Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators

We present a simple global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators. Nonlinear control structures based on the classical PID controller, which assure global asymptotic stability of the closed-loop system, have emerged. Some works that deal with global nonlinear PID regulators based on Lyapunov theory have been reported by (Arimoto, 1995a), (Kelly, 1998) and (Santibanez & Kelly, 1998). Recently, a particular case of the class of nonlinear PID global regulators originally proposed in (Santibanez & Kelly, 1998) was presented by (Sun et al., 2009). Few saturated PID controllers (that is, bounded PID controllers taking into account the actuator torque constraints) have been reported: for the case of semiglobal asymptotic stability, a saturated linear PID controller was presented in (Alvarez et al., 2003) and (Alvarez et al., 2008); for the case of global asymptotic stability, saturated nonlinear PID controllers were introduced in (Gorez, 1999), (Meza et al., 2005), (Santibanez et al., 2008). The work introduced by (Gorez, 1999) was the first bounded PID-like controller in assuring global regulation; the latter works, introduced in (Meza et al., 2005) and (Santibanez et al., 2008), also guarantee global regulation, but with the advantage of a controller structure which is simpler than that presented in (Gorez, 1999). A local adaptive bounded regulator was presented by (Laib, 2000). Recently a new saturated nonlinear PID regulator for robots has been proposed in (Santibanez et al., 2010), the controller structure considers the saturation phenomena of the control computer, the velocity servo-drivers and the torque constraints of the actuators. The work in (Orrante et al., 2010) presents a variant of the work presented by (Santibanez et al., 2010), where now the controller is composed by a saturated velocity proportional (P) inner loop, provided by the servo-driver, and a saturated position proportional-integral (PI) outer loop, supplied by the control computer. In this chapter we use a passivity based approach to explain the results of global regulation of a class of nonlinear PID controllers proposed by (Santibanez & Kelly, 1998), that include the particular cases reported by (Arimoto, 1995a) and (Kelly, 1998). At the end of the 80’s, it was established in (Kelly & Ortega, 1988) and (Landau & Horowitz, 1988) that the nonlinear Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators


Introduction
We present a simple global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators. Nonlinear control structures based on the classical PID controller, which assure global asymptotic stability of the closed-loop system, have emerged. Some works that deal with global nonlinear PID regulators based on Lyapunov theory have been reported by (Arimoto, 1995a), (Kelly, 1998) and (Santibáñez & Kelly, 1998). Recently, a particular case of the class of nonlinear PID global regulators originally proposed in (Santibáñez & Kelly, 1998) was presented by (Sun et al., 2009). Few saturated PID controllers (that is, bounded PID controllers taking into account the actuator torque constraints) have been reported: for the case of semiglobal asymptotic stability, a saturated linear PID controller was presented in (Alvarez et al., 2003) and (Alvarez et al., 2008); for the case of global asymptotic stability, saturated nonlinear PID controllers were introduced in (Gorez, 1999), (Meza et al., 2005), . The work introduced by (Gorez, 1999) was the first bounded PID-like controller in assuring global regulation; the latter works, introduced in (Meza et al., 2005) and , also guarantee global regulation, but with the advantage of a controller structure which is simpler than that presented in (Gorez, 1999). A local adaptive bounded regulator was presented by (Laib, 2000). Recently a new saturated nonlinear PID regulator for robots has been proposed in , the controller structure considers the saturation phenomena of the control computer, the velocity servo-drivers and the torque constraints of the actuators. The work in (Orrante et al., 2010) presents a variant of the work presented by , where now the controller is composed by a saturated velocity proportional (P) inner loop, provided by the servo-driver, and a saturated position proportional-integral (PI) outer loop, supplied by the control computer. In this chapter we use a passivity based approach to explain the results of global regulation of a class of nonlinear PID controllers proposed by (Santibáñez & Kelly, 1998), that include the particular cases reported by (Arimoto, 1995a) and (Kelly, 1998). At the end of the 80's, it was established in (Kelly & Ortega, 1988) and (Landau & Horowitz, 1988) that the nonlinear

Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators
dynamics of rigid robots describe a passivity mapping from torque input to velocity output. This property is know as the passive structure of rigid robots (Ortega & Spong, 1989). The controller design methodology for robot manipulators introduced by (Takegaki & Arimoto, 1981), also called energy shaping plus damping injection technique, allows to naturally split the controller tasks into potential energy shaping for stabilization at the desired equilibrium, and damping injection, to make this equilibrium attractive (Ortega et al., 1995b). As a feature of this kind of controllers, it can be shown that the passivity property, from a new input torque to output velocity, is preserved for robots in closed-loop with the energy shaping term and damping injection term of the controller. Furthermore, considering that corresponding feedback to the integral action define a passive mapping, then it is possible to use a passivity theorem of interconnected systems to explain the stability of a class of nonlinear PID global regulators for robots. The theorem used allows to conclude global asymptotic stability of the origin of an unforced feedback system, which is composed by the feedback interconnection of state strictly passive dynamic systems with a passive and zero state observable system. The objective of this chapter is to present in a simple framework the global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators. The remainder of this chapter is organized as follows: Section 2 summarizes the dynamics for rigid robots and also recalls some of their important properties. The rationale behind the energy shaping plus damping injection technique for rigid robots are given in Section 3. The class of nonlinear PID regulators is given in Section 4. In Section 5 we recall the definition of the passivity concepts for dynamical systems and we present the passivity theorem useful for asymptotic stability analysis of interconnected systems. In Section 6 we present a passivity analysis and application of passivity theorem to conclude global asymptotic stability. An Evaluation in simulation to verify the theoretical results is presented in Section 7. Finally, our conclusions are shown in Section 8. Throughout this chapter, the norm of a vector x is defined as x = √ x T x and that of a matrix A is defined as the corresponding induced norm A = λ M {A T A}. L n 2 and L n 2e denote the space of n-dimensional square integrable functions and its extension, respectively.
where K(q,q) y U (q) represent the kinetic and potential energy of the system respectively. The equation of motion of Euler -Lagrange for a manipulator of n degrees of freedom are given by: where τ ∈ IR n is the vector of external generalized forces acting on each joint of robot, q ∈ IR n represents the vector of generalized coordinates of the system,q = d dt q is the vector of generalized velocities. In the case of a robot manipulator of n degrees of freedom, kinetic energy is a quadratic function of the velocity vectorq of the form: where M(q) ∈ IR n×n is the manipulator inertia matrix, which is symmetric and positive definite. The equations of motion of Euler -Lagrange provide the following dynamic model (Spong et al., 2006): or in compact form, the dynamics of a serial n-link rigid robot can be written as: where C(q,q)q and g(q) are given by: (3) C(q,q)q ∈ IR n is the vector of centrifugal and Coriolis forces, and g(q) ∈ IR n is the vector of gravitational forces or torques obtained as the gradient of the potential energy of robot U (q).
In terms of the state vector [q TqT ] T , the robot dynamics given by equation (2) can be written as: (4)

Planar robot of 2 degrees of freedom
The equation of motion of Euler-Lagrange (1) for a robot of n degrees of freedom can be equivalently written as: In the particular case of a planar robot of n = 2 degrees of freedom rotational joints, consider the diagram shown in Fig. 1 (Reyes & Kelly, 2001). It is known that the compact form (2) of the dynamics of a robot for two degrees of freedom with rigid links can be written as:  The meaning of the parameters of the prototype planar robot is shown in Table 1.

Properties of the robot dynamics
Although the equation of motion (2) is complex, it has several fundamental properties which can be exploited to facilitate the analysis of stability. Three important properties of the robot dynamics are the following: Property 1. (Koditschek, 1984) The matrix C(q,q) and the time derivativeṀ(q) of the inertia matrix satisfy:q Property 2. (Tomei, 1991) There exists a positive constant k g such that Property 3. (Passive structure of rigid robots) In relation to the dynamic model (2). The operator : τ →q is passive (Kelly & Ortega, 1988;Landau & Horowitz, 1988;Ortega & Spong, 1989), i.e.: where V a (t) is the total energy of robot plus a suitable constant which is introduced so that V a (t) is a nonnegative function (Ortega et al., 1995b): being 1 2q T (t)M(q(t))q(t) energy kinetic, U (q(t)) the potential energy of robot due to gravity, y k u = min q U (q(t)).

Energy shaping methodology
Energy is one of the fundamental concepts in control of mechanical systems with multidegrees-of-freedom. The action of a controller can be understood in energy terms as a dynamical system called "actuator" that supplies energies to the controlled system, upon interconnection, to modify desirably the behavior of the closed-loop (interconnected) system. This idea has its origin in (Takegaki & Arimoto, 1981) and is later called the "energy-shaping" approach, which is now known as a basic controller design technique common in control of mechanical systems. Its systematic interpretation is called "passivity-based control" (Arimoto, 2009). The main idea of this methodology is to reshape the robot system's natural energy and inject damping via velocity feedback, for asymptotic stabilization purposes, such that a regulation objective is reached . This is achieved by choosing a controller structure such that, first, the total potential energy function of the closed-loop system due to gravity and the controller is radially unbounded function in the position error with a unique and global minimum at zero position error, and second, it injects damping via velocity feedback. The resulting closed loop system is an autonomous one which has the nice property that zero position error and zero velocity form always the unique equilibrium point. By using the total energy, i.e., the kinetic plus total potential energy, as a Lyapunov function, it follows that this equilibrium is stable. In order to prove that the equilibrium is in fact globally asymptotically stable, the final key step is to exploit the autonomous nature of the closed loop system to invoke the Krasovskii-LaSalle's theorem. This approach has been continued by their colleagues and several researchers (Arimoto, 1995a;Nijmeijer & Van der Schaft, 1990;Wen & Bayard, 1988), who have offered extensions and improvements (Ailon & Ortega, 1993;Berghuis & Nijmeijer, 1993b;Kelly, 1993;Ortega et al., 1998) and (Ortega & Garcia-Canseco, 2004;Ortega et al., 2008;Sepulchre et al., 1997;Vander, 1999). As a feature of this kind of controllers, it can be shown that the passivity property, from a new input torque to output velocity, is preserved either for rigid robots or elastic joint robots in closed loop with the energy shaping term of the controller. The damping injection term, via velocity feedback or by way of a suitable filtering of position, defines an input, output or state strictly passive mapping from velocity input to damping output, and asymptotically stabilizes the desired equilibrium of the closed loop system. We broach the regulation problem whose goal is to find τ(t) such that where q d ∈ IR n is a vector of constant desired joint displacements. The features of the system can be enhanced by reshaping its total potential energy. This can be done by constructing a controller to meet a desired energy function for the closed-loop system, and inject damping, via velocity feedback, for asymptotic stabilization purposes (Nijmeijer & Van der Schaft, 1990).
To this end, in this section we consider controllers whose control law can be written by where F (q) is some kind of dissipation function from which the damping force can be derived, an example is the so called Rayleigh dissipative function F (q)= 1 2q T K vq , where K v is the matrix of coefficient of viscous friction,q = q d − q ∈ IR n denotes the joint position error and U a (q d ,q) is some kind of artificial potential energy provided by the controller whose properties will be established later. The first right hand side term of (9) corresponds to the energy shaping part and the other one to the damping injection part. We assume the dissipation function F (q) satisfies the following conditions: The closed-loop system equation obtained by substituting the control law (9) into the robot dynamics (2) leads to d dt where (3) has been used. If the total potential energy U T (q d ,q) of the closed-loop system, defined as the sum of the potential energy U (q) due to gravity plus the artificial potential energy U a (q d ,q) introduced by the controller is radially unbounded inq, andq = 0 ∈ IR n is an unique minimum, which is global for all q d , then the origin q TqT T = 0 ∈ IR 2n of the closed-loop system (13) is global and asymptotically stable (Takegaki & Arimoto, 1981).

Classical PID regulators
Conventional proportional-integral-derivative PID regulators have been extensively used in industry due to their design simplicity, inexpensive cost, and effectiveness. Most of the present industrial robots are controlled through PID regulators (Arimoto, 1995a). The classical version of the PID regulator can be described by the equation: where K p , K v and K i are positive definite diagonal n × n matrices, andq = q d − q denotes the position error vector. Even though the PID controller for robot manipulators has been very used in industrial robots (Arimoto, 1995a), there still exist open problems, that make interesting its study. A open problem is the lack of a proof of global asymptotic stability (Arimoto, 1994). The stability proofs shown until now are only valid in a local sense (Arimoto, 1994;Arimoto et al., 1990;Arimoto & Miyazaki, 1983;Arimoto, 1996;Dorsey, 1991;Kelly, 1995;Kelly et al., 2005;Rocco, 1996;Wen, 1990) or, in the best of the cases, in a semiglobal sense (Alvarez et al., 2000;Meza et al., 2007). In (Ortega et al., 1995a), a so-called PI 2 D controller is introduced, which is based on a PID structure but uses a filter of the position in order to estimate the velocity of the joints, and adds a term which is the integral of such an estimate of the velocity (this added term motivates the name PI 2 D); for this controller, semiglobal asymptotic stability was proved. To solve the global positioning problem, some globally asymptotically stable PID-like regulators have also been proposed (Arimoto, 1995a;Gorez, 1999;Kelly, 1998;Santibáñez & Kelly, 1998), such controllers, however, are nonlinear versions of the classical linear PID. We propose a new global asymptotic stability analysis, by using passivity theory for a class of nonlinear PID regulators for robot manipulators. For the purpose of this chapter, it is convenient to recall the following definition presented in (Kelly, 1998).

A class of nonlinear PID controllers
The class of nonlinear PID global regulators under study was proposed in (Santibáñez & Kelly, 1998). The structure is based on the gradient of a C 1 artificial potential function U a (q) satisfying some typical features required by the energy shaping methodology (Takegaki & Arimoto, 1981). The PID control law can be written by ( see Fig. 2). • U a (q) is a kind of C 1 artificial potential energy induced by a part of the controller.
• K v and K i are diagonal positive definite n × n matrices •q is the velocity error vector • α is a small constant, satisfying (Santibáñez & Kelly, 1998) By defining z as: we can describe the closed-loop system by which is an autonomous nonlinear differential equation whose origin q TqT z T T = 0 ∈ IR 3n is the unique equilibrium.

Some examples
Some examples of this kind of nonlinear PID regulators • (Kelly, 1998) This controller has associated an artificial potential energy U a (q) given by • (Arimoto et al., 1994a) (16). This controller has associated a C 2 artificial potential energy U a (q) given by where K pa is a diagonal positive definite n × n matrix whose entries are k pai and sat(q) = tanh[q]=[tanh(q 1 ) tanh(q 2 ) . . . tanh(q n )] T . This controller has associated a C ∞ artificial potential energy U a (q) given by where K pa is a diagonal positive definite n × n matrix whose entries are k pai and sat(q) = Sat[q]=[ Sat(q 1 ) Sat(q 2 ) . . . Sat(q n )] T . This controller has associated a C 1 artificial potential energy U a (q) given by where Sat(x; λ) stands for the well known hard saturation function Following the ideas given in (Santibáñez & Kelly, 1995) and (Loria et al., 1997) it is possible to demonstrate, for all above mentioned regulators, that U a (q) leads to a radially unbounded virtual total potential function U T (q d ,q).

Passivity concepts
In this chapter, we consider dynamical systems represented bẏ where u ∈ IR n , y ∈ IR n , x ∈ IR m , f (0, 0)=0 and h(0, 0)=0. Moreover f , h are supposed sufficiently smooth such that the system is well-defined, i.e., ∀ u ∈ L n 2e and x(0) ∈ IR m we have that the solution x(·) is unique and y ∈ L n 2e .
Definition 2. (Khalil, 2002) The system (22)-(23) is said to be passive if there exists a continuously differentiable positive semidefinite function V(x) (called the storage function) such that where ǫ, δ, and ρ are nonnegative constants, and ψ(x) :I R m → IR is a positive definite function of x. The term ρψ(x) is called the state dissipation rate. Furthermore, the system is said to be • lossless if (24) is satisfied with equality and ǫ = δ = ρ = 0; that is, u T y =V(x) • input strictly passive if ǫ > 0 and δ = ρ = 0, • output strictly passive if δ > 0 and ǫ = ρ = 0, • state strictly passive if ρ > 0 and ǫ = δ = 0, If more than one of the constants ǫ, δ, ρ are positive we combine names.
Now we recall the definition of an observability property of the system (22)-(23).
Definition 3. (Khalil, 2002) The system (22) Right a way, we present a theorem that allows to conclude global asymptotic stability for the origin of an unforced feedback system, which is composed by the feedback interconnection of a state strictly passive system with a passive system, which is an adaptation of a passivity theorem useful for asymptotic stability analysis of interconnected system presented in (Khalil, 2002). Theorem 1. Consider the feedback system of Fig. 3 where H 1 and H 2 are dynamical systems of the formẋ for i = 1, 2, where f i :I R m i × IR n → IR m i and h i :I R m i × IR n → IR n are supposed sufficiently smooth such that the system is well-defined. f 1 (0, e 1 )=0 ⇒ e 1 = 0, f 2 (0, 0)=0,y h i (0, 0)=0.

Fig. 3. Feedback connection
The system has the same number of inputs and outputs. Suppose the feedback system has a well-defined state-space modelẋ f and h are sufficiently smooth, f (0, 0)=0, and h(0, 0)=0. Let H 1 be a state strictly passive system with a positive definite storage function V 1 (x 1 ) and state dissipation rate ρ 1 ψ 1 (x 1 ) and H 2 be a passive and zero state observable system with a positive definite storage function V 2 (x 2 ); that is, Then the origin is asymptotically stable. If V 1 (x 1 ) and V 2 (x 2 ) are radially unbounded then the origin of (25) will be globally asymptotically stable.
Proof. Take u 1 = u 2 = 0. In this case e 1 = −y 2 and e 2 = y 1 . Using V(x)=V 1 (x 1 )+V 2 (x 2 ) as a Lyapunov function candidate for the closed-loop system, we havė which shows that the origin of the closed-loop system is stable. To prove asymptotic stability we use the LaSalle's invariance principle and the zero state observability of the system H 2 .I t remains to demonstrate that x = 0 is the largest invariant set in Ω = {x ∈ IR m 1 +m 2 :V(x)= Now, as x 1 ≡ 0 ⇒ẋ 1 = f 1 ≡ 0 and in agreement with the assumption about f 1 in the sense that f 1 (0, e 1 )=0 ⇒ e 1 = 0, we have e 1 ≡ 0 ⇒ y 2 ≡ 0. Also x 1 ≡ 0, e 1 ≡ 0 ⇒ y 1 ≡ 0 (owing to assumption h 1 (0, 0)=0). Finally, y 1 ≡ 0 ⇒ e 2 ≡ 0, and e 2 ≡ 0 and y 2 ≡ 0 ⇒ x 2 ≡ 0 in agreement with the zero state observability of H 2 . This shows that the largest invariant set in Ω is the origin, hence, by using the Krasovskii-LaSalle's theorem, we conclude asymptotic stability of the origin of the unforced closed-loop system (25). If V(x) is radially unbounded then the origin will be globally asymptotically stable. ∇∇∇

Analysis via passivity theory
In this section we present our main result: the application of the passivity theorem given in Section 5, to prove global asymptotic stability of a class of nonlinear PID global regulators for rigid robots. First, we present two passivity properties of rigid robots in closed-loop with energy shaping based controllers. Property 4. Passivity structure of rigid robots in closed-loop with energy shaping based controllers ( see Fig. 4). The system (2) in closed-loop with is passive, from input torque τ ′ to output velocityq, with storage function where U a (q d ,q) is the artificial potential energy introduced by the controller with properties requested by the energy shaping methodology and U T (q d ,q) is the total potential energy of the closed-loop system, which has an unique minimum that is global. Furthermore the closed-loop system is zero state observable.

Proof.
The system (2) in closed-loop with control law (26) is given by where (3) and (14) have been used. In virtue of Property 1, the time derivate of the storage function (27) along the trajectories of the closed-loop system (30) yieldṡ

V(q(t),q(t)) =q T τ ′
where integrating from 0 to T, in a direct form we obtain (28), thus, passivity from τ ′ toq has been proved. ♦ The zero state observability property of the system (30) can be proven, by taking the output as y =q and the input as u = τ ′ , becausė q ≡ 0, τ ′ ≡ 0 ⇒q ≡ 0.
The robot passive structure is preserved in closed-loop with the energy shaping based controllers, because this kind of controllers also have a passive structure. Passivity is invariant for passive systems which are interconnected in closed-loop, and the resulting system is also passive. ♦ Property 5. State strictly passivity of rigid robots in closed-loop with the energy shaping plus damping injection based regulators (see Fig. 5). The system (2) in closed-loop with is state strictly passive, from input torque τ ′′ to output (q − α sat(q)), with storage function where 1 2q T M(q)q is the kinetic energy, U T (q d ,q) is the total potential energy of the closedloop system, and α sat(q)M(q)q is a cross term which depends on position error and velocity,   (Santibáñez & Kelly, 1998). In this case K vq is the damping injection term. The State dissipation rate is given by : Consequently the inner product of the input τ ′′ and the output y =(q − α sat(q)) is given by: (q − α sat(q)) T τ ′′ ≥V(q,q)+ϕ(q,q), Proof. The closed-loop system(2) with control law (31) is where (3) and (14) have been used. In virtue of property 1, the time derivate of the storage function (32) along the trajectories of the closed-loop system (36) yields tȯ from which we get (34), so state strictly passivity from input τ ′′ to output (q − α sat(q)) is proven. ♦ The robot dynamics enclosed loop with the energy shaping plus damping injection based controllers defines a state strictly passive mapping, from torque input τ ′′ to output y =(q − α sat(q)) y T τ ′′ ≥V 1 (q(t),q(t)) + ϕ(q(t),q(t)), where ϕ(q,q) is called the state dissipation rate given by (33) which is positive definite function and radially unbounded (Santibáñez & Kelly, 1995). The integral action defines a zero state observable passive mapping with a radially unbounded and positive definite storage function By considering the robot dynamics in closed loop with the energy shaping plus damping injection based control action, in the forward path and the integral action in the feedback path (see Fig. 6), then, the feedback system satisfies in a direct way the theorem 1 conditions and we conclude global asymptotic stability of the closed loop system. • A1. The system in the forward path defines a state strictly passive mapping with a radially unbounded positive definite storage function.
• A2. The system in the feedback path defines a zero state observable passive mapping with a radially unbounded positive definite storage function.
Besides, the equilibrium q TqT z T T = 0 ∈ IR 3n of the closed-loop system (20) is globally asymptotically stable.

Simulation results
Computer simulations have been carried out to illustrate the performance of a class of nonlinear PID global regulators for robot manipulators. A example of this kind of nonlinear PID regulators is given by (Kelly, 1998) where artificial potential energy is given by U a (q)= 1 2q The manipulator used for simulation is a two revolute joined robot (planar elbow manipulator), as show in Fig. 1. The meaning of the symbols is listed in Table 2 whose numerical values have been taken from (Reyes & Kelly, 2001).  Table 2. Physical parameters of the prototype planar robot with 2 degrees of freedom

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The PID tuning method is based on the stability analysis presented in (Santibáñez & Kelly, 1998). The tuning procedure for the PID controller gains can be written as: where K p denotes a diagonal positive definite n × n gain matrix resulting of the artificial potential energy U a (q)= 1 2q T K pq of the controller. The PID tuning requires to compute k g . Using property 2 and the above expressions of the gravitational torque vector, we obtain that k g = 80.578 [kg m 2 /sec 2 ]. The gain was tuned as . With the end of supporting the effectiveness of the proposed controller we have used a squared signal whose amplitude is decreased in magnitude every two seconds. More specifically, the robot task is coded in the following desired joint positions degrees if 4 ≤ t < 6 sec 0 degrees if 6 ≤ t < 8 sec Above position references are piecewise constant and really demand large torques to reach the amplitude of the respective requested step. In order to evaluate the effectiveness of the proposed controller. The proposed Nonlinear PID control scheme has been tuned to get their best performance in the presence of a step input whose amplitude is 45

Conclusions
In this chapter we have given sufficient conditions for global asymptotic stability of a class of nonlinear PID type controllers for rigid robot manipulators. By using a passivity approach, we have presented the asymptotic stability analysis based on the energy shaping methodology. The analysis has been done by using an adaptation of a passivity theorem presented in the literature. This passivity theorem, deals with systems composed by the feedback interconnection of a state strictly passive system with a passive system. Simulation results confirm that the class of nonlinear PID type controllers for rigid robot manipulators