2. Robot Formations Motion Dynamics Based on Scalar Fields 1.Introduction to non-holonomic physical problem 2.New Interaction definition as a computational tool  2.1 Schematic picture of the team in a non-holonomic scenary  2.2 Mathematical model  2.3 Further details 3.The method 4.Example of Application 5.Conclusions

3. 1. Introduction to non-holonomic physical problem  Lagrange’s Equation for a multibody system: Constraints: a. Holonomic b. Non-holonomic Second Newton Law Generalized coordinates (Yun and Sarvar, 1998) (General Case)

4. 2.1Schematic picture of the team in a non-hlonomic scenary

5. 2.2Mathematical Model  We stablish the following mechanical law motion: “The time variable of the team is generated by the team itself”  That law suggest the correspondance:  We are looking for a one to one correspondance between the second Newton law and (2).  Properties of the robots (p 1,p 2,....p N ), could be any of interest (masses, inertial tensors, etc).

6.  A single interaction  over the team is defined as follows:  The term  is due to the team itself and  is due to the interaction over the team.  The function  is still undefined but will be defined for the correspondance (3).  The robots will be considered in the sequel as punctual masses.  Notice the interactions are formed by scalar fields  and such fields result symmetric (  (q,t)=  (-q,t)), because of the quadratic characteristic of t.  The time variable is the same for every interaction applied in the same team.

7. 2.3 Further details of the definition  The function  is obtained taking into account the term  Finally the entire definition is:  The scalars r and l could be settled in each particular problem for the initial condition.  The definition results in a algebraic equation but for the final method we need to get the first and second derivates. Such a term appears if we use:

8. The Method  We can rewritte our definition as follows:  Taking temporal derivates on the above relation:

9.  Notice each force acting over the team has a one to one correspondance with our definition:  Now we do the same in the newtonian mechanics, we calculate each interaction Q k (related with each  i ) by superposition.  In order to get a computational method we separate the problem into External and Internal fields (interactions)

10.  Scheme of the interactions in the team’s scenary: External Interaction Internal Interaction

11.  The separation into External and Internal fields, suggests:  The next step consists of calculating the External fields directly from the external forces (which are assumed known):  For the Internal fields we need the Constraints of the team.

12.  Solving the null space of our definition:  Incorpopring the null space of the trajectories into the constraints:  Notice for solving the Internal fields we need to solve a system of partial differential equations with one (  int ) unknown variable. Holonomic Case Non-Holonomic Case

13.  Example of application Holonomic Constraints

14.  External Interaction considered (Obstacle)  The last equation is the one to solve in order to get the External fields

15.  For the Internal fields we rewritte the constraints:  With the solution for the External fields we do:  The last equation is a system of partial differential equations with one unknown function  int

16.  Conclusions  A novel definition for a team of robots was introduced  The main advantage lies in the computational way to incorpore the constraints  The time-space behavior of the definition becomes usefull for mobile frameworks.  It is a open issue to tackle the problem to solve the system of partial differential equations from the constraints  Finally we can change the parameters in the model (masses, inertia tensors,etc)