Control Engineering Practice

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,

aDepartment of Electrical Engineering, University of Sao Paulo at S~ ao Carlos, Brazil~

bDepartment of Mechanical Engineering, University of S~ao Paulo at S~ao Carlos, Av. Trabalhador S~ao-carlense, 400, CEP 13566-590, S~ao Carlos, SP, Brazil

a r t i c l e i n f o

Article history:

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

Space robotics H1control Adaptive control Neural networks Fuzzy systems

a b s t r a c t

This paper aims to formulate and investigate the application of various nonlinearH1control 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 fixed- base planar manipulator which is, following the DEM concept, dynamically equivalent to a free-floating space manipulator.

&2011 Elsevier Ltd. All rights reserved.

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 representa- tive types of space robotic systems considering some of their specific planning and control problems: free-flying space manip- ulators 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 dis- turbances 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

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 free- floating 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 free- floating space manipulator. Online trajectory tracking is evaluated considering a predefined trajectory.

In virtue of the complexity of these systems and the environ- ment 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 differ- ent 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 environ- ment 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. There- fore, robustness is a significant challenge to the motion control of space manipulators. Two approaches to control uncertain systems Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/conengprac

Control Engineering Practice

0967-0661/$ - see front matter&2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conengprac.2010.12.011

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

E-mail address:siqueira@sc.usp.br (A.A.G. Siqueira).

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 success- fully applied in the literature to universally approximate mathema- tical 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) andPazelli 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 Nomenclature

SM parameters and coordinates

m_iu mass of linki Ciu mass center of linki Iiu inertia tensor of linki

Jiu joint connecting linkði1Þto linki Li vector connectingJiutoCiu

Ri vector connectingC_iutoJu_iþ1 Miu total mass

yiu rotation of linkiaround jointJu_i

ð

j

^u,y^u,cuÞEuler angles (Z-Y-Z) w.r.t. the spacecraft attitude qu generalized coordinates

DEM parameters and coordinates

mi mass of linki Ci mass center of linki Ii inertia tensor of linki

Ji joint connecting link (i1) to linki Wi vector connectingJitoJi+ 1

l_ci vector connectingJ_itoC_i Mt total mass

yi rotation of linkiaround jointJi

ð

j

,y,cÞ Euler angles (Z-Y-Z) w.r.t. the base attitude (J₁) q generalized coordinates

DEM model

M(q) symmetric positive definite inertia matrix Cðq,qÞ_ matrix of Coriolis and centrifugal forces

t

torque vector acting upon the joints of the DEM b index w.r.t. the passive spherical joint (base) m index w.r.t. the active joints (manipulator)

Problem formulation

q_m^d desired reference trajectory for manipulator joints x~m state tracking error

NonlinearH1Control

g

desired disturbance attenuation level u control variable

Q symmetric positive definite weighting matrix w.r.t.

x~m

R symmetric positive definite weighting matrix w.r.t.u

Adaptive nonlinearH1control

X matrix of known functions F vector with uncertain components

Z symmetric positive definite weighting matrix w.r.t.F_

Adaptive neural network nonlinearH1control

Hðxe,FÞ set ofnneural networks

Fk 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 G¼tanh activation function for the neurons in the

hidden layer

Adaptive fuzzy nonlinearH1control

Hðxe,FÞ set ofnfuzzy systems

Fk vector of adjustable parameters in the output func- tional consequences

u^ki fuzzified input variables Aij linguistic variables

yl output functional consequences

m

l d.o.f. ofl-th rule

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 phenom- enon 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.

InPathak 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 theH1criterion, has been widely used among robotic applications for guaranteeing good disturbance rejection properties. TheH1norm 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 space- crafts (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 inves- tigate the application of nonlinear H1 controllers to a free- floating 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 nonlinearH1 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 fromSiqueira 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. TheH1

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 consid- ered 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

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 quanti- tative description of the results is presented in the form of performance indexes which express the error of trajectory track- ing and the energy consumption.

In summary, the contribution of this paper consists of an experimental investigation on the motion control of a free- floating space manipulator subject to parametric uncertainties and external disturbances, performed by different methods of adaptive nonlinearH1controllers.

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 parame- trization 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 ann-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 experi- mentally 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 ton+1 andJiis the joint connecting the (i1)-th link andi-th link,yiis the rotation of thei-th link around jointJi, and the Z-Y-Z Euler anglesð

j

,y,cÞrepresent the SM base attitude and the DEM

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

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 theZ- axis by

j

; a rotation of the XuYuZu-system about theXu-axis byy, and a rotation of the X⁰⁰Y⁰⁰Z⁰⁰-system about the Z⁰⁰-axis byc.

LetCibe the center of mass of thei-th link,Libe the vector connectingJu_i andCu_i,R_ibe the vector connectingCu_iandJu_iþ1,l_ci be the vector connectingJiandCi, andWibe the vector connecting JiandJi+ 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

mi¼ M²_tmu_i Pi1

k¼1mu_kPi k¼1mu_k

,

I1¼Iu₁, Ii¼Iu_i,

W1¼R1

Mt

Xⁱ

k¼1

mu_k,

Wi¼ Ri

Mt

Xⁱ

k¼1

mu_kþ Li

Mt

Xⁱ¹

k¼1

mu_k,

lc1¼0,

lci¼ Li

Mt

Xⁱ¹

k¼1

mu_k, ð1Þ

wherei¼2,y,n+1 andMtis 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

MðqÞq€þCðq,qÞ_ q_¼

t

, ð2Þ

whereq¼ ½

j

y c y2 ynþ1^TAR^ðnþ3Þare generalized coordi- nates,MðqÞAR^{ðnþ3Þðnþ3Þ}is the symmetric positive definite iner- tia matrix, Cðq,qÞ_ AR^{ðnþ3Þðnþ3Þ} is the matrix of Coriolis and centrifugal forces, and

t

¼ ½0 0 0

t

2

t

nþ1^TAR^ðnþ3Þ 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

t

dand modeling uncertainties can be introduced dividing the parameter matrices M(q) and Cðq,qÞ_ into a nominal and a perturbed part:

MðqÞ ¼M0ðqÞ þDMðqÞ,

Cðq,qÞ ¼_ C0ðq,qÞ þ_ DCðq,qÞ,_

where M0(q) and C0ðq,qÞ_ are nominal matrices and DMðqÞ and DCðq,qÞ_ are the model uncertainties.

2.2. Problem formulation

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

passive joint dynamics intervene with the control of the manip- ulator active joints.

Letqbe partitioned asq¼ ½q^T_b q^T_m^T, where the indexesbandm represent the passive spherical joint (base) and the active joints (manipulator), respectively. Defined¼ ½d^T_b d^T_m^Tas a vector repre- senting the sum of parametric uncertainties of the system and

t

d¼ ½

t

^T_d

b

t

^T_dm^T as the finite energy external disturbance also introduced. Eq. (2) can be rewritten as

0 tm

" # þ db

dm

" # þ td_b

tdm

" #

¼ Mbb Mbm

M_mb Mmm

" # q€_b q€_m

" #

þ Cbb Cbm

C_mb Cmm

" # q_b

q__m

" # , ð3Þ whereMbbðqmÞAR³³,MbmðqmÞAR³ⁿ,MmbðqmÞARⁿ³,MmmðqmÞA Rⁿⁿ,Cbbðqm,qÞ_ AR³³,Cbmðqm,qÞ_ AR³ⁿ,Cmbðqm,qÞ_ ARⁿ³,Cmmðqm, q_mÞARⁿⁿ are nominal matrices and

t

mARⁿ. 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 symmetric positive definite and the matrixNmmðqm,q__mÞ ¼M_mmðqm, q__mÞ2Cmmðqm,q__mÞis skew-symmetric.

Remark 1. The controller is designed for the SM considering the DEM model.

Letq^d_mARⁿandq_^d_mARⁿbe the desired reference trajectory and the corresponding velocity for the controlled joints, respectively.

The state tracking error is defined as

x~m¼ q_mq_^d_m qmq^d_m

" #

¼ q_~m

q~m

" #

: ð4Þ

The variablesqmd

,q_^d_m, andq€^d_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

u¼ ½Mmm Cmm z_~1

z~1

" #

¼MmmT1x_~mþCmmT1x~m, ð5Þ

withz~1andT1being introduced by the transformation

z~¼ z~1

z~2

" #

¼T0x~m¼ T1

T2

" #

x~m¼ T11 T12

0 I

q_~m

q~m

" #

, ð6Þ

whereT11,T12ARⁿⁿare 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 nonlinearH1control 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_~m¼ATðqm,q_mÞ~xmþBTðqmÞT11ð

t

mFðxemÞEðxe_bÞ þdmþ

t

dmÞ, ð7Þ where

ATðqm,q_mÞ ¼T₀¹ M¹_mmCmm 0 T₁₁¹ T₁₁¹T12

" #

T0,

BTðqmÞ ¼T₀¹ M¹_mm 0

" # ,

Fðxe_mÞ ¼Mmmðq€^d_mT₁₁¹T12q_~_mÞ þCmmðq_^d_mT₁₁¹T12q~_mÞ,

and

EðxebÞ ¼Mmbq€_bþCmbq__b,

withxem¼ ½q^T_m q_^T_m ðq^d_mÞ^T ðq_^d_mÞ^T ðq€^d_mÞ^TTandxe_b¼ ½_q^T_b q€^T_b^T.

2.3. NonlinearH1control

Given a desired disturbance attenuation level of

g

40, the state feedback nonlinearH1 control method aims to attenuate the disturbance in the system through a control law of the form u¼Kð~xmÞx~min order to satisfy the performance criterion

uðÞminAL2

0amaxoðÞAL2

R1

0 1

2x~^T_mðtÞQx~mðtÞ þ¹₂u^TðtÞRuðtÞ

dt R1

0 1

2

o

^TðtÞ

o

ðtÞ

dt r

g

², ð8Þ

whereQandRare symmetric positive definite weighting matrices defined by the designer,x~mð0Þ ¼0 and

o

¼T11ðdmþ

t

dmÞrefers to the disturbance term in (7).

Following the game theory, a solution of thisminimaxproblem is given byChen et al. (1994), in a simplified form, in terms of the algebraic equation

0 K K 0

T^T₀B R¹1

g

^2I

B^TT0þQ¼0, ð9Þ

withB¼ ½I 0^T. This simplification in the synthesis of the non- linearH1controller is the main consequence of the transforma- tion (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 pro- blem (8) subject to (7) has an optimal solution

u¼ R¹B^TT0x~m, ð10Þ

if there are matricesK40 and a non-singularT₀solutions of (9), withRo

g

²I.

Considering that the matrixR1is obtained through a Cholesky decomposition of theR-dependent term in (9)

R^T₁R1¼ R¹1

g

^2I

1

, andQis factorized as

Q¼ Q₁^TQ1 Q12

Q₁₂^T Q₂^TQ2

" #

,

the solution of (9) is given by

T0¼ R^T₁Q1 R^T₁Q2

0 I

" #

and K¼1

2ðQ₁^TQ2Q₂^TQ1Þ1

2ðQ₂₁^T þQ12Þ:

Thus, from (7), the applied torques which guarantee the desired H1performance (8) can be computed by

t

_m¼T₁₁¹uþFðxemÞ þEðxe_bÞ: ð11Þ

Remark 2.This nonlinearH1 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.

3. Adaptive robust controller design

The adaptive control designs presented in the following apply different learning methods to estimate uncertain parameters and also the behavior of unmodeled dynamics. TheH1control law is applied to attenuate the effects of estimation errors and external disturbances.

Hence, given a desired level of attenuation

g

40 and matrices Q¼Q^T40,R¼R^T40,P0¼P^T₀40 and Z0¼Z₀^T40, the adaptive

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

Z T 0

ðx~^T_mQx~_mþu^TRuÞdtrx~^T_mð0ÞP₀x~_mð0Þ þF~^Tð0ÞZ₀F~ð0Þ þg² Z T

0

ðo^ToÞdt, ð12Þ whereF~ ¼FFdenotes the parameter estimation error.

3.1. Adaptive nonlinearH1control

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 determineM(q) andCðq,qÞ._

Assume the linear parametrization property of the term FðxemÞdmin (7):

FðxemÞdmHðxe,FÞ ¼XF, ð13Þ where xe is the input vector of Xðqm,q__m,q_^d_mT11T12q~_m,q€^d_m T₁₁¹T12q_~mÞ, being X a ðnpÞ regression matrix of known func- tions, andFis ap-dimensional vector with uncertain components depending on manipulator parameters.

Let the optimal approximation parameters vector be F¼arg min

FAOF

xmaxeAO_xeJHðxe,FÞðFðxemÞdmÞJ²,

whereJJ²is the Euclidean norm.

Define

t

¼

t

mEðxebÞ and the modified error Eq. (7) may be rewritten as

x_~m¼ATðqm,q__mÞ~xmþBTðqmÞT11ð

t

FðxemÞ þdmþ

t

dm

þHðxe,FÞHðxe,FÞÞ

¼ATðqm,q__mÞx~mþBTðqmÞT11ð

t

Hðxe,FÞÞ þBTðqmÞT11ðHðxe,FÞFðxe_mÞ þdmþ

t

dmÞ

¼ATðqm,q__mÞx~mþBTðqmÞuþBTðqmÞ

o

ð14Þ

with

u¼T11ð

t

Hðxe,FÞÞ, ð15Þ

o

¼T11ðHðxe,FÞFðxemÞ þdmþ

t

dmÞ, ð16Þ where

o

refers to the estimation error from the adaptive system and external disturbances. Using u¼u the control law provided by the nonlinearH1controller in (10),

t

can be computed by

t

¼Hðxe,FÞ þT¹₁₁u: ð17Þ

Considering the solvability analysis developed by Chen et al.

(1997), a dynamic state feedback controller given by

F_ ¼

a

ðt,x~mÞ ¼ Z¹X^TT11B^TT0x~m, ð18Þ

t

¼

t

ðt,F,x~mÞ ¼XFT₁₁¹R¹B^TT0x~m, ð19Þ with Z¼Z^T40, 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 uncer- tainties and disturbances – and substituting the result back on its second line, leads to

M¼MmmMmbM¹_bbMbm,

h¼ ðCmbMmbM¹_bbCbbÞq__bþ ðCmmMmbM¹_bbCbmÞq__m:

The relation between

t

and the torques applied upon the manipulator joints,

t

m, is given by

q€_m¼ M^¹_mmðC^mmq__m

t

Þ, ð20Þ

t

m¼M^q€_mþ^

h, ð21Þ

whereM^mm,C^mm,M^ andh^ are the matricesMmm,Cmm,Mandh computed with the estimated valuesF.

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 ðFðxemÞ þEðxe_bÞdmÞ in (7), the solution would also need the acceleration of the free-floating base q€_b which is not easy to obtain in practice.

3.2. Adaptive neural network nonlinearH1control

Define a set ofnneural networksH_kðxe,F_kÞ,k¼1,y,n, where xe is the input vector andFk are the adjustable weights in the output layers. The single-output neural networks are of the form

Hkðxe,FkÞ ¼ X^pk

i¼1

f_kiG X^qk

j¼1

w^k_ijx_ejþb^k_i 0

@

1

A¼x^T_kFk, ð22Þ

whereqkis the size of vectorxeandpkis the number of neurons in the hidden layer. The weightsw_ij^kand the biasesb_i^kfor 1rirp_k, 1rjrqkand 1rkrnare 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 ¼tanhð:Þ.Fig. 2illustrates the structure ofHk.

The complete neural network is denoted by

Hðxe,FÞ ¼

H1ðxe,F1Þ H2ðxe,F2Þ

^ Hnðxe,FnÞ 2

66 66 4

3 77 77 5¼

x^T₁ 0 . . . 0

0 x^T₂ ^{^} 0

^ ^ & ^

0 0 . . . x^T_n

2 66 66 64

3 77 77 75

F1

F2

^ Fn

2 66 64

3 77

75¼XF, ð23Þ

with

x^T_k¼ tanh X^qk

j¼1

w^k_1jx_ejþb^k₁ 0

@

1

A tanh X^qk

j¼1

w^k_p

kjx_ejþb^k_p

k

0

@

1 A 2

4

3 5,

Fk¼ ½f_k1 f_kpk^T:

Consider a first approach where the term FðxemÞ þEðxe_bÞdm

in (7) is completely unknown regarding its structure and para- meter values. The neural network defined in (23) is applied to learn the dynamic behavior of the robotic system:

FðxemÞ þEðxebÞdmHðxe,FÞ ¼XF, ð24Þ where the input vectorxeshould be defined as

xe¼ ½q^T_m q_^T_m q_^T_b q€^T_b ðq^d_mÞ^T ðq_^d_mÞ^T ðq€^d_mÞ^TT: ð25Þ Note that this definition ofxetakes into account the values of q__bandq€_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

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

xe¼_ xem¼ ½q^T_m q_^T_m ðq^d_mÞ^T ð_q^d_mÞ^T ðq€^d_mÞ^TT, ð26Þ 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 F¼arg min

FAO_Fmax

x_eAO_xeJHðxe,FÞðFðxemÞ þEðxe_bÞdmÞJ², the modified error Eq. (7) may be rewritten as

x_~m¼ATðqm,q__mÞ~xmþBTðqmÞuþBTðqmÞ

o

ð27Þ with

u¼T11ð

t

mHðxe,FÞÞ, ð28Þ

o

¼T11ðHðxe,FÞFðxemÞEðxe_bÞ þdmþ

t

d_mÞ, 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

F_ ¼

a

ðt,x~mÞ ¼ Z^TX^TT11B^TT0x~m, ð30Þ

t

m¼

t

mðt,F,x~mÞ ¼XFT₁₁¹R¹B^TT0x~m, ð31Þ withZ¼Z^T40, is solution to the adaptive neural network non- linearH1 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 termFðxemÞare 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):

Eðxe_bÞdmHðxe,FÞ ¼XF: ð32Þ Similarly, xe¼_ xem, the optimal approximation parameters vector is given by

F¼arg min FAO_Fmax

xeAO_xeJHðxe,FÞðEðxebÞdmÞJ²,

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

x_~m¼ATðqm,q_mÞ~xmþBTðqmÞuþBTðqmÞ

o

ð33Þ with

u¼T11ð

t

mFðxemÞHðxe,FÞÞ, ð34Þ

o

¼T11ðHðxe,FÞEðxe_bÞ þdmþ

t

dmÞ, 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 byChang and Chen (1997) and applied to (12). A dynamic state feedback controller chosen as F_ ¼

a

ðt,x~mÞ ¼ Z^TX^TT11B^TT0x~m, ð36Þ

t

m¼

t

mðt,F,x~mÞ ¼Fðxe_mÞ þXFT₁₁¹R¹B^TT0x~m, ð37Þ withZ¼Z^T40, is solution to the adaptive neural network non- linearH1control problem subject to the distinct formulation (33) and satisfies (12) for any initial condition.

3.3. Adaptive fuzzy nonlinearH1control

The Adaptive Network-based Fuzzy Inference System (ANFIS) defined byJang (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 G_kpk

x_e

W^k

G_k1

G_k2

G_k...

Φk

ξk

b₁^k b₂^k b...k

b_pk^k

kpk

k...

k2

k1

H_k(x_e, Φk)

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

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

IFu1isA11andu2isA12yanduqk isA1qk, THENy1¼f₁₀þf₁₁u1þf₁₂u2þ. . .þf_1qkuq_k,

^

IFu1isAp_k1andu2isAp_k2yanduqk isApkqk, THENyp_k¼f_pk0þf_pk1u1þf_pk2u2þ. . .þf_pkqkuq_k,

whereAij,i¼1,y,pkandj¼1,y,qk, are linguistic variables referred to fuzzy sets defined on the input spacesU1,U2,y,Uq_k;u1AU1, u2AU2,. . .,uq_kAUq_k are input variables values;p_kis the number of fuzzy rules andqkis 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 outputsylfrom each linear subsystem implied as

Hkðxe,FkÞ ¼ Ppk

l¼1

m

_lyl

Ppk

l¼1

m

_l

¼ Pp_k

l¼1

m

_lðf_l0þf_l1u1þf_l2u2þ. . .þf_lqkuq_kÞ Pp_k

l¼1

m

l

¼x^T_kF_k, ð38Þ Table 2

DEM parameters.

Body mi(kg) Ii(kgm²) Wi(m) lci(m)

Link 1 1.932 0.0153 0.203 0

Link 2 0.850 0.0075 0.203 0.096

Link 3 0.625 0.0060 0.203 0.077

Fig. 3.UnderActuated Robot Manipulator.

Table 3

Selected weighting matrices.

g¼2 R Q1 Q2 Z

NonlinearH1 0:9 0

0 2:5

0:64 0

0 0:64

9 0

0 7

–

AdaptiveH1 0:9 0

0 2:5

0:64 0

0 0:64

9 0

0 7

10 0

0 10

Adaptive NeuralH1(1) 0:9 0

0 2:5

0:64 0

0 0:64

9 0

0 7

10 0

0 10

Adaptive NeuralH1(2) 0:9 0

0 2:5

4 0

0 4

9 0

0 7

100 0

0 100

Adaptive FuzzyH1(1) 0:9 0

0 2:5

1 0

0 1

9 0

0 7

0:8 0

0 0:8

Adaptive FuzzyH1(2) 0:9 0

0 2:5

4 0

0 4

25 0

0 7

20 0

0 20

0 0.5 1 1.5 2 2.5 3 3.5

−60

−40

−20 0 20 40 60 80

Time (s)

Joint position (º)

Base Joint 2 Joint 3 Reference

0 0.5 1 1.5 2 2.5 3 3.5

−30

−20

−10 0 10 20 30 40 50

Time (s)

Joint velocity (º/s)

Fig. 4.Joints and base positions and velocities—nonlinearH1control.

Table 1 SM parameters.

Body mu_i(kg) Iu_i(kgm²) Ri(m) Li(m)

Base 4.816 0.0153 0.253 0

Link 2 0.618 0.0075 0.118 0.120

Link 3 0.566 0.0060 0.126 0.085

where 1rkrn and

m

l is the degree of freedom of l-th rule, defined as the minimum among the grade of membership asso- ciated to the entries in the activated fuzzy sets by thel-th rule,

m

l¼A_l1ðu1Þ4A_l2ðu2Þ4. . .4A_lq

kðuq_kÞ: ð39Þ

Thus, a set of fuzzy inference systems based on the T-S method is defined as

Hðxe,FÞ ¼

H1ðxe,F1Þ H2ðxe,F2Þ

^ Hnðxe,FnÞ 2

66 66 4

3 77 77 5¼

x^T₁ 0 . . . 0

0 x^T₂ ^{^} 0

^ ^ & ^

0 0 . . . x^T_n

2 66 66 64

3 77 77 75

F1

F2

^ Fn

2 66 64

3 77

75¼XF, ð40Þ

with

x^T_k¼ 1 Pp_k

l¼1

m

^k_l^½

m

^k1

m

^k1x^T_e

m

^k2

m

^k2x^T_e

m

^kp_k

m

^kp_kx^T_e, F_k¼ ½f^k₁₀ f^k₁₁ . . . f^k_1q

k f^k₂₀ f^k₂₁ . . . f^k_2q

k f^k_p

k0 f^k_p

k1 . . . f^k_p

kqk^T, wherexeAR^qk is the input vector,Xis anðnpÞmatrix depen- dent on the input values and on the values of fuzzy rules degrees of freedom and the vector F^AR^{nðpkðqkþ1ÞÞ} is a p-dimensional vector representing the adjustable parameters in the output functional consequences, withp ¼n(pk(qk+ 1)).

Consider a first approach where the term FðxemÞ þEðxe_bÞdm

in (7) is completely unknown regarding its structure and para- meter values. The T-S fuzzy system defined in (40) is applied to

learn the dynamic behavior of the robotic system:

FðxemÞ þEðxe_bÞdmHðxe,FÞ ¼XF: ð41Þ For the proposed application, the input vector is defined as xe¼x~m¼ ½_~q^T_m q~^T_m^T and Að~xmÞ ¼ ½A1ð_~q_mÞ A2ðq~_mÞ 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

F_ ¼

a

ðt,x~mÞ ¼ Z^TX^TT11B^TT0x~m, ð42Þ

t

m¼

t

mðt,F,x~mÞ ¼XFT₁₁¹R¹B^TT0x~m, ð43Þ with Z¼Z^T40, 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:

EðxebÞdmHðxe,FÞ ¼XF: ð44Þ As in (36)–(37), a dynamic state feedback controller given by F_ ¼

a

ðt,x~mÞ ¼ Z^TX^TT11B^TT0x~m, ð45Þ

t

m¼

t

mðt,F,x~mÞ ¼FðxemÞ þXFT₁₁¹R¹B^TT0x~m, ð46Þ with Z¼Z^T40, 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.

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 coordinatesqb,q__b and q€_b are not included among the entries ofFðxemÞ,Hðxe,FÞandu.

4. Results

For validation and comparison purposes, the proposed adap- tiveH1 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,qm¼ ½q2 q3^T are the joints to be controlled. The

0 0.5 1 1.5 2 2.5 3

−0.1

−0.05 0 0.05 0.1 0.15

Time (s)

Torque (N.m)

Joint 2 Joint 3

Fig. 5.Torques—nonlinearH1control.

0 0.5 1 1.5 2 2.5 3

−60

−40

−20 0 20 40 60 80

Time (s)

Joint position (º)

Base Joint 2 Joint 3 Reference

0 0.5 1 1.5 2 2.5 3

−30

−20

−10 0 10 20 30 40

Time (s)

Joint velocity (º/s)

Fig. 6.Joints and base positions and velocities—adaptive nonlinearH1control.