1/24 Romansy’06 , Warsaw University of Technology, June 24-26 2006 A Decoupled Approach to Optimal Time-Energy Trajectory Planning of Parallel Kinematic Machines Amar Khoukhi, Luc Baron and Marek Balazinski Mechanical Engineering Dept. École Polytechnique of Montréal C. P. 6079, Succ. CV, Montréal, Canada, H3C 3A7 Tel. (514) 340-4711/ ext. 4271 Fax (514) 340-5867 (amar.khoukhi, luc.baron, marek.balazinski)@polymtl.ca
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1/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
A Decoupled Approach to Optimal Time-Energy Trajectory Planning of
13/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
The control law :
)( ) ,( )( qGqqNvqM ccc−⋅−−
++=Γ
allows the robot to have a linear and decoupled behavior as:
vq ..=
where is an auxiliary input.v
[ ] [ ] 16
66
66
2
6666
66661 2
×
×
×
××
××+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎦
⎤
⎢⎢⎣
⎡= k
k
k
kk
k vIh
Ih
XIOIhIX
This gives Linear Simple Dynamics
Decoupled Formulation
14/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
Transfer Non Linearity to Objective Function
kkcT
kckkckkc
N
kv
Dd vXMXGXXNvXME
kh
)((U))() ,()(([[Min 11211
1
0
−−−−−
=∈
++⎪⎩
⎪⎨⎧
= ∑∈ΗC
])( 1kc XG−
+ ] }kkTk hQXX 22 ι++
Constraints remain the same, Except for actuator torques:
MaxkckkckkcMin XGXXNvXM ττ 1211 )() ,( )( <++<−−−
Decoupled Formulation
) ,( 21 kkc XXN−
+
15/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
Set: initial and final state positions, Limits on strut lengths, torques, accelerations, and sampling periods, Lagrange multipliers. Feasible tolerance, Infeasible tolerance, Convergence tolerances, Total number of sampling periods, and iterations.
21/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
AL (Minimum Energy)AL (Minimum Time Energy)
Numerical SimulationsExample: 2-dof PKM
22/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
AL with initial mass mp =200 kg
Numerical Simulations
AL with modified mass mp =300 kg
Example: 2-dof PKM
23/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
Numerical Simulations
Remarks
Initial solution : kinematically feasible,
Augmented lagrangian, smooth and monotonous increasing of energy consumption variations
Multi-criteria approach Minimum time is 15% faster than for the trajectorycomputed with only minimum energy criterion.
pm kg=300Needed actuator torques & necessary energy & time to achieve thesame task are bigger. Nonetheless, the algorithm converges and gets to the target with an acceptable precision of order of 10-3.
Modified mass
Example: 2-dof PKM
Corresponding torques: Outside the admissible domain.
24/24Romansy’06 , Warsaw University of Technology, June 24-26 2006
Conclusions
Multi-objective offline programming system
Under several constraints related to the robot-task-workspaceRobot dynamic model with Lagrange formalism includingmoving platform and struts
Optimal time-energy trajectory planning
Validation through a 2 dof planar parallel robot
Variational Calculus ApproachNon linear and Non convex optimal control problem.Solution: Augmented Lagrangian with Decoupling
Short term PerspectiveInclude Obstacle Avoidance in the Optimal Trajectory Planning System