Feasible Trajectories for Mobile Robots with Kinematic and Environment ConstraintsBackgroundPaper’s GoalPaper OrganizationHolonomic SystemsNon-holonomic Systems2-dof Mobile Robot in 3D C-spaceSlide 8Reachable Configurations2 Primary ManeuversSlide 11Lemma 1Lemma 2Lemma 3TheoremPath ExamplesSlide 17Slide 18Slide 19Slide 20Conclusions101/13/19Feasible Trajectories for Mobile Robots with Kinematic and Environment ConstraintsJean-Paul LaumondPresented at Intelligent Autonomous Systems Conf.in the Netherlands in 1986Paper Presentation By Chris Enedah01/13/192BackgroundNon-holonomic constraints mean that robots can not always follow a free path to reach a goal configurationWe would like to be able to:Generate a collision-free path without considering the robot’s constraintsModify the path (if necessary) to compensate for the robot’s constraints01/13/193Paper’s GoalTo show that a mobile robot with two degrees of freedom and a 3-dimensional C-space can (at the expense of maneuvers) follow any trajectory within a connected region of the configuration free space01/13/194Paper Organization1st Section – Introduction2nd Section – Discussion on topology and connectivity of configuration space4th Section – Automatic generation of maneuvers in pre-defined contextsFocus of presentation is on section 3 and the appendix01/13/195Holonomic SystemsNumber of degrees of freedom of system is greater than or equal to dimension of the configuration spaceAny infinitesimal motion in Cfree-space can be achievedThus, any collision-free path is feasible01/13/196Non-holonomic SystemsFor non-holonomic systems the number of degrees of freedom is less than the dimension of the configuration spaceNot all collision-free paths are feasible01/13/1972-dof Mobile Robot in 3D C-space01/13/1982-dof Mobile Robot in 3D C-space2 independent driving wheels located on a single axisConfiguration given by q = (x,y,)Unable to move sidewaysAdditionally constrained so that it can’t rotate on itself i.e. driving wheels can’t have opposite velocitiesMotion is just like a car’s01/13/199Reachable ConfigurationsSet of configurations reachable from center of cylinder, CYL(x,y,/2,,/2)01/13/19102 Primary ManeuversType 1 ManeuverAllows translation in direction of constraintOrientation does not change01/13/19112 Primary ManeuversType 2 ManeuverOrientation change with same initial and final position01/13/1912Lemma 1Let q = (x,y,) and q’ = (x’,y’,) be two configurations in CYL(x,y,,,) such that the line passing through (x,y) and (x’,y’) is perpendicular to the main axis of the car. There exists a feasible path from q to q’ that consists of a finite sequence of manuevers of type 1 that are all completely contained in CYL(x,y,,,).01/13/1913Lemma 2Let q = (x,y,) and q’ = (x,y,’) be two configurations in CYL(x,y,,,). There exists a feasible path from q to q’ that consists of a finite sequence of manuevers of type 2 that are all completely contained in CYL(x,y,,,).01/13/1914Lemma 3Let q and q’ be any two configurations in CYL(x,y,,,). There exists a feasible path from q to q’ that is completely contained in CYL(x,y,,,).01/13/1915TheoremLet q and q’ be two configurations in a connected region of A’s free space Cfree. Then there exists a feasible collision-free path for A from q to q’.01/13/1916Path Examplesqinit = (x,y,)qgoal = (x’,y’, ’)01/13/1917Path Examplesqinit = (x,y,)qgoal = (x’,y’, ’)01/13/1918Path ExamplesProblem01/13/1919Path ExamplesPossible Solution 101/13/1920Path ExamplesPossible Solution 201/13/1921ConclusionsA collision free-path generated for a holonomic robot A can be transformed to a new path that satisfies the kinematic constraints of a non-holonomic robot B with the same geometry as AHowever, transformed path may require an unreasonable number of primary maneuvers 1 &