The IDA-PBC Methodology Applied to a Gantry Crane

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1 Outline Methodology Applied to a Gantry Crane Ravi Banavar 1 Faruk Kazi 1 Romeo Ortega 2 N. S. Manjarekar 1 1 Systems and Control Engineering IIT Bombay 2 Supelec Gif-sur-Yvette, France MTNS, Kyoto, 2006

2 Outline Outline 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

3 Outline Outline 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

4 Outline Outline 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

5 Outline Outline 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

6 Outline Model Crane Mechanism 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

7 Model The Crane Mechanism Crane Mechanism x ments F x F M m l The control objective is: To move the payload from any positionq T i i x i i l i to the desired position specified asq D 0 x D l D T Figure: Overhead gantry crane

8 Model The Dynamic model Crane Mechanism ements F r l Assumptions The cable is massless and inelastic. Dissipative forces on the cart and at the winch are negligible Figure: Pulley and cable schematic m No slipping occurs at the point of contact between the winch and the cable

9 Outline Model Crane Mechanism 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

10 Model The Lagrangian of the System Crane Mechanism The Lagrangian of the system is where M q V q mgl cos q q 1 2 qt M q q V q ml 2 ml cos 0 0 ml cos M m 0 m sin 0 0 I p 0 0 m sin 0 m

11 Model Crane Mechanism Holonomic Constraint The no-slip constraint at the pulley implies r is the radius of the pulley l where r The constraint at the velocity level can be written as 0 0 r 1 q 0 The codistribution 0 0 r 1 is expressed as h q where h q l r This implies the integrable nature of the constraint and reduces the dimension of the configuration manifold to 3

12 Model Crane Mechanism The Lagrangian in the new coordinates We perform a linear transformation of coordinates as q Aq where q x l T and q x l r T Here A The Lagrangian in the new coordinates: r 1 q 1 1 q A q T M A 1q 1 A q V A 1q 2

13 Model Crane Mechanism The Lower Dimensional Manifold q r x T Inertia matrix M r q ml 2 0 ml 0 cos 0 ml 0 cos m M mr sin 0 m sin r mr 2 I p Potential energy V r q mgl 0 cos The Hamiltonian H H r q r p r 1 2 pt 1 r M r q p r V r q r (1) The desired equibrium - q D 0 x D l D l 0 r T

14 Outline Model 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

15 Model Port-Hamiltonian systems A major generalization of the class of Hamiltonian systems: x J x y g T x H x H x x g x u x x y IR m The system (2) with J satisfying condition J x J T x, is called as a port-hamiltonian system with structure matrix J x (2)

16 Model Mechanical systems Hamiltonian system representation: q q p H p H q p q p B q u u IR m y B T q q p B T q q y IR m H p Here, the Hamiltonian H q p 1 2 pt M 1 q p P q is the total energy of the system. In the port-hamiltonian form q p The matrix G 0 I n I n 0 qh ph 0 G q (3) u (4) IR n m is invertible in the case the system is

17 Model Methodology Basic Steps in IDA-PBC Energy shaping: Modify the total energy function of the system to assign the desired equilibrium Damping injection: To achieve asymptotic stability Synthesize the control input as: u u es q p u di q p

18 Outline Model 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

19 Model The Obtaining the energy shaping term: q p 0 I n I n 0 qh ph 0 G q u es 0 M 1 M d M d M 1 J 2 q p qh d ph d The first row is clearly satisfied For underactuated case second row gives: u es G T G 1 G T qh M d M 1 qh d J 2 M d 1 p

20 Outline Model 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

21 Model The Matching Equation gives rise to two PDES - Kinetic Energy (KE) and Potential Energy (PE) : 1 G q p T M 1 1 p M d M G q p T M d 1 p J 2 M d 1 p 0 qv M d M 1 qv d 0 2 Since the desired equilibrium position is already stable, we propose to shape potential energy only leaving kinetic energy unchanged. This makes M d M and J Using the original kinetic energy makes one matching condition trivially satisfied and simplifies the remaining one considerably

22 Model Solving the Potential Energy PDE Potential Energy (PE) PDE is given as: G qv M d M 1 qv d 0 Solving PE PDE gives: V d mgl 0 cos Choosing x x to be a quadratic function yields V d q mgl 0 cos K p 2 D 2 K px 2 x x D 2

23 Outline Model Simulations 1 Dynamic Model of an Overhead-Gantry Crane The Crane Mechanism 2 Stabilization of the Gantry Crane using IDA-PBC Methodology 3 Simulations 4

24 Model Simulations Simulation 1:m 1 Kg, M 6 Kg θ(t)(deg) F x x(t)(m) Time(sec) Time(sec) Time(sec) θ (rad per sec) l(m) F α θ (rad) Time(sec.) Time(sec) Initial ( 10, 0 8 m, 0 Initial:1.5 m Desired:0.5 m 5m) Desired Initial:30 deg Desired:0 deg configuration- configuration- ( 0, 0 5 m, 0 7m)

25 Model Simulations Simulation 2 :m 1 Kg, M 6 Kg θ(t)(deg) F x x(t)(m) Time(sec) Time(sec) Time(sec) F α θ (rad per sec) l(m) θ (rad) Time(sec.) Time(sec) Initial configuration - (30, 1 5 m, 0 9m) Desired configuration - ( 0, 0 5 m, 0 7m)

26 Model Simulations Simulation 3 :m 2 Kg, M 15 Kg θ(t)(deg) x(t)(m) F x (N) Time(sec) Time(sec) Time(sec) θ (rad per sec) l(m) F alpha (Nm) θ (rad) Time(sec.) Time(sec) Initial Initial:30 deg Desired:0 deg configuration - (30, 1 5 m, 0 9m) Desired Initial:1.5 m Desired:0.5 m configuration - ( 0, 0 5 m, 0 7m)

27 Model Presented a controller design procedure for an overhead crane that minimizes the swing of the payload using IDA-PBC methodology Introduced a no-slip constraint as a holonomic constraint Employed coordinate transformation to reduce the configuration space for design

28 Model Future Work Investigating the effect of cable elasticity and friction Introducing the constraint for positive cable tension Incorporating kinetic energy shaping for transient performance

Minimizing Cable Swing in a Gantry Crane Using the IDA-PBC Methodology

3 th National Conference on Mechanisms and Machines (NaCoMM7), IISc, Bangalore, India. December -3, 7 NaCoMM-7-65 Minimizing Cable Swing in a Gantry Crane Using the IDA-PBC Methodology Faruk Kazi, Ravi