Robot Formations Motion Dynamics Based on Scalar Fields

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Robot Formations Motion Dynamics Based on Scalar
Fields

• Introduction to non-holonomic physical problem
• 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
• The method
• Example of Application
• Conclusions

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1. Introduction to non-holonomic physical
problem

• Lagranges Equation for a multibody system

Second Newton Law
Generalized coordinates

• Constraints

a. Holonomic
b. Non-holonomic
(Yun and Sarvar, 1998)
(General Case)
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2.1 Schematic picture of the team in a
non-hlonomic scenary
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2.2 Mathematical 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

• Properties of the robots (p1,p2,....pN), could be
  any of interest (masses, inertial tensors, etc).

• We are looking for a one to one correspondance
  between the second Newton law and (2).

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• 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.

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2.3 Further details of the definition

• The function ? is obtained taking into account
  the term

Such a term appears if we use

• 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.

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The Method

• We can rewritte our definition as follows

• Taking temporal derivates on the above relation

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• 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 Qk (related with each
  ?i) by superposition.

• In order to get a computational method we
  separate the problem into External and Internal
  fields (interactions)

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• Scheme of the interactions in the teams scenary

External Interaction
External Interaction
Internal Interaction
External Interaction
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• 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.

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• Solving the null space of our definition

• Incorpopring the null space of the trajectories
  into the constraints

Holonomic Case
Non-Holonomic Case

• Notice for solving the Internal fields we need to
  solve a system of partial differential equations
  with one (?int) unknown variable.

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• Example of application

Holonomic Constraints
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• External Interaction considered (Obstacle)

• The last equation is the one to solve in order to
  get the External fields

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• 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

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• 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)