Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 8 doc

127 Figure 21a. Simulation results for τ j i = 0, i, j = 1, 2, 3 Mathematical Models of Cooperative System s Figure 21b. Simulation results for τ j i = 0, i, j = 1, 2, 3 128 Multi-Arm Cooperating Robots 129 Figure 22a. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] Mathematical Models of Cooperative System s Figure 22b. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] 130 Multi-Arm Cooperating Robots 131 Figure 22c. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] Mathematical Models of Cooperative System s Figure 22d. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] 132 Multi-Arm Cooperating Robots 133 Figure 22e. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] Mathematical Models of Cooperative System s Figure 22f. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] 134 Multi-Arm Cooperating Robots 135 Figure 22g. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] Mathematical Models of Cooperative System s part referring to the physical quantity used in modeling and the number indicat- ing the ordinal number of the physical quantity vector. Thus, Q13 is the sym bol for the internal coordinate q 3 1 , whereas QS13 and SS13 are the symbols for its first and second derivatives ˙q 3 1 and ¨q 3 1 . Diagrams of the dependent variable and its derivatives are always gi ven one below the other. The symbol Tij, i, j = 1, 2, 3 is associated to the dri ving moments τ j i . The symbols of the quantities at the ma- nipulated object M C ∗0, ∗0S, ∗SS, ∗=X, Y, F I and at contact points &i#, & = Y,FI,F,M,#= X, Y , denote respectively the linear and angular displacements of the manipulated object MC ∗ 0 , ˙ ∗ 0 , ¨ ∗ 0 , ∗=X, Y, ϕ, linear and angular displacements of the contact points & # i ,&= Y ,#= X, Y and ϕ i , i = 1, 2, 3, are the forces and moments at the contact points & # i ,&= F ,#= X, Y and M i , i = 1, 2, 3. For example, Y 1X, Y 1Y and FI1 are the symbols of the dis- placement components Y x 1 , Y y 1 and ϕ 1 of the first contact point, while F 1X, F 1Y and M1 are the symbols of the components of the forces F x 1 and F y 1 and moment M 1 in the direction of the displacements Y x 1 , Y y 1 and ϕ 1 of the first contact point. 136 Multi-Arm Cooperating Robots [...]... object gripping and manipulation The result is a set of nominal quantities (states and inputs) defining different nominal 142 Multi-Arm Cooperating Robots motions of the cooperative system From this set, the desired inputs to the cooperative system control are selected Thus, the quantities are selected that are directly tracked, i.e the quantities that close the feedback loops of the control system of... motion in contact with a body that can move in three-dimensional space without any constraint (Figure 1) The manipulators and object are all assumed to be rigid apart from the neighborhoods of the contact points, whereby the resulting manipulator-object contact is elastic and the manipulator tip cannot move over the object surface The manipulated object and the neighborhood of its contact points with the... to have six DOFs For the elastic system, gravitational and contact forces are the external forces acting at the MCs of these bodies By contact forces is understood the six-dimensional vector of generalized force formed from the three-dimensional vector of axial force (dimension [N]) and three-dimensional vector of torques (dimension [Nm]) The dynamics of a cooperative system thus defined is modeled... 4.12) Nominal motion is determined on the basis of the model given by Equations (102) and (175) for gripping, and by Equations (115) and ( 181 ) for the general motion in the form (211) The model characteristics presented in Sections 4.12 and 4.13 show that there is a functional dependence between the kinematic configuration and elastic system load This property makes the problem of the synthesis of nominals... manipulation establishes a functional dependence between the kinematic configuration and the elastic system load The model of the elastic system establishes a relation between 6m active forces and 6m+6 kinematic quantities and their derivatives As only six dynamic conditions are defined, the problem is how 140 Multi-Arm Cooperating Robots to define the rest 6m + 6m quantities in order to get the desired... involving differential constraints Such a system is solved by taking the left-hand side of the equality being given and seeking the right-hand one, or vice versa, or by giving additional conditions until the task becomes closed in a mathematical sense The problem is how to set out the conditions that are given in advance and, when these conditions are being fulfilled, how to find the solution of the... the cooperative 137 1 38 Multi-Arm Cooperating Robots system nominals The procedure comes from the solution of the problem of coordinated motion of an elastic structure, taking into account the specific features of cooperative manipulation The procedure has been defined on the basis of the mathematical model of the dynamics of the cooperative manipulation of the object by the non-redundant manipulators... cooperative manipulation, one cannot simultaneously prescribe the arbitrary trajectories of the object (6 quantities) and manipulator (6m quantities) and seek active forces (6m quantities), as there can appear excessive contacts and internal stress of the object and manipulators On the other hand, active forces (contact forces or driving torques) in the course of cooperative system’s motion are not known... links of the natural cooperative system and, on the basis of these records, determine the nominal contact forces and check the system stresses If this is not possible, it is necessary to determine the contact forces first and then, based on them as driving torques and known position in space, by solving the differential equations that describe elastic system dynamics (102) or (115), determine the nominal... mathematical model, but without force uncertainty This model faithfully describes the dynamics and statics of the cooperative system The model of a rigid manipulated object is expanded by equations of elastic connections This yields a dynamic model of the separated elastic system, composed of a model of rigid body dynamics and a set of equations to describe the elastic interconnections Depending on the . τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] 134 Multi-Arm Cooperating Robots 135 Figure 22g. Simulation results for τ 1 1 = 50 [Nm] and τ 1 2 =−50 [Nm] Mathematical Models of Cooperative System s part referring. F x 1 and F y 1 and moment M 1 in the direction of the displacements Y x 1 , Y y 1 and ϕ 1 of the first contact point. 136 Multi-Arm Cooperating Robots 5 SYNTHESIS OF NOMINALS We understand cooperative. non-optimal motion. The problem of force uncertainty is solved by considering the cooperativ e sys- tem as an elastic system. In Section 3.3, we showed that the problem of force 1 38 Multi-Arm Cooperating