2010 11th International Workshop on Variable Structure Systems Mexico City, Mexico, June 26 - 28, 2010
Feedback Stabilization and Force Control Using Sliding Modes in a Mechanical System Subject to Unilateral Constraints Raul Rasc´on, Joaqu´ın Alvarez and Luis T. Aguilar
Abstract—Force feedback sometimes produces an undesirable chattering behavior, where the mechanical system repeatedly makes and breaks contact with the constraint surface this have been shown in experiments on single degree of freedom (1-DOF). This behavior is an example of a limit cycle, and is likely caused by the nonlinearity in the system dynamics introduced by the unilateral (i.e. one-sided) constraint. Yet most published stability analyses of force-controlled robots assume that the constraint is bilateral, meaning the mechanical system never loses contact with the constraint. In this note, we analyze the stability of a 1DOF force-controlled mechanical system in contact with a rigid unilateral constraint applying a sliding mode control algorithm, with the force sensor modeled as a stiff spring. It is proved that the nonlinear system is globally asymptotically stable and achieves zero steady-state force error.
I. I NTRODUCTION In most constrained mechanical systems, the nature of the constraint is best described by its unilateral behavior. The constraint, almost always, divides the workspace of a mechanical system into two regions. A region where the constraints are strictly satisfied and the mechanical system behaves freely and a region defining the violation of the constraint. Unilateral nature of the constraint is clearly depicted in this example, by impacting robotic systems we mean devices involving robot manipulators that collide with some other mechanical system [1]. This kind of systems can always be written in the general form of lagrangian system subject to unilateral constraints [2]. Roughly speaking, it suffices to define the distance between the different bodies as being the constraint. In particular, mechanical systems are subject to unilateral constraints. They belong to the class of nonsmooth dynamical systems [4]. From a general point of view, controllability, observability, and stabilizability of such systems have not yet been understood, except in particular cases [3]–[5]. This is typically the case of mechanical systems which form a subclass of complementarity dynamical systems. Many applications in industry involve mechanical systems interacting with environment. Examples of such systems can be found in manufacturing automation, material handling by robots and space applications. An important task is to model its complete behavior such that the system and its constraint are presented in a natural way [6]. The formulation given in this paper systematically describes the complete behavior of a mechanical system interacting with an environment.
Considerable research has been done in control of robots in constrained motion; see [7] and the references therein. Much of this research has been based under the assumption that the robot is already in contact with the external environment. In many industrial applications the mechanical system is in free motion before constrained motion starts. It is known in literature that switching from constrained motion to free motion poses no problem when compared to switching from free motion to constrained motion. Due to nonzero impact velocity, the transition from free motion to constrained motion leads to impulsive forces on the system because that we use a spring as a force sensor (flexible joint) to avoid this impulsive forces in the mechanical system. Goldsanith [8] analyzed the stability of a 1-DOF force-controlled robot in contact with a unilateral constraint with proportional force feedback applied. In this paper, we develop a sliding mode force error control algorithm [9], [10] which is proposed for a typical regulation trajectory. Asymptotic stability of this algorithm is shown using Lyapunov methods [11], [12]. The sliding mode controller as well as the stability analysis of the closed-loop system will be our main contribution to the solution of this problem. The proposed control scheme is tested for a mechanical system interacting with an environment. Numerical experiments results prove the validity of the theoretical analysis. The rest of the paper is outlined as follows: In Section II we describe the behavior of a mechanical system subject to unilateral constraint. The state feedback design and its stability analysis, using nonsmooth Lyapunov stability theory, is presented in Section III. Section IV presents numerical simulations performed with MATLAB. Section V gives conclusions. II. P ROBLEM S TATEMENT The main concern of the present paper is the force control design and its stability analysis of a mechanical system subject to unilateral constraints (see Figure 1). The equations of motion of the open-loop constrained mechanical system with one degree of freedom can be expressed in joint coordinates space as
R. Rasc´on and J. Alvarez are with CICESE Research Center, Electronics and Telecommunication Department, P.O. BOX 434944, San Diego, CA 92143-4944, (emails: rrascon{jqalvar}@cicese.mx). L.T. Aguilar is with Instituto Polit´ecnico Nacional, Centro de Investigaci´on y Desarrollo de Tecnolog´ıa Digital, Avenida del parque 1310 Mesa de Otay, Tijuana M´exico 22510 (email:
[email protected]).
978-1-4244-5831-8/10/$26.00 ©2010 IEEE
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u = m¨ x + bx˙ + F + w
F =
(
0 kx
if x < 0 if x ≥ 0
(1)
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2010 11th International Workshop on Variable Structure Systems Mexico City, Mexico, June 26 - 28, 2010
error e1 = kx − Fd e2 = k x˙
(6)
where Fd > 0 is the desired contact force. We can write the system (4), (5) in terms of e1 and e2 variables as e˙ 1 = k x˙ = e2 e˙ 2 = k¨ x=− Fig. 1.
Mechanical system and constraint
b k k e2 − g(e1 ) − w m m m
(7)
where where m ∈ R is the mass, joint friction is assumed proportional to velocity, with the friction coefficient denoted by b ∈ R, a force sensor mounted on the tip of the mechanical system is modelled as a spring with stiffness k ∈ R, the mass is driven by an actuator force u ∈ R. In order to account discrepancies in the model, an unknown external disturbances w(t) ∈ R has been introduced and its upper bound W is assumed known a priori sup |w(t)| ≤ W.
(3)
t
The position x ∈ R of the mechanical system is zero when the force sensor is in contact with the wall and applying zero force. Substituting (2) into (1) gives m¨ x + bx˙ + g(x) + w = 0,
(4)
( −u g(e1 ) = e1 − u + Fd
if e1 < −Fd if e1 ≥ −Fd .
Reordering equation (7) into its matricial form, we have e˙ =
0 0 1 0 ) − e − g(e 1 b k k w. 0 −m m m
b − m(µ + λ) mλµ m u = Fd + 1 − e1 + e2 − M sign(s) k k k (10) where M , µ, and λ are positive tunable gain parameters, ensures the displacement of the trajectories toward the sliding surface s = µe1 + e2 .
if x < 0 if x ≥ 0.
(5)
The function g(x) represents the combined control and contact force acting on the mass. Since this quantity is zero when x = u/k, the system (4) has an equilibrium point at (x = u/k, x˙ = 0). Due in system (4) position feedback is used instead of force feedback, the position of the wall and the stiffness of the sensor must be known precisely in order to achieve the desired contact force. For this reason, force feedback is highly preferred over position feedback, provided that closed-loop stability is achieved.
III. C ONTROL D ESIGN
(9)
The following control law
where ( −u g(x) = kx − u
(8)
(11)
A. Stability Analysis
We will analyze the stability of the closed-loop system (9), (10) in its free-motion phase (e1 < −Fd ) and in its constrained-motion phase (e1 ≥ −Fd ). Case 1 (e1 < −Fd ): By substituting (10) and (8) into (9), the closed-loop system takes the form 0 0 0 1 e˙ = ! k e+ k − k w − (µ + λ) m − λµ m Fd − M sign(s) m (12) Now, we need to ensure the existence of sliding modes by verifying ss˙ < 0. To this end, notice that
First, we shift the equilibrium point of (4) to the origin by defining the following state transformation based on the force
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ss˙ = s(µe˙ 1 + e˙ 2 )
2010 11th International Workshop on Variable Structure Systems Mexico City, Mexico, June 26 - 28, 2010
k k − λµ e1 − (µ + λ)e2 + Fd =s µe2 + m m k −s M sign(s) + w m k k k =s − λµ e1 − λe2 + Fd − M sign(s) − w m m m k k =s −λ(µe1 + e2 ) + (e1 + Fd ) − M sign(s) − w m m k k 2 =−λs + s(e1 + Fd ) − M |s| − ws m m k k 2 ≤−λs + s(e1 + Fd ) − M − W |s| (13) m | {z } m negative
The condition ss˙ < 0 is satisfied when s > 0 and M > When s < 0, we can see that
k mW.
k k ss˙ = −λs2 − M − W + (e1 + Fd ) |s|. m m Therefore, the sliding surface is reached when k k (e1 + Fd ) < M − W. m m
(14)
which is a positive definite and radially unbounded. The time derivative of V (s) along the solution of the closed-loop system yields V˙ (s) = ss˙ = s(−λs − M sign(s)) = −λs2 − M |s| < 0 (18) which is negative definite. Then, we conclude that the origin is an asymptotic stable equilibrium point. IV. N UMERICAL S TUDY Performance issues and robustness properties of the proposed sliding mode controller are additionally tested in numerical experiments. In the simulations, performed with MATLAB, the mechanical model depending on the error of force (9) is studied with the parameters Fd = 7.5084, b = 15, k = 375.42, and m = 2.06238 [Kg]. The controller feedback gain were set to λ = 50 and the sliding surface parameter µ = 6. All these values were taken from real measurements performed in the control laboratory at CICESE. The initial position and the initial velocity are set to e1 (0) = −7.5084 that represent 2 cm mass-spring real displacement and e2 (0) = 0, respectively. A comparison of the controller’s performance (10) is made against a proportional force feedback controller of the form u = Fd − kf e1 (see [8]), where kf = 1 is the force feedback gain that must meet the condition kf ≥ 0. We assume that the system is affected by Coulomb friction force w = sign(x) ˙ + 10 sin(t).
Case 2 (e1 ≥ −Fd ): By substituting of (10) and (8) into (9) gives
0 0 1 0 e˙ = e− − k w. −λµ −(µ + λ) M sign(s) m
(15)
V. C ONCLUSIONS
Analyzing the existence of sliding modes by verifying ss˙ < 0, yields ss˙ = s (µe˙ 1 + e˙ 2 ) k = s µe2 − λµe1 − (µ + λ)e2 − M sign(s) − w m k = −λ(µe1 + e2 ) − M sign(s) − ws m k ≤ −λs2 − M − W |s| < 0. m
(16)
Then, we concluded the existence of sliding modes on the surface s = µe1 + e2 . Given the above equation we demonstrate that the origin of the system (15) is asymptotic stable. For this purpose, we introduce the Lyapunov function V (s) =
1 2 s 2
Good performance and desired robustness properties of the system (9) under sliding mode control law (10) are concluded. from Figures 2, 3, and 5.
(17)
An asymptotic force feedback stabilization problem is studied for a 1-DOF mechanical prototype, operating under constrained conditions. A general framework of resolving such problem is proposed. The framework consists of the problem decomposition and output feedback synthesis, involving the second order sliding mode state feedback design. Due to the controller structure, the signum term is multiplied by a small constant which generate a diminished chattering effect on the output of the system. The experimental verification, made for a laboratory prototype, demonstrates the effectiveness of the developed approach. Although the solution of the problem is relatively simple the importance of the problem lies on the basis for some hard problems that we can address on future such synchronization and coordination between two-arm pushing to perform tasks of grasping and transporting large objects by pushing them from two ends or synchronization of juggling systems. Short-term efforts will be focus on the experimental validation of the proposed sliding mode algorithm, based on that evidence some comparisons will be made using supertwisting and quasicontinuous controllers.
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2010 11th International Workshop on Variable Structure Systems Mexico City, Mexico, June 26 - 28, 2010
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[1] N. Mansard and O. Khatib, “Continuous control law from unilateral constraints”, in 2008 IEEE Int. Conf. on Robotics and Automation, pp. 3359–3364, 2008. [2] N. Mansard, O. Khatib and A. Kheddar, “A unified approach to integrate unilateral constraints in the stack of tasks”, IEEE Trans. on Robotics, vol. 25, no. 3, pp. 670–685, June 2009. [3] B. Brogliato, M. Mabrouk, A. Zavala Rio, “On the controllability of linear juggling mechanical systems,” Systems and Control Letters, vol. 55, pp. 350–367, 2006. [4] B. Brogliato, Nonsmooth Mechanics: second edition. London: Springer, 1999. [5] B. Brogliato, “Some perspectives on the analysis and control of complementarity systems,” IEEE Trans. Automat. Control, vol. 48, no. 6, pp. 918–935, 2003. [6] B. Ben Amor, N.K. Haded and F. Mnif, “Controllability analysis of 1DOF linear juggling system”, in 6th International Multi-Conference on Systems, Signals and Devices, pp. 1–6, 2009. [7] D. Wang and N.H. McClamroch, “Position and force control for constrained manipulator motion: Lyapunov’s direct method,” IEEE Transactions on Robotics and Automation, vol. 9, no. 3, pp. 308–312, 1993. [8] P.B. Goldsanith, “Stability of robot force control applied to unilateral constraints,” in 1996 IEEE Canadian Conf. on Electrical and Computer Engineering, pp. 498–501, 1996.
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2010 11th International Workshop on Variable Structure Systems Mexico City, Mexico, June 26 - 28, 2010 [9] V. Utkin, Sliding modes in control optimization. Berlin: Springer-Verlag, 1992. [10] Y. Orlov, Discontinuous systems – Lyapunov analysis and robust synthesis under uncertainty conditions. London: Springer-Verlag, 2009. [11] B.E. Paden and S.S. Sastry, “A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators,” IEEE Transactions on Circuits and Systems, vol. 34, no.1, pp. 73–81, 1987. [12] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth systems”, IEEE Transactions on Automatic Control, vol. 39, no. 9, pp. 1910–1914, 1994.
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