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							                    Formation Motion Planning for
                    Payload Transport by Modular
                    Wheeled Mobile Manipulators




 Rajankumar Bhatt
 Advisor : Dr. Venkat Krovi
 Mechanical and Aerospace Engineering Department
 State University of New York at Buffalo.

Rajankumar Bhatt
December 12, 2003
Slide 1 of 36                               Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                            Agenda



           Introduction
           Implementation
           Results
           Conclusion
           Future Work




Rajankumar Bhatt
December 12, 2003
Slide 2 of 36                        Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                 Robot Collectives




                                                                                              The CMU Millibots

   Smaller robots to perform task of single large robot
          – Cheaper, less powerful but better performance
   Issues
          – Control many robots
          – Coordinate their actions                      Loosely
                                                          coupled
   Types of Cooperation
          – Information based cooperation                                                    Tightly
          – Physical cooperation                                                             coupled

Rajankumar Bhatt      Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 3 of 36                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                             Formation with Payload




         A Wheeled                                             Formation of Wheeled
       Mobile Manipulator
    • Increased workspace
                                                                Mobile Manipulators
    • Redundancy                                              Transporting a Payload

                    • Material handling Applications
                    • Cooperatively performs tasks that cannot be performed
                      by single mobile robot
                    • Re-configurability and accommodation for disturbances
Rajankumar Bhatt          Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 4 of 36                                                         Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                     State-of-The-Art
   Behavior based approach
   •    Specify simple behaviors for different goals
   •    Emergent behavior by combination of multiple simple behaviors
   •    Issue is “How to decompose global behaviors into component modules?”
   •    Advantages are decentralization, limited communication
        (Balch, T. and Arkin, R. C., 1998; )

   Virtual Leader (Includes Leader follower/Virtual structures)
   •    Motion of followers specified with respect to that of leader
   •    Problem reduces to control of single robot
         • Robots considered as point objects
             (Shahidi, R.; Shayman, M.; Krishnaprasad, P.S., 1991)
         • Robots considered with nonholonomic constraints
             (Young, Beard and Kelsey, 2001)




Rajankumar Bhatt      Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 5 of 36                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                              State-of-The-Art                           (Cont’d)




   Manipulation
   • Behavior-based multiple robot system
          – (Wang, Z., Nakano, E., Matsukawa, T. and Hanada, K., 1996; Sugar and
            Kumar, 1999)


   • Cooperative manipulation of object by two robots
          – (Abou-Samah M. and Krovi, V., 2002)




                                                                                    Abou-Samah M. and Krovi, V., 2002

Rajankumar Bhatt      Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 6 of 36                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                Mathematical Modeling
   Homogeneous Transformation

                      F RM            d
                                       F
                AM                       SE  2 
            F

                      00              1 
   Spatial Twist
                                                                                                               Moving Frame
                                                 1
               TM   AM  AM   se  2 
          F     F       F         F
                            
    Body Fixed Twist                     Preferred
                                       representation
          M                            1 F
                 TM    AM 
                
                 F
                      
                            F
                                             AM

   Similarity Transform
                                                                    Inertial Frame
          N                            F                              1
               TM    AF   TM   AF 
              
                 F
                    
                            N
                                      
                                             F              N



Rajankumar Bhatt        Introduction       Implementation       Results    Conclusion   FutureWork
December 12, 2003
Slide 7 of 36                                                                Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                Mathematical Modeling                                        (Cont’d)




  More about Twists
  •    Twist as linear operator
                               v
                     TM   
                    
                        F
                          0 0 0
                                 
                               0                          vx 
                                                       v=  
                               0                         vy 
                                                            T
  Twist Vector              tM = vx
                            F
                                               vy     
                                                        

       0                 vx     vx 
                  0       vy   vy 
                                  
       0
                   0       0     
                                    
Rajankumar Bhatt                Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 8 of 36                                                               Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                       Mathematical Modeling (Cont’d)

   Nonholonomic Constraint                                       No side ways slip
   can be characterized as:
                                                                              x sin   y cos   0
      q  x        y 
                                                           T
                                 q  x        y 
                       T
                                                  •                Does not constrain configuration
                                                    •                Ability to control 3-dof by 2 inputs
                                                    •                However, poses difficulty in control
                                                    •                Complex maneuvers may be
                                                                     required


                                                     cos 
                                                                                                       r                  
                                                                            0 M
                                                                                                                   r
                                                                                                        2 cos      cos  

                                                                              v
                                                                                                                   2

                                                 q   sin 
                                                                                                                          
                                                                                                                              
                                                                                                   q   sin        sin    L 
                                                                                                         r         r
                                                                           0  M                    2                   R 
                                                                                  
                                                                                                                   2

                                                                              
                                                                                                                     r 
                                                      0
                                                                           1                          r
                                                                                                       
                                                                                                        d            d 
                                                                                                                           
                                                                                                                           


Rajankumar Bhatt           Introduction   Implementation       Results   Conclusion   FutureWork
December 12, 2003
Slide 9 of 36                                                              Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                        Mathematical Modeling                                                                               (Cont’d)




Body Fixed Twist based approach

                   1 F                          1                            1 F
        F AE             AE   M AE   F AM                                                                       
                                                                                               M
                                                                                   AM        AE      f  L , R
            E                                                      M
                 F TE                                                    F TM 
                                                                              

                                                    1                         1 M
                                 B AE   M AB 
                                                                                         AB B AE
                                                                         B M
                                                                                 TB 
                                                                                    

                                                 1                        1
                                I1 AE   B AI1                                      AI1 I1 AE
                                                                                                            
                                                                                   B
                                                                                                     f 1
                                                                  I1 B
                                                                          TI1 
                                                                              
                                           1                    1
                                I2 AE   I1 AI2                   I1           I2


                                                                                                         
                                                                       AI 2         AE
                                                         I2
                                                                                                    f 2
                                                               I1TI 
                                                                   2




                                                                                          
                                               1 I
                                 I 2 AE 
                                                      AE
                                                                                        f 3
                                                    2



                                       E
                                            I 2 TE 
                                                   




                                                                                                                                     General Mobile Manipulator
Rajankumar Bhatt                            Introduction                           Implementation             Results   Conclusion   FutureWork
December 12, 2003
Slide 10 of 36                                                                                                              Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                               Mathematical Modeling                                                   (Cont’d)




                                                                        R 
   Body Fixed Twist                                                     
                     Ev  E                                            L 
                     E    t1                                   5  1
                                    E   t2   E   t3      E   t4   Et       
                                                                      
                                              J                         2 
                                                                        
                                                                        3

   •    Simple equations in end-effector frame
   •    Invariant with respect to selection of global
        frame
   Modules
   •    Type I: RRR Mobile Manipulator                                   a b0
   •    Type II: RR Mobile Manipulator                                 a  b  l3  0
   •    Type III: R Mobile Manipulator                               a  b  l2  l3  0
                                                                                                 General Mobile Manipulator
Rajankumar Bhatt               Introduction           Implementation      Results   Conclusion   FutureWork
December 12, 2003
Slide 11 of 36                                                                        Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                         What do we want to do?

  Given path and a number of mobile robots

            Determine optimal motion plans for
              each mobile robot

    Subject to

              Nonholonomic constraints
              Other regional constraints


     Using
               Metrics defined on se(2)

                                                                  Point P moving on arbitrary curve
Rajankumar Bhatt       Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 12 of 36                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                      Team-Fixed Frame

 Without payload

  Olfati-Saber and Murray, 2002
         •       Any robot location can be origin
         •       Align x-axis in the direction of nearest neighbor




    Belta and Kumar, 2002
             •    Origin located at center of mass
             •    Axis directed along principal inertial directions


 With payload
                        Center of mass of payload
                        Point of reference on payload
Rajankumar Bhatt            Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 13 of 36                                                          Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Parameterization

 Configuration of Frame            M0                                                        Orientation of
                                                                                               mobile robots
                          g0  ( FRM0 , F p0 )  SE(2)                                        not considered

                           Configuration of Entire Formation
                                        q  ( g0 , p1 , p2 ,..., pn )  Q  SE(2)                      2
                                                                                                             ...    2

                                                                                                                n
                               An Alternate Parameterization of Formation

                                                     q  ( g0 , r ,  )  Q  SE(2)                        n
                                                                                                                Tn
                                                                             (Polar)

                                                       Relation between
                                                       Two Parameterizations

                                                       r
                                                       i
                                                                 M0
                                                                      pi      and         i   M pi       0




Rajankumar Bhatt    Introduction    Implementation    Results   Conclusion   FutureWork
December 12, 2003
Slide 14 of 36                                                    Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                          Metrics (Objective Function)
 Manipulability (Jacobian) based Performance Measures

      J mn  U V T                      Σ  diag 1,2 ,...k ,0,..0 V  SO  n 
                                          U  SO  m
 •        Yoshikawa’s Measure of Manipulability     ....
                                                 y   1 2      k
 •        Condition Number   c  1 k
 •        Isotropicity Index i   k 1

 Energy (Riemannian metric) based Performance Measures (Zefran, 1996)
 •        family of left invariant metrics on se(2) (specialized from                             se(3) )

            T1 , T2        t1TWt2                    T1, T2  se 3
                      I

                                                                       1       Klein                         0
             11                  012                                                            ,
          W 
                                    I 22 
                                                                       2       Park                           arbitrary
             021                                                    3       Energy                   I zz ,   m

Rajankumar Bhatt           Introduction    Implementation   Results   Conclusion    FutureWork
December 12, 2003
Slide 15 of 36                                                             Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                            Motion Parameterization

 Path Parameterization

                        A( s ) :[0, s0 ]  SE (2)

 Time Parameterization for                 s
                         s (t ) :[t1 , t2 ]  [0, s0 ]

  Motion Parameterization


                    R(s(t ))    p(s(t )) 
     A  s(t )                           SE  2 
                    0             1 



Rajankumar Bhatt          Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 16 of 36                                                        Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                           Implementation




                                                                                                          vx 
                                                                                                   F
                                                                                                    tM = vy 
                                                                                                          
                                                                                                          
                                                                                                          

  T1 , T2    I
                  t1TWt2


Rajankumar Bhatt            Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 17 of 36                                                          Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                Adaptive Arc-length Parameterization




        x( s )  ax  bx s  cx s 2  d x s 3
        y(s)  ay  by s  cy s 2  d y s3

Rajankumar Bhatt         Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 18 of 36                                                       Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Geometric Motion Planning Strategy

 Curvature
 • Rate of change of tangential angle

                                        x ' y '' y ' x ''
                               
                                       x'  y' 
                                                            3
                                            2           2       2


 Radius of Curvature
                                  r 1 
 Geometric Motion Planning Strategy
   (GMPS) requires Team-fixed frame
   be aligned to Serret-Frenet frame all
   times



Rajankumar Bhatt        Introduction   Implementation       Results   Conclusion   FutureWork
December 12, 2003
Slide 19 of 36                                                          Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                      Geometric Motion Planning Strategy (Cont’d)

 Velocity of Team-fixed Frame                                                                        In this thesis, we
 • Referenced to Team-fixed frame                                                                    consider s  1 .
                                                                                 s
                      M0
                           v  et s
                               ˆ                           M0
                                                                  et 
                                                                    ˆ                    s
                                                                              1  
 Velocity of Individual Modules


M0                                  0     ( s) 1   ri cos i 
     vi                    ri  
                                                  0   ri sin i 
             M0
     0          FTM 0      ( s )
                          1               0
                                                               
                             0               0  1 
                                           0                   

                    i       M0
                                   vi      Oi Oc

Rajankumar Bhatt                Introduction   Implementation    Results   Conclusion   FutureWork
December 12, 2003
Slide 20 of 36                                                               Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
      Geometric Motion Planning Strategy                                                               (Cont’d)



Formation of Mobile Robots • Klein Metric

                                             i  1x1 01x 2  i 
                                                         T

                                                                      i2
                                             vi  02 x1 02 x 2   vi 
                                                                  

                                                    • Energy Metric

                                             i   I zz
                                                         T
                                                                           012  i 
                                                                                       I zzi2  mvi2
                                            vi  021                    mI 22   vi 
                                                                                   



                                             For Screw Motion
                                             Klein Metric = constant

Rajankumar Bhatt    Introduction   Implementation   Results   Conclusion    FutureWork
December 12, 2003
Slide 21 of 36                                                  Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
             Geometric Motion Planning Strategy                                                       (Cont’d)




 Determination of orientation of Individual
      modules is secondary stage


 • Orientation and Location are decoupled
 • Select orientation to be aligned with the
      velocity field
 • Visually:
       – Direction of nonholonomic constraint =
            Direction of line joining Origin of frame of
            individual modules and ICR.



Rajankumar Bhatt         Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 22 of 36                                                       Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                        Results and Validation

 GMPS applied to formation of three mobile robots
       – Screw motion
       – Non-screw (Sinusoidal) motion
 Objective function
       – Klein Form
       – Energy Form
 Validation using analytical results


Rajankumar Bhatt    Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 23 of 36                                                  Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                       Formation Motion Planning                                                 (Cont’d)




Energy Metric (Analytical) – Screw Motion


                                                         i   I zz
                                                                       T
                                                                                     012  i 
                                                                                                 I zzi2  mvi2
                                                        vi  021                  mI 22   vi 
                                                                                             



                                                                                     2                 
                                                              v     2
                                                                   1   2 r 2  2r  cos  2   
                                                                       i                               m
                                                                                                         
                                                                            i      i              i     

Optimal Values (Analytical)
                                                                          
                                                                 cos    
                    *                                     r*      2    
                             2
                                                                               
Rajankumar Bhatt             Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 24 of 36                                                             Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Motion Planning                                                                               (Cont’d)



• Kinetic Energy Metric (Analytical) – Screw Motion
                                          Desired Trajectory                                               Curvature
                               2                                                         2



                           1.5                                                      1.5




                                                                           (1/m)
                     y (m)
                               1                                                         1

 For Screw Motion
                              0.5                                                      0.5


     const                   0
                                -1      -0.5          0       0.5   1
                                                                                         0
                                                                                              0      2         4        6     8
                                                   x (m)                                                 Arc Length (m)

                                             Optimal Radius                                                Optimal 
                               1                                                       -0.5

                              0.9
                                                                                        -1



             
                              0.8
                                                                       * (radians)
                                                                                       -1.5
                                                                                                                                                
     cos    
                     R* (m)




                              0.7
                                                                                                                                       *  
r*      2                  0.6
                                                                                        -2
                                                                                                                                                2
                             0.5
                                                                                       -2.5


                              0.4                                                       -3
                                    0    2           4        6     8                         0      2         4        6     8
                                               Arc Length (m)                                            Arc Length (m)



Rajankumar Bhatt       Introduction              Implementation     Results               Conclusion          FutureWork
December 12, 2003
Slide 25 of 36                                                                                    Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Motion Planning                                             (Cont’d)




• Energy Metric (Numerical) – Screw Motion




    Regional Constraints
                                                                Optimal Solution

Rajankumar Bhatt      Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 26 of 36                                                    Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Motion Planning                                                                           (Cont’d)



• Energy Metric (Analytical) – Sinusoidal Motion
                                                              Desired Trajectory                                                   Curvature
                                                     1                                                         1


                                                    0.5                                                      0.5



        a 2 sin(t )




                                                                                               (1/m)
                                       y (m)
  
                                                     0                                                         0



     1  a 2 2 cos 2 (t )                    -0.5                                                          -0.5


                                                     -1                                                       -1
                                                          0   2          4         6     8                          0       2         4        6   8
                                                                      x (m)                                                     Arc Length (m)

                                                                  Optimal Radius                                                   Optimal 
                                                                                                             -0.5
                                                     2
                                                                                                              -1
                                                     1

                        



                                                                                              * (radians)
                                                                                                             -1.5
                cos    
                                           R* (m)




                                                     0


           r*      2                              -1
                                                                                                              -2


                                                    -2
                                                                                                             -2.5


                                                                                                              -3
                                                          0   2         4        6       8                          0       2         4        6   8
                                                                  Arc Length (m)                                                Arc Length (m)


Rajankumar Bhatt        Introduction   Implementation               Results    Conclusion    FutureWork
December 12, 2003
Slide 27 of 36                                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                   Formation Motion Planning                                                       (Cont’d)




• Energy Metric (Numerical) – Sinusoidal Motion
                                                                          Curvature
                                  1.5
                                                                                                         Numerical
                                                                                                         Analytical
                                    1



                                  0.5
                    Kappa (1/m)




                                    0



                                  -0.5



                                   -1



                                  -1.5
                                         0        1         2       3         4        5          6        7           8
                                                                        Arc Length (m)

Rajankumar Bhatt                             Introduction   Implementation   Results   Conclusion     FutureWork
December 12, 2003
Slide 28 of 36                                                                             Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Payload Transport

Type III: R Mobile Manipulator
• Forms a Rigid Virtual Structure
• Relative locations cannot be
     changed
• Screw Motion
      – Initial Optimal Placement is adequate

• Non-Screw Motion
      – No Optimal configuration possible




Rajankumar Bhatt     Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 29 of 36                                                   Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                    Formation Payload Transport                     (Cont’d)




                               Type II: RR Mobile Manipulator
                               • Allows some flexibility in locating M i 
                                   with respect to      M 0 
                               • Still restrictive
                               •   M i  constrained to lie in a circle
                                   centered around M 0 




Rajankumar Bhatt
December 12, 2003
Slide 30 of 36                              Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                       Formation Payload Transport                                                       (Cont’d)




Type I: RRR Mobile Manipulator
• Task frame based parameterization


       q  ( g0 , r ,  )  Q  SE(2)                              n
                                                                        T n

• RPR Linkage


           1
 AM 0 

  F
       
                F
                    AM i    M0
                                  AJ1i 1i  J1i AJ 2 i  ri  J 2 i AM i  2i 




Rajankumar Bhatt
December 12, 2003
Slide 31 of 36                                                                   Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                        Formation Payload Transport                                                        (Cont’d)




Type I: RRR Mobile Manipulator
• Module frame based parameterization


              g  ( F RM , F p, 1 ,  2 ,  2 )  SE (2)  T 3
• RRR Linkage



            1
   F AMi 
         
                 F
                     AM0    Mi
                                  AE1i 1i  E1i AE2 i 2i  E2 i AM0 3i 




Rajankumar Bhatt                   Introduction   Implementation    Results      Conclusion   FutureWork
December 12, 2003
Slide 32 of 36                                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                     Formation Payload Transport                                                   (Cont’d)



Equivalence of Parameterization

                    1                        1                           1
        AM0 
       
         F
             
                         F
                             AMi  AMi 
                                 
                                   F
                                       
                                                   F
                                                       AM0   AM0  I F AM0  I
                                                             
                                                                F
                                                                   
• Below 3 equations can be solved for 3 unknowns


           1i  2i  3i  1i  2i  0

        l3 cos   l2 cos    3i   l1 cos 1i  2i  1i   r cos 1i  0
                                                                    i



        l3 sin   l2 sin    3i   l1 sin 1i  2i  1i   r sin 1i  0
                                                                    i




• Results of the unconstrained formation planning can be directly applied

Rajankumar Bhatt               Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 33 of 36                                                             Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                          Conclusion


         • Modular, reconfigurable system
         • Nonholonomic constraints in motion planning
         • Optimal motion plans for formation of mobile
              robots
         • Validation of numerically obtained results with
              analytical results
         • Equivalence between the unconstrained case and
              constrained (payload transport) case.


Rajankumar Bhatt       Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 34 of 36                                                     Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                                      Future Work



         • Jacobian based Objective Function
         • Higher order curves rather than Cubic Splines
         • Velocity of Serret-Frenet frame being any
              constant or even more general case of being
              variable.




Rajankumar Bhatt     Introduction   Implementation   Results   Conclusion   FutureWork
December 12, 2003
Slide 35 of 36                                                   Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
                     Thank You




                    Thank You



Rajankumar Bhatt
December 12, 2003
Slide 36 of 36              Automation, Robotics and Mechatronics Lab, SUNY at Buffalo