Experimental evaluation of a cascade control technique with friction compensation for a flexible joint robot

Controle em Cascata com Compensação de Atrito para Robôs com Juntas Flexíveis: Teoria e Resultados Experimentais

ROBÓTICA

Experimental evaluation of a cascade control technique with friction compensation for a flexible joint robot

Controle em Cascata com Compensação de Atrito para Robôs com Juntas Flexíveis: Teoria e Resultados Experimentais

A. R. G. Ramires^I; E.R. De Pieriy^II; R. Guentherz^III; J. F. Golinz^III

^IUniversity of Vale de Itajaí, Computation Engineering, Campus São José Rodovia SC 407, Km 4 88122-000 São José-SC, Brazil. [e-mail]

^IIDepartment of Automation and Systems, Federal University of Santa Catarina 88040-900 Florianópolis, SC, Brazil. [e-mail]

^IIIDepartment of Mechanical Engineering, Federal University of Santa Catarina 88040-900 Florianópolis, SC, Brazil. [e-mail]

ABSTRACT

This paper focuses on the experimental results of the position tracking control of a flexible joint robot manipulator. The applied control strategy is based on a well know cascade control methodology that divides the robot model into two subsystems: a link and a rotor subsystem. Thanks to this division a friction compensation based on the LuGre model is performed to the mechanical system. The main goal of the present work is to apply a systematic procedure to identify the friction model parameters and to analyse the control strategy performance when implemented to a robot prototype specially built for practical tests.

Keywords: Robot control, joint flexibility, friction observer, friction compensation, cascade control.

RESUMO

Neste trabalho são apresentados resultados experimentais de seguimento de trajetória de posição de um robô manipulador com flexibilidade nas juntas. A estratégia de controle é baseada na metodologia do controle em cascata, dividindo o modelo do robô em dois subsistemas: o subsistema dos elos e o subsistema do rotor. Essa abordagem facilita a compensação do problema de atrito do subsistema mecânico ou dos elos. Usou-se para a compensação de atrito o modelo de LuGre. O principal objetivo da abordagem apresentada é fornecer um procedimento sistemático para identificação dos parâmetros do modelo de atrito e a análise de desempenho da estratégia proposta quando aplicada a um protótipo especialmente construído para testes experimentais.

Palavras-chave: Controle de Robôs, Robôs Flexíveis, Observador de Atrito, Controle em Cascata

1 INTRODUCTION

Undesirable elastic deformations may appear in fast robots equipped with gear-boxes, harmonic drives, chains or belts. The elastic deformation due to joint flexibility might cause positioning errors, increase the settling time and even produce torsional resonance and instability (Spong, 1987). Also, the elastic coupling introduces an additional degree of freedom at each joint. In particular, a robot with flexible transmissions has the double of degrees of freedom with respect to its rigid equivalent. Thus, this robot is a partially actuated system, since it has 2n variables to be controlled through only n control torques. Therefore, the usual methods for tracking control, such as the inverse dynamics formulation and the control based on Lyapunov, cannot be directly applied to that system (Readman, 1994).

Different control strategies for elastic joint robots have been developed in the literature. Among those strategies, one may cite PD control (Tomei, 1991; Dupont, 1994), dynamic feedback linearization methods, singular perturbation theory, robust and adaptive schemes (Nicosia and Tomei, 1988; Tomei, 2000), integrator backstepping based methodologies (Nicosia and Tomei, 1992) and cascade methodologies (Guenther and Hsu, 1993).

There are some important differences between those techniques. A basic comparative study can be found in (Brogliato et al., 1995). PD controllers, which work well for industrial robots, add some damping for the joint elasticity but their performance and robustness are very limited. The controller bandwidth has to be reduced until robustness against highly nonlinear dynamics is reached. Stability proofs for such controllers are more complicated than for the controllers using extensive model information. The singular perturbation is a simple two stage method, referred as corrective control, but it only works well in the case of weak flexibility. Adaptive versions of that control technique have been developed with good results (Ghorbel and Spong, 2000). An extension to consider the problem of position and force control of uncertain constrained flexible joint robots was presented in (Huang et al., 2006). The dynamic feedback linearization methods are quite simple to be used but in the most general cases the dynamics of flexible joint robots are not feedback linearizable. Also, the main disadvantage of feedback linearization is the necessity of the exact knowledge of the robot parameters and the need of joint acceleration and jerk measurements.To overcome these difficulties, adaptive and passivity based control techniques have been proposed (Fantoni and Lozano, 2000; Albu-Schäffer and Hirzinger, 2001). The backstepping schemes use the cascade decomposition property of the robot model (Kokotovic and Sussmann, 1989). The backstepping is a systematic procedure to control nonlinear systems but the designer should pay attention to over-parametrization and time consuming computations. These latter are more involved than compared to the traditional cascade control procedure, for example. Also, in some algorithms, the measurement of joint accelerations is necessary. Using the cascade control methodology, a robust tracking of a reference model, avoiding overparametrization, was achieved (Wang and Khorrami, 2000). The cascade control schemes also use the system decompositions in small subsystems. The overall system is partitioned in such a way that the states of the first subsystem are the control variables to the following subsystem. In this technique, the model uncertainties can be taken into account by using an adaptive control scheme. With this methodology, the problem of measuring joint accelerations can be circumvented.

In robots with joint flexibilities, some important phenomena like actuator dynamics and friction are not accurately included in the design of control algorithms (Lozano et al., 1997; Ramirez et al., 2003). In fact, friction occurs in all mechanical systems,e.g. bearings, transmissions, hydraulic and pneumatic cylinders, valves, brakes and wheels (Canudas De Wit et al., 1995; Canudas De Wit and Lischinsky, 1997; Guenther et al., 2006; Lischinsky et al., 1999; Guenther and Perondi, 2004). In electrical driven robots the friction is always present in transmission mechanisms, rotor and joints (Canudas De Wit et al., 1995). Friction can lead to steady state or trajectory tracking errors (Dupont, 1994; Guenther and Perondi, 2002; Ramirez et al., 2002) or can lead to cycle limites in flexible robots as shown in (Jeon and Tomizuka, 2005). Friction compensation is desired to circumvent these problem and it is particularly difficult to be performed using a non-model based compensation (Lozano et al., 1997; Tomei, 2000) or even using a model-based compensation (Armstrong-Hélouvry, 1993; Canudas De Wit and Lischinsky, 1997) or using adaptive techniques (Xie, 2007).

Several models have been used to describe the friction phenomenon dynamics. In (Gomes and Santos da Rosa, 2003) a model is proposed to compensate friction effects. In a different structure (Canudas De Wit et al., 1995; C. Canudas de Wit, 2007) develop a friction estimation scheme based in the LuGre model to build a model-based fiction compensation. In (Gandhi et al., 2002) an extension of the LuGre model is proposed to compensate friction in harmonic drives. Recently, a Generalized Maxwell-Slip model has been proposed by (Al-Bender et al., 2005) to describe the friction dynamics.

This paper proposes an experimental setup to analyse a cascade control methodology in which the mechanical system is interpreted as an interconnected system: a link subsystem driven by the elastic torque generated by the rotor subsystem. The cascade control strategy allows to introduce friction compensation in the mechanical system. For this purpose, a model-based friction compensation is adopted. It uses a friction estimation scheme based on the LuGre model (Canudas De Wit et al., 1995). The estimated friction torque is added to the reference torque generated by the position controller. This compensation assumes that the actuator has a fast and accurate torque response. The convergence of the tracking errors is demonstrated using the Lyapunov direct method, for the case where all the system parameters are known and there are not external forces. Experimental results illustrate the main properties of the proposed controller with the friction compensation model.

This paper is organized as follows. In Section 2, the model of the flexible joint robot manipulator with friction is recalled. The cascade control with friction compensation is proposed in Section 3. In Section 4, the stability of the proposed controller is demonstrated. Section 5 describes the off-line procedure for the identification of friction parameters. In Section 6, the experiment is presented.

2 MODEL OF RIGID AND FLEXIBLE ROBOTS WITH N LINKS

A mathematical model of a robot with n rigid links connected by n rotational or translation rigid joints is given by (Canudas De Wit et al., 1996)

where the vectors q₁, and represent, respectively, the link position, velocity and acceleration, M(q₁) is the inertia matrix, C(,q1) represents the centrifuges and Coriolis terms, G(q₁) represents the gravitational torques and τ is the control input. As pointed out in (Canudas De Wit et al., 1996), the elastic effects are small and can be neglected, the inertia matrix M(q₁) is symmetric and positive definite and both M(q₁) and M^-1(q₁) are uniformly bounded for all q1, and a convenient choice of C(,q₁) leads to an skew-symmetric matrix - 2C. This system is considered completely actuated in the sense that one torque is provided for each degree of freedom.

In order to obtain the mathematical model of a flexible joint robot, the links are supposed to be rigid, the joint flexibility is modeled as a torsional spring for rotational joints and as a linear spring for prismatic joints, and the inertia matrix takes into account the individual inertia of the links and rotors (Spong and Vidyasagar, 1989). Then, the dynamical model of a flexible joint robot can be written as

where q1 is associated to the position on the link, q2 is associated to the position on the rotor and . The elements , G(q1) and are described in (Canudas De Wit et al., 1996).

With the hypothesis introduced by (Tomei, 1991), (Canudas De Wit et al., 1996) and (Spong, 1987), it is possible to obtain a simplified model, given by

with J = diag(J₁,J₂,...,J_n), Jj being the inertia associated to the rotor j, and K = diag(K₁,K₂,...,K_n),Kj being the spring coefficient associated to joint j. Some robot configurations can be adequately modelled by (3), like manipulators with one rotational joint and one link or manipulators with two rotational joints and two links with perpendicular axes. In general, when the transmission reductions are important (for example when harmonic drivers are used) the simplified model (3) is valid.

The mathematical model (3) can be interpreted as two coupled subsystems: the first one represents the rigid robot model and the second one corresponds to the flexible joint dynamics. The coupling between the two models is given by the elastic torque represented by the term K(q₂ - q₁). The cascade interconnection of the two subsystem is shown in Fig. (1).

From the control point of view, the simplified model (3) describes a non negligible elastic displacement between each link and its corresponding rotor. On the other hand, the whole system becomes partially actuated, since for 2n equations describing the dynamical behavior of the system there are only n torque inputs. The main consequence of this behavior is to limit the application of classical controllers (as P, PD and PID) normally used to control rigid manipulators. Furthermore, for an effective control of flexible joint robots it is necessary to use more elaborated control strategies (possibly with friction compensation) and to take into account some robustness requirements.

2.1 Friction Model

In this paper, the LuGre model (Canudas De Wit et al., 1995) is used to represent the friction. This model can describe complex friction behavior, such as stick-slip motion, pre-sliding displacement, Dahl and Stribeck effects and frictional lag.

The friction force is given by

where z is an internal state that describes the average elastic deflection of the contact surfaces during the stiction phases, Γ₀ is a diagonal matrix of the stiffness coefficient of the microscopic deformations z during the pre-sliding displacements, Γ₁ is a diagonal matrix of the damping coefficients associated with and Γ₂ is a diagonal matrix that represents the viscous friction

The dynamics of the internal state z is expressed by

with

where g(

_i ) for i =1,...,n is a positive function that describes part of the steady state characteristics of the model for constant velocity motions. This function is given by

where F_ci is the Coulomb friction, Fsi is the static friction and v_si is the Stribeck velocity and as stated in (Canudas De Wit et al., 1995), each function g(_i) is upper bounded, i.e., 0 <g(_i) < a_i, for i =1,...,n.

The simplified model (3) is modified to take into account the friction. Then,

3 CASCADE CONTROL WITH FRICTION COMPENSATION

In this section, the cascade control with friction compensation is proposed for systems that may be represented by the model (8). Initially, the following errors are defined

where

₁ and₂ are diagonal positive definite matrices of gains, q_1d is the desired link position, q_2d is the desired rotor position and _r1 is a reference for the link velocity.

The system (8) can be transformed into two cascade subsystems with inputs v and q2 using a suitable change of coordinates given by

where v is the control applied to the rotor subsystem and u_d is the control for the link subsystem.

Substituting (10) and (11) in (8), the system is rewritten as

The robot system is then interpreted as an interconnected system like the one presented in Fig. 2.

The control project is performed through two steps: link control and rotor control. In addition, a friction observer is designed.

3.1 Tracking Control of the Link Subsystem

The control u_d is designed using the property of passivity in robot manipulators(Canudas De Wit et al., 1996). Thus, u_d is given by

where K_D1 is a diagonal positive definite matrix of gains and is the estimated friction torque.

The closed loop form of this subsystem is obtained substituting (13) in the first equation of (12)

3.2 Tracking Control of the Rotor Subsystem

The control v is chosen as

where K_D2 is a diagonal positive definite matrix of gains. Then, the control torque π is given by

The closed loop form of this subsystem is obtained substituting (15) in the second equation of (12)

3.3 Friction Observer

According to (Canudas De Wit et al., 1995) the estimated friction is

where is the estimated internal state.

In (Guenther and Perondi, 2002), the following expression is proposed to model the dynamic of ,

where Ke is diagonal positive definite matrix of gains.

The observer in (19) requires a slight modification to be used within the cascade control scheme. This modification is necessary because the derivative of the estimated friction is used to calculate the control signal in the cascade control strategy. Therefore, the function |

_1i| has to be smoothed by a function m(_1i), like

where k_vi is a positive constant (Guenther and Perondi, 2002) taken as k _vi =2.0 in the applications. This gives a new matrix (₁), defined as

and a new model for the dynamic of zˆ, which is used in this work,

Thus, the difference between the friction internal state z and the estimated state denoted by , has the time derivative

where , defined as, is a residual difference caused by the use of m(q1i) instead of |_1i|.

4 STABILITY ANALYSIS

The closed loop is constituted by the robot and the cascade controller, considering the friction observer. In the ideal case, in which all the system parameters are known, the convergence properties of the tracking errors are stated below.

Consider the nonnegative energy function

where V₁ and V₂ are nonnegative functions related to the link and the rotor subsystems, respectively, given by

with and .

The time derivative of V1, substituting by the expression obtained from (14) and by the expression in (23) and using the property of skew symmetry of , is given bye

The time derivative of V2, substituting by the expression obtained from (17), is given by

Substituting P₁, P₂, s₁ and s₂in equations (27) and (28), the time derivative of V may be written as¹

with

where the matrix is defined as:

In fact matrix is the inverse of except for ₁ = 0 where the inverse in not defined. Equation (30) with a predefined tolerance ε > 0 is used to avoid numerical difficulties when ₁ tends to zero and consequently λ_max(N2) tends to .

The gains K_e, K_D1, K_D2,

₁ and ₂ may be chosen to make N₂ positive definite. A possible way of achieving this is to choose K_e, K_D1, K_D2,

₁and ₂ such that N₂ becomes diagonal dominant.

Using the Rayleigh-Ritz theorem² and the concept of uniform ultimate boundedness (Corless and Leitmann, 1991), it follows that

According to (Canudas De Wit et al., 1995), due to the fact that then . In fact there exists an invariant set so that all solutions of z(t) starting in remains in this set.On the other hand, the approximation error: is a limited function. Due to these facts it is possible to define a upper bound such that , then a condition for having < 0 is

Since < 0, the norm kk decreases until

This causes < 0 and forces the norm ∞λ∞ to increase. Eventually, the condition (32) is reached and < 0 again.

The conclusion is that V (t) is a decreasing function that converges to a fixed limit, implying that V (t) is bounded and

₁, ₁,₂,₂, and are bounded. The tracking errors converge to a residual set which depends on the friction coefficients and the controller gains.

5 OFF-LINE IDENTIFICATION OF THE FRICTION PARAMETERS

For the LuGre model, 6 parameters must be estimated for each link of the robot, that is, for the link i, the parameters Γ_0i, Γ_1i, Γ_2i, F_ci, F_si, v_si must be identified.

The estimation is done for each link at a time. All other joints are blocked and only the joint i moves. The estimation proposed in (Canudas De Wit and Lischinsky, 1997) is done in two steps: the estimation of the static parameters Γ_2i, F_ci, F_s and vsi and of the dynamic parameters Γ_0i and Γ_1i.

5.1 First Step: Static Parameters

The static parameters of the LuGre model may be estimated using a static map. This static map³ is obtained from a series of experiments, where in each experiment the robot joint moves with a constant velocity and the friction torque is measured indirectly through the rotor current. Then, the static parameters are adjusted to the experimental data using nonlinear techniques.

One of these techniques is to solve a nonlinear least squares problem. In this case, supposing that m experiments are made and that for each velocity vj a friction torque Fss(vj ) is measured (for j =1,. ..,m), the objective is to find the parameters that minimize a cost function, that is, to solve

where F^ˆss(vj ) are the estimated values for the friction torque, calculated according to the static friction model

5.2 Second Step: Dynamic Parameters

In order to determine the dynamic parameters of the friction model, it is important to perform an experiment in which the effects of these parameters stand out. In particular, the stick-slip motion and the transient motions due to velocity reversals depend highly on Γ_0i and Γ_1i. Therefore, these 2 parameters may be obtained from open-loop experiments that induce stick-slip motion and with velocity reversals. The dynamic parameters are adjusted to the experimental data using a nonlinear optimization method.

Supposing that an experiment is made and that p samples are collected, that is, for each instant t(k) a link angular position x(t(k)) is measured (for k =1,.. .,p), the objective is to find the parameters that minimize a cost function in the form

where

(t(k)) is the estimated value of the link angular position obtained by numerical integration from the model

where J_x represents the sum of the inertia in the rotor and in the link and u is the torque applied to the link (the values of the static parameters Γ_2i, F_ci,F_si and vsi identified in the first step are used here). It is important to notice that, since the robot is approximated by a rigid model (37), the experiment must be performed in low velocities, in order to not excite the flexible transmission dynamics.

6 EXPERIMENT: CASCADE CONTROL OF A FLEXIBLE JOINT ROBOT

In this section, it is described the experiment performed, which consists in the implementation of the cascade control with friction compensation proposed in Section (3) to a planar flexible joint robot with one link and one transmission. A drawing of the robot considered in this work is shown in Figure 3. Although it has two links, here only the first link is studied.

6.1 Hardware Description

The hardware set-up is shown in Figure 4.

In this application, the dSPACE DS1102 system is used to control the manipulator. The controller is implemented in a personal computer using the Simulink/MATLAB interface. The controller reads the rotor angular positions in real-time from two Hewlett Packard HEDS-6010 high resolution incremental encoders. Comparing the actual position and the reference, the controller calculates the control torque that must be sent back to the robot using the 12 bits Digital to Analog Converter outputs. The 12A PWM Advanced Motion Controls servo amplifier drives the necessary current to the motor. In this application the servo amplifier is used in torque mode (voltage input to current output). Figure 5 shows the structure of the controller.

The control block represents the cascade control algorithm. Link.position and Rotor.position blocks represent the link and rotor actual positions, respectively. The Trajectory block generates the desired trajectory. The desired link position q1d and its derivatives, up to 3^rd order, are uniformly bounded. The DAC unit and the Encoder Phase Lines unit are dSPACE drivers that perform the digital to analog conversion and the digital interpretation of the encoder values, respectively.

6.2 Model of the Planar Flexible Joint Robot with One Linkk

Using the simplified model (8), the planar robot with one link and one flexible transmission can be represented by

where I is the inertia of the link and the load (a constant value), q₁ is the joint position, q₂is the rotor position, J is the rotor inertia, K is the spring constant, F is the friction ^1.5 torque and is the control torque. The inertia and spring constant were obtained using indirect measurements. The spring constant, in particular, was obtained using a suitable Input Torque (Nm) experiment (Seto, 1964).

The nominal parameters for the first link of the manipulator are given in Table 1.

6.3 Identification of Friction Parameters

The identification of the friction parameters was done according to the procedure described in Section 5. The encoder resolution was 0.72 degrees. The velocity resolution for measuring static values was 0.1 rad/s. The static map obtained for the first link of the robot manipulator is shown in Fig. 6.

Using the Matlab function lsqcurvefit to solve the optimization problem (34), the static parameters are estimated as F_s1 =0.15 Nm, F_c1 =0.11 Nm, v_s1 =0.05 rad/s and Γ_2i, =0.2 Nms/rad. Fig. 6 shows a non symmetric friction with velocity, but in this paper symmetric values were used, like in (Canudas De Wit and Lischinsky, 1997).

Fig. 7 and Fig. 8 illustrate the open-loop experiment performed for the determination of the dynamic parameters.

Fig. 7 shows the torque input and Fig. 8 the measured angular position. By solving the problem optimization problem 36, the dynamic parameters are estimated as Γ₀₁ = 100

Nm/rad and Γ₁₁ =0.1 Nms/rad. A simulation of the link angular position using these estimated parameters is also shown in Fig. 8.

The estimated values of the 6 parameters of the LuGre model for the first link are given in Table 2.

6.4 The Cascade Control Strategy

In this experiment, it is used the cascade control proposed in ^0.04 Section 3, adapted to the model (40) of the planar flexible joint robot considered here.

From (13) (18) and (19), the control ud is given by

Fromand, from (16), the control torque is

Equations (41) and (42) use only accelerations and jerks of the reference signal. The control gains were chosen to obtain a response without actuator vibrations and with a sufficiently smooth control signal. Table 3 shows the controller gains.

6.5 Experimental Results

The tests were performed using two different trajectories. The link and rotor control were given by equations (41) and (42), respectively. The velocities were obtained using a filter and a numeric derivative process.

Figure 9 shows the system response using the cascade controller without friction compensation.

In order to outline the friction effect, Figure 10 shows the improved system response when using friction compensation.

Figure 11 shows that the control torque strongly depends on the estimated friction torque.

Figure 12 shows the actual and desired link positions using a polynomial trajectory without friction compensation.

Figure 13 shows the improved system response when using the cascade controller with friction compensation.

The tracking errors for both cases are shown in Figure 14.

Figure 15 shows a faster dynamic motion experiment, which allows to better assess the performance of the cascade control algorithm with respect to the joint flexibilities. It is possible to see that the results remain good.

Obviously, since friction varies with temperature, load and wear, besides the direction of motion, friction compensation results cannot be assured in other tests, without refining parameter identification.

7 CONCLUDING REMARKS

In this paper, a systematic procedure to identify the friction model parameters and to experimentally analyse the position tracking control strategy performance, when implemented to a flexible joint robot prototype, has been presented. The proposed cascade control methodology uses the LuGre friction model without any assumption about the actuator’s torque response. The convergence of the tracking errors was theoretically proved as well as the experimental results demonstrate the effectiveness of the friction compensation. Further works will consider adaptive procedures to update the parameters of the friction observer, the problem of design control gains to meet performance, the influence of perturbations and the study of the force control problem in robots with elastic joints.

APPENDIX A. SYMBOL NOTATION AND MODEL PARAMETERS

In this appendix the simplified model of a flexible joint robot with friction terms and a friction force model are revisited and the parameters described in table 4 and table 5. The flexible robot model with the friction parameters is given by:

The friction force model is given by:

Artigo submetido em 20/12/2007 (Id.: 00841)

Revisado em 01/04/2008, 15/09/2008, 03/12/2008, 18/04/2009

Aceito sob recomendação da Editora Associada Profa. Anna Helena Reali Costa

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1

Recalling that the matrices

K_D1 and

K_D2 are diagonal.

2

For an hermitian matrix

3

An example of a static map is given in

Fig. 6.

Publication Dates

• Publication in this collection
  29 Jan 2010
• Date of issue
  Dec 2009

History

• Accepted
  18 Apr 2009
• Received
  20 Dec 2007