Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 5 pdf

67 the close neighborhood of the unloaded state 0. To form the equations of motion in this field, it is necessary to know the accumulated potential energy in the system of bodies. If the potential gravitational forces (they constantly act in the direction of the Oz axis of the absolute coordinate frame) in Lagrange’s equations are associ- ated with non-potential forces and are considered as a system of unknown external forces, then it remains only to determine the potential energy of the elastic forces (deformation energy). The purpose of the introduced assumptions and proposed modeling procedure is to avoid solving a system of equations that describes the deformation of the elastic system and, using approximate methods, derive the model of the coopera- tive system only on the basis of known absolute coordinates of the MCs and their derivati ves, along with gripping points at the initial moment, i.e. the contact points (tips of the manipulators) and the MC of the manipulated object. The idea of modeling in the system of absolute coordinates is based on the follo wing. As it is assumed that all the mass is concentrated at the elastic system nodes, inertial and e x ternal forces (represented by gravitational and contact forces) act at these nodes. The links between particular nodes are massless, so that the dissipation forces of the elastic system are also associated with the forces at other nodes. As we do not deal with manipulation in a resistive environment, there are no surface resistance forces. Hence, the forces acting at each node can be replaced with one resulting force. These resulting forces act at the nodes of the elastic system. To each deformed state corresponds only one system of node forces. The instantaneous deformed state can be obtained by static deformation of the unloaded state 0 involving the same system of forces, which enables one to calculate defor- mation ener gy by using static procedures. The work of the external forces is equal to the work of the internal forces, i.e. to the deformation energy. Components of the balancing elastic forces are equal to the deriv atives of deformation work with respect to the corresponding coordinate and are equal to the components of the resulting node forces. The resulting forces are decomposed along the axis of the absolute coordinate frame, so that deformation energy also has to be expressed in the same coordinate frame as a global frame for the elastic system. Deformation energy is a function of the properties of the concrete shape and elastic system material, and can be determined using exact or approximate meth- ods. When adopting assumptions needed to form the mathematical model, cooper- ative manipulation should be considered as a system with a finite number of DOFs and, hence, the deformation energy (i.e. the stiffness matrix) should be determined by some approximate methods. The basic notion of describing the deformation energy via the absolute coordi- nates will be illustrated using the finite-element method. The theoretical basis of the finite-element method is the principle of mini- Mathematical Models of Cooperative Systems mum energy for varying displacements (the principle of virtual displacements), whereby the increments of the works of the external and internal forces are the same [6, 7, 23, 24]. Generally, the method consists of decomposing the structure into characteristic elementary fi n ite elements, separately forming stiffness equa- tions for each of the finite elements in the local coordinate frame, and forming equations of global stiffness of the overall structure in the joint (global) coordinate frame for all the elements, whereby it is necessary to take into account the condi- tions of interconnection of the finite elements into a whole (the conditions of force equilibrium and compatibility of displacements). The procedures of forming equations of individual stif fnesses ha ve been de- scribed in detail in [6, 7, 23, 24]. The equation of individual stiffness of the ith finite element is of the form F ei = K ei  ei ,K ei = (K ei ) T , (47) where F ei is the vector of node forces acting on a finite element; K ei is the square matrix of individual stiffness of the finite element, and  ei is the v ector of node displacements of the finite element that defines the number of DOFs of the finite- element motion in the direction of the node force F ei . The number of DOFs of the finite-element motion depends on the choice of the type of load or displacement that is to be taken into consideration. For a given choice of displacements  ei , the matrix K ei for one finite element is determined only once. If the stiffness matrix also contains the motion modes of the finite element as a rigid body, then it is singular. If this equation is given in the local coordinate frame of the finite element attached to the element position in the elastic structure, an orthogonal transformation has to be applied to transpose it into the global coordinate frame. By uniting all the equations of the finite elements, one obtains the following system of equations F e = K e  e ,K e = (K e ) T , (48) with the disassembled stiffness matrix K e = diag(K e1 ,K e2 ,K e3 , ),andex- panded vectors of the force F e = column(F e1 ,F e2 ,F e3 , ) and displacement  e = column( e1 , e2 , e3 , ). If the equations of stiffness of each finite ele- ment are given in a global coordinate frame, the conditions of structure assembly are reduced to equating the forces and displacements of the finite elements at the common mode and eliminating redundant rows and columns from the disassem- bled stiffness matrix (method of direct stiffness [7]). On the contrary, one seeks the matrix a of the global kinematic conditions of the connection of node displace- ments (continuity) of the elastic structure  in the common (global) coordinate frame of the node displacements of finite elements  e which, for the statically de- termined systems (a = a 0 ), is represented by the algebraic relation (displacement 68 Multi-Arm Cooperating Robots 69 method)  e = a 0 . (49) This relation defines how the finite elements are assembled in the structure and it is easily obtained for the statically determined systems. Elements of the ma- trix a 0 are obtained by considering the geometry of the relation between the node displacements of finite elements  e and individual unit displacements in the direc- tion of each displacement  i as known, whereby all the other displacements  j , j = i are zero. If the system is statically indeterminate, then from the viewpoint of kinematics, the system is indeterminate too. Then, it is not possible to define the abov e relation on the basis of kinematic observations but is necessary to also take into account the equilibrium conditions from which kinematically indeterminate quantities are associated with the displacements . If the kinematically uncertain quantities are denoted by  k , the preceding relation will acquire the form  e = a 0  + a 1 ·  k , (50) where a 1 denotes the matrix by which the function describing how finite elements are connected in the structure is supplemented by kinematically indeterminate quantities  k . The forces F e act at the location and in the direction of the displace- ment  e . These forces must act on the overall structure because of the e xistence of node displacements of the ov erall structure ,andtheyare F e = K e  e = K e a 0  + K e a 1  k . (51) Using the principle of virtual displacements and considering the equations of vari- ation of the unknown and independent kinematically indeterminate quantities  k for the given constant displacements , one obtains a T 1 F e = a T 1 K e a 0  + a T 1 K e a 1  k = 0, (52) because the work of external forces is realized only on the given displacements . From this we have  k =−(a T 1 K e a 1 ) −1 a T 1 K e a 0 , (53) so that the relationship between the node displacements  e and predetermined dis- placements  for the statically indeterminate (kinematically indeterminate) system is  e = a, a = a 0 − (a T 1 K e a 1 ) −1 a T 1 K e a 0 . (54) Taking into account that the node forces F  acting in the direction of displacement  are related to the forces F e via the transformation matrix b (it can be determined by the force method) F  = bF e , (55) Mathematical Models of Cooperative Systems the total deformation energy in the deformed elastic structure generated from the moment of the beginning of deformation  e = 0 to the final deformation state, caused by the given displacements  and resulting node displacements  e ,isde- termined by A d = 1 2 (F e ) T  e = 1 2 (F e ) T a = 1 2  T a T K e a = 1 2  T a T K e a = 1 2  T K  , K  = a T K e a, (56) where K  is the stiffness matrix of the overall elastic structure defined in the global coordinate frame . The work of the node forces F  on the displacements  is determined by A F = 1 2 (F  ) T  = 1 2 (F e ) T b T . (57) As the works of the forces are the same, then a = b T and 1 2 (F  ) T  = 1 2  T a T K e a, ⇒ F  = a T K e a = K  . (58) If the work expressions are known, i.e. if the stiffness matrices and node displace- ments are determined, the forces F e along the displacement  e and the forces F  along the displacement  are F e = ∂A d ∂ e = K e  e ,F  = ∂A d ∂ = 2 ∂A F ∂ = a T K e a = K  . (59) The matrix K e is a constant diagonal block matrix. The matrix K  = a T K e a is also a constant matrix because the elastic system deformation is considered with respect to the immobile non-deformed state of the system. This means that if the stiffness matrix of the system is known, the forces along the given values of the displacement can be uniquely determined. Also, if the forces along the given dis- placements are known, the work of internal forces (as a scalar) is uniquely de- termined. As the strains of elastic structures are considered with respect to the immobile unloaded state, it is customary to measure the components of the vector  in the global coordinate frame attached to that state. It is also assumed that the states of the elastic structure before and after the deformation are kinematically determined (the positions of the supports and their displacements are known). The above discussion has been related to the modeling of an elastic system for the immobile unloaded state 0. Detailed standardized procedures of forming a stiffness matrix for concrete elastic systems can be found in the literature con- cerning the theory of elastic systems [6, 7]. If the displacements  are expressed 70 Multi-Arm Cooperating Robots 71 in external coordinates of the cooperative system, then the results of the theory of elasticity can be used without any alteration to model those phases of the co- operative system’s motion in which the unloaded state 0 of the elastic system is immobile. These phases are the gripping and releasing of the immobile object. In the case of the immobile unloaded state 0, it is necessary to find an expres- sion for the deformation energy in the system of absolute coordinates, w h ich are global coordinates for the elastic system. In cooperative manipulation, the distances of the nodes ρ ij 0 , i, j = 0, 1, ,m are kno wn, as well as the relative orientation of the bodies with the MC at the nodes A ij 0 , i, j = 0, 1, ,m of the elastic system before deforma- tion and absolute coordinates of the nodes after deformation Y . Let the relation between the given displacements  and these quantities be defined as  = (Y, ρ ij 0 | i,j =0, ,m , A ij 0 | i,j =0, ,m ) = (Y ), ρ ij 0 =const, A ij 0 = const,i,j= 0, ,m. (60) Deformation work of the elastic system determined in the coordinate frame  by the relation (56), in the ne w coordinate frame Y , will be A d = 1 2 (F  ) T  = 1 2  T (Y )K  (Y ) = A d ((Y )) = A d (Y ). (61) As the derivativ e of the scalar function A d with respect to the vector argument, Y is the v ector of the function  as an argument of the scalar function A d determined by ∂A d ((Y )) ∂Y =  ∂ ∂Y  T ∂A d ∂ , (62) the resulting node forces F Y along the displacement Y are F Y = ∂A d ∂Y =  ∂ ∂Y  T F  . (63) If the coordinates Y are expressed as a function of the coordinates  and transfor- mation matrix c(Y ) by the relation  = c(Y ) · Y ⇒  ∂ ∂Y  T =  ∂c(Y) ∂Y Y  T + c T (Y ), (64) the deformation work will be A d (Y ) = 1 2 Y T c T (Y )K  c(Y )Y = 1 2 Y T π(Y)Y, (65) Mathematical Models of Cooperative Systems where π(Y) = c T (Y )K  c(Y ). The force F Y in the direction of the displacement Y is obtained by introducing (64) and (59) into (63) or by differentiating (65) F Y =   ∂c(Y) ∂Y Y  T + c T (Y )  K  c(Y )Y = 1 2 Y T ∂π(Y) ∂Y K  c(Y )Y + c T (Y )K  c(Y )Y = 1 2 Y T ∂ ∂Y (c T (Y )K  c(Y ))Y + c T (Y )K  c(Y )Y = 1 2 Y T ∂π(Y) ∂Y Y + π(Y)Y (66) or, in a shorter form, F Y = ∂A d (Y ) ∂Y = K(Y)·Y, K(Y) =   ∂c(Y) ∂Y Y  T + c T (Y )  K  c(Y ), (67) where K(Y) is the generalized stiffness matrix, dependent of the generalized coor- dinates Y . Therefore, to form the expression for deformation work (65) or for the resulting node forces of the elastic system in the absolute coordinates (66), it is necessary to determine the relationship between the elastic displacements of the elastic system nodes and absolute coordinates (60), as well as the stiffness matrix of the assembled system K  . The stiffness matrix of assembled system K  is identical to the stiffness matrix of the elastic system considered with respect to the immobile unloaded state. This matrix is determined by the usual methods of the theory of elasticity. If the method of finite elements is used, it is necessary to divide the elastic system into character- istic finite elements, choose for each of them the local representative displacements  ei , determine individual stiffness matrices K ei , and finally, determine the relation  e = a for connecting the finite elements into a unique elastic system. The pro- cedure can be carried out only for a concrete kno wn structure of the elastic system. This problem in cooperative manipulation can be overcome if the elastic properties are assigned to the tips of the manipulator grippers only. Then, it is possible to choose in advance the suitable forms of the elastic tips of the grippers as finite ele- ments and determine the matrices of individual stiffness for them in advance. T he synthesis of the stiffness matrix of the composed system would reduce to forming 72 Multi-Arm Cooperating Robots 73 efficient on-line algorithms for connecting such finite elements and manipulated object into a unique whole, which can be the subject of future research. It is not simple to establish a relationship between the elastic displacements of elastic system nodes and absolute coordinates (60). If that relation is of the form (64), the deformation work and forces at the nodes will be of the form (65), (66), (67). It is necessary to describe the method of forming deformation work (65). The basic goal of the methods of the theory of elasticity is to establish a relationship between the kno wn load of the elastic system and unknown elastic displacements, or between the known elastic displacements and unknown acting load of the elastic system. After establishing these relations, the internal strain and support reactions are determined. To model the general motion of the cooperati ve system, it is necessary to ex- press deformation work and/or node forces as a function of absolute coordinates of the loaded elastic system in the form of (65), (66), (67). This can be done without finding the transformations (60) or (64). The basic idea in describing deformation energy with the aid of absolute coor- dinates is the following. On the basis of the known instantaneous positions of the nodes of a loaded elastic system and positions of the nodes at the moment of object gripping, instantaneous relative displacements of nodes are found. For the known v alues of instantaneous relati ve displacements of nodes, internal forces betw een them are determined. The deformation energy of the elastic system is determined as one-half of the sum of the products of internal forces and the corresponding relative displacements of the nodes. Namely, deformation work is the work of the internal forces (strains) and is a function of the relative displacements of nodes as known quantities (see relations (27), (28), (29) and (31)). A d = 1 2  T F int = 1 2  T π  , (68) where F int = F int (Y ) and  = (Y) are the vectors of internal forces and relative displacements of nodes (deformation) of the elastic system, and π  is the constant diagonal matrix in the direction of the action of the internal forces. Members of the matrix π  are uniquely calculated as a function of the stiffness matrix K ˙ and spatial characteristics of the unloaded elastic system. Characteristics of the elastic system depend on the type of contact, geometric configuration of con- tact points, and elastic properties of the object and tips of the manipulators. To each different elastic system corresponds a different stiffness matrix and, consequently, a different matrix π  . Mathematical Models of Cooperative Systems The procedure to calculate stiffness members of the matrix π  asafunctionof the members of the stiffness matrix K ˙ is the subject of the theory of elasticity. In deriving the model of cooperative system dynamics, it is essential that this rela- tionship is unique and that stiffness members of the matrix π  are constant for a concrete elastic system. To illustrate the modeling procedure the adopted members of the matrix π  (given in Appendix B) represent the stiffness of linear and torsion springs between any two nodes of the elastic system, without determining their v alues by the procedures of the theory of elasticity for the concrete elastic system. The relationship between the mutual displacements of elastic system nodes  and their absolute coordinates Y are relatively easily established (see (27)). As a result, one obtains the expression for the deformation energy whose mathematical form is identical to expression (65) (see (33), i.e. the same ef fect is achieved as in determining the transformation of coordinates (60). 4.4 Elastic System Deformations as a Function of Absolute Coordinates Let the elastic system be driven out of the state 0 and let the corresponding dis- placements of the nodes y i be given as (Figure 13) y i =  δ i A i  =  r ia − r i0 A ia − A i0  ∈ R 6×1 ,i= 0, 1, ,m, (69) where r ia and r i0 are the respective position vectors of the instantaneous MC and the MC in the state 0 of the ith body in the three-dimensional Cartesian space, while A ia and A i0 are the orientation vectors of the instantaneous state and state 0 of the ith body measured by the angular displacement of the coordinate frame with the origin at the MC and axes directed along the main inertia axes with respect to the Cartesian coordinate frame. The subscript i = 0 relates to the rigid object handled by the manipulators, whereas the subscripts i = 1, ,m refer to the elastic interconnection. According to (69), it is obvious that there are two different ways of deforming an elastic system: • deforming the elastic system around its immobile unloaded state r i0 = const A i0 = const ⇒ ˙r i0 = 0 ˙ A i0 = 0, (70) • deforming the elastic system around its mobile unloaded state r i0 = const A i0 = const ⇒ ˙r i0 = 0 ˙ A i0 = 0. (71) 74 Multi-Arm Cooperating Robots 75 The state 0 can be represented by only one coordinate frame which is adopted as the O 0 x 0 y 0 z 0 frame, with the origin attached to the manipulated object MC, whose orientation at the given moment is given by the v ector A 0 = A 0 0 = A 00 = col(ψ 00 θ 00 φ 00 ), and the position of the coordinate frame origin is r 00 . The other coordinate frames O i0 x i0 y i0 z i0 , whose origins are placed at the MCs of elastic interconnections, i.e. at contact points, are rotated with respect to O 0 x 0 y 0 z 0 by a constant value of orientation A 0 i during all the time of the motion of the unloaded state 0 (Figure 14). This means that the coordinate fram es O i0 x i0 y i0 z i0 for the unloaded state of the elastic system may have a different orientation A 0 0 , A 0 0 + A 0 1 , ,A 0 0 + A 0 m . According to (69) and Figures 13 and 14, the following v ector relations hold r ia = r i0 + δ i , ˙r ia =˙r i0 + ˙ δ i ,i= 0, 1, ,m, A ia = A i0 + A i , ˙ A ia = ˙ A i0 + ˙ A i ,i= 0, 1, ,m, (72) where r i0 , r ia and A i0 , A ia are the MC positions and orientations of the ith body in the unloaded and loaded states; δ i and A i are the MC displacements and change of orientation of the ith body in the case of deformation. Th e dot over a quantity denotes the time deri vati ve of that quantity. For the unloaded state we have r i0 = r 00 + ρ i0 , ˙r i0 =˙r 0 +˙ρ i0 ,i= 1, ,m, A i0 = A 0 0 + A 0 i = A 0 | A 0 i =0 = A 00 | A 0 i =0 , ˙ A i0 = ˙ A 0 0 + ˙ A 0 i = ˙ A 0 | A 0 i =0 = ˙ A 00 | A 0 i =0 , (73) where ρ i0 is the position vector of the MC of the ith object of the elastic system in the unloaded state 0, given with respect to the coordinate origin of the space O 0 x 0 y 0 z 0 . From this follow the relations for the coordinates and their deriv atives of the state 0: r 00 = r 0a − ρ 00 − δ 0 = r 1a − ρ 10 − δ 1 = = r ma − ρ m0 − δ m ,ρ 00 = 0, A 0 = A 0a − A 0 = = A ia − A 0 i − A i = = A ma − A 0 m − A m , ˙r 00 =˙r 0a − ˙ δ 0 =˙r 1a −˙ρ 10 − ˙ δ 1 = =˙r ma −˙ρ m0 − ˙ δ m , ˙ A 0 = ˙ A 0a − ˙ A 0 = = ˙ A ia − ˙ A 0 i − ˙ A i = = ˙ A ma − ˙ A 0 m − ˙ A m . (74) Mathematical Models of Cooperative Systems Figur e 14. Angular displacements of the elastic system Relativ e angular displacement of two arbitrary bodies is defined by the differ - ence of absolute values of their orientation (Figure 14). Let the bodies before elas- tic system deformation have the same orientation A 0 for A 0 i = 0, i = 1, ,m, or let their orientations differ by a constant value A 0 ij = A 0 i − A 0 j , = A i0 − A j0 , A 0 i = A 0 j . Starting from the state with the orientation A i0 , three (and four if A 0 ij = 0), successive changes of orientation yield the same state. Let the initial state have the orientation A i0 . By changing the orientation to A i = A ia − A i0 , the state with the absolute orientation A ia = A i0 + (A ia − A i0 ) is attained. By changing the ori- entation to −A ij a =−(A ia − A ja ), the attained orientation of the jth body is A ja = A i0 +(A ia − A i0 ) −(A ia − A ja ). After a further change of orientation to −A j =−(A ja − A j0 ) and A ij 0 = (A i0 − A j0 ) = (A 0 i − A 0 j ), the resulting states will be of the orientation A j0 = A i0 + (A ia − A i0 ) − (A ia − A ja ) − (A ja − A j0 ) and A fin = A i0 +(A ia −A i0 )−(A ia −A ja )−(A ja −A j0 )+(A i0 −A j0 ), respectively. Simple adding gives A fin = A i0 , i.e. we return to the state with the initial value of orientation. Hence, it comes out that the change of relative orientation of two arbi- trary bodies attained in the loaded state is defined by the difference of the absolute v alues of their orientations A ij = A i −A j = A ia −A i0 −(A ja −A j0 ) = A ia −A ja −(A i0 − A j0 ) = A ia − A ja −A ij 0 = A ia − A ja | A 0 i =A 0 j . To achieve a more legi- ble presentation, we assume that all coordinate frames O i0 x i0 y i0 z i0 have the same orientation for the unloaded state of the elastic system, A 0 i = 0, i = 1, ,m. Relativ e displacements of the points of a loaded elastic system are defined by 76 Multi-Arm Cooperating Robots [...]... = Aia − Aj a , ˙ Aij (78) By joining ( 75) and (78), we obtain the relation for displacements and displacement rates between the points of arbitrary MCs of the elastic interconnections and manipulated object (Figure 15) yij D R (δij + δij ) Aij = δij Aij = ria − rj a − ρij 0 Aia − Aj a = = ρij a − ρij 0 Aia − Aj a , (79) 78 Multi-Arm Cooperating Robots Figure 15 Displacements of the elastic system yij... 6×1 , ( 95) where Fe = col(Fe0 , Fe1 , , Fem ) = col(Fe0 , Fec ) ∈ R 6m+6 are the expanded vector of generalized forces (forces and moments); y = col(y0 , y1 , , ym ) = col(y0 , yc ) ∈ R 6m+6 is the expanded vector of displacements; K ∈ R (6m+6)×(6m+6) is the stiffness matrix; y0 and Fe0 are the vectors of displacements and forces at the manipulated object MC, whereas yc and Fec are the expanded... for relative displacements of nodes of the ‘linear’ R elastic system By its form, the expression ( 85) for the component yij differs only by the term col(ρij 0 , 03 ) 82 Multi-Arm Cooperating Robots 4 .5 Model of Elastic System Dynamics for the Immobile Unloaded State In this section, all the models of dynamics are derived using Lagrange’s equations ∂T ∂D ∂ d ∂T − − + = Qi , i = 0, 1, , m, dt ∂ gi... of dimensions of the particular bodies and overall elastic system are such that the continuity of the first derivative of elastic hypersurface is preserved, i.e one part of the smooth continual elastic hyper-surface (part of the elastic line of a linear body) can be replaced with a hyper-chord that is sufficiently close to the hyper-tangent to the elastic hyper-surface Displacements of the elastic system... are by assumption solid and rigid, that case is not possible For each passage of the elastic structure through the unloaded state D 0, the vector yij becomes zero In that case ρij 0 = ρij a and Aia = Aj a = A0 , from which (1 − ρij 0 / ρij a ) = 0 and Aia − Aj a = A0 − A0 = 0 80 Multi-Arm Cooperating Robots R R The component yij contains only the term δij By combining (82) and (79), we obtain ρij... respect to the state 0 form an elastic hyper-surface To each point of the elastic hyper-surface corresponds the deflection and slope angle of the hyper-tangent For the case of the introduced assumption, the elastic system deflection at the MC of a concrete object corresponds to the translation of the object MC, whereas the slope of the hyper-tangent to the elastic hyper-surface at the MC corresponds to the... description of the motions of elastic interconnections and manipulated object are separated ¨ ˙ Wc (yc )yc + wc (y, y) = Fc , ¨ ˙ W0 (y0 )y0 + w0 (y, y) = 0, (102) where the subscript c denotes the quantities related to the contact points and the subscript 0 stands for the quantities related to the manipulated object At that (see 86 Multi-Arm Cooperating Robots Appendix A) yc = col(y1 , y2 , , ym... (86) ˙R ˙T ˙ In the case of the colinearity of δij and ρij a , δij = ρij 0 × ω0 = 0 and (δij · ˙ ˙ ˙ ρij a )/ ρij a = δij and, as the unit vectors are identical, δij / δij = ρij a / ρij a , D ˙ ˙ ˙ ˙ then δij = δij = ria − rj a For small displacements of the elastic system, the vectors ρij a = ria − rj a and ρij 0 × ω0 are approximately normal (Figure 15) , so that (ρij 0 × ω0 )T · (ria − rj a ) ≈ 0,... [col(w1 (y, y), , wm (y, y))T ∈ R 6m, (103) where yc denotes the expanded vector of positions of contacts in a 6m-dimensional space and Fc is the expanded vector of contact forces associated with this vector It should be noticed that there is no contact force acting directly at the manipulated object MC, so that F0 = 0 Equations (100) and (102) represent the final form of the equations of motion of the... unloaded state 0 This means that the only unknown quantities are the instantaneous orientation A0 and angular velocity ω0 of the unloaded state 0 D The vector δij can be decomposed into the component δij that is colinear to R the vector ria − rj a = ρij a and the component δij non-colinear to this vector (FigD ure 15) The component δij reflects the linear change of the distance between the R nodes, whereas . joining ( 75) and (78), we obtain the relation for displacements and displace- ment rates between the points of arbitrary MCs of the elastic interconnections and manipulated object (Figure 15) . y ij =  δ ij A ij  =  (δ D ij +. finite ele- ments and determine the matrices of individual stiffness for them in advance. T he synthesis of the stiffness matrix of the composed system would reduce to forming 72 Multi-Arm Cooperating. ,m, (72) where r i0 , r ia and A i0 , A ia are the MC positions and orientations of the ith body in the unloaded and loaded states; δ i and A i are the MC displacements and change of orientation