On variational integrators for optimal control of mechanical control systems. Leonardo Colombo, David Martín de Diego and Marcela Zuccalli
|
|
- Erick Gregory
- 6 years ago
- Views:
Transcription
1 RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. Falta Falta), Falta, pp Falta Comunicación Preliminar / Preliminary Communication On variational integrators for optimal control of mechanical control systems Leonardo Colombo, David Martín de Diego and Marcela Zuccalli Abstract. In this paper we study discrete second-order vakonomic mechanics, that is, constrained variational problems for second-order lagrangian systems. One of the main applications of the presented theory will be optimal control of underactuated mechanical control systems. We derive geometric integrators which are symplectic and preserve the momentum map. Additional, we show the applicability of the proposed theory in an example, the planar rigid body. 1. Introduction One of the most important problems in engineering are related with planning and control of robotic devices. The applications include the optimal design of autonomous vehicles and of minimalist manipulators. In this sense, for fully actuated mechanical systems a broad range of powerful techniques were developed in the last decades. These techniques are possible because fully actuated systems possess a number of strong properties that facilitate control design, such as feedback linearizability, passivity, matching conditions, and linear parametrizability. For underactuated systems one or more of the above structural properties are usually lost. For these reasons, control design becomes much more difficult and there are correspondingly less results available. The class of underactuated mechanical systems include spacecraft, underwater vehicles, satellites, mobile robots, helicopters, wheeled vehicles, mobile robots, underactuated manipulators, etc. With this motivation in mind, we will develop in this paper the study of discrete second-order vakonomic mechanics showing its applicability for optimal control of underactuated mechanical systems. This study is motivated by the importance of the design of structure-preserving algorithms for numerical simulation of controlled systems. A main idea for this paper will be the design of variational integrator using discrete variational calculus adapted to second-order theories. For this type of systems it is possible to construct discretization schemes which preserve an appropriate symplectic form and the momentum map associated with a Lie group action. The variational integrators have shown, in explicit examples, an exceptionally good long-time behaviour and this research is of great interest from numerical and geometrical considerations see [9, 4]). The paper is organized as follows. In Section 2, we consider an optimal control problem for an underactuated mechanical system and we reduce it to an equivalent second-order variational problem using standard Presentado por Falta. Recibido: Falta. Aceptado: Falta. Palabras clave / Keywords: Underactuated mechanical system, constrained variational calculus, Levi-Civita connection, optimal control, vakonomic mechanics, Lagrangian mechanics, Higher-order mechanics, Discrete mechanics, Variational integrators Mathematics Subject Classifications: Falta c Falta Real Academia de Ciencias, España. 1
2 Leonardo Colombo, David Martín de Diego and Marcela Zuccalli tools from riemannian geometry. In Section 3, we recall some basic elements of discrete vakonomic mechanics [2]. In Section 4, we develop discrete variational calculus for second-order dynamics. Using this formulation we derive variational integrators for optimal control problems for underactuated mechanical systems. As an example, we use this geometric integrator to the optimal control of the planar rigid body. 2. Optimal control of underactuated mechanical systems In this section we consider a particular class of mechanical control systems, underactuated system; that is, systems in which the number of control inputs is less than the dimension of the configuration space. We assume that these systems are controllable, that is, for any two points x 0 and x f in the configuration space, there exists an admissible control ut) defined on some time interval [0, T ] such that the system with initial condition x 0 reaches the point x f in time T see for more details [5, 4]). Let Q be a differentiable manifold of dimension n with local coordinates q A ) and G a riemannian metric specifying the kinetic energy of the mechanical system. The metric is locally written as G = G AB dq A dq B. Denote also by G : T Q T Q the induced vector bundle isomorphism and by # G : T Q T Q the inverse isomorphism. We can construct the Levi-Civita connection G on Q as the unique affine connection which is torsion-less and metric with respect to G. It is determined by the standard formula for all X, Y, Z XQ). 2G G XY, Z) = XGY, Z)) + Y GX, Z)) ZGX, Y )) +GX, [Z, Y ]) + GY, [Z, X]) GZ, [Y, X]) Fixed a potential function V : Q R, the mechanical system is defined by the mechanical Lagrangian L : T Q R, and the corresponding equations of motion are Here, grad G V is the vector field on Q characterized by Lv q ) = 1 2 Gv q, v q ) V q), where v q T q Q 1) Ġ ct)ċt) + grad GV ct)) = 0. 2) Ggrad G V, X) = XV ), for every X XQ). In local coordinates the Equations 2) are d 2 q C dt 2 = Γ C ABqt)) dqa dq B AB V G dt dt q C. 3) where G AB ) are the entries of the inverse matrix of G AB ) and where Γ C AB are the Christoffel components of the Levi-Civita connection. Adding control forces in our picture, given by {θ m+1,..., θ n }, where θ a are independent 1-forms on Q, m + 1 a n; then if we denote by Y a = G θ a the corresponding input sections, then equations 2) are modified as follows: Ġ ct)ċt) + grad G V ct)) = u at)y a ct)). 4) where u m+1,..., u n ) U R n m. Denote by D = span{y a }, m + 1 a n, and by {Z α }, 1 α m a basis of D. An alternative way of writing Equations 4) is the following one C ab G Ġ ct)ċt) + grad G V ct)), Y b) = u a, m + 1 a n 5) G Ġ ct)ċt) + grad G V ct)), Zα) = 0, 1 α m 6) 2
3 On Variational Integrators for optimal control of mechanical control systems where C ab = G Y a, Y b). Given a cost function C : T Q U R, a solution of the optimal control problem consists on a trajectory q A t), u a t)) of the state variables and control inputs satisfying equations 5) and 6) with given initial and final conditions q A t 0 ), q A t 0 )) and q A t f ), q A t f )) respectively, extremizing the cost functional A = tf t 0 Cq A, q A, u a ) dt. This optimal control problem is equivalent to the following second order constrained problem see [6] for an intrinsic point of view): Extremize the functional: Ã = tf t 0 Lq A t), q A t), q A t)) dt subjected to the second-order constraints given by Φ α q A, q A, q A ) = G Ġ ct)ċt) + grad G V ct)), Zα) = 0, with α = 1,..., m, where L : T 2) Q R is defined as Lq A, q A, q A ) = C q A, q A, C ab G Ġ ct)ċt) + grad G V ct)), Y b)). Notice that T 2) Q denotes the second order tangent bundle of Q see [10]). 3. Dicrete vakonomic mechanics In this section we recall some basic elements for discrete vakonomic mechanics using discrete variational calculus with constraints see [2]). The solutions of the vakonomic problem are the critical sequences of a discrete action sum subjected to some constraint functions. Consider a continuous vakonomic system defined by a Lagrangian L : T Q R and a constraint submanifold M of T Q locally defined by the vanishing of m-independents constraints Φ α : T Q R with 1 α m. For a discretization of this systems, we substitutes the velocity space T Q by the cartesian product Q Q, and then the lagrangian L by a discrete lagrangian function L d : Q Q R this discrete Lagrangian may be considered as an approximation of the continuous Lagrangian L). In the same way, the discrete constraint submanifold M d Q Q defined, locally, by the vanishing of m-independent constraints functions Φ α d : Q Q R, 1 α m. Fixed q 0, q N Q for some integer N > 2) consider the sequences q 0, q 1,...,, q N ) Q N+1. We also assume that the sequences verify the discrete constraints if Φ α d q k, q k+1 ) = 0, k = 0,..., N 1. Define the discrete action sum by A d q 0, q 1,..., q N ) = N 1 k=0 L d q k, q k+1 ). The discrete constrained variational problem is defined by { ext Ad q 0, q 1,..., q N ) with q 0 and q N fixed suject to Φ α d q k, q k+1 ) = 0, 1 α m and 0 k N 1. 7) 3
4 Leonardo Colombo, David Martín de Diego and Marcela Zuccalli We construct the augmented Lagrangian L d : Q Q R m R: L d x, y, λ) = L d x, y) + λ α Φ α d x, y). From the classical Lagrange multiplier Lemma [1], we deduce that the solutions of the problem 7) coincides with the solutions of the discrete variational problem { ext Ad q 0, q 1,..., q N, λ 0, λ 1,..., λ N 1 ) with q 0 and q N fixed, q k Q, λ k R m 8) k = 0,..., N 1, q N Q, where A d q 0, q 1,..., q N, λ 0, λ 1,..., λ N 1 ) = and λ k is a m-vector with components λ k α with 1 α m. N 1 k=0 L d q k, q k+1, λ k ) Therefore, applying standard discrete variational calculus we deduce that the solutions of problem 7) verify the following set of difference equations { D1 L d q k, q k+1 ) + D 2 L d q k 1, q k ) + λ k αd 1 Φ α d q k, q k+1 ) + λ k 1 α D 2 Φ α d q k 1, q k ) = 0, 1 k N 1, Φ α d q k, q k+1 ) = 0, 1 α m and 0 k N 1 where D 1 L d and D 2 L d denote the derivatives of the discrete lagrangian L d respect to the first and the second arguments, respectively. This equations are called discrete vakonomic equations. For all function F C 2 ) F Q Q) we denote by D 12 F the n n-matrix x A y B. Then, if the matrix D 12 L d x, y) + λ α D 12 Φ α Φ α d d x, y) x, y) ) x Φ α T d x, y) 0 m m y M n+m) n+m)r) is regular, then, by a direct application of the implicit function theorem, we deduce that, in a neighborhood of the point x, y, λ), exists a unique local application Υ d : M d R m M d R m x, y, λ) y, z, Λ), such that for all solutions q 0, q 1,..., q N, λ 0, λ 1,..., λ N 1 ) of the discrete vakonomic equations, we have that locally Υ d q k 1, q k, λ k 1 ) = q k, q k+1, λ k ). The application Υ d is called the discrete flow of the discrete vakonomic problem. Remark 1 In [2], the authors have shown that, under the regularity condition, the discrete flow Υ d preserves a symplectic form naturally defined on M d R m given locally by [ 2 L d ω d x, y, λ) = x A x, y) + λ 2 Φ α ] d yb x A x, y) dx A dy B + Φα d yb x A x, y)dλ α dx A and if the discrete Lagrangian L d and the constraint Φ α d are invariant under the action of a Lie group G on Q, the flow preserves the momentum map, that is where J d : Q Q R m g is given by x, y, λ) J d x, y, λ) : J d Υd q k 1, q k, λ k 1 ) ) = J d q k 1, q k, λ k 1 ). g R ξ D 2 L d x, y) + λ α D 2 Φ α d x, y), ξ Q y), where ξ Q is the infinitesimal generator of ξ and g is the Lie algebra of G 4
5 On Variational Integrators for optimal control of mechanical control systems 4. A variational integrator for optimal control of underactuated mechanical systems In this section we extend the results of the previous section to the case of constrained second-order lagrangian dynamics. As a consequence of the previous sections, the derived numerical methods are directly applied to our initial objective: geometric integrators for optimal control problems for underactuated mechanical systems. In particular, we design numerical methods preserving an appropriate symplectic form and they are momentum preserving Discrete second-order vakonomic mechanics Given a second order vakonomic mechanical system with Lagrangian L : T 2) Q R and a second order constraint submanifold M of T 2) Q, locally defined by the vanishing of m independent constraints Φ α : T 2) Q R. A natural discretization of this system consists on a discrete Lagrangian function L d : Q Q Q R and a constraint submanifold M d of Q Q Q which is locally defined by the vanishing of m-constraint functions Φ α d : Q Q Q R. Fixed q 0, q 1 and q N 1, q N for some integer N > 4), we consider the discrete sequences on Q, q 0, q 1,...,, q N ) Q N+1 verifying the discrete constraints Φ α d q k 1, q k, q k+1 ) = 0 k = 1,..., N 1 by Define the discrete action sum by A d q 0,..., q N ) = N 2 k=0 L d q k, q k+1, q k+2 ). We are looking for solutions of the following discrete variational problem with constraints ext A d q 0, q 1,..., q N ), with q 0, q 1 and q N 1, q N fixed subject to Φ α q k, q k+1, q k+2 ) = 0 with 1 α m and 0 k N 2. As, in the previous section, consider the augmented Lagrangian L d : Q Q Q R m R defined L d x, y, z, λ) = L d x, y, z) + λ α Φ α d x, y, z). The solutions of the problem 9) coincide with the solutions of the following problem { extremize Ad q 0, q 1,..., q N, λ 0, λ 1,..., λ N 2 ) with q 0, q 1 and q N 1, q N fixed where q k Q, λ k R m, k = 0,..., N, A d q 0, q 1,..., q N, λ 0, λ 1,..., λ N 2 ) = and λ k is a m-vector with components λ k α with 1 α m. Hence, the extremality conditions are N 2 k=0 L d q k, q k+1, q k+2, λ k ) 0 = D 3 Ld q k 2, q k 1, q k ) + D 2 Ld q k 1, q k, q k+1 ) +D 1 Ld q k, q k+1, q k+2 ) + λ k 2 α D 3 Φ α d q k 2, q k 1, q k ) +λ k 1 α D 2 Φ α d q k 1, q k, q k+1 ) + λ k αd 1 Φ α d q k, q k+1, q k+2 ), 2 k N 2 0 = Φ α d q k 2, q k 1, q k ) 0 = Φ α d q k 1, q k, q k+1 ) 0 = Φ α d q k, q k+1, q k+2 ). 9) xx 10) 5
6 Leonardo Colombo, David Martín de Diego and Marcela Zuccalli If the matrix D13 Ld x, y, z) + λ α D 13 Φ α d x, y, z) D 3 Φ α d x, y, z) D 1 Φ α d x, y, z)) T 0 ) regularity condition) 11) is regular for all x, y, z) M d and λ α R, 1 α m, then by a direct application of the implicit function theorem, we deduce that there exists a local) unique application Υ d : M d R 2m M d R 2m q 0, q 1, q 2, q 3, λ 0 α, λ 1 α) q 1, q 2, q 3, q 4, λ 1 α, λ 2 α) which univocally determines q 4 and λ 2 α, 1 α m from the initial conditions q 0, q 1, q 2, q 3, λ 0 α, λ 1 α). Here, M d denotes the submanifold of Q 4 = Q Q Q Q M d = {q 0, q 1, q 2, q 3 ) Q 4 Φ α d q 0, q 1, q 2 ) = 0, Φ α d q 1, q 2, q 3 ) = 0, 1 α m}. The mapping Υ d will be called the discrete second-order vakonomic flow. Remark 2 See [7]). Using similar techniques than in [2] and [3] it is possible to show that, under the regularity assumption, the discrete second-order vakonomic flow is symplectic and preserves momentum when we have an action preserving the discrete lagrangian L d and the constraint submanifold M d. Let Θ d, Θ+ d be the discrete canonicals one forms on Q4 R 2m given by Θ + d q 0, q 1, q 2, q 3, λ 0, λ 1 ) = 1 i+1 D j L d q i j+1, q i j+2, q i j+3 ) + λ i j+1 α D j Φ α d q i j+1, q i j+2, q i j+3 ) dq i i=0 j=1 Θ + d q N 3, q N 2, q N 1, q N, λ N 3, λ N 2 ) = N 3 D j L d q i j+1, q i j+2, q i j+3 ) + λ i j+1 α D j Φ α d q i j+1, q i j+2, q i j+3 ) dq i i=n 1 j=i N+3 We define the discrete Poincaré-Cartan 2-form, Ω d on Q 4 R 2m, Ω d = dθ + d = dθ d Ω Md Ω d q 0, q 1, q 2, q 3, λ 0, λ 1 ) = [ D 21 L d q 0, q 1, q 2 ) + λ 0 αd 21 Φ α d q 0, q 1, q 2 ) ] dq 1 dq 0 + [ D 31 L d q 0, q 1, q 2 ) + λ 0 αd 31 Φ α d q 0, q 1, q 2 ) ] dq 2 dq 0 + D 1 Φ α d q 0, q 1, q 2 )dλ 0 α dq 0 + [ D 11 L d q 1, q 2, q 3 ) + λ 1 αd 11 Φ α d q 1, q 2, q 3 ) + D 12 L d q 0, q 1, q 2 ) + λ 0 αd 12 Φ α d q 0, q 1, q 2 ) ] dq 0 dq 1 + [ D 31 L d q 1, q 2, q 3 ) + λ 1 αd 31 Φ α d q 1, q 2, q 3 ) + D 32 L d q 0, q 1, q 2 ) + λ 0 αd 32 Φ α d q 0, q 1, q 2 ) ] dq 2 dq 1 +D 1 Φ α d q 1, q 2, q 3 )dλ 1 dq 1 + D 2 Φ α d q 0, q 1, q 2 )dλ 0 dq 1 If we define j : M d R 2m Q 4 R 2m, the canonical inclusion, this give rise the two form = j Ω d. From the definition of Υ d it is obvious that it applies M d R 2m onto itself. Moreover, Υ Md R 2m ) Ω Md = Ω Md Assume that additionally we have an action of a Lie group G on the space Q. It naturally induces and action on the Q Q Q by g, q 0, q 1, q 2 )) gq 0, gq 1, gq 2 ) where g G. Construct now the discrete momentum map : Q4 R 2m g J ± d 6
7 On Variational Integrators for optimal control of mechanical control systems by J ± d q 0, q 1, q 2, q 3, λ 0, λ 1 ) : g R ξ Θ ± d q 0, q 1, q 2, q 3, λ 0, λ 1 ), ξ Q 4q 0, q 1, q 2, q 3 ) where ξ Q 4 is the infinitesimal generator corresponding to an element of the Lie algebra ξ g, that is, ξ Q 4q 0, q 1, q 2, q 3, λ 0, λ 1 ) = ξ Q q 0 ), ξ Q q 1 ), ξ Q q 2 ), ξ Q q 3 )). If the discrete lagrangian L d and the discrete constraints Φ d are invariant under this action then L d = L d + λ α Φ α d is also invariant, and, as a consequence, J + d = J d = J d. Under these conditions we can show that JΥd q 0, q 1, q 2, q 3, λ 0, λ 1 ), ξ = Jq 0, q 1, q 2, q 3, λ 0, λ 1 ), ξ. Therefore, from the symmetry invariance we deduce the preservation of the momentum map, that is, we have deduced a discrete version of the classical Noether theorem for this particular kind of systems Application to optimal control for underactuated mechanical systems In this section, we show that the discrete vakonomic approach of a second-order problem is and appropriate framework for the treatment of the discrete versions of optimal control problems for underactuated mechanical system considered in Section 3. The main application is the construction of geometric numerical integrators for this type of systems. A reasonable discretization of the Euler-Lagrange equations with controls is D 2 L d q k 1, q k ) + D 1 L d q k, q k+1 ) + hu a ) k G Y a ) q k= 0, 1 k N 1. 12) where h is the fixed time step. Using the same idea that in Section 2, we can rewrite Equation 12) as C ab q k ) D 2 L d q k 1, q k ) + D 1 L d q k, q k+1 ), Y a qk = hu a ) k, m + 1 a n 13) D 2 L d q k 1, q k ) + D 1 L d q k, q k+1 ), Z α qk = 0, 1 α m. 14) The optimal control problem is determined prescribing the discrete cost functional A d = N 1 k=1 hcq k, q k+1, u a ) k ) with initial and final conditions q 0, q 1 and q N 1, q N, respectively. Since the control variables appear explicitly in 12), the previous discrete optimal control problem is equivalent to the second-order discrete vakonomic problem determined by L d q k 1, q k, q k+1 ) = C q k, q k+1, 1 ) h C abq k ) D 2 L d q k 1, q k ) + D 1 L d q k, q k+1 ), Y a qk Φ α d q k 1, q k+1, q k+2 ) = D 2 L d q k 1, q k ) + D 1 L d q k, q k+1 ), Z α qk. As in the previous subsection, we define L d : Q Q Q R m R as L d q k, q k+1, q k+2, λ α,k ) = L d q k, q k+1, q k+2 ) + λ α,k Φ α d q k, q k+1, q k+2 ), obtaining a second-order vakonomic system and from the results deduced on the previous section we can derive the discrete equations of motion which give us a geometric integrator preserving symplecticity and momentum map if it is the case) for optimal control problems for underactuated mechanical systems. 7
8 Leonardo Colombo, David Martín de Diego and Marcela Zuccalli Example 1 Planar Rigid Body: The configuration space for this system in Q = R 2 S 1 and it can be considered as the simplest example in the category of rigid body dynamics, but adequate to test our method. The three degrees of freedom describe the translations in R 2 and the rotation about its center of mass. The configuration is given by the followings variables: θ describes the relative orientation the body reference frame with respect to the inertial reference frame. The vector x, y) denotes the position of the center of mass measured with respect to the inertial reference frame. The lagrangian is of kinetical type L = 1 2 qt M q, where M = m m J and where m is the mass of the body and J is its moment of inertia about the center of mass. For simplicity we assume that the body moves in a plane perpendicular to the direction of the gravitational force, and then the potential energy is zero. Additionally, we assume that there are not acting frictional forces. For the planar body, the control forces that we consider are applied to a point on the body with distance l > 0 from the center of mass, along the body x axis see [5] for more details about this example). The equations of motion are The control vector fields are mẍ = u 1 cos θ u 2 sin θ mÿ = u 1 sin θ + u 2 cos θ J θ = lu 1 Y 2 = cos θ m Y 3 = sin θ m x + sin θ m x + cos θ m y l J y and element Z 1 generating the orthogonal D of D = span {Y 2, Y 3 }, θ Z 1 = l cos θ x + l sin θ y + θ. The equations of motion are now modified as follows, mj ) J + ml 2 cos θẍ + sin θÿ l θ = u 1 m sin θẍ + m cos θÿ = u 2 ml cos θẍ + l sin θÿ + J θ = 0. Consider the following cost function C = u2 1 + u 2 2, then this optimal control problem is equivalent to 2 the following second-order constrained problem with second-order constraints: Extremize à = tf t 0 Lq A t), q A t), q A t)) dt subject to the second-order constraints given by Φ α q A, q A, q A ) = ml cos θẍ + l sin θÿ + J θ = 0. 8
9 On Variational Integrators for optimal control of mechanical control systems Here L : T 2) Q R is defined by Lq A, q A, q A ) = m2 J 2 ) 2 2 J + ml 2 ) 2 cos θẍ + sin θÿ l θ + sin θẍ + cos θÿ) 2) Now, we develop a discrete version of this higher-order constrained problem. Consider the discretization L d : Q Q R, with Q = R 2 S 1, of the lagrangian L : T Q R: and L d x k, y k, θ k, x k+1, y k+1, θ k+1 ) = 1 mxk+1 x k ) 2 + my k+1 y k ) 2 + Jθ k+1 θ k ) 2) 2h G Y 2 ) q k = cos θ k dx + sin θ k dy l dθ G Y 3 ) q k = sin θ k dx + cos θ k dy G Z 1 ) q k = ml cos θ k dx + ml sin θ k dy + J dθ where the control forces are applied to a point on the body with distance l > 0 from the center of mass, along the body x axis. Denoting by 2 [q k ] = q k+2 2q k+1 +q k h, then the discrete control equations are 2 m 2 [x k ] = u 1 ) k cos θ k u 2 ) k sin θ k m 2 [y k ] = u 1 ) k sin θ k + u 2 ) k cos θ k J 2 [θ k ] = lu 1 ) k Therefore we obtain the following constrained second-order discrete variational problem: L d q k, q k+1, q k+2 ) = m2 h J 2 cos θk 2 J + ml 2 ) 2 2 [x k ] + sin θ k 2 [y k ] l 2 [θ k ] ) 2 + sin θ k 2 [x k ] + cos θ k 2 [y k ] ) ) 2 Φ α d q k, q k+1, q k+2 ) = ml cos θ k 2 [x k ] + l sin θ k 2 [y k ] + J 2 [θ k ] = Conclusions and future work In this paper, we have introduced a discrete geometric framework for optimal control of mechanical systems. Our procedure is strongly based on the formulation of second-order discrete vakonomic systems. Moreover, the construction allows us to design variational integrators sharing some qualitative properties with the continuous case, as for instance, symplectic and momentum preserving, and, as a consequence a good energy behavior associated with these systems. In a future paper, we will study the performance of the proposed numerical methods, preferably, for long-time integration and the extension of backward error analysis to these methods. Acknowledgement. This work has been supported by MICINN Spain) Grant MTM C02-01, MTM E, project Ingenio Mathematica i-math) No. CSD Consolider- Ingenio 2010) and IRSES-project Geomech L.Colombo also wants to thank CSIC for a JAE- Pre grant. The authors wish to thank the referee for his helpful comments. References [1] Abraham R, Marsden JE. 1978). Foundations of Mechanics. Second Edition, Benjamin/Cummings, New York. 9
10 Leonardo Colombo, David Martín de Diego and Marcela Zuccalli [2] Benito R, Martín de Diego D. 2005). Discrete Vakonomic Mechanics. Journal of Mathematical Physics, 46, [3] Benito R, de León M, Martín de Diego D. 2006). Higher order discrete Lagrangian mechanics. International Journal of Geometric Methods in Modern Physics, 3, [4] Bloch A.M. 2003). Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York. [5] Bullo F, Lewis A. 2005). Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. Texts in Applied Mathematics, Springer Verlag, New York. [6] L. Colombo, D. Martin de Diego and M. Zuccalli. 2010) Optimal Control of Underactuated Mechanical Systems: A Geometrical Approach. Journal Mathematical Physics 51, [7] L. Colombo, D. Martin de Diego and M. Zuccalli. Higher-order Discrete vakonomic mechanics. Preprint [8] Cortés J. 2002). Geometric, Control and Numerical Aspects of Nonholonomic Systems. Lec. Notes in Math., 1793, Springer-Verlag, Berlin [9] Hairer E, Lubich C, Wanner G. 2002). Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations Springer Series in Computational Mathematics, 31, Springer-Verlag Berlin [10] de León M and Rodrigues PR. 1985). Generalized Classical Mechanics and Field Theory, North-Holland Mathematical Studies 112, North-Holland, Amsterdam. Colombo Leonardo: leo.colombo@icmat.es; Instituto de Ciencias Matemáticas ICMAT-CSIC- UAM-UC3M-UCM) C/ Serrano 123, Madrid, Spain Martín de Diego David: david.martin@icmat.es; Instituto de Ciencias Matemáticas ICMAT- CSIC-UAM-UC3M-UCM) C/ Serrano 123, Madrid, Spain Zuccalli Marcela: marce@mate.unlp.edu.ar; Departamento de Matemáticas, Facultad de Ciencias Exactas, Universidad Nacional de La Plata. Calle 50 y 115, 1900 La Plata, Buenos Aires, Argentina 10
arxiv:math-ph/ v1 22 May 2003
On the global version of Euler-Lagrange equations. arxiv:math-ph/0305045v1 22 May 2003 R. E. Gamboa Saraví and J. E. Solomin Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationGeometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics
Geometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics Harris McClamroch Aerospace Engineering, University of Michigan Joint work with Taeyoung Lee (George Washington University) Melvin
More informationTHEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009
[under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was
More informationHamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics. Manuel de León Institute of Mathematical Sciences CSIC, Spain
Hamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics Manuel de León Institute of Mathematical Sciences CSIC, Spain joint work with J.C. Marrero (University of La Laguna) D.
More informationDISCRETE VARIATIONAL OPTIMAL CONTROL
DISCRETE VARIATIONAL OPTIMAL CONTROL FERNANDO JIMÉNEZ, MARIN KOBILAROV, AND DAVID MARTÍN DE DIEGO Abstract. This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian
More informationA GEOMETRIC HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES. Contents
1 A GEOMETRIC HAMILTON-JACOBI THEORY FOR CLASSICAL FIELD THEORIES MANUEL DE LEÓN, JUAN CARLOS MARRERO, AND DAVID MARTÍN DE DIEGO Abstract. In this paper we extend the geometric formalism of the Hamilton-Jacobi
More informationInvariant Lagrangian Systems on Lie Groups
Invariant Lagrangian Systems on Lie Groups Dennis Barrett Geometry and Geometric Control (GGC) Research Group Department of Mathematics (Pure and Applied) Rhodes University, Grahamstown 6140 Eastern Cape
More informationarxiv: v5 [math-ph] 1 Oct 2014
VARIATIONAL INTEGRATORS FOR UNDERACTUATED MECHANICAL CONTROL SYSTEMS WITH SYMMETRIES LEONARDO COLOMBO, FERNANDO JIMÉNEZ, AND DAVID MARTÍN DE DIEGO arxiv:129.6315v5 [math-ph] 1 Oct 214 Abstract. Optimal
More informationOn the geometry of generalized Chaplygin systems
On the geometry of generalized Chaplygin systems By FRANS CANTRIJN Department of Mathematical Physics and Astronomy, Ghent University Krijgslaan 281, B-9000 Ghent, Belgium e-mail: Frans.Cantrijn@rug.ac.be
More informationIntrinsic Vector-Valued Symmetric Form for Simple Mechanical Control Systems in the Nonzero Velocity Setting
008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-3, 008 Intrinsic Vector-Valued Symmetric Form for Simple Mechanical Control Systems in the Nonzero Velocity Setting
More informationSolutions to the Hamilton-Jacobi equation as Lagrangian submanifolds
Solutions to the Hamilton-Jacobi equation as Lagrangian submanifolds Matias Dahl January 2004 1 Introduction In this essay we shall study the following problem: Suppose is a smooth -manifold, is a function,
More informationNonholonomic Constraints Examples
Nonholonomic Constraints Examples Basilio Bona DAUIN Politecnico di Torino July 2009 B. Bona (DAUIN) Examples July 2009 1 / 34 Example 1 Given q T = [ x y ] T check that the constraint φ(q) = (2x + siny
More informationNoether s Theorem. 4.1 Ignorable Coordinates
4 Noether s Theorem 4.1 Ignorable Coordinates A central recurring theme in mathematical physics is the connection between symmetries and conservation laws, in particular the connection between the symmetries
More informationVideo 3.1 Vijay Kumar and Ani Hsieh
Video 3.1 Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Dynamics of Robot Arms Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Lagrange s Equation of Motion Lagrangian Kinetic Energy Potential
More informationROLLING OF A SYMMETRIC SPHERE ON A HORIZONTAL PLANE WITHOUT SLIDING OR SPINNING. Hernán Cendra 1. and MARÍA ETCHECHOURY
Vol. 57 (2006) REPORTS ON MATHEMATICAL PHYSICS No. 3 ROLLING OF A SYMMETRIC SPHERE ON A HORIZONTAL PLANE WITHOUT SLIDING OR SPINNING Hernán Cendra 1 Universidad Nacional del Sur, Av. Alem 1253, 8000 Bahía
More informationReview of Lagrangian Mechanics and Reduction
Review of Lagrangian Mechanics and Reduction Joel W. Burdick and Patricio Vela California Institute of Technology Mechanical Engineering, BioEngineering Pasadena, CA 91125, USA Verona Short Course, August
More informationSome aspects of the Geodesic flow
Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques
More informationStabilization of Relative Equilibria for Underactuated Systems on Riemannian Manifolds
Revised, Automatica, Aug 1999 Stabilization of Relative Equilibria for Underactuated Systems on Riemannian Manifolds Francesco Bullo Coordinated Science Lab. and General Engineering Dept. University of
More informationSolving high order nonholonomic systems using Gibbs-Appell method
Solving high order nonholonomic systems using Gibbs-Appell method Mohsen Emami, Hassan Zohoor and Saeed Sohrabpour Abstract. In this paper we present a new formulation, based on Gibbs- Appell method, for
More informationCDS 205 Final Project: Incorporating Nonholonomic Constraints in Basic Geometric Mechanics Concepts
CDS 205 Final Project: Incorporating Nonholonomic Constraints in Basic Geometric Mechanics Concepts Michael Wolf wolf@caltech.edu 6 June 2005 1 Introduction 1.1 Motivation While most mechanics and dynamics
More informationEquivalence, Invariants, and Symmetry
Equivalence, Invariants, and Symmetry PETER J. OLVER University of Minnesota CAMBRIDGE UNIVERSITY PRESS Contents Preface xi Acknowledgments xv Introduction 1 1. Geometric Foundations 7 Manifolds 7 Functions
More informationCOSYMPLECTIC REDUCTION OF CONSTRAINED SYSTEMS WITH SYMMETRY
Vol. XX (XXXX) No. X COSYMPLECTIC REDUCTION OF CONSTRAINED SYSTEMS WITH SYMMETRY Frans Cantrijn Department of Mathematical Physics and Astronomy, Ghent University Krijgslaan 281 - S9, 9000 Ghent, Belgium
More informationThe only global contact transformations of order two or more are point transformations
Journal of Lie Theory Volume 15 (2005) 135 143 c 2005 Heldermann Verlag The only global contact transformations of order two or more are point transformations Ricardo J. Alonso-Blanco and David Blázquez-Sanz
More informationGlobal Formulations of Lagrangian and Hamiltonian Dynamics on Embedded Manifolds
1 Global Formulations of Lagrangian and Hamiltonian Dynamics on Embedded Manifolds By Taeyoung Lee, Melvin Leok, and N. Harris McClamroch Mechanical and Aerospace Engineering, George Washington University,
More informationReduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad
More informationHybrid Routhian Reduction of Lagrangian Hybrid Systems
Hybrid Routhian Reduction of Lagrangian Hybrid Systems Aaron D. Ames and Shankar Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 94720
More informationarxiv: v1 [math.ds] 29 Jun 2015
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X doi:10.3934/xx.xx.xx.xx pp. X XX SECOND-ORDER VARIATIONAL PROBLEMS ON LIE GROUPOIDS AND OPTIMAL CONTROL APPLICATIONS arxiv:1506.08580v1
More informationBACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More informationAutonomous Underwater Vehicles: Equations of Motion
Autonomous Underwater Vehicles: Equations of Motion Monique Chyba - November 18, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/2016 - Control of Unmanned Vehicles
More informationON COLISSIONS IN NONHOLONOMIC SYSTEMS
ON COLISSIONS IN NONHOLONOMIC SYSTEMS DMITRY TRESCHEV AND OLEG ZUBELEVICH DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA, 119899, MOSCOW,
More informationSteering the Chaplygin Sleigh by a Moving Mass
Steering the Chaplygin Sleigh by a Moving Mass Jason M. Osborne Department of Mathematics North Carolina State University Raleigh, NC 27695 jmosborn@unity.ncsu.edu Dmitry V. Zenkov Department of Mathematics
More informationNonholonomic systems as restricted Euler-Lagrange systems
Nonholonomic systems as restricted Euler-Lagrange systems T. Mestdag and M. Crampin Abstract We recall the notion of a nonholonomic system by means of an example of classical mechanics, namely the vertical
More informationConification of Kähler and hyper-kähler manifolds and supergr
Conification of Kähler and hyper-kähler manifolds and supergravity c-map Masaryk University, Brno, Czech Republic and Institute for Information Transmission Problems, Moscow, Russia Villasimius, September
More informationSeminar Geometrical aspects of theoretical mechanics
Seminar Geometrical aspects of theoretical mechanics Topics 1. Manifolds 29.10.12 Gisela Baños-Ros 2. Vector fields 05.11.12 and 12.11.12 Alexander Holm and Matthias Sievers 3. Differential forms 19.11.12,
More informationExponential Controller for Robot Manipulators
Exponential Controller for Robot Manipulators Fernando Reyes Benemérita Universidad Autónoma de Puebla Grupo de Robótica de la Facultad de Ciencias de la Electrónica Apartado Postal 542, Puebla 7200, México
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationThe Geometry of Euler s equation. Introduction
The Geometry of Euler s equation Introduction Part 1 Mechanical systems with constraints, symmetries flexible joint fixed length In principle can be dealt with by applying F=ma, but this can become complicated
More informationRemarks on Quadratic Hamiltonians in Spaceflight Mechanics
Remarks on Quadratic Hamiltonians in Spaceflight Mechanics Bernard Bonnard 1, Jean-Baptiste Caillau 2, and Romain Dujol 2 1 Institut de mathématiques de Bourgogne, CNRS, Dijon, France, bernard.bonnard@u-bourgogne.fr
More informationChoice of Riemannian Metrics for Rigid Body Kinematics
Choice of Riemannian Metrics for Rigid Body Kinematics Miloš Žefran1, Vijay Kumar 1 and Christopher Croke 2 1 General Robotics and Active Sensory Perception (GRASP) Laboratory 2 Department of Mathematics
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationLinear weakly Noetherian constants of motion are horizontal gauge momenta
Linear weakly Noetherian constants of motion are horizontal gauge momenta Francesco Fassò, Andrea Giacobbe and Nicola Sansonetto (September 20, 2010) Abstract The notion of gauge momenta is a generalization
More informationInvariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups
Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,
More informationSOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.1. SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM BY N. PAPAGHIUC 1
More informationDynamics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Dynamics Semester 1, / 18
Dynamics Basilio Bona DAUIN Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Dynamics Semester 1, 2016-17 1 / 18 Dynamics Dynamics studies the relations between the 3D space generalized forces
More informationSquirming Sphere in an Ideal Fluid
Squirming Sphere in an Ideal Fluid D Rufat June 8, 9 Introduction The purpose of this wor is to derive abalytically and with an explicit formula the motion due to the variation of the surface of a squirming
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationComplex Hamiltonian Equations and Hamiltonian Energy
Rend. Istit. Mat. Univ. Trieste Vol. XXXVIII, 53 64 (2006) Complex Hamiltonian Equations and Hamiltonian Energy M. Tekkoyun and G. Cabar ( ) Summary. - In the framework of Kaehlerian manifolds, we obtain
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationThe Astrojax Pendulum and the N-Body Problem on the Sphere: A study in reduction, variational integration, and pattern evocation.
The Astrojax Pendulum and the N-Body Problem on the Sphere: A study in reduction, variational integration, and pattern evocation. Philip Du Toit 1 June 005 Abstract We study the Astrojax Pendulum and the
More informationExamples of nonholonomic systems 1
Examples of nonholonomic systems 1 Martina Stolařová Department of Mathematics, Faculty of Science, University of Ostrava, 30. dubna, 701 03 Ostrava, Czech Republic, e-mail: oint@email.cz Abstract. The
More informationOn the homogeneity of the affine connection model for mechanical control systems
2 F. Bullo and A. D. Lewis On the homogeneity of the affine connection model for mechanical control systems Francesco Bullo Andrew D. Lewis 29/2/2 Abstract This work presents a review of a number of control
More informationOn mechanical control systems with nonholonomic constraints and symmetries
ICRA 2002, To appear On mechanical control systems with nonholonomic constraints and symmetries Francesco Bullo Coordinated Science Laboratory University of Illinois at Urbana-Champaign 1308 W. Main St,
More informationGroup Actions and Cohomology in the Calculus of Variations
Group Actions and Cohomology in the Calculus of Variations JUHA POHJANPELTO Oregon State and Aalto Universities Focused Research Workshop on Exterior Differential Systems and Lie Theory Fields Institute,
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More information5 Constructions of connections
[under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M
More informationA CRASH COURSE IN EULER-POINCARÉ REDUCTION
A CRASH COURSE IN EULER-POINCARÉ REDUCTION HENRY O. JACOBS Abstract. The following are lecture notes from lectures given at the Fields Institute during the Legacy of Jerrold E. Marsden workshop. In these
More informationTowards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity
Towards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity Melvin Leok Mathematics, University of Michigan, Ann Arbor. Joint work with Mathieu Desbrun, Anil Hirani, and Jerrold
More informationA DISCRETE THEORY OF CONNECTIONS ON PRINCIPAL BUNDLES
A DISCRETE THEORY OF CONNECTIONS ON PRINCIPAL BUNDLES MELVIN LEOK, JERROLD E. MARSDEN, AND ALAN D. WEINSTEIN Abstract. Connections on principal bundles play a fundamental role in expressing the equations
More informationSUBTANGENT-LIKE STATISTICAL MANIFOLDS. 1. Introduction
SUBTANGENT-LIKE STATISTICAL MANIFOLDS A. M. BLAGA Abstract. Subtangent-like statistical manifolds are introduced and characterization theorems for them are given. The special case when the conjugate connections
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationCase Study: The Pelican Prototype Robot
5 Case Study: The Pelican Prototype Robot The purpose of this chapter is twofold: first, to present in detail the model of the experimental robot arm of the Robotics lab. from the CICESE Research Center,
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More informationLeonhard Frerick and Alfredo Peris. Dedicated to the memory of Professor Klaus Floret
RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (), 003, pp. 57 61 Análisis Matemático / Mathematical Analysis Comunicación Preliminar / Preliminary Communication Leonhard Frerick and Alfredo Peris Dedicated
More informationNoether Symmetries and Conserved Momenta of Dirac Equation in Presymplectic Dynamics
International Mathematical Forum, 2, 2007, no. 45, 2207-2220 Noether Symmetries and Conserved Momenta of Dirac Equation in Presymplectic Dynamics Renato Grassini Department of Mathematics and Applications
More informationDirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems
Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems Hiroaki Yoshimura Mechanical Engineering, Waseda University Tokyo, Japan Joint Work with Jerrold E. Marsden
More informationGauge Fixing and Constrained Dynamics in Numerical Relativity
Gauge Fixing and Constrained Dynamics in Numerical Relativity Jon Allen The Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation is reviewed. Gauge freedom is discussed and
More informationOn the Linearization of Second-Order Dif ferential and Dif ference Equations
Symmetry, Integrability and Geometry: Methods and Applications Vol. (006), Paper 065, 15 pages On the Linearization of Second-Order Dif ferential and Dif ference Equations Vladimir DORODNITSYN Keldysh
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationPhysics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text:
Physics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text: 1.3 1.6 Constraints Often times we consider dynamical systems which are defined using some kind of restrictions
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationOrientation transport
Orientation transport Liviu I. Nicolaescu Dept. of Mathematics University of Notre Dame Notre Dame, IN 46556-4618 nicolaescu.1@nd.edu June 2004 1 S 1 -bundles over 3-manifolds: homological properties Let
More informationEN Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 2015
EN53.678 Nonlinear Control and Planning in Robotics Lecture 2: System Models January 28, 25 Prof: Marin Kobilarov. Constraints The configuration space of a mechanical sysetm is denoted by Q and is assumed
More informationPart II. Classical Dynamics. Year
Part II Year 28 27 26 25 24 23 22 21 20 2009 2008 2007 2006 2005 28 Paper 1, Section I 8B Derive Hamilton s equations from an action principle. 22 Consider a two-dimensional phase space with the Hamiltonian
More informationMultibody simulation
Multibody simulation Dynamics of a multibody system (Newton-Euler formulation) Dimitar Dimitrov Örebro University June 8, 2012 Main points covered Newton-Euler formulation forward dynamics inverse dynamics
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationarxiv: v2 [math-ph] 12 May 2017
On the Geometry of the Hamilton Jacobi Equation and Generating Functions arxiv:166.847v2 [math-ph] 12 May 217 Sebastián Ferraro 1, Manuel de León 2, Juan Carlos Marrero 3, David Martín de Diego 2, and
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationarxiv:math/ v1 [math.dg] 29 Sep 1998
Unknown Book Proceedings Series Volume 00, XXXX arxiv:math/9809167v1 [math.dg] 29 Sep 1998 A sequence of connections and a characterization of Kähler manifolds Mikhail Shubin Dedicated to Mel Rothenberg
More informationIn most robotic applications the goal is to find a multi-body dynamics description formulated
Chapter 3 Dynamics Mathematical models of a robot s dynamics provide a description of why things move when forces are generated in and applied on the system. They play an important role for both simulation
More informationReduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 049, 28 pages Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group David IGLESIAS, Juan Carlos
More informationA MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS
A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationAppell-Hamel dynamical system: a nonlinear test of the Chetaev and the vakonomic model
ZAMM Z. Angew. Math. Mech. 87, No. 10, 692 697 (2007) / DOI 10.1002/zamm.200710344 Appell-Hamel dynamical system: a nonlinear test of the Chetaev and the vakonomic model Shan-Shan Xu 1, Shu-Min Li 1,2,
More information4.7 The Levi-Civita connection and parallel transport
Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves
More informationSymmetry Preserving Numerical Methods via Moving Frames
Symmetry Preserving Numerical Methods via Moving Frames Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver = Pilwon Kim, Martin Welk Cambridge, June, 2007 Symmetry Preserving Numerical
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationHow big is the family of stationary null scrolls?
How big is the family of stationary null scrolls? Manuel Barros 1 and Angel Ferrández 2 1 Departamento de Geometría y Topología, Facultad de Ciencias Universidad de Granada, 1807 Granada, Spain. E-mail
More informationWilliam P. Thurston. The Geometry and Topology of Three-Manifolds
William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed
More informationRelativistic simultaneity and causality
Relativistic simultaneity and causality V. J. Bolós 1,, V. Liern 2, J. Olivert 3 1 Dpto. Matemática Aplicada, Facultad de Matemáticas, Universidad de Valencia. C/ Dr. Moliner 50. 46100, Burjassot Valencia),
More informationM3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011
M3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011 Text for the course: Professor Darryl D Holm 25 October 2011 Imperial College London d.holm@ic.ac.uk http://www.ma.ic.ac.uk/~dholm/ Geometric Mechanics
More informationEuler-Poincaré reduction in principal bundles by a subgroup of the structure group
Euler-Poincaré reduction in principal bundles by a subgroup of the structure group M. Castrillón López ICMAT(CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid (Spain) Joint work with Pedro L. García,
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationLagrangian Description for Particle Interpretations of Quantum Mechanics Single-Particle Case
Lagrangian Description for Particle Interpretations of Quantum Mechanics Single-Particle Case Roderick I. Sutherland Centre for Time, University of Sydney, NSW 26 Australia rod.sutherland@sydney.edu.au
More informationDiscrete Dirac Mechanics and Discrete Dirac Geometry
Discrete Dirac Mechanics and Discrete Dirac Geometry Melvin Leok Mathematics, University of California, San Diego Joint work with Anthony Bloch and Tomoki Ohsawa Geometric Numerical Integration Workshop,
More informationCurves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,
Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal
More informationSmooth Dynamics 2. Problem Set Nr. 1. Instructor: Submitted by: Prof. Wilkinson Clark Butler. University of Chicago Winter 2013
Smooth Dynamics 2 Problem Set Nr. 1 University of Chicago Winter 2013 Instructor: Submitted by: Prof. Wilkinson Clark Butler Problem 1 Let M be a Riemannian manifold with metric, and Levi-Civita connection.
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More information