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In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

Pontryagin calculus in Riemannian geometry

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application/pdf Pontryagin calculus in Riemannian geometry Francois Dubois, Danielle Fortune, Juan Antonio Rojas Quintero

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        <identifier identifierType="DOI">10.23723/11784/14340</identifier><creators><creator><creatorName>Francois Dubois</creatorName></creator><creator><creatorName>Danielle Fortune</creatorName></creator><creator><creatorName>Juan Antonio Rojas Quintero</creatorName></creator></creators><titles>
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In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

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Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Pontryagin calculus in Riemannian geometry Fran¸cois Dubois 1 , Danielle Fortun´e 2 , Juan Antonio Rojas Quintero 3 , Claude Vall´ee Geometric Science of Information - GSI 2015 Palaiseau (France), Friday 30 october 2015 1 Department of Mathematics, University Paris Sud, Orsay, France, CNAM Paris, Structural Mechanics and Coupled Systems Laboratory. 2 21, rue du Hameau du Cherpe, 86280 Saint Benoˆıt, France. 3 School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Claude Vall´ee (1945 - 2014) Th`ese 3`eme cycle (1972) : “Sur l’axiomatique de la thermodynamique de R. Giles et la concavit´e de l’entropie” supervised by Bernard Nayroles. Th`ese d’Etat (1987) : “Lois de comportement des milieux continus dissipatifs compatibles avec la physique relativiste” supervised by Jean-Marie Souriau (1922 - 2012) Professor at Poitiers University (Mechanics department) Leader of the “Colloque International de Th´eories Variationnelles” (1996 - 2011) Member of the scientific committee of the GSI conference Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Claude Vall´ee (1945 - 2014) Claude Vall´ee at CITV, Aix en Provence, 28 august 2012. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Outlook Introduction 1) Pontryagin framework for differential equations 2) Pontryagin hamiltonian 3) Riemannian metric for robotics applications 4) Optimal dynamics 5) Intrinsic evolution of the generalized force Conclusion Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Introduction Dynamics of articulated systems The choice of the Lagrangian is directly linked to the conservation of energy. Euler-Lagrange methodology applied to the kinetic and potential energies. System of second order ordinary differential equations for the motion. These equations are identical to those deduced from the fundamental principle of dynamics. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Introduction (ii) Configuration parameters: their choice does not affect the energy value. The kinetic energy is a positive definite quadratic form with respect to the configuration parameters derivatives. Its coefficients are ideal candidates to define and create a Riemannian metric structure on the configuration space. The Euler-Lagrange equations have a contravariant tensorial nature and highlight the covariant derivatives with respect to time with the introduction of the Christoffel symbols. Control of articulated robot How to choose a Hamiltonian and a cost function ? Here the presence of the Riemann structure is sound. It enables a cost function invariant when coordinates change. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion References M. Lazrak and C. Vall´ee. “Commande de robots en temps minimal”, Revue d’Automatique et de Productique Appliqu´ees (RAPA), volume 8, issue 2-3, p. 217-222, 1995. J.A. Rojas Quintero. Contribution `a la manipulation dextre dynamique pour les aspects conceptuels et de commande en ligne optimale, Thesis Poitiers University, 31 October 2013. J.A. Rojas Quintero, C. Vall´ee, J.P. Gazeau, P. Seguin, M. Arsicault. “An alternative to Pontryagin’s principle for the optimal control of jointed arm robots”, Congr`es Fran¸cais de M´ecanique, Bordeaux, 26 - 30 August 2013. C. Vall´ee, J.A. Rojas Quintero, D. Fortun´e, J.P. Gazeau. “Covariant formulation of optimal control of jointed arm robots: an alternative to Pontryagin’s principle”, arXiv:1305.6517, 28 May 2013. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Pontryagin framework for differential equations • Dynamical system : state vector y(t ; λ(•)) System controlled by a set of variables λ(t) First order ordinary differential equation: (1) dy dt = f (y(t), λ(t), t) . Initial condition: y(0 ; λ(•)) = x . Optimal solution : minimize the following cost function J : J(λ(•)) ≡ T 0 g y(t), λ(t), t dt . • Pontryagin’s main idea : consider the differential equation (1) as a constraint satisfied by the variable y. Lagrange multiplier p = p(t) associated to the constraint (1). This new variable is a covariant vector function of time Global Lagrangian functional : L(y, λ, p) ≡ T 0 g(y, λ, t) dt + T 0 p(t) dy dt −f (y, λ, t) dt . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Adjoint equations • Proposition 1. If the Lagrange multiplier p(t) satisfies the adjoint equations, dp dt + p ∂f ∂y − ∂g ∂y = 0 and the final condition : p(T) = 0 , the variation δJ of the cost function for a given variation δλ of the paramater is given by the relation δJ = T 0 ∂g ∂λ − p ∂f ∂λ δλ(t) dt . At the optimum this variation is identically null : Pontryagin optimality condition: ∂g ∂λ − p ∂f ∂λ = 0 . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Pontryagin hamiltonian • Hamiltonian : H(p, y, λ) ≡ p f − g Optimal Hamiltonian H(p, y) ≡ H(p, y, λ∗ ) for λ(t) = λ∗(t) equal to the optimal value associated to the optimal condition J(λ∗ ) ≤ J(λ) ∀λ • Proposition 2. Symplectic form of the dynamic equations. the “forward” differential equation dy dt = f (y(t), λ(t), t) . and the “backward” adjoint differential equation dp dt + p ∂f ∂y − ∂g ∂y = 0 take the symplectic form: dy dt = ∂H ∂p , dp dt = − ∂H ∂y . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Riemannian metric • Dynamical system parameterized by a finite number of functions qj (t) Set of all states Q : q ≡ {qj } The kinetic energy K is a positive definite quadratic form of the time derivatives ˙qj for each state q ∈ Q. The coefficients of this quadratic form define a mass tensor M(q). The mass tensor is composed by a nonlinear regular function of the state q ∈ Q. It contains the mechanical caracteristics of mass and inertia of the articulated system. • We have K(q, ˙q) ≡ 1 2 k Mk (q) ˙qk ˙q . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Riemannian metric (ii) • The mass tensor M(q) is symmetric and positive definite for each state q. Consider the Riemannian metric g defined by the mass tensor M. (Lazrak and Vall´ee (1995), Siebert (2012)) We set: gk (q) ≡ Mk (q) . • Riemannian manifold structure for the space of states Q classical geometrical tools of Riemannian geometry: Covariant space derivation ∂j ≡ ∂ ∂qj Contravariant space derivation ∂j : < ∂j , ∂k > = δj k Component j, of the inverse mass tensor M−1: Mj , Mij Mj = δi Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Riemannian metric (iii) • Connection Γj ik = 1 2 Mj ∂i M k + ∂kM i − ∂ Mik , Γj ki = Γj ik, d ∂j = Γjk dqk ∂ , d ∂j = −Γj k dqk ∂ , Relations between covariant components ϕj and contravariant components ϕk of a vector field: ϕj = Mjk ϕk, ϕk = Mkj ϕj Covariant derivation of a vector field ϕ ≡ ϕj ∂j : dϕ = ∂ ϕj + Γj k ϕk dq ∂j Covariant derivation of a covector field ϕ ≡ ϕ ∂ : dϕ = ∂kϕ − Γj k ϕj dqk ∂ Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Riemannian metric (iv) • Ricci identities ∂j Mk = Γp jk M p + Γp j Mkp ∂j Mk = −Γk jp Mp − Γjp Mpk Gradient of a scalar field: dV = ∂ V dq = < V , dqj ∂j > V = ∂ V ∂ Gradient of a covector field ϕ = ϕ ∂ : dϕ ≡ < ϕ , dqj ∂j > ϕ = ∂kϕ − Γj k ϕj ∂k ∂ Second order gradient of a scalar field V : 2V = ( V ) 2 V = ∂k∂ V − Γj k ∂j V ∂k ∂ • Components Rj ik of the Riemann tensor: Rj ik ≡ ∂ Γj ik − ∂kΓj i + Γp ik Γj p − Γp i Γj pk Anti-symmetry of the Riemann tensor: Rj ik = −Rj i k. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Riemannian form of the Euler-Lagrange equations • Proposition 3. In the presence of an external potential V = V (q), the Lagrangian L(q, ˙q) = K(q, ˙q) − V (q) allows to write the equations of motion in the classical Euler-Lagrange form: d dt ∂L ∂ ˙qi = ∂L ∂qi These equations take also the Riemannian form: Mk ¨q + Γij ˙qi ˙qj + ∂kV = 0 . • Equations of motion when a mechanical forcing control u is present (forces and torques typically): ¨qj + Γj k ˙qk ˙q + Mj ∂ V = uj . with the contravariant components of the control u in the right hand side. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Optimal dynamics • Cost function The space of states Q has a natural Riemannian structure. Intrinsic and invariant cost function non sensible to the change of coordinates. (2) J(u) = 1 2 T 0 Mk (q) uk u dt . (Rojas Quintero’s thesis, 2013) • The controlled system ¨qj + Γj k ˙qk ˙q + Mj ∂ V = uj with the cost function (2) is of type dY dt = f (Y (t), λ(t), t) , J(λ(•)) ≡ T 0 g Y (t), λ(t), t dt with Y = {qj , ˙qj } , f = {Y j 2 , −Γj k ˙qk ˙q − Mj ∂ V + uj } , λ = {uk } , g = 1 2 Mk (Y1) uk u . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Optimal dynamics (ii) • Lagrange multipliers of the Pontryagin method : P = {pj , ξj } • Hamiltonian H(Y , P, λ) function of state Y , adjoint P and control variable λ = {uk} We have: H(Y , P, λ) = pj ˙qj + ξj − Γj k ˙qk ˙q − Mj ∂ V + uj − 1 2 Mk (Y1) uk u . • Proposition 4. Interpretation of one adjoint state. When the cost function J(u) ≡ 1 2 T 0 Mk (q) uk u dt is stationary, the adjoint state ξj is exactly equal to the applied force (and torque!) uj : ξj = uj . Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Intrinsic evolution of the generalized force • Covector ξ : ξ = ξj ∂j . We have the following result: • Proposition 5. Covariant evolution equation of the optimal force. With the above notations and hypotheses, the forces and torques u satisfy the following time evolution: d2u dt2 j + Ri k j ˙qk ˙q ui + 2 jkV uk = 0 . (Rojas Quintero’s thesis, and Vall´ee et al., 2013). Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Conclusion The control optimization of a robotic system shows how important is the introduction of an appropriate geometric structure. Riemannian geometry favors the metric associated to the kinetic energy. The systems have a well-defined Riemannian tensorial nature: contravariant for the equation of motion covariant for the equation of the control variables. Mechanical interpretation of the Pontryagin’s adjoint states. Second order covariant derivatives and Riemann curvature tensor for the equation of forces and torques. Numerically stable method when discretization is considered. Future numerical developments: juggle between two coupled systems of second-order ordinary differential equations. Claude outlook introduction Pontryagin Hamiltonian Riemannian metric optimal dynamics force evolution conclusion Thank you for your attention !