Transverse Linearization for Controlled Mechanical Systems with Several Passive Degrees of Freedom (Application to Orbital Stabilization)

Transcription

1 Transverse Linearization for Controlled Mechanical Systems with Several Passive Degrees of Freedom (Application to Orbital Stabilization) Anton Shiriaev 1,2, Leonid Freidovich 1, Sergey Gusev 3 1 Department of Applied Physics and Electronics Umeå University, Sweden (http : // 2 Department of Engineering Cybernetics Norwegian University of Science and Technology, Norway (http : // 3 Department of General Mathematics and Informatics St. Petersburg State University, Russia (http : // American Control Conference, June 11, 2009, St. Louis, Missouri

2 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

3 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

4 Underactuated Mechanical Systems We consider the class of Euler-Lagrange systems ( ) d L(q, q) L(q, q) = B(q) u dt q q [ ] M(q) q + C(q, q) q + G(q) = B(q) u where q R n and q R n are vectors of generalized coordinates and velocities, u R m is a vector of control inputs, L(q, q) = 1 2 qt M(q) q V(q) is the Lagrangian, M(q) is a positive-definite matrix of inertia, V(q) is a potential energy, and B(q) is a matrix of full-rank. Challenges: underactuated, i.e. 0 dim u = m < n = dim q, not feedback linearizable and non minimum phase. It is not clear what kinds of trajectories are possible and which can be stabilized.

5 Underactuated Mechanical Systems We consider the class of Euler-Lagrange systems ( ) d L(q, q) L(q, q) = B(q) u dt q q [ ] M(q) q + C(q, q) q + G(q) = B(q) u where q R n and q R n are vectors of generalized coordinates and velocities, u R m is a vector of control inputs, L(q, q) = 1 2 qt M(q) q V(q) is the Lagrangian, M(q) is a positive-definite matrix of inertia, V(q) is a potential energy, and B(q) is a matrix of full-rank. Challenges: underactuated, i.e. 0 dim u = m < n = dim q, not feedback linearizable and non minimum phase. It is not clear what kinds of trajectories are possible and which can be stabilized.

6 Goals of Analysis and/or Control Design 1 Motion generation: Design a control transformation u = v + U(q, q) such that there exist a nontrivial periodic solution q = q (t) = q (t + T), t 0, (T > 0) of the closed-loop system with v 0, satisfying certain specifications (such as amplitude and period). 2 Motion stabilization: Design an exponentially orbitally stabilizing controller v = f(q, q), f ( q (t), q ) 0

7 Goals of Analysis and/or Control Design 1 Motion generation: Design a control transformation u = v + U(q, q) such that there exist a nontrivial periodic solution q = q (t) = q (t + T), t 0, (T > 0) of the closed-loop system with v 0, satisfying certain specifications (such as amplitude and period). 2 Motion stabilization: Design an exponentially orbitally stabilizing controller v = f(q, q), f ( q (t), q ) 0

8 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

9 Geometric relations and restriction of the dynamics A periodic motion q = q (t) = [ q 1 (t),..., q n (t) ] T can be represented in a time-independent form ( q 1 (t) = φ 1 θ (t) ) (,..., q n (t) = φ n θ (t) ) where θ is a parameterizing variable, describing a desired behavior of one of the generalized coordinates. The induced family of functions { } φ 1 ( ),..., φ n ( ) is called virtual holonomic constraints. Let us consider a restriction of the dynamics consistent with these constraints, i.e. with q 1 φ 1 (θ),..., q n φ n (θ) where θ is treated as a new generalized coordinate.

10 Geometric relations and restriction of the dynamics A periodic motion q = q (t) = [ q 1 (t),..., q n (t) ] T can be represented in a time-independent form ( q 1 (t) = φ 1 θ (t) ) (,..., q n (t) = φ n θ (t) ) where θ is a parameterizing variable, describing a desired behavior of one of the generalized coordinates. The induced family of functions { } φ 1 ( ),..., φ n ( ) is called virtual holonomic constraints. Let us consider a restriction of the dynamics consistent with these constraints, i.e. with q 1 φ 1 (θ),..., q n φ n (θ) where θ is treated as a new generalized coordinate.

11 Computing the restricted dynamics The dynamics consistent with q 1 φ 1 (θ),..., q n φ n (θ) describes possible evolutions of θ and can be computed: Lemma (Constrained dynamics) Independently on the generating control law, θ is simultaneously a solution of (n m) differential equations α i (θ) θ + β i (θ) θ 2 + γ i (θ) = 0, i = 1,..., n m some of which can be of lower order or even trivial. One of the solutions is θ = θ (t). For each i such that α i (θ) 0, there is ) a conserved quantity I (θ(0), (i) θ(0), θ(t), θ(t).

12 Using the restricted dynamics The projected dynamics is α i (θ) θ + β i (θ) θ 2 + γ i (θ) = 0, i = 1,..., n m or in an algebraic form I (i)( ) θ(0), θ(0), θ(t), θ(t) = 0, i = 1,..., n m provided α i (θ) 0. 1 All the functions α i ( ), β i ( ), γ i ( ), and I (i) ( ) can be computed analytically from the expression for the Lagrangian L(q, q). 2 Since existence of a periodic solution for the restricted dynamics is equivalent to the existence of a periodic solution for the Euler-Lagrange equations consistent with the constraints, it can be used for motion planning, not only for stability analysis/stabilization to be discussed.

13 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

14 Transverse dynamics to linearize We want to find new variables in a vicinity of a target periodic trajectory such that the system s states are decomposed into: 1 A scalar variable representing position along the trajectory. 2 The remaining (2n 1) variables representing the dynamics transverse to the trajectory and defining a moving Poincaré section S (i) (t), t [0, T]. Figure: Each surface S (i) (t) transversal to the flow (locally, around the intersection with the cycle) is called a Poincaré section.

15 Looking for good conserved quantities In order to find transverse coordinates, one just needs (2n 1) independent quantities conserved along the periodic trajectory. However, they should be chosen in such a way that it is possible to compute linearization of the transverse dynamics along the periodic trajectory analytically. Suppose virtual holonomic constraints are given. They are conserved along the cycle! So, let us introduce the following change of coordinates ( ) ( ) q 1,..., q }{{ n, q } 1,..., q }{{ n y } 1,..., y n 1, ẏ 1,..., ẏ n 1, θ, θ }{{}}{{} q q y ẏ y 1 = q 1 φ 1 (θ),..., y n 1 = q n 1 φ n 1 (θ).

16 Looking for good conserved quantities In order to find transverse coordinates, one just needs (2n 1) independent quantities conserved along the periodic trajectory. However, they should be chosen in such a way that it is possible to compute linearization of the transverse dynamics along the periodic trajectory analytically. Suppose virtual holonomic constraints are given. They are conserved along the cycle! So, let us introduce the following change of coordinates ( ) ( ) q 1,..., q }{{ n, q } 1,..., q }{{ n y } 1,..., y n 1, ẏ 1,..., ẏ n 1, θ, θ }{{}}{{} q q y ẏ y 1 = q 1 φ 1 (θ),..., y n 1 = q n 1 φ n 1 (θ).

17 Dynamics in deviations from the constrained manifold Theorem (Rewritten dynamics) The Euler-Lagrange dynamics can be locally rewritten as α i (θ) θ + β i (θ) θ 2 + γ i (θ) = w, w = g (i) y (θ, θ, θ,y,ẏ) y + g (i) ẏ (θ, θ, θ,y,ẏ) ẏ + g (i) v (θ, θ,y,ẏ) v ÿ = F(θ, θ, y, ẏ) + N(θ, y) v. where F(θ, θ, 0, 0) 0 and i is such that α i (θ) 0. For the rewritten dynamics the following quantities y 1,..., y n 1, ẏ 1,..., ẏ n 1, I (1) ( ),..., I (n m) ( ) are conserved along the target trajectory as well as their combinations and derivatives. We need to choose those independent (2n 1) of them dynamics of which can be linearized analytically.

18 Dynamics in deviations from the constrained manifold Theorem (Rewritten dynamics) The Euler-Lagrange dynamics can be locally rewritten as α i (θ) θ + β i (θ) θ 2 + γ i (θ) = w, w = g (i) y (θ, θ, θ,y,ẏ) y + g (i) ẏ (θ, θ, θ,y,ẏ) ẏ + g (i) v (θ, θ,y,ẏ) v ÿ = F(θ, θ, y, ẏ) + N(θ, y) v. where F(θ, θ, 0, 0) 0 and i is such that α i (θ) 0. For the rewritten dynamics the following quantities y 1,..., y n 1, ẏ 1,..., ẏ n 1, I (1) ( ),..., I (n m) ( ) are conserved along the target trajectory as well as their combinations and derivatives. We need to choose those independent (2n 1) of them dynamics of which can be linearized analytically.

19 Dynamics in deviations from the constrained manifold Theorem (Rewritten dynamics) The Euler-Lagrange dynamics can be locally rewritten as α i (θ) θ + β i (θ) θ 2 + γ i (θ) = w, w = g (i) y (θ, θ, θ,y,ẏ) y + g (i) ẏ (θ, θ, θ,y,ẏ) ẏ + g (i) v (θ, θ,y,ẏ) v ÿ = F(θ, θ, y, ẏ) + N(θ, y) v. where F(θ, θ, 0, 0) 0 and i is such that α i (θ) 0. For the rewritten dynamics the following quantities y 1,..., y n 1, ẏ 1,..., ẏ n 1, I (1) ( ),..., I (n m) ( ) are conserved along the target trajectory as well as their combinations and derivatives. We need to choose those independent (2n 1) of them dynamics of which can be linearized analytically.

20 A choice of transverse coordinates The dynamics for y and ẏ ÿ = F(θ, θ, y, ẏ) + N(θ, y) v is easy to linearize since F I (i) = ( F θ θ F θ θ ) /(2 θ θ 2 ) For the rest, the following identity is useful ) d dt I(i)( θ (0), θ (0), θ, θ = 2 θ ( ) w β(θ) I (i) ( ) α(θ) Introduce another (conceptual) change of coordinates ( ) y, ẏ, θ, θ (y, ẏ, I (i), ψ (i)) The target periodic trajectory transforms as follows { } θ = θ (t) ψ (i) = ψ (t) q = q (t) θ = θ (t) I (i) = 0 q = q (t) y = 0 y = 0 ẏ = 0 ẏ = 0

21 A choice of transverse coordinates The dynamics for y and ẏ ÿ = F(θ, θ, y, ẏ) + N(θ, y) v is easy to linearize since F I (i) = ( F θ θ F θ θ ) /(2 θ θ 2 ) For the rest, the following identity is useful ) d dt I(i)( θ (0), θ (0), θ, θ = 2 θ ( ) w β(θ) I (i) ( ) α(θ) Introduce another (conceptual) change of coordinates ( ) y, ẏ, θ, θ (y, ẏ, I (i), ψ (i)) The target periodic trajectory transforms as follows { } θ = θ (t) ψ (i) = ψ (t) q = q (t) θ = θ (t) I (i) = 0 q = q (t) y = 0 y = 0 ẏ = 0 ẏ = 0

22 A choice of transverse coordinates The dynamics for y and ẏ ÿ = F(θ, θ, y, ẏ) + N(θ, y) v is easy to linearize since F I (i) = ( F θ θ F θ θ ) /(2 θ θ 2 ) For the rest, the following identity is useful ) d dt I(i)( θ (0), θ (0), θ, θ = 2 θ ( ) w β(θ) I (i) ( ) α(θ) Introduce another (conceptual) change of coordinates ( ) y, ẏ, θ, θ (y, ẏ, I (i), ψ (i)) The target periodic trajectory transforms as follows { } θ = θ (t) ψ (i) = ψ (t) q = q (t) θ = θ (t) I (i) = 0 q = q (t) y = 0 y = 0 ẏ = 0 ẏ = 0

23 Linear deviations A possible choice of transverse coordinates: x = [ I (i), y T, ẏ T ] T R 2n 1 The variable ψ (i) defines location along the periodic trajectory and the moving Poincaré section: { } S (i) (t) := [θ, θ, y; ẏ] R 2n : ψ (i) (θ, θ) = ψ (i) (t) Let I(i) Y 1 Y 2 TS (i) (t) := TS (i) denotes linearization for I (i) y ẏ S (i) { [q T, q T ] T : ( q q (t) )T q (t)+ ( q q (t) )T q } (t) = 0

24 Transverse linearization Keeping the first-order terms in the Taylor multiseries expansions, the linearization of the dynamics for the transverse coordinates takes the form of the time-periodic linear system d dt X (t) = A(t) X (t) + B(t) V with the following structure: d dt I(i) = a(i) 11 (t) I(i) + a(i) [ ] d Y1 = dt Y 2 12 (t) Y 1 + a (i) 13 (t) Y 2 + b (i) 1 (t) V [ ] 0(n 1) 1 0 (n 1) (n 1) 1 (n 1) (n 1) I(i) Y A 21 (t) A 22 (t) A 23 (t) 1 Y 2 + [ 0(n 1) 1 B 2 (t) ] V How to use it for orbital stabilization?

25 Using transverse linearization for stabilization Theorem (Exponential orbital stabilization) The following two statements are equivalent. 1 There is a C 1 -smooth K(t) such that [ ] V = K(t) I (i) T, Y1 T, Y 2 T, K(t) = K(t + T) stabilizes the origin of the transverse linearization. 2 There exists a C 1 -smooth v = f(q, q) that ensures exponentially orbitally stability. Furthermore, two possible choices for v are v(t) = K (τ) [ I (i), y T, ẏ ] T T τ = s : [ q T (t), q T (t) ] S (i) (s) T or O ε (q ) TS (i) (s) where O ε (q ) is a small tube around the trajectory.

26 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

27 Stable oscillations of a spherical pendulum on a puck Achieved goal: exponentially orbitally stable oscillations of the pendulum around the upright equilibrium. Figure: A spherical pendulum on a puck. The coordinates x 1 and x 2 represent the position of the puck in the horizontal plane; the angles ε 1 and ε 2 give the orientation with respect to the inertia frame. Dynamics can be described as follows d dt [ ] L( ) ε {1,2} L( ) ε {1,2} = 0, d dt [ ] L( ) ẋ {1,2} L( ) x {1,2} = u {1,2},

28 Synchronization of oscillations of pendula on carts Goal: synchronous exponentially orbitally stable oscillations of the cart-pendulum systems around their upright equilibria. Figure: Three identical cart-pendulum systems. The coordinates x 1, x 2, and x 3 represent positions of the carts along the horizontal; θ 1, θ 2, and θ 3 give the angles of the pendula with respect to the vertical. Dynamics can be described as follows 2 ẍ i + cos(θ i ) θ i sin(θ i ) θ 2 i = u i, cos(θ i ) ẍ i + θ i g sin(θ i ) = 0, i = 1,..., N

29 Synchronization of oscillations: numerical simulations 0.4 θ (t) 1 θ (t) θ (t) [θ 1 (t), θ 2 (t), θ 3 (t)] time (sec) Figure: Synchronization of oscillations for 3 cart-pendulum systems: An evolution of the angles the θ 1, θ 2, θ 3 -variables along the solution of the closed-loop system. They are synchronized in about one period 5 sec and have reached after transition the target amplitude of oscillations of 0.2 [rad].

30 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Outline 1 Problem Formulation (Euler-Lagrange systems) 2 Key idea: virtual holonomic constraints 3 Main result: transverse linearization and orbital stabilization 4 Examples: spherical pendulum & synchronization of oscillations 5 Summary Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems

31 Problem Formulation (Euler-Lagrange systems) Key idea: virtual holonomic constraints Main result: transverse linearization and orbital stabilization Examples: spherical pendulum & synchronization of oscillations Summary Summary Class of models: underactuated Euler-Lagrange systems. Goal: to create orbitally exponentially stable periodic motions. Approach: finding linearizable transverse coordinates and stabilizing their linearized dynamics. Key: using deviations from geometric relations along the target periodic motion and a conserved quantity of the restricted dynamics. Possibility: creating synchronous oscillations. Anton Shiriaev, Leonid Freidovich, Sergey Gusev Transverse Linearization for Underactuated Mechanical Systems