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

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 with six DOFs, in whi

Figure 21a Simulation results for τ j = 0, i, j = 1, 2, 3

Mathematical Models of Cooperative Systems

Figure 21b Simulation results for τ j = 0, i, j = 1, 2, 3

Figure 22a Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Mathematical Models of Cooperative Systems

Figure 22b Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Figure 22c Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Mathematical Models of Cooperative Systems

Figure 22d Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Figure 22e Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Mathematical Models of Cooperative Systems

Figure 22f Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Figure 22g Simulation results for τ1= 50 [Nm] and τ1= −50 [Nm]

Mathematical Models of Cooperative Systems

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 symbol for the internal coordinate q13, whereas QS13 and SS13 are the symbols for its

first and second derivatives ˙q3

1 and ¨q3

1 Diagrams of the dependent variable and its

derivatives are always given one below the other The symbol T ij , i, j = 1, 2, 3

is associated to the driving moments τ i j The symbols of the quantities at the ma-nipulated object MC∗0, ∗0S, ∗SS, ∗ = X, Y, F I and at contact points &i#,

& = Y, F I, 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 F I1 are the symbols of the dis-placement components Y1x , Y1y and ϕ1of the first contact point, while F 1X, F 1Y and M1 are the symbols of the components of the forces F1x and F1y and moment

M1in the direction of the displacements Y1x , Y1y and ϕ1of the first contact point

5 SYNTHESIS OF NOMINALS

We understand cooperative system trajectory as a line described by the state vector

in the state space during cooperative system motion, or the image of this line in some other space of the same dimension From the point of view of mathematics, the trajectory represents a hodograph of the time-dependent vector, characterized

by the number of coordinates corresponding to the state space dimension Each motion of the cooperative system that takes place without external perturbations is

unperturbed motion The trajectory described in the state space during the unper-turbed motion is called an unperunper-turbed trajectory Nominal motion of the coopera-tive system is any of its unperturbed motions satisfying a certain set of conditions.

The maximum number of independent conditions that can be imposed on the nomi-nal motion of a cooperative system is equal to the number of its independent inputs Constraints can be imposed either on the input or states of the cooperative system

A nominal trajectory is an unperturbed trajectory that is realized by the coopera-tive system during its nominal motion The nominal input is the vector of external

actions under which the nominal motion is performed The input to a cooperative

system is represented by the vector of the manipulator driving torques By the nom-inal of a cooperative system is understood the nomnom-inal input and its corresponding

nominal trajectory From the mathematical point of view, a nominal trajectory is the solution of the system of differential equations describing the cooperative sys-tem dynamics which is obtained by the action of the nominal input The problem of determining the nominal motion considered in this section is to define a procedure for the synthesis of the nominal vector, i.e the vector of the nominal trajectories and the vector of nominal inputs, so to ensure that the differential equations de-scribing the cooperative system dynamics are identically satisfied, provided part

of the nominal vector has been given in advance Such an approach ensures that the determined nominals are realizable under the condition that the mathematical model describes well enough the system dynamics

In this chapter we present a procedure for the synthesis of the cooperative

137

system nominals The procedure comes from the solution of the problem of co-ordinated 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 with six DOFs, in which the problem of force uncertainty is solved by introducing elastic properties into part of the cooperative system

5.1 Introduction – Problem Definition

Part of the cooperative system nominals represent the inputs to its control Hence, the first step in solving the task of cooperative system control is to determine its nominal motion, be it considered as a system of either rigid or elastic bodies Generally, the task of the synthesis can be formulated as follows The cooper-ative system is primarily used to manipulate objects The user chooses and defines the manipulation object and its characteristics (dimensions, shape and maximal al-lowed gripping intensity) The desired object motion is defined by defining the trajectory of a chosen point on the object (e.g the MC or some other reference point) Starting from the data thus defined, the problem is how to determine the driving torques that are to be introduced at the manipulator’s joints to ideally real-ize the preset requirement in the case of the absence of any disturbance At that, the stress of the object and manipulator must be within the allowed limits

When the problem of cooperative system operation is approached from the point of view of the mechanics of a rigid body, which is the usual procedure in the available literature, there appears the problem of force uncertainty The problem of determining nominals for a rigid cooperative system has not been discussed in the literature The problem has been reduced to determining the nominals of driving torques to drive rigid manipulators As we know the kinematic relations between the internal and external coordinates for a known load and tip position of non-redundant manipulators, the problem is easily solvable The problem of planning (optimal) trajectories of the cooperative system in the work space with or without obstacles has been treated in a number of works [26, 42–46]

As the models used do not faithfully describe the cooperative system’s dy-namics, the nominal motion of cooperative systems in the available literature has not been determined as its realizable motion, even when the maximum possible number of preset requirements (equal to the number of driving torques) has been ideally realized, irrespective of whether the fulfillment of these conditions results

in its optimal or non-optimal motion

The problem of force uncertainty is solved by considering the cooperative sys-tem as an elastic syssys-tem In Section 3.3, we showed that the problem of force

uncertainty in a cooperative system can be solved by introducing the assumption

of the elastic properties of the entire cooperative system or part of it The

cooper-ative system was approximated by m non-redundant manipulators with six DOFs

of motion in contact with a body that can move in three-dimensional space with-out 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 manipulators are approximated by an elastic system of m+ 1 elastically interconnected solid rigid bodies Each body is allowed 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 as a general mo-tion of an elastic structure Such expansion yields a complex 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 ex-panded 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 introduced assumptions on the characteristics of elastic connections, these equations are differ-ential (if neither mass nor damping are neglected) or algebraic (see Section 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 elas-tic system load This property makes the problem of the synthesis of nominals of the elastic cooperative system essentially more complex

The problem of determining the nominal motion of an elastic cooperative sys-tem can be interpreted in the following way In the cooperative syssys-tem’s motion, the nominal trajectory and its derivatives of the MC or some other reference point

of the manipulated object (one node of the elastic system), is prescribed In this way, six kinematic conditions for describing an elastic system in space are de-fined It is assumed that in the course of the cooperative system nominal motion the prescribed trajectory is realized in an ideal way The mathematical model of cooperative manipulation establishes a functional dependence between the kine-matic 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

Synthesis of Nominals

to define the rest 6m + 6m quantities in order to get the desired nominal motion

of the cooperative system At that, any motion of the cooperative system can be chosen as the nominal motion, including the one that corresponds to the resonance states of the elastic system

From a mathematical point of view, after introducing the desired quantities (as

if they were ideally fulfilled) the mathematical model is transformed into a non-homogeneous system of differential equations 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 obtained system of equations

In 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 However, even if the values of contact forces F c (6m quantities) are known,

because of the singularity of the elastic system stiffness matrix, it is not possible to

simply give and solve the system of equations and obtain the remaining 6m unique

nominal trajectories Such an approach does not ensure a unique description of the cooperative system in space

A consistent solution of the cooperative system nominals assumes a solution that ensures its unique position in space and a unique load of the elastic structure (object), or the values of contact forces in that position

The problem of determining nominal motion can be solved by introducing ad-ditional conditions, specific to the cooperative manipulation Namely, in the case when the cooperative system’s kinematic configuration represents a copy of some existing natural kinematic configuration, it is possible to record the nominal trajec-tories of all the 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 differen-tial equations that describe elastic system dynamics (102) or (115), determine the nominal trajectories of all the elastic system nodes After that, the determination of the nominals of rigid non-redundant manipulators with the aid of (166) is a simple and uniquely solvable problem

Several approaches can yield the solution of the values of contact forces ap-pearing during the motion

1 A first approach starts from the a priori definition of the optimal motion,

yielding the extremal values of the acting load [26]

2 The second approach starts from the condition that the vector of contact forces always remains inside the friction cone and the force intensity en-sures permanent contact of the manipulators and the manipulated object The stress state of the elastic system is determined by its total load, ob-tained as a resultant of the inertial, damping, contact, and gravitation forces Hence, the fulfillment of the condition for contact force does not guarantee non-violation of the system’s permitted stresses

3 The third approach starts from the condition required of the elastic system stress state (equivalent to the elastic force condition), without taking explicit care of the contact maintaining conditions Several versions can be distin-guished in the scope of this approach

• The first version is based on the requirement that there is a certain rela-tionship between the ratio and magnitude of the elastic forces and dis-placement (as with a pilot, see [47]) Because the work phase schedule

in cooperative manipulation is known, the remaining variants are based

on the requirement that the motions of the object and manipulators are coordinated

• The second version relies upon the possibility that the coordination is achieved by presetting the motion conditions either to the MC or to one contact point of the manipulated object and permitting elastic dis-placements of the elastic system nodes after the gripping step, due to a change of dynamic forces, orientation during the motion and, possibly, the required changes in gripping conditions

• The third version starts from the assumption that the coordination is achieved by setting the motion conditions to one contact point and pre-serving in the motion the shape of the geometric figure formed by the contact points at the end of the gripping phase

In this chapter we consider the second and third variants of determining the nominal coordinated motion of the cooperative system on the basis of the condi-tions of manipulated object MC and one contact point In the proposed procedure,

we first analyze the nominal motion of the separated elastic system and then, on the basis of this analysis, determine the nominal motion of the manipulators The nom-inals are determined only for the phases of object gripping and manipulation The result is a set of nominal quantities (states and inputs) defining different nominal

Synthesis of Nominals