Intelligent Control and Automation, 2011, 2, 823 doi:10.4236/ica.2011.21002 Published Online Febru ar y 2011 (http://www.SciRP .o rg/journal/ica) Copyright © 2011 SciRes. ICA Unified Modeling Approach of Kinematics, Dynamics and Control of a FreeFlying Space Robot Interacting with a Target Satellite Murad Shibli Mechanical Engineering Department, College of Engineering, United Arab Emirates Unive rsity, AlAin , UAE Email: malshibli@uaeu.ac.ae desired Received November 28, 2010; revised December 12, 2010; accepted December 13, 201 0 Abstract In this paper a unified controloriented modeling approach is proposed to deal with the kinematics, linear and angular momentum, contact constraints and dynamics of a freeflying space robot interacting with a target satellite. This developed approach combines the dynamics of both systems in one structure along with holo nomic and nonholonomic constraints in a single framework. Furthermore, this modeling allows considering the generalized contact forces between the space robot endeff ecter and the target satellite as internal forces rather than external forces. As a result of this approach, linear and angular momentum will form holonomic and nonholonomic constraints, respectively. Meanwhile, restricting the motion of the space robot endeffector on the surface of the target satellite will impose geometric constraints. The proposed momentum of the combined system under consideration is a generalization of the momentum model of a freeflying space robot. Based on this unified model, three reduced models are developed. The first reduced dynamics can be considered as a generalization of a freeflying robot without contact with a target satellite. In this re duced model it is found that the Jacobian and inertia matrices can be considered as an extension of those of a freeflying space robot. Since control of the base attitude rather than its translation is preferred in certain cases, a second reduced model is obtained by eliminating the base linear motion dynamics. For the purpose of the controller development, a third reducedorder dynamical model is then obtained by finding a common solution of all constraints using the concept of orthogonal projection matrices. The objective of this approach is to design a controller to track motion trajectory while regulating the force interaction between the space robot and the target satellite. Many space missions can benefit from such a modeling system, for example, autonomous docking of satellites, rescuing satellites, and satellite servicing, where it is vital to limit the con tact force during the robotic operation. Moreover, Inverse dynamics and adaptive inverse dynamics control lers are designed to achieve the control objectives. Both controllers are found to be effective to meet the spe cifications and to overcome the unactuation of the target satellite. Finally, simulation is demonstrated by to verify the analy tica l result s. Keywords: FreeFlying Space Robot, Target Satellite, Servicing Flying Robot, Adaptive Control, Inverse Dynamic Control, Hubble Telescope 1. Introduction Freeflying space robots and freefloating space robots have been under intensive consideration to perform many space missions such as: inspection, maintenance, repair ing and servicing satellites in earth orbit. Particularly, servicing satellite equipped with robot arms can be em ployed for recovering the attitude, charging the exhaust ing batteries, attaching new thrusters, and replacing the failed parts l ike gyros, sola r panels o r antennas o f anoth er satellite. There are two major classes of space robots can be classified: 1) freeflyi ng sp ace rob ots a nd 2) freefloating space robots. The manipulator system of the first type is a system in which the reaction jets (thrusters) are kept active so as to control the position and attitude of the systems’ spacecraft. In opposition to the free flying robot, a freefloating space robo tic system is a system in whic h
M. SHIBLI Copyright © 2011 SciRes. ICA the spacecrafts’ reaction thrusters are shut down to con serve attitude c ontrol fuel. Comprehensive understanding of the kinematics and momentum of space robots and their interaction with a floating object is considered as a very essential part in designing an efficient multibody system with effective control techniques of contact forces and motion trajecto ries. Many techniques in dynamic modeling of space robots have been developed in [110]. Kinematics mo tion of a space robot system are developed based on the concept of a Virtual Manipulator (VM) [1014]. It as sumes imaginary mechanical links and it does not model the angular momentum, then the attitude motion of the base satellite has to be considered by other means. One body of the space robotic system is used as the reference fr a me wi t h a point on it to represent the tran sitional DOF of t he s ys tem [2 ,8,9]. A tree topology of open chain mul tibody system with the system center of mass as the translational DOF is proposed in [6,7]. Many techniques in dynamic modeling of space robots have been reviewed in [2,4,5]. NewtonEuler dynamic approach of multibody systems is proposed in [6,7]. This approach is characterized the use of a tree topology of open chain multibod y system with the syste m Center of Mass as the translational DOF. Barycenters are used efficiently to formulate the kinematics and dynamics of freefloating space robots. Another approach is called the direct approach and it uses one body of the system to be the reference frame with a point on it to represent the transitional DOF of the system [2,8 ,9]. This approach is simpler but results in coupled equations. A virtual mani pulator is proposed in [1014] and used to simplify the system dynamic s of sp ace robots. It decoupl es the s ystem Center of Mass transactional DOF. Freeflying space robots dedicated for maintenance or rescue operations are involved in contact tasks. Many studies on spacebased robotic systems have assumed zero external applied forces. Dynamics of space robots by using what is socalled the virtual manipulator (VM) is proposed in [1014]. Multibody systems approach based on NewtonEuler dynamic is proposed in [6,7]. Achievements in Space Robotics are presented in [15]. In this article three p arts are intro duced. I n the first part, the achievements of orbital robotics technology in the last decade are reviewed, highlighting the Engineering Test Satellite (ETSVII) and Orbital Express flight demon strations. In the second part, some of the selected topics of planetary robotics from the field robotics research point of view are described. Finally, technological chal lenges to asteroid robotics are discussed. In work [16] three dynamical models of a two link space robot are developed. One model treats the gravita tional field as constant over the volume of the robot and another model uses 0th order Taylor series expansions of a continuous gravitational field over the volume of the robot. A third model neglects the effects of gravity. The dynamics of a dualarm space robot system was syste matically studied, and a dynamic model based on KaneHuston’s method and screw theory was presented in [17]. The numerical example shows that acting mo ment of a composition unit of the robot can be solved for given value of motion parameters with the exploitation of the dynamic model, vice versa. A simulation system of a three layer structure based on ADAMS, MATLAB and VC++ is present in [18], which can simulate and analyze the kinematics and dynamics of space robot in the process of capturing and releasing space object. Veri fication results show that this system can well explain space robot’s dynamic and kinetic characteristic in cap turing and releasing task under the space circumstances. In research [19], the kinematics and dynamics of free floating coordinated space robotic system with closed kinematic constraints are developed. An approach to position and force control of freefloating coordinated space robots with closed kinematic constraints is pro posed for the first time. Unlike previous coordinated space robot control methods which are for open kine matic chains, the method presented here addresses the main difficult problem of control of closed kinematic chains. The controller consists of two parts, position controller and internal force controller, which regulate, respectively, the object position and internal forces be tween the object and endeffec tors. The inve rse kinemat ic control based on mutual mapping neural network of freefloating dualarm space robot system without the basepsilas control is discussed in [20]. With the geome trical relation and the linear, angular momentum conser vation of the system, the generalized Jacobian matrix is obtained. Based on the above result, a mutual mapping neural network control scheme employing Lyapunov functi ons is d esi gned to co ntr ol the end effectors to trac k the desired trajectory in workspace. The control scheme does not require the inverse of the Jacobian matrix. A planar dualarm space robot system is simulated to verify the proposed control scheme. In [21], the kinematics of the FFSR is introduced firstly. Then the null space ap proach is used to reparameterize the path: the direction and magnitude are decoupled and no direction error is introduced. And the Newton iterative method is adopted to find the optimal magnitude of the joint velocity. A planar FFSR with a 2 DOFs manipulator is selected to test the al gorithm and simula tion results illustrate that the path following is realized precisely. The genetic algo rithm with wavelet approximation is applied to nonho lonomic motion planning in [2225]. The problem of nonholonomic motion planning is formulated as an op
M. SHIBLI Copyright © 2011 SciRes. ICA timal control problem for a drift. The problem of position control of robotic manipula tor s b ot h nonr e dund a nt a nd r e dund a nt i n t he ta s k sp ac e is addressed in [26]. A computat ionall y simple class of task space regulators consisting of a transpose adaptive Jaco bian controller plus an adaptive term estimating genera lized gravity forces is proposed. The Lyapunov stability theor y is used to d er ive t he c ontr ol sc he me. I n [27 ] glo b al randomized jointspace path planning for articulated robots that are subjected to taskspace constraints is ex plored. This paper describes a representation of con strained motion for jointspace planners and develops two simple and efficient methods for constrained sam pling o f j o int co n fi gur a tio ns : tange nt space sa mpling (TS) and firstorder retraction (FR). In work [28], control moment gyroscopes (CMGs) are proposed as actuators for a spacecraftmounted robotic arm to reduce reaction forces and torques on the spacecraft base. With the es tablished kinematics and dynamics for a CMG robotic system, numerical simulations are performed for a gen eral CMG system with an added payload. In [29] the problem of dynamic coupling and control of a space ro bot with a free fl yin g ba se is discussed, which could be a spacecraft, space station, or satellite. The dynamics of the system systematically and demonstrate nonlinearity of parameterization of the dynamics structure is formu lated. The dynamic coupling of the robot and base sys tem is studeid, and propose a concept, i.e., coupling fac tor, to illustrate the motion and force depe nd e ncies. Dynamics and control of a flexible space robot cap turing a static tar get was pre sented in [30] . The dynamics model of the robot system is derived with Lagrangian formulation. The control method of flexible space during capturing target was discussed. Work [31] proposes an adaptive controller for a fully freefloating space robot with kinemat ic and d ynamic mod el uncertai nty. In a dap tive control design for the space robot, because of high dynamical coupling between an actively operated arm and a passively moving endpoint, two inherent difficul ties exist, such as nonlinear parameterization of the dy namic equation and both kinematic and dynamic para meter uncertainties in the coordinate mapping from Car tesian space to joint space. Research [32] addresses modeling, simulation and controls of a robotic servicing system for the hubble space telescope servicing missions. The simulation models of the robotic system include flexible body dynamics, control systems and geometric models of the contacting bodies. These models are in corporated into MDA’s simulation facilities, the multi body dynamics simulator “space station portable opera tions training s imulator (SPO TS)”. Most previous studies describing the dynamics of a space robo t neglect the coupled d ynamics with a floati ng environment or consider only abstract external forces/ moments or impulse forces. The target has its own iner tial and nonlinear forces/moments that significantly in fluence the ones of the space robot and cannot be ignored. Applying improper forces at the constraint surface may cause a severe damage to the target and/or to the space robot and its base satellite or cause the target to escape away. To accomplish a capture in practice is not instan taneous, because the endeffecter needs to keep moving and applying a force/moment on the surface of the target until the target is totally captured. Moreover, from tra jectory planning point of view, not all trajectories and displacements (velocities) are allowed due to the con servation of momentum and geometric constraints. In this work, a unified controloriented modeling approach is proposed to deal with the kinematics, constraints and dynamics of a freeflying space robot interacting with a target satellite. This model combines the dynamics of both systems together in one structure and handles all holonomic and nonholnomic constraints in a single framework. Moreover, this approach allows considering the generalized constraint forces between the space robot endeffecter and the target satellite as internal forces ra ther than external forces. Most of the adaptive control algorithms assume the absence of external forces acting on space robot. As it can be seen most s tudies ignored considering constraints imposed by linear momentum, angular momentum, and contact constraints all together. The kinematics, dynam ics, the uncertainty of parameters of a freeflying space robot and that of the target are considered separately. In this paper the uncertainty of the combined system as a whole is considered which gives more global results. In this paper a unified controloriented dynamics model is developed by unifying dynamics of the space robot and the target satellite together along with all ho lonomic and nonholonomic constraints. Many space missions can benefit from such a control system, for example, autonomous docking of satellites, rescuing sa tellites, and satellite servicing, where it is vital to limit the contact force during the robotic operation. It worthy to monition that the advantage of this approach is consi dering the contact forces between the space robot endeffector and the target as internal forces rather than external forces. In this paper, inverse baseddynamics and an adaptive inverse baseddynamics controllers are proposed to handle the overall combined coupled dy namics of the basedsatellite servicing robot and the tar get satellite all together with geometric and momentum constraints imposed on the system. A reducedorder dy namical model is obtained by finding a common solution of all constraints using the concept of orthogonal projec tion matrices. The proposed controller does not only
M. SHIBLI Copyright © 2011 SciRes. ICA show the capability to meet motion and contact forces desired specifications, but also to cope with the under actuation problem [33,34]. The paper is organized as follows: In Section 2, mod eling of kinematics, linear and angular momentum, and contact constraints are derived, and then a common solu tion for all co nstraints is proposed. In Section 3 , a n ove r all dyna mics model is develope d. In Section 4 an inverse dynamics controller is proposed. An adaptive inverse dynamics controller is presented In Section 5. Mean while, in Section 6, simulation results are demonstrated to verify the analytical results, and finally in 7 summary is concluded. 2. Kinematics and Momentum Modeling 2.1. Nomenclature All generalized coordinates are measured in the inertial frame unless ano ther frame is mentioned as follo ws : the mass of t he ith body : the inertia of the ith body : the robot joint variable vecto r : the position vector of the centroid of the base : the position vector o f the target satellite cen troid : the position vector of the ith joint : the relative position vector of the target satellite centroid with respect to the endeffecter (EE) : the linear velocity of the base : the base angular velocity vector : the 3 × 3 identit y matrix : the joint torque vector : forces and moments act on the centroid o f ba se sate llite . 2.2. Kinematics The purpose of this part is to model the kinematics of a freeflying space robotic manipulator in contact with a captured satellite as a whole. In this model the contact between the space robot and the target satellite is as sumed established and not escaped. Our combined system can be modeled as a multibody chain system composed of n + 2 rigid bodies. While the manipulator links are numbered from 1 to n, the base satellite (body 0) is denoted by b, in particular, and the body (the target satellite) by T. Moreover, This multibody system is connected by n + 1 joints, which are given numbers from 1 to n + 1. Where the endeffecter is represented as the joint as sho wn in Figure 1. We assume that all system bodies are rigid, the contact surfaces are frictionless and known. Also the effect of gravity gradient, solar radiation and aerodynamic forces are weak and neglected. It is assumed also that the base satellite is rea c tionwheel actuated. Referring to Figure 1, the position vector of the ith body centroid with respect to the inertial frame can be expressed as (1) where the relative vector is the position of the ith body centroid with respect to the base frame [35,36]. Upon differentiating both sides of (1) with respect to time, the relationship between the ith body velo c ity (2) where is the linear velocities of the ith body in base coordinates. Now in the case of any ith body of the ma nipulator, the velocity can be expressed in terms of the linear Jacobian matrix as (3) where () ()() 1 122 ,, ,,0, ,0 i Liii ii Jz RrzRrzRr =×− ×−×− (4) Figure 1. F reefl o ating space robot in contact with a target satellite.
M. SHIBLI Copyright © 2011 SciRes. ICA The endeffecter tip velocity is given by (5) Additionall y, the velocit y of the target satellite i n the reference frame can be obtained by der iving Equa tion (1) as T TbbT bLTTEET V VRJqRv ω =+Ω ×++×+ (6) Since the target satellite is not stationary, (6) shows the relative linear and angular velocities T , be tween the end effecter and the target satellite and meas ured in the base frame. Another relationship is needed between the ith body angular ve locity and joint angular velocity (7) where i ω is the angular velocities of the ith body in base coordinates and i ω in case of the manipulator is given by (8) where the angular Jacobian 12 ,, ,,0, ,0 i J zz z= (9) While in the case of the target satellite, the absolute angular velocity of can be expressed as T (10) Former analysis will be used in the next analysis to derive the mome ntum of s freeflying space robot. 2.3. Linear and Angu lar Momentu m The linear and angular momentum of a multibody sys tem is a key par t in under stand ing the moti on of the s ys tem when it is not subjected to external forces. They may impose kinematiclike constraints when the system is free of any external force. The linear momentum P and angular momentum L of the whole s ystem is give n b y (11) ( ) 1 0 nBi iiii i LImR V + = ≡Ω+ × ∑ (12) By means of (210), linear and angular momentum in (1112) can then be represented in a compact form as b bbb Vb b bb b TbT b TbT VV Vq b Tbq VVv T T v MM M V Pq MM LM MM v MM ω ω ω Ω Ω ΩΩ ΩΩ = + Ω + (13) where each block of the matrix is defined as follows ( 14) / 1 0, bb i b n Vi i ib MmR R +× Ω= ≠ ≡−×∈ ∑ (15) 13 0, bL i nn Vq i i ib MmJ R +× = ≠ ≡∈ ∑ (16) / 133 0, () b n ii b i ib MImD RIR +× Ω=≠ ≡ ++∈ ∑ (17) { } / 13 0, bA bL n qi ii i ib MIJmRJR +× Ω= ≠ ≡+ ×∈ ∑ (18) 1/ b TnEE M mRR ω + × ≡−× ∈ (19) / 1 () bT T EE b MmD RIR ω × ≡ +∈ (20) 31 bT Vv n MUm R× + (21) [ ] 1 33 1 bT n vn MmRR + × Ω+ ≡−× ∈ (22) Note that the matrix function for a vector is defined as [ ] 0 0 0 zy yx RR R RRR RR × − ×≡− ∈ − (23) and [ ][ ] 22 2 233 22 () T y zxyxz xyx zyz xzyzx y DR RR R RRRRR RRR RRRR RRRRRR × ≡× × +− − =−+− ∈ −− + (24) and the submatrices of the Jacobian of the ith body representing the linear and angular parts are defined be for e . Note that as in (13) the system is subjected to a non holonomic (nonintegrable) constraint because of con servation of angular momentum in the absence of exter nal forces. Note that the momentum constraints are not purely kinematical because of the inertial characteristics it carries in. Thus, this constraint is called kinematics like. The physical meaning behind these constraints is that they restrict the kinematically possible displace ments (possible velocities) of the individual parts of the system. On the contrast, the linear momentum results i n a holonomic (integrable) co nstraint. Now assuming zero initial conditions then linear and angular momentum is give by
M. SHIBLI Copyright © 2011 SciRes . ICA 0 0 b bbb Vb b bb b TbT b TbT VV Vq b Tbq VVv T T v MM M Vq MM M MM v MM ω ω ω Ω Ω ΩΩ ΩΩ = + Ω + (25) Then it is possible that the relative linear and angular velocities of the target satellite can calculated as 1 1 b bb b TbT b TbTbb b TbTb b TbTb VV V Vvb TTb Tv V VvVq vq MM MM V MM vMM MM M q MM M ω ω ω ω ω Ω − Ω Ω ΩΩ − ΩΩ Ω = − Ω − (26) Equations (26 ) enables us to calculate the targe t veloc ities v ω witho ut me as ur e me nt s. 2.4. Contact Constraints We assume that the endeffecter moves on a subsurface of the target satellite S and the profile of this surface is known so that it can be d efined as ( ) :,,, , ,.SFxy zcconst αβγ = = (27) Let be the vector of generalized coordinates of the robot endeffecter in the target frame. The endeffecter, as a result of the contact with the target, is subjected to holo no mic kinemati c c on st ra i nts de fi ned i n the c o nstr aint frame as (28) where is twice differe ntiable . T he robo t joint and target coordinates are related through the for ward kienmatic function χ = (29) Now differe ntiating (28) with r e sp e c t to time gives ( ) 0 χχ χ ∂Φ = ∂ (30) Also differentiating (29) with respect to time (31) Substituting (31) into (30) (chain rule) yields to (32) where the matrix is the Jacobian matr ix 2.5. Common Solution of the Constraints In this section we study the constraints on a space robot in contact with a target satellite in one form. This entire system is subjected to holonomic and nonholonomic constraints at the same time. These Holonomic constraint are usually given in algebraic form relating the genera lized variables (28). Now differentiating the holonomic constraint at the velocity level a s in (3032) leads to 0J θ θθ = (33) where is the Jacobian of the holonomic con straint as defi ned in (32). On the other hand, the conservation of momentum holds two types of constraints: linear momentum which is holnomic; and nonhol nomic co nstraints come into play as a result of the conservation of the angular momentum. These momentum constraints are not given in algebraic form, but there are given in kinematicallike form as in (13) and can be rewritten in a compact form as (34) where 0 c represents the vector of the initial conditions of the momentum. Equation (34) has k momentum con straint equations with , where N is number of generalized coordinates. The purpose of representing holonomic constraints in the form (33) is to treat both holonomic and nonholonmic constraints at the same dif ferential level. But a difference exists in the matter of initial conditio ns. Holono mic constraints are r estricted to position initial conditions, but nonholonomic are only restricted to their momentum conditions. Now, all holonomic and nonholonomic constraints can be combined together as ( ) 0 0 J c B θ θθ θ = (35) where the ne w combined matrix is of dimension . This implies a set of linear equations with θ as vector of the gene ralized variables. Since matrices and have the same number of columns, we now seek for their common solutions, if exist, expressing them in terms of the solutio ns of (33) and (34). T he common solutions of (33) and (34) are the solutions of the combined con straints (35). From the theory of linear algebra, the solu tions of Equations (33) and (34) constitute the intersec tion manifold (36) where and are the null space of θ and B, respectively. And the upper right script +
M. SHIBLI Copyright © 2011 SciRes . ICA represents the Pseudoinverse. Equation (36) is the set of solutions of (33) and (34) is consistent if (36) is non empty. The n the common solution [ 36] is the manifolds (a) ( )( ) ( ) ( ) ( ) ( ) 0 NB NJ NJ PPPBcNJN B θθ θ ++ + ++ (37) (b) ( ) ( ) ( ) ( ) ( ) 0 ()0 NB NB NJ Bc PPPBc NJ NB θ θ + ++ =−+ ++ (38) (c) ( ) ( ) ( ) 0 JJBBBcN JN B θθ θ + + ++ =++ (39) where is the projection matrix on the null space of a given matrix ()⋅. Since each of the manifolds given in (37)(39) give a solution of the combined system (35), these expressions can also be used to get the generalized (pseudoinverse) of the combined matrices. Each of the following expres sions i s a inverse of the combined matrix : (d) ( )( ) ( ) 0NB NJ NJ XJP PPJB θθ + + ++ = ++− (40) (e) ( ) ( ) ( ) ( ) 0NB NB NJ YBP PPJB θ θ + + ++ = −+− (41) (f) ( ) ZJJ BBJB θθ θ + + +++ = + (42) Moreover, if (43) then each expressions of (4042) is the MoorePenrose inverse of . 3. Generalized Dynamics Modeling To drive the dynamic equation of a space robot interact ing with a target, the total system kinetic energy as the total summation of the transitional and rotational energy of each body in the system can be expressed as ( ) 1 0 1 2 i nT Tb iii ii i TmV VI + = ≡+Ω Ω ∑ (44) where and is the transitional and rotational velocities of ith body , respectively, or it can be rear ranged [ ] 1 b bTT T V qv ω =Ω bbbbb TbT Vbbb TbT bb VqqT T bb VqT TT bT bTT V vvqvvT bT bTT TT VVVq VVvb Tqv b TT q qqv TT TT v T TTTT v MM MMMV MMMM M M MMMMq MM MMM v M MMMM ω ωω ω ω ω ω ωω ω Ω Ω Ω Ω Ω ΩΩΩ Ω Ω × (45) where the block matrix in (45) is the inertia matrix and the submatrices are defined previously in (1422) and also { } ( )() 111 0, ii nnn T BT qiLi LiAiA i ib MmJJIJ JR ++× + = ≠ ≡⋅+ ∈ ∑ (46) / 11 EE nn MIm DRR ω × ++ ≡+ ∈ (47) ( ) 31 11 TT EE n T TT qn ATn LT MIJmJ RR ω ×+ ++ ≡ +×∈ (48) ( ) 31 1 T n T qvnLT Mm JUR ×+ + ≡∈ (49) 33 1 TT vnT EE M mRR ω × + ≡−×∈ (50) (51) Note that the inertia matrix M defined in (45) is sym metric positive definite. Now define (52) Then the total kinetic energy can be expresses in a compact fro m (53) From the kinetic energy formulation, the dynamics equations can be derived by using the Lagrangian ap proach. Since there is no potential energy accounted in our system, the Lagrange function L is equal to the ki netic energy T then becomes (54) where is the vector of unknown Lagrangian multip liers. The holonomic constraints are behind the generalized constraint forces as a result of the contact between the manipulator endeffecter and the surface of the target satellite. The combined system dynamics model can be represented as (assuming the target satellite is unac tauted)
M. SHIBLI Copyright © 2011 SciRes . ICA 0 bbbbb TbT Vbbb TbT bb b VqqT T bb T VqT TT bT bTT T V vvqvvT bT bTT TT VVVq VVvV b Tqv b TT q qqvq TT Tw T v T TTTTv v MM MMMC V MMMM MC M MMMMC q C MM MMM vC M MMMM ω ωω ω ω ω ω ωω ω Ω Ω Ω Ω Ω ΩΩΩ ΩΩ Ω + 0 0 T T T bL bL T bA bA T T Tw T Tv J F J F J J J θ λ τ =+ (55) The dynamic developed in (55) along with the com bined constraints in (35) completes the overall modeling of a space ro bot interacting with a target satellite. 4. Inverse Dynamics Control The basic idea of inverse dynamics control is to seek a nonlinear dynamics control law that cancels exactly all nonlinear terms in the system dynamics (55) so that the closed loop dynamics is linear and decoupled [6]. Now as su ming zero initial conditions in (13) and (35), the overall dynamics subjected to the constraints can be expressed in a compact form as (56) where the inertia matrix M is defined in (55), the nonli near vector is the centrifugal/Coriolos forces the generalized constraint forc es are , is the vector of unknown Lagrangian multipliers, and are defined, respectively, as t t T bL T bA T Tq T Tw T Tv J J J J J J = , 0 0 L A b b F F ττ = as in (55), and finally the constraint matrix ( ) ( ) J AB θ θ θθ = as given in ( 35). In the constraint Equation (35) there are li near equations and N of the generalized velocities θ . It clear that there are fewer equations than unknowns, this implies the existence of infinite solutions. From the theory of linear algebra, the solution of (35) can be given by (57) where is an orthogonal projector of full rank and belong to the null space of and the vector can be chosen arbitrary. It implies that 0 T SA=. [37] Now differentiating (57) at the acceleration level with respect to time yields (58) Upon substituting the velocities (57) and the accelera tion (58) into the dynamics (55) we obtain ( ) c MSMS CSF ν ντ ++ =+ (59) Let us define the controller as ( ) ( ) T dD Pc HSCSHSKeK eJ τ ννλ =++++ − (60) where the position tracking error is defined as and c λ is defined as (61) where = − and the gain matrices P , , and are chosen as diagonal with positive ele ment s . Note that the input to the proposed controller (60) are the joint angles and velocities, angular velocity of the base, relative velocities of both satellites, contact forces and the output o f the controller is the joint torques. Note also that TNkm sR τ −− ∈ has the advantage of overcom ing the underactuation of the syste m as a res ult of target satellite jet shutdown or failure, and the inputs provided by the robot and the base is enough to control the whole system. This is because the number of constraints the number of passive inputs of the target satel lite. Now let us substitute the control law (60) into the dy namics (59), then, the closed loop dynamics is given by ( ) ( ) 0 TpDpPp TT FFI F SMSeKeKe SJKeKedt ++ =−+ = ∫ (62) Since J belongs to the null space of S, that is, 0 TT SJ = and since by the virt ue of (5 7), the p rojectio n matrix and its transpose are of full rank, and the inertia matrix is symmetric positive definite, then T SMS is also a positive definite. Now we need to verify the terms inside the brackets in (60) are zero. This condi tion can be guaranteed by choosing the proper positive gains D K and P K such that as . If the gain matrices and are chosen as diagonal with positive diagonal elements, then the resulted closed loop dynamics is linear, decoupled and exponentially stable. Global stability can then be guaranteed. The closed loop dynamics natural frequency and damping ration can be chosen to meet specific requirements. Also
M. SHIBLI Copyright © 2011 SciRes . ICA by inspecting the right hand side we can see that the can be guaranteed to be zero by choosing the suitable gain matrices and . Now we can readily summarize the hybrid inverse dynamics controller in the following theorem: Theor em 1 : Fo r the d ynamic s ystem given in (5 9 ) and subjected to constraints (35), the inverse dynamics con trol law defined by (60)(61) is globally stable and guar antees zero steady state and force tracking errors. To further improve the dynamic response in case of system parameters uncertainty, an adaptive controller would serve that objective as in the nest section. 5. Adaptive Inverse Dynamics Control Similar to the analysis followed in the previous section and recalling (60) ( ) T HSHS CSJ νντλ ++ =+ (63) Now we assume that there are some uncertainties in the syst e m p a ramet er s s uc h a s mas se s a nd i nertias a nd fo r this reason an adaptive control approach will be investigated. The dyna mics (63) can be represented b y benefiti ng fro m the prope rty of l inearity in parameters as [38,39] (64) where , , Y is an matrix of known functions and known is the regressor, and is an dimensional vector of the sys tem parameters. After examining the structure of dy namics (55), three properties are obtained: Property 1: The modified inertia matrix ( )()( ) 2 T HSH S = is symmetric po sitive definite. Property 2: If Proper ty 1 is verified, then is skewsymmetric matrix where . Property 3: The d ynamics (64) is linear in its parame ters. The nonlinear co ntrol l aw is proposed to have th e fo rm ( ) ˆ ˆT dD Pc HKeKeCJ τ ννλ =+++− (65) where c dFF λλ = − and is a positive definite diagonal matrix for the force control feedback gain and = −, and where and are the estimates of and 1 , respectively. Note that the geometric of the Jacobian in (56) is assumed to be determined. Then the dynamics (55) can be modified to (66) where is the estimated vector o f parameters . Upon substituting (65) into (63), and by adding and subtracting at the same time the term on the left hand side of (65) we get ()( ) 1 1 11 11 ˆˆ ˆ ˆ TT dD Pc HHHC HKeKeCSJ ν ν νν νν λλ −++ =+++−− (67) Rearranging (58) and canceling out the similar terms, yields ( ) 1 111 11 ˆ ˆ ˆˆ dDPPP HHCC HKe KeH Y ν ννν νν ν α − +− =−++− = (68) or, ( ) 11 1 ˆPDPPP H CHeKeKeY νν α +=+ += (69) where Pd e = − , and . The closed loop dynamics error can be written as ( ) 1PDPPP eKeKeY α ++ = (70) where . It is possible now to express the error dynamics (70) in a state space form as (71) where 1 , , p pPD eOI O xA B eKK Y = == −− (72) where A is a Hurwitz matrix, that is, the real parts of its eigenvalues are negative, which guarantees globally ex ponentially stability. Based on the state space formula tion and Lyapunov techniques an adaptive control law can be chosen as (73) where and s ym met ri c , a nd P is a un ique po sitive definite solution to the Lyapunov equation where Q is a positive definite symmetric. Proof: Let the Lyapo nuv candidate func tion chosen as (74) Now if take the time derivative of V along the trajec tories of (71) and by using the adaptation law (65), one gets ( )( ) ( ) ( ) 11 11 11 11 1 11 1 1 11 TT TT TT T TTT TT TTTTT TT TT TTTT T TTTT TTTTT VxPxxPx Ax BYPx xPAx BY YB PxYB Px xAPxYB PxxPAxx PBY x P BYYB Px xA PPAxYBPx x P BYx P BYY αααα αα αα αα αα α α αα −− −− =+ +Γ+Γ =+ ++ − ΓΓ−ΓΓ = +++ −ΓΓ−ΓΓ = ++ +−− TT B Px (75)
M. SHIBLI Copyright © 2011 SciRes . ICA By canceling out equivalent terms, this reduces to 0 T VxQx (76) Since is negative semidefinite with regard to x and the parameter error, and V is lower bounded by zero, V remains bounded in the time interval . This fact can be stated as 0Vdt ∞− <∞ ∫ (77) Now if we assume that is bounded then from (74) is bonded. If is bounded then is uniformly continuous. If is uniformly continuous and has a fi nite integral as given in (76) then by Barbalat’s lemma as which i mplie s as . Substi tuting the control law (65) and (73) into the d y namics (55) yields (78) where is a bounded function. T hus (79) and the force tracking error is bounded and can be adjusted by changing the feedback gain . Thus, the previous adaptive algorithm can be concluded in the following theor em: Theorem 2: For t he dyna mic s yst e m gi ven in (6 3 ) a nd subjected to constraints (35), the adaptive control law defined by (69) and (73) is globally stable and guarantees zero steady state and force tracking errors. 6. Simulation Results This section will demonstrate the kinematics, dynamics and controller presented in this paper as follows. Part A (PD controller): A 6DOF space robot arm mounted on a base satellite is used to demonstrate the analytical results. We assume that the endeffecter estab lished a co ntac t with a target satellite. This target satellite is assumed to be totally floating and unactuated due to the thrusters’ failure. The mass of the base servicing sa tellite is chosen as 30 0 kg, the masses o f the 6robot arm as [20 20 15 10 10 10] kg, and 1000 kg for the target satellite. T he initial linear velocities of both the base and the target are assumed to be 10 m/sec to keep a constant linear relative velocity while conducting the task and to avoid any damage. Two different PD controllers are used, to control the base satellite reaction wheels and robot arm as: ( ) ,rwPbb desDb kAA k τ =− +Ω , , respectively, where , , , . The simulation results are shown in Figures 26. Figures 2 and 3 shows a very slight varia tion in the base sate llite attitude and s mall increase i n its linea r velocity over 3 minutes. O n the contrast, Figures 5 and 6 show that t he target dr ifts away, b ut a slig ht rise i n its linear ve locity. The drift is due to the as sumption that the target satellite is unactuated (passive) and there is no regular control over its linear and angular motion. More complicated control techniques other than the PD con troller should be investigated to cope with the unactua tion problem. Finally, Figure 4 shows that most robot arm links approaches their desired values. Several simu lations are run and show that linear forces are preferable on angular forces, and as long as the forces are relatively small comparing to the target mass/inertia, its linear ve locities slig htly change. Part B (Inv erse Dynamics Controller): For simulation we assume that the endeffecter of the servicing space robot manipulator has established a contact with a target satellite. The robot arm is composed of 6DOF and mounted on base satellite is used to demonstrate the Figure 2. YRPangles of the base satellite (10 e3). Figure 3. Linear velocity of the base satellite.
M. SHIBLI Copyright © 2011 SciRes . ICA Figure 4. Robot arm angles. 0.1 Figure 5. YRPangles of the target satellite. Figure 6. Linear velocity of the target satellite. analytical results. This target satellite is assumed to be totally floating and unactuated due to the thrusters’ fail ure. The mass of the base servicing satellite is chosen as 300 kg, the masse s of t he 6robot arm as [10 10 10 10 10 10] kg, and 1500 kg for the target satellite. The initial linear velocities of both the base and the target are as sumed to be 20 m/sec to keep a constant linear relative velocity while conducting the task and to avoid any damage. All other initial conditions are assumed to be zero. The desired values for the robot angular position are chosen as [ ] 0.3,0.2,0.4,0.45,0.5,0.0 des q= . The contact forces are assumed to be linear and only in the xdirection. T he motio n and force gain diagonal matrices , , , are chosen as ( ) 30,30,30,30,30,30,30,30,30,30 P K diag= , ( ) 25,25,25,25,25,25,25,25,25,25 D K diag= , ( ) 50,50,50,50,50,50 I K diag= , 10,10,10,10,10,10 F K diag= The simulation is used to verify the analytical results and whether the proposed controller can track the desired motion and the specified contact forces and, moreover, overcome the underactuation (passivity) in the target satellite. The simulation results are shown in Figures 714. Figures 7 and 8 show, respectively, a fast response for both linear and angular velocities of the base. Fig ures 9 and 10 represent, respectively, the error response of robot arm angular position and velocities. The con troller is able to bring the links to the steady state posi tion at around 25 sec. Figure 11 shows the joints actua tors response which approaches zero after 25 sec. In Figures 12 and 13, the linear and angular velocities error response of the target satellite present a noticeable fast response. Finally, Figure 14 shows the error in the La grangian force multiplier. The error gets close to zero at time 20 sec. Base Satellite Angular velocity error, rad/s ec Figure 7. Base satellite Ang ul ar velocity err or.
M. SHIBLI Copyright © 2011 SciRes . ICA Base Satellite linear velocity error, m/sec Figure 8. Base satellite linear velocity error. Rob ot arm pos ition er r o r, rad/sec Figure 9. Spac e r obot arm angular position error. Robot arm velocity error, rad/sec 0.3 0.1 Figure 1 0. Space rob ot arm angular velocity error. Part C (Adaptive Inverse Dynamics Controller): For simulation we assume that the endeffecter of the servic ing space robot manipulator has established a contact Figure 1 1. Space robot arm actuation torque. Target Satellite Angular velocity error, rad/s ec Figure 1 2. Target satellite angular velocity error. Target Satell it e li ne ar vel oc it y er r or, m/sec Figure 1 3. Target satellite l inear velocity error. with a target satellite. The robot arm is composed of 6DOF and mounted on base satellite is used to demon strate the analytical results. This Hubble Telescope is
M. SHIBLI Copyright © 2011 SciRes . ICA assumed to be totally floating and unactuated due to the thrusters’ shutdown. The mass of the base servicing sa tellite is chosen as 3000 kg, the masses of the 6robot arm as [100 100 50 50 20 10] kg, and 11000 kg for the Hubble Telescope as shown in Table 1. The initial rela tive linear velocities of both the base and the target are as sumed to be zero m/sec to keep a constant linear relative veloci ty while conducting the task a nd to avoid any damage. All other i nitial conditio ns are assumed to be zero. The d e sired values for the robot angular position are chos en as [ ] 0.3 0.2 0.4 0.45 0.5 0.0 des q= . The contact forces are assumed to be linear and only in the ydirection and with desired value as 0 des λ = . The motion and force gain diagonal matrices , , are in the simulation as: ( ) 20,20,50,50,50,50,50,50,30,30 P K diag= ( ) 20,20,50,50,50,50,50,50,40,40 D K diag= ( ) 80,80,80,80,80,80 F K diag= . Figure 1 4. Lagrang ian error. Table 1. Simulated Combined system parameters. Link i Mass (kg) (kg.m2) (kg.m2) (kg.m2) Base Sat. 3000 1000 1000 1000 Hubble 11000 3000 3000 3000 Link 1 100 30 30 30 Link 2 100 30 30 30 Link 3 50 15 15 15 Link 4 50 15 15 15 Link 5 20 7 7 7 Link 6 10 3 3 3 We assumed that the space robot endeffector move on the surface of a Hubble telescope in the zdirection as sho wn in Figure 15. The simulation is used to verify the analytical results and whether the proposed controller can track the desired motion and the specified contact forces and, moreover, overcome the underactuation (passivity) in the target satellite. The simulation results are shown in Figures 1623. Figures 16 and 17 show, respectively, a fast re sponse for both linear and angular velocities of the base. Figures 18 and 19 represent, respectively, the error re sponse of robot arm angular position and velocities. The controller is able to bring the links to the steady state position at around 40 sec. Figure 20 shows the joints actuators response which approaches zero after 40 sec. In Figures 21 and 22, the linear and angular velocities error response of the target satellite present a noticeable fast response. Finally, Figure 23 shows the error in the La grangian force multiplier. The error gets close to zero at time 40 sec. Figure 15. A freeflying space robot conducting a main ten ance task on the surface of the Hubble Space Telescope. Base Satellite Angular velocity error, rad/sec Figure 1 6. Base satellite Angular velocity error.
M. SHIBLI Copyright © 2011 SciRes . ICA Base Satellite linear velocity error, m/sec Figure 1 7. Base satellite linear veloci ty error. Rob ot arm pos ition er r o r, rad/sec Figure 1 8. Space robot ar m a ngular position error. Robot arm velocity error, rad/sec Figure 1 9. Space robot arm angular velocity error. 7. Conclusions An ove rall controloriented modeling approach is devel oped to deal with the kinematics, constraints and dy namics of a freeflying space robot interacting with a target satellite. Treating kinematic constraints at the dif ferential level together with the constraints of linear and angular momentum, a common solution is proposed. Finally, based on the Lagrangian approach, a generalized dynamical model suitable for control algorithms is de veloped. This framework allows considering the genera lized constraint forces between the endeffecter and the target satellite as internal forces rather than external forces. Future work will focus on designing a controller using inverse dynamic and adap t ive/robust te chni ques. The hybrid inversedynamics based controller pro posed in this paper is capable to track the desired motion values and contact force specifications. The reduced order dynamics by using the orthogonal projector tech niques does not suffer of passivity or underactuation. Robot ar m position err or, rad/sec Figure 2 0. Space robot ar m a c tuation torq ue . Target Satellite Angular velocity error, rad/sec Figure 2 1. Hubble Telescop e angular velocity error.
M. SHIBLI Copyright © 2011 SciRes . ICA Target Sate ll it e li ne ar vel oc it y err or, m/sec Figure 2 2. Hubble Telescop e linear velocit y error. Figure 2 3. Lagrangian multiplier error. This controller deals with all geometric constraints and momentum constraints. Future work will focus on de signing a controller using adaptive control approach. An adaptive inversedynamics based controller proposed in this paper is cap able to track the desired motion values and contact force specifications and moreover, to over come the combined system parameters uncertainty. Moreover, the reducedorder dynamics by using the or thogonal projector techniques does not suffer of passivity or underactuation. The results of the approach proposed in this paper is advantageous comparing to the most studies which de scribe the dynamics of a space robot and neglect the coupled dynamics with a floating environment or they consider only abstract external forces/moments or im pulse forces. Most of the adaptive control algorithms assume the absence of external forces acting on space robot. As it can be seen most studies ignored considering constraints imposed by linear momentum, angular mo mentum, and contact constraints all together. This paper work introduces a unified controloriented modeling ap proach is proposed to deal with the kinematics, con straints and dynamics of a freeflying space robot inte racting with a target satellite along with parameters un certainty. 8. Acknowledgements The authors would like to acknowledge the Canadian Space Agency (CSA) for supporting this research. 9. Referen ces [1] S. Dubowsky, E. E. Vance and M. A. Torres, “Th e Con trol of Space Manipulators Subject to Spacecraft Attitude Control Saturation Limits,” Proceedi ng of NASA Confe rence on Space Telerobotics, Pasadena, 31 February2 March 1989, pp. 409418. [2] A. Ellery, “An Introduction to Space Robotics,” Springer, New York, 2000. [3] W. Fehse, “Automated Rendezvous and Docking of Spacecraft,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511543388 [4] P. C. Hughes, “Spacecraft Attitude Dynamics,” Wiley, New York, 1986. [5] P. W. Likins, “Analytical Dynamics and Nonrigid Space craft Simulation,” JPL Technology Report, 3215 93, July 1974. [6] W. W. Hooker and G. Margulies, “The Dynamical Atti tude Equations for an NBody Satellite,” Journal of As tronaunt Sciences, Vol. 12, No. 4, 1965, pp. 123128. [7] R. E. Roberson and J. Wittenburg, “A Dynamical Fo r malism for an Arbitrary Number of Rigid Bodies, with Reference to the Problem of Satellite Attitude Control,” Proceeding on International Federation of Automatic Control Congress, Londo n, 19 66, London, 1968. [8] J. Y. L. Ho, “Direct Path Method for Flexible Multibody Spacecraft Dynamics,” Journal of Spacecraft and Rock ects, Vol. 14, 19 97 , pp. 102110. doi:10.2514/3.571 67 [9] W. Hooker, “Equations of Motion for Interconnected Rigid and Elastic Bodies: A Derivation Independent of Angular Momentum,” Celestial Mechanics, Vol. 11, 1975, pp. 337 359. doi:10.1007/BF01228811 [10] Z. Vafa and S. Dubbowsky, “Minimization of Spacecraft Disturbances in Space Robotic Systems,” 11th Proceed ing on AAS Guidance and Control, Advances in the As tronautical Sciences, San Diego, Vol. 66, 1988, pp. 91108. [11] Z. Vafa and S. Dubbowsky, “On the Dyna mics o f Mani pulators in Space using the Virtual Manipulator Ap proach,” IEEE Proceding on International Conference Robotics Automation, Raleigh, March 1987, pp. 579585.
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