Experimental investigation on adaptive robust controller designs applied to a free-floating space manipulator

Control Engineering Practice 19 (2011) 395–408

Contents lists available at ScienceDirect

Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Experimental investigation on adaptive robust controller designs applied to a free-floating space manipulator Tatiana F.P.A.T. Pazelli a, Marco H. Terra a, Adriano A.G. Siqueira b, a b

~ Paulo at Sao ~ Carlos, Brazil Department of Electrical Engineering, University of Sao ~ Paulo at Sao ~ Carlos, Av. Trabalhador Sao-carlense, ~ Department of Mechanical Engineering, University of Sao 400, CEP 13566-590, Sa~ o Carlos, SP, Brazil

a r t i c l e i n f o

abstract

Article history: Received 13 October 2009 Accepted 18 December 2010 Available online 3 March 2011

This paper aims to formulate and investigate the application of various nonlinear H1 control methods to a free-floating space manipulator subject to parametric uncertainties and external disturbances. From a tutorial perspective, a model-based approach and adaptive procedures based on linear parametrization, neural networks and fuzzy systems are covered by this work. A comparative study is conducted based on experimental implementations performed with an actual underactuated fixedbase planar manipulator which is, following the DEM concept, dynamically equivalent to a free-floating space manipulator. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Space robotics H1 control Adaptive control Neural networks Fuzzy systems

1. Introduction Considerable research efforts have been directed to some primary functions of manipulator robots for in-space operations, such as assembly, inspection and maintenance. A number of dynamic and control problems are unique to this area due to the distinctive and complex dynamics found in many of these applications. The main feature of space robots is the dynamic coupling between the spacecraft and the robotic arm. It causes the coordinated rotation of the main body with the motions of the arm. Dubowsky and Papadopoulos (1993) identified representative types of space robotic systems considering some of their specific planning and control problems: free-flying space manipulators and free-floating space manipulators. In a free-flying space manipulator the spacecraft attitude is actively controlled by reaction wheels and/or jets. However, the use of these mechanisms increases the consumption of electrical power and fuel, also adding more weight, complexity and disturbances to the system. The consumption of relatively large amounts of fuel can limit the system’s useful life in space. Thus, to deal with this limitation, trajectory planning control approaches for this type of system are formulated in order to optimize energy consumption (Sakawa, 1999; Torres and Dubowsky, 1992). Differently, free-floating space manipulators (SM) are systems that allow the spacecraft to move freely in response to manipulator

 Corresponding author. Tel.: + 55 16 3373 9398; fax: + 55 16 3373 9402.

E-mail address: [email protected] (A.A.G. Siqueira). 0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2010.12.011

motions with the purpose of conserving energy (Papadopoulos and Dubowsky, 1991). Considering this configuration, coordinated spacecraft/manipulator motion control has been addressed in case of redundant manipulators (Caccavale and Siciliano, 2001; Dubowsky and Torres, 1991). Trajectory planning algorithms have also been developed in order to minimize the reaction of the freefloating base while the manipulator task is executed (Huang and Xu, 2006; Liu et al., 2009; Papadopoulos et al., 2005; Torres and Dubowsky, 1992; Tortopidis and Papadopoulos, 2006). Although each of these types of space robotic systems demands a specific controller design, the combination of both can be employed during different phases of a mission. This paper focuses on the optimization of the motion control of a freefloating space manipulator. Online trajectory tracking is evaluated considering a predefined trajectory. In virtue of the complexity of these systems and the environment where they operate, the choice of a control system to this class of manipulators assumes an important role to deal with parametric uncertainties and external disturbances. When different payloads are being handled during different tasks, mass and inertial characteristics are difficult to be estimated precisely. The hostile work environment can also generate model uncertainties due to wear of mechanic and electronic components. Moreover, most space manipulators are designed to be light weight and low power systems, what is consistent with the zero gravity environment and energy efficiency concerns. As a result, joint friction, damping, and other system nonlinear uncertainties are much more expressive in space robots than in industrial robots. Therefore, robustness is a significant challenge to the motion control of space manipulators. Two approaches to control uncertain systems

396

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

Nomenclature SM parameters and coordinates miu Ciu Iiu Jiu Li Ri Miu yiu ðju, yu, cuÞ qu

mass of link i mass center of link i inertia tensor of link i joint connecting link ði1Þ to link i vector connecting Jiu to Ciu vector connecting Cui to Jui þ 1 total mass rotation of link i around joint Jui Euler angles (Z-Y-Z) w.r.t. the spacecraft attitude generalized coordinates

DEM parameters and coordinates mi Ci Ii Ji Wi lci Mt

yi ðj, y, cÞ q

mass of link i mass center of link i inertia tensor of link i joint connecting link (i 1) to link i vector connecting Ji to Ji + 1 vector connecting Ji to Ci total mass rotation of link i around joint Ji Euler angles (Z-Y-Z) w.r.t. the base attitude (J1) generalized coordinates

Nonlinear H1 Control

g u Q R

Adaptive nonlinear H1 control

X F Z

M(q) _ Cðq, qÞ

t b m

symmetric positive definite inertia matrix matrix of Coriolis and centrifugal forces torque vector acting upon the joints of the DEM index w.r.t. the passive spherical joint (base) index w.r.t. the active joints (manipulator)

matrix of known functions vector with uncertain components _ symmetric positive definite weighting matrix w.r.t. F

Adaptive neural network nonlinear H1 control Hðxe , FÞ

set of n neural networks vector of adjustable weights in the output layer wk vector of constant weights in the input layer bk vector of constant biases in the hidden layer G ¼ tanh activation function for the neurons in the hidden layer

Fk

Adaptive fuzzy nonlinear H1 control Hðxe , FÞ

Fk DEM model

desired disturbance attenuation level control variable symmetric positive definite weighting matrix w.r.t. x~ m symmetric positive definite weighting matrix w.r.t. u

uki Aij yl

ml

set of n fuzzy systems vector of adjustable parameters in the output functional consequences fuzzified input variables linguistic variables output functional consequences d.o.f. of l-th rule

Problem formulation qdm x~ m

desired reference trajectory for manipulator joints state tracking error

subject to external disturbances are commonly used in controller designs: adaptive control and robust control. Adaptive controllers estimate and compensate the uncertain dynamics of a system, (Craig, 1988). The adaptive procedure is generally based on the linear parametrization property, which demands a precise knowledge of the model structure. It considers that the unknown parameters are constant or vary slowly. Besides, unmodeled dynamics are usually present and their effects decrease the performance of this approach. This method has been addressed to the motion control of free-floating space manipulators in several works in the literature (e.g., Gu and Xu, 1993; Parlaktuna and Ozkan, 2004; Woerkom et al., 1996). However, most of them require the measurement of orientation, velocity and acceleration of the free-floating base, which are difficult measures to be obtained in practice. Another line of research on adaptive control is defined by intelligent systems, which exhibit interesting abilities of learning and interpretation. Approaches based on fuzzy and neural network systems are able to learn and approximate real-world concepts, building a knowledge base that may be interpreted and modified by the user. Moreover, these approaches have high potential to provide mechanisms for building systems subject to rapidly varying

unknown parameters. Thus, intelligent systems have been successfully applied in the literature to universally approximate mathematical models of dynamic systems, see for instance (Hornik et al., 1989; Jang, 1993; Narendra and Parthasarathy, 1990; Polycarpou, 1996; Takagi and Sugeno, 1985). Within the scope of this work, applications of intelligent systems to identify and control free-floating space manipulators can be found in Sanner and Vance (1995); Taveira et al. (2006); Guo and Chen (2008); Huang and Chen (2008) and Pazelli et al. (2008). On the other hand, robust controllers have been designed to deal with parametric uncertainties and external disturbances. In the case of space robots, disturbances may be generated, for example, by spacecraft movements, microgravity effects and sensor and actuator noises due to extremes of temperature and glare. Impact effects during the docking or rendezvous process are also representative disturbances for this kind of robot. Sliding mode control is one of the main approaches that deals with disturbance effects. However, this method is usually accompanied by a phenomenon called chattering, which can increase energy consumption and damage the actuators. In addition, due to high frequency content of chattering, it can easily stimulate flexible modes which in turn may cause instability. This issue is extensively worked throughout the literature, and different

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

procedures have already been applied to diminish this effect. Feng et al. (2008) proposed a robust sliding mode controller with a radial basis function neural network to eliminate the chattering phenomenon in the tracking control problem of a free-floating space robot. Guo and Chen (2006) have presented a robust control scheme for a dual-arm space robot system based on the augmentation approach proposed by Gu and Xu (1993). Zhongyi et al. (2008) employed a disturbance observer in each joint of the arm to estimate plant uncertainties, unmodeled dynamics and external disturbances within the execution of a free-floating space manipulator task. In this scheme, a low pass filter plays a significant role in ensuring the robustness and disturbance suppression performance of the system. In Pathak et al. (2008), robust trajectory control of free-floating space robots is performed by a set of linear time invariant controllers coupled to the robotic manipulator through a set of high feed-forward gains and suitably defined feedback paths. In this reference, the robustness of the control method is guaranteed in virtue of the controller not demand the knowledge of the manipulator parameters. A different method of robust control, based on the H1 criterion, has been widely used among robotic applications for guaranteeing good disturbance rejection properties. The H1 norm is a measure of the robustness applied by the control system to the plant. It defines the level of attenuation in the input/output relationship between the disturbance and the controlled output. Applications of this approach include the control of position and attitude of satellites and spacecrafts (Chen et al., 2000; Dalsmo and Egeland, 1997; Kang, 1995; Skullestad and Gilbert, 2000; Yang and Sun, 2002). Concerning the motion control of free-floating manipulators, however, works that have resorted to this method are difficult to find. Considering the presented scenario, this paper aims to investigate the application of nonlinear H1 controllers to a freefloating space manipulator subject to parametric uncertainties and external disturbances. To conduct a comparative study, this work covers the following methods of robust control based on the H1 criterion: (1) Nonlinear H1 control based on game theory; (2) Adaptive nonlinear H1 control; (3) Adaptive neural network nonlinear H1 control; and (4) Adaptive fuzzy nonlinear H1 control. To the best of the authors’ knowledge, a comparison among adaptive nonlinear H1 controllers has not been made throughout the literature. Method (1), derived from Siqueira and Terra (2004), and method (2), based on linear parametrization, demand a complete knowledge of the nominal model structure. The intelligent procedures, (3) and (4), are proposed in two different approaches. The first one applies the intelligent system to learn the dynamic behavior of the robotic system, which is considered totally unknown. The second approach considers a well-defined nominal model structure for the arm and the intelligent system is applied to estimate only the behavior of uncertain dynamics. It must be noted that these controllers do not demand any kinematic or dynamic data from the spacecraft. The H1 criterion is applied to all the proposed techniques to attenuate the effect of estimation errors and external disturbances. An important issue of this comparative study is that the analysis of controllers requires accurate simulations of plant behavior. However, it is difficult to recreate space conditions on Earth. Thus, a modeling approach that relates space and terrestrial systems is opportune. An analytical method of modeling free-floating space robots, called Virtual Manipulator (VM), was proposed in Vafa and Dubowsky (1990). The VM is an inertial fixed-base robot whose first joint is a passive spherical one, representing the free-floating nature of the space manipulator. However, only kinematic equivalence is considered in this representation. Based on the VM concept, Liang et al. (1997) mapped a SM to a conventional fixed-base manipulator and showed that both kinematic and dynamic properties of the space manipulator system are preserved in this mapping. This manipulator is called Dynamically Equivalent Manipulator (DEM). The DEM goes

397

beyond the VM concept since it models the free-floating space manipulator system both kinematically and dynamically. Thus, the DEM can be physically built using a conventional manipulator system. It allows to experimentally study the dynamic performance and task execution of a space manipulator system without having to resort to complex experimental set-ups to simulate the space environment. Nevertheless, experimental applications using this resource are not found in literature. Therefore, in order to validate and compare the proposed control procedures in this paper, experimental implementations are performed with an actual underactuated fixed-base planar manipulator. Based on the DEM concept, this robot is dynamically equivalent to a free-floating space manipulator. A qualitative analysis of the results is developed concerning the ability of attenuating the effects of the disturbance. In addition, a quantitative description of the results is presented in the form of performance indexes which express the error of trajectory tracking and the energy consumption. In summary, the contribution of this paper consists of an experimental investigation on the motion control of a freefloating space manipulator subject to parametric uncertainties and external disturbances, performed by different methods of adaptive nonlinear H1 controllers. This paper is organized as follows: the kinematic and dynamic equivalences between the free-floating space manipulator and a fixed-base manipulator are presented in Section 2; the solutions for the nonlinear H1 control problems based on the linear parametrization property of the model, neural networks and fuzzy systems are presented in Section 3; and, finally, experimental results for a two-link free-floating space manipulator are presented in Section 4.

2. Model description and problem formulation 2.1. Free-floating space manipulator mapped by a dynamically equivalent fixed-base manipulator Consider an n-link serial-chain rigid manipulator mounted on a free-floating base and that no external forces and torques are applied on this system. Consider also the Dynamically Equivalent Manipulator (DEM) approach (Liang et al., 1997). The DEM is a (n + 1)-link fixed-base manipulator with its first joint being a passive spherical one and whose model is both kinematically and dynamically equivalent to the SM dynamics. Since it is a conventional manipulator, it can be physically built and experimentally used to study control algorithms for space manipulators. Fig. 1 shows the representation and parameter notation for both SM and DEM. Let the SM parameters be identified by apostrophe (0 ), the links of the manipulators are numbered from 2 to n +1 and Ji is the joint connecting the (i 1)-th link and i-th link, yi is the rotation of the i-th link around joint Ji, and the Z-Y-Z Euler angles ðj, y, cÞ represent the SM base attitude and the DEM

Fig. 1. The space manipulator and its corresponding DEM (Liang et al., 1997).

398

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

first passive joint orientation. The Z-Y-Z convention of the Euler angles is used in this paper as in Liang et al. (1997). This convention represents a rotation of the XYZ-system about the Zaxis by j; a rotation of the XuYuZu-system about the Xu-axis by y, and a rotation of the X00 Y00 Z00 -system about the Z00 -axis by c. Let Ci be the center of mass of the i-th link, Li be the vector connecting Jui and Cui , Ri be the vector connecting Cui and Jui þ 1 , lci be the vector connecting Ji and Ci, and Wi be the vector connecting Ji and Ji + 1. Considering that the DEM operates in the absence of gravity and that its base is located at the center of mass of the SM, the kinematic and dynamic parameters of the DEM can be described from the SM parameters as Mt2 mui , Pi k ¼ 1 muk k ¼ 1 muk

I1 ¼ Iu1 , Ii ¼ Iui ,

m

i R1 X mu , Mt k ¼ 1 k

q_ m Þ2Cmm ðqm , q_ m Þ is skew-symmetric. Remark 1. The controller is designed for the SM considering the DEM model.

i i1 R X L X mu þ i mu , Wi ¼ i Mt k ¼ 1 k Mt k ¼ 1 k

lc1 ¼ 0, lci ¼

ð3Þ where Mbb ðqm Þ A R33 , Mbm ðqm Þ A R3n , Mmb ðqm Þ A Rn3 , Mmm ðqm Þ A _ A R33 , Cbm ðqm , qÞ _ A R3n , Cmb ðqm , qÞ _ A Rn3 , Cmm ðqm , Rnn , Cbb ðqm , qÞ nn n _ qm Þ A R are nominal matrices and tm A R . For simplicity of notation, the 0 index referring to the nominal system was suppressed. It must also be noted that (3) is formulated so that the inertia matrix related to the controlled joints, Mmm(qm), is _ mm ðqm , symmetric positive definite and the matrix Nmm ðqm , q_ Þ ¼ M

mi ¼ Pi1

W1 ¼

passive joint dynamics intervene with the control of the manipulator active joints. Let q be partitioned as q ¼ ½qTb qTm T , where the indexes b and m represent the passive spherical joint (base) and the active joints T T (manipulator), respectively. Define d ¼ ½db dm T as a vector representing the sum of parametric uncertainties of the system and td ¼ ½tTdb tTdm T as the finite energy external disturbance also introduced. Eq. (2) can be rewritten as " # " # " # " #" # " #" # tdb q_ b Mbb Mbm q€ b Cbb Cbm 0 db þ þ tdm ¼ Mmb Mmm q€ m þ Cmb Cmm q_ m , tm dm

i1 Li X mu , Mt k ¼ 1 k

ð1Þ

where i¼2,y,n +1 and Mt is the total mass of the SM. Observe that the mass of the passive joint, m1, is not defined by the equivalence properties. From Lagrange theory, dynamic equations of the DEM are given by a set of nonlinear coupled differential equations as _ q_ ¼ t, MðqÞq€ þCðq, qÞ

ð2Þ T

ðn þ 3Þ

where q ¼ ½j y c y2    yn þ 1  A R are generalized coordinates, MðqÞ A Rðn þ 3Þðn þ 3Þ is the symmetric positive definite iner_ A Rðn þ 3Þðn þ 3Þ is the matrix of Coriolis and tia matrix, Cðq, qÞ centrifugal forces, and t ¼ ½0 0 0 t2    tn þ 1 T A Rðn þ 3Þ is the torque vector acting upon the joints of the DEM. From a practical point of view, the SM must handle different payloads whose mass and inertial characteristics can be difficult to obtain. Also, the dynamics of joint friction, damping, vibration or other nonlinear uncertainties are not easy to model or measure, producing effects that cannot be neglected. Moreover, the hostile work environment and the floating base are sources of disturbances which can also generate trajectory errors in the SM. Due to this fact, a finite energy torque disturbance may be defined by td and modeling uncertainties can be introduced dividing the _ into a nominal and a parameter matrices M(q) and Cðq, qÞ perturbed part: MðqÞ ¼ M0 ðqÞ þ DMðqÞ,

Let qdm A Rn and q_ dm A Rn be the desired reference trajectory and the corresponding velocity for the controlled joints, respectively. The state tracking error is defined as " # " # q_~ m q_ m q_ dm x~ m ¼ ¼ : ð4Þ q~ m qm qdm d

The variables qdm, q_ dm , and q€ m (desired acceleration) are assumed to be within the physical and kinematic limits of the control system and no reference trajectory exists for the base. In order to minimize x~ m acting with minimal torque and energy consumption, a control input is chosen according to Johansson (1990) as " # z_~ 1 ¼ Mmm T1 x~_ m þCmm T1 x~ m , u ¼ ½Mmm Cmm  ð5Þ z~ 1 with z~ 1 and T1 being introduced by the transformation " # " #  " _ # T1 z~ 1 q~ m T11 T12 z~ ¼ ¼ T0 x~ m ¼ x~ m ¼ , T2 z~ 2 q~ m 0 I

where T11, T12 A Rnn are constant matrices to be determined. The defined control input (5) includes selective torques that affect only the kinetic energy, the reference trajectories (4) and a state space transformation (6) that eliminates redundancies among position and velocity coordinates while keeping full state space order. Consequences of the choice of this transformation are that the synthesis of the nonlinear H1 control depends on an algebraic Riccati equation, which is easily solved, and the resulting gain will be constant. From (3)–(6), state space representation of the DEM is given by x_~ m ¼ AT ðqm , q_ m Þx~ m þ BT ðqm ÞT11 ðtm Fðxem ÞEðxeb Þ þ dm þ tdm Þ,

_ ¼ C0 ðq, qÞ _ þ DCðq, qÞ, _ Cðq, qÞ

where

_ are nominal matrices and DMðqÞ and where M0(q) and C0 ðq, qÞ _ are the model uncertainties. DCðq, qÞ

AT ðqm , q_ m Þ ¼ T01

"

"

2.2. Problem formulation BT ðqm Þ ¼ Since a free-floating space manipulator is being considered, it is assumed that only the active joints of the DEM are controlled, with the passive spherical joint not locked. In this case, the

ð6Þ

T01

1 Mmm Cmm

0

1 T11

1 T11 T12

1 Mmm

0 d

# T0 ,

# ,

1 1 T12 q_~ m Þ þCmm ðq_ dm T11 T12 q~ m Þ, Fðxem Þ ¼ Mmm ðq€ m T11

ð7Þ

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

nonlinear H1 control problem is defined for the proposed application by the following performance criterion

and Eðxeb Þ ¼ Mmb q€ b þ Cmb q_ b ,

Z

d T with xem ¼ ½qTm q_ Tm ðqdm ÞT ðq_ dm ÞT ðq€ m ÞT T and xeb ¼ ½q_ Tb q€ b T .

0

T

~ ð0Þ þ g2 ~ ð0ÞZ0 F ðx~ Tm Q x~ m þ u T RuÞ dt r x~ Tm ð0ÞP0 x~ m ð0Þ þ F

Z

T

ðoT oÞ dt,

0

~ ¼ F F denotes the parameter estimation error. where F

Given a desired disturbance attenuation level of g 4 0, the state feedback nonlinear H1 control method aims to attenuate the disturbance in the system through a control law of the form u ¼ Kðx~ m Þx~ m in order to satisfy the performance criterion  R 1 1 T 1 T ~ ~ 0 2 x m ðtÞQ x m ðtÞ þ 2 u ðtÞRuðtÞ dt  R 1 1 max ð8Þ r g2 , min T uðÞ A L2 0 a oðÞ A L2 0 2 o ðtÞoðtÞ dt where Q and R are symmetric positive definite weighting matrices defined by the designer, x~ m ð0Þ ¼ 0 and o ¼ T11 ðdm þ tdm Þ refers to the disturbance term in (7). Following the game theory, a solution of this minimax problem is given by Chen et al. (1994), in a simplified form, in terms of the algebraic equation    0 K 1 ð9Þ T0T B R1  2 I BT T0 þQ ¼ 0, g K 0 with B ¼ ½I 0T . This simplification in the synthesis of the nonlinear H1 controller is the main consequence of the transformation (6). The resulting gain will be constant in a similar way to the state feedback linearization approaches. Hence, for the proposed application, the H1 control problem (8) subject to (7) has an optimal solution ð10Þ

if there are matrices K 4 0 and a non-singular T0 solutions of (9), with R o g2 I. Considering that the matrix R1 is obtained through a Cholesky decomposition of the R-dependent term in (9)  1 1 , RT1 R1 ¼ R1  2 I

g

and Q is factorized as " T # Q12 Q1 Q1 Q¼ , T Q12 Q2T Q2

3.1. Adaptive nonlinear H1 control Adaptive nonlinear H1 control presented in this section assumes that the model structure of the system is well-defined but its formulation is based on uncertain (or unknown) para_ meters which determine M(q) and Cðq, qÞ. Assume the linear parametrization property of the term Fðxem Þdm in (7): Fðxem Þdm  Hðxe , FÞ ¼ XF,

ð13Þ d

where xe is the input vector of Xðqm , q_ m , q_ dm T11 T12 q~ m , q€ m  1 T11 T12 q_~ m Þ, being X a ðn  pÞ regression matrix of known functions, and F is a p-dimensional vector with uncertain components depending on manipulator parameters. Let the optimal approximation parameters vector be

F ¼ arg min max JHðxe , FÞðFðxem Þdm ÞJ2 , F A OF xe A Oxe

where J  J2 is the Euclidean norm. Define t ¼ tm Eðxeb Þ and the modified error Eq. (7) may be rewritten as x_~ m ¼ AT ðqm , q_ m Þx~ m þ BT ðqm ÞT11 ðt Fðxem Þ þ dm þ tdm þHðxe , FÞHðxe , FÞÞ ¼ AT ðqm , q_ m Þx~ m þBT ðqm ÞT11 ðt Hðxe , FÞÞ þBT ðqm ÞT11 ðHðxe , FÞFðxem Þ þ dm þ tdm Þ ¼ AT ðqm , q_ m Þx~ m þBT ðqm Þu þ BT ðqm Þo

ð14Þ

with u ¼ T11 ðt Hðxe , FÞÞ,

ð15Þ

o ¼ T11 ðHðxe , FÞFðxem Þ þ dm þ tdm Þ,

ð16Þ

where o refers to the estimation error from the adaptive system and external disturbances. Using u ¼ u the control law provided by the nonlinear H1 controller in (10), t can be computed by

the solution of (9) is given by " # RT1 Q1 RT1 Q2 1 1 T and K ¼ ðQ1T Q2 Q2T Q1 Þ ðQ21 þ Q12 Þ: T0 ¼ 2 2 0 I Thus, from (7), the applied torques which guarantee the desired H1 performance (8) can be computed by 1  tm ¼ T11 u þ Fðxem Þ þ Eðxeb Þ:

T

ð12Þ

2.3. Nonlinear H1 control

u  ¼ R1 BT T0 x~ m ,

399

ð11Þ

1 t ¼ Hðxe , FÞ þT11 u:

ð17Þ

Considering the solvability analysis developed by Chen et al. (1997), a dynamic state feedback controller given by _ ¼ aðt, x~ m Þ ¼ Z 1 XT T11 BT T0 x~ m , F

ð18Þ

1 1 T t ¼ t ðt, F, x~ m Þ ¼ XFT11 R B T0 x~ m ,

ð19Þ

T

Remark 2. This nonlinear H1 control method assumes that the model structure is completely known and represents parameter uncertainties as internal disturbances, modeling them in the same way as external disturbances.

with Z ¼ Z 40, is solution to the adaptive nonlinear H1 control problem subject to (14) and satisfies (12) for any initial condition. Factoring out q€ b in the first line of (3) – disregarding uncertainties and disturbances – and substituting the result back on its second line, leads to

3. Adaptive robust controller design

1 M ¼ Mmm Mmb Mbb Mbm ,

The adaptive control designs presented in the following apply different learning methods to estimate uncertain parameters and also the behavior of unmodeled dynamics. The H1 control law is applied to attenuate the effects of estimation errors and external disturbances. Hence, given a desired level of attenuation g 4 0 and matrices Q ¼ Q T 40,R ¼ RT 4 0,P0 ¼ P0T 40 and Z0 ¼ Z0T 4 0, the adaptive

1 1 h ¼ ðCmb Mmb Mbb Cbb Þq_ b þ ðCmm Mmb Mbb Cbm Þq_ m : The relation between t and the torques applied upon the manipulator joints, tm , is given by

^ 1 ðC^ mm q_ t Þ, q€ m ¼ M m mm

ð20Þ

tm ¼ M^ q€ m þ h^ ,

ð21Þ

400

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

^ and h^ are the matrices M , C , M and h ^ mm , C^ mm , M where M mm mm computed with the estimated values F. Remark 3. This solution uses only the velocity of the free-floating base, q_ b . If it was tried to linearly parametrize the complete term ðFðxem Þ þ Eðxeb Þdm Þ in (7), the solution would also need the acceleration of the free-floating base q€ b which is not easy to obtain in practice.

j¼1

where qk is the size of vector xe and pk is the number of neurons in the hidden layer. The weights wkij and the biases bki for 1r i rpk , 1 rj r qk and 1 r kr n are assumed to be constant and specified by the designer. Thus, the adjustment of neural networks is performed only by updating the vectors Fk . The activation function for the neurons in the hidden layer is chosen to be Gð:Þ ¼ tanhð:Þ. Fig. 2 illustrates the structure of Hk. The complete neural network is denoted by 3 2 3 2 T x1 0 . . . 0 2 F1 3 H1 ðxe , F1 Þ 7 6 7 6 T 6 7 6 H2 ðxe , F2 Þ 7 6 0 x2 ^ 0 7 76 F 2 7 7¼6 Hðxe , FÞ ¼ 6 ð23Þ 7 ¼ XF, 6 76 6 7 6 ^ ^ ^ 4 5 7 ^ & ^ 4 5 4 5 T Hn ðxe , Fn Þ Fn 0 0 . . . xn with 2

0

¼ 4tanh@

qk X

1 wk1j xej þ bk1 A

0    tanh@

j¼1

qk X

13

avoiding the necessity of any data from the free-floating base. Experimental results described in Section 4 show the feasibility of this assumption. Defining the following optimization problem

the modified error Eq. (7) may be rewritten as x_~ m ¼ AT ðqm , q_ m Þx~ m þ BT ðqm Þu þBT ðqm Þo u ¼ T11 ðtm Hðxe , FÞÞ,

ð28Þ

o ¼ T11 ðHðxe , FÞFðxem ÞEðxeb Þ þ dm þ tdm Þ,

ð29Þ

where o refers to the estimation error from the neural network system and external disturbances. Considering the results achieved by Chang and Chen (1997), a dynamic state feedback controller given by _ ¼ aðt, x~ m Þ ¼ Z T XT T11 BT T0 x~ m , F

ð30Þ

1 1 T tm ¼ tm ðt, F, x~ m Þ ¼ XFT11 R B T0 x~ m ,

ð31Þ

T

with Z ¼ Z 4 0, is solution to the adaptive neural network nonlinear H1 control problem subject to (27) and satisfies (12) for any initial condition. Consider now a second approach where the model structure and nominal values for the term Fðxem Þ are well-defined and available for the controller. In this case, the neural network is applied to estimate only the behavior of parametric uncertainties and spacecraft dynamics (considered as an unmodeled disturbance):

Fk ¼ ½fk1    fkpk  :

F ¼ arg min max JHðxe , FÞðEðxeb Þdm ÞJ2 , F A OF xe A Oxe

Consider a first approach where the term Fðxem Þ þ Eðxeb Þdm in (7) is completely unknown regarding its structure and parameter values. The neural network defined in (23) is applied to learn the dynamic behavior of the robotic system:

and the modified error Eq. (27) may be rewritten as

Fðxem Þ þEðxeb Þdm  Hðxe , FÞ ¼ XF,

with

ð24Þ

where the input vector xe should be defined as T d xe ¼ ½qTm q_ Tm q_ Tb q€ b ðqdm ÞT ðq_ dm ÞT ðq€ m ÞT T :

ð25Þ

Note that this definition of xe takes into account the values of q_ b and q€ b , which was inspired in the mathematical model of the space manipulator aforementioned. However, these values are not easy to be measured in practice. Considering that a neural k

k

k

Wk

ξk Gk1

Φk

Gk2 Gk...

k2

x_~ m ¼ AT ðqm , q_ m Þx~ m þ BT ðqm Þu þBT ðqm Þo

ð33Þ

u ¼ T11 ðtm Fðxem ÞHðxe , FÞÞ,

ð34Þ

o ¼ T11 ðHðxe , FÞEðxeb Þ þ dm þ tdm Þ,

ð35Þ

where o refers to the estimation error from the neural network system and external disturbances. This is an original proposal that follows the stability analysis developed by Chang and Chen (1997) and applied to (12). A dynamic state feedback controller chosen as _ ¼ aðt, x~ m Þ ¼ Z T XT T11 BT T0 x~ m , F

ð36Þ

1 1 T tm ¼ tm ðt, F, x~ m Þ ¼ Fðxem Þ þ XFT11 R B T0 x~ m ,

ð37Þ

T

with Z ¼ Z 4 0, is solution to the adaptive neural network nonlinear H1 control problem subject to the distinct formulation (33) and satisfies (12) for any initial condition.

k1

xe

ð32Þ

_ xem , the optimal approximation parameters Similarly, xe ¼ vector is given by

T

k

ð27Þ

with

Eðxeb Þdm  Hðxe , FÞ ¼ XF:

wkpk j xej þ bkpk A5,

j¼1

bpk b... b2 b1

ð26Þ

F A OF xe A Oxe

Define a set of n neural networks Hk ðxe , Fk Þ, k¼ 1, y, n, where xe is the input vector and Fk are the adjustable weights in the output layers. The single-output neural networks are of the form 0 1 pk qk X X T Hk ðxe , Fk Þ ¼ fki G@ wkij xej þbki A ¼ xk Fk , ð22Þ

xTk

d xe ¼ _ xem ¼ ½qTm q_ Tm ðqdm ÞT ðq_ dm ÞT ðq€ m ÞT T ,

F ¼ arg min max JHðxe , FÞðFðxem Þ þEðxeb Þdm ÞJ2 ,

3.2. Adaptive neural network nonlinear H1 control

i¼1

network-based approach is in general used when it is not possible to supply all the variables values to the system model, the vector xe can be redefined as

Hk(xe, Φk)

k...

3.3. Adaptive fuzzy nonlinear H1 control

kpk

The Adaptive Network-based Fuzzy Inference System (ANFIS) defined by Jang (1993) introduces a training procedure for Takagi– Sugeno (T-S) fuzzy inference systems. Considering an input/output dataset, ANFIS builds a fuzzy inference system whose membership functions parameters are adjusted by a hybrid learning rule which

Gkpk Fig. 2. Structure of a single-output neural network.

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

combines the gradient method and the least squares estimate. An adaptive network architecture, similar to neural networks structure, consists of nodes and directional links through which nodes are connected. Part or all of the nodes are adaptive, which means their outputs depend on the parameters pertaining to these nodes. The learning rule specifies how these parameters should be changed to minimize a prescribed error measure. In this paper, an ANFIS was applied to an input/output dataset within an off-line procedure before the task execution. Thus, it provides a set of input membership functions to be used during the control action. The T-S fuzzy model is characterized by a fuzzy rule base with functional consequences instead of fuzzy consequences, as

Table 1 SM parameters. Body

mui (kg)

Iui (kgm2)

Ri (m)

Li (m)

Base Link 2 Link 3

4.816 0.618 0.566

0.0153 0.0075 0.0060

0.253 0.118 0.126

0 0.120 0.085

Body

mi (kg)

Ii (kgm2)

Wi (m)

lci (m)

Link 1 Link 2 Link 3

1.932 0.850 0.625

0.0153 0.0075 0.0060

0.203 0.203 0.203

0 0.096 0.077

401

Table 2 DEM parameters.

IF u1 is A11 and u2 is A12 y and uqk is A1qk , THEN y1 ¼ f10 þ f11 u1 þ f12 u2 þ . . . þ f1qk uqk , ^ IF u1 is Apk 1 and u2 is Apk 2 y and uqk is Apk qk , THEN ypk ¼ fpk 0 þ fpk 1 u1 þ fpk 2 u2 þ . . . þ fpk qk uqk , where Aij, i¼1,y,pk and j¼1,y,qk, are linguistic variables referred to fuzzy sets defined on the input spaces U1, U2, y, Uqk ; u1 A U1 , u2 A U2 , . . . ,uqk A Uqk are input variables values; pk is the number of fuzzy rules and qk is the size of an input vector. The inferred output from the T-S method is crisp (hence, it does not demand a defuzzifier) and it is defined by the weighted average of outputs yl from each linear subsystem implied as Ppk m yl Hk ðxe , Fk Þ ¼ Pl p¼k 1 l

ml m ðfl0 þ fl1 u1 þ fl2 u2 þ . . . þ flqk uqk Þ T l¼1 l ¼ xk Fk , Ppk m l¼1 l l¼1

Ppk ¼

ð38Þ Fig. 3. UnderActuated Robot Manipulator.

Table 3 Selected weighting matrices.

g¼2

R

Nonlinear H1



Adaptive H1

0  0:9

2:5  0

0  0:64

0:64  0

Adaptive Neural H1 (1)

0  0:9

2:5  0

0  0:64

0:64  0

Adaptive Neural H1 (2)



0 0:9

2:5  0



Adaptive Fuzzy H1 (1)



0 0:9 0  0:9

2:5  0 2:5  0

Adaptive Fuzzy H1 (2)

Q1 0:9

0

0





Q2

0:64

0 4

0 1 0  4 

2:5

0





0:64

–

0  9

7  0



0  9

7  0

0  10

0 9

7  0





4  0 1  0

Z 

9

0

0 7  9 0 0 7   25 0

4

0

Base Joint 2 Joint 3 Reference

7

40

0

20 0 −20

30 20 10 0 −10

−40

−20

−60

−30 0

0.5

1

1.5 2 Time (s)

2.5

3

3.5

0

0.5

1

1.5 2 Time (s)

Fig. 4. Joints and base positions and velocities—nonlinear H1 control.

2.5

3

3.5

0



10  0

0 10  100 0



40 Joint velocity (º/s)

60

10

0 0:8 0  20



50

80 Joint position (º)

0

0



100  0 0:8  0 20

402

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

where 1 r kr n and ml is the degree of freedom of l-th rule, defined as the minimum among the grade of membership associated to the entries in the activated fuzzy sets by the l-th rule,

ml ¼ Al1 ðu1 Þ4Al2 ðu2 Þ4 . . . 4Alqk ðuqk Þ:

ð39Þ

Thus, a set of fuzzy inference systems based on the T-S method is defined as 3 2 3 2 T x1 0 . . . 0 2 F1 3 H1 ðxe , F1 Þ 6 7 6 7 T 6 7 6 H2 ðxe , F2 Þ 7 6 0 x2 ^ 0 7 76 F 2 7 7¼6 ð40Þ Hðxe , FÞ ¼ 6 7 ¼ XF, 6 76 6 7 ^ 4 ^ 5 ^ & ^ 7 4 5 6 4 ^ 5 T Hn ðxe , Fn Þ Fn 0 0 . . . xn

Fðxem Þ þ Eðxeb Þdm  Hðxe , FÞ ¼ XF:

ð41Þ

For the proposed application, the input vector is defined as T T xe ¼ x~ m ¼ ½q~_ m q~ m T and Aðx~ m Þ ¼ ½A1 ðq_~ m Þ A2 ðq~ m Þ comprises the fuzzy sets defined for the fuzzified inputs. Observing the similar formulation in Section 3.2 and based on the analysis presented in Chang and Chen (1997) and Chang (2005), a dynamic state feedback controller given by _ ¼ aðt, x~ m Þ ¼ Z T XT T11 BT T0 x~ m , F

ð42Þ

1 1 T tm ¼ tm ðt, F, x~ m Þ ¼ XFT11 R B T0 x~ m ,

ð43Þ

T

with 1

xTk ¼ Ppk

mk l¼1 l

½mk1 mk1 xTe mk2 mk2 xTe    mkpk mkpk xTe ,

Fk ¼ ½fk10 fk11 . . . fk1qk fk20 fk21 . . . fk2qk    fkpk 0 fkpk 1 . . . fkpk qk T , where xe A Rqk is the input vector, X is an ðn  pÞ matrix dependent on the input values and on the values of fuzzy rules degrees of freedom and the vector F A Rnðpk ðqk þ 1ÞÞ is a p-dimensional vector representing the adjustable parameters in the output functional consequences, with p ¼ n(pk(qk + 1)). Consider a first approach where the term Fðxem Þ þ Eðxeb Þdm in (7) is completely unknown regarding its structure and parameter values. The T-S fuzzy system defined in (40) is applied to

with Z ¼ Z 4 0, satisfies (12) for any initial condition. It is solution to the adaptive fuzzy nonlinear H1 control problem subject to (27), where, in this case, o refers to the estimation error from the T-S fuzzy system and external disturbances. The second approach proposed in Section 3.2 may also be considered for the fuzzy-based controller. In this case, the T-S fuzzy system is applied to estimate only the behavior of parametric uncertainties and spacecraft unmodeled dynamics: Eðxeb Þdm  Hðxe , FÞ ¼ XF:

ð44Þ

As in (36)–(37), a dynamic state feedback controller given by _ ¼ aðt, x~ m Þ ¼ Z T XT T11 BT T0 x~ m , F

ð45Þ

1 1 T tm ¼ tm ðt, F, x~ m Þ ¼ Fðxem Þ þ XFT11 R B T0 x~ m ,

ð46Þ

with Z ¼ Z T 4 0, satisfies (12) for any initial condition. It is solution to the adaptive fuzzy nonlinear H1 control problem subject to (33), where, in this case, o refers to the estimation error from the T-S fuzzy system and external disturbances.

0.15 Joint 2 Joint 3

Remark 4. Within both neural and fuzzy control methods, measured values for orientation, velocity and acceleration of the free-floating base are not necessary. The intelligent systems are applied to estimate the spacecraft dynamics as well as parametric uncertainties and unmodeled dynamics. It can be seen by the control laws defined by (30)–(31), (36)–(37), (42)–(43) and (45)–(46) that the spacecraft coordinates qb, q_ b and q€ b are not included among the entries of Fðxem Þ, Hðxe , FÞ and u.

0.1

0.05

0

4. Results

−0.05

For validation and comparison purposes, the proposed adaptive H1 control solutions are applied to a free-floating, planar, two-link space manipulator system, whose nominal parameters are given in Table 1. The corresponding DEM is a fixed-base, three-link, planar manipulator whose first joint is configured as passive, that is, qm ¼ ½q2 q3 T are the joints to be controlled. The

−0.1 0.5

1

1.5

2

2.5

3

Time (s) Fig. 5. Torques—nonlinear H1 control.

80

40

60

Base Joint 2 Joint 3 Reference

40 20 0 −20

Joint velocity (º/s)

0

Joint position (º)

Torque (N.m)

learn the dynamic behavior of the robotic system:

30 20 10 0 −10

−40

−20

−60

−30 0

0.5

1

1.5

2

Time (s)

2.5

3

0

0.5

1

1.5

2

Time (s)

Fig. 6. Joints and base positions and velocities—adaptive nonlinear H1 control.

2.5

3

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

experimental fixed-base manipulator UARM (UnderActuated Robot Manipulator), whose nominal parameters are given in Table 2, is used in the application of the controllers, Fig. 3. The experimental manipulator UARM is composed of a DC motor in each joint, a break and an optical encoder with quadrature decoding used to measure joint positions. Joint velocities are obtained by numerical differentiation and filtering (Wijngaard, _ are 1996). Modeling matrices for the DEM, M(q) and Cðq, qÞ, defined in the Appendix. A trajectory tracking task is defined for the space manipulator joints, characterized by initial conditions qm ðt0 Þ ¼ ½203 203 T and desired final position qm ðtf Þ ¼ ½803 603 T , with tf ¼3 s. The reference trajectory, qdm, is a fifth degree polynomial given by qdm ðti Þ ¼ a0 þ a1 ti þ a2 ti2 þ a3 ti3 þ a4 ti4 þ a5 ti5 , with a4 ¼

a0

a1 ¼ a2 ¼ ½0 0T ,

qm(t0),

¼

15ðqm ðtf Þqm ðt0 ÞÞ tf4

ð47Þ

and a5 ¼

6ðqm ðtf Þqm ðt0 ÞÞ . tf5

10ðqm ðtf Þqm ðt0 ÞÞ , tf3

a3 ¼

The coefficients in (47)

were defined under the following conditions: qdm ðt0 Þ ¼ qm ðt0 Þ, qdm ðtf Þ ¼ qm ðtf Þ, q_ dm ðt0 Þ ¼ q_ dm ðtf Þ ¼ ½0 0T , d d q€ m ðt0 Þ ¼ q€ m ðtf Þ ¼ ½0 0T :

The time variable ti is defined by ti þ 1 ¼ ti þ dt, where t0 ¼ 0 and dt refers to the time spent during the i-th control loop. This interval

0.15

is measured at each control loop and, in the experimental implementation, the average value obtained was about dt ¼ 3 ms. During the experiments, a limited disturbance was introduced at ti ¼ 1 s in the following form " # 0:075e5ti sinð6pti Þ td ¼ : ð48Þ 0:05e5ti sinð6pti Þ The disturbance td is equivalent to approximately 75% of the torque applied in case that no disturbance is inserted. The level of disturbance attenuation defined for the applied controllers is g ¼ 2. The weighting matrices R and Q are adjusted empirically. The matrix R weighs the strength of the control law and was settled to target the best performance of the nonlinear H1 controller (Section 2.3) whose results are used as a basis of comparison. Thus, R is kept constant for all the evaluated controllers. Matrices Q1 and Q2, which compound the weighting matrix Q, are adjusted empirically for each controller aiming their best performance in terms of trajectory tracking errors (which is the objective of the control system), observing the appearance of the obtained graphics and the values of the L2 norm of the state vector. The matrix Z is selected in order to optimize the learning speed of the unknown dynamics. The selected weighting matrices are shown in Table 3. To establish a basis for comparison, experimental results for the nonlinear H1 controller, joint positions, velocities and torques, are shown in Figs. 4 and 5. Concerning the application of the adaptive nonlinear H1 control proposed in Section 3.1, the uncertain vector F is defined to obtain the linear parametrization of Fðxem Þ by a combination of dynamic parameters (link mass, moment of inertia, etc.) as

F ¼ ½I2 I3 m2 lc22 m3 lc23 m3 W22 m3 W2 lc3 T : Joint 2

The correspondent regression matrix X is computed as " # x11 x11 þ x12 x11 x11 þ x12 x11 x13 X¼ , 0 x11 þ x12 0 x11 þ x12 0 x23

Joint 3

0.1

with

0.05

1 x11 ¼ q€ d2 T11 T12 q_~ 2 , 1 x12 ¼ q€ d3 T11 T12 q_~ 3 ,

0

1 x21 ¼ q_ d2 T11 T12 q~ 2 ,

−0.05

1 x22 ¼ q_ d3 T11 T12 q~ 3 ,

x13 ¼ 2x11 cosðq3 Þ þ x12 cosðq3 Þx21 q_ 3 sinðq3 Þx22 ðq_ 2 þ q_ 3 Þsinðq3 Þ,

−0.1 0.5

1

1.5

2

2.5

3

x23 ¼ x11 cosðq3 Þ þ x21 q_ 2 sinðq3 Þ:

Time (s)

Considering the definition of F, it is interesting to note that the adaptive control law was effective in estimating the effect of

Fig. 7. Torques—adaptive nonlinear H1 control.

80

40

60

Base Joint 2 Joint 3 Reference

40 20 0 −20

Joint velocity (º/s)

0

Joint position (°)

Torque (N.m)

403

30 20 10 0 −10

−40

−20

−60

−30 0

0.5

1

1.5 Time (s)

2

2.5

3

0

0.5

1

1.5

2

2.5

Time (s)

Fig. 8. Joints and base positions and velocities—adaptive neural network nonlinear H1 control (1).

3

404

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

these parameter combinations when dealing with the trajectory tracking problem. In order to achieve the actual values of the uncertain parameters, it would be necessary to apply a persistently exciting reference signal, which is not the case in this paper. Experimental results for the adaptive nonlinear H1 control are shown in Figs. 6 and 7. For the nonlinear H1 controllers via neural network proposed in Section 3.2, let n¼2 be the number of joints of the space manipulator (active joints in DEM). Define Hðxe , FÞ ¼ ½H1 ðxe , F1 Þ H2 ðxe , F2 ÞT with p1 ¼p2 ¼ 7 neurons in the hidden layer, the bias vector b1 ¼ b2 ¼ ½3 2 1 0 1 2 3 and the weighting matrix for the first layer Wi1 ¼ Wi2 ¼ ½w1ij  ¼ ½w2ij  ¼ ½1 1 1 1 1 1 1 1 1 1: The uncertain vector F is defined as F ¼ ½FT1 FT2 T , with

FT1 ¼ ½f11 f12 f13 f14 f15 f16 f17 , FT2 ¼ ½f21 f22 f23 f24 f25 f26 f27 , T

T

and the matrix X ¼ ½x1 x2 T can be computed with

xT1

For the proposed adaptive fuzzy nonlinear H1 controllers, from Section 3.3, a set of fuzzy systems may be defined by 1

2

1 2 Hðxe , FÞ ¼ ½H1 ð½q_~ m q~ m , F1 Þ H2 ð½q_~ m q~ m , F2 ÞT ,

where H1(.,.) and H2(.,.) correspond to the estimate of the uncertain part of the dynamic behavior of joints 2 and 3, respectively. The fuzzy sets Aðx~ m Þ shown in Fig. 12 were defined by ANFIS off-line learning using a set of input/output data from the results presented in Taveira et al. (2006). They are applied to both H1(.,.) and H2(.,.) for the universe of discourse of velocity errors, u11 ¼ u21 ¼ q_~ m A U11 ¼ U12 ¼ U1 , and for the universe of discourse of position errors, u12 ¼ u22 ¼ q~ m A U21 ¼ U22 ¼ U2 . The number of linguistic variables in U1 and U2 are defined by the designer, r1 ¼ r2 ¼3. During the control loop these graphics are used to determine the grade of membership associated to xe, which defines the value of m in (38) and, consequently, the computation of X. A fuzzy rule base is defined with pk ¼r1r2 ¼9 rules. Thus, the vector of adjustable components F is defined as F ¼ ½FT1 FT2 T , with

FT1 ¼ ½f110 f111 f112    f190 f191 f192 ,

¼ ½x11 x12 x13 x14 x15 x16 x17 ,

FT2 ¼ ½f210 f211 f212    f290 f291 f292 ,

xT2 ¼ ½x21 x22 x23 x24 x25 x26 x27 ;

and the matrix X ¼ ½x1 x2 T is computed with

as in Eq. 23 Experimental results for the adaptive neural network nonlinear H1 controller are shown in Figs. 8 and 9. For the second approach, neural network plus nominal model, experimental results are shown in Figs. 10 and 11.

xT1 ¼ ½x110 x111 x112    x190 x191 x192 ,

T

T

xT2 ¼ ½x210 x211 x212    x290 x291 x292 ; 0.15

0.15 Joint 2

Joint 3

0.1

Torque (N.m)

0.1

0.05

0

−0.05

0.05

0

−0.05

−0.1

−0.1 0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

80 Base Joint 2 Joint 3 Reference

40 20 0 −20 −40 −65

Joint velocity (º/s)

40

60

30 20 10 0 −10 −20 −35

0.5

1

3

Fig. 11. Torques—adaptive neural network nonlinear H1 control (2).

Fig. 9. Torques—adaptive neural network nonlinear H1 control (1).

0

2.5

Time (s)

Time (s)

Joint position (º)

Torque (N.m)

Joint 2

Joint 3

1.5

2

Time (s)

2.5

3

0

0.5

1

1.5

2

2.5

3

Time (s)

Fig. 10. Joints and base positions and velocities—adaptive neural network nonlinear H1 control (2).

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

1

0.8

0.8

0.6

0.6 μ

μ

1

405

0.4

0.4

0.2

0.2

0

−20

−10 0 10 Velocity error (°/s)

0

20

−2

−1.5

−1 −0.5 0 Position error (°)

0.5

Fig. 12. Fuzzy sets A1 ðq_~ m Þ and A2 ðq~ m Þ.

80

40 Base Joint 2 Joint 3 Reference

40 20

Joint velocity (º/s)

Joint position (°)

60

0

−20

30 20 10 0 −10 −20

−40 −65

−35 0

0.5

1

1.5 2 Time (s)

2.5

3

3.5

0

0.5

1

1.5 2 2.5 Time (s)

3

3.5

Fig. 13. Joints and base positions and velocities—adaptive fuzzy nonlinear H1 control (1).

L2 ½x~ m  ¼



1 ðtr t0 Þ

Z

tr t0

Jx~ m ðtÞJ22 dt

1=2

and the sum of the applied torques E½tm  ¼

2 Z X i¼1

tr t0

jtmi ðtÞj dt ,

,

0.15 Joint 2 Joint 3

0.1 Torque (N.m)

as in Eq. 38 Experimental results for the adaptive fuzzy nonlinear H1 controller are shown in Figs. 13 and 14. For the second approach, fuzzy system plus nominal model, experimental results are shown in Figs. 15 and 16. The nonlinear H1 controller, the adaptive nonlinear H1 controller and both the adaptive neural network nonlinear H1 procedures are implemented online in these experiments. On the other hand, the two adaptive fuzzy nonlinear H1 approaches are also composed of an off-line procedure that generates the set of input membership functions used during the control action. The disturbances included are sufficiently strong to check the limits of the robustness of the controllers, and put in evidence the effects of the nonlinearities of the system. The graphical results illustrate that all the applied controllers reject disturbance efficiently and attenuate its effect in the trajectory tracking task. These results clearly demonstrate the robustness of the H1 criterion, specially if the graphics of joints positions are analyzed. Looking at velocity curves it can be inferred that the control approach based on adaptive techniques provide more robust signals than the approach based only on the nominal model. Considering that the same value of g was applied to all the proposed controllers, two performance indexes are used to compare the nonlinear H1 controllers applied: the L2 norm of the state vector

0.05 0 −0.05 −0.1 0

0.5

1

1.5 2 2.5 Time (s)

3

3.5

Fig. 14. Torques—adaptive fuzzy nonlinear H1 control (1).

where tr is the time spent for all the joints to achieve the desired position. The L2 norm of the trajectory tracking error is a performance index widely used in literature (Berghuis et al., 1995; Jaritz and Spong, 1996; Reyes and Kelly, 2001; Whitcomb et al., 1993). The analysis of the sum of the applied torques is important in the sense that it is directly related to energy consumption, which is a concern in space applications. Several experiments were conducted with each of the proposed control techniques applied to the UARM and the mean value of five results obtained for each controller is shown in Table 4. The quantitative analysis focused on Table 4 endorses the better tracking performance of the adaptive approaches over the nonlinear H1 controller based only on the mathematical model. Adaptive intelligent methods, neural network and fuzzybased approaches, proved to be more effective to the proposed tracking task than the adaptive H1 controller based on linear parametrization, in terms of L2 norm of x~ m . Also, the complexity

406

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

80 Joint position (º)

Base Joint 2 Joint 3 Reference

40 20 0 −20

Joint velocity (º/s)

40

60

30 20 10 0 −10 −20

−40 −65

−35 0

0.5

1

1.5 2 2.5 Time (s)

3

3.5

0

0.5

1

1.5 2 2.5 Time (s)

3

3.5

Fig. 15. Joints and base positions and velocities—adaptive fuzzy nonlinear H1 control (2).

0.15 Joint 2 Joint 3

Torque (N.m)

0.1

0.05

0

−0.05

−0.1 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

Fig. 16. Torques—adaptive fuzzy nonlinear H1 control (2).

Table 4 Performance indexes. Controller

L2 ½x~ m 

E½tm  (Nms)

Nonlinear H1 Adaptive H1 Adaptive Neural H1 (1) Adaptive Neural H1 (2) Adaptive Fuzzy H1 (1) Adaptive Fuzzy H1 (2)

0.0778 0.0379 0.0317 0.0292 0.0340 0.0318

0.1419 0.1244 0.1322 0.1406 0.1563 0.1788

of a parametrization scheme must be compared to the structure of the intelligent system depending on the application. It is noted that the use of the nominal model within the control law formulation leads to an increase of energy consumption. Therefore, a negotiation between error tolerance and energy consumption should be considered concerning the application requirements.

5. Conclusion In this paper, various solutions are established to the problem of tracking control with a guaranteed H1 performance for free-

floating space manipulator systems. The Dynamically Equivalent Manipulator concept is considered to model the free-floating space manipulator. Nonlinear H1 control techniques are developed according to the knowledge and availability of the parameter matrices for the controllers. An experimental tracking task is proposed for the free-floating space manipulator joints considering the existence of parametric uncertainties in the model, the application of torque disturbances of considerable magnitude and assuming unknown spacecraft dynamics. The adaptive nonlinear H1 controller based on the linear parametrization procedure demands a precise knowledge of the model structure. Thus, a different formulation of the problem was proposed to avoid the use of the unknown spacecraft dynamics in the control law. Intelligent methods of the adaptive procedure are performed in two different approaches. Once the neural network structure is defined, the implementation of this method, considering the online learning procedure, is simple and direct. Fuzzy system implementation demands more work to be configured. Despite that, ANFIS technique inserted in this method significantly reduces the need of human expertise to set up reasonable input membership functions. Note that the intelligent methods do not demand any orientation, velocity or acceleration value from the free-floating base, which is a very interesting result since values as the spacecraft velocity or acceleration are not easy to obtain. Results obtained by each controller were compared based on well-defined metrics and the intelligent approaches presented better performances. Moreover, the results achieved by the proposed controllers are very attractive for the intended application in space. The ability to attenuate the effect of disturbances is an important factor for these systems. In addition to the obvious benefits of this property of robustness, it may be emphasized that maintaining positive task execution performances even when subject to modeling uncertainties and external disturbances, the space manipulator can maintain its free-floating status. Therefore, an unplanned action in the spacecraft during the task is avoided, saving fuel and electrical power.

Acknowledgments The authors would like to thank the anonymous reviewers for their suggestions for improving the quality of the manuscript. This work was supported by Grants from the Fundac- a~ o de Amparo a Pesquisa do Estado de Sa~ o Paulo (FAPESP-03/12001-0 and 06/03951-2).

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

Appendix Dynamic modeling matrices for the three-link planar DEM used in Section 4. Let the auxiliary variables be t1 ¼ m2 W1 lc2 sinðq2 Þ, t2 ¼ m3 W1 W2 sinðq2 Þ, t3 ¼ m3 W1 lc3 sinðq2 þq3 Þ, t4 ¼ m3 W2 lc3 sinðq3 Þ: The inertia matrix is given by 2 3 M11 ðqÞ M12 ðqÞ M13 ðqÞ 6 M ðqÞ M ðqÞ M ðqÞ 7 M0 ðqÞ ¼ 4 21 22 23 5 M31 ðqÞ M32 ðqÞ M33 ðqÞ 2

2

M11 ðqÞ ¼ I1 þ I2 þ I3 þ m1 lc1 þ m2 ðW1 þ lc2 þ 2W1 lc2 cosðq2 ÞÞ 2 þ m3 ðW12 þ W22 þ lc3 þ2W1 W2 cosðq2 Þ þ 2W2 lc3 cosðq3 Þ

þ 2W1 lc3 cosðq2 þ q3 ÞÞ, 2

M12 ðqÞ ¼ I2 þ I3 þ m2 ðlc2 þ W1 lc2 cosðq2 ÞÞ 2 þ m3 ðW22 þ lc3 þ W1 W2 cosðq2 Þ þ 2W2 lc3 cosðq3 Þ

þ W1 lc3 cosðq2 þq3 ÞÞ, 2

M13 ðqÞ ¼ I3 þ m3 ðlc3 þ W2 lc3 cosðq3 Þ þW1 lc3 cosðq2 þ q3 ÞÞ, M21 ðqÞ ¼ M12 ðqÞ, 2

2

M22 ðqÞ ¼ I2 þ I3 þ m2 lc2 þ m3 ðW22 þ lc3 þ 2W2 lc3 cosðq3 ÞÞ, 2

M23 ðqÞ ¼ I3 þ m3 ðlc3 þ W2 lc3 cosðq3 ÞÞ, M31 ðqÞ ¼ M13 ðqÞ, M32 ðqÞ ¼ M23 ðqÞ, 2

M33 ðqÞ ¼ I3 þ m3 lc3 , and the matrix of Coriolis and centrifugal forces is written as 2 3 _ _ _ C11 ðq, qÞ C12 ðq, qÞ C13 ðq, qÞ 6 _ _ _ 7 C22 ðq, qÞ C23 ðq, qÞ _ ¼ 4 C21 ðq, qÞ C0 ðq, qÞ 5 _ C31 ðq, qÞ

_ C32 ðq, qÞ

_ C33 ðq, qÞ

_ ¼ 2ðt1 þt2 þ t3 Þq_ 2 þ 2ðt3 þt4 Þq_ 3 , C11 ðq, qÞ _ ¼ ðt1 þ t2 þ t3 Þq_ 2 þ 2ðt3 þ t4 Þq_ 3 , C12 ðq, qÞ _ ¼ ðt3 þ t4 Þq_ 3 , C13 ðq, qÞ _ ¼ ðt1 þt2 þ t3 Þq_ 1 þ 2t4 q_ 3 , C21 ðq, qÞ _ ¼ t4 q_ 3 , C22 ðq, qÞ _ ¼ t4 ðq_ 2 þ q_ 3 Þ, C23 ðq, qÞ _ ¼ ðt3 þt4 Þq_ 1 2t4 q_ 2 , C31 ðq, qÞ _ ¼ t4 q_ 2 , C32 ðq, qÞ _ ¼ 0, C33 ðq, qÞ where mi, Ii, Wi and lci are given in Table 2. References Berghuis, H., Roebbers, H., & Nijmeijer, H. (1995). Experimental comparison of parameter estimation methods in adaptive robot control. Automatica, 31(9), 1275–1285. doi:10.1016/0005-1098(95)00046-Y. Caccavale, F., & Siciliano, B. (2001). Quaternion-based kinematic control of redundant spacecraft/manipulator systems. In IEEE international conference

407

on robotics and automation (ICRA) (Vol. 1, pp. 435–440). Seoul, Korea, 21–26 May, doi:10.1109/ROBOT.2001.932589. Chang, Y. C. (2005). Intelligent robust control for uncertain nonlinear time-varying systems and its application to robotic systems. IEEE Transactions on Systems, Man and Cybernetics—Part B: Cybernetics, 35(6), 1108–1119. doi:10.1109/ TSMCB.2005.850149. Chang, Y. C., & Chen, B. S. (1997). A nonlinear adaptive H1 tracking control design in robotic systems via neural networks. IEEE Transactions on Control Systems Technology, 5(1), 13–29. doi:10.1109/87.553662. Chen, B. S., Chang, Y. C., & Lee, T. C. (1997). Adaptive control in robotic systems with H1 tracking performance. Automatica, 33(2), 227–234. doi:10.1016/ S0005-1098(96)00147-1. Chen, B. S., Lee, T. S., & Feng, J. H. (1994). A nonlinear H1 control design in robotic systems under parameter perturbation and external disturbance. International Journal of Control, 59(2), 439–461. doi:10.1080/00207179408923085. Chen, B. S., Wu, C. S., & Jan, Y. W. (2000). Adaptive fuzzy mixed H2 =H1 attitude control of spacecraft. IEEE Transactions on Aerospace and Electronic Systems, 36(4), 1343–1359. doi:10.1109/7.892682. Craig, J. (1988). Adaptive control of mechanical manipulators. Reading, MA: AddisonWesley. Dalsmo, M., & Egeland, O. (1997). State feedback H1 suboptimal control of a rigid spacecraft. IEEE Transactions on Automatic Control, 42(8), 1186–1191. doi:10.1109/9.618253. Dubowsky, S., & Papadopoulos, E. (1993). The kinematics dynamics and control of free-flying and free-floating space robotic systems. IEEE Transactions on Robotics and Automation, 9(5), 531–543. doi:10.1109/70.258046. Dubowsky, S., & Torres, M. A. (1991). Path planning for space manipulators to minimize spacecraft attitude disturbances. In IEEE international conference on robotics and automation (ICRA) (Vol. 3, pp. 2522–2528). Sacramento, USA, 9–11 April, doi:10.1109/ROBOT.1991.132005. Feng, B., Ma, G., Wen, Q., & Wang, C. (2008). Robust chattering-free sliding mode control of space robot in task space. Journal of Control Theory and Applications, 6(2), 146–152. doi:10.1007/s11768-008-6058-6. Gu, Y. L., & Xu, Y. (1993). A normal form augmentation approach to adaptive control of space robot systems. In IEEE international conference on robotics and automation (ICRA) (Vol. 2, pp. 731–737). Atlanta, USA, 2–6 May, doi:10.1109/ ROBOT.1993.291872. Guo, Y., & Chen, L. (2006). Robust control of dual-arm space robot system with two objects in joint space. In IEEE/RSJ international conference on intelligent robots and systems (pp. 5091–5095). Beijing, China, 9–15 October, doi:10.1109/IROS. 2006.282598. Guo, Y., & Chen, L. (2008). Adaptive neural network control of coordinated motion of dual-arm space robot system with uncertain parameters. In Seventh world congress on intelligent control and automation (WCICA) (pp. 4842–4846). Chongqing, China, 25–27 June, doi:10.1109/WCICA.2008.4594534. Hornik, K., Stinchcombe, M., & White, H. (1989). Multilayer feedforward networks are universal approximators. Neural Networks, 2(5), 359–366. doi:10.1016/ 0893-6080(89)90020-8. Huang, D., & Chen, L. (2008). Inverse kinematic control of free-floating space robot system based on a mutual mapping neural network. In Seventh world congress on intelligent control and automation (WCICA) (pp. 8666–8670). Chongqing, China, 25–27 June, doi:10.1109/WCICA.2008.4594293. Huang, P., & Xu, Y. (2006). Pso-based time-optimal trajectory planning for space robot with dynamic constraints. In IEEE international conference on robotics and biomimetics (ROBIO) (pp. 1402–1407). Kunming, China, 17–20 December, doi:10.1109/ROBIO.2006.340134. Jang, J. S. R. (1993). ANFIS—adaptive network-based fuzzy inference system. IEEE Transactions on Systems, Man and Cybernetics, 23(3), 665–685. doi:10.1109/ 21.256541. Jaritz, A., & Spong, M. W. (1996). An experimental comparison of robust control algorithms on a direct drive manipulator. IEEE Transactions on Control Systems Technology, 4(6), 627–640. doi:10.1109/87.541692. Johansson, R. (1990). Quadratic optimization of motion coordination and control. IEEE Transactions on Automatic Control, 35(11), 1197–1208. doi:10.1109/9.59805. Kang, W. (1995). Nonlinear H1 control and its application to rigid spacecraft. IEEE Transactions on Automatic Control, 40(7), 1281–1285. doi:10.1109/9.400476. Liang, B., Xu, Y., & Bergerman, M. (1997). Dynamically equivalent manipulator for space manipulator system: Part 1. In IEEE international conference on robotics and automation (ICRA) (Vol. 4, pp. 2765–2770). Albuquerque, USA, April, doi:10.1109/ROBOT.1997.606705. Liu, Z., Huang, P., Yan, J., & Liu, G. (2009). Multi-objective genetic algorithms for trajectory optimization of space manipulator. In Fourth IEEE conference on industrial electronics and applications (ICIEA) (pp. 2810–2815). Xi’an, China, 25–27 May, doi:10.1109/ICIEA.2009.5138722. Narendra, K. S., & Parthasarathy, K. (1990). Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, 1(1), 4–27. doi:10.1109/72.80202. Papadopoulos, E., & Dubowsky, S. (1991). On the nature of control algorithms for free-floating space manipulators. IEEE Transactions on Robotics and Automation, 7(6), 750–758. doi:10.1109/70.105384. Papadopoulos, E., Tortopidis, I., & Nanos, K. (2005). Smooth planning for freefloating space robots using polynomials. In IEEE international conference on robotics and automation (ICRA) (pp. 4272–4277). Barcelona, Spain, 18–22 April. Parlaktuna, O., & Ozkan, M. (2004). Adaptive control of free-floating space manipulators using dynamically equivalent manipulator model. Robotics and Autonomous Systems, 46(3), 185–193. doi:10.1016/j.robot.2003.11.007.

408

T.F.P.A.T. Pazelli et al. / Control Engineering Practice 19 (2011) 395–408

Pathak, P. M., Kumar, R. P., Mukherjee, A., & Dasgupta, A. (2008). A scheme for robust trajectory control of space robots. Simulation Modelling Practice and Theory, 16(9), 1337–1349. doi:10.1016/j.simpat.2008.06.011. Pazelli, T. F. P. A. T., Siqueira, A. A. G., & Terra, M. H., (2008). Mixed model based/ anfis adaptive robust controller with H1 criterion applied to free-floating space manipulators. In Ninth international symposium on artificial intelligence, robotics and automation in space (iSAIRAS). JPL - NASA, Los Angeles, USA, 25–29 February. Polycarpou, M. M. (1996). Stable adaptive neural control scheme for nonlinear systems. IEEE Transactions on Automatic Control, 41(3), 447–451. doi:10.1109/ 9.486648. Reyes, F., & Kelly, R. (2001). Experimental evaluation of model-based controllers on a direct-drive robot arm. Mechatronics, 11(3), 267–282. doi:10.1016/S09574158(00)00008-8. Sakawa, Y. (1999). Trajectory planning of a free-flying robot by using the optimal control. Optimal Control Applications and Methods, 20(5), 235–248. doi:10.1002/(SICI)1099-1514(199909/10)20:5235::AID-OCA6583.0.CO;2-I. Sanner, R. M., & Vance, E. E. (1995). Adaptive control of free-floating space robots using neural networks. In American control conference (ACC) (Vol. 4, pp. 2790–2794). Seattle, USA, 21–23 June. Siqueira, A. A. G., & Terra, M. H. (2004). Nonlinear and Markovian H1 controls of underactuated manipulators. IEEE Transactions on Control Systems Technology, 12(6), 811–826. doi:10.1109/TCST.2004.833626. Skullestad, A., & Gilbert, J. M. (2000). H1 control of a gravity gradient stabilized satellite. Control Engineering Practice, 8(9), 975–983. doi:10.1016/S09670661(00)00044-7. Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics, 15, 116–132.

Taveira, T. F. P. A., Siqueira, A. A. G., & Terra, M. H. (2006). Adaptive nonlinear H1 controllers applied to a free-floating space manipulator. In IEEE international conference on control applications (CCA) (pp. 1476–1481). Munich, Germany, October, doi:10.1109/CCA.2006.286094. Torres, M. A., & Dubowsky, S. (1992). Minimizing spacecraft attitude disturbances in space manipulators systems. AIAA Journal of Guidance, Control, and Dynamics, 15(4), 1010–1017. doi:10.2514/3.20936. Tortopidis, I., & Papadopoulos, E. (2006). Point-to-point planning: Methodologies for underactuated space robots. In IEEE international conference on robotics and automation (ICRA) (pp. 3861–3866). Orlando, USA, 15–19 May, doi:10.1109/ ROBOT.2006.1642293. Vafa, Z., & Dubowsky, S. (1990). The kinematics and dynamics of space manipulators: The virtual manipulator approach. International Journal of Robotics Research, 9(4), 3–21. doi:10.1177/027836499000900401. Whitcomb, L. L., Rizzi, A. A., & Kodistschek, D. E. (1993). Comparative experiments with a new adaptive controller for robot arms. IEEE Transactions on Robotics and Automation, 9(1), 59–70. doi:10.1109/70.210795. Wijngaard, W. (1996). An adaptive filter for low frequency encoder applications. Measurement, 18(1), 1–7. doi:10.1016/0263-2241(96)00036-X. Woerkom, P. L. M., Guelman, M., & Ehrenwald, L. (1996). Integrated adaptive control for space manipulators. Acta Astronautica, 38(3), 161–174. doi:10.1016/0094-5765(96)00067-7. Yang, C. D., & Sun, Y. P. (2002). Mixed H2 =H1 state-feedback design for microsatellite attitude control. Control Engineering Practice, 10(9), 951–970. doi:10.1016/ S0967-0661(02)00049-7. Zhongyi, C., Fuchun, S., & Jing, C. (2008). Disturbance observer-based robust control of free-floating space manipulators. IEEE Systems Journal, 2(1), 114–119. doi:10.1109/JSYST.2007.914871.