Thông tin tài liệu
87 where (see Appendix B) M is the inertial matrix and Y i = col(r ia , A ia ) ∈ R 6×1 ,Y= col(Y 0 , ,Y m ) ∈ R 6m+6 L va (Y ) = diag(L v0a , ,L vma ) ∈ R (6m+6)×(6m+6) , L via = diag(I 3×3 ,L ωia (A ia )) ∈ R 6×6 , v ia = col(˙r ia ,ω ia (A ia )) = L via (Y i ) ˙ Y i ∈ R 6×1 , ω ia = L ωia (A ia ) ˙ A ia ∈ R 3×1 , Wa(Y) = diag(W 0a , ,W ma ) = L T va (Y )ML va (Y ) ∈ R (6m+6)×(6m+6) . (105) The overall potential energy due to linear and rotational displacements of the body is defined by a = m i=0 m j=i+1 1 2 y T ij K ij a y ij = m i=0 m j=i+1 1 2 (Y i − Y j ) T π ij (Y i − Y j )| y ij =y D ij , (106) where (see Appendix B) det π ij = 0and π ij = π ji = ij (Y i ,Y j ))K ij a ij (Y i ,Y j )) (107) = diag(c x ij β, c y ij β, c z ij β, c ψ ij ,c ζ ij ,c ϕ ij ) ∈ R 6×6 ,β= 1 − ρ ij 0 r ia − r ja 2 . In combined form, we have 2 a = Y T π a (Y )Y, det π a = 0, rank π a = 6m, (108) where π a (Y ) is a symmetric matrix, π a (Y ) = π T a (Y ),definedby π a (Y ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ m k=0,k=0 π 0k −π 01 −π 02 ··· −π 0m −π 01 m k=0,k=1 π 1k −π 12 ··· −π 1m ··· ··· ··· ··· ··· −π 0m −π 1m −π 2m ··· m k=0,k=m π km ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ R (6m+6)×(6m+6) . (109) Mathematical Models of Cooperative Systems The overall dissipation ener gy exchanged in the course of linear and rotational displacements of the body is defi ned by D a =− m i=0 m j=i+1 1 2 ( ˙ Y i − ˙ Y j ) T D ij ( ˙ Y i − ˙ Y j ) =− 1 2 ˙ Y T D a (Y ) ˙ Y, (110) where D ij = D T ij = D ji = diag(G ij a (r ia ,r ja )D δ ij G ij a (r ia ,r ja ), D A ij ) ∈ R 6×6 ,D δ ij ,D A ij ) is the damping matrix of elastic interconnections between the ith and j th nodes, and, as D ij = D ji D a (Y ) = D T a (Y ) (111) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ n k=0,k=0 D 0k −D 01 −D 02 ··· −D 0m −D 01 n k=0,k=1 D 1k −D 12 ··· −D 1m ··· ··· ··· ··· ··· −D 0m −D 1m −D 2m ··· n k=0,k=m D km ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Generalized forces for the individual components Y j i of the vector Y i are Q j ia = m k=0 ∂Y k ∂Y j i (G k (m k g) + F ck ) = G j i (m i g) + F j ci ,i= 0, 1, ,m, j = 1, ,6. (112) By introducing (104), (108), (110) and (112) into the Lagrange equations (93) and after uniting all 6m + 6 equations, the general form of the model of the elastic system that under the action of the system of external contact forces F c , performs a macro motion (the mobile unloaded state 0) will be W a (Y ) ¨ Y + w a (Y, ˙ Y) = F, (113) where (see Appendix B) W a (Y ) = W T a (Y ) = diag(W 0a (Y 0 ), ,W ma (Y m )) ∈ R (6m+6)×(6m+6) , det W a (Y ) = 0, w a (Y, ˙ Y) = col(w 0a (Y, ˙ Y), ,w ma (Y, ˙ Y)) ∈ R 6m+6 . (114) 88 Multi-Arm Cooperating Robots 89 Of the 6m + 6 equations (113), only rank K equations are independent. Equation (113) can be presented so that the description of the motion of con- nections and manipulated object are separated W ca (Y c ) ¨ Y c + w ca (Y, ˙ Y) = F c , W 0a (Y 0 ) ¨ Y 0 + w 0a (Y, ˙ Y) = 0, (115) where the subscript c denotes the quantities related to the contact points, and the subscript 0 denotes the quantities related to the manipulated object. At that (see Appendix B), Y c = col(Y 1 , ,Y m ) ∈ R 6m ,Y 0 ∈ R 6×1 , F c = col(F 1 , ,F m ) ∈ R 6m ,F 0 = 0 ∈ R 6 , W ca (Y c ) = W T ca (Y c ) = diag(W 1a (Y 1 ), ,W ma (Y m )) ∈ R 6m×6m , detW ca (Y c ) = 0, w ca (Y, ˙ Y) = (w T 1a (Y, ˙ Y) w T ma (Y, ˙ Y)) T ∈ R 6m×1 , (116) where Y c denotes the expanded vector of contact position in the 6m-dimensional space and F c stands for the expanded vector of contact forces acting at the contact points. It should be noticed that no force is acting at the manipulated object MC, so that F 0 = 0. Equations (113) and (115) represent the final form of equations describing the behavior of the elastic system that under the action of the system of external forces F c , performs a general motion around the unloaded state 0, which also performs a general motion. 4.7 Properties of the Potential Energy and Elasticity Force of the Elastic System Denote by S y ∈ R (6m+6)×(6m+6) the coordinate frame whose unit vectors coincide with the unit vectors of the generalized coordinates Y = col(Y 0 , ,Y m ), Y i = col(r ia A ia ) ∈ R 6×1 ,i= 0, ,m of the manipulated object, MC position, and elastic interconnections. Assume that with respect to the state characterized by the absence of any dis- placement ( y ij = 0), a certain displacement of the nodes, defined by (79), takes place (i.e. y ij = 0) and, to this new position of elastic system nodes (henceforth, loaded state) let correspond the coordinates Y . Let this displacement be kept con- stant. It is necessary to determine the properties of the potential energy and force Mathematical Models of Cooperative Systems of the elastic system with the change of nodes coordinates in the adopted coordi- nate frame, i.e. at the translation and rotation of the loaded state without relative displacements of the nodes. We will briefly repeat the relations needed for the analysis. According to (108), for y R ij = 0, the potential (i.e. deformation) energy of the elastic system at an arbitrary point Y of the system S y is 2 a (Y ) = Y T π a (Y )Y ∈ R 1 , detπ a = 0, rank π a = 6m, (117) where π a (Y ) = π T a (Y ) is given by π a (Y ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ n k=0,k=0 π 0k −π 01 −π 02 ··· −π 0m −π 01 n k=0,k=1 π 1k −π 12 ··· −π 1m ··· ··· ··· ··· ··· −π 0m −π 1m −π 2m ··· n k=0,k=m π km ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ R 6(m+1)×6(m+1) , (118) since π ij = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ c ij (1 − ρ ij 0 r ia −r ja ) 2 00000 0 c ij (1 − ρ ij 0 r ia −r ja ) 2 0000 00c ij (1 − ρ ij 0 r ia −r ja ) 2 000 000c ψ ij 00 0000c θ ij 0 0000c ϕ ij ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = π ij (Y i ,Y j ) = π ji (Y j ,Y i ) = diag c ij (1 − ρ ij 0 r ia − r ja 2 I 3×3 ,c ψ ij ,c θ ij ,c ϕ ij ), det π ij = 0. (119) The potential energy derivative with respect to the coordinate defines the elasticity force decomposed along the coordinates Y of the system S y by F e (Y ) = ∂ a (Y ) ∂Y = 1 2 ∂ ∂Y (Y T ¯π a (Y )Y ) 90 Multi-Arm Cooperating Robots 91 = 1 2 ∂Y T ¯π a (Y )Y ∂Y + π a (Y )Y = K(Y) · Y ∈ R (6m+6)×1 . (120) An arbitrary component F ei of the assembled vector F e is defined by F ei (Y ) = ∂ a ∂Y i = 1 2 ∂ ∂Y i (Y T ¯π a (Y )Y ) = 1 2 ∂Y T ¯π a (Y )Y ∂Y i + π ia (Y )Y ∈ R 6×1 , (121) where π ia (Y ) ∈ R 6×(6m+6) are the submatrices composed of the rows starting from the (6i + 1)th to (6i + 6)th row i nclusive, of the matrix π a (Y ),and∂(Y T ¯π a Y)/∂Y i is the vector of the derivati ve of the quadratic form (scalar) Y T π a Y with respect to the vector Y i , whereby the macron denotes that partial derivation is carried out ov er the matrix π a . It should be noticed that the potential energy of the elastic system is equal to the sum of the internal forces works. When deriving the expression for poten- tial energy in the adopted generalized coordinates Y , a linear relationship between nodes displacements and elasticity force has been implicitly built in. Studies of the properties of potential energy and elastic system elasticity force in the loaded state motion will be reduced to the study of the behavior of the displacement vector of nodes of the elastic system connected to the mobile loaded state in the fix ed frame of adopted coordinates Y . 4.7.1 Properties of potential energy and elasticity force of the elastic system in the loaded state translation Let the elastic system be translated from the point Y by the vector η, defined by the expression η = ⎡ ⎣ η 0 ··· η m ⎤ ⎦ = ⎡ ⎣ ¯η ··· ¯η ⎤ ⎦ ∈ R (6m+6)×1 , ¯η = η i = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ¯η 1 ¯η 2 ¯η 3 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ˆη 0 ∈ R 6×1 ,i= 0, ,m. (122) Let us define the potential energy and elastic force at a point Y I = Y + η ∈ R (6m+6)×1 , Mathematical Models of Cooperative Systems Y Ii = r ia A ia + ˆη 0 = Y i + η i = Y i +¯η, i = 0, ,m. (123) The overall potential energy at that point is 2 a (Y I ) = 2 a (Y + η) = (Y + η) T · π a (Y + η) · (Y + η), (124) whence 2 a (Y I ) = Y T π a (Y + η)Y + Y T π a (Y + η)η + η T π a (Y + η)Y + η T π a (Y + η)η. (125) The product π a (#) · η is π a (#) · η = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎝ m k=0,k=0 π 0k − π 01 − − π 0m ⎞ ⎠ ¯η ··· ⎛ ⎝ m k=0,k=m π km − π 0m − − π (m−1)m ⎞ ⎠ ¯η ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎣ 0 ··· 0 ⎤ ⎦ = 0 ∈ R (6m+6)×1 , (126) so that the quadratic form is (##) · π a (#)η = 0, which gi ves 2 a (Y I ) = Y T π a (Y + η)Y. (127) The matrix π a (Y + η) is a function of the submatrices π ij : π a (Y + η) = π a (π 01 (Y 0 +¯η, Y I +¯η), , π ij (Y i +¯η, Y j +¯η), ,π (m−1)m (Y m−1 +¯η, Y m +¯η) . (128) Having in mind the expressions for π ij (Y i ,Y j ), Y Ii and Y Ij , t he stiffness m atrix π ij (Y Ii ,Y Ij ) between these nodes is π ij (Y Ii ,Y Ij ) = diag c ij 1 − ρ ij 0 r ia +ˆη − (r ja +ˆη) 2 I 3×3 ,c ψ ij ,c θ ij ,c ϕ ij = diag c ij 1 − ρ ij 0 r ia − r ja 2 I 3×3 ,c ψ ij ,c θ ij ,c ϕ ij = π ij (Y i ,Y j ) (129) 92 Multi-Arm Cooperating Robots 93 or, in a shorter form, π ij (Y i +¯η, Y j +¯η) = π ij (Y i ,Y j ), (130) so that π a (Y + η) = π a (π ij (Y i +¯η, Y j +¯η)) = π a (π ij (Y i ,Y j )) = π a (Y ), (131) whereas the overall potential energy is a (Y + η) = 1 2 Y T π a (Y + η)Y = 1 2 Y T π a (Y )Y = a (Y ). (132) Let us conclude that the o verall potential energy does not change in the course of the parallel displacement of the elastic system, and that this regularity in the system of selected coordinates is described by (132). The elasticity force F e (Y I ) at the point Y I , analogously to (120), is defined by the expression F e (Y I ) = ∂ a (Y I ) ∂Y I . (133) Since ∂ a (Y I ) ∂Y = ∂Y I ∂Y T T · ∂ a (Y I ) ∂Y I = ∂Y T I ∂Y · ∂ a (Y I ) ∂Y I , (134) then ∂ a (Y I ) ∂Y I = ∂Y T I ∂Y −1 · ∂ a (Y I ) ∂Y . (135) Substituting the last expression into (133), in view of (132) and (120), we obtain F e (Y I ) = ∂Y T I ∂Y −1 · ∂ a (Y ) ∂Y = ∂Y T I ∂Y −1 · F e (Y ). (136) From expression (123), for Y I we have ∂Y T I /∂Y = (∂Y T I /∂Y ) −1 = I 3×3 ,sothat we can finally conclude that F e (Y + η) = F e (Y ), (137) i.e. in the parallel displacement of the elastic system by the vector η defined by (122), elasticity force does not change. Mathematical Models of Cooperative Systems 4.7.2 Properties of potential energy and elasticity force of the elastic system during its rotation in the loaded state We seek the form of relations that hold in the system of adopted generalized co- ordinates Y during the rotation of the loaded elastic system without a change of relative distances of the nodes. It is known [31] that every orthogonal transformation of three-dimensional space coordinates µ = Rζ, R T R = RR T = I, R T = R −1 , (138) retains • the v ector module, • the angle between the vectors, and if also det R = 1, then all basic vectors in the coordinates transformation (138) preserve their mutual orientation (orientation of coordinate frames, vector product and mixed vector product) and such a transformation is called a characteristic ro- tation. For example, to describe rotation in terms of Euler angles, it is possible to have 12 systems of angles and six variants of the matrix R. For the case of a certain choice of the angles β 1 , β 2 , β 3 , the matrix R will be obtained as a product of three matrices that describe three successive rotations by the selected angle R = R(β 1 ,β 2 ,β 3 ) = R(β 1 ) · R(β 2 ) · R(β 3 ). Let the loaded state before the rotation be at the point Y of (6m + 6)-dimensional space. Let the orientation of each body w ith MC i , i = 0, 1, ,m,i.e. ofthe ov erall loaded state, change by the rotation a = (a ψ a θ a ϕ ) T ∈ R 3×1 . (139) The new orientation of an arbitrary body with the MC i will be A Iia = A ia + a, (140) b ut t he mu tual orientation of the bodies i and j will remain unchanged, A Iij = A Iia − A Ija = A ia + a − (A ja + a) = A ia − A ja = A ij , (141) i.e. in the orientation subsystem of six-dimensional space, the change of orientation by a constant vector a means the translation of the coordinates of this subsystem. 94 Multi-Arm Cooperating Robots 95 Figur e 1 7. Rotation of the loaded elastic system Let the loaded state after rotation by the orientation (139) be at the point Y I (Figure 17). Let P be the instantaneous pole of rotation. Since the loaded state moves a s a rigid body, the following relations will hold: r Ija = r ja + ν j ,j= 0, 1, ,m. (142) In view of (138) ρ Irj = ρ rj + ν j , ρ Irj = A(a)ρ rj , ⇒ ν j = ρ Irj − ρ rj = (A(a) − I 3×3 )ρ rj , (143) after the substitution, one gets r Ija = r ja + (A(a) − I 3×3 )ρ rj = r r + A(a)ρ rj (144) where r r is the vector of the instantaneous position of the rotation pole; ρ r is the position vector of the points of instantaneous rotation pole P whose positions are not known, and A(a) is the matrix of coordinates transformation in the rotation. Since ρ ij a = ρ ri − ρ rj = r ia − r ja , then from ρ Iija = r Iia − r Ija = r ia − r ja + (A(a) − I 3×3 )(ρ ri − ρ rj ) = A(a)ρ ij a , (145) Mathematical Models of Cooperative Systems one gets r Ija = r Iia − A(a)ρ ij a . (146) If the positions of all the nodes r Ija , j = 0, 1, ,m, are expressed as a func- tion of the positions r Iia of the point i and distance ρ ij a of these points from the point i (ρ iia = 0), then, in view of (140) and (141), we obtain Y i = r ia A ia , Y Ij = r Ija A Ija = r Iia − A(a)ρ ij a A ia − A ij + a = Y Ii − A(a)ρ ij a A ij . (147) For the case of the absence of rotation, a = 0, we have A(0) = I and A(a)ρ ij a = ρ ij a . From (147), we w ill have Y Ij = Y Ii − ¯ A r (a)Y ij (= Y Ii − Y ij | a=0 ), Y ij = ρ ij a A ij , ¯ A r = diag(A(a), I 3×3 ) ∈ R 6×6 . (148) Therefore, for the known absolute coordinates of a node Y Ii and vector of relati ve positions Y ij , in the absence of rotation and for the known value of the change of orientation a in the rotation space, it is possible to uniquely determine the coordi- nates of all the nodes of the mobile elastic system in which relative distances of the nodes do not change (fictitious rigid body). The relations (147) and (148) hold for an arbitrary position of the instantaneous rotation pole P and, on the basis of the requirement for, e.g., a manipulated object MC, they will allow the finding of the nominal motion conditions of the other nodes. If we consider pure rotation about the instantaneous rotation pole (which may also be a node) then, by placing the coordinate frame origin at the instantaneous rotation pole in view of r r = 0, we obtain Y j = ρ rj A j , (149) Y Ij = ρ Irj A Ij = A(a)ρ rj A j + a = ¯ A r (a)Y j + 0 a ,j= 0, 1, ,m. After coupling all m + 1 relations for pure rotation, the coordinates in the rotated state and deri vative with respect to the previous coordinates will be Y I = A r (a)Y + a r (a), ∂Y I ∂Y = A r (a), 96 Multi-Arm Cooperating Robots [...]... φ(z, z) = δτ , z ˙ where ⎡ 0 0 (z) = ⎣ N(q) (183) ⎤ W (Y0 ) φ(z, z) = ˙ ⎦ ∈ R (6m +6) ×(6m +6) , ˙ n(q, q, Y0 , Y0 ) ˙ ˙ w(q, q, Y0 , Y0 ) ˙ ∈ R 6m +6 , δτ = col(τ, 0, , 0) ∈ R 6m +6 , τ ∈ R 6m×1 (184) 1 06 Multi-Arm Cooperating Robots Figure 18 Block diagram of the cooperative system model Obviously, the two model forms, (177) and (183), are identical From this form of the model we get the essential characteristic... manipulated object MC Y0 ∈ R 6 1 and their derivatives, or the position vector of the MCs of elastic interconections and of manipulated object Y ∈ R 6m +6 and their derivatives, or the internal coordinates vector of the leader qv ∈ R 6 1 and vector of positions of the MCs of the remaining elastic interconnections Ys ∈ R (6m 6) ×1 and the MC of the manipulated object Y0 ∈ R 6 1 and their derivatives The... coordinates vector q ∈ R 6m×1 and position vector of the manipulated object MC y0 ∈ R 6 1 and their derivatives, or the position vector of the MCs of elastic interconnections yc ∈ R 6m×1 and of manipulated object y0 ∈ R 6 1 and their derivatives, or the vector of the leader’s internal coordinates qv ∈ R 6 1 and position vector of the MCs of the remaining elastic interconnections ys ∈ R (6m 6) ×1 and their derivatives,... that the positions of the contact points and the forces there are determined according to (148) and ( 163 ) 100 Multi-Arm Cooperating Robots In addition to the source vector Y , with the structure ⎤ ⎡ ⎤ ⎡ Y0 Y0 ⎢ Y1 ⎥ ⎥ ⎣ ⎦ = Y0 ∈ R (6m +6) ×1, Y = ⎢ ⎣ · · · ⎦ = Yv Yc Ys Ym ⎤ ⎡ Y2 Ys = ⎣ · · · ⎦ ∈ R (6m 6) ×1, Ym ⎡ ⎤ Y1 Yv ∈ R 6m×1 , Yc = ⎣ · · · ⎦ = Ys Ym Yv = Y1 ∈ R 6 1 , use is also made of the vector with... , 0) ∈ R 6m +6 , τ ∈ R 6m×1 (178) 104 Multi-Arm Cooperating Robots 4.11 Model of Cooperative System Dynamics for the Mobile Unloaded State Equations (113) or (115), ( 167 ) and (172) define the model of cooperative work of m rigid manipulators with six DOFs handling a rigid object whose general motion in three-dimensional space is unconstrained, whereby the connections between the object and manipulators... blockdiag(J1 , , Jm ) ∈ R 6m×6m , fc = col(fc1 , , fcm ) = col(−F1 , , −Fm ) ∈ R 6m×1 , q = col(q1 , , qm ) ∈ R 6m×1 , ˙ q = col(q1 , , qm ) ∈ R 6m×1 ˙ ˙ ( 168 ) The vector equation ( 167 ) determines 6m connections 4.9 Kinematic Relations By assumption, the contact of the manipulator and object is stiff and rigid The contact position is determined by the position and orientation of the manipulator... internal coordinates, is given by [32– 36] ¨ ˙ Hi (qi )qi + hi (qi , qi ) = τi + JiT fci , i = 1, , m, ( 166 ) where Hi (qi ) ∈ R 6 6 is a positively determined inertia matrix of the ith manipu˙ lator; hi (qi , qi ) ∈ R 6 1 is the vector taking into account the effect of gravitation, Coriolis acceleration, and friction; τi ∈ R 6 1 is the vector of joint drives; Ji ∈ R 6 6 is the transformation matrix of... (6m +6) ×1 ⎢ ⎥ Y0 ⎣ Ym ⎦ Y0 Y0 ( 164 ) ( 165 ) For both vectors, the transformation matrix Ar (a) and the vectors ar (a) and η from (150) and (122) are the same, so that all the previous conclusions, derived for the source vector, also hold for the vector with a transformed structure 4.8 Model of Manipulator Dynamics The model of motion of a non-elastic manipulator with six DOFs with noncompliant joints and. .. internal and absolute coordinates Y of contact points is expressed as Yi = i (qi ) ∈ R 6 1 , i = 1, , m, ( 169 ) then the relation between their velocities and accelerations will be [32–34] ∂ ˙ Yi = i (qi ) ∂qi · qi = Ji (qi )qi ∈ R 6 1 , i = 1, , m, ˙ ˙ ¨ ˙ ¨ Yi = J˙i (qi )qi + Ji (qi )qi ∈ R 6 1 , i = 1, , m, (170) (171) 102 or in united form Multi-Arm Cooperating Robots Yc = (q) ∈ R 6m×1 ,... connections between the object and manipulators are elastic and the motion takes place around the immobile unloaded state 0 The number of inputs into the model is 6m, whereas the number of independent state quantities (positions and velocities) is 2 · (6m + 6) , of which 2 · rank K are dictated by the elastic system and 2 · (6m + 6 − rank K) are dictated by the leader’s dynamics State quantities can . diag(W 0a (Y 0 ), ,W ma (Y m )) ∈ R (6m +6) ×(6m +6) , det W a (Y ) = 0, w a (Y, ˙ Y) = col(w 0a (Y, ˙ Y), ,w ma (Y, ˙ Y)) ∈ R 6m +6 . (114) 88 Multi-Arm Cooperating Robots 89 Of the 6m + 6 equations (113), only. inertial matrix and Y i = col(r ia , A ia ) ∈ R 6 1 ,Y= col(Y 0 , ,Y m ) ∈ R 6m +6 L va (Y ) = diag(L v0a , ,L vma ) ∈ R (6m +6) ×(6m +6) , L via = diag(I 3×3 ,L ωia (A ia )) ∈ R 6 6 , v ia = col(˙r ia ,ω ia (A ia )). (177) where (z) = ⎡ ⎣ N(q) 0 0 W(y 0 ) ⎤ ⎦ ∈ R (6m +6) ×(6m +6) , φ(z, ˙z) = n(q, ˙q,y 0 , ˙y 0 ) w(q, ˙q,y 0 , ˙y 0 ) ∈ R 6m +6 , δ τ = col(τ, 0, ,0) ∈ R 6m +6 , τ ∈ R 6m×1 . (178) Mathematical Models of
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