜ π ) = ( β, n , n , p ) of theclosed loop dynamics is globally stable, and the trackingerror e ( t ) converges to zero. The proof is given in Appendix A. The main differencebetween the above formulation and the one of [15] is that ˙ ξ is viewed as an auxiliary control input. Observe that theinitial condition ξ (0) can be arbitrary chosen thanks to theadditional term Λ e in (4c). This term plays the role ofan integral action in the expression of (3a), and does notaffect stability and convergence. For Lemma 1 to hold, itis assumed that System (1) is fully actuated. The followingsection proposes an extension of Lemma 1 to the case whereSystem (1) is underactuated.II. M AIN THEORETICAL RESULTS
Assume that System (1) endows only m < n torquecontrol inputs so that the first k := n − m rows on theright hand side of Eq. (1) are identically equal to zero, i.e. Y ( q, ˙ q, ˙ q, ¨ q ) π = (cid:18) k ¯ τ (cid:19) , (5)with ¯ τ ∈ R m . Now, partition the generalized coordinatevector q as follows q := (cid:18) q n q c (cid:19) , (6)where q n ∈ R k , q c ∈ R m , and the suffixes “ n ” and “ c ” standfor noncollocated and collocated , respectively. Also, assumethat the control objective is the asymptotic stabilization of thecollocated joint coordinates q c about reference trajectories r ( t ) ∈ R m , i.e. the stabilization of the tracking error e := q c − r (7)to zero. As before, we want to design control laws forthis control objective without knowledge of the inertialparameters π .Next theorem basically states that modulo a redefinitionof the auxiliary control input ˙ ξ , the laws (4) ensure theasymptotic stabilization of only the collocated joint variables. Theorem 1.
Assume that Properties 1-5 and Assumption 1hold true. Partition the variables in (3a) and the regressor Y ( · ) as follows: s := (cid:18) s n s c (cid:19) , ξ := (cid:18) ξ n ξ c (cid:19) , Y ( · ) := (cid:18) Y n ( · ) Y c ( · ) (cid:19) , (8) where s n , ξ n ∈ R k , s c , ξ c ∈ R m , Y n ∈ R k × p , Y c ∈ R m × p .Apply the following control laws to System (5) ¯ τ = Y c ( q, ˙ q, ξ, ˙ ξ )ˆ π − Ks c , (9a) ˙ˆ π = − Γ Y (cid:62) ( q, ˙ q, ξ, ˙ ξ ) s, (9b) ˙ ξ = (cid:18) ˙ ξ n ˙ ξ c (cid:19) = (cid:32) (cid:99) M − n (cid:104) K n s n − Y n (cid:16) q, ˙ q, ξ, (cid:0) k ˙ ξ c (cid:1)(cid:17) ˆ π (cid:105) ¨ r − Λ ˙ e − Λ e (cid:33) (9c) with K, K n , Λ , Λ ∈ R m × m diagonal, constant positivedefinite matrices, and the matrix (cid:99) M n defined as the k th orderleading principal minor of the mass matrix M evaluated withestimated base parameters, i.e. (cid:99) M n := SM ( q, ˆ π ) S (cid:62) (10) where the selector S is given by S := (cid:0) I k k × m (cid:1) . (11) Then, the following results hold.i) There exists a constant vector β ∈ R m such that theequilibrium point ( (cid:82) t e ( s ) ds, e, s, ˜ π ) = ( β, m , n , p ) of the associated closed loop dynamics is locally stable.ii) Assume that the noncollocated joint velocities remainbounded, i.e. ∃ δ > such that | ˙ q n | < δ ∀ t . Then, inaddition to the local stability, the tracking error e ( t ) also converges to zero. The proof is given in Appendix B. The interest of the in-voked reformulation of classical adaptive schemes presentedin Lemma 1 lies in the similarity between the control laws (4)and (9). In particular, in both cases, the evolution of thevariable ξ can be obtained by numerical integration of itsdynamics ˙ ξ . Also, when the system endows m unactuateddegrees of freedom, it suffices to modify the first m elementsof this dynamics – see Eq. (9c) – to still ensure stability andconvergence of the collocated joint coordinates. However,convergence is guaranteed when the noncollocated jointvelocities | ˙ q n | remain bounded. This requirement, whichfollows from the application of Barbalat’s Lemma, reflectsphysical limitations, mostly due to the energy exchangedbetween noncollocated and collocated joint variables. As amatter of fact, simulations we have performed suggest thatfriction effects play a role in guaranteeing the boundednessof | ˙ q n | , and consequently the convergence of the trackingerror e ( t ) to zero.The local nature of the controls (9) is due to the fact thatthe matrix (10) may not be invertible far from the point ˜ π =0 p . Observe that the invertibility of (10) in a neighborhoodof this point is guaranteed by Property 1, which implies thateach leading principal minor of the mass matrix M ( q, π ) ispositive definite, and therefore invertible.The non-invertibility of the matrix (10) is related to the standard parameters associated with the estimated baseparameters ˆ π . In particular, when the standard parameters associated with an estimate ˆ π are not physical consistent (e.g.a negative mass of a rigid body composing the underlyingmechanical system) the inertia matrix M ( q, ˆ π ) may not bepositive definite [19]. This problem would be avoided ifthe adaptation law ˙ˆ π guaranteed an evolution ˆ π ( t ) suchthat the associated standard parameters were always physicalconsistent. Such an adaptation law is under preparation andwill be presented and discussed in a forthcoming paper.Now, let us remark that if det (cid:16) (cid:99) M n ( q ( t ) , ˆ π ( t )) (cid:17) > ∀ t (12)independently of the initial conditions, the laws (9) ensureglobal stability. However, this is not always the case. To avoida possible ill-conditioning of the laws (9), a desingularizationpolicy must be defined when the above determinant getsclose to zero. We present below the policy used in this paper.Observe that the time derivative of (12) depends upon theadaptation law ˙ˆ π . Consequently, it is theoretically possible tomodify the law (9b) in order to ensure that the determinant of (cid:99) M n never decreases below a certain threshold. Next Lemmapresents such a modification of the adaptation law ˙ˆ π . Lemma 2.
Consider the laws (9) with the adaptation lawredefined as follows ˙ˆ π = − Γ (cid:104) Y (cid:62) ( q, ˙ q, ξ, ˙ ξ ) s − ηδ (cid:105) , (13) For example, the standard parameters of a rigid body consists in a ten-dimensional vector composed of the mass, the three first moments of mass,and the six elements of the inertia matrix [18]. ig. 1: iCub model in the Gazebo environment. with η ∈ R and δ ∈ R p given by: η := if tr (cid:16) (cid:99) M − n Υ (cid:17) ≥ or det (cid:16) (cid:99) M n (cid:17) >ε − tr( (cid:99) M − n Υ) δ T Γ δ otherwise (14a) δ := k (cid:88) i =1 Y (cid:62) M n ( q, e i ) (cid:99) M − n e i , (14b) where ε ∈ R + , Y M n := S (cid:2) Y (cid:0) q, n , n , (cid:0) e i m (cid:1)(cid:1) − Y ( q, n , n , n ) (cid:3) , (15a) Υ := ( υ , . . . , υ i , . . . , υ k ) , (15b) υ i := ∂∂q (cid:104) Y M n ˆ π (cid:105) ˙ q − Y M n Γ Y (cid:62) ( q, ˙ q, ξ, ˙ ξ ) s, (15c) and e i ∈ R k denotes a vector of k zeros except for the i th coordinate, which is equal to one.Then, the following results hold.i) If det (cid:16) (cid:99) M n (cid:17) > , then | δ | > .ii) Assume that det (cid:16) (cid:99) M n (cid:17) (0) > ε . Then, det (cid:16) (cid:99) M n (cid:17) ( t ) ≥ ε ∀ t. The proof is in Appendix C. This Lemma states that itis always possible to maintain the determinant of (cid:99) M n abovea certain threshold ε . In fact, the desingularizing term (14)would be ill-conditioned only at | δ | = 0 , but this never occurswhen det ( (cid:99) M n ) > – see the result i ) . Fig. 2: Experimental setup. The two-link pendulum is de-picted in overlay. Blue circle: underactuated joint ( q n );yellow circle: actuated joint ( q c ).Clearly, the larger the threshold ε , the larger the influenceof the desingularizing term ηδ on the stability result ofTheorem 1. Consequently, this threshold must be tuned de-pending on the specific application, which defines the valueof det ( (cid:99) M n ) at ˆ π = π . Simulations and experimental resultspresented next show that the influence of this desingularizingterm does not significantly affect the practical stability andboundedness of the tracking error e ( t ) .IV. S IMULATIONS AND E XPERIMENTAL R ESULTS
In this section, we test the control laws (9a)-(9c)-(13) firstthrough simulations, and then through experiments carriedout on a two-link manipulator with rotational joints. In partic-ular, the robot consists in the hip (nonactuated joint) and the knee (actuated joint) of the iCub humanoid robot, while therobot’s ankle is kept fixed (see Figure 2). The underactuatedsystem is then simulated in the “Gazebo” environment.Furthermore, the YARP middle-ware interfaces the robot,either real or simulated, to our controller implementation .The main aim of the simulation campaign is to determine aset of gains to use during the experiments.The laws (9a)-(9c)-(13) require to compute the regressor Y ( · ) of a two-link manipulator [20, p. 149]. This regressor Gazebo is the official simulator of the Darpa Robotic Challenge, and isdeveloped by the Open Source Robotics Foundation (OSRF). The officialwebsite is The implementation is available at the CoDyCo repository
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Time (s) J o i n t P o s i t i on [ deg ] (a) Knee angle q c (blue) and its reference value (red). Time (s) T o r que [ N m ] (b) Desired torque for the actuated joint. Fig. 3: Simulation results.is computed with only viscous friction terms, i.e. F ( · ) ≡ .The main reasons why are the following ones.i) The simulator “Gazebo” currently supports only viscousfriction.ii) The iCub humanoid robot is equipped with a low-leveltorque control loop that is in charge of stabilizing anydesired joint torque [21], [22]: this loop is supposed tocompensate for friction effects, but this compensation isnever perfect. Then, the terms of viscous friction left inthe regressor account for imperfect friction compensa-tions during adaptive control.Figure 3 depicts typical simulation results obtained withthe laws (9a)-(9c)-(13) evaluated with • Λ = 5 , • Λ = 1 , • K = 0 . I , • Γ = 0 . I , • ε = 5 , • ˆ π (0) = (1 . , − . , . , , − . , . , . , . , • ξ (0) = 0 , β = 0 .The desired trajectory for the knee’s joint was piecewiseconstant, and more precisly equal to: r = − ◦ , ≤ t ≤ s , r = − ◦ , s ≤ t ≤ s . Figure 3 shows that convergenceof the tracking error is quickly achieved, while the actuationtorque remains within the physical limits. By using the abovegains, we went one step further and applied the laws (9a)-(9c)-(13) to the aforementioned two-link robot.Two experiments with two different reference trajectories r ( t ) were conducted on the real robot. They are:i) a piecewise constant reference defined by r ( t ) = − ◦ ≤ t < , − ◦ ≤ t < , − ◦ ≤ t < (16) ii) a time varying reference r ( t ) defined by (in degrees): r ( t ) = −
60 + 35 sin(2 π . t ) 0 ≤ t < , −
60 + 35 sin(2 π . t ) 53 ≤ t < , −
60 + 35 sin(2 π . t ) 90 ≤ t < . (17)Figure 4 depicts the results of the experiment i). Observethat the tracking error converges to zero with a settling timeapproximately equal to that of the simulation result (compareFigures 3a and 4a). However, unmodeled friction effectsand imperfect tracking of the low-level torque control loopreflect in reduced overshoots and zero hip velocity close tozero tracking error (see Figure 4b). Note also that the initialconditions for ˆ π render the determinant of (cid:99) M n at t = 0 smaller than the chosen threshold ε = 5 (see Figure 4d).Then, this determinant can only increase because of thedesingularization of Lemma 2, and once it goes beyond thethreshold, it never decreases ε = 5 . It is important toobserve that this experiment is basically quasi-static, so thecoupling effects between the collocated and noncollocatedjoints are not preponderant.Figure 5 shows the experimental results obtained by apply-ing the aforementioned laws with the time-varying referencetrajectory given by (17). Observe that despite rapid variationsof the desired control torque (see Figure 5c), the trackingerror remains bounded and slowly converges to zero. Also,note that coupling effects between the hip and knee joints areno longer negligible, since the hip velocity achieves peaksof about
60 [ deg/s ] (see Figure 5b). These rapid variationsof the joint coordinates q cause large oscillations of the de-terminant of the matrix (cid:99) M n ( q, ˆ π ) (see Figure 5d). However,the desingularization of Lemma 2 forces the determinant toremain above the threshold ε = 5 .V. C ONCLUSION
We presented an extension of the adaptive controlmethod [15], which was developed for fully actuated ma-nipulators, to the case of underactuated mechanical systems.
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Time (s) J o i n t P o s i t i on [ deg ] (a) Knee angle (blue) and its reference value (red) (as Equation 16). Time (s) H i p v e l o c i t y [ r ad / s ] (b) Velocity of the hip ˙ q n . Time (s) T o r que [ N m ] (c) Desired knee joint torque to be stabilized by the low-level torquecontrol loop. Time (s) D e t e r m i nan t (d) Determinant of the matrix (cid:99) M n (blue) and threshold ε = 5 (red). Fig. 4: Experimental results for constant reference value (16).Local stability and convergence of the collocated variableswere demonstrated by using Lyapunov and Barbalat argu-ments. Compared to existing results, our approach does notmake use of any acceleration measurements, thus avoid-ing altogether causality concerns. The control results werevalidated with both simulations performed in the Gazeboenvironment, and with an implementation on a two-linkmanipulator obtained from the iCub humanoid robot. It wasbeyond the scope of this work to address the classical, andwell known, drawbacks of adaptive control schemes [23].Although the control laws presented in this paper are con-tinuous, the associated basin of attraction of the equilibriumpoint is local. This local nature is due to the fact that thepresented control laws rely on the invertibility of the system’sinertia matrix along the estimated system’s model. Thismatrix may not be invertible for physical inconsistent baseparameters [19]. Then, our goal is to design an estimationdynamics such that the associated base parameters are alwaysphysical consistent. In this case, the control laws presented in this paper would guarantee global stability and convergence.A
PPENDIX
A. Proof of Lemma 1
First, from Eq. (4c) observe that the variable ξ ( t ) can beobtained by integration, i.e. ξ ( t ) = ˙ r − Λ e − Λ y, with y := (cid:82) t e ( s ) ds − β , and β ∈ R n a properly chosenconstant. Now, consider the following candidate Lyapunovfunction V := 12 (cid:2) s (cid:62) M s +˜ π (cid:62) Γ − ˜ π +2 e (cid:62) K Λ e + 2 y (cid:62) Λ K Λ y (cid:3) . (18)Note that since K, Λ , and Λ are diagonal matrices, then theproducts K Λ and Λ K Λ are diagonal and positive definitematrices. In view of Properties 2, 5 and the controls (4a)-(4b), one easily verifies that the time derivative of (18) yields, ˙ V = − s (cid:62) Ks − s (cid:62) F v s + 2 e (cid:62) K Λ ˙ e + 2 y (cid:62) Λ K Λ e. (19)
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Time (s) P o s i t i on E rr o r [ deg ] (a) Knee angle (blue) and its reference value (red) (as Equation 17). Time (s) H i p v e l o c i t y [ r ad / s ] (b) Velocity of the hip ˙ q n . Time (s) T o r que [ N m ] (c) Desired knee joint torque to be stabilized by the low-level torquecontrol loop. Time (s) D e t e r m i nan t (d) Determinant of the matrix (cid:99) M n (blue) and threshold ε = 5 (red). Fig. 5: Experimental results for time-varying reference trajectory (17).Recall that F v is a positive definite matrix. By substituting s = ˙ q − ξ = ˙ e + Λ e + Λ y in the first term on the right hand side of (19) one has: ˙ V = − (cid:0) ˙ e (cid:62) y (cid:62) Λ (cid:1) ¯ K (cid:18) ˙ e Λ y (cid:19) − s (cid:62) F v s − e (cid:62) Λ K Λ e, (20)with ¯ K := (cid:18) K KK K (cid:19) . (21)One easily verifies that ¯ K is positive semi-definite. Asa consequence, ˙ V ≤ and the global stability of ( (cid:82) t e ( s ) ds, e, s, ˜ π ) = ( β, n , n , p ) follows. By using As-sumption 1 and the fact that the variables (cid:82) t e ( s ) ds, e, s, and ˜ π are bounded, one deduces that ¨ V is bounded, which in turnimplies that ˙ V is uniformly continuous. Then, the applicationof Barbalat’s Lemma ensures that ˙ V tends to zero. In viewof (20), this implies that the error e ( t ) converges to zero. B. Proof of Theorem 1
Thanks to the formulation of the control result ofLemma 1, this proof is similar to that above. In particular, ξ c ( t ) = ˙ r − Λ e − Λ y, with e given by (7), y := (cid:82) t e ( s ) ds − β , and β ∈ R m a properly chosen constant. Now, reconsider the candidateLyapunov function (18). In view of Properties 2, 5 andthe partitioning (8), the application of the controls (9a)-(9b)renders the time derivative of V as follows: ˙ V = − s T (cid:18) Y n ( q, ˙ q, ξ, ˙ ξ )ˆ πKs c (cid:19) − s (cid:62) F v s + 2 e (cid:62) K Λ ˙ e + 2 y (cid:62) Λ K Λ e. (22)Now, in view of the Property 1, note that the auxiliary controlinput ˙ ξ n is locally well defined since each leading principalminor of the mass matrix M ( q, π ) is invertible when ˆ π liesn a neighborhood of π . As a consequence, the choice of theauxiliary control input ˙ ξ n in (9c) implies that (cid:99) M n ˙ ξ n + Y n (cid:16) q, ˙ q, ξ, (cid:0) k ˙ ξ c (cid:1)(cid:17) ˆ π = Y n ( q, ˙ q, ξ, ˙ ξ )ˆ π = K n s n . In view of (8) and of the above equation, the expression of ˙ V in (22) becomes ˙ V = − s Tn K n s n − s c Ks c − s (cid:62) F v s +2 e (cid:62) K Λ ˙ e +2 y (cid:62) Λ K Λ e. Analogously to the proof of Lemma 1, by substituting s c = ˙ q c − ξ c = ˙ e + Λ e + Λ y in the second term on the right hand side of ˙ V one obtains ˙ V = − s Tn K n s n − (cid:0) ˙ e (cid:62) y (cid:62) Λ (cid:1) ¯ K (cid:18) ˙ e Λ y (cid:19) − s (cid:62) F v s − e (cid:62) Λ K Λ e ≤ . Consequently, the local stability of the equilibrium point ( (cid:82) t e ( s ) ds, e, s, ˜ π ) = ( β, m , n , p ) follows. In addition,under the Properties 1-4, Assumption 1 and the boundednessof ˙ q n , it is possible to verify that ¨ V is bounded, which impliesthat ˙ V is uniformly continuous. Then, analogously to theproof of Lemma 1, one shows that the error e ( t ) convergesto zero. C. Proof of Lemma 2
Proof of i) . If det (cid:16) (cid:99) M n (cid:17) > , then the matrix (cid:99) M − n exists.By multiplying Eq (14b) times ˆ π (cid:62) , one obtains: ˆ π (cid:62) δ = k (cid:88) i =1 ˆ π (cid:62) Y (cid:62) M n ( q, e i ) (cid:99) M − n e i = k (cid:88) i =1 e (cid:62) i (cid:99) M (cid:62) n (cid:99) M − n e i = k .Observe that M ( q, · ) is symmetric by construction. Now,since the system is underactuated, then k ≥ . Consequently, | δ | can never be zero.Proof of ii) . Consider the following storage function V d := 12 det ( (cid:99) M n ) .It is possible to verify that the time derivative of V d is: ˙ V d = det ( (cid:99) M n ) tr (cid:16) (cid:99) M − n ˙ (cid:99) M n (cid:17) = det ( (cid:99) M n ) (cid:104) tr (cid:16) (cid:99) M − n Υ (cid:17) + ηδ (cid:62) Γ δ (cid:105) ,where η as in Eq (14a), δ as in Eq (14b) and Υ as defined inEq (15b). By choosing the adaptation law defined in Eq (13),one obtains that ˙ V d ≥ if det (cid:16) (cid:99) M n (cid:17) ≤ ε . As a consequence det( (cid:99) M n ) ≥ ε ∀ t if det (cid:16) (cid:99) M n (cid:17) (0) > ε .ACKNOWLEDGMENTThe authors thank Silvio Traversaro for the useful discus-sions during the development of this project. R EFERENCES[1] R. Olfati-Saber, “Nonlinear Control of Underactuated MechanicalSystems with Application to Robotics and Aerospace Vehicles,” Ph.D.dissertation, 2000.[2] P. L. Hera, “Underactuated Mechanical Systems: Contributions totrajectory planning, analysis, and control,” Ph.D. dissertation, 2011.[3] Y. Liu and H. Yu, “A survey of underactuated mechanical systems,”
Control Theory & Applications, IET , vol. 7, no. February, pp. 921–935, 2013.[4] M. W. Spong, “Underactuated Mechanical Systems,” in
Lecture notesin control and information sciences, Control problems in robotics andautomation , 1998, pp. 135–150.[5] K. J. Astrom and B. Wittenmark,
Adaptive Control , 2nd ed. Boston,MA, USA: Addison-Wesley Longman Publishing Co., Inc., 1994.[6] M. Reyhanoglu, “Dynamics and control of a class of underactuatedmechanical systems,”
Automatic Control, . . . , vol. 44, no. 9, pp. 1663–1671, 1999.[7] R. Brockett,
Asymptotic Stability And Feedback Stabilization , 1983.[8] A. De Luca, R. Mattone, and G. Oriolo, “Dynamic mobility of redun-dant robots using end-effector commands,”
Robotics and Automation,. . . , no. April, pp. 1760–1767, 1996.[9] M. Park and D. Chwa, “Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method,”
Indus-trial Electronics, IEEE Transactions on , vol. 56, no. 9, pp. 3541–3555,2009.[10] J. Ghommam, F. Mnif, and N. Derbel, “Global stabilisation andtracking control of underactuated surface vessels,”
IET Control Theory& Applications , vol. 4, no. 1, pp. 71–88, Jan. 2010.[11] A. De Luca and G. Oriolo, “Motion Planning and Trajectory Controlof an Underactuated Three-Link Robot via Feedback Linearization,”
Robotics and Automation, 2000. . . . , 2000.[12] R. Santiesteban and T. Floquet, “Secondorder sliding mode con-trol of underactuated mechanical systems II: Orbital stabilizationof an inverted pendulum with application to swing up/balancing,” . . . Nonlinear Control , no. April 2007, pp. 544–556, 2008.[13] M. Spong, “Energy based control of a class of underactuated mechan-ical systems,” , 1996.[14] Y.-l. Gu and Y. Xu, “Under-actuated robot systems: dynamic in-teraction and adaptive control,”
Proceedings of IEEE InternationalConference on Systems, Man and Cybernetics , vol. 1, pp. 958–963.[15] J.-J. Slotine, “Adaptive manipulator control: A case study,”
IEEETransactions on Automatic Control , vol. 33, no. 11, pp. 995–1003,1988.[16] M. Spong, “Comments on ”Adaptive Manipulator Control: A Casestudy”,”
IEEE Transactions on Automatic Control , vol. 35, pp. 761–762, 1990.[17] B. Siciliano and O. Khatib,
Handbook of Robotics , 2008, vol. 15.[18] W. Khalil and E. Dombre,
Modeling, identification and control ofrobots . Butterworth-Heinemann, 2004.[19] K. Yoshida and W. Khalil, “Verification of the Positive Definiteness ofthe Inertial Matrix of Manipulators Using Base Inertial Parameters,”
The International Journal of Robotics Research , vol. 19, no. 5, pp.498–510, May 2000.[20] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo,
Robotics: Mod-elling, Planning and Control , 2009.[21] M. Fumagalli, M. Randazzo, F. Nori, L. Natale, G. Metta, andG. Sandini, “Exploiting proximal F/T measurements for the iCubactive compliance,” in . Ieee, Oct. 2010, pp. 1870–1876.[22] M. Fumagalli, S. Ivaldi, M. Randazzo, L. Natale, G. Metta, G. San-dini, and F. Nori, “Force feedback exploiting tactile and proximalforce/torque sensing,”
Autonomous Robots , vol. 33, no. 4, pp. 381–398, Apr. 2012.[23] B. D. O. Anderson, “Failures of adaptive control theory and theirresolution,”